Dark photon limits: A handbook

The dark photon is a massive hypothetical particle that interacts with the Standard Model by kinetically mixing with the visible photon. For small values of the mixing parameter, dark photons can evade cosmological bounds to be a viable dark matter candidate. Due to the similarities with the electromagnetic signals generated by axions, several bounds on dark photon signals are simply reinterpretations of historical bounds set by axion haloscopes. However, the dark photon has a property that the axion does not: an intrinsic polarization. Due to the rotation of the Earth, accurately accounting for this polarization is nontrivial, highly experiment dependent, and depends upon assumptions about the dark photon’s production mechanism. We show that if one does account for the dark photon polarization, and the rotation of the Earth, an experiment’s discovery reach can be enhanced by over an order of magnitude. We detail the strategies that would need to be taken to properly optimize a dark photon search. These include judiciously choosing the location and orientation of the experiment, as well as strategically timing any repeated measurements. Experiments located at $\pm 35°$ or $\pm 55°$ latitude, making three observations at different times of the sidereal day, can achieve a sensitivity that is fully optimized and insensitive to the dark photon’s polarization state, and hence its production mechanism. We also point out that several well-known searches for axions employ techniques for testing signals that preclude their ability to set exclusion limits on dark photons, and hence should not be reinterpreted as such.

Figure 1

Current constraints on the DP’s mass, $m_X$, and kinetic mixing parameter with the SM photon, $\chi$. The general color scheme is cosmological bounds in blue, experimental bounds in red, and astrophysical bounds in green. The thick white line that divides the parameter space in two is the upper limit for which DPs are a viable candidate for 100% of the DM. The focus of this work are the experimental bounds that reach below this line. Descriptions of each bound are given in Sec. 2.

Figure 2

Schematic of the two categories of experiment that we structure this study around. They are distinguished by their sensitivity to the direction of the DP polarization. On the left, “axial” experiments sensitive to a single direction of polarization are represented via a cylindrical cavity. In the middle, “planar” experiments, sensitive to a plane of polarization, are shown using a simple sketch of a dish antenna. We express the DP polarization here in the lab-centered north-west-zenith basis shown in the far right image. To introduce some nomenclature that will be important later: the cavity in this example would be sensitive to electric fields along the $\widehat{\mathcal{Z}}$ axis, so we refer to this as “zenith pointing.” On the other hand, the dish antenna is facing north, so it is sensitive polarizations in the $\widehat{\mathcal{Z}}-\widehat{\mathcal{W}}$ plane, which we refer to as “north facing.”

Figure 3

A diagram of the coordinate systems used in this paper. On the left, a geocentric equatorial coordinate system in which we fix the DP polarization vector, $\widehat{\mathbf{X}}$, defined by angles $({\theta }_X,{\varphi }_X)$. The ${\widehat{\mathbf{x}}}_e$ direction is conventionally chosen to point towards the vernal equinox, with ${\widehat{\mathbf{y}}}_e$ pointing 90° of right ascension to the east. The Earth rotates anticlockwise in the ${\widehat{\mathbf{x}}}_e-{\widehat{\mathbf{y}}}_e$ plane. On the right, the detector-centric Cartesian coordinate system using axes pointing towards the north, west, and zenith. We also show three planes in the coordinate system which we will need when describing those experiments that are sensitive to the component of $\widehat{\mathbf{X}}$ projected onto a two-dimensional plane. The zenith-facing plane is horizontal, whereas the north-facing and west-facing planes are both vertical in the lab.

Figure 4

We define $⟨{\cos }^2\theta {⟩}_T^{\mathrm{excl}}$ in Eq. (34) to parametrize how much a 95% C.L. DP exclusion limit is impacted by the distribution of possible polarization angles. This parameter is an effective conversion factor for recasting prior exclusion limits into the fixed polarization scenario. Here, we show the latitude dependence of this factor when $T$ is an integer number of sidereal days. The left-hand panel is for axial (1D) experiments, and the right-hand panel is for planar (2D) experiments. Each panel has three lines for the three possible orientations of those experiments. This figure is to demonstrate the preferential latitudes derived in Eqs. (46) and (49) for integer-day-long measurements. The factor plotted here depends upon the shape of the distribution of $⟨{\cos }^2\theta {⟩}_T$ and is typically at the 15th–30th percentile.

