Kinetic Mixing, Dark Photons and an Extra Dimension: I
Thomas G. Rizzo
hep-ph







SLAC-PUB-17178

December 30, 2023










Kinetic Mixing, Dark Photons and an Extra Dimension: I








Thomas G. Rizzo [rizzo@slac.stanford.edu]






SLAC National Accelerator Laboratory 
2575 Sand Hill Rd., Menlo Park, CA, 94025 USA









 


Extra dimensions (ED) can provide a useful tool for model-building. In this paper we introduce a single, flat ED extension of the kinetic-mixing/dark photon (DP) portal  for dark matter (DM) 
interactions with the Standard Model (SM) assuming a compactification `radius' of order R^-1∼10-1000 MeV and examine the resulting modifications to and augmentation of the 
usual DP phenomenology. In the present scenario, both the DP and DM experience the full 5-D while the SM fields are constrained to lie on a 4-D brane at the boundary of the ED. Such a 
setup can naturally yield the observed value of the DM relic density and explain the required rough degeneracy of the DM and DP masses needed to obtain it. Gauge symmetry breaking can 
occur via boundary conditions without the introduction of an additional singlet Higgs scalar thus avoiding all constraints associated with the coupling of such a field to the usual SM Higgs 
field in 5-D. The self-consistency in the removal of the kinetic mixing terms is found to lead to a brane localized kinetic term for the 5-D gauge field on the SM brane.  Multiple variations of 
this scenario are found to be possible which are consistent with current experimental constraints but which predict very different phenomenologies. In this paper,  we discuss the case of 
a complex scalar 5-D DM field, consistent with constraints arising from the CMB, which may or may not obtain a vacuum expectation value (vev). This approach can lead to interesting and 
distinctive signatures while being constrained by a wide array of existing measurements but with the details being dependent upon the model specifics.




empty









§ INTRODUCTION




The nature of Dark Matter (DM) is one of the greatest mysteries in particle physics: the Standard Model (SM) provides us with no candidates for such particles and 
forces us to entertain new physics scenarios for a possible explanation.  Weakly Interacting Massive Particles (WIMPS)<cit.>,  particularly in the 
form of the Higgsino/gaugino Supersymmetric partners (as well as axions<cit.>) have generally been the most popular of the DM 
candidates. One reason for this is that the UV complete scenarios wherein such particles might arise were motivated to solve other problems and the existence of a potential DM 
candidate within them was a welcome bonus. However, 
as the predicted WIMP signatures have failed to show themselves at the LHC or in either direct (DD) or indirect detection (ID) experiments<cit.>, it 
behooves us to widen our theoretical viewpoint as well as our experimental  search windows. This is particularly true for the case of lighter DM masses, below that of the 
traditional WIMP mass scale, where many of the conventional searches clearly falter. This is the strong message contained in the white papers from both the Dark 
Sectors Workshop<cit.> and the U.S. Cosmic Visions Workshop<cit.>. Of course, without the guidance of more UV complete 
theories, such as SUSY, it is difficult to focus, a priori, on any  particular mass range without some input from experiment as to where we can or should look given our existing 
and potential future capabilities. One possibility is, roughly,  the ∼10 MeV to 1 GeV mass range where the standard WIMP-targeted DD experiments involving 
nuclear targets are found to suffer due to detector energy thresholds as well as from potentially serious neutrino backgrounds but which can be accessed 
by other means<cit.>. In this mass range the DM can still be in thermal contact with the SM in the early universe and a modified version of the WIMP paradigm 
associated with thermal relic freeze-out can still go through, albeit with new, non-SM interactions responsible for achieving the observed relic density.
 
Theories employing extra spatial dimensions (ED) at the ∼ TeV scale<cit.> have provided us with very useful tools for both model building and as means  to 
attempt addressing the outstanding issues within the SM, in particular, the gauge hierarchy<cit.> and flavor problems<cit.>. Such theories can also provide 
a potential origin for a TeV scale, non-SUSY version of WIMP DM<cit.>.  It is perhaps possible that ED can also open a window into non-WIMP DM in the ∼ 
10 MeV to 1 GeV mass range which we will consider below.  What is immediately clear is that if EDs exist at the ∼ 10 -1000 MeV scale then, 
to avoid conflict with many experimental constraints, none of the SM fields can experience these EDs and thus the SM must be confined to a 4-D brane within this larger 
bulk space.[Interestingly, EDs of this (inverse) mass scale have arisen previously in discussions of the ADD model<cit.> when the number of 
EDs is chosen to be 6 or 7 and the low-energy Planck scale is not far above the current limits from the LHC ∼ 5-10 TeV.] Here we imagine that only the 
DM and the mediator field  for the DM interactions with the brane-localized SM particles are allowed to exist in this higher dimensional bulk, , the 
EDs are Dark. For simplicity we will consider here the case of a single, flat, extra dimension (although generalizations of this framework can easily be 
constructed<cit.>) in the spirit of Universal Extra Dimensions (UED)<cit.> while simultaneously ignoring the effects of gravity in the current discussion of this setup. 
However, it is interesting to speculate on the embedding of the approach that we will discuss here into a more complex structure that does include gravity and addresses the 
hierarchy problem; here the ADD model<cit.> is the natural choice as the ED in ADD are flat. Within this scenario, if the number of additional dimensions n=5(6) and 
the ED reduced Planck scale is taken to be ∼ 10 TeV, just above the current LHC constraints, then the ED would have an inverse size of roughly R^-1∼ 100(1000) MeV 
which, as we'll see,  is comparable to that of interest to us here. (This result assumes, of course, that all these ED have the same size.) Thus it may be possible to embed the model 
setups that follows into an ADD-like scenario which may lead to some very additional interesting phenomenology. This is, however, beyond the scope of the present work.


One self-consistent and phenomenologically interesting scenario for DM at these mass scales is the Dark Photon (DP) model<cit.> wherein a new 'dark' U(1)_D 
gauge field, acting as the DM mediator, kinetically mixes<cit.> (, via the `kinetic mixing portal') with the SM U(1)_Y hypercharge gauge field at the 
renormalizable level. Such a mixing can be generated, ,  by loops of fields having both types of gauge charges. In such a setup, the SM fields will only interact 
with DM, which is an SM singlet but carries a U(1)_D `dark' charge, via this kinetic mixing. In more realistic 4-D versions of these models, the DP and DM masses are 
generally uncorrelated, independent parameters. For example, while the mass of the DP is usually generated through spontaneous symmetry breaking, , via the 
vacuum expectation value (vev) of a dark SM singlet Higgs field which also carries a U(1)_D charge, the DM field, also being a SM singlet, can have a U(1)_D-invariant 
mass term whose value is generally unrelated to the dark Higgs vev. Why these two mass scales should be similar, as they must be to satisfy several 
phenomenological requirements, , the value of the DM relic density, is somewhat of a mystery. Furthermore, in such models, there is no way to avoid some coupling of the dark 
scalar with the SM Higgs through a renormalizable term. This leads to an additional Higgs portal interactions<cit.> between the DM and the SM of 
some significance making the resulting physics more complex and can lead to too large of a branching fraction for exotic SM Higgs decays 
unless the quartic coupling linking the dark and SM Higgs sectors is tuned to a very small value. Fortunately, as we will see, we can employ ED boundary conditions to completely 
turn off this Higgs portal coupling. In this paper we will also see that the similarity of the DP and DM masses can be a natural outcome of a scenario 
with ED. Furthermore, the same choice of boundary conditions in the ED can be employed to break the U(1)_D symmetry and generate 
a DP mass without the need to introduce the additional dark Higgs field, though such fields may find other applications. In order to satisfy important CMB 
constraints on the DM annihilation cross section for masses in our range of interest, we consider the DM to originate from a complex bulk scalar field that may or may 
not obtain a vev. This choice naturally splits this general setup into two distinct model classes with very different phenomenologies that we will separately explore in detail. The 
specific pair of Abelian model classes that we present here are to be thought of as only toy models-proofs of principle and are not fully detailed, UV-complete scenarios.
 
The outline of this paper is as follows: In Section 2 we provide an overview of the essential ingredients of the U(1)_D gauge sector and some of the important model building 
constraints that need to be addressed in our ED constructions. This includes an analysis of the 5-D kinetic and mass mixing between the 5-D DP Kaluza-Klein (KK) tower 
states with those of the SM where we demonstrate the need for a brane-localized kinetic term (BLKT)<cit.> required to bring the effective 4-D action to the usual canonical 
form while avoiding tachyons and ghosts in the spectrum. The constraints imposed by measurements of the CMB on the nature of the DM field and the relative 
DM/DP mass spectra are also presented in this Section. In Section 3 we provide detailed discussions of a pair of model classes wherein the DM is a assumed to be a complex scalar 
which does or does not obtain a vacuum expectation value. The phenomenology of these two model classes is discussed and shown to be quite distinctive and leading 
to interesting signatures in future experiments. A discussion and our conclusions can be found in Section 4.











§ ESSENTIALS OF THE DP FRAMEWORK IN ONE EXTRA DIMENSION



This Section provides an overview of the essential elements of the ED models that we consider below as well as some general model-building requirements on such theories.





 §.§ Kinetic and Mass Mixing of the KK Dark Photon
 


We begin our analysis by considering a straightforward generalization of the usual DP portal/kinetic-mixing (KM) models to the case of one flat ED. Here we write the SM  
plus pure U(1)_D gauge field parts of the full 5-D action as 


    S=∫ d^4x  ∫_y_1^y_2 dy  [-1/4V̂_ABV̂^AB +(-1/4B̂_μνB̂^μν 
    +ϵ_5/2c_wV̂_μνB̂^μν  + L_SM)  δ(y-y_SM) ]  ,


where we consider the extra dimensional co-ordinate, y, to take on values on an interval bounded by two branes at y=y_1,2, , y_1  ≤ y ≤  y_2; the SM is assumed to 
be confined to one of these two branes, , y_SM. Here L_SM  is the remainder of SM Lagrangian, apart from the hypercharge field strength piece, written explicitly 
above, V̂ is the 5-D DP gauge field while ϵ_5 is a 5-D kinetic mixing parameter, normalized in a familiar manner<cit.> with c_w=cosθ_w.  Note 
that since all the SM fields are localized to one of the branes, the kinetic mixing itself must also be localized there. Following the usual procedure, we expand the the 5-D DP gauge 
field into a Kaluza-Klein (KK) tower of states, separating the 4-D and 5^th components: 


    V̂^μ [5](x,y) = ∑_n   f_n^[5](y) V̂_n^μ [5] (x)


where the functions f_n^[5](y) are determined by the equations of motion and the specific boundary conditions (BCs) as usual. (For 
convenience we will work in the V^5=0 gauge in this Section.) We remind the reader that in performing this very familiar procedure, an integration by parts 
is necessarily performed and that the BC  f_m ∂_y f_n |_y_1^y_2 =0 
for all values of n,m, is imposed. Here we  employ the common shorthand notation for the difference of the values at either end of the interval. In what follows 
we will consider specific BCs which will satisfy this requirement; recall that in the most typical discussed scenario employing orbifold BCs that this requirement is trivially satisfied. 
Integration over the ED then produces the standard results except for some distinct aspects, the most important being that the single 5-D KM term above now becomes 
an infinite sum of KM terms in 4-D given by


    ∑_n ϵ_n/2 c_wV̂_n^μνB̂_μν


where we identify ϵ_n = ϵ_5 f_n(y_SM), , in principle each member of the DP KK tower can experience a different amount of KM depending upon the value of its wavefunction 
on the SM brane. As we will see, consistency of the field redefinitions introduced to remove the KM requires that this be so.  As usual, we can think of the non-zero ϵ's as 
arising from, , the existence of a pair of SM color- and iso-singlet fields with slightly different masses localized on the SM brane and having hypercharge(dark charge) Y(Q_D)=1(1) 
and 1(-1), respectively<cit.>. 

