Kinetic Mixing, Dark Photons and Extra Dimensions III: Brane Localized Dark Matter

Extra dimensions have proven to be a very useful tool in constructing new physics models. In earlier work, we began investigating toy models for the 5-D analog of the kinetic mixing/vector portal scenario where the interactions of dark matter, taken to be, e.g., a complex scalar, with the brane-localized fields of the Standard Model (SM) are mediated by a massive $U(1)_D$ dark photon living in the bulk. These models were shown to have many novel features differentiating them from their 4-D analogs and which, in several cases, avoided some well-known 4-D model building constraints. However, these gains were obtained at the cost of the introduction of a fair amount of model complexity, e.g., dark matter Kaluza-Klein excitations. In the present paper, we consider an alternative setup wherein the dark matter and the dark Higgs, responsible for $U(1)_D$ breaking, are both localized to the 'dark' brane at the opposite end of the 5-D interval from where the SM fields are located with only the dark photon now being a 5-D field. The phenomenology of such a setup is explored for both flat and warped extra dimensions and compared to the previous more complex models.


Introduction
The nature of dark matter (DM) and its possible interactions with the fields of the Standard Model (SM) is an ever-growing mystery. Historically, Weakly Interacting Massive Particles(WIMPs) [1], which are thermal relics in the ∼ few GeV to ∼ 100 TeV mass range with roughly weak strength couplings to the SM, and axions [2,3] were considered to be the leading candidates for DM as they naturally appeared in scenarios of physics beyond the Standard Model (BSM) that were constructed to address other issues. Important searches for these new states are continuing to probe ever deeper into the remaining allowed parameter spaces of these respective frameworks. However, the null results so far have prompted a vast expansion in the set of possible scenarios [4,5] which span a huge range in both DM masses and couplings. In almost all of this model space, new forces and, hence, new force carriers must also exist to mediate the interactions of the DM with the SM which are necessary to achieve the observed relic density [6]. One way to classify such interactions is via "portals" [7] of various mass dimension that result from integrating out some set of heavy fields; at the renormalizable level, the set of such portals is known to be quite restricted [7,8]. In this paper, we will be concerned with the implications of the vector boson/kinetic mixing (KM) portal, which is perhaps most relevant for thermal DM with a mass in the range of a ∼ few MeV to ∼few GeV and which has gotten much attention in the recent literature [8]. In the simplest of such models, a force carrier, the dark photon (DP), the gauge field corresponding to a new gauge group, U (1) D , under which SM fields are neutral, mediates the relevant DM-SM interaction. This very weak interaction is the result of the small KM between this U (1) D and the SM hypercharge group, U (1) Y , which is generated at the one-(or two)-loop level by a set of BSM fields, called portal matter (PM) [8,9], which carry charges under both gauge groups. In the IR, the phenomenology of such models is well-described by suitably chosen combinations of only a few parameters: the DM and DP masses, m DM,V , respectively, the U (1) D gauge coupling, g D and , the small dimensionless parameter describing the strength of the KM, ∼ 10 − (3−4) . Frequently, and in what follows below, this scenario is augmented to also include the dark Higgs (DH) boson, whose vacuum expectation value (vev) breaks U (1) D thus generating the DP mass. This introduces two additional parameters with phenomenological import: the DH mass itself and the necessarily (very) small mixing between the DH and the familiar Higgs of the SM. Successfully extending this scenario to a completion in the UV while avoiding any potential issues that can be encountered in the IR remains an interesting model-building problem.
Extra dimensions (ED) have proven themselves to be a very useful tool for building interesting models of new physics that can address outstanding issues that arise in 4-D [10]. In a previous pair of papers [11,12], hereafter referred to as I and II respectively, we considered the implications of extending this familiar 4-D KM picture into a (flat) 5-D scenario where it was assumed that the DM was either a complex scalar, a Dirac fermion or a pseudo-Dirac fermion with an O(1) mass splitting. In all cases we found some unique features of the 5-D setup, e.g., the existence of strong destructive interference between the exchanges of Kaluza-Klein (KK) excitations of the DP allowing for light Dirac DM, which is excluded by CMB [13,14] constraints in 4-D, new couplings of the split pseudo-Dirac states to the DP that avoids co-annihilation suppression found in 4-D [15], or the freedom to choose appropriate 5-D wave function boundary conditions, etc, all of which helped us to avoid some of the model building constraints from which the corresponding 4-D KM scenario can potentially suffer.
The general structure of the model setups considered previously in I and II followed from some rather basic assumptions: (i) The 5-D space is a finite interval, 0 ≤ y ≤ πR, that is bounded by two branes, upon one of which the SM fields reside while the DP lives in the full 5-D bulk. This clearly implies that the U (1) D − U (1) Y KM must solely occur on the SM brane. The (generalization of) the usual field redefinitions required to remove this KM in order to obtain canonically normalized fields then naturally leads to the existence of a very small, but negative brane localized kinetic term (BLKT) [16] for the DP which itself then leads to a tachyon and/or ghost field in its Kaluza-Klein (KK) expansion. We are then led to the necessary conclusion that an O(1) positive BLKT must already exist to remove this problem; the necessity of such a term was then later shown to be also very useful for other model building purposes. (ii) A simple way to avoid any significant mixing between SM Higgs, H, and the dark Higgs, S which is employed in 4-D to generate the DP mass, is to eliminate the need for the dark Higgs to exist. This then removes the necessity of fine-tuning the parameter λ HS in the scalar potential describing the ∼ S † SH † H The outline of this paper is as follows: In Section 2, we present the construction of this model while remaining agnostic to the specific geometry of the extra dimension, while in Section 3 we specialize our discussion to the case in which the extra dimension is flat and present a detailed analysis of this scenario. In Section 4, we present an analogous discussion of the model in the case of a warped extra dimension, with appropriate comparison to the results from the flat case scenario. Section 5 contains a summary and our conclusions.

