Probing Dark Photons with Plasma Haloscopes

Dark photons (DPs) produced in the early Universe are well motivated dark matter (DM) candidates. We show that the recently proposed tunable plasma haloscopes are particularly advantageous for DP searches. While in-medium effects suppress the DP signal in conventional searches, plasma haloscopes make use of metamaterials that enable resonant absorption of the DP by matching its mass to a tunable plasma frequency and thus enable efficient plasmon production. Using thermal field theory, we confirm the in-medium DP absorption rate within the detector. This scheme allows to competitively explore a significant part of the DP DM parameter space in the DP mass range of $6-400$ $\mu$eV. If a signal is observed, the observation of a daily or annual modulation of the signal would be crucial to clearly identify the signal as due to DP DM and could shed light on the production mechanism.


I. INTRODUCTION
Uncovering the nature of dark matter (DM), the predominant constituent of matter in the Universe, remains an elusive target and subject of significant scientific efforts (see e.g. Ref. [1] for a review). While a lot of attention has been devoted to exploring DM composed of Weakly Interacting Massive Particles (WIMPs), another generic possibility is DM composed of very light ( O(eV)) bosonic particles that exhibit classical fieldlike behavior.
A central pillar in experimental efforts to search for the DM are direct detection experiments that attempt to measure the energy deposited within a detector by interactions of DM in the Milky Way halo passing through it. Light DM with mass significantly smaller than O(GeV) will not deposit sufficient nuclear recoil energy within conventional ton-scale direct detection experiments (e.g. [18][19][20]) with keV-level thresholds optimized towards WIMP searches, focusing on electro-weak mass scales. Hence, other types of searches are required to explore it.
In-medium effects within detectors themselves suppress DP interactions and thus present a major limiting factor for experimental searches. If the DM interaction is on resonance with quasi-particle production within the material, such as plasmons, these effects can be mitigated. In a recent study [46], a novel tunable plasma haloscope for axion detection has been proposed. While the more conventional axion haloscope cavity searches for axions coupled to photons rely on resonant axion-photon conversion associated with cavity frequency modes, the plasma haloscope takes advantage of novel metamaterial properties to consider axion-plasmon resonant conversion. The tunable plasma frequency that resonantly matches the axion mass is not directly associated with the dimensions of the detector in plasma haloscopes and allows to efficiently cover a large axion parameter space.
In this work we show that plasma haloscopes constitute particularly advantageous testing grounds for DPs. Plasmons have been extensively discussed in relation to dark sectors within astrophysical environments (e.g. [4,[47][48][49]), the early Universe (e.g. [50]) and more recently in the context of "inelastic" nuclear recoils from heavy ∼MeV-GeV DM within the more conventional direct detection experiments [51,52].
While DP-plasmon resonances have been considered in astrophysical environments (e.g. [48,49]), they have not been discussed in laboratory settings. Utilizing a DP-plasmon resonance with a tunable frequency, plasma haloscopes allow to take full advantage of in-medium effects that often plague DP searches, and thus significantly enhance detection sensitivity.

II. DARK PHOTON IN-MEDIUM RESONANCE
The low-energy effective Lagrangian due to the presence of a gauge boson X of a hidden sector U(1) symmetry that kinetically mixes (e.g. [10,21,53]) with the ordinary photon A is where F µν , X µν denote the fields strengths of the SM photon and the DP, respectively, J µ EM is the electromagnetic current, m X is the DP mass 1 after neglecting terms of order α 2 (see below) and sin α is the kinetic mixing parameter. The kinetic mixing term can be removed by diagonalization throughÃ = A cos α,X = X − sin αA. In the interaction (Ã,X) basis, the effective Lagrangian is where χ ≡ tan α andÃ,X denote the photon produced in electromagnetic interactions and the DP sterile state, respectively. In the interaction basis the electromagnetic coupling gets renormalized to (e/ cos α) and there arẽ A −X photon-DP oscillations due to mass-mixing.