At this point in the standard 4-D treatment, we would `undo' the KM via a set of field redefinitions to bring the action into canonical form; we must do the same here but now face an infinite KK 
tower of states. This `undoing' naturally leads to a BLKT for the DP field; the simplest way to see this is the following. Suppressing Lorentz indices and performing a shift 
B̂→ B+ϵ_5/c_wV̂ on the the SM brane removes the KM but leaves a term ϵ_5^2/4c_w^2 V̂^2 which corresponds to a negative 
(dimensionless) BLKT δ_0=-ϵ_5^2/Rc_w^2. A negative BLKT is well-know to lead to tachyonic and/or ghost states in the KK mass spectrum 
so that this negative BLKT must be necessarily compensated by already having in the setup an additional positive BLKT on the SM brane. Another way to see this makes direct use of the KK 
decomposition and generalizes the treatment of the KM of two new U(1)'s with the SM hypercharge<cit.>.  Writing B̂ = B+∑_n α_n V_n and defining Σ_i= 
(1- ∑_a=1^i ϵ_a^2/c_w^2)^1/2, some algebra informs us that 


    α_n=ϵ_n/c_w 1/Σ_nΣ_n-1 ,


and we now suggestively re-write


    Σ_n^2= 1-ϵ_1^2/c_w^2∑_a=1^n ϵ_a^2/ϵ_1^2 .


Note as we ascend further up the KK tower the sums appearing in the Σ's extend to larger and larger values of n, and eventually to infinity. Clearly if all the ϵ's had the 
same value or were to grow in magnitude with increasing n, then at some point the Σ's would become undefined. However, a BLKT on the SM brane for 
the DP field leads to the well-known effect of suppressing the values of the the KK wavefunctions ever more strongly with increasing n and this leads to a convergence in the 
sums encountered above provided that the BLKT is of sufficiently magnitude.  We note that once this convergence is assured, for typical values of ϵ_1 ≤ 10^-(3-4), we can 
work to leading order in the ϵ_n and employ the approximate result that Σ_n ≃ 1 in our subsequent numerical calculations. In that case, to leading order in the 
ϵ_n, we find  that we can replace V̂→ V<cit.> as non-trivial terms only enter here at second order in these parameters. 





 §.§ Mixing Phenomena



 Once we go to the canonically normalized basis for the hypercharge and DP KK fields we next consider the mass mixing that results between the SM Z and this DP KK tower whose 
members now couple weakly to hypercharge (and thus to the SM Higgs) via the ϵ_n ≠ 0. Similarly, since the Z mixes with all of the V_n KK tower members it also picks up a 
small coupling to all of the dark sector fields. This mixing leads to a number of interesting effects even in the 4-D case. We note that at this point, to be quite general, we 
will not commit ourselves with respect to any specific ED origin of the unmixed  DP KK mass terms or their specific 
values other than to note that they will generally be assumed to be of order of the inverse size of the compactification scale, R^-1∼ 10-1000 MeV. We will denote these KK masses as 
M_n which is sufficiently general for present purposes. We can schematically write the resulting neutral gauge boson mass-squared matrix, M^2, in terms of M_Z^2, 
the set of ϵ_n's and the corresponding unmixed DP KK squared masses,  M_n^2, obtaining  the general form (with each element to leading order in the ϵ_n's)


    M^2     =  ( [                    M_Z^2            -t_wϵ_1 M_Z^2           -t_w ϵ_2 M_Z^2                      ...;           -t_w ϵ_1 M_Z^2  M_1^2+t_w^2 ϵ_1^2 M_Z^2      t_w^2 ϵ_1 ϵ_2 M_Z^2                     ....;           -t_w ϵ_2 M_Z^2      t_w^2 ϵ_1 ϵ_2 M_Z^2 M_2^2+ t_w^2 ϵ_2^2 M_Z^2                     ....;                      ...                      ...                      ...                      ... ])  .


where t_w =tanθ_w and reproduces the 4-D result to this order. (See, , the last paper in Ref. <cit.>.)  To leading order in the ϵ's this matrix can 
be diagonalized by the small rotations:


    V_i    →    V_i +t_w ϵ_i M_Z^2/M_i^2-M_Z^2 Z
    
    Z    →    Z -t_w ∑_i ϵ_i M_i^2/M_i^2-M_Z^2  V_i  ,


which lead to the corresponding mass shifts for the Z and DP KK states given by<cit.>


    M_i^2    →    M_i^2[1 + t_w^2ϵ_i^2 M_Z^2/M_i^2-M_Z^2]
     
     M_Z^2     →     M_Z^2 [ 1- t_w^2M_Z^2 ∑_n  ϵ_n^2/M_n^2-M_Z^2]


We note the appearance of infinite KK sums in the above expressions, ,  for the Z mixing and mass shift. As will be seen below, the presence of the BLKT will allow these sums 
to converge due to the rapidly decreasing values of the ϵ_n^2's. Even with such convergence one may worry that the shift in the Z mass will be small enough as to not be in 
conflict with the agreement of the SM with the current electroweak data<cit.>.  Another concern is the possibility that  Z-KK mixing may significantly alter the various Z couplings 
to the SM fields and, since the V_n couple to the dark sector,  lead to a sizable contribution to the Z → invisible decay width (and possibly for 
the SM Higgs field as well). We will see below that these issues are all under control within specific model frameworks. 
Whatever the detailed nature of the DM sector (scalars or fermions), the V_i will couple to pairs of DM tower states, DM_n  DM_m, with a strength which we will denote as 
g_D c̃_nm^i below where here g_D is the 4-D dark gauge coupling and are given by integrals over y co-ordinate of the product of the relevant gauge and DM KK tower 
wavefunctions.  The mixing of the Z with the DP KK modes also induces a corresponding coupling of the Z to pairs of DM states, DM_n  DM_m, which is given by


    -g_D ϵ_1 t_w ∑_i (ϵ_i/ϵ_1) c̃_nm^i M_Z^2/M_i^2-M_Z^2 .

 
If the ϵ's and c̃'s were independent of their indices this sum would not be well-behaved; BLKTs, as we will see, insure convergence.
We note that for numerical  purposes g_D will here always be taken to be not far from O(0.1-1) (like e or the SM weak charge) so that g_D^2>>(eϵ_i)^2. 
Since the unmixed V_i couple to hypercharge, their mass mixing with the Z alters their couplings as well as those of the Z. To leading order in the ϵ's this mixing  
produces V_i couplings to the SM fields of the form 
 
 
    g/c_w  t_w ϵ_i [ T_3LM_i^2/M_Z^2-M_i^2 + Q c_w^2 M_Z^2-M_i^2/M_Z^2-M_i^2] ,

 
 where Q(T_3L) is the SM electric charge (third component of weak isospin). For the lighter members of the KK tower with masses below ∼ a few GeV, where the ratios 
 M_i^2/M_Z^2 can be safely neglected, the V_i couple to the combination ≃ eQϵ_i since e=gs_w. Thus lighter members of the U(1)_D 
 tower will all couple to the SM like the usual 4-D dark photons but with a decreasing strength with increasing i since the ϵ's must decrease as i increases.  However, in the opposite 
 limit where the ratios M_Z^2/M_i^2 are now small, the V_i couple to ≃g/c_w t_wϵ_i Y where Y=Q-T_3L is the SM hypercharge. Note that as we ascend the 
 gauge KK tower, once M_i^2/M_Z becomes significant the tower field picks up a parity-violating interaction with the SM fields which can potentially lead to additional constraints. 
 
 In the case of the Z, to lowest order in the mixing, employing the tree-level relationship e=gs_w as above, the Z couplings including mixing effects can be written as
 
 
    e/s_wc_w[(1+F)T_3L-(s_w^2+F)Q] ,

 
 where we have defined the dimensionless quantity, F, as   
 
 
    F= ∑_i (t_wϵ_i)^2 M_Z^2/M_Z^2-M_i^2 ,

 
 which is related to the Z mass shift obtained above: δ M_Z^2/M_Z^2 =1-F.  Defining α_w=α_QED(M_Z^2) we can use this expression and that 
 for the (null) shift in the W boson mass to determine the (tree-level) values of the STU oblique parameters<cit.>  to lowest order in the parameter F:
 
 
    T   =   2F/α_w
    
     S   =   4c_w^2F/α_w
    
     U   =   0 ,
 
 
 where we see that S and T are related as S=2c_w^2T. With values of ϵ_1 below ∼ 10^-(3-4) we will have no tree-level conflict with the usual electroweak data. 
 Lastly, in a similar manner, the mixing of Z with the gauge KK states (the off-diagonal terms in the mass matrix) induces a set of HZV_i couplings, where H denotes the SM 
 Higgs field, and which are given to LO in this mixing by
 
 
    K_HZV_i=2M_Z^2/v_H [t_wϵ_i M_i^2/M_Z^2-M_i^2]  ,

 
 while the corresponding HV_iV_j coupling is given by
 
 
    K_HV_iV_j≃2M_Z^2/v_H [t_wϵ_i M_i^2/M_Z^2-M_i^2] [i → j]  ,

 
where in both cases we have normalized to the SM HZZ coupling; here v_H is the usual SM Higgs vev ∼ 246 GeV. These expressions show that these couplings have an unusual 
dependence on the location of the relevant V_i within the gauge KK tower in that M_i^2 (clearly) grows with i while the ϵ_i's will decreases with increasing i.    
  
 



 §.§ Cosmology: Planck and CMB Constraints



In the 4-D DP scenario, it is well known that the measurements of the CMB power spectrum by Planck<cit.> place constraints on the nature of the DM and the relative 
DM/DP mass spectrum when these states are light, ,  below a few GeV in mass. This remains true in the 5-D extensions of these models that we consider. These 
constraints essentially place bounds on the DM annihilation cross section into various electromagnetically interacting final states that can lead to re-ionization in the early universe with the 
sensitivity peaking near z ∼ 600<cit.>.  For example, for 50-100 MeV DM that has an s-wave annihilation into the e^+e^- final state (which we might expect to be the dominant 
mode in this mass range), the annihilation cross-section is constrained to roughly satisfy the bound  <σ v>   (1-3)· 10^-29  cm^3  s^-1 
This is ∼ 3 orders of magnitude smaller than the canonical thermal cross section needed at freeze-out to achieve the observed relic density<cit.>,
,   <σ v>≃ 4.5· 10^-26  cm^3  s^-1 for (self-conjugate) DM in this mass range.  This apparent 
conflict is only a serious one if the value of the DM annihilation cross section, <σ v>, does not vary significantly with temperature or, more precisely, with velocity. At z ∼ 600 the 
DM is moving far slower than it would be at freeze-out which occurs at far higher temperatures so that a significant velocity-dependence in the annihilation rate can significantly soften or even 
eliminate this as a serious constraint. A straightforward solution to this problem is to consider only those models wherein the DM annihilation to SM particles via the spin-1 DP 
tower is a p-wave process so that <σ v> has an overall v^2 suppression factor. In 4-D, following this approach 
excludes the possibility of Dirac fermionic DM as these lead to an s-wave annihilation process without some additional physics. However, the DM can still be, ,  a complex scalar or a 
Majorana fermion and the possibility of co-annhilation opens up further possibilities. 5-D models are similarly constrained by these same considerations.  Furthermore, one finds in the 4-D case 
that an additional problem arises if  M_DM  > M_DP. If 
this occurs the DM can annihilate into pairs of DPs which is found to be an s-wave process for both fermionic and bosonic DM in the initial state<cit.> and is thus constrained 
by the CMB. In the 5-D case, this tells us that the lightest mode in the DM KK tower (, the actual DM) must be lighter than the corresponding lightest KK state in the 
DP mediator tower which can be a significant model-building constraint. This is particularly true in the case of the 5-D models we consider below as both of these 
masses will have a related ED origin of order R^-1.  Note that here we are particularly interested in scenarios wherein the lightest DP KK and DM masses are 
naturally similar in value as this produces the most successful phenomenology particularly with respect to the relic density. Our goal is to avoid putting this mass relationship into the model 
`by hand' as is generally done in 4-D where these masses can a priori be vastly different. We note that the choice of p-wave annihilation implies, apart from some unusual circumstances, that the 
indirect detection of DM annihilation today will be very unlikely due to the small value of the present-day annihilation cross section arising from this strong v^2 suppression. 