General Setup Overview
Before beginning the analysis of the current setup, we will very briefly review the formalism from I that remains applicable, generalizing it slightly to incorporate either a flat or warped extra dimension. As noted in the Introduction, we consider the fifth dimension to be an interval 0 ≤ y ≤ πR bounded by two branes; for definiteness we assume that the SM is confined to the y = 0 brane whereas the DM and dark Higgs are constrained to the opposite brane at y = πR. For a flat extra dimension, this assignment of branes is arbitrary, however we shall see that it has some physical motivation in the warped case. For now, we shall describe the metric as following the form, where f (y) is simply some function of the bulk coordinate y: For a flat extra dimension, f (y) = 1, while for an RS setup, f (y) = e −ky , where k is a curvature scale. The DP, described by a gauge fieldV A (x, y) lies in the full 5-D bulk, and kinetically mixes with the 4-D SM hypercharge gauge fieldB µ (x) on the SM brane via a 5-D KM parameter 5 as described (before symmetry breaking) by the action where c w = cos(θ w ), the weak mixing angle, Greek indices denote only the 4-dimensional vector parts of the gauge fieldV , andV y denotes the fifth component of this field. Here we see immediately that shifting the fields in the usual manner on the SM brane to remove the KM produces the small negative BLKT ∼ − 2 5 Rc 2 w mentioned above. The presence of such a negative brane term is highly suggestive of the necessity of introducing a (larger) positive BLKT to the model (for example, in the case of a flat extra dimension, negative BLKT's such as this are well known to lead to tachyonic KK modes or ghost-like states). We shall later see more explicitly that such a positive BLKT is in fact necessary in both the cases of a warped and a flat extra dimension, working from the treatment of kinetic mixing in the effective 4-dimensional theory with an infinite tower of KK modes. Since spontaneous symmetry breaking takes place on the dark brane via the vev of the dark Higgs, S, we know [24,25] that in the KK decomposition the 5th component ofV A (which does not experience KM) and the imaginary part of S combine to form the Goldstone bosons eaten byV to become the corresponding longitudinal modes. So, we are free in what follows to work in the V y = 0 gauge, at least for the flat and Randall-Sundrum-like geometries that we are considering here. Then the alluded to the relevant KK decomposition for the 4-D components of V is given by 1V where we have factored out R −1/2 in order to render v n (y) dimensionless. To produce a Kaluza-Klein tower, then we will require that the functions v n (y) must satisfy the equation of motion in the bulk, where here the m n are the physical masses of the various KK excitations. Defining the KK-level dependent quantity n = 5 v n (y = 0), which we see explicitly depends on the values of the DP KK wavefunctions evaluated on the SM brane, we see that the 5-D KM becomes an infinite tower of 4-D KM terms given by n n 2c wV µν nBµν .
As discussed in I, the intuitive generalization of the usual kinetic mixing transformations,B µν = B µν + n n cw V µν n ,V µν → V µν , will be numerically valid in scenarios in which the infinite sum n 2 n / 2 1 is approximately < ∼ O(10), and 1 1. Otherwise, terms of O( 2 1 ) (at least) become numerically significant and can't be ignored in the analysis, even if each individual n remains small. In both the cases of a warped and flat extra dimension, the sum n 2 n / 2 1 is within the acceptable range as long as there is a sufficiently large positive BLKT on the same brane as the SM-DP kinetic mixing, as was shown for flat space in I and will be demonstrated for warped space in Section 4. So, by selecting 1 ∼ 10 − (3−4) , within our present analysis we can always work to leading order in the n 's, and thus the transformationŝ B µν = B µν + n n cw V µν n ,V µν → V µν will be sufficient for for our purposes in removing the KM. Finally we note that the sum of the brane actions corresponding to the usual (positive) BLKT term for V on the SM brane and the corresponding dark Higgs generated mass term for V on the dark brane is given by where factors of R have been introduced to make τ dimensionless as usual and for m V to have the usual 4-D mass dimension. We note that one of the main advantages of our present setup is that the dark Higgs which generates the brane mass term m V is isolated from any mixing with the SM Higgs, and as a result, its phenomenological relevance in this construction is quite limited. As such, for our analysis we can ignore this scalar and instead simply assume the existence of the brane-localized mass term m V without further complications. We will define the 4-D gauge coupling of the DP to be that between the DM and the lowest V KK mode as evaluated on the dark brane. The action S branes supplies the boundary conditions, as well as the complete orthonormality condition, necessary for the complete solutions of the v n . These are for the boundary conditions, and for the orthonormality condition. At this point, once the function f (y) is specified, as we shall do in Sections 3 and 4 for a flat and a warped extra dimension respectively, it is possible to uniquely determine the bulk wave functions v n (y) for all n given the parameters R, τ , m 2 V , and whatever additional parameters are necessary to uniquely specify f (y).
Beyond discussing characteristics of individual KK modes, we shall find it convenient at times in our analysis to speak in terms of summations over exchanges of the entire DP KK tower. In particular, the sum shall appear repeatedly in our subsequent discussion, where for our purposes here s is simply a positive number, but in our actual analysis shall denote the Mandelstam variable of the same name. To evaluate this sum, we can perform an analysis similar to that of [25,26]. First, we note that the orthonormality condition of the KK modes in Eq.(8) requires that where the sum in the second line of this equation is over all KK modes n. We then note that the equation of motion Eq.(4) and the y = 0 boundary condition of Eq. (7) can be recast in an integral form as Using this integral form of the equation of motion for v n (y), we can now compute the sum F (y, y , s). Combining Eqs. (10) and (11), we can write the integral equation Eq. (12) can be straightforwardly rewritten as a differential equation, ∂ y [f (y) 2 ∂ y F (y, y , s)] = Rδ(y − y ) − sF (y, y , s), where the y = 0 boundary condition is explicitly in the integral equation Eq. (12), while the second is easily derivable from the y = πR boundary condition on v n (y) given in Eq. (7). Once a function f (y) (and therefore a metric) has been specified, the function F (y, y , s) is then uniquely specified by Eq. (13).
With equations of motion for the KK modes' bulk profiles v n (y) and the summation F (y, y , s) specified, it is now useful to discuss some general aspects of our construction's phenomenology before explicitly choosing a metric. First, we note that the effective couplings of the n th KK mode of the DP to the DM on the dark brane are given by g DM n = g 5D v n (y = πR)/ √ R, where g 5D is the 5-dimensional coupling constant appearing in the theory, while recalling that the effective kinetic mixing parameters n are similarly given by n = 5D v n (y = 0)/ √ R. In terms of the value of these parameters for the least massive KK mode, g D ≡ g DM 1 and 1 , we can then write Armed with these relationships, our subsequent analysis will treat 1 , g D , and the mass of the least massive KK DP excitation, m 1 (which we trade for R), as free parameters and identify them with the corresponding quantities that appear in the conventional 4-D KM portal model.
