PHYSICAL REVIEW D 99, 095028 (2019)
Moduli portal dark matter
† ‡ Debtosh Chowdhury,1,2,* Emilian Dudas,1, Maíra Dutra,2,3, and Yann Mambrini2,§ 1Centre de Physique Th´eorique, École Polytechnique, CNRS, Universit´e Paris-Saclay, 91128 Palaiseau Cedex, France 2Laboratoire de Physique Th´eorique (UMR8627), CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France 3Ottawa-Carleton Institute for Physics, Carleton University, 1125 Colonel By Drive, Ottawa, K1S 5B6 Ontario, Canada
(Received 19 November 2018; published 23 May 2019)
We show that moduli fields as mediators between the Standard Model and the dark sector can naturally lead to the observed relic abundance. Indeed, even if moduli are very massive, the nature of their couplings with matter and gauge fields allows to produce a sufficiently large amount of dark matter in the early Universe through the freeze-in mechanism. Moreover, the complex nature of the moduli fields whose real and imaginary parts couple differently to the thermal bath gives an interesting and unusual phenomenology compared to other freeze-in models of that type.
DOI: 10.1103/PhysRevD.99.095028
I. INTRODUCTION these models, it has been shown that the effects of non- instantaneous reheating [15] and noninstantaneous ther- Despite indirect but clear evidence [1] of the presence malization [16] should be considered with care. It then of a large amount of dark matter (DM) in our Universe, its seemed interesting to study constructions with massive nature still remains elusive. The absence of any signal in scalar moduli fields which are abundantly present in direct-detection experiments like XENON [2],LUX[3], supergravity (SUGRA) and string theory extensions, and PANDAX [4] questions the weakly coupled dark and to check if they can play the role of a mediator matter paradigm. The simplest extensions—such as 0 between the dark sector and the Standard Model (SM). Higgs-portal [5], Z-portal [6], or even Z -portal models Indeed, moduli fields couple generically to Standard Model [7]—have a large part (if not all) of their parameter space fields through higher-dimensional operators (mostly with excluded when combining direct, indirect, and accelerator derivative interactions). As a consequence, mechanisms searches (for a review on weakly interacting massive implying moduli are important at high energies/temper- particle searches and models, see Ref. [8]). This uncom- atures, being therefore potentially relevant for the freeze-in fortable situation justifies the need to look for different mechanism. scenarios, allowing feeble couplings, or the possibility of Moduli fields appear in many extensions of the Standard dark matter production at the very early stages of reheating. Model. Indeed, in any higher-dimensional supergravity or Such mechanisms are usually dubbed “feebly interacting string theory extension of the Standard Model there are massive particles” [9] (see Ref. [10] for a review). scalar fields coming from the compactification of the In this context, several models have been studied and it higher-dimensional metric, dilaton, or various antisymmet- has been confirmed that dark matter production is naturally feasible in different setups like SO(10) unified construction ric tensors. In particular, internal volumes and shapes and [11],Uð1Þ0 anomaly-free models [12], the spin-2 portal their axionic partners are abundant in such constructions. [13], or high-scale supersymmetry (SUSY) [14]. In all off Most of them are flat directions at tree level and get potentials and therefore masses from various perturbative and nonperturbative effects. Their resulting masses and *[email protected] † vacuum expectation values (VEVs) are model dependent [email protected] ‡ [email protected]; [email protected] and will be taken as free parameters in what follows. Their §[email protected] VEVs determine the values of four-dimensional parame- ters: the gauge and Yukawa couplings, the wave functions Published by the American Physical Society under the terms of of various fields, and the Planck mass. If one assumes that the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the low-energy theory (obtained after their decoupling) is the author(s) and the published article’s title, journal citation, the Standard Model or a phenomenologically motivated and DOI. Funded by SCOAP3. extension of it, then their couplings can be obtained by