In principle, there may be additional constraints on a set of short-lived DP KK states in the the ∼ 0.1-1 GeV mass range arising from other cosmological considerations. In 4-D there has been 
only a limited examination of the impact of a single DP for such masses<cit.> where it has been shown that most of the sensitivity is at very small values of ϵ below our range of 
interest. This subject is certainly deserving of further study. 




§ MODELS WITH COMPLEX SCALAR DM



Since scalar DM allows for a simple means to obtain a p-wave annihilation cross section for DM we will concentrate on this possibility and entertain other possibilities elsewhere<cit.>.  This 
scalar, S, is naturally a complex field as it must carry a dark change for it to couple to the DP. Within this general framework there are two possible scenarios which depend upon whether 
or not this complex scalar obtains a vev. In 4-D such a vev is required in order to break the U(1)_D symmetry and give a mass to the DP. In 5-D, however, this need not be the case as we can break 
this gauge symmetry by the appropriate choice of BCs as will be seen in detail below. The phenomenology of these two possible paths is quite different and we consider them separately in the 
following. In either case one may be concerned that S (or its real part h) will mix with the SM Higgs through a generalization of the (brane localized) quartic term


    S_HS =∫ d^4x  ∫_y_1^y_2 dy   λ_HS H^† H S^† S δ(y-y_SM)


and possibly lead to significant Higgs physics alterations without fine-tuning the coupling. In our ED scenarios, as the SM Higgs would then couple with the entire KK tower of dark Higgs 
states this would likely lead to unacceptable changes in the SM Higgs properties and decay modes.  Fortunately in a 5-D setup we can avoid this coupling completely by requiring the appropriate 
BCs so that  the KK tower of states with which the Higgs would mix have vanishing values for their 5-D wavefunctions on the SM brane. The implementation of this condition will 
differ in the two scenarios we consider below.





 §.§ Model 1: Complex Scalar DM Without a VEV


The simplest possible 5-D DM matter within the KM scenario previously described is to add a complex 5-D scalar, which is a SM singlet field, to the action above. Here we 
assume that its bulk potential is such that it does not produce a 5-D vev and for simplicity we set its bulk mass term to zero as it will play no essential role in what follows. Ordinarily such mass terms 
for scalars would be of significant importance as one normally applies orbifold BCs, , either S or ∂_y S vanishes simultaneously on both branes, so that a massless zero-mode is 
present with the bulk mass being the only source of mass for the lightest KK scalar mode. This will not be the case here.

The first step in is to perform a KK decomposition of the 5-D part of the action; this is given by the expression 


    S_5D=∫ d^4x  ∫_y_1^y_2 dy  [-1/4  V_AB  V^AB  + (D_A S)^† (D^A S) -V(S^† S) ]


where D_A is the 5-D covariant derivative, D_A=∂_A +ig_5D Q_D V_A, and where Q_D is the dark charge of S with g_5D being the U(1)_D coupling. 
Note that we will drop consideration of any of the potential terms in this subsection appearing in V(S^† S). The next step is to KK decompose both S and V; for the moment 
let us ignore the possible existence of any BLKTs. In that case both S and V can be expanded in a similar manner. In thus setup it is convenient to adopt the unitary, 
V_5=0, gauge in our analysis below.  We further assume that the boundaries at y_1=0 and y_2=π R define the interval of interest and that for model building purposes 
we place the SM fields at y_SM=π R.

Generally we can write the forms of the V,S  5-D KK wavefunctions for these fields in suggestive notation as 


    v_n(y)   =   A cosκ y + B sinκ y
    
    s_n(y)   =   C cosσ y + D sinσ y  ,


with all of the parameters A-D,κ,σ generally being n-dependent, and determined by both the BCs and the normalization conditions.  What are our BCs? 
We first require that s_n(π R)=0 (in order to avoid the Higgs portal) while the v_n cannot vanish there or the corresponding fields would fail as mediators. Remembering the integration 
by-parts constraint above and that we wish to avoid the possibility of massless modes for any of these KK states we choose the following BCs: ∂_y v_n(π R)=v_n(0)=0 while 
∂_y s_n(0)=s_n(π R)=0. Note that if we had chosen the usual orbifold BCs for the vectors a massless zero-mode would 
appear. Here, the (degenerate) masses of both KK towers of states are given by m_n=(n+1/2)/R, where n=0,1,.., and where 
v_n ∼sin x_ny/R and s_n ∼cos x_ny/R  with x_n=n+1/2. [Since the lowest mass state, V_1, corresponds to the choice n=0 with this counting, the 
index n=0 will sometimes by applied as a lowest mode label.]  
We observe that there are no massless modes and that we have broken the gauge symmetry without an explicit Higgs vev. 
Of course in the V_5=0 gauge employed here, the V_5 field has been eaten level-by-level to become the longitudinal Goldstone component of all of the massive vector KK tower fields. 

While this appears promising, we've not been entirely successful since at this point all of the ϵ_n's will have the same value and the two sets of gauge and scalar KK states are mass 
degenerate while from our discussion above we require that m_S_1 < m_V_1 to avoid there being an s-wave DM annihilation. Since these two effects are directly related 
to the values of the v_n(π R), in order to reduce these quantities as n increases, we introduce the anticipated BLKT,  whose coefficient we denote by a dimensionless 
parameter δ_A,  for the v_n on the SM brane. (The presence of a corresponding BLKT on the y=0 brane would have no effect as the v_n vanish there.) This adds a term


    S_BLKT=∫ d^4x ∫_0^π R dy  [-1/4  V_μν  V^μν·δ_AR δ(y-π R) ]


to the gauge part of the action.  We know from our previous discussion that such a term must be present a priori and, further, that δ_A≥ |δ_0| encountered above. 
This BLKT<cit.> will introduce a discontinuity in the derivative of the v_n wavefunctions at y=π R and then one finds that the gauge 
KK tower masses m^V_n=x^V_n/R will now be determined by the roots of the equation 


    π x^V_n= δ_A/2 x^V_n


and the v_n for y≤π R are now given by v_n(y)=N^V_n sin(x^V_n y/R) where the wave-function normalization is given by 


    (N^V_n)^2 = 2/π R [1+(δ_A x^V_n/2)^2+δ_A/2π]^-1 ( sin^2π x^V_n)^-1


The values of the ϵ_n can be easily read off from this normalization factor; here we take the convention that the ϵ_n's will be defined to be positive which 
results in additional powers of -1 appearing in some of the expressions below. 
δ_A≠ 0 is observed to have multiple effects: As required, as n increases the values of the ratio ϵ_n/ϵ_1 
decrease roughly as ∼ 1/n as is shown in Figs. <ref> and  <ref>.






In these Figures we see that for fixed δ_A the value of ϵ_n falls rapidly with n and also that, for a given KK mode, n, the value of ϵ_n decreases as 
δ_A  increases.  
There is, however a further effect; when δ_A>0 the entire KK mass spectrum of the field is lowered as seen in Fig. <ref>. [We remind the reader 
that  if negative values of δ_A are employed this leads to ghosts and/or 
tachyons<cit.> in the KK mass spectrum so that this possibility is excluded.] The specific decrease of the values of the roots x^V_n are (for 
δ_A>0)  shown in Fig. <ref> as functions of δ_A. 



Here we see that the action of the BLKT for large δ_A for n≥ 1 is to effectively reduce the root spectrum values, x_n, from ≃ (n+1/2) → ≃ n. 

Since the ϵ_n's can now be explicitly calculated, we can evaluate some of the quantities discussed earlier, , the oblique 
parameters were shown to depend upon a parameter F, defined above. Taking, , δ_A=0.5, R^-1=100 MeV and ϵ_1=10^-3 we obtain a value of 
F < 10^-5 demonstrating that this setup is `safe', at least at tree-level, with respect to the electroweak constraints. Similarly, we note that as in the 4-D case for values of ϵ_1 
of this magnitude or smaller, the anomalous g-2 value of the muon<cit.> will still remains unexplained.

From the above discussion we know that δ_A ≥ |δ_0|; can we also see that a minimum value is required from the KK decomposition. This we find by noting 
that the infinite sum, ∑_n  ϵ_n^2/ϵ_1^2, encountered above, must be below some fixed  a priori value. Clearly, 
as δ_A decreases(increases) this sum will correspondingly increase(decrease) in magnitude and it will diverge completely as δ_A → 0. To address this 
question,  Fig. <ref> shows the value of this sum as a function of  δ_A ≤ 1. We observe in this Figure is that for any value of δ_A  0.25 this sum is 
O(10) or less. However,  as δ_A becomes very small the sum grows rapidly as an inverse power of δ_A. We will restrict ourselves to the `natural' range of 
δ_A corresponding to values not too far from unity in the discussion that follows in which case the value of the sum is below O(10) and thus well-behaved. We note that since 
δ_A ≥ |δ_0| and ϵ_1^2=ϵ_5^2 (N^V_1)^2 we can obtain the rather weak bound δ_A > πϵ_1^2/2c_w^2.




Although this gauge BLKT solves the KM consistency issue, the reduction in the the gauge KK masses, , the values of the roots, seen in Fig. <ref>  now 
leads to a further issue since this BLKT, level by level, make these gauge fields lighter than the corresponding ones for the scalar DM fields. This 
we need to avoid in order to obtain a purely p-wave DM annihilation process. The natural solution  is to introduce a corresponding  BLKT, described by the 
corresponding dimensionless parameter δ_S, for the scalar field S on the y=0 brane (as the s_n wavefunctions vanish at y=π R).  This new piece of the 
action is similar to the one for the gauge fields above:


    S_BLKT'=∫ d^4x ∫_0^π R dy  [ (D_μ S)^† (D^μ S)  ·δ_SR δ(y) ]


The equation governing the corresponding roots x^S_n determining the scalar KK masses is exactly 
the same as that for the vector KKs above but with the trivial δ_A →δ_S replacement. In this case, we can write the S wavefunctions as (noting that they still vanish 
on the y=π R brane via the root equation)


    s_n(y) =N^S_n [cosx^S_ny/R -δ_S/2x^S_n sinx^S_ny/R]


where N^S_n is given by


    (N^s_n)^2 = 2/π R [1+(δ_S x^S_n/2)^2+δ_S/2π]^-1 .


The behavior of the roots x^S_n as functions of δ_S are identical to those of x^V_n as functions of δ_A since they result from identical root equations. To 
obtain, , m_S_1 < m_V_1, one needs only require so that δ_S>δ_A.  Specifically, for a given value of δ_A (and thus for a specific x^V_1), to obtain a mass 
ratio  m_S_1/m_V_1=λ <1 we can obtain this result uniquely by choosing a specific value of δ_S given by 


    δ_S = 2πλ x^V_1/λ x^V_1


as is shown in Fig. <ref>. We observe that for small δ_A where the lowest gauge KK root hardly differs from 0.5, the required values of δ_S for any fixed value 
of λ is essentially independent of δ_A and for values of δ_A ∼ 1 the required value of δ_S to obtain λ=0.5 grows significantly large , , 
≥ 10 or so. Further, we see that if we allow x^V_1 to decrease significantly then, correspondingly, very large values of δ_S are required to obtain the typical 
λ values of interest. 