Finally, we also note that it is useful to give symbolic results for two cross sections that are of particular phenomenological interest for dark matter within the mass range we are considering, and which can be expressed in a manner agnostic to the specific functional form of the bulk wavefunctions v n (y) and the sum F (y, y , s). First, we note that in the limit where the mass of the DM particle m DM is far greater than the mass of an electron, we can approximate the DM-electron scattering cross section for direct detection as To ensure that our DM candidate produces the correct relic abundance, we also must compute the thermally averaged annihilation cross section for DM into SM particles (which we shall denote by the symbol σ), weighted by the Møller velocity of the DM particle pair system v M øl in the cosmic comoving frame [27]. We are careful to note that σ here refers to the Lorentz-invariant cross section. To find this average, we must integrate σv Møl weighted by the two Bose-Einstein energy distributions, f (E), of the complex DM fields in the initial state. As noted in [27], if the freeze-out temperature, T F satisfies x F = m DM /T F > ∼ 3 − 4 as it will below, we can approximate these Bose-Einstein distributions with Maxwell-Boltzmann ones, and can employ the following formula to express the thermal average as a one-dimensional integral, where K n (z) denotes the modified Bessel function of the second kind of order n, v lab is the relative velocity of the two DM particles in a frame in which one of them is at rest, and ε ≡ (s − 4m 2 DM )/(4m 2 DM ), i.e., the kinetic energy per unit mass in the aforementioned reference frame. This integral can be performed numerically; in our numerical evaluations here we will assume x F = 20 but note that other values in the 20-30 range give very similar results. We can proceed now by computing the cross-section for the annihilation of a DM particle-antiparticle pair into a pair of SM fermions of mass m f and electric charge Q f , in which case we invoke the following expression for the cross-section of a 2 → 2 process, where s is the standard Mandelstam variable, m DM is the mass of the DM particle, Ω is the center-ofmass scattering angle and M is the matrix element for the annihilation process we are considering. When s is far from any KK mode resonances, we arrive at the result σv lab = 1 3 In practice, for both of the specific cases we shall consider in our analysis, we shall find it necessary to consider regions of parameter space such that DM annihilation through the first KK mode enjoys some resonant enhancement [28]. In order to accommodate this scenario, we have to modify Eq. (18) slightly, arriving at where Γ i is the total width of V i which we need to calculate as a function of m i . Physically, we have simply subtracted the contribution of the lowest-lying KK mode from the sum F (0, πR, s), where its propagator appears with its pole mass, and added this contribution again with the Breit-Wigner mass instead. Since the annihilation of two complex scalars into a pair of fermions through a vector gauge boson is p-wave process, and so is v 2 rel suppressed at later times (i.e., at lower temperatures when the DM is moving slowly), we are safe from the previously mentioned strong constraints on DM annihilation during the CMB at z ∼ 10 3 [23]. We further note that if m DM > m 1 , then we would expect the s-wave process φφ † → 2V 1 to be dominant for unsuppressed values of g D . In order to avoid this possibility, we must then require that m DM < m 1 and this will be reflected in our considerations below. We note that if m 1 > 2m DM then the O(g 2 D ) decay V 1 → φφ † will dominate, otherwise, V 1 will decay to SM fermions with a suppressed O(e 2 2 1 ) decay partial width. It should be noted that in computing the cross sections of Eqs. (15) and (19), we have treated the system as though all KK tower members couple to the SM like normal dark photons, namely, V n couples to SM fermions with a coupling strength of eQ n . As noted in I, this is not strictly true: If the full computation for the various model couplings is done, the DP coupling of the n th KK mode to SM fermions is only well-approximated by this expression if m n , the KK mode's mass, is much less than the mass of the Z boson, which if we sum over the entire infinite tower of KK modes breaks down for sufficiently large n. However, numerically we find that the cross sections in Eqs. (15) and (19) are heavily dominated by exchanges of the lighter KK modes in which this approximation is valid, rather than the exchanges of the heavier modes with m n > ∼ m Z for which the approximation breaks down. Similarly, the dominance of the lighter KK modes with m n m Z in these cross sections numerically overwhelms the contribution from Z boson exchanges, even though the Z boson also possesses non-zero couplings to both the SM and DM fields once mixing is fully accounted for.

Flat Space Model Setup
In order to further explore the phenomenology of our construction, we must now specify the geometry of the extra dimension, namely by selecting a specific function f (y) in Eq. (1). With this determined, we can then straightforwardly find the spectrum of KK gauge bosons V n , their bulk wavefunctions v n (y), and concrete expressions for the cross sections of Eqs. (15) and (19). Initially, we shall consider the case of a flat extra dimension, i.e., f (y) = 1. The equation of motion for the bulk profile v n (y) is then straightforward; from the generic case given in Eqs. (4) and (7), we quickly arrive at which when combined with the orthonormality condition Eq. (8) quickly yields the expressions where we have defined the dimensionless quantities x F n and a F from combinations of dimensionful parameters for the sake of later convenience 2 . The allowed values of x F n (and hence the mass spectrum of the KK tower) are given by the solutions to the equation, Given the results of Eqs. (21) and (22), we can now examine the behavior of a number of phenomenologically relevant quantities. To begin, it is useful to get a feel for the numerics of x F 1 , the lowest-lying root of the mass eigenvalue equation Eq. (22). Since we are free to choose m 1 within the ∼ 0.1 − 1 GeV mass range of interest, the lowest root x F 1 = m 1 R tells us the value of the compactification radius R within this setup, hence, the value of x F 1 (a F , τ ) is important to consider. In I, where boundary conditions were used to break U (1) D , the parameter a F is, of course, absent. However, it was found that x F 1 in that case was a decreasing function of τ , as is typical for the effect of BLKTs, with x F 1 (τ = 0) = 1/2. Here, on the other hand, it is the value of a F = 0 that generates a mass for the lowest lying DP KK state so that we expect x F 1 → 0 as a F → 0 and thus to grow with increasing a F . The top and bottom panels of Fig. 1 show that, indeed, the values of x F 1 follow this anticipated behavior. For a fixed value of a F , x F 1 decreases as τ increases and for a fixed value of τ , x F 1 increases with the value of a F . Beyond the position of the lowest-lying root of Eq. (22), the particular spectrum of the more massive KK modes are obviously of significant interest. A clear phenomenological signal for the types of models we are considering is the experimental observation of the DP KK excitations, perhaps most importantly that of the second DP KK excitation. Hence, knowing where the 'next' state beyond the lowest lying member of the KK tower may lie is of a great deal of importance, i.e., where do we look for the DP KK excitations if the lowest KK state is discovered? In Fig. 2 we display the ratio as a functions of a F , τ and we see that for a reasonable variation of these parameters this mass ratio lies in the range 3 − 4. Note that for fixed a F this ratio increases with increasing τ (mostly since since x F 1 is pushed lower). Meanwhile, for any fixed value of τ , this ratio sharply declines with increasing a F in the region a F 1 (largely because x F 1 itself decreases sharply in this regime), while for a F > ∼ 1 the ratio slowly increases with increasing a F . Non-zero values of a F , τ particularly influence the low mass end of the DP KK mass spectrum as, e.g., a F = 0 provides the mass for the lightest KK mode in the present case. However, beyond the first few KK levels the masses of the DP KK states, in particular the ratio m n /m 1 grows roughly linearly with increasing n with a slope that is dependent on the values of the parameters a F , τ as is shown in Fig. 3. It is actually straightforward to see the eventual linear trend of the lines in Fig. 3 analytically, using the root equation Eq. (22). In particular, note that as x F n → ∞, as a function of τ for a F =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Bottom) As in the previous panel, but now as a function of a F assuming τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively.
