String scale interacting dark matter from $\pi_1$

We show that in a wide class of string derived models of particle physics, heavy string modes with masses around the GUT scale can serve as a viable dark matter candidate. These heavy string modes wind around specific cycles in the extra-dimensional space, closely related to the fundamental group $\pi_1$. As a consequence of a non-trivial $\pi_1$, there is an exact discrete symmetry that stabilizes such winding strings. The dark matter candidate couples to the Standard Model via gravity and via the exchange of heavy string states. We find that, for reasonable values of the string coupling, our dark matter candidate can be produced in sizable amounts via freeze-in. Our scheme applies to many string constructions, including Calabi-Yau compactifications, and can be tested against constraints from the CMB.


Introduction
Successful models for particle dark matter consist of two main ingredients: the dark matter candidate that interacts with the Standard Model (SM) only very weakly, and a mechanism that ensures its relative longevity. In most instances, a Z 2 symmetry is invoked to keep the dark matter particle from decaying. Moreover, if the model is supposed to explain the presently observed dark matter relic density, one needs to make sure that it is produced in sufficient quantities in the early universe. The prime example for a dark matter candidate is the weakly interacting massive particle (WIMP), which is in thermal equilibrium with the thermal plasma and "freezes out" after dropping out of equilibrium [1,2]. However, it has been demonstrated that a dark matter species may also be produced thermally in sizable quantities even if it never attains thermal equilibrium ("freeze-in") [3]. More specifically, it has become clear that freeze-in production can work with extremely heavy dark matter candidates (with masses above the GUT scale) and couplings that are suppressed by 1/m 2 Pl , a framework that is known as Planckian interacting dark matter (PIDM) [4,5]. This observation is intriguing from the viewpoint of string model building, for a number of reasons. On the one hand, it is expected that the lightest massive string states have masses around the string scale, and that there are, apart from gravity, stringy interactions between these states and the massless states of the Standard Model that are suppressed by 1/m 2 s . Furthermore, there is a wide class of string models that have a stabilizing Z 2 symmetry built in [6]. The goal of this paper is to show that this class of string models can yield viable models of dark matter. As an explicit example, we consider the framework of heterotic string theory with six extra dimensions compactified on a special class of orbifolds but our scheme is valid more generally. In particular, our dark matter candidate is given by a closed string state that winds around a certain cycle in the extra-dimensional compact space. Then, its stability can be ensured via an exact discrete symmetry that originates from the topological property of certain compactification spaces to be non-simply connected. Consequently, the discrete symmetry is classically exact and can potentially be broken only non-perturbatively due to a discrete anomaly [7,8]. This stable string state is generically very heavy (with a mass at the compactification or GUT scale) and interacts with the massless states in the plasma not only via gravity, but also via the exchange of other heavy winding modes.

Interactions between dark matter and the Standard Model
In this section, we will examine which couplings between the dark matter candidate and the particles of the Standard Model can arise at the renormalizable level. To be specific, we consider a string model compactified on the so-called Z 2 × Z 2 -5-1 orbifold in the classification of ref. [9].
However, our findings easily carry over to other orbifold geometries and also to other string compactifications, cf. appendix A for further details. In this model, the dark matter candidate is stabilized by an exact Z 4 symmetry that originates from string selection rules [6], which in turn are related to topological properties of the compact orbifold space. It should be noted that all massless strings, especially those for the Standard Model particles, have even Z 4 charges, while the DM particles carry odd charges. Hence, the Z 4 has a Z 2 subgroup with precisely the charge assignment needed for dark matter parity and the lightest string with odd Z 2 charge is stable. Now, one has to examine the model further, in order to identify all allowed stringy couplings of the DM particle to the Standard Model. Since we are dealing with a supersymmetric setup, one needs to distinguish between couplings from the Kähler potential and those from the superpotential.
In most instances, the coupling of dark matter to the SM is dominated by 2 → 2 scattering.
Hence, due to the constraining Z 4 symmetry, the most general coupling looks like the left diagram in figure 1. At tree level, one can boil that down to the exchange of mediator fields M , cf. the right diagram in figure 1. What are possible mediator fields? One example are gravitational interactions, where the exchanged particle is the graviton. This case has been studied extensively in the PIDM program [4,5] and, as we will see, will give in our setup a contribution that is superfield type of strings Z 4 charge