This Figure also leads us to contemplate the region where λ<0.5 not directly considered in these plots and which requires substantially large values of δ_S implying 
that it might be disfavored. For these λ values the decay mode V_1→ S_1^† S_1 is open so that the lightest DP gauge KK state can decay to stable DM. Since 
we assume that g_D ∼ 0.1-1 while ϵ_1 < 10^-3, the branching fraction for this decay is very close to unity. A short consideration of this parameter  
space reveals that now all of the scalar DM and DP gauge KK states will eventually cascade down into a combination of the fields S_1,S_1^† and V_1. This implies that if 
λ <0.5 the decays of all of the tower fields will eventually lead to invisible final states.[We note that once the values of δ_A, λ and R^-1, to set 
the overall mass scale, are chosen the entire phenomenology of this scenario is completely fixed.]  This is quite different than the region 0.5 ≤λ≤ 1 wherein these KK cascade 
decays can be rather complex, as will be discussed below. Since the production and decay of the gauge KK tower fields will end up producing 
missing energy in its various manifestations, it is interesting to ask whether the presence of such excitations would be observable relative to the corresponding signals anticipated in the 
4-D DP model with a similar mass hierarchy. While when λ <0.5 (but not very small) we might not expect any significant qualitative difference in the results 
for the thermal DM relic density or for the spin-independent scattering cross section, when the gauge KK states are directly produced at accelerators some 
differences with 4-D expectations might be observable. 

One of the best ways to probe this `all invisible' decay scenario is to generalize the 4-D analysis for the direct production of the invisibly decaying DP in, , a fixed target 
experiment. To be specific, we consider the (approximately forward) production of a DP KK tower at the proposed LDMX experiment<cit.> which will scatter an 
E_0= 4-8 GeV e^- beam off of a Tungsten target. In such a case, once δ_A and R are fixed, a tower of all of the kinematically accessible, on-shell, DP KK states is produced with a 
known rate all of which cascade down to invisible DM.  For example, the electron recoil energy spectrum in such a situation is obtained by complete analogy with the 4-D case and, in the 
Improved Weizsacker-Williams Approximation (IWW)<cit.>, is given by 


    dσ/dy =8α^3 χ ∑_n [ϵ_n^2β_n y+(1-y)^2/3/(m^V_n)^2 y/(1-y) +m_e^2(1-y)^2]


where here χ is a nuclear form factor<cit.>, β_n^2=1-(m^V_n/E_0)^2 and y=E_e/E_0 with m_e(E_e) being the electron mass and recoil energy. Similarly  
one can generalize the expression for the electron p_T distribution to the case of multiple contributing DP KK states. Since the effect of EDs will be the most significant when 
the KK couplings expressed through the ϵ_n's are large we will assume that δ_A=0.5. Similarly, we expect that the ED effects will be largest the greater the number of KK states  
that can contribute to the above sum for a fixed value of E_0; this will happen when we chose R^-1 to be small. In Fig. <ref>, we pair-wise compare, for three different 
values of R^-1, the results of a 4-D IWW calculation for both the electron recoil spectrum and the scattered electron p_T distribution with those obtained using the present 5-D model 
employing the cuts in the LDMX study<cit.>.  The 4-D model will be defined to be the same as the 5-D model when truncated to only the single lowest KK contribution to the cross section. 
Here, only pairs of curves for each R^-1 should be compared and the overall normalizations ignored since we are examining only the relative shape differences in these distributions as 
normalization shifts can be accounted for by a corresponding change in ϵ_1. As expected, when R^-1 is 500 MeV, the 4-D and 5-D distributions for both observables are completely 
indistinguishable as only very few KK modes can contribute. As R^-1 decreases, to 200 MeV and then to 50 MeV, the KK contributions become more visible in the case of the p_T distribution 
whereas the electron recoil energy spectrum appears to show little if any sensitivity to EDs.  Clearly, for even larger values of R^-1 there will be no ED sensitivity while, for fixed R, the 
sensitivity will grow with the value of the beam energy, E_0. Similarly, lowering δ_A will increase the KK tower couplings and result in an increase in sensitivity. It is beyond the scope of the 
present work to determine whether or not experiments such as LDMX could be sensitive to these differences as this will require a much more sophisticated study including detector simulation.




As is well-known<cit.>, significant additional sensitivity to gauge KK production with invisible decays is possible from radiative production in the decay of mesons, 
, π,η→γ V_i and in e^+e^- collisions at, . BELLE-II<cit.>, although in both cases multiple gauge KK states may now become accessible via the photon recoil 
spectrum. Within our framework, the lightest gauge KK state may or may not decay to SM fields while the next set of KK's will decay solely to DM and its KK excitations as will be 
discussed below. This production of multiple photon peaks in the decay or recoil spectrum would provide a rather unique signal for the present scenario.

The direction production of an extended DP sector may also be observable using other techniques such as in the mixing-induced SM Higgs decays H→ ZV_i,V_iV_j as 
discussed above. Once we chose the values of δ_A and R^-1 these decay rates can be calculated employing the toolbox we have developed. However, 
we immediately find that the current rough constraint B(H→ ZV_i,V_iV_j)<0.25, corresponding to a partial width < 1 MeV, is very easily satisfied for the suggestive choices 
δ_A=0.5, R^-1=100 MeV and ϵ_1=10^-3. In the case of H→ ZV_i, this parameter choice yields a partial width 
of ≃ 0.16 eV while for the double mixing suppressed mode H→ V_iV_j, we obtain a partial width of only 
≃ 1.5· 10^-5 eV. Note that increasing the value of δ_A makes the ϵ_n's fall even faster so the KK sums will decrease while changing R^-1 sets 
the mass scale of KK modes contributing to these sums; ϵ_1 remains as an overall scaling factor. We can safely conclude that these Higgs decay modes are far beyond the 
reach of experimental detection for reasonable parameter choices.

Once the wavefunctions and masses of the vector and scalar KK states are determined one can calculate all of the couplings of interest, in particular, the explicit values of 
the couplings of the KK scalars to the KK gauge tower states as given by the quantities


    c_mn^i = ∫_0^π R s_n(y)s_m(y)v_i(y) dy


as described earlier.  As noted the actual physical couplings, in terms of 4-D coupling g_D=g_5Dc_11^1 (with α_D=g_D^2/4π),  are given by the ratio 
c̃_nm^i=c_nm^i/c_11^1.  With these in-hand we now turn to some DM phenomenology. Of particular interest is how the lightest scalar (and 
its complex conjugate) which forms the DM interacts with the DP KK tower, , the couplings c̃_11^i. Knowing these quantities 
and the KK masses we can calculate, , the DM annihilation rate which determines the relic density as well as the DM spin-independent direct detection (SI-DD) cross section 
up to a common overall factor related to the specific KK mass scale. In both cases, when we calculate the amplitude for the relevant process, we must perform a KK sum over an 
infinite tower of the gauge KK states; we first turn to the SI-DD process.[We note that in a general gauge the unphysical V_5 field does not mediate any interaction between the 
scalar sector and the SM fields in the present setup.] 


As in 4-D, the SI scattering process occurs via t-channel exchange of dark photons, but now with the full KK tower of states participating. For the light DM that we consider 
here, ∼ 10-100 MeV, scattering off of electrons is likely to provide the greatest sensitivity<cit.>. Assuming a form factor of unity since m_DM^2>> m_e^2 , 
this cross section can be expressed as (for either S or S^† scattering)<cit.> 


    σ_e = 4αμ^2 g_D^2 ϵ_1^2/(m^V_1)^4 [∑_n  (-1)^n+1ϵ_n/ϵ_1 c̃_11^n  (m^V_1)^2/(m^V_n)^2]^2


where μ=m_em_DM/(m_e+m_DM) is the reduced mass ∼ m_e  for the DM masses of interest to us; here, m_DM=m^S_1. Numerically this yields the result 


    σ_e ≃ 3.0· 10^-40cm^2  (100 MeV/m^V_1)^4  (g_Dϵ_1/10^-4)^2  ×Sum


whereby the quantity `Sum' represents the squared KK summation of the previous expression which we expect to be ∼ O(1)  as the series converges rapidly and whose numerical value 
we will return to shortly. Note that here `Sum' isolates the only difference between the prediction of this ED scenario and the more familiar one in the 4-D case. For representative parameter 
values SuperCDMS is likely to be able to probe this range of cross sections in the future but it now lies a few orders of magnitude below the current constraints<cit.>.

The calculation of the thermal DM annihilation cross section into final state electrons (the most likely possible final state for typical DM masses) can be expressed in a similar 
fashion by writing σ v_rel=b̃v_rel^2, where the detailed kinematic information, including the sub-leading terms in the velocities, and (away from any resonances for simplicity) 
is contained in the parameter b̃ which is given in the limit of a zero electron mass by<cit.>


    b̃=g_D^2 e^2 ϵ_1^2/192π m_DM^2 γ^4/γ^2-1 ∑_n,m (-1)^n+m[(ϵ_nϵ_m/ϵ_1^2) c̃_11^nc̃_11^m/(γ^2-r_n)(γ^2-r_m)]


where here the double sum is over the gauge KK tower states, γ^2=s/4m_DM^2 is the usual kinematic factor determined by the DM velocities employing the standard 
Mandelstam variable and  r_n=(m^V_n)^2/4m_DM^2. To be specific below we will assume the freeze-out temperature to be x_F=m_DM/T≃ 20 so that at freeze-out 
<v_rel^2>≃ 0.3. In the case where other final states, such as muons and light hadrons, can also contribute, this cross section must be corrected by the well-known 
R-ratio factor. As we'll see from the mass spectra of our KK states in Table <ref> below, for our benchmark models we not only have  2m^S_1 > m^V_1 but also that 
2m^S_1 is significantly below m^V_2 implying that the thermal DM annihilation cross section is dominated by phase space regions far from any of the narrow s-channel 
KK resonances.

To go further in our numerical evaluations of both the SI cross section and σ v_rel, we need to choose some specific benchmark models (BM) forcing us into some  
particular parameter choices. Here we give two examples both of which have δ_A=0.5. For BM1, we take m^V_1/m_DM=0.8 implying δ_S≃ 2.38, while 
for BM2, we assume that m^V_1/m_DM=0.6 implying δ_S≃ 6.03. Apart from the overall mass scale set by R^-1, these quantities determine the 
complete model phenomenology.  In the upper panel of Fig. <ref> we find the value of the quantity `Sum' defined above for our two benchmark points as a function of the 
number of contributing gauge KK tower states n. Here we see that (i) the results for these two BM points are essentially identical, (ii) the KK summation converges very 
rapidly, roughly by the time the n∼ 5 KK state is reached, to its final value. (iii) The value of this sum is less than unity due to the destructive interference among the gauge KK 
exchanges, , `Sum' ≃ 0.852(0.849) for BM1(BM2). This means that the entire KK tower above the lowest level only makes a ∼ 7% contribution to the amplitude. 
Finally, (iv) we see that the 4-D and 5-D predictions are numerically quite close. It is important to emphasize 
the very rapid convergence of these sums and the essentially negligible contributions of the higher KK states here.





The lower panel of Fig. <ref> shows the values of a quantity b for both BM points; in this Figure we have rescaled the quantity b̃ above by an 
overall factor so that this quantity b  as shown here is both dimensionless and is roughly O(10):  


    b̃ =b  [g_D^2e^2ϵ_1^2/m_DM^2 (GeV^2)]  10^-20cm^3 s^-1 .


In this panel we see that BM1(BM2)  leads to a value of b ≃ 7.9(15.9) which differ by roughly a factor of ∼ 2 due to the mass spectrum and various coupling 
variations. Note that, roughly, for g_Dϵ_1 ∼ 10^-4 and m_DM∼ 10-100 MeV we can easily obtain a thermal cross section of ∼ 9· 10^-26 cm^3s^-1 
as needed to reproduce the observed relic density for light complex DM masses<cit.>[We remind the reader that for DM that is not self-conjugate the 
required annihilation cross section is twice that of the the canonical value.]  It is again important to emphasize 
the very rapid convergence of these sums and the essentially negligible contributions of the higher KK states beyond n∼ 5 here.