Eq. (22) approaches where tanc(z) ≡ tan(z)/z. It is well known that the difference between consecutive solutions of tanc(z) = C, for some constant C, approaches π for very large z. So, we can see that for high-mass KK modes, the difference between consecutive solutions of Eq. (22) will approach 1. Hence, the slope of the lines in Fig. 3 can be easily approximated as ∼ (x F 1 ) −1 , and will therefore exhibit the inverse of the dependence of x F 1 on the parameters τ and a F , which we have already observed in Fig. 1. In addition, we can note that without taking the ratio of x F n to x F 1 , any large-n solution of Eq. (22) eventually follows the pattern x F n ≈ n. The next quantities of phenomenological relevance are the relative values of the KM parameters, n / 1 , and the couplings of the DP KK tower states to DM, g n DM /g D ; note that these latter quantities are found to oscillate in sign. Before exploring the numerics in detail here, it is useful to note that one can get a feel for the behavior of these ratios by purely analytical methods. In particular, by invoking Eqs. (14), (21), and (22), it is possible to derive the expressions .
From Eq. (24), we can readily take the limits of ( n / 1 ) 2 and (g n DM /g D ) 2 at large n (and hence large x F n ≈ n). We arrive at the result that as n → ∞ From the first expression in Eq. (25), we see that the ratio ( n / 1 ) falls roughly as 1/n for large n; this result is readily borne out numerically in the top panel of Fig. 5, where we also see that even for small n, n never significantly exceeds the value of 1 , offering encouraging evidence that the small-KM limit we took in Section 2 was valid. More rigorously demonstrating this validity, however, will require the use of sum identities we shall derive later in this section.
In contrast to the behavior of the effective kinetic mixing terms n / 1 , the ratio |g n DM /g D | approaches a constant non-zero value as n → ∞. The precise value of this asymptotic limit of the ratio |g n DM /g D | is naturally of quite significant phenomenological interest: If |g n DM /g DM | is large, one might be concerned that even for a reasonable value of g D 1, the DM particle may experience some non-perturbative couplings to the various KK modes. 3 In Fig. 4, we explore the τ and a F dependence of this asymptotic coupling limit numerically; notably, we find that the coupling ratio increases sharply as a F increases. For comparison's sake, in both panels of Fig. 4, we have depicted as a dashed line the maximum |g n DM /g D | that would be allowed such that all couplings would remain perturbative (that is, have a structure constant (g n DM ) 2 /(4π) < 1) given a choice of g D = 0.3, that is, assuming that the coupling of DM to the first KK mode of the dark photon field has approximately the same coupling constant as the electroweak force. In the figure then, we see that such a choice of g D is only permitted when a F < ∼ 3/2; much larger and the DM interactions with large-n KK modes become strongly coupled. In both Figs. 4 and 5, however, we see that limiting our choice of a F to a F < ∼ 3/2 leads to substantially more modest asymptotic values of |g n DM /g D |, of < ∼ 10. Because |g n DM /g D | rises quadratically (or more accurately, the square of this ratio rises quartically) with increasing a F , these conditions would be only slightly less restrictive if a somewhat smaller value of g D , e.g., g D = 0.1 were chosen.
To continue our discussion of the phenomenology of our construction, we must now also find the sum F (y, y , s), which we remind the reader is defined in Eq. (9), for the flat space case, which we can accomplish by inserting f (y) = 1 into Eq.(13), yielding ∂ 2 y F (y, y , s) = Rδ(y − y ) − sF (y, y , s), ∂ y F (y, y , s)| y=0 = −sτ RF (0, y , s), ∂ y F (y, y , s)| y=πR = −m 2 V RF (πR, y , s), from which the solution y > ≡ max(y, y ), y < ≡ min(y, y ) can be straightforwardly derived. We see that, as expected, the sum F (y, y , s) has poles whenever s = m 2 n , as can be seen from the mass eigenvalue condition Eq. (22); in other words, our sum of propagators possesses poles exactly where the individual propagators have poles. Additionally, equipped with this sum, it is possible to derive in closed form the sum n 2 n / 2 1 , which we recall from I and Section 2 must be < ∼ 10 in order for our assumption of small KM to be valid. Taking the limit of F (y, y , s) as s → 0, we arrive at the result The limit of the ratio |g n DM /g D | as n → ∞, as a function of τ for a F =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. The dashed line denotes the largest possible ratio such that the couplings of the DM particle to the gauge boson KK modes remain perturbative for all KK modes in the theory, assuming g D = 0.3 (Bottom) As in the previous panel, but now as a function of a F assuming τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. Differentiating this sum with respect to y at y = 0 and applying the SM-brane boundary condition given in Eq. (20), we rapidly arrive at The form of the sum in the second line of Eq. (29) already then confirms what has previously been observed in I, namely, that a nontrivial positive BLKT is necessary for the consistency of our KM analysis. The sum sharply increases to infinity as τ → 0, indicating that an insufficiently large τ will result in the sum being unacceptably large, namely > ∼ O (10). Furthermore, a negative τ would suggest a still more worrying scenario, indicating the need for at least one KK state to be ghost-like (have a negative norm squared). To determine if our kinetic mixing treatment is valid for the full parameter space we consider, we depict the sum n 2 n / 2 1 in Fig. 6. Our results here explicitly confirm those observed in I, namely, that for selections of (τ, a F ) such that τ > ∼ 1/2, the summation n 2 n / 2 1 remains small enough not to vitiate our treatment of kinetic mixing: The sum remains < ∼ O(10).