2470-0010=2019=99(9)=095028(11) 095028-1 Published by the American Physical Society CHOWDHURY, DUDAS, DUTRA, and MAMBRINI PHYS. REV. D 99, 095028 (2019)
SM 2 2 ¯ 2 ¯ 4 starting from the low-energy theory and expanding the low- LT ⊃ ZHjDμHj − μ ðT ; T ÞjHj − λðT ; T ÞjHj energy parameters in a power series. If moduli fields are 1 þ ðZ ¯ þ Z ¯ þ Þ heavier than the reheating temperature, then they can be 2 LfLi=DfL RfRi=DfR H:c: safely replaced by their VEVs and the low-energy theory is 1 just the Standard Model or an appropriate extension. If they − Z μν − Z0 ˜ μν ð Þ 4 GGμνG GGμνG ; 2 are lighter, however, they can lead to various physical – α β effects. This strategy was used in early papers [17 23] in αH L;R L;R where ZHð¼1 þ Λ tÞ, ZL;Rð¼1 þ Λ t þ i Λ aÞ, order to study various low-energy effects of the moduli α β Z ð¼1 þ G Þ Z0 ð¼ G Þ fields. Recent works have considered moduli as dark matter G Λ t , and G Λ a are the wave functions candidates [24] or the production of dark matter from of the scalar (H), fermionic (f), and (Abelian and non- moduli decay in the early Universe [25–29]. The present Abelian) gauge fields (Gμ) of the Standard Model, respec- work is intended to complete the list of interesting tively. In the above equation, Gμν is the field-strength tensor ˜ μν 1 μνρσ consequences of moduli fields by studying their possible of the gauge field (Gμ) and G ð¼ 2 ϵ GρσÞ is its dual role as mediators between the Standard Model and the dark field-strength tensor. matter sector. From the first line of Eq. (2) we see that the scalar The paper is organized as follows. In Sec. II, we describe potential depends on the mass parameter μ, which is also a the model under consideration. In Sec. III, we discuss function of the moduli fields. Parametrizing the contribu- bounds on the moduli masses coming from cosmology. In tion of the moduli to the μ parameter in a similar fashion as Sec. IV, we outline the dark matter relic abundance through in Eq. (1), we can write the freeze-in mechanism in the early Universe, taking into αμ account noninstantaneous reheating. Section V is devoted μ2 ¼ μ2 1 þ t ; ð3Þ to the computation of the dark matter production rate in the 0 Λ early Universe in our model and we delineate the parameter μ μ space for the model in consideration. In Sec. VI,we with 0 being the SM parameter that reproduces the Λ summarize the main results of our work and conclude observed Higgs mass at the electroweak scale. As is the by highlighting the new aspects that emerged from our highest scale in the theory, the contribution to the Higgs analysis. mass due to the moduli is small. On the other hand, there is a secondpffiffiffiffiffiffiffi possibility: the μ parameter gets generated at a scale ( hFi) close to the Planck scale. In this case, the II. THE MODEL effective μ parameter can be written as Let Λ be the new physics scale (which can be the string 2 2 hFi scale, unification scale, or SUSY/SUGRA breaking scale, μ ¼ μ0 þ t; ð4Þ for instance). The consistency of the effective field theory MP Λ — requires that is the largest mass scale of the theory in where hFi is the VEVof the “spurion” field. In this case, one particular, larger than the dark matter or mediator masses needs a considerable amount of cancellation or fine-tuning and the maximum temperature after reheating. One can between the two contributions in Eq. (4) to reproduce the 1 T then define the couplings of the complex modulus field observed Higgs mass. In contrast to Eq. (3), in this case the T ≡ þ (decomposed as t ia) to the Standard Model field k coupling of t to the Higgs is quite large. This leads to the fact Z by expanding the wave functions k: that the width of t could be larger than the mass of t unless we demand that the width be at most the mass of t. This sets an c d α β upper bound on the spurion VEV, hFi ≲ m M , where m is Z ðT ; T¯ Þ ≈ 1 þ k T þ k T¯ ≡ 1 þ k t þ i k a; ð1Þ t P t k Λ Λ Λ Λ the mass of t. Throughout our analysis, we will consider the case of Eq. (3) unless otherwise stated