Next, we consider the decay properties of the various S_n and V_n states. By construction, for both BM points, S_1 and S_1^† are stable states forming 
the DM while V_1 decays only into SM final states as the decay V_1→ S_1^† S_1 is kinematically forbidden; furthermore, V_2 only decays into 
S_1^† S_1 since g_D^2>>(eϵ_1)^2.  In this 5-D model, V_1 acts like the 4-D DP decaying to only SM states while V_2 acts like the 4-D model 
where the DP decays only to DM. In a similar fashion, the decay S_2→ S_1V_1 occurs with a 100% branching fraction.  The decays of 
the higher KK states are found to be somewhat sensitive to the BM choice due to the differences in couplings and phase space although the gauge KK masses 
are the same for both BMs. To see these mass differences explicitly, Table <ref> summarizes the masses of the lowest lying KK states for both BMs in units of R^-1. Here 
we see that as we ascend up the scalar KK tower the mass differences between the two chosen BM points vanishes as might have been expected based. In addition, we observe 
that the gauge and scalar KK states also eventually will become degenerate for large n, as expected. 



Finally,  we examine the branching fractions for the various gauge mediated decay modes of the heavier KK states for the two BMs; these are found in Table <ref>. 
The decay width for S_n(m)→ S_m(m')V_l(m_V) (with the relevant masses in the parentheses) is given by


    Γ_S=g_D^2 (c̃_nm^l)^2 m^5/16π m_V^4 [1-2m_V^2+m'2/m^2+(m_V^2-m'^2)^2/m^4]^3/2 .


Correspondingly, for the decay V_i(m_V)→ S^†_j(m_j)S_k(m_k)+h.c., we find a similar expression


    Γ_V=g_D^2 (c̃_jk^i)^2 m_V/24(1+δ_jk)π [1-2m_j^2+m_k^2/m_V^2+(m_j^2-m_k^2)^2/m_V^4]^3/2 .



In the Table we see that there can be quite significant differences in how the various KK states decay based on the small differences in masses and the variations in the c̃_nm^i 
couplings. Clearly, searches for these more massive KK states will be influenced by these parametric variations. The fact that these two BMs can show such differences suggests 
that even greater variations are likely possible as we scan over the full parameter space.  As noted, once decays of these light KKs into other dark sector states are kinematically 
allowed the corresponding lifetimes are generally controlled by the coupling factors ∼ g_D^2 × O(1)  so that such decays are quite rapid. Of course the lightest KK gauge 
state, which decays to SM fields via (eϵ_1)^2 can be long-lived as has been often discussed in the literature<cit.> for the 4-D case with typical cτ 
values of order 100 μm for ϵ_1 ∼ 10^-4 and masses of ∼ 100 MeV. As we progress up the various KK towers, decay widths will increase due to the usual phase 
space and mass factors although in most cases these will be compensated for by shrinking values of the relevant parameters c̃_nm^i and possible phase space suppressions. 

In any given experiment where multiple KK states V_n's are produced on-shell by interactions, cascade decays through all of the lower mass KK states will occur with 
various BFs which we've seen are parameter dependent. However, in all cases where m^S_1/m^V_1 is in the 0.5-1 range considered here, this cascade will produce a rather 
complex final state: multiple DM scalar pairs, appearing as as missing energy, as well as potentially numerous V_1's,  which all produce relatively soft-lepton pairs, possibly 
with displaced vertices, depending upon the precise values of R^-1 and ϵ_1.  This complex cascade phenomena is a rather unique signature of the present scenario 
and warrants further detailed study.





As a final note, we now calculate the partial width of the Z into dark sector fields<cit.>,  , the tower of scalars, 
Z→∑_n,m S^†_nS_m+h.c.,  which is induced by the mixing of the Z with the V_i as previously described.  As discussed above this may provide an important 
additional constraint on the model parameter space particularly if all the subsequent decays end up as missing energy. Here one must also assume a value for the 
(inverse) compactification radius, R^-1 as mentioned earlier since this sets the overall mass scale of the KK states. The smaller R^-1 is chosen to be the greater the 
number of KK states that are kinematically accessible in this decay process. As noted, in much of the parameter space these modes will appear as additional contributions to the 
partial width for Z→ invisible (since the cascade products are quite soft or will explicitly be missing energy) which is bounded to be below ∼ 1-2 MeV[<cit.>. To be definitive, 
we choose  representative values of R^-1=100 MeV, g_D ϵ_1=10^-4 and the couplings and mass spectra corresponding to BM2 for our numerical example. Here we 
sum over the set of kinematically accessible final states although those higher up in the tower make only an infinitesimal contributions due to their highly suppressed couplings. 
With these input values we find a partial width of ∼ 0.02 MeV which is more than a factor of ∼ 50-100 below the current experimental 
constraint and so presents no phenomenological issues.  Here we see that the suppression from the shrinking values of the ϵ_n and c̃^i_jk as the KK towers are ascended
allows predictions to be easily compatible with experiment. 





Another example of this is very strong suppression of couplings due to the BLKTs is provided by the incoherent production of multiple gauge KK states at the the LHC in the `monojet'  
channel<cit.> assuming that they all eventually decay to DM or soft decay products as is described above. If R^-1=100(200) MeV then at √(s)=13 TeV many 
KK states are kinematically accessible although their couplings rapidly grow infinitesimal making for a vanishingly small contribution to the signal.
We follow the ATLAS monojet analysis<cit.> here and employ the mass dependence of the gauge KK couplings to quarks as obtained above, , ∼ Q for low masses and 
∼ Y for larger masses although this does not have a 
qualitative impact on our numerical results. Without any BLKT-induced suppression the expected rates for this process could be quite large. Fig. <ref> shows a 
comparison of the current ATLAS 13 TeV limit limit on this `monojet' signal as a function of the minimum p_T of the leading jet. Also shown are the predictions of the present model in 
the absence of the BLKTs assuming ϵ_1=10^-4 and with R^-1=100(200) MeV. Even with this small value of ϵ_1 we see that the predictions would not lie very 
far from the current ATLAS upper limit. However, assumimg that δ_A=0.5, then the predicted signal rate falls by 
a factor of 10^3-4, , it lies very far from the current limit. Increasing our chosen value of δ_A will result in a greater suppression of the rate 
for this process since the ϵ_n's will fall off more quickly with increasing n. 






 §.§ Model 2: Complex Scalar DM With a VEV



When the SM bulk singlet field S acquires a non-zero vev, v_s, the phenomenology of the model becomes more complex as there are more distinct physical states in 
the spectrum. In 4-D, the scalar vev is a necessary ingredient in order to break the U(1)_D gauge symmetry and to give the DP a mass. As we saw in the preceding subsection, 
this vev was not needed for this purpose as, in the unitary gauge, the vector fields eat their 5th components, V_5, to acquire masses with these 
BCs. However, if it is non-zero, this vev will contribute to the DP and scalar KK tower masses. When this vev is non-zero, the complex field S also decomposes 
(assuming CP conservation) into a set of CP-even (h) and CP-odd (χ) fields; this CP-odd component is eaten by the Higgs mechanism in 4-D. Here 
one linear combination of the χ and V_5 towers becomes the KK Goldstone bosons generating the KK gauge masses whereas the other 
linear combination is realized as a physical CP-odd scalar KK tower. The magnitude of this admixture of these weak eigenstates is level-dependent and is partially 
determined by the dimensionless product g_Dv_sR. This scenario is very similar to that of the familiar 5-D Abelian Higgs model, described in detail in<cit.> and whose analysis we 
will follow and whose results we directly quote. However,  the present scenario differs in detail from this presentation in several ways: (i)  the the location of the SM fields is at 
y=π R, (ii) the ED interval is restricted to 0≤ y ≤π R, (iii) our choice of BCs will produce non-zero masses even for the lightest KK 
modes, and, finally, (iv) the presence of a gauge BLKT on the SM brane is introduced as in the previous subsection to render the `undoing' of the KM on the SM brane physical.

The simplest version of this setup as is described below has a significant flaw in that the DM field, the lightest CP-even scalar,  can have too short of a lifetime to be the true 
DM; we will investigate how this problem can be evaded without any significant phenomenological changes in the next subsection.

The first step in the present analysis is to examine the impact once S obtains a non-zero vev, , once S→ (v_s+h+iχ)/√(2) at the 5-D level, due to minimizing the potential:  


    V_S= -μ_S^2 S^† S +λ_S (S^† S)^2 ,


which we re-express it in terms of the (almost) physical fields h,χ that will be KK expanded after application of the appropriate BCs. 
At quadratic order, the h field obtains a 5-D bulk mass, M_h^2=2λ_S v_s^2, while the χ field (the would-be Goldstone in 4-D) remains massless in 5-D. The gauge 
field also obtain a bulk mass term, M_V^2=(g_D Q_D v_s)^2 where Q_D is the dark charge of S and is taken to be unity without loss of generality. 
Note that with our choice of BCs,  the bulk gauge mass is not the mass of the lightest gauge KK mode.  We assume that h and χ, satisfy the BCs 
∂_y (h_n,χ_n)=0 at y=0 and that h_n,χ_n=0 at y=π R, similar to that of S in the previous subsection, and for simplicity without any associated 
BLKT (but which we will return to below); thus h_n,χ_n ∼cosσ_n y. Assuming CP conservation, the h_n are realized as physical states with masses given by 
(m^h_n)^2=(n+1/2/R)^2+2λ_Sv_s^2 and with the wavefunctions h_n(y)= √(2/π R)cos(n+1/2) y/R. 
As mentioned earlier, the V_5n, which also experience the bulk gauge mass, and χ_n fields mix to form the KK Goldstone bosons, G_n, and the physical CP-odd scalars, 
a_n as given by<cit.>  


    G_n    =   σ_n V_5n+g_Dv_s χ_n/(σ_n^2+g_D^2v_s^2)^1/2
    
    a_n    =   σ_n χ_n-g_Dv_s V_5n/(σ_n^2+g_D^2v_s^2)^1/2 ,


where σ_n=(n+1/2)/R [It is occasionally useful to write these expressions in the form employing KK level dependent mixing angles: 
a_n=cosθ_n χ_n-sinθ_n V_5n, , as will be employed below.]. The Goldstones are, of course, absent in the unitary gauge in which we will work, while the a_n 
KK tower fields acquire physical masses given by<cit.>  (m^a_n)^2=(n+1/2/R)^2+g_D^2v_s^2 and the a_n wavefunctions are given by 
a_n(y)=√(2/π R)sin(n+1/2)y/R. The functional change in the y-dependence is a result of the fact that a_n ∼∂_y χ_n. 
Note the V_5 fields do not experience the effects of the gauge BLKT.

For the gauge KK fields we assume that the v_n satisfy the same BCs as in the previous subsection, including the BLKT described by δ_A. However, in the present 
case the KK masses are altered due to the presence of the bulk mass contribution and are now of the form


    (m^V_n)^2=(x^V_n/R)^2+g_D^2v_s^2 ,


where the roots x^V_n are given by the solutions of the equation 


    π x^V_n = δ_A/2x^V_n[ (x^V_n)^2+(g_Dv_sR)^2] =Ω_n


The corresponding gauge tower wavefunctions are given by an expression very similar to that found in the last section except for a minor but significant difference in the 
normalization factor due to the gauge bulk mass:


    (N^V_n)^2 = 2/π R [1+Ω_n^2+δ_A/π-Ω_n/π x^V_n]^-1 ( sin^2π x^V_n)^-1 ,


which reduces to the previous result in the absence of the bulk gauge mass. Note that these gauge roots are still found to satisfy x^V_n ≤ n+1/2 for all n, 
so that we always have  m^V_n < m^a_n. The remainder of the (relative) mass spectrum is set by the size of ratio of couplings r=2λ_S/g_D^2 which is naturally expected to be 
of ∼ O(1) but can be freely chosen.  Note that when r<1 the h_n KK states are lighter than the corresponding a_n (which we will assume here) and vice-versa. We also define 
the dimensionless ∼ O(1) quantities h=8λ_Sv_s^2R^2 and a=4g_D^2v_s^2R^2, not to be confused with the 5-D fields. Concentrating on the important lightest modes in each 
KK tower we also define the specific mass ratios m^V_1/m^h_1= Σ and m^a_1/m^h_1 = 1+δ. These few parameter ratios control the mass spectra, the associated 
couplings and consequently determine the various model signatures. Note that in units of (2R)^-2, the squared masses of the h_n,V_n and a_n states are given by 
(2n+1)^2+h, (2x^V_n)^2+a and (2n+1)^2+a, respectively.