Next, we apply the results of Eqs. (21), (22), and (27) to find the DM-e − scattering cross section, to explore the possibility of direct detection of the DM. Inserting Eq. (27) into Eq. (15) yields where in the second line we have substituted the parameter m 1 , the mass of the lowest-lying KK mode of the dark photon field, for the compactification radius R. We can now suggestively rewrite this expression as Note here that the quantity Σ F φe depends only on the model parameters (τ, a F ), while the rest of the expression above is independent of them. While the closed form of Σ F φe is convenient for calculation, we have also included an explicit expression for this quantity in terms of an infinite sum over KK modesnotably, because the quantity g n DM n (or alternatively, v n (πR)v n (0)) alternates in sign and decreases sharply with increasing n, we can see in Fig. 7 that the sum rapidly converges, coming within O(10 −2 ) corrections to the value of the closed form of Σ F φe even when the sum is truncated at n = 10. Looking at the numerical coefficient of Σ F φe in Eq. (31), meanwhile, we see that for m 1 ∼ O(100 MeV) and g D 1 ∼ 10 −4 , the DM-e − scattering cross section easily avoids current direct detection constraints as long as the quantity Σ F φe is O(1) or smaller [29][30][31], although it does lie within the possible reach of future experiments such as SuperCDMS [31]. We can see that this is the case for a broad range of parameters in Fig. 8; for every choice of (τ, a F ) that we are considering here, Σ F φe lies between 0.6 and 0.9 implying that the KK states lying above the lightest one do not make critical contributions to this cross section. Hence, at present, this model can easily evade present DM direct detection constraints for reasonable choices of m 1 ∼ 100 MeV and g D 1 ∼ 10 −4 .
Our brief phenomenological survey of the flat space scenario now concludes with a discussion of the thermally averaged annihilation cross section at freeze-out, that is, demonstrating that this construction is capable of producing the correct relic density of DM in the universe. To begin, we insert Eq. (27) into the expression for the φ † φ → ff (where f is some fermion species) velocity-weighted annihilation cross  Figure 6: (Top) The sum n 2 n / 2 1 over all n, as a function of τ for a F =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Bottom) As in the previous panel, but now as a function of a F assuming τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. section of Eq. (19). This yields the result σv lab = 1 3 We can then use this expression in the single integral formula for a thermally averaged annihilation cross section given in Eq. (16), and compare the results to the approximate necessary cross section to reproduce the (complex) DM relic density with a p−wave annihilation process, namely 7.5 × 10 −26 cm 3 /s [32]. We note that this quantity is the only one in our analysis which has any direct dependence on the mass of the DM, m DM , (assuming, as we do, that the DM particle's mass is substantially greater than that of the electron). In fact, because we must rely on resonant enhancement in order to achieve the correct relic density, we see that with all the other parameters fixed our results for the thermally averaged cross section are extremely sensitive to m DM and largely agnostic to differing choices of (τ, a F ). In Fig. 9, we depict the thermally averaged velocity-weighted cross section as a function of the DM mass m DM , requiring, as we have argued must be the case in Section 2, that m DM < m 1 . For demonstration purposes, we have selected that m 1 = 100 MeV, x F = m DM /T = 20, g D = 0.3, (g D 1 ) = 10 −4 , and have included only the possibility of the DM particles annihilating into an e + e − final state.
Notably, the cross sections depicted are largely independent of the choices of (τ, a F ) near values of m DM /m 1 that produce the correct relic abundance (that is, relatively near the m 1 resonance of the cross section). In fact, for all parameter space points we depict here, it is possible to produce the correct cross section when m DM ∼ 0.36m 1 or m DM ∼ 0.54m 1 ; however, other values would be required if we  (31), as a function of τ for a F =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Bottom) As in the previous panel, but now as a function of a F assuming τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. also varied m 1 or g D 1 By leveraging the resonance, therefore, our model is clearly able to reproduce the observed relic abundance for a wide variety of reasonable points in parameter space.

Warped Space Model Analysis
We now consider the possibility that the extra dimension is not flat, but rather has a Randall-Sundrumlike geometry with a curvature scale k. In this case, f (y) in the metric of Eq. (1) shall be e −ky , but our analysis closely follows that of the flat space scenario. The warped geometry does, however, necessitate additional care in certain aspects of model construction, which we should address before moving forward with our discussion.
First, in the warped space scenario, because f (y) is non-trivial, we need two parameters to describe the metric rather than the single parameter, R, that we used in the flat-space analysis. We shall find the most convenient parameters with which to describe our metric are kR, the product of the curvature scale and the compactification radius, and the so-called "KK mass", M KK ≡ k exp(−kRπ). Second, unlike the flat-space case, our choice to place the SM on the y = 0 brane and the DM on the y = πR brane is no longer arbitrary. Specifically, we note that naturalness suggests that ∼ M KK is a natural scale for mass terms localized on the y = πR brane, and that the lowest-mass Kaluza-Klein modes of any bulk fields should also in general be O(M KK ), while the natural scale for mass terms localized on the y = 0 brane should be ∼ M KK exp(kRπ), which is exponentially larger [20,33]. In our construction, then, naturalness suggests that we localize the higher-scale physics (the SM, with a scale of roughly O(250 GeV)) on the y = 0 brane, and the lower-scale DM sector with a scale of O(0.1 − 1 GeV) localized on the y = πR brane. Furthermore, the hierarchy between the two scales roughly sets the value of the product kR, namely, we must require that e −kRπ ∼ O(0.1 − 1 GeV)/O(250 GeV). Thus we will require that kR ≈ 1.5 − 2. We note in passing, therefore, that in contrast to the flat space model, the warped space construction offers the aesthetically appealing characteristic of explaining the mild hierarchy between the brane-localized vev of the SM Higgs and the brane-localized mass parameters of the DM and DP fields appearing on the opposite brane.