Kähler potential terms
If (at least part of) the massless Standard Model matter is localized in the extra-dimensional compact space, there is a set of mediators that can couple to both, the dark matter candidate and the Standard Model. In our concrete setting, there are three mediator strings, where each of them potentially couples to a different subset of SM matter. In what follows, we will make the simplifying assumption that all three have the same mass and couplings, 1 such that one can effectively work with one mediator V (M) . Moreover, we shall see later on that small variations of the coupling strengths do not have a massive impact on our results. Let us now examine the coupling of dark matter to a localized Standard Model particle Φ i . Then, the most general Kähler potential consistent with gauge invariance and the stringy Z 4 symmetry for the coupling of the Standard Model to the mediator reads Here, we observe that only the coupling to the vector field V (M) = (V µ , λ) with coupling g 1 is renormalizable, hence all other terms will be dropped. Therefore, we consider only the coupling of the mediator vector superfield V (M) to the dark matter candidates Φ DM and Φ DM in the dark matter Kähler potential Let us parameterize the SM chiral multiplets as Φ i = (f i ,f i ) and the dark matter multiplets as Φ DM = (χ, ϕ) and Φ DM = (χ , ϕ ). The relevant Lagrangian for the 2 → 2 production of dark matter from the D-terms of the Kähler potentials reads Additionally, there is a four-scalar vertex coming from the kinetic term of the mediator multiplet. The Lagrangian coming from the auxiliary field in V (M) reads which, upon setting the auxiliary field D M on-shell, yields and, hence, we obtain a four-scalar vertex with a coupling g 1 g 2 . Then, we find that at tree level the relevant channels for the non-gravitational interactions of dark matter with the Standard Model are given by the processes shown in fig.s 2-7. There, we present the production channels for the dark matter multiplet Φ DM , analogous diagrams exist also for its partner multiplet Φ DM .

Superpotential terms
In addition to the Kähler terms, there can also arise couplings from the superpotential. In particular, it is possible to couple a mediator, residing in a chiral multiplet, to the Standard Model via Higgs portal and neutrino-portal-like terms. However, in order for the stringy couplings to exist, the Higgs field must be localized. For the neutrino portal, the Higgs field has to live at the same fixed point as the lepton doublet it is supposed to couple to; if these fields live at different fixed points in the same sector the couplings are suppressed, and if they live in another twisted sector the couplings are completely absent. Similar terms then exist for the coupling of the mediator to the dark matter candidate. If we again make the assumption that the mediator couples to all SM states with the same coupling constant, the terms containing the mediator in the corresponding superpotential read where we used the SM superfields H u,d = ( H u,d , H u,d ) for the Higgses and L = ( ,˜ ) for the lepton doublet. For processes involving the exchange of a mediator, we are interested in the 3-point interactions that arise from this choice for W. Additionally, there is also a four-scalar interaction of two dark matter scalars and two SM scalars. It turns out that the impact of the superpotential couplings to the dark matter production rate is subleading when compared to the contribution from the Kähler potential. Moreover, as discussed above, the existence of the neutrino portal couplings requires an specific localization of the lepton and Higgs fields, which is model dependent. For these reasons, and because models with localized Higgs pairs have a less appealing phenomenology, we will not push any further in this direction and consider Kähler terms only by putting the Higgs field in the bulk.