Figure 5

Distributions of $⟨{\cos }^2\theta (t){⟩}_T$ after sampling the DP polarization $({\theta }_X,{\varphi }_X)$ isotropically across the sky. For each value of $T$ we use the logarithmic color scale to show the value of the distribution, normalized by its maximum value. The three columns correspond to three different latitudes, ${\lambda }_{\mathrm{lab}}=35°$, 45°, and 55°, from left to right. The two rows correspond to north-pointing experiments (upper panels) and zenith-pointing experiments (lower panels). Notice that at integer values of $T$ in sidereal days, the distribution approaches a single point at a value of $1/3$ in the upper right-most panel, and the lower left-most panel. These correspond to the peaks of the north-pointing and zenith-pointing lines shown in the left-hand panel of Fig. 4. The white lines correspond to our two conversion factors which we use to parametrize how much DP limits are impacted by the distribution of $⟨{\cos }^2\theta {⟩}_T$ in the fixed polarization scenario, as described in Sec. 5f. The dashed line is $⟨{\cos }^2\theta {⟩}_T^{\mathrm{excl}}$, as defined in Eq. (34), and can be used to rescale exclusion limits. Whereas the dot-dashed line is defined in Eq. (36), and can be used to rescale discovery limits.

Figure 6

As in Fig. 5, but now for planar experiments, i.e., those where the DP signal depends upon the angle between the DP polarization and a plane. We show the distributions of $⟨{\cos }^2\theta (t){⟩}_T$ after sampling the DP polarization $({\theta }_X,{\varphi }_X)$ isotropically across the sky. For each value of $T$ we use the color-scale to display the distribution, which we have normalized by its maximum value. The three columns correspond to three different latitudes, ${\lambda }_{\mathrm{lab}}=35°$, 45°, and 55°, from left to right. The two rows correspond to north-facing (upper panels) and zenith-facing (lower panels) experiments. Notice that at integer values of $T$ in sidereal days, the distribution approaches a single point at a value of $2/3$ in the upper right-most panel, and the lower left-most panel. These correspond to the peaks of the north-facing and zenith-facing lines in the right-hand panel of Fig. 4.

Figure 7

Closeup of bounds on DPs in the radio-microwave frequency regime. The limits set by axion cavity haloscopes are in shades of red, whereas the limits from devoted DP experiments operating in the same range of frequencies are shown in shades of green. We impose an upper limit on DPs as DM from the bounds of Refs. [39, 121] (the same upper limit as shown in Fig. 1). All of the experimental bounds shown here have been rescaled from the original sources. Firstly, they have been rescaled such that they all assume the same DM density of ${\rho }_0=0.45\mathrm{GeV}{\mathrm{cm}}^{-3}$. Secondly, we have rescaled them such that they all consider the fixed DP polarization scenario. This relies on the factor $⟨{\cos }^2\theta {⟩}_T^{\mathrm{excl}}$, which is different for each experiment. The method of deriving these factors is detailed in Sec. 5g, and the result for each experiment is listed in the final columns of Tables 1 and 2.