With these concerns addressed, we can now move on to determining the bulk profiles and sums of KK modes required for our analysis. First, we note that the equations of motion for the bulk profile v n (y) become 4 The solution to these equations can be written, where A n is a normalization factor, and the function ζ (n) ν (z) is given by with J ν , Y ν denoting order-ν Bessel functions of the first and second kind, respectively. Notice that v n (y) then automatically satisfies its boundary condition at the brane y = 0, while the allowed values of x W n (and hence the masses of the KK tower modes m n ) are then found with the boundary condition at y = πR, which can be simplified to The normalization constant A n can be found using the orthonormality relation of Eq. (8), yielding Using Eqs. (34) and (36), we can now continue on to an exploration of the phenomenology of various KK modes, much as we have done in Section 3 for the scenario with a flat extra dimension. We begin, as in the case of flat space, by determining the dependencies of the lowest-lying root of Eq.(36), x W 1 , as a function of the parameters (τ, a W ), depicted in Fig. 10. Note that in Fig. 10 and subsequent calculations, we have elected to specify the parameter (kR)τ (that is τ scaled by the quantity kR) rather than τ . This is because in practice, expressions featuring the brane term τ in this setup will always do so through the quantity (kR)τ ; we therefore find, as has been the case in other work with RS brane terms [16], that (kR)τ is the more natural parameter to use.
Qualitatively, we observe largely similar behavior for the root x W 1 in Fig. 10 as we observed in x F 1 in Fig. 1, namely that x W 1 < ∼ 1 for the range of (τ, a W ) parameters we probe, and that x W 1 increases with increasing a W and decreases with increasing τ . It is interesting to note that the specific values of x W 1 are somewhat sensitive to the specific value of kR: In particular, when kR = 2.0, the values of x F 1 for a given The same as the bottom left, but assuming kR = 2.0 choice of (kR)τ and a W is approximately 15% lower than these values in a scenario where kR = 1.5.
Next, we discuss the quantity m 2 /m 1 , the ratio of the mass of the second KK mode of the dark photon field to that of the first KK mode; as in our discussion of this ratio in the flat space scenario, this quantity continues to possess substantial phenomenological importance due to the potential of the second KK mode to be an experimental signal for the existence of extra dimensions. In Fig. 11, we depict this mass ratio's dependence on the quantities τ and a W . The most salient difference between the results here and those for the flat space case discussed in Section 3 lies in the typical magnitude of the ratio itself: With a flat extra dimension, we found that reasonable selections for τ and a F resulted in ratios m 2 /m 1 ∼ 3 − 4. In the warped setup, we find that the same ratio now typically lies within the range of m 2 /m 1 ∼ 6 − 16. This represents one of the primary distinctions between the warped and flat constructions, namely, that for a given mass of the lightest KK mode of the dark photon, m 1 , the mass of the second KK mode m 2 is significantly greater in the case of a warped extra dimension than it is in the case of a flat one. Beyond this observation, we also note that changing kR in our computations below has an effect roughly in line with what we might expect from the results depicted in Fig. 1, namely, that a larger value of kR slightly increases the ratio m 2 /m 1 , likely because the value of the root x W 1 is somewhat reduced.
We complete our exploration of the relative masses of various KK modes just as we have in the flat space scenario, namely, by exploring the growth of m n as n increases. We depict the results in Fig. 12 for both kR = 1.5 and kR = 2.0, for various selections of (kR)τ and a W . The most salient contrast between these results and those in the flat space analysis again lies in the magnitude of the mass ratio: In the warped setup, m n /m 1 increases significantly more sharply with n than it does in the flat space, such that at large n, typical values of m n /m 1 are approximately three times larger for a warped extra dimension than they are for a flat one. The dominant share of this discrepancy is determinable from the mass eigenvalue equation Eq.(36)-numerically, it can be readily seen that the difference between successive roots of this equation approaches π as n becomes large, so the eventual slope of the line depicted in Fig. 12 should be roughly π(x W 1 ) −1 . This is compared to the analogous slope in the flat space scenario, which, as discussed in Section 3, should be approximated by (x F 1 ) −1 . Because the typical values of x F 1 and x W 1 are roughly comparable, this in turn suggests that the slope of the lines in Fig. 12 should be steeper by roughly a factor of O(π) than their flat space counterparts in Fig. 3. Before moving on, we also note that the same behavior with increasing kR that we observed in the ratio m 2 /m 1 appears again as we consider more massive KK modes, namely, that increasing kR will increase the value of the ratios of heavier KK mode masses to that of the lightest mode.
Having addressed the masses of the various dark photon KK modes, we now move on to discuss the effective kinetic mixing and DM coupling terms that arise in this construction. In Fig. 13, we depict the behavior of the ratios n / 1 and |g n DM /g D | as a function of the KK mode n (we note that once again, as in the flat space scenario, the values of g n DM oscillate in sign). The results are qualitatively quite similar to the flat space scenario depicted in Fig. 5. In particular, we find once again that while n / 1 consistently decreases for large n, |g n DM /g D | again approaches a non-zero asymptotic value. This asymptotic value for |g n DM /g D |, much like its flat space analogue, can be explored further by semi-analytical means. By using Eqs. (34) and (36), as well as the identities, it is possible to determine that as n becomes very large, the ratio |g n DM /g D | becomes well-approximated by the expression . The mass ratio of the lowest two DP KK states, m 2 /m 1 = x W 2 /x W 1 assuming kR = 1.5, as a function of (kR)τ for a W =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Top Right) As in the top left, but now assuming kR = 2.0. (Bottom Left) As in the top left, but now as a function of a W assuming (kR)τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Bottom Right) As in the bottom left, but assuming kR = 2.0. In Fig. 14, we depict the dependence of this approximate asymptotic value on τ and a W . The behavior of this quantity is quite similar to the analogous results Fig. 4 for the flat space scenario, in particular, we observe a sharp increase in the ratio here as a W increases, just as the corresponding ratio in the flat space case increases sharply with increasing a F . We note that the typical maximum values that we observe in Fig. 14, however, are roughly a factor of 2 smaller than those we observed in Fig. 4, however, as a F and a W are not directly comparable quantities, the significance of this diminished range is not obvious. Again, as in Fig. 4, we have included a dashed line which denotes the maximum value that this ratio can attain such that all g n DM remain perturbative for the choice g D = 0.3; in this case, we see that such a requirement effectively excludes choices of a W > ∼ 2. The ratio g n DM /g D , assuming kR = 1.5, as a function of n for various choices of ((kR)τ, a W ) =(1/2,1/2) [red], (1/2,1) [blue], (1/2,3/2) [green], (1,1/2) [magenta], (3/2,1/2) [cyan] and (1,1) [yellow], respectively. (Bottom Right) The same as the bottom left, but assuming kR = 2.0 Just as in our analysis of the flat space setup, we can now move on from discussing individual KK modes' masses and couplings to the basic predictions of phenomenologically important processes. In order to do this, we must first evaluate the sum F (y, y , s) (defined in Eq. (9) for the warped metric, by The approximate asymptotic value of |g n DM /g D | given by Eq.(39) for large n, assuming kR = 1.5, as a function of (kR)τ for various choices of a W =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), 1/2(yellow). (Top Right) The same as the top left, but assuming kR = 2.0 (Bottom Left) The approximate asymptotic value of |g n DM /g D | given by Eq.(39) for large n, assuming kR = 2.0, as a function of a W for various choices of (kR)τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), 1/2(yellow). The dashed line represents the maximum value that this ratio can obtain and still have all KK couplings remain perturbative for g D = 0.3. (Bottom Right) The same as the bottom left, but assuming kR = 2.0 solving Eq. (13) with f (y) = e −ky inserted. We arrive at the differential equation ∂ y [e −2ky ∂ y F (y, y , s)] = Rδ(y − y ) − sF (y, y , s), ∂ y F (y, y , s)| y=0 = −sτ RF (0, y , s), ∂ y F (y, y , s)| y=πR = −m 2 V Re −2kRπ F (πR, y , s).
By defining the variables z ≡ ( √ s/M KK )e k(y−πR) and z ≡ ( √ s/M KK )e k(y −πR) , we can solve Eq.(40) in terms of Bessel functions, yielding We note that in this form, it is readily apparent that F (y, y , s) has poles wherever √ s is equal to the mass of a KK mode m n , just as we would expect given the components of its sum and just as we previously observed in the flat-space sum Eq. (27).
With a solution for F (y, y , s) in hand, we can then replicate our analysis in Section 3 to determine whether or not our kinetic mixing treatment is valid in the parameter space we're probing, this time applied to the warped space scenario not considered in I. Through analogous steps to those taken in Section 3, we find that the sum n 2 n / 2 1 in the case of warped spacetime is also given by the only difference from the flat-space result here being the form of the function v 1 (0). The τ −1 dependence of this sum suggests the same requirements as the identical flat space result, then: The BLKT τ must still be large enough so that its magnitude remains < ∼ 10 and positive so that the result does not require the existence of ghost states. In Fig. 15, we depict the sum n 2 n / 2 1 for different values of τ and a W . Notably, while the sum is generally within reasonable < ∼ 10 limits, when (kR)τ ≈ 1/2, the sum becomes quite close to, and even somewhat exceeds, 10. While the largest values of n 2 n / 2 1 achieved among the region of parameter space we have explored still aren't quite large enough to render 2 1 terms in our analysis numerically significant (at least for the 1 ∼ 10 −(3−4) terms we consider here), the sharp rate of increase they enjoy with decreasing τ near (kR)τ = 1/2 suggests that probing significantly below this value is unlikely to yield valid results. On the surface, this may seem to contrast slightly with our results in Section 3, in which we found that restricting τ to values larger than 1/2 stayed roughly < ∼ 6. Closer inspection indicates that this discrepancy can largely be attributed to the use of (kR)τ as the variable we are employing instead of τ : If one compares the maximum value obtained by the warped sum at (kR)τ = 3/4 (for kR = 1.5) and (kR)τ = 1 (for kR = 2.0), for which the variable τ itself is simply 1/2, the results for the sum with both kR values very closely matches that which was observed in the flat space construction of Section 3. Hence, in both the flat and warped space cases, our setup's treatment of kinetic mixing easily remains valid for τ > ∼ 0.5, although it should be noted that as kR increases, any boundary from these perturbativity concerns on the more natural warped-space parameter (kR)τ , which is often used instead of τ for warped setups [16], will become increasingly stringent.
Moving on, it is then straightforward to find the DM-e − scattering cross section by inserting our n / 2 1 over all n assuming kR = 1.5, as a function of (kR)τ for a W =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Top Right) As in the top left, but now assuming kR = 2.0. (Bottom Left) As in the top left, but now as a function of a W assuming (kR)τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Bottom Right) As in the bottom left, but assuming kR = 2.0. results for F (y, y , s) given in Eq.(43) into Eq. (15), arriving at Notably, this is the same result (up to a normalization convention of the parameter a W and, of course, different bulk wave functions v 1 (y)) that we derived for the flat-space case Eq. (30). In particular, the sum F (0, πR, 0) has identical results (again, up to normalization of a W ) for the flat-and warped-space scenarios. Just as in the flat space case, the numerical coefficient in front of the quantity Σ W φe , which now encapsulates all of the cross section's dependence on the parameters τ and a W , indicates that as long as Σ W φe < ∼ 1, the resultant cross section is not constrained by current experimental limits, although we remind the reader that such cross sections may lie within reach of near-term future direct-detection experiments. In Fig. 16, we depict the dependence of Σ W φe on various choices of τ and a W ; we find that just as for the flat space case, this requirement is easily satisfied for every τ and a W we consider.
We also note that the sum over individual KK modes in the computation of Σ W φe quickly converges to the closed form expression even when truncated for very low n; as depicted in Fig. 17, Σ W φe , just like its flat space analogue, converges to within O(10 −2 ) corrections to its exact value even when truncated at n ≈ 10. Hence, just as in the flat space scenario, exchanges of the lightest few DP KK modes dominate the direct detection signal.
Finally, we can conclude our discussion of the warped space scenario by considering the thermally averaged annihilation cross section of DM particles into SM fermions. Inserting the relevant value of F (y, y , s) into Eq. (19) allows us to derive the DM annihilation cross-section, σv lab , for the warped space scenario, yielding σv lab = 1 3 where we remind the reader that the functions ω 0,1 (z) are defined in Eq.(43). Inserting this result into Eq. (16), we can straightforwardly obtain the thermally averaged DM annihilation cross section via numerical integration. Just as in the flat space case, we specify that m DM = 100 MeV, x F = (m DM /T ) = 20, g D = 0.3, and g D 1 = 10 −4 , and consider DM annihilation into an e + e − final state. Our results, depicted in Fig. 18 along with a dashed line marking < σv >= 7.5 × 10 −26 cm 3 /s, the approximate necessary cross section to produce the observed DM relic abundance, exhibit substantial similarity with the results for the flat space scenario given in Fig. 9; in particular, in both cases the dependence of the cross section on the BLKT τ and the brane-localized mass parameter m V ∝ a F,W is extremely limited, and the correct relic abundance is obtained when m DM ≈ 0.36m 1 or m DM ≈ 0.53m 1 . Of course, as we vary the DM mass and g D 1 , other values of m 1 will also be allowed. In short, for the annihilation cross section at freeze-out, we observe qualitatively similar behavior in the warped space setup as we do in the flat space scenario: For our choice of parameters resonant enhancement is necessary in order to realize the correct dark matter relic density, and the cross section is largely agnostic to specific selections for the brane-localized kinetic and mass terms for the DP field.

Summary and Conclusions
In this paper, we have discussed a modification to our previous setup in I and II. In lieu of imparting mass to the lightest dark photon KK modes via dark photon boundary conditions, which necessitates a bulk DM particle with corresponding KK modes, our current construction simplifies this structure  (45), assuming kR = 1.5, as a function of (kR)τ for a W =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. (Top Right) As in the top left, but now assuming kR = 2.0. (Bottom Left) As in the top left, but now as a function of a W assuming (kR)τ =3(cyan), 5/2(magenta), 2(green), 3/2(blue), 1(red), and 1/2(yellow), respectively. by reinstating the dark Higgs as a scalar localized on the opposite brane in the theory from the brane containing the SM, preventing mixing between the SM and dark Higgs scalars. The DM particle can then be placed on the same brane as the dark Higgs, removing the additional complication of a KK tower of DM particles and resulting in substantially simpler phenomenology while still removing the effects of the dark and SM Higgses mixing.
We then briefly explored the model-building possibilities for this setup in two scenarios, one with a flat extra dimension and the other with a warped RS metric, in particular considering the behavior of the DP tower's mass spectrum, couplings, and mixing parameters with SM fields, as well as briefly touching on the predictions for SI direct detection experiments and thermally averaged annihilation cross sections at freeze-out for various points in parameter space. Exploring the case of a warped extra dimension in addition to that of a flat one affords us significant additional model-building freedom; for example, given the same choice for the lightest DP KK mode mass, subsequent KK modes for the warped scenario are approximately ∼ 3 times heavier than they are in the flat scenario, demonstrating a qualitatively different KK spectrum. The ability for warped extra dimensions to generate hierarchies, meanwhile, can be straightforwardly exploited to naturally explain the mild O(10 2−3 ) hierarchy that exists between the SM Higgs scale and the characteristic mass scales of the dark brane, namely the masses of the DM and the lightest DP KK modes ∼ 0.1 − 1 GeV.
With this model, we find few parameter space restrictions in either the warped or flat space constructions. The requirement that every DP KK mode's coupling to DM remain perturbative provides an upper limit on the DM-brane-localized mass term m V , in particular, we find that for the flat construction, m V < ∼ 1.5R −1 , where R is the compactification radius of the extra dimension, while for warped space, m V < ∼ 2M KK / √ kR, where M KK is the KK mass in the model and kR ∼ 1.5 − 2.0. We also find, in agreement with I for the flat space scenario and novelly for the case of warped space, that a positive O(1) value for the SM-brane-localized kinetic term (referred to here as τ ) is necessary in order to ensure the validity of our kinetic mixing analysis (in particular to ensure that O( 2 1 ) and higher order terms can in fact be safely neglected). For both the flat and warped space scenarios, however, this constraint is quite mild; requiring τ ≥ 1/2 is sufficient to satisfy it.
Regarding possible experimental signals, we explicitly consider that of SI direct detection from scattering with electrons. We find that selecting g D 1 ∼ 10 −4 and m 1 ∼ 100 MeV still places the SI direct detection cross sections in both the flat and warped space constructions at ∼ 10 −40 cm 2 , below current experimental constraints. However, we note that such signals are roughly within the order of magnitude of the possible reach of near-term future experiments, and are not especially sensitive to variations in the brane-localized kinetic and mass terms of the particular extra dimensional model (in the flat scenario, we see reasonable variation in these parameters producing at most an approximately 25% change in the value of the DD cross section, while for the warped scenario this variation is approximately 5%). As such, experiments such as SuperCDMS may place meaningful constraints on DP KK mode masses, couplings, and mixings in the near future.
The requirement that the thermally averaged annihilation cross section for the DM gives rise to the correct DM relic density, meanwhile, substantially constrains our selection of the relative DM particle mass m DM /m 1 . In particular, for natural selections of the other model parameters we see in both the flat and warped scenarios the DM annihilation cross section must enjoy some resonant enhancement of the contribution from the exchange of the lightest DP KK mode in order to attain a sufficiently large value. Given the sharpness of the resonance peak, this requirement places a significant constraint on the m DM ; for the choices g D 1 = 10 −4 , m 1 = 100 MeV, and g D = 0.3, m DM must lie near 0.36 or 0.54 of m 1 for flat space and 0.36 or 0.53 of m 1 for warped space. This cross section is also notably largely insensitive to differing choices of the brane-localized DP mass m V and the BLKT τ provided m 1 , g D , and 1 are kept fixed, indicating that the exchange of the lightest KK mode is, somewhat unsurprisingly given its resonant enhancement, of paramount importance as contributors to this process.
Overall, we find that constructing this model within a flat or warped space framework results in little qualitative difference in our results. The most salient potential phenomenological difference lies in the differing relative masses of DP KK modes (in particular, the ratio of the second-lightest dark photon mass to that of the lightest is in general 3-4 times larger in the RS-like metric we consider than in the flat space case), which would have considerable effect on experimental searches for dark photons in colliders. Otherwise, however, we note that a wide range of natural and currently phenomenologically viable parameter space is available for both constructions.
As we move forward to explore the possibilities of kinetic mixing in theories of extra dimensions, we continue to find alternate constructions that allow for phenomenologically viable models. Here, following the work of I and II, we have presented another, simpler, construction that utilizes the additional modelbuilding freedom afforded by extra dimensions to ameliorate phenomenological concerns that arise in 4D kinetic mixing theories.