Dark matter production
Although the dark matter candidate is too heavy to be in thermal equilibrium, it can still be produced thermally via freeze-in. The production of dark matter is then governed by the Boltzmann equatioṅ Here, n is the number density of all states in the dark matter sector, in other words n = n χ + n ϕ + n χ + n ϕ † + n χ + n ϕ + n χ + n ϕ † . On the right hand side of equation (9), σ eff v is the effective thermally averaged cross section for the various 2 → 2 dark matter production channels, taking also coannihilations into account [10,11]. Using m i = m χ it reads Here, g i counts the internal degrees of freedom of each species i (where g i = 2 for a Weyl fermion and g i = 1 for a real scalar), and the summation indices i and j run over all fields in the dark sector. Furthermore, p = s/4 − m 2 χ , and K 1 is the modified Bessel function of the second kind of order 1. The equilibrium density n eq is given by where K 2 is the modified Bessel function of the second kind of order 2. The cross sections σ ij correspond to the various possible scattering processes shown in figures 2-7, and are given by Here, and M ij (t) denote the matrix elements for the respective process. In what follows, we focus on the case with bulk Higgs fields and, hence, there are no contributions from the superpotential. Then, the non-vanishing cross sections σ ij are given by plus the corresponding terms for χ , ϕ . Moreover, it holds that σ ij = σ ji . With these preparations in place, one can perform the thermal averaging eq. (10) numerically (cf. figure 8) and turn one's attention to the Boltzmann equation. As the actual density in the freeze-in case is always much smaller than the equilibrium one, the full Boltzmann equation (9) can be approximated to sufficient accuracy by neglecting n 2 compared to n 2 eq on the right hand side of equation (9) and hence usinġ n + 3H(t) n = σ eff v n 2 eq .
Proceeding like in ref. [4], one can now simplify the discussion by introducing the dimensionless abundance X = na 3 /T 3 rh in terms of the scale factor a and the reheating temperature T rh , such that equation (17) can be integrated to yield Here, we used the fact that the scale factor at the end of inflation can be chosen to be 1, and that the abundance of dark matter immediately after inflation vanishes. For simplicity, we consider a non-perturbative reheating scenario with instantaneous reheating [12] with where H i is the Hubble rate at the end of inflation and Γ the inflaton decay rate. Then, the reheating temperature coincides with the highest temperature reached during reheating and is given by The non-observation of tensor modes in the cosmic microwave background (CMB) by the Planck satellite combined with constraints from Bicep2 and Keck requires a tensor-to-scalar ratio r < 0.056 [13]. This gives an upper limit on H i and therefore on the reheating temperature Note that this bound is believed to become more stringent in the near future [14]. Upon adopting the convention that the scale factor after inflation a i = 1, the dependence of the temperature and the Hubble rate on the scale factor for the radiation dominated phase after reheating is Thus, the abundance eq. (18) can be seen as a function X f (H i , g 1 g 2 , m χ , m M ) of the Hubble rate at the end of inflation H i , the couplings g 1 g 2 , the dark matter mass m χ and the mediator mass m M . In order to compare to the observed dark matter relic density Ω X h 2 = 0.12 [15], one can use (cf. [5]) Hence, for a GUT scale dark matter particle (m χ ∼ 10 16 GeV), the critical abundance is of order 10 −23 . It is interesting to notice that the Hubble rate H i required to obtain this abundance remains relatively stable even if vectorlike SM exotics are added, owing to the nature of freeze-in production. To see this, note that if the couplings of all contributing chiral multiplets are roughly equal, the Hubble rate needed to match the correct final abundance is determined by the contribution x f = X f /N Φ of a single multiplet to the final abundance. Then, the critical contribution per chiral multiplet be written as where g * counts the number of degrees of freedom in the thermal bath at T rh and N Φ is the number of contributing chiral multiplets (for the case of the MSSM with three right handed neutrinos, g * = 240 and N Φ = 48). Adding n V vectorlike pairs of exotics now changes these figures according to Hence, adding an arbitrary number of vectorlike exotics lowers x crit. f by at most 25%. This change requires an even smaller adjustment in the Hubble rate H i , and therefore our results are largely insensitive to the full particle content of a given model.