Neglecting O(χ 2 ) terms and dropping the tilde, we have which gives the wave equation in vacuum (in momentum space) where we defined the four momentum K = (ω, k). Here we have used the Fourier expansion for a free field with 1 The DP mass can be generated via the Higgs or the Stueckelberg mechanisms.
the energy ω = + |k| 2 + m 2 X . We treat the fields as complex, X µ c (t, x), of which the actual fields constitute the real part X µ = Re{X µ c }. As in Ref. [54], we include a volume V in the definition of the Fourier transform, which will later simplify the definition of the DM density in Eq. (15), Our use of a classical field description is justified as the state occupation number is very large (e.g. [41]). For a DP that kinetically mixes with the SM photon, effects of in-medium propagation can be significant. Inmedium interactions can be accounted for by including a linear response (e.g. [27,47,55]), where Π µν is a polarization tensor. Hence, the total current coupling to photon is eJ µ EM + J µ ind . Let us first assume an isotropic and parity invariant medium, and later discuss the extremely anisotropic wire metamaterial scenario. In the Lorentz gauge, the polarization tensor is described by two polarization functions Π T and Π L , for transverse and longitudinal excitations (e.g. [21]) where e L and e +,− are the longitudinal and the transverse polarization vectors, respectively, and P µν i are projectors. Assuming k parallel to theẑ axis, the polarization vectors are where e x , e y are orthogonal unit vectors perpendicular to the unit vector k/k. The wave equation for the interaction eigenstate A = (A 0 , A), in the absence of a J µ EM reads The relationship between the dielectric tensor and the polarization tensor components is given by [21,56] so that for small k (small velocity of the DM) T = L = . The dielectric function has real and imaginary parts, just like the polaritazion tensor. The real part is a phase shift in a travelling photon wave function, while the imaginary part is due to absorption of photons travelling in the medium. In the Drude model, typically used to describe metals, can be written as where Γ is the damping within the medium. For small k we have that ω m χ and the zero-components of the fields are negligible, A 0 (k · A/ω) 0 and X 0 0 (see Appendix of Ref. [41]). Thus, the ordinary electric field produced by DP (assuming an isotropic medium) is given by

III. OUTPUT POWER
We would like to relate the X field in Eq. (12) with the DM energy density. First, we note that the detected electric field, as produced by DP in Eq. (12), should be the time derivative of the interaction eigenstate 2 .
Another remark is related to the field X in Eq. (5), which is the sterile interaction eigenstate. In contrast, the DP DM is residing in a vacuum propagation eigenstate (i.e. a mass eigenstate), after a nontrivial cosmological evolution [10,[57][58][59][60]. The vacuum propagation eigenstates (A , X ), found by diagonalizing the mass term of Eq. (2), can be related to (A, X) via where χ has been assumed to be small. We observe that the DM (vacuum propagation eigenstate) consists primarily of the sterile interaction eigenstate, which is the source in Eq. (12). At this point, we can relate X in Eq. (12) with the DM energy density. The Hamiltonian of the DP is that of a massive photon (i.e. the Proca field), where E X = ∂ 0 X + ∇X 0 and B X = ∇ × X. Thus, the energy density is [10,61] where we used the Fourier expansion in Eq. (5) and indicated the only term in H whose integral is not zero. 2 We have a different result with respect to the one quoted in Ref. [40] (see the discussion after their Eq. (9)), in which what is called "electric field" is in our case (1 − )E. However, the final results of Ref. [40] do not depend on this expression (nor do ours, as we tune the haloscope to obtain → 0).