Interesting mass spectra can be found in multiple ways, given that there are now several free parameters to consider, based on the constraints and discussion above; the approach 
we follow is to again choose two interesting benchmark points as in the previous subsection. As we will see, one feature of this set up is that masses associated with the 
3 KK towers will generally be more degenerate than those found in the previous subsection since we are no longer introducing a separate BLKT for the 5-D scalar. Note that since the 
V_n are lighter than the a_n, due to the gauge BLKT,  the lightest member of the h_n KK tower, a real scalar field is to be identified with the DM. This implies that we must require 
that m^h_1<m^V_1, , 1+h < (2x^V_1)^2+a, to avoid the CMB constraint when choosing our BM points.  This constraint is powerful in that for fixed values of a, x^V_1 falls with i
ncreasing δ_A and for fixed δ_A, x^V_1 also falls with increasing a. Thus small values of δ_A and not too large values of a are preferred. The upper panel in 
Fig. <ref> shows how the numerical value of this root falls with increasing a for two representative values of δ_A. As might be expected, since the x^V_n depend on a, 
so will the ϵ_n's; this is shown in the lower panel of Fig. <ref>. 




The couplings between the dark scalar and gauge fields are determined by the integrals over the products of the 5-D wavefunctions which takes the form (where the cosθ_m mixing 
factor defined above) 


    c_mn^i = ∫_0^π R dy cosθ_m a_m(y)h_n(y)v_i(y)


and, correspondingly, the normalized c̃_mn^i can also be defined as above. A few obvious differences that we see immediately are:  (i) that instead of an S^† SV 
4-D interaction we now have an haV interaction, with a tensor structure of the form ∼ (a∂_μ h-h∂_μ a)V^μ and (ii) the appearance of an overall factor of 
cosθ_m. This factor occurs because it is actually the weak-basis fields χ_m which are entering into this interaction and 
so we must project out this part of the a_m fields.  This interaction structure has many important implications for DM physics. First, in DD experiments, at tree-level, the basic 
scattering process is now inelastic and of the form h_1 e→ a_1 e for scattering off of electrons via DP tower exchange. However, the elastic h_1e → h_1 e process does occur 
at the1-loop level but is suppressed by the electron mass as well as the usual loop factor leading to an effective operator given by 


    O= α/2πm_eg_D^2ϵ_1^2 L e̅eh_1^2


where L is a product of ϵ_i's, c̃_nm^i's and loop functions summed over intermediate states of the a_n and V_i's. (Note that the KK sums here are well-behaved 
since the falling ϵ_n's suppress the higher KK contributions.) 
This suppression, even with 
g_D^2ϵ_1^2  10^-6 and m_h_1=25 MeV, leads to cross sections ∼ 10^-50 cm^2 so this process is not likely to be accessible soon. We note that for any 
interesting, non-tuned value of δ, the a_1-h_1 mass splitting will always be sufficiently large as to prevent the tree-level inelastic process from occurring. Since h_1 is 
lighter than V_1 by construction, the DM obtains its thermal relic abundance via co-annihilation, , via the s-channel process h_1a_1→ V_n → SM and so this is again sensitive to
the particular value of δ (exponentially through the Boltzmann suppression factor). Note that since the a_1 is heavier than V_1, the process 
a_1a_1 → V_1V_1 via t,u- channel h_n exchange is barely kinematically allowed and is also doubly Boltzmann suppressed. Third, the annihilation of DM today will likely be 
unobservable, not only due to the p-wave, v^2-suppressed cross section but also due to the fact that all of the a_1 states needed for co-annihilation will have long since decayed away 
via the 3-body mode a_1→ h_1V_N^*→ e^+e^-. The lifetime for this process is also found to be rather sensitive to the value of δ as expected.  Lastly, while the a_1a_1 initial 
state can annihilate into the h_n in the s-channel, the h_n themselves do not couple to any of the SM fields 
at tree level since their wavefunctions vanish on the SM brane as discussed above. Lastly, we note that the V_5 content of the physical fields cannot 
mediate any interactions with those of the SM.  

We now turn to our specific BMs for this scenario. Since the spectrum is fixed, apart from an overall scale, once we choose our parameters and since the DM mass, m_h_1, is 
the smallest one we will express all our masses in units of the DM mass below for convenience. For BM1, we will assume δ_A=0.5 as above, a=1, δ=0.2 which leads to 
Σ=m_V_1/m_h_1=1.109 so that h=7/18≃ 0.389 while for BM2, we take δ_A=0.6, a=1, δ=0.14, h=0.4 and thus Σ=m_V_1/m_h_1=1.096, 
respectively. Tables <ref> and  <ref> display the masses of the lowest lying KK states for these two BM models. Once the values of 
δ_A and a are fixed all of the couplings c̃_mn^i are also fixed and these do not depend upon the numerical values of h,δ. 
We note that we can switch the parameter ordering in generating further BM points. Choosing a pair of Σ,δ values, h 
becomes fixed and we can determine the value of the lowest gauge KK root as a function of a. Using the equation for the gauge roots then gives us δ_A. For example, 
with Σ=1.1,δ=0.2 and a=1, we obtain δ_A≃ 0.554, x^V_1≃ 0.4125 and h=7/18.









As in the previous subsection, V_1 can decay only to the SM, most likely e^+e^- pairs. However, as noted, a_1(m_a) can decay 
to h_1(m_h) via an off-shell V_i(m_V_i) KK tower, materializing as e^+e^- as well since this mass splitting is likely to be greater than 1 MeV. The partial width for this 3-body 
process employing<cit.> the traditional variables m_ij^2=(p_i+p_j)^2 is given by


    Γ_3=(eg_Dϵ_1)^2/128π^3m_a^3∫  dm_12^2 ∫ dm_23^2  S[ (m_a^2-m_h^2)^2-(m_23^2-m_13^2)^2 +m_12^2
    (m_12^2-2(m_h^2+m_a^2))]


where S symbolizes the KK sum (away from resonances) given by 


    S=∑_ij (-1)^i+j(ϵ_iϵ_j/ϵ_1^2) c̃_11^i c̃_11^j/(m_12^2-m_V_i^2)(m_12^2-m_V_j^2)


Here, 4m_e^2≤ m_12^2≤ (m_a-m_h)^2 is the square of the e^+e^- pair mass, , and m_12^2+m_13^2+m_23^2=2m_e^2+m_a^2+m_h^2 with  
p_1,2,3 being identified with the momenta of e^±, a, respectively. The top panel of Fig. <ref> shows the resulting unboosted decay lengths of the a_1 in our two BM 
scenarios as a function of the h_1 mass. Here we see that the typical a_1 decay length lies in the ∼10-1000 cm range but this scales as 
(10^-3/g_Dϵ_1)^2 for other choices of these parameters.  To get a feel for the δ-dependence of these decay lengths, we fix m_h_1=100 MeV, δ_A=1 and 
take various choices of h=0.1-0.5 in the lower panel and determine cτ vs δ; note that the h-dependence itself is rather weak. In this parameter range cτ is found 
to scale roughly as ∼δ^-3.6. 




The heavier h,a and V KK states decay in a manner similar to that in the previous subsection except that the existence of three distinct towers adds complexity. However, given that 
the haV vertex is the only one relevant for any gauge mediated two-body decays, these are necessarily of the form V→ ah, a→ Vh, h→ Va, . The expressions 
for the various partial widths are similar to those encountered in the last subsection. A short consideration tells us that the BFs for the processes V_2→ h_1a_1,  h_2→ V_1a_1 
and a_2→ V_1h_1 are all ∼100% for both BMs. The corresponding BFs for the next set(s) of KK states can be found in Table <ref>. Here we see that for many of the modes 
there are no large differences between the two BMs; this is a result of the rather similar, compressed spectra in both cases. The most significant differences are visible in the decay modes 
with relatively small BFs for the higher gauge KK states. In this setup we note that when the heavier gauge KK states are directly produced from SM fields they will cascade 
through various channels down to (potentially numerous) combinations of the three h_1,a_1 and V_1 fields, similar to what we described in the last subsection. While the h_1 will produce 
missing energy and V_1 will decay (possibly with a long lifetime) into an e^+e^- pair, the a_1 will produce both missing energy and a soft e^+e^- pair after traveling ∼ 1 m. 
These final states would be interesting to observe. Unlike in the scenario discussed in the previous subsection, if V_1 can decay to pairs of DM particles, 
long-lived a_1's will still be produced in the cascade decays of the more massive states. In such a case, missing energy plus multiple displaced e^+e^- production vertices will yield a 
rather clean signature.





We now turn our attention to the DM relic density; the relevant process for this is h_1(m_h)a_1(m_a) → V_i^*(m_V_i) → e^+e^-, plus other possible SM final 
states if phase space is available; here we limit ourselves to the e^+e^- mode. In the m_e→ 0 limit we obtain, away from the narrow V_i,j resonance peaks: 


    σ v=(g_Deϵ_1)^2 s/12π(1-(m_a+m_h)^2/s)(1-(m_a-m_h)^2/s) ∑_i,j (-1)^i+j(ϵ_iϵ_j/ϵ_1^2) 
    c̃_11^i c̃_11^j/(s-m_V_i^2)(s-m_V_j^2)


There are two interesting aspects of this DM co-annihilation process<cit.>, one being the mass splitting between the a_1 and h_1, controlled by 
δ, that leads to significant Boltzmann suppression of the annihilation rate. As is well-known<cit.>, this modifies the DM co-annihilation rate by a factor of


    σ v_eff= 2α/1+α^2 σ v_ha


where σ v_ha is the the h_1-a_1 annihilation cross section above and α =(1+δ)^3/2e^-xδ with x=m_h_1/T≃ x_F∼ 20-30. If this was the sole 
effect, the annihilation cross section would fall off drastically with increasing δ as can be seen in the top panel of Fig. <ref> where all other quantities are held fixed and we  
vary δ for three specific values of x_F. 






The second aspect of relevance is the proximity of the sum of the initial state masses, m_a_1+m_h_1, to that of the next lowest lying gauge KK mass, 
m_V_2.[A similar situation can arise in the case of UED<cit.>.]  By construction, the sum of the a_1,h_1 masses lies above that of V_1; however, we note that 
this sum is, for both BMs, not far below that of V_2 implying 
that partial resonant enhancement may play a role in off-setting the Boltzmann suppression. This is particularly true as the thermal velocities of h_1,a_1 can raise 
the center of mass energy in the collision process closer to the V_2 resonance region. To probe the potential importance of the V_2 resonance, we examine the scaled 
annihilation cross sections for the BM models at fixed temperature (, fixed x_F) and with the corresponding fixed values of the velocities but allowing for one of the parameters 
h,δ to differ from their chosen BM values; this is shown in Figs. <ref> and  <ref>. 
While varying the value of h simply moves the h_1,a_1 spectrum for fixed δ so that the V_2 resonance is encountered, raising δ both lets us encounter this 
resonance while also simultaneously suppressing the overall cross section. Note that for the true BM parameters, taking (g_Dϵ_1/m_DM(GeV))^2 ∼ 10^-5 would roughly 
yield the desired value of the relic density. Thus the nearby resonance can, at least locally, appear to compensate for the overall Boltzmann suppression. 