Results
We have solved the integral (18) numerically. If we use the simplified reheating scenario and a fixed value of the dark matter mass m χ , the resulting abundance X f depends only on the Hubble rate H i after inflation (which sets the reheating temperature), and the product of the two involved couplings. In principle, there is also a light dependence on the mediator mass m M , however as one observes, varying the mediator mass shows only little effect on the final abundance, especially for larger values of the couplings. Our results are displayed in figure 9. There, we varied the couplings g 1 g 2 over a broad range, and determined the value of H i needed to produce the critical dark matter relic abundance X crit. f , eq. (23). One observes that for very small couplings, values for the Hubble rate that exceed the current CMB bounds [13] are needed in order to produce the right amount of dark matter. The bounds are more strict for lower mediator masses. Likewise, the critical Hubble rate changes for less than an order of magnitude for a wide range of coupling strengths, roughly from g 1 g 2 = 10 −8 to g 1 g 2 = 1.
Moreover, one observes that for values of g 1 g 2 greater than 10 −6 , the production of dark matter via the stringy operators largely dominates the graviton exchange presented in ref. [4,5] and therefore neglecting this gravitational channel is a good assumption. For lighter dark matter masses it is impossible to get near the CMB bound, even the projected ones, without encountering overproduction by graviton exchange first. However, starting from m χ ∼ 3·10 16 GeV, our approach becomes sensitive to at least the projected CMB bound in ranges for the stringy couplings for which graviton exchange can still be safely neglected. We also observe that for any value of the couplings, our Hubble rate lies in ranges where -given the DM mass -gravitational production [16] can be neglected.

Conclusions
We have shown that generic string constructions can accommodate a candidate for dark matter. Opposed to other studies of dark matter in string theory (cf. [17,18,19]), we focused on the dark matter candidate. Specifically, the dark matter candidate is a heavy string state with no charge under the Standard Model gauge group and a mass at or above the GUT scale. It is stabilized against decay by stringy selection rules, closely related to the topological property of a nontrivial fundamental group π 1 of the compactification space. Because of its high mass, the dark matter particle never attains thermal equilibrium, and therefore it must be produced by freeze-in rather than freeze-out. We find that generically, the dark matter candidate interacts with the thermal bath only via gravity, and by the exchange of heavy mediators arising in the massive string spectrum. For not too small string couplings the latter ones dominate over graviton exchange. For definiteness, we considered an explicit model in heterotic orbifolds, but we believe that our results carry over very well to many other string constructions, for example in Calabi-Yau constructions with freely-acting Wilson lines [20,21,22]. In our setup, we chose a small-radius limit, where winding strings are the lightest extra states, in order to identify the stringy selection rules. There exists a T -dual large-radius picture where the winding states are exchanged by Kaluza-Klein excitations. In our picture, the dark matter candidate is a string with winding around a particular non-contractible cycle on the orbifold, thereby ensuring its stability. An analysis on the level of the orbifold space group reveals that this winding state can couple to the Standard Model -apart from gravity -via the exchange of heavy winding strings that are SM singlets. Going to an N = 1 supersymmetric field theory, we identified the relevant terms for 2 → 2 production of the heavy dark matter candidate from the Kähler potential and the superpotential. We find that in order to obtain the correct dark matter relic density, one needs values for the Hubble rate at the end of inflation that range up to 10 12 GeV, if the string coupling is perturbative. This way, one is able to constrain the allowed parameter space of the model by bounds on the tensor-to-scalar ratio in the CMB.
We observe that our results generalize very well to generic string models: The most prominent influence comes only from the mass of the dark matter candidate itself, which is constrained to lie around the GUT scale. Other model-dependent parameters, such as the mass of the mediator contribute only at subleading order. Furthermore, the required Hubble rate after inflation remains within the same order of magnitude for the entire sensible range of string couplings. Finally, we observe that adding vectorlike matter that is charged under the SM and contributes to the production of dark matter, changes the critical final abundance only by a small amount, and leaves the required Hubble rate invariant up to the percent level.
Acknowledgments. This work was supported by the Deutsche Forschungsgemeinschaft (SFB1258). We would like to thank Mathias Garny and Kai Urban for useful discussions.

A An explicit realization in string theory
We consider the E 8 × E 8 heterotic string theory compactified on the Z 2 × Z 2 -5-1 orbifold geometry [23,24,25], see also refs. [26,27,9] and refs. [28,29] for MSSM-like string models based on this orbifold geometry. This orbifold geometry can be constructed in three steps. First, one defines a six-torus via a six-dimensional lattice that is spanned by six basis vectors e i , i = 1, . . . , 6.