The space average of the complex DP field is corresponding to a plane wave with frequency m X whose amplitude X µ 0 = (X 0 0 , X 0 ) we defined as X µ 0 = X µ (k = 0)/ √ V Equation (15) allow us to make contact between the two different formulations of DP DM as a classical field and as particles, the latter given by the local velocity distribution of DM particles in the laboratory frame f lab (v), through Thus, taking k = m X v in Eq. (15) the DM velocity distribution is identified with [54] f The kinetic energy of DM particles in the dark halo of our Galaxy is of O(10 −6 )m X . Neglecting terms of this order, we can identify ω(k) 2 = m 2 X in Eq. (15). Then, using the inverse Fourier transform of Eq. (5) we obtain where the brackets indicate the space average of the square magnitude of the DP field. Further, neglecting the DM velocity that is of O(10 −3 ), i.e. using in the Fourier expansion Eq. (5) (which means approximating the local DP DM velocity distribution by , allows us to identify the DP field with the plane wave in Eq. (16). Therefore, This result immediately follows from Eq. (19) for a field constant in space and can be obtained directly using Eq. (20) and We can also make contact through the expressions we obtained for the DM energy density ρ with the momentum density function f X (k) that we employ in Appendix A. Writing the energy density as we can use Eq. (15) and obtain Alternatively, if we neglect the DM velocity, i.e. using Eq. (20), we find The power P in a plasma is given by where U is the stored energy. The damping is given by Γ = ω/Q, where Q is the quality factor and (ω/Q) denotes the full-width-half-maximum of the signal. For DP, the on resonance output power, when put in a suggestive form reminiscent of cavity haloscopes [10], is thus given by where in the first line we used the definition of energy density of the electromagnetic field within a dispersive medium [62] and in the second line we used Eq. (21) and defined the "geometric factor" (27) which is typically of O(1). Here, ρ = 0.45 GeV/cm 3 is the local DM density (the value assumed in e.g. Ref. [63] and compatible with observations [64][65][66]), V d is the fiducial volume of the detector, and κ is the signal coupling efficiency factor. In Appendix A, we employ the machinery of thermal field theory to directly compute the power absorbed by the detector in the "propagator approach" [49]. Our results confirm the expressions obtained with classical electrodynamics in the main text, as well as the expressions obtained with quantum field theory (with in-medium corrections), as used in e.g. Refs. [21,26].

IV. THIN WIRE METAMATERIALS
Recent advances in material science have devoted significant attention to metamaterials, composite materials with periodically or randomly distributed artificial micro-structure with size and spacing smaller than the wavelength of interest. Their unique properties can be exploited for new physics searches.
We briefly describe wire metamaterials, which have recently been proposed for the realization of tunable plasma haloscopes [46]. As demonstrated in Ref. [67], an extended network of thin wires comprises a composite dielectric that behaves like a metal with a plasma frequency in the GHz range. We envision a configuration based on a system of aligned wires in theẑ-direction. Since such a system constitutes an extremely anisotropic medium, the longitudinal and transverse modes of polarization tensor are mixed. This can be encoded through a matrix (see e.g. Ref. [27]) where the A and B indices run over the longitudinal and transverse polarization vectors. A non-magnetic material has a symmetric polarization tensor and we can thus diagonalize K. In the z-direction of the wires the effective dielectric constant can be described by the standard Drude model for metals [67,68], with eigenvalues For a rectangular array of wires, with a wire radius d and an inter-wire spacing in each direction perpendicular to the wires of a and b, the plasma frequency is given by [68] where s = √ ab, r = a/b and allowing for ω p ∼ GHz when s ∼ cm spacing. Since the plasma frequency is predominantly determined by the inter-wire spacing, this setup allows to realize a tunable plasma haloscope. The resulting electric field is thus where θ is Envisioning a finite cylindrical experimental configuration, the discussion of bounded plasma solutions for the field propagation in plasma haloscopes can be found in Ref. [46]. Unlike axions, DPs can have electric fields in directions other than theẑ-direction of wire alignment. If the boundary of the plasma is a conducting cylinder, for electric fields polarised in the non-axial directions the device behaves like a resonant cavity. Thus, in principle, DPs can excite transverse electric modes (i.e. modes with E z = 0) if the mode frequency matches the DP mass. However, as the DP interacts with cavities analogously to the axion, the rate is proportional to the the overlap integral of the DP wave function and the cavity mode. In the case of transverse electric modes in a cylinder, this overlap is zero [69]. Thus, we only consider E z .  27), obtained using finite plasma field description within a wave-guide as derived in Ref. [46], describing a system of radius 30 cm with a quality factor Q = 100 as a function of plasma frequency νp.
[Right] Plasma frequency νp = ωp/(2π), given in Eq. (30), of a wire metamaterial configuration composed of aligned 10 µm copper wires as a function of inter-wire spacing in orthogonal directions perpendicular to the wires, a = 1 mm and b. As the spacing b → 0, the plasma frequency saturates to a maximal value. This indicates both the maximum reachable plasma frequency as well as tunability of a given system.