In order to full understand the combined impact of both the resonances and Boltzmann suppression acting simultaneously on the thermally averaged cross section we return to the 
basic formalism and compute (numerically) the velocity-weighted cross section (with x_F=20 assumed here): 


    <σ v>= ∫ d^3p_a ∫ d^3p_h e^-E_a/Te^-E_h/T σ v_ha/∫ d^3p_a ∫ d^3p_h e^-E_a/Te^-E_h/t


with E_a,h being the incoming, velocity-dependent energies of a_1,h_1 in the collision process, respectively. Note that BM2 has a 
slightly smaller value of δ, 0.12 vs. 0.15,  so that the Boltzmann suppression is reduced and also the sum of the a_1,h_1 masses is closer to that of V_2 to that resonance effects 
should also be larger. However, since the ϵ's are somewhat smaller for BM2 since δ_A is greater, there is for BM2 a small suppression of the couplings 
of the gauge KK states above the lightest mode relative to those of BM1. Explicitly for BM1(BM2) we obtain the cross sections 
<σ v>= 2.5(2.8)· 10^-26cm^3 s^-1 (g_Dϵ_1/10^-8)^2  (500  MeV/m_h_1)^2.  Here we see that the reduced Boltzmann suppression and resonance 
proximity can compensate for each other and the reduced KK gauge couplings and that reasonable parameter choices can lead to the observed relic density for both BMs. These results are 
shown more generally as functions of δ in Fig. <ref>. Here we see in particular that the resonance is sufficiently Doppler broadened to increase the cross sections for these BMs 
by roughly an order of magnitude or more compared to naive expectations. 





 §.§ Model 2':  The DM Lifetime Problem



As mentioned above, the setup in the previous subsection has an important flaw in that the state h_1 can decay to SM fields (two pairs of e^+e^-) via double off-shell V tower  
exchange and this decay happens too rapidly for h_1 to be the DM. However, there is an easy solution to this problem which leaves the previously discussed phenomenology 
intact with only very minor numerical changes. The first step is to make a_1 the potential DM state and to do that we must decrease the a_1,h_1 masses relative to V_1; this 
can be accomplished by introducing a scalar BLKT, δ_S> δ_A, as was done in an earlier subsection above. Next, by requiring r>1, we make h_1 heavier than a_1 
while simultaneously maintaining the constraint m_h_1+m_a_1>m_V_1. This essentially just interchanges the roles of a_1 and h_1 in the previous section without any 
other phenomenological impact. Of course, a_1 remains unstable but, since it is CP-odd, it decays via a triple off-shell decay chain yielding a 6-body final state: 
a_1 → V^*h_1^*, h_1^*→ V^*V^* followed by V^* → e^+e^-.  A very rough estimate of this partial decay width given by  


    Γ_6/m_a_1∼α_D^3 α_em^3 ϵ_1^6/(3!)^2  (2π)^14 2π^8/2^6Γ(8) (m_a_1/m_V_1)^12 ,


which follows from the coupling constants, six-particle final state phase space factors, the appearance of two triplets of identical particles in the final state, the assumption that 
h_1 is only slightly more massive than the a_1, and the dominance of V_1 in the off-shell gauge boson propagator summations.  
Taking α_D ≃α_em and ϵ_1=10^-4 as above, m_a_1/m_V_1=1/2, and m_a_1=50 MeV one obtains 
Γ_6 ∼  3 · 10^-51 MeV corresponding to a lifetime of ∼ 2 · 10^29s which is a factor of ∼ 10^4 longer than the lower bound from the 
CMB<cit.> and ∼ 100 times longer than the tentative new limits from the 21 cm line<cit.> for this mass. Clearly a more detailed calculation of this partial 
width is warranted but this rough estimate indicates that the a_1 is very likely to be sufficiently long-lived in this setup to act as the DM without much impact on the analysis 
presented in the previous subsection.





§ DISCUSSION AND CONCLUSIONS



In this paper we have considered an extension of the familiar 4-D kinetic-mixing portal/dark photon model to 5-D by adding a single, flat extra dimension of inverse `radius' 
R^-1∼ 10-1000 MeV, which is treated as an interval allowing for more general boundary conditions. In the simple models we construct, while the gauge mediator and 
the dark matter experience the full 5-D, the SM fields are localized to a 4-D brane at one end of this interval. To avoid constraints from CMB measurements we have considered 
the case of complex scalar dark matter with the lightest scalar Kaluza-Klein state lying below that of the corresponding lightest dark photon KK state to insure p-wave annihilation. 
In these models, the consistency of the field redefinitions needed to undo the effects of KM and the avoidance of ghosts and/or tachyons requires the 
existence of a BLKT on the brane where the 4-D SM fields are localized. We constructed two distinct scenarios depending upon whether or not the scalar dark matter field 
obtained a vev in the 5-D bulk. The presence of this vev was shown not to be required in order to break the dark U(1)_D gauge symmetry as this was accomplished by the 
choice of BCs with the fifth component of the gauge fields then acting as the Goldstone bosons.  Constraints arising from precision electroweak measurements, from rare Z 
and Higgs decays as well as from monojet searches at the LHC were shown to be satisfied even in the presence of multiple KK excitations. We also found that we can 
use these BCs to remove the possibility of a Higgs portal that induces a coupling of the SM Higgs with the KK tower scalar fields thus avoiding potential conflicts 
with constraints on exotic Higgs decays. If the bulk scalar does not obtain a vev, the dark matter remains as a complex field whereas when the vev is present the complex 
scalar decomposes into CP-even and CP-odd scalar towers. When the vev is absent the required ordering of the scalar vs gauge mass spectrum necessitates the addition of 
a BLKT for the scalar field on the non-SM brane. In the setup where a vev occurs the CP-even scalars are mass eigenstates while the CP-odd field mixes with the 
fifth component of the gauge field to form both Goldstone KK and physical CP-odd scalar towers. Without a scalar BLKT, level-by-level these CP-odd scalar fields 
were determined to be more massive than the corresponding gauge states due to the gauge BLKT; a BLKT for the scalar can change this hierarchy. Indirect detection of dark 
matter today in either scenario was shown to be unlikely: in the first setup, due to the p-wave nature of the annihilation process, the cross section is suppressed due to the low 
velocity of the dark matter. In the second scenario, the relic density is achieved via p-wave co-annihilation with the opposite CP scalar. In the first scenario, a small but potentially 
observable direct detection, spin-independent scattering cross section off of bound atomic electrons was found whereas in the second such a cross section was determined to be 
unobservably small due to loop suppression with the corresponding tree-level, now inelastic, scattering process being kinematically forbidden for non-relativistic dark matter velocities. 

The overall allowed parameter space of the pair of models that we have constructed is somewhat difficult to visualize. In attempting this and to be as model-independent as possible 
is likely best to focus on the DP KK masses and their couplings to SM matter as represented by the ϵ_n.  Semi-quantitatively, the simplest picture to keep in mind is to 
imagine the oft-shown<cit.> ϵ^2-m_V mass plane as this occurs in all the types of models that we consider; see, , Fig. 22 in  Ref.<cit.>. 
Once an (experimentally allowed) pair of values for ϵ_1,m_V_1 is chosen, the model can be very crudely represented by this point  plus the (asymptotically correct) locus 
of all the subsequent set of pairs of points m_V_n≃ (2n-1) m_V_1 and ϵ_n ≃ϵ_1/(nδ_A)  in this plane.  Detailed models will of course differ from 
this crude picture by O(1) effects but this image roughly captures the main features of the allowed  parameter space. To give a specific example and provide a clearer understanding 
of this, consider Model 1 above taking R^-1=100 MeV, ϵ_1=3· 10^-4 and δ_A=0.5, 1, 2, which are values typical of the previous discussions above. Fig. <ref> 
then shows the loci of these KK points in the ϵ^2-m_V plane that can be directly compared with Fig. 22 in Ref.<cit.>. Presently, for these chosen parameter 
values, the KK's are seen to lie in an allowed region between experimentally excluded areas. Clearly, we could just as easily have chosen an different R^-1,ϵ_1 `origin' point 
for this loci but the overall shape of the predicted parameter space would be reasonably similar. It is important to remember, however, that these KK states will generally not decay in the 
same manner as the 4-D states of the same mass as represented in Ref.<cit.>.





A unique feature of these 5-D models is the simultaneous production of several members of the dark photon tower each of which cascade decays to pairs of 
kinematically accessible scalar tower states. As in the 4-D case several production mechanisms are possible with the final signals being determined by the relative values of the 
masses of the lightest KK tower states.  When the DM is less than half of the mass of the lightest gauge state, this cascade results in a missing energy signature. In associated 
production experiments, such as in meson decays or in inclusive e^+e^- →γ +X reactions, the presence of the individual KK tower states may be reconstructed. In electron 
fixed-target scattering experiments where only the outgoing electron is measured, on the other hand, only the cumulative effect of production and decay of the gauge KK towers is 
observable which may in many cases be difficult to differentiate from the conventional 4-D signal depending on the detailed parameters values. Perhaps even more interesting is the 
situation when the DM is somewhat heavier than the lightest gauge KK state; in such cases the cascades not only produce missing energy but also multiple pairs of light 
charged SM fields, , electrons and/or muons. However, when the DM scalar obtains a vev, then numerous long-lived states are also automatically produced as part of these 
cascades. In either case these are rather striking and distinctive experimental signatures for the 5-D scenario.

The theoretical landscape of DM models continues to broaden and the experimental searches must continue to widen if we are eventually going to corner DM.








§ ACKNOWLEDGEMENTS




The author would like to thank J.L. Hewett, D. Rueter and G. Wojcik for valuable discussions. The author would like to thank D. Morrissey for bringing  the related work in 
Ref. <cit.> to his attention after the first version of the present paper appeared. The author would also like to thank T. Slatyer for discussions on the CMB  constraints for 
light dark matter masses. This work was supported by the Department of Energy, Contract DE-AC02-76SF00515.




99


Arcadi:2017kky 
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susy17
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  [hep-ph/9804398];
  


  N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali,
  
  Phys. Lett. B 429, 263 (1998)

  [hep-ph/9803315];
  

  L. Randall and R. Sundrum,
  
  Phys. Rev. Lett.  83, 3370 (1999)
  [hep-ph/9905221];
  


  T. Appelquist, H. C. Cheng and B. A. Dobrescu,
  
  Phys. Rev. D 64, 035002 (2001)

  [hep-ph/0012100].
  

flavor
For some early work on this subject in the RS model, see, for example, 


  K. Agashe, G. Perez and A. Soni,
  
  Phys. Rev. D 71, 016002 (2005)

  [hep-ph/0408134];
  
  



  S. J. Huber,
  
  Nucl. Phys. B 666, 269 (2003)

  [hep-ph/0303183];
  
  


  A. L. Fitzpatrick, G. Perez and L. Randall,
  
  Phys. Rev. Lett.  100, 171604 (2008)

  [arXiv:0710.1869 [hep-ph]].
  
 
ued2
 G. Servant and T. M. P. Tait,
  
  Nucl. Phys. B 650, 391 (2003)

  [hep-ph/0206071];
  
  H. C. Cheng, J. L. Feng and K. T. Matchev,
  
  Phys. Rev. Lett.  89, 211301 (2002)

  [hep-ph/0207125].
  

morrissey
For earlier related work employing a modified warped RS setup, see 


  K. L. McDonald and D. E. Morrissey,
  
  JHEP 1005, 056 (2010)

  [arXiv:1002.3361 [hep-ph]] and 
  
  



  
  JHEP 1102, 087 (2011)

  [arXiv:1010.5999 [hep-ph]].
  
  
  
  keith
See, for example,   
  

  K. R. Dienes and B. Thomas,
  
  Phys. Rev. D 85, 083523 (2012)
 
  [arXiv:1106.4546 [hep-ph]] 
  
  
 and  
 

 
  
  Phys. Rev. D 85, 083524 (2012)

  [arXiv:1107.0721 [hep-ph]] and subsequent works.
  
  
 

 vectorportal 
 There has been a huge amount of work on this subject; see, for example, 
 

  D. Feldman, B. Kors and P. Nath,
  
  Phys. Rev. D 75, 023503 (2007)
  
  [hep-ph/0610133];
  
  


  D. Feldman, Z. Liu and P. Nath,
  
  Phys. Rev. D 75, 115001 (2007)
 
  [hep-ph/0702123 [HEP-PH]].;
  
  
 

  M. Pospelov, A. Ritz and M. B. Voloshin,
  
  Phys. Lett. B 662, 53 (2008)

  [arXiv:0711.4866 [hep-ph]];
  
  


  M. Pospelov,
  
  Phys. Rev. D 80, 095002 (2009)

  [arXiv:0811.1030 [hep-ph]]; 
  


  H. Davoudiasl, H. S. Lee and W. J. Marciano,
  
  Phys. Rev. Lett.  109, 031802 (2012)

  [arXiv:1205.2709 [hep-ph]] and 
  
  



  
  Phys. Rev. D 85, 115019 (2012)
  doi:10.1103/PhysRevD.85.115019
  [arXiv:1203.2947 [hep-ph]];
  
  
  

  R. Essig et al.,
  
  arXiv:1311.0029 [hep-ph];
  
  


  E. Izaguirre, G. Krnjaic, P. Schuster and N. Toro,
  
  Phys. Rev. Lett.  115, no. 25, 251301 (2015)

  [arXiv:1505.00011 [hep-ph]];
  
  
 For a general overview and introduction to this framework, see  
 

  D. Curtin, R. Essig, S. Gori and J. Shelton,
  
  JHEP 1502, 157 (2015)

  [arXiv:1412.0018 [hep-ph]].
  
 
 KM
 

  B. Holdom,
  
  Phys. Lett.  166B, 196 (1986) and

  


 
  
  Phys. Lett. B 178, 65 (1986); 

  


  K. R. Dienes, C. F. Kolda and J. March-Russell,
  
  Nucl. Phys. B 492, 104 (1997)

  [hep-ph/9610479];
  
 

  F. Del Aguila,
  
  Acta Phys. Polon. B 25, 1317 (1994)
  [hep-ph/9404323];
  
  


  K. S. Babu, C. F. Kolda and J. March-Russell,
  
  Phys. Rev. D 54, 4635 (1996)

  [hep-ph/9603212];
  
  


  T. G. Rizzo,
  
  Phys. Rev. D 59, 015020 (1998)

  [hep-ph/9806397].
  
  
  
 higgsportal
 See, for example, 
  

  B. Patt and F. Wilczek,
  
  hep-ph/0605188;
  
  


  S. Gopalakrishna, S. Jung and J. D. Wells,
  
  Phys. Rev. D 78, 055002 (2008)

  [arXiv:0801.3456 [hep-ph]];
  
  


  J. D. Clarke, R. Foot and R. R. Volkas,
  
  JHEP 1402, 123 (2014)

  [arXiv:1310.8042 [hep-ph]];
  
  




  J. Liu, X. P. Wang and F. Yu,
  
  JHEP 1706, 077 (2017)

  [arXiv:1704.00730 [hep-ph]];
  
  
L. Darm, S. Rao and L. Roszkowski,
  
  arXiv:1710.08430 [hep-ph];
  


  J. L. Feng, I. Galon, F. Kling and S. Trojanowski,
  
  arXiv:1710.09387 [hep-ph];
  
  
 
 blkts
 

  G. R. Dvali, G. Gabadadze and M. A. Shifman,
  
  Phys. Lett. B 497, 271 (2001)

  [hep-th/0010071];
  
  
 

  M. Carena, E. Ponton, T. M. P. Tait and C. E. M. Wagner,
  
  Phys. Rev. D 67, 096006 (2003)

  [hep-ph/0212307];
  
  


  M. Carena, T. M. P. Tait and C. E. M. Wagner,
  
  Acta Phys. Polon. B 33, 2355 (2002)
  [hep-ph/0207056];
  
  


  F. del Aguila, M. Perez-Victoria and J. Santiago,
  
  Acta Phys. Polon. B 34, 5511 (2003)
  [hep-ph/0310353] and 
  
  


 
  
  JHEP 0302, 051 (2003)

  [hep-th/0302023];
  
  


  H. Davoudiasl, J. L. Hewett and T. G. Rizzo,
  
  Phys. Rev. D 68, 045002 (2003)

  [hep-ph/0212279] and 
  
  


 
  
  JHEP 0308, 034 (2003)

  [hep-ph/0305086].
  
  

stuff
See, for example, the last paper in Ref.<cit.>. 
 

Heeck:2011md 
  J. Heeck and W. Rodejohann,
  
  Phys. Lett. B 705, 369 (2011)

  [arXiv:1109.1508 [hep-ph]].
  
  

pdg
C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016) and 2017 update.
 

Ade:2015xua 
  P. A. R. Ade et al. [Planck Collaboration],
  
  Astron. Astrophys.  594, A13 (2016)

  [arXiv:1502.01589 [astro-ph.CO]].
  


Liu:2016cnk 
  H. Liu, T. R. Slatyer and J. Zavala,
  
  Phys. Rev. D 94, no. 6, 063507 (2016)

  [arXiv:1604.02457 [astro-ph.CO]].
  
  See also, 
  

  M. Dutra, M. Lindner, S. Profumo, F. S. Queiroz, W. Rodejohann and C. Siqueira,
  
  arXiv:1801.05447 [hep-ph].
  
  

Steigman:2015hda 
  G. Steigman,
  
  Phys. Rev. D 91, no. 8, 083538 (2015)

  [arXiv:1502.01884 [astro-ph.CO]].
  
  
  
pair


See, for example, 
  J. Berger, K. Jedamzik and D. G. E. Walker,
  
  JCAP 1611, 032 (2016)
 
  [arXiv:1605.07195 [hep-ph]];
  
 

  A. Fradette, M. Pospelov, J. Pradler and A. Ritz,
  
  Phys. Rev. D 90, no. 3, 035022 (2014)
 
  [arXiv:1407.0993 [hep-ph]].
  

wip


  T. G. Rizzo,
  
  arXiv:1805.08150 [hep-ph].
 


Hagiwara:2017zod 
  K. Hagiwara, A. Keshavarzi, A. D. Martin, D. Nomura and T. Teubner,
  
  Nucl. Part. Phys. Proc.  287-288, 33 (2017).

  
  

peskin
M. Peskin and T. Takeuchi,
  
  Phys. Rev. D. 46, 381 (1992)

  

maksymyk
I. Maksymyk, C.P. Burgess, D. London,
  
  Phys. Rev. D. 50, 529 (1994)


LDMX
For more information about the LDMX experiment, see  <https://confluence.slac.stanford.edu/display/MME/Light+Dark+Matter+Experiment> 
and also 


  T. Raubenheimer et al.,
  
  arXiv:1801.07867 [physics.acc-ph].
  



Mans:2017vej 
  J. Mans [LDMX Collaboration],
  
  EPJ Web Conf.  142, 01020 (2017).

  


Bjorken:2009mm 
  J. D. Bjorken, R. Essig, P. Schuster and N. Toro,
  
  Phys. Rev. D 80, 075018 (2009)

  [arXiv:0906.0580 [hep-ph]].
  
 
 Belle
 

  G. Inguglia,
  
  PoS DIS 2016, 263 (2016)
  [arXiv:1607.02089 [hep-ex]].
  
  
 
 
Liu:2017jzn 
  Z. C. Liu, C. X. Yue and Y. C. Guo,
  
  arXiv:1703.00153 [hep-ph].
  
  
 
mono


  M. Aaboud et al. [ATLAS Collaboration],
  
  arXiv:1711.03301 [hep-ex];
  
  
  

  A. M. Sirunyan et al. [CMS Collaboration],
  
  arXiv:1712.02345 [hep-ex].
  
  


Chen:2016tdz 
  C. S. Chen, G. L. Lin, Y. H. Lin and F. Xu,
  
  Int. J. Mod. Phys. A 32, no. 31, 1750178 (2017)

  [arXiv:1609.07198 [hep-ph]]. 
  
  
See also, 


  T. Emken, C. Kouvaris and I. M. Shoemaker,
  
  Phys. Rev. D 96, no. 1, 015018 (2017)

  [arXiv:1702.07750 [hep-ph]]  and 
  
  


  M. J. Dolan, F. Kahlhoefer and C. McCabe,
  
  arXiv:1711.09906 [hep-ph].
  
  

 
Essig:2017kqs 
  R. Essig, T. Volansky and T. T. Yu,
  
  Phys. Rev. D 96, no. 4, 043017 (2017)

  [arXiv:1703.00910 [hep-ph]].
  
  
 
 
Essig:2015cda 
  R. Essig, M. Fernandez-Serra, J. Mardon, A. Soto, T. Volansky and T. T. Yu,
  
  JHEP 1605, 046 (2016)

  [arXiv:1509.01598 [hep-ph]].
  
  

  
  
Berlin:2014tja 
See, for example, 
  A. Berlin, D. Hooper and S. D. McDermott,
  
  Phys. Rev. D 89, no. 11, 115022 (2014)

  [arXiv:1404.0022 [hep-ph]].
  
  


5d


  A. Muck, A. Pilaftsis and R. Ruckl,
  
  Phys. Rev. D 65, 085037 (2002)

  [hep-ph/0110391];
  
  

  T. Flacke, A. Menon and D. J. Phalen,
  
  Phys. Rev. D 79, 056009 (2009)

  [arXiv:0811.1598 [hep-ph]].
  
  


Davoli:2017swj 
See, for example,
  A. Davoli, A. De Simone, T. Jacques and V. Sanz,
  
  JHEP 1711, 025 (2017)

  [arXiv:1706.08985 [hep-ph]].
  

nikita


See, for example, 
  N. Blinov, E. Izaguirre and B. Shuve,
  
  Phys. Rev. D 97, no. 1, 015009 (2018)
  doi:10.1103/PhysRevD.97.015009
  [arXiv:1710.07635 [hep-ph]].
  
  



Feng:2017drg 
We follow the analysis for light dark matter as presented in  J. L. Feng and J. Smolinsky,
  
  Phys. Rev. D 96, no. 9, 095022 (2017)

  [arXiv:1707.03835 [hep-ph]] .
  
See also, 


  B. Li and Y. F. Zhou,
  
  Commun. Theor. Phys.  64, no. 1, 119 (2015)

  [arXiv:1503.08281 [hep-ph]] 
  
  
and  


  B. Li and Y. F. Zhou,
  
  Commun. Theor. Phys.  64, no. 1, 119 (2015)

  [arXiv:1503.08281 [hep-ph]].
  
  


Slatyer:2016qyl 
  T. R. Slatyer and C. L. Wu,
  
  Phys. Rev. D 95, no. 2, 023010 (2017)
 
  [arXiv:1610.06933 [astro-ph.CO]].
  
  


21club
See, for example, 


  H. Liu and T. R. Slatyer,
  
  arXiv:1803.09739 [astro-ph.CO];
  
  


  S. Clark, B. Dutta, Y. Gao, Y. Z. Ma and L. E. Strigari,
  
  arXiv:1803.09390 [astro-ph.HE];
  
  


  A. Mitridate and A. Podo,
  
  arXiv:1803.11169 [hep-ph].
  