Figure 8

We display the improvement that can be made to both exclusion limits (top row) and discovery limits (bottom row) on the DP’s kinetic mixing parameter, $\chi$, as a function of the measurement time, $T$, and for several experiment geometries and orientations. The left-hand panels correspond to axial experiments, whereas the right-hand panels correspond to planar experiments. In all cases we display the improvement made to the limit relative to the most conservative possible assumption (${\chi }_{\text{conserv}}$.): ignoring daily modulation and taking the instantaneous results for $⟨{\cos }^2\theta {⟩}_T^{\mathrm{excl}}$ and $⟨{\cos }^2\theta {⟩}_T^{\mathrm{disc}}$ when $T\to 0$. In the scenario that the DP has a random polarization every coherence time, $⟨{\cos }^2\theta {⟩}_T$ effectively takes on a single value of $1/3$ (axial) or $2/3$ (planar), and is the best the experiment could do. What this figure shows is that by properly accounting for the daily modulation and how that changes the distribution of $⟨{\cos }^2\theta {⟩}_T$, the sensitivity in the fixed-polarization scenario can be significantly improved—and for certain cases it can even match the random-polarization scenario. The potential improvement could be up to a factor of $\sim 3.75$ or $\sim 1.36$ in the exclusion limits, and $\sim 10.5$ or $\sim 2.37$ in the discovery limits.

Figure 9

In each panel the color scale refers to the value of $⟨{\cos }^2\theta {⟩}_{3T}^{\mathrm{disc}}$ obtained from three measurements, each of duration $T$, and each separated by time $T_{\mathrm{wait}}$. The left column corresponds to axial experiments, and the right column to planar experiments. The top row is for north-pointing/facing experiments located at their optimal latitude of $\sim 55°$, the middle row is for west-facing/pointing experiments which can be located at any latitude, and finally the bottom row is for zenith-pointing/facing experiments at the optimal latitude of $\sim 35°$. Note that in all cases the distribution is periodic in $T_{\mathrm{wait}}$ over one sidereal day, so there is no need to extend the plots vertically. The orange contours with labels enclose the region of the $T-T_{\mathrm{wait}}$ space where that choice of $T_{\mathrm{wait}}$ leads to an enhancement $⟨{\cos }^2\theta {⟩}_{3T}^{\mathrm{disc}}/⟨{\cos }^2\theta {⟩}_T^{\mathrm{disc}}$ by at least the amount shown.

Figure 10

Projected DP discovery limits for planned experiments, and future runs of existing experiments: DM-Radio [191], dark $E$ field radio [35], ALPHA [38, 159], MADMAX [160], LAMPOST [36], SuperCDMS [94] (assuming a Ge target), and LZ [192]. We have plotted projections for two scenarios, the “standard” projection is the most pessimistic and assumes that no timing or directional information is taken into account when running the experiment, or calculating the limit. Our “optimized” projection would be obtained if the experiment followed some simple changes to the data taking that are outlined in the text of Sec. 6e. For the rest of the bounds, rather than show every existing limit individually as in Fig. 1, we have combined the limits into three categories based on the level of assumption involved. The red region encloses all constraints that are based on $\mathrm{photon}\to \mathrm{DP}$ transitions (e.g. light-shining through walls) which in a sense are the least model dependent. In green, we show bounds invoking stellar cooling arguments, which also rely on photon-DP conversion, but could be circumvented through similar model-dependent arguments as those used for axions [97, 193, 194, 195, 196, 197, 198]. Then in blue we show the bounds that rely on DPs comprising the majority of DM. In gray, we show the direct detection bounds from devoted DP experiments, namely, SHUKET, WISPDMX, and dark $E$ field radio. Note that the future projections vary in their levels of optimism about the future, hence this is not the fairest comparison; however it serves to demonstrate what might be possible.

Figure 11

Three mock examples of the reconstruction accuracy of the DP polarization axis. The true polarization axis is given by the black stars. In each case we assume that the experiment has been able to discriminate the signal from background at the 95% C.L. in one day, and the value of $T$ corresponds to the additional amount of data used to test for the daily modulation. The size of each contour corresponds to the expected $2\sigma$ contour around $({\theta }_X,{\varphi }_X)$ that the experiment could set given the measurement time $T$. We display these angles in galactic longitude and latitude $(l,b)$ with the galactic plane running horizontally through $b=0$, and the galactic centre at (0,0). The celestial equator is shown with a purple line, and the projected directions of the north and south poles are shown with purple circles.