Thermal Relic Targets with Exponentially Small Couplings

If dark matter was produced in the early Universe by the decoupling of its annihilations into known particles, there is a sharp experimental target for the size of its coupling. We show that if dark matter was produced by inelastic scattering against a lighter particle from the thermal bath, then its coupling can be exponentially smaller than the coupling required for its production from annihilations. As an application, we demonstrate that dark matter produced by inelastic scattering against electrons provides new thermal relic targets for direct detection and fixed target experiments.

Introduction-Dark Matter (DM) was produced in the early Universe by unknown dynamics. If the production of DM is tied to its measurable interactions, experiments can help disentangle its cosmological origin. The most studied example is the Weakly Interacting Massive Particle (WIMP) [1][2][3][4], with abundance set by the decoupling of its annihilations into Standard Model (SM) particles. In this case the DM annihilation rate is predicted and testable by current experiments.
A well-known variant of the WIMP is DM with a mass in the MeV to GeV range that annihilates into electrons and positrons (left of Fig. 1) [5,6]. This scenario requires additional force mediators beyond the SM [1,[5][6][7][8]. The requirement that DM has the observed abundance fixes the size of its coupling to electrons, implying a sharp target for direct detection and fixed target experiments [9][10][11][12][13][14]. The WIMP and its variants are now driving a large experimental effort. Are there alternate scenarios on the same theoretical footing as the WIMP that experiments should also target?
We identify a new example that satisfies the above assumptions: DM, χ, is produced by scattering against electrons. Elastic scattering, χe ± → χe ± , does not change the number of DM particles, but the abundance can be set by inelastic scattering, χe ± → ψe ± (right of n w Y 9 k e g 8 f n + S v 2 D a c q e 7 m S U P 5 Z J 7 V i r f n R c r V 3 l v a + S Q H J E T 4 p I L U i E 3 5 J Z U C S d P 5 J m 8 k j f r x X q 3 P q 2 v y e i C l e / s k 3 + w v n 8 B X 0 y p 4 g = = < / l a t e x i t > e < l a t e x i t s h a 1 _ b a s e 6 4 = " t s D k s u g i 4 x Z L 9 a 7 9 W l 9 z U Z X r H T n i P y D 9 f 0 L I J i q V A = = < / l a t e x i t > FIG. 1. The process setting the relic density for the WIMP (left) and coscattering (right), where χ is DM and ψ is a dark partner. Fig. 1), where ψ is a dark partner that experiences its own rapid annihilations. This is an example of coscattering [34] as studied recently by Refs. [35][36][37][38][39]. See also Refs. [40][41][42] for related prior work. Here we introduce a mechanism for extending any WIMP-like model to include a coscattering phase, and we show how this opens up new parameter space with smaller DM coupling.
The rate for DM to coscatter (annihilate) is proportional to the electron (DM) number density n e (n χ ) multiplied by the relevant cross section. Suppose that m e < T f < m χ , where T f is the temperature at which annihilations or coscatterings decouple. Then the number density of electrons, which are relativistic, is larger than the number density of DM, which is non-relativistic, by the exponentially large factor n e /n χ ∝ e mχ/T f . Therefore, the rate of coscattering can exceed annihilation despite an exponentially smaller cross section, opening up vast new parameter space for thermal relic targets. Below we focus on a particular model with coscattering against electrons, but our observation applies to many more models and to replacing the electron with other SM particles. Another scenario that satisfies the above assumptions and leads to a smaller DM coupling is anni- hilation through a pole [43], requiring a special relation between the mass of DM and another particle.
The rest of this letter is organized as follows. We begin by comparing coscattering to the more familiar WIMP, and we derive the exponential factor that enlarges the parameter space for coscattering. We then show how any model with annihilating DM can be extended to include coscattering, taking as an example scalar DM that couples to electrons through a dark photon. We explore the detailed phenomenology of this example, highlighting the experimentally testable parameter space. Our Appendix contains a map of the phase space of this model, charting where the the DM abundance is set by coscattering or alternate mechanisms.