V. EXPERIMENTAL SETUP
We now discuss a setup configuration for DP detection with a plasma haloscope. The output power is immediately read from Eq. (26) as For an unknown DP field direction at the experiment, our limits are obtained by averaging over all the possible directions, with cos 2 θ = 1/3. Such an average occurs when the the polarisation of the DP fluctuates over short timescales, so that a measurement samples many different polarisations. In the case of a fixed polarisation, for a measurement longer than a day the signal will still be averaged over to give an O(1) number, but the exact average will depend on the angle of the DP with the Earth and the experiment's location on Earth.
For simplicity we will follow the specifications of Ref. [46]. We consider a cylindrical structure enclosing copper wires with diameter of ∼ 10 µm, which are readily commercially available. We take Q = 10 2 , κ = 0.5 3 .
Unlike the axion case [46], a magnetic field is not required for DP search and hence we are not restricted by the geometric considerations related to the magnet bore size. However, such a device will still need to be cooled, which limits the volume. For a quantum limited detection at low frequencies a dilution refrigerator is required [70], which is unlikely to be possible for V > O(m 3 ). ADMX, which has a cavity volume of 136 L, has the largest dilution refrigerator currently used in axion detection [71]. However, one could also envision a larger, warmer system aiming at lower frequencies. Another limitation for considered volume is that heavier DPs with large de Broglie wavelength λ DB will lose coherence over the experiment's dimensions, which both reduces the expected signal and increases the signal's sensitivity to the (unknown) DP velocity dispersion, similar to cavities and dielectric haloscopes [54]. Hence, each dimension of the experiment should not exceed O(10)% of the DP's de Broglie wavelength.
Employing the physical dimensions for the experiment as proposed in Ref. [46], we consider a cylinder of volume V = 0.8 m 3 and diameter 60 cm. For a lower limit on detection frequency, we envision a minimum of 300 wires, which in a 60 cm diameter cylinder gives a wire spacing of 3 cm 4 . The lower limit on the number of wires stems from the requirement that the medium behaves as a plasma, which was confirmed to occur within a square array of 200 − 400 wires [72]. From Eq. (30), this translates into a lower limit on detection sensitivity to plasma frequency ν p = 1.5 GHz, corresponding to a DP mass of m DP 6 µeV. These considerations also applied to Ref. [46], signifying that plasma haloscopes are more sensitive to light DM at smaller mass range than originally thought. Note that when the wavelength of light in the medium becomes larger than the experiment the power becomes suppressed [73]. For us, this translates into geometry factor approaching zero as R/ √ z ω → 0, where R is the radius of the cylinder. We illustrate this suppression explicitly in the left panel of Fig. 1, where at 1.5 GHz G = 1.7 × 10 −2 -leading to a hundredfold suppression of the power. Thus, if even lower frequencies are desired, one must increase the radius of the experiment, at the possible expense of a sub Kelvin cooling system.
For considerations of a (soft) upper limit on detection frequency, we note that mechanically tuning such a device would prove very challenging for sub-mm wire spacings. However, as can be seen from Eq. (30), only one direction needs to be tuned in order to change ω p [46]. Thus, one can compensate for limitations in the tuning direction by manufacturing wires to be more densely packed in an orthogonal direction. Hence, if the closest possible spacing is a ∼ 1 mm, one could still achieve a plasma frequency of ν p = 100 GHz with an orthogonaldirection wire spacing of b ∼ 0.5 mm. We note that employing an arbitrarily small spacing in the direction that is not being tuned would not prove beneficial. To illustrate this, we depict in the right panel of Fig. 1 the plasma frequency of Eq. (30) as a function of spacing b for a set of 10 µm wires of spacing a = 1 mm. We observe that ω p is saturated as b → 0 (though the formalism breaks down when the wire radius becomes comparable to the spacing). Hence, if one is restricted to mechanical tuning, the minimum spacing allowed in the tuning direction determines the maximal plasma frequency. To get to 100 GHz would require a minimum spacing 1.5 mm. However, we note that wire metamaterials can be tuned via the Josephson effect [74], which would evade such constraints.

VI. SIGNAL DETECTION
We envision detecting the signal via an antenna coupled to the system. While the full design of such an antenna would be the subject of a detailed technical proposal, if multiple elements are needed to extract power from the full volume a summing network could be employed, as was proposed for axion detection in Ref. [75]. As the refractive index is almost purely imaginary on resonance, waves decay over a distance 1/ √ z ω in transverse (non-z) directions. Thus, systems with R 1/ √ z ω require some antenna elements to be placed inside the medium for a full readout. Under these assumptions, we can obtain the scan rate via Dicke's radiometer formula for signal-to-noise ratio (S/N ) (see discussion in e.g. Ref. [76]) where T sys is the system noise temperature, ∆ν DP 10 −6 ν DP is the DP signal line width, ∆t is the measurement time that covers a frequency range ∼ Q/ω. Here, frequency is ν DP = ω/(2π), where ω = m DP on resonance. We consider quantum limited detection 5 , taking T sys = m DP . 5 At low frequencies ( 10 GHz) detection near quantum limit has been demonstrated with Josephson parametric amplifiers operating at low temperatures [70].
For our parameters, the power at a given frequency is At higher masses, this equation is modified by the restriction that experimental size does not exceed ∼ 30% of λ DB . As we are considering a narrow aspect ratio cylindrical cavity whose height h exceeds the diameter, we impose h < 0.3 λ DB . At lower frequencies, the geometric factor G of Eq. (27), displayed on Fig. 1, is suppressed due to the finite size of the wave-guide [46].
We can obtain a rough estimate of the experimental livetime required to scan over a region of parameter space by treating the resonance as a rectangular spectrum with a width given by the full width half maximum (ω/Q) and height given by the half maximum (P out /2). This estimate assumes that a relatively rapid frequency tuning ( ∆t) without disturbing the system is possible, although these effects could be included [77]. Given that a low Q is assumed, each measurement is relatively long, so this is not a significant restriction on the tuning time. Thus, integrating Eq. (35), the total scanning time between frequencies ν 1 and ν 2 is given by Requiring a signal-to-noise ratio of (S/N ) ≥ 3, a kinetic mixing parameter value down to χ 7 × 10 −16 can be probed across our whole parameter range of 6 − 400 µeV within ∼ 5 years of experimental livetime.
For the above input parameters and experimental livetime of 5 years, in Fig. 2 we display the projected sensitivity for DP parameter range in green along with existing constraints 6 . We note that while axion haloscope limits are often converted in the literature into limits on DPs in the vein of Ref. [10], many of the published limits have employed magnetic field as a veto on signals (e.g. Refs. [80,81]). Thus, any potential DP signals were rejected, signifying that one cannot directly reinterpret such studies as DP searches. In Ref. [71] such a procedure was also used, however only injected simulation candidate events were present before this was done. Hence, while it is uncertain if a DP would have been observed if there had been a candidate event, one can use the lack of candidates as a limit. In the cases which did not explicitly employ such a procedure, we assume that no veto was employed. If the analysis is done allowing for DP signals, cavity searches such as ADMX [33], HAYSTAC [82], CULTASK [34], OR-GAN [35], KLASH [36] and RADES [37,38] are potentially very sensitive to DPs, particularly for ν 10 Ghz. As multiple axion haloscope searches perform such vetoing, we only display published DP projected sensitivities along with existing limits, the most relevant being dielectric haloscopes like MADMAX [39,83]. With a small enough mass-gap, Dirac materials could also be sensitive in the range of our parameter space of interest [26].
To plot limits, we have assumed that the DP field direction changes relatively rapidly, so the averaged angle can be used for all experiments. As individual measurements are long, plasma and dielectric haloscopes would obtain similar results for a fixed DP direction. However, cavity haloscopes generally measure for very short periods, and so would only sample a single value of cos 2 θ in each measurement. Further, usually multiple measurements, presumably occurring at different times of the day, are combined to get the final limits. To get a limit on a time varying signal would require time and location information for each measurement when combined, in order to take into account signals potentially vanishing at different times of the day. As detailed timing information is usually not reported, it is non-trivial to rigorously turn existing limits into DP limits in such a scenario.
While for axions both quantum limited and constant noise temperatures were explored in Ref. [39], the limits plotted for DP in Ref. [83] assumed a more modest noise temperature of 8 K. Thus for a direct comparison we plot also a more conservative system with T sys = 8 K (dashed green line). We note, however, that as near quantum limited detection is currently available below ∼ 10 GHz, such a plot is somewhat pessimistic in the low mass regime 7 . As shown in Fig. 2 even with conservative parameters plasma haloscopes are capable of searching a large fraction of the DP parameter space, being complementary to dielectric haloscopes like MADMAX. In particular, if both types of experiments use mechanical tuning, the smaller spacings between elements in plasma haloscopes for a given frequency make it more suitable for somewhat lower frequencies.

VII. SIGNAL MODULATION
Due to Earth's motion with respect to the Galaxy there is a time modulation of a DM signal, since for a NFW profile the DM distribution is constant in time in the 7 Recently an alternatively designed dielectric haloscope, DALI [84], was proposed. However, Ref. [84] neglected the quantum limit of linear amplification. HAYSTAC is exploring using squeezed states with Josephson parametric amplifiers at lower frequencies to evade the quantum limit [85], however Ref. [84] assumed linear amplifiers would be used, desiring commercially available technology.

Plasma Haloscope HAYSTAC
Projected sensitivity reach of plasma haloscopes for DP searches assuming an 0.8 m 3 volume of plasma with a quality factor Q = 100. We assume a 5 year livetime, and display both the optimistic quantum limited (solid green) as well as 8 K noise temperature (dashed green) cases. Existing limits (dark gray) from haloscope cavity searches of ADMX [63,[86][87][88] and HAYSTAC [82] are shown. We display the projected DP reach of MADMAX (blue) dielectric haloscope, which assumes a constant 8 K noise temperature [83].
Galactic rest frame during the duration of any experiment. Eq. (18), where v is the velocity of a DM particle in the detector's rest frame, relates the classical field and the particle formulations of DP DM. Aside from the velocity and location of the detector, the orientation of the direction of the wiresẑ in Eq. (33) also changes periodically in the Galactic rest frame. This leads to three distinct time modulation effects (see e.g. [76]): 1) the periodic change of the detector speed with respect to the Galaxy, 2) a possible change in DM density during data taking, and 3), the daily change in orientation of the detector with respect to the Galaxy (due to Earth's rotation about itself). This last effect occurs only for DP, but not axions, in some DM generation models and makes use of the strong directionality of the detector we consider.
The time modulation effects due to the periodic change in the velocity of the detector in the Galactic rest frame are very different when the DM particle is absorbed compared to when it scatters off and deposits only a portion of its kinetic energy within the detector.
When the DM particle is absorbed, the energy deposited in the detector is the particle mass m X plus its kinetic energy of O(10 −6 )m X , since the characteristic DM speed is O(10 −3 )c. The DM signal obtains a finite width due to the spread of the kinetic energy. The peak frequency of the DM line is at ω peak = m X (1 + v 2 Sun /2), where v Sun , the speed of the Sun with respect to the Galaxy, is about 240 km/s (see e.g. [65] for a discussion of the uncertainties in this speed). The orbital motion of Earth, with speed close to 30 km/s, produces an annual modulation of amplitude ∆v of the detector's speed with respect to the Galaxy, and thus a periodic shift (v Sun ∆v)ω peak of O(10 −7 )ω peak in the DM line peak frequency. Hence, detecting this modulation would require an experimental energy resolution of O(10 −7 )m X .
The existence of a escape speed v esc of about 550 km/s from the Galaxy at Earth's location introduces a high frequency cutoff ω max = m X (1 + (v esc + v Sun ) 2 /2) of the DM line. The escape speed has been measured by the RAVE and GAIA surveys, with values between 480 km/s and 640 km/s [89][90][91] with respect to the Galaxy (considering the 90% C.L. intervals of each measurement). The cutoff could be difficult to measure due to the fast decrease of the DM speed distribution with increasing speed, but it is also annually modulated with an ampli- The daily rotation of Earth around itself, whose surface speed at the equator is O(10 −6 )c, would induce an even smaller daily modulation of the peak frequency ω peak (and also of ω max ). This would consequently require an even finer energy resolution of O(10 −9 )m X .
The extremely good energy resolution required to detect the annual modulation of DM line peak, and possibly its daily modulation, could be achievable with a long enough measurement time once a signal has been observed. The potential detection of the time modulation of the peak frequency, as well as other features of the DM line shape, due e.g. to Sun's gravitational lensing, DM streams or a dark disk, have been studied at length, primarily in the context of axion detection, see e.g. [54,76,[92][93][94][95][96].
A seemingly erratic time modulation in the amplitude of the signal could occur due to the existence of pervasive very dense and small (relative to the size of the Solar System) clumps of the DM. These clumps are predicted if DP DM is produced due to fluctuations of the vector field during inflation [12], as described in Ref. [76], e.g. if the experiment passes through a clump about every day and spends several seconds within it.
An important daily modulation of the DP signal could arise due to the strong dependence of the output power of Eq. (34) on the orientation of the DP vector field X with respect to the direction of the wires in the experiment, A daily modulation would only arise if the direction of the DP field is fixed in a large enough region of the Galaxy near Earth, such that the experiment spends at least several days within it. This could be possible if the DP DM is produced through a misalignment mechanism after inflation, depending on details of structure formation in the Universe (see e.g. Ref. [10]). The possibility of observing a directional modulation has been studied for other detectors for DP detection (see e.g. Ref. [61]) and also for axion detection (see e.g. Ref. [54]).
In the most favorable case in which our detector is located at 45 o latitude and the direction of X coincides with the direction of the wires, assumed to be in vertical position in the detector, at one time of the day (so that cos θ = 1 at this instant) the daily modulation would be maximal. The output power would be maximal P out = P MAX out at one time of the day, and 12 hours later (when the azimuth at the location of the experiment would be at 90 o with respect to the original position, thus cos θ = 0) the power would be P out = 0. Thus, in this most favorable case, the daily modulation of the output power would be 100%. Even just splitting the daily observation into two equal time interval bins in this most favorable case, the total signal in the half day centered at the instant of maximum power would contain 92.4% of the daily signal (and the other half just the remaining 7.6%), which would make for a very distinguishable daily modulation. Instead, in the worst case scenario, if the X direction coincides with Earth's rotation axis, cos θ would be a constant (and related to the latitude of the experiment, always assuming vertical wires) and the daily modulation would be zero.
The observation of this directional daily modulation of the signal would not only allow to reconstruct the direction of the field X with respect to the detector, and thus with respect to the Galaxy (any direction in the experiment frame can be easily expressed in the Galactic rest frame and vice-versa -see e.g. Appendix A of Ref. [97], and Ref. [98]), but could also point towards the DP DM production through the misalignment mechanism.
If the DP DM is instead produced through quantum fluctuations of the vector field during inflation, the direction of the field could change many times during the duration of the experiment, and even during a day. As already mentioned in Ref. [76], in this latter case an optimal setup using simultaneously multiple experiments aligned in different directions would be require to determine the instantaneous direction of X.
If a signal is observed, the observation of a daily or annual modulation of the signal would be crucial to clearly identify the signal as due to DP DM.

VIII. CONCLUSIONS
In this work we have shown that recently proposed plasma haloscopes are particularly well suited for DP DM searches. Plasma haloscopes take advantage of inmedium effects, which suppress the signal in conventional searches. Using description based on thermal field theory as well as classical electrodynamics, we have confirmed the DP absorption rate at the experiment. By employing metamaterials, the plasma frequency in plasma haloscopes can be tuned to match the DP mass, which allows to competitively probe the region of the parameter space with DP masses of 6 − 400 µeV. Once detected, analysis of signal modulation will allow for a definitive test that DM has been observed and could shed light on the production mechanism. Here we employ thermal field theory to obtain the DP DM absorption rate, following the discussion of Refs. [49,99]. The emission and absorption rates of a boson by a medium is related to the self-energy of the particle in the medium itself as [100,101] where Γ = Γ abs − Γ prod is the rate with which the considered particle distributions approach thermal equilibrium. Thus, the absorption rate is obtained by calculating the in-medium DP self-energy. This statement corresponds to the optical theorem in the framework of thermal field theory. We need to generalize the approach used for dark photon production by an electromagnetic plasma as in Ref. [49] to the case in which a non-thermal population of cold DM is absorbed by a detector. Let us use the momentum distribution f X (k) so that the DP number density n is For a boson with absorption rate Γ abs and production rate Γ prod one finds [100] ∂f ∂t We consider that only A and X interconvert in the medium so that the production rate of one is equal to the absorption rate of the other. Since the plasma haloscope is at cryogenic temperatures there is no separate population of photons in the medium. For the DP masses considered here the occupation number is large, f X 1.
FIG. 3. Feynman diagram representing the DP (single line) self energy contribution due to the mixing with the plasmon (double line), through the vertex χm 2 X (crosses). The plasmon propagator is cut by the diagonal dashed line to indicate that its frequency is taken to be equal to the DP mass as its momentum goes to zero, i.e. ω = mX = ωp.
Hence, the DP and photon evolve respectively as where Γ X = Γ X abs −Γ X prod . Here, we are assuming that the passage from vacuum to the haloscope is strongly nonadiabatic, so that the components in the basis of interaction eigenstates of the vacuum propagation eigenstate constituting the DM are conserved after the interface. This means that the active component is small right after passing through the medium interface (as it is proportional to χ, not χ/ ), while the sterile component is large. This is analogous to the case of neutrinos oscillating in a matter potential in the "slab approximation" [102,103].
Using ∂f A /∂t = −∂f X /∂t, one has where in the last approximation we have assumed that f X f A and used Eq. (A1). At lowest order the DP self-energy, shown in Fig. 3, is given by [49] Π L,T X = m 2 X + m 2 X χ where K = (ω, k) is the four-momentum of the external DP and Π L,T are the self-energy of the longitudinal and transverse plasmons respectively (we drop A in the notation). Both longitudinal and transverse excitations can be produced by the DP population, which can be absorbed both when with longitudinal and transverse polarization. From Eq. (A6), it is seen that we need the plasmon self-energy to obtain the DP absorption rate. The real part, which modifies the dispersion relations of photons in the medium, in the nonrelativistic approximation is given by [49] Re Π T = ω 2 p , (A8a) where ω p is the plasma frequency. These are Eqs. (6.32) and (6.38) of Ref. [47], considering that they are written for the case in which Im Π T,L = 0. We see that the dispersion relation for the transverse plasmon gives ω 2 − k 2 = ω 2 p , with the usual interpretation of transverse excitations as particles with mass ω p . The latter is given in a system of wires by Eq. (30). The longitudinal plasmon, on the other hand, has a peculiar dispersion relation, so that in the nonrelativistic limit ω is independent from k. The imaginary part of the photon self-energy is related to the rate Γ by which plasmons thermalize in the medium -i.e., the damping in the classical description.
When treating longitudinal plasmons one defines the vertex renormalization constant Z L [48] relevant for the coupling of external photons or plasmons to electrons in the medium. We then interpret Enforcing the on-shell condition K 2 = m 2 X , the resulting absorption rate of DPs is where we have defined the longitudinal and transverse DP populations (f L X and f T X , respectively, and f L X + f T X = f X ), and we have used Eq. (A1) to define Γ T = −Im Π T /ω. This expression closely resembles the production rate of X bosons obtained in Ref. [104]. Moreover, interpreting as an effective coupling, we find the same expression as Eq. (4.4) of Ref. [26] (see also Refs. [21,27]). The longitudinal and transverse photons are indistinguishable in the zero momentum limit, Γ T = Γ L ≡ Γ, so that for non-relativistic DPs on resonance (ω = m X = ω p ) the number absorption rate per unit volume is As long as the dark matter distribution is narrower than the line width of the resonance, the DP momentum distribution can be written as where n = ρ/m X is the DM number density, which coincides with Eqs. (24) and (21). Thus, the power absorbed in a homogeneous and isotropic detector with volume V d is given by where Q = ω/Γ is the quality factor and ρ is the local DM density. Equation (A15) is equivalent to Eq. (26) in the main text in the limit where boundary conditions are negligible so that the "geometric factor" goes to unity.