A.1 Strings on orbifolds
Closed strings on orbifolds are characterized by their boundary conditions that specify which transformation is needed such that the string is closed. In more detail, for a string (i.e. a worldsheet boson) X(σ 0 , σ 1 ) as a function of worldsheet time and space coordinates σ 0 and σ 1 ∈ [0, 1] the boundary condition reads X(σ 0 , σ 1 + 1) = θ k ω X(σ 0 , σ 1 ) + n i e i + n τ τ ⇔ g = (θ k ω , n i e i + n τ τ ) ∈ S , (28) where k, ∈ {0, 1}, n i ∈ Z, n τ ∈ {0, 1} and summation over i = 1, . . . , 6 is implied. Strings with k = 0 or = 0 are called twisted strings, in contrast to the case k = = 0 which gives rise to so-called untwisted strings. One can encode the boundary condition (28) into group elements g ∈ S of the so-called space group S. Then, g is called the constructing element of the string (28). If g ∈ S has a fixed point, i.e. if there is a point x g such that θ k ω x g + n i e i + n τ τ = x g , the string with constructing element g is localized in the extra dimensions at x g . The freely-acting nature of τ becomes evident by noticing that constructing elements with fixed points necessarily have n τ = 0. Furthermore, strings with boundary conditions (1, n i e i + n τ τ ) ∈ S live in the orbifold bulk. They are winding strings if n i = 0 or n τ = 0, where the mass of a winding string is proportional to the radius and the winding number of its winding direction. Hence, in general only bulk strings with constructing element (1, 0) are massless.
The Z 2 ×Z 2 -5-1 orbifold geometry has the important property of having a cycle that generates a non-trivial fundamental group π 1 [24] and, hence, renders the orbifold geometry non-simply connected. In fact, this cycle is generated by the freely-acting shift τ . The existence of the freely-acting shift has two important consequences for our discussion: 1. There are heavy string modes with constructing elements (1, n τ τ ) ∈ S that wind around the freely-acting τ -direction and 2. There is an exact Z 4 symmetry [6], where a string with general constructing element eq. (28) carries a discrete charge q = n τ + 2(n 2 + n 4 + n 6 ) mod 4 such that q ∈ [0, 1, 2, 3] , where n τ ∈ {0, 1} and n 2 , n 4 , n 6 ∈ Z are the integer winding numbers. It turns out that all massless strings (those from the bulk and those that are localized at orbifold fixed points) have n τ = 0 and, therefore, carry even Z 4 charges, while there exist massive strings with odd Z 4 charges.
Consequently, there exists a lightest winding string from the bulk with winding numbers n τ = 1 and n i = 0, i.e. with constructing element (1, τ ), that has odd Z 4 charge. Hence, it is stable and we can identify it as our dark matter candidate Φ DM . Its mass partner Φ DM has constructing element (1, −τ ) and therefore Z 4 charge 3.

A.2 String interactions
In order to find the three point couplings allowed by the space group selection rule [30,31], one needs to fulfill for each coupling that where [g i ] denotes the conjugacy class of the constructing element g i . The calculation is the same for the Kähler and the superpotential. In the Kähler potential, one looks for terms of the form ΦΦ † V (M) , where Φ † has inverted quantum numbers and hence has the inverse constructing element. In the superpotential, one looks for terms of the form Φ 1 Φ 2 Φ M , where Φ 2 is either the mass partner of Φ 1 (for the dark matter particle and the Higgs portal), or it is another field localized appropriately (for the neutrino portal).
In any case, we observe that there exist several winding states with trivial Z 4 charge, most prominently those with n 2 + n 4 + n 6 = 0 mod 2. These states are particularly interesting candidates for mediators: 1. On the level of space group elements, they couple to both, DM and twisted strings. Let us work out for the coupling of dark matter to the θ-twisted sector (i.e. k = 1 and = 0 in eq. (28)).
• It is evident that 1, 2. Their local shift is a lattice vector, cf. ref. [28]. Hence, these states have P sh = 0 and the corresponding couplings are not forbidden by gauge invariance.
It turns out that the construction shown above not only works for the θ-, but also for the ω-and θω-twisted sector. In summary, we have the winding strings V (M) and Φ M that mediate between the dark matter strings (Φ DM and Φ DM ) and the twisted sector sector of SM θ ω θω g ∈ S of mediator (1, e 4 + e 6 ) (1, e 2 + e 6 ) (1, e 2 + e 4 )