Coscattering vs. Annihilations-A WIMP with mass m χ begins in equilibrium with the SM thermal bath when T > m χ . Its abundance is depleted through annihilations when T < m χ . When the rate of DM annihilations becomes slower than the expansion of the Universe the total number of DM particles is fixed (freezes-out) determining the observed relic density today [2,3]. Here we have used the superscript "eq" to denote particles in thermal equilibrium with the SM bath. The sudden freeze-out approximation described above points to annihilation cross sections comparable to those induced by the SM weak interactions: where x f = T f /m χ describes the temperature when Eq. 1 is satisfied. If annihilations are too small, and no other process removes DM, the relic density is too large. In this regime, coscattering is an alternative mechanism for setting the DM abundance [34]. DM still begins in equilibrium with the SM thermal bath when T > m χ . The relic density is set when T < m χ and inelastic scattering becomes slower than Hubble. In the following we assume that DM scatters against an electron or positron, as on the right of Fig. 1, but other choices are possible [34]. DM scatters into a dark partner ψ with a heavier mass, m ψ ≥ m χ , whose annihilations leave equilibrium after the inelastic scattering. DM is depleted by χe ± → ψe ± followed by ψ annihilations. When χe ± → ψe ± freezes-out, so does the χ abundance. The small component of ψ left over from the later freeze-out of ψ annihilations is converted into χ by decays, or else persists until today.
The main qualitative difference between coscattering and the WIMP is that we can obtain the correct relic density with χ interactions that are exponentially smaller than required for annihilations. If we assume that momentum transfer via χe ± → χe ± is efficient until after the coscattering diagram freezes-out, we can write a Boltzmann equation for the total χ number density In this case freeze-out occurs when where p ≈ 10 is chosen to match numerical solutions to Eq. 3. If we use detailed balance to write the thermal average for the endothermic process χ → ψ in terms of the thermal average for the inverse process [34], we can write the χ relic density as where ∆m ≡ m ψ − m χ and g χ,ψ counts the number of degrees of freedom of χ and ψ. In order to reproduce Ω χ = Ω DM we require σ ψ→χ v ∼ e −x f (1−∆m/mχ) σ WIMP . This corresponds to a significant exponential suppression, as long as ∆m/m χ < 1, because x f ∼ 20 is needed to match the observed abundance. (The necessary freezeout temperature is universal across different types of thermal relics including coscattering and the WIMP).
The Model-The coscattering phase can be added to any WIMP model in a modular way. Consider a WIMP with an annihilation channel, such as the left of Fig. 1, and add a mass mixing between DM and a dark partner, ψ, with its own rapid annihilations. The coscattering diagram on the right of Fig. 1 is generated by rotating the annihilation diagram and inserting the mixing.
As an example we consider dark scalar QED [5,6,[9][10][11][12][13][14] with DM coupled to the SM via a U (1) d massive dark photon, A d , kinetically mixed [44] with the ordinary photon, A: where F (d) is the ordinary (dark) photon field strength, and is a dimensionless measure of the mixing. DM is a complex scalar χ with charge 1 under U (1) d : L ⊃ |D µ χ| 2 −m 2 χ |χ| 2 . DM can annihilate: χχ * → γ * d → e + e − . We add a complex scalar, ψ, that is neutral under U (1) d and mixed with χ: Coscattering, χe → ψe, is generated by the mixing. Quartic couplings for χ and ψ can also be included without modifying our discussion. The mass mixing δm 2 and the dark photon mass m 2 A can arise from a dark Higgs coupled to χ and ψ. Note that we use m 2 χ,ψ for mass parameters in the Lagrangian and m 2 χ,ψ for mass eigenvalues. The angle that rotates from the Lagrangian basis to the mass eigenstate basis is θ ∼ δm 2 /m 2 χ . For the rest of this letter we (slightly) abuse notation by calling the lightest eigenstate χ and the heaviest ψ.
We take ψ to annihilate to a dark state (this is a key difference versus Refs. [35,37,38], where the dark partner has a large coupling to the SM), ψψ * → SS, where S is a real scalar lighter than ψ, χ couples to S only through its mixing with ψ. We take S to decay rapidly to the thermal bath via a small Yukawa coupling to electrons. There is a large range of values for this coupling (10 −4 y eS 10 −9 ) such that S remains in equilibrium with the SM bath during freeze-out but otherwise y eS does not enter the relic density calculation.
We consider parameters where χχ * → e + e − annihilations are too feeble to set the relic density. However ψψ * → SS annihilations are efficient and the last process to freeze-out. The coscattering process χe → ψe leaves equilibrium after χχ * → e + e − , but before ψψ * → SS. It is the fastest process converting χ to ψ and its freeze-out determines the χ abundance. This hierarchy of thermal rates is depicted in Fig. 2. The figure also shows that χ would have too large of an abundance in the WIMP limit where only χ annihilations are active, while the addition of ψ leads to the correct relic density via coscattering. The freeze-out hierarchy described in the previous paragraph can be realized with the spectrum: m A > m ψ > m S > m χ √ δm 2 and the hierarchy of couplings: , λ ψS . Furthermore, we fix r S ≡ (m S − m χ )/(m ψ − m χ ) = 0.75 in the following. More details on these choices and the freeze-out phases of this theory can be found in the Appendix.
Phenomenology-The main qualitative feature of coscattering with SM states is the exponentially small coupling to SM particles, compared to the WIMP. In Fig. 3 we plot current constraints and future probes in terms of the scattering cross section relevant to electron recoil experiments:σ e ≈ e 2 g 2 D 2 m 2 e c 4 θ /(πm 4 A ). Coscattering can reproduce the observed relic density in the region between the solid orange line (where it reduces to WIMP freeze-out) and the solid blue line. In the coscattering region we find that χ is in kinetic equilibrium with the SM during freeze-out due to rapid energy exchange from χe ± → χe ± [45,46]. This is distinct from previous studies of coscattering where DM kinetically decouples during freeze-out [34][35][36][37][38].
In Fig. 3 we fix α d = 0.5, m A = 3m χ , r S = 0.75, and ∆ ≡ (m 2 ψ − m 2 χ )/m 2 χ = 0.1. Within the coscattering region is determined by the value ofσ e on the y-axis, once θ is fixed at each point to give the right relic density. In this region the quartic λ ψS is chosen at each point to select the coscattering regime, as discussed in the Appendix. Below the coscattering region the mixing angle is set to its maximum value θ = 0.45 that is reached on the blue line at the boundary of the coscattering parameter space.
The reason why coscattering provides a range of viable couplings as opposed to the WIMP relic density line can be understood with the help of Fig. 4. The relic density is set by processes that interchange χ and ψ, so is sensitive to the mixing angle, θ, as shown in the right panel of Fig. 4. Larger θ allows for a smaller coupling of χ to electrons when fixing the relic density. So the smallest possible coupling of DM to the SM is achieved when θ = O (1). Smaller values of θ span the region between the blue and the orange lines of Fig. 3. When θ becomes too large we enter a coannihilation or WIMP regime, de-pending on the parameters in the dark sector, as shown in Fig. 4 and discussed in the Appendix.
We also comment on the χ − ψ mass splitting, ∆. The coscattering process is endothermic (m ψ ≥ m χ ) and the relic density depends exponentially on ∆. This is shown in the left panel of Fig. 4 and in Eq. (5). When ∆ goes to zero there is no suppression from the thermal average andσ e can be small, while for larger ∆ a larger coupling of χ to electrons has to compensate the kinematical suppression. So ∆ ≈ 0 gives the smallest direct detection cross section compatible with the observed relic density. In Fig. 3 we have chosen ∆ = 0.1; the smallest possibleσ e obtained for ∆ = 0 is approximately a factor of 5 below the solid blue line in the plot.
We would like to draw attention to the relative smoothness of the line bounding the coscattering parameter space from below in Fig. 3, compared to the WIMP relic density line. The WIMP line is determined by s-channel DM annihilations and reflects the structure of SM resonances that occur when 2m χ ≈ m SM . Coscattering is t-channel and receives contributions from all SM states that are relativistic at freezeout, T f m SM . As the DM mass increases, new SM states contribute smoothly. There is theoretical uncertainty when freeze-out happens near the QCD phase transition, T f ∼ 100 − 200 MeV. In this regime we show both coscattering off quarks (upper curve) and pions/kaons (lower curve).
Conclusions-In this paper we introduced a ther-mal relic that shares the attractive theoretical properties of the WIMP, but reproduces the DM relic density with an exponentially smaller coupling to the SM. We have shown that coscattering [34] can be realized by extending any model of WIMP DM, opening up orders of magnitude of new parameter space. We have explored the phenomenology of one concrete example model, providing a new benchmark for future light DM experiments.
Our work provides further motivation for direct detection experiments that probe sub-GeV DM scattering with nucleons or electrons, and missing momentum searches at fixed target experiments. Coscattering motivates extending the sensitivity of these programs beyond traditional WIMP targets. where we define Notice that χ annihilations to electrons are p-wave suppressed and we assumed ∆ × x f 1 in the equation for χe → ψe. In the opposite limit, ∆ × x f 1, the zero appearing in Eq. 9 at ∆ = 0 is lifted by the temperature dependence of the thermal average and In the following we will keep using Eq. 9 which is however only valid for ∆ × x f 1.
We have only included the process affecting the number density of χ and which are exothermic (in the parameter space discussed in the paper). Assuming thermal equilibrium the thermal average of the inverse endothermic processes are obtained by using detailed balance with X = e, S. Coscattering occurs when various conditions are realized: the DM particle χ is in kinetic equilibrium with the SM (I), there is no chemical potential for χ (II), and the last reactions to decouple which changes the χ number density are exchange reactions χ ↔ ψ (III). While this is also the setup of Ref. [34], here we focus on the situation in which the dominant coscattering is χe → ψe, thanks to the enhancement coming from the large number density of relativistic electrons (IV). All the above conditions have to be enforced at the freeze-out temperature T f when the exchange reactions in (III) leave equilibrium.
The exchange of χ ↔ ψ from inverse decay is subdominant to χe → ψe when where we used detailed balance to rewrite the endothermic reactions in terms of exothermic ones. Using Eq. 9, Eq. 14 requires For ∆ = 0.1 (∆m ≈ 0.05), as used for our numerical results, the contribution from inverse decay is always negligible. For much smaller values of ∆, Eq. 9 is not accurate and instead Eq. 12 should be used, which leads to similar conclusions. Finally, we verified numerically that inverse decay is negligible for ∆ × x f ∼ 1.
In order for (II) to be satisfied, it is sufficient for dark annihilation of ψ, ψψ * → SS, to be faster than the coscattering reaction, χe → ψe, at T f : Using detailed balance and the expression for relativistic and non-relativistic number densities in kinetic equilibrium we obtain the following relation with c II ≈ 0.35 ∆ 2 /x 3/2 f , expressing a lower bound on σ ψ /σ χ .
In order for (III) to be satisfied, that is to be in the coscattering regime, all annihilations for χ should be slower than χe → ψe at T f . Starting with χχ * → e + e − , this requires with c This relation can also be interpreted as the fact that coscattering allows the usual thermal annihilation cross section χχ * → e + e − to be exponentially smaller than its thermal freeze-out value, as it is apparent by rewriting Eq.
The two additional reactions χχ * → SS and χψ → SS impose constraints that depend on the relation between m S and m χ,ψ . We discuss these constraints assuming that both reactions are endothermic, m S ≥ (m χ +m ψ )/2. This is the relevant situation for the parameter choice made in the paper (r S = 0.75). From this choice it follows that the exponential suppression of the two thermal averages is the same, but χψ → SS is only suppressed by one power of the mixing angle θ 2 . Requiring χψ → SS to be out of equilibrium at T f is thus the leading constraint and it requires with c (2) . Finally the requirement that coscattering occurs by scattering on electrons (IV) requires χS → ψS to be out of equilibrium at T f : where c IV ≈ 0.72 Notice that if m S < m ψ , which we assumed, this constraint is subleading to Eq. (20).
Putting all the conditions together, the ratio σ ψ /σ χ is constrained to lie in the interval For fixed values of the masses, this can be interpreted as a constraint on θ, θ 2 c Notice that for a fixed value of r S there is a maximal value of ∆m/m χ such that B > 1.
Freeze-out phases Fig. 6 shows how the scalar QED model described in the paper may exhibit different DM freeze-out mechanisms depending on the choice of parameters. We display the phase diagram in the (λ ψS ,σ e ) plane. At each point in the plane we solve the coupled Boltzmann equations for the number densities of χ and ψ, and we calculate the final abundance of χ. A freezeout temperature T f is defined as the SM temperature at which n χ /n eq χ = 2.5. At this temperature the rates of the various processes are compared to determine which mechanism determines the abundance. On the black line the final χ abundance corresponds to the observed DM density.
Notice that everywhere on the plane the ratio Γ χe→ψe /Γ χχ→e + e − is fixed to be the left hand side of Eq. (18) and thus only depends on T f . In particular the ratio is greater than one (meaning that χe → ψe is faster than χχ → e + e − at T f ) everywhere in the plane for the given choice of parameters.
Starting from the left side of the plot, at small λ ψS , the first reaction to decouple is ψψ * → SS. This implies that the χ abundance is fixed as soon as χ * χ → e + e − annihilations decouple, which is the standard WIMP scenario. Notice that the black line representing the observed DM Left panel: freeze-out phase diagram for the scalar QED model. In each point of the plane a freeze-out temperature T f is defined to be the temperature at which nχ/n eq χ = 2.5. A freeze-out phase is then assigned by comparing the rate of the various reactions involving χ as described in the text. On the black line the observed relic abundance is reproduced. The blue star corresponds to λ ψS = √ λminλmax where λmin and λmax correspond, for fixed θ, to the minimal and maximal values of λ ψS in the coscattering phase, respectively. In the main body of the text we choose λ ψS = λ ψS in order to reside in the coscattering phase. Right panel: for every point on the plane, θ is selected to reproduce the observed relic density. In the blue wedge, this occurs in the coscattering phase. The dashed blue line corresponds to the locus of points for which λ ψS = λ ψS .
abundance is approximately horizontal in this region as a consequence of the fact that the final abundance only depends on the χχ * → e + e − cross section.
Moving to the right, for fixedσ e , we hit the boundary after which ψψ * → SS decouples after χχ * → e + e − but before χe → ψe. This is the classical coannihilation scenario [74,75] in which the abundance of the inert state χ is depleted by the existence of another state ψ with larger interactions and with which χ shares a chemical potential (this condition being enforced by fast χe → ψe exchange reactions). Notice that the black line is approximately vertical in this region as a consequence of the fact that freeze-out is determined by the decoupling of ψψ * → SS.
Increasing λ ψS even more, ψψ * → SS becomes faster than both χχ * → e + e − and χe → ψe, which leads to the coscattering regime. In the white part of the left of Fig. 6, the final abundance is determined by inelastic scattering on electrons through χe → ψe, implying that iso-countours of χ abundance are independent of λ ψS . Finally moving to even larger λ ψS , in the brown shaded region, the reaction χS → ψS becomes the dominant one and sets the final abundance; this is the regime of secluded coscattering discussed in Ref. [34].
Notice that moving from left to right, points on the black line (an iso-contour of χ relic abundance) have monotonically decreasing direct detection cross section.
Notice also that moving to larger values ofσ e , the boundaries between the various phases all move to larger values of λ ψS , as a consequence of the fact that for larger σ e , both Γ χe→ψe and Γ χχ→e + e − are larger, shifting the transition values of λ ψS to larger coupling sizes. The blue star shows the value of λ ψS chosen in the main body of the text in order to reside within the coscattering phase. In the right panel of Fig. 6 we show the phases of the model in the same plane, but varying θ point by point to fix the right relic density. As θ increases coscattering and annihilations of χ and ψ with the hidden sector state S grow, reducing the available parameter space for coscattering off electrons. At the same time the coupling to electrons required to reproduce the measured relic density decreases. When this coupling becomes large we exit the coscattering regime to enter a WIMP phase where χχ * → e + e − sets the relic density.
Direct detection The direct detection scattering cross section of χ on electrons,σ e , is given by [14] σ e e 2 g 2 where we assumed m χ m e . Eqs. (24) and (8) define a one-to-one relation, σ WIMP e (m χ ), between the mass of a WIMP annihilating to electrons and the size of its direct detection cross section. For a coscattering DM candidate,σ WIMP e (m χ ) is a strict upper bound on the size of the direct detection cross section and, in particular,σ e (m χ )/σ WIMP e ≈ 1/B. For a given mass, m χ , the upper bound on B, described in the previous section, corresponds to the smallest possible value of the direct detection cross section (see Fig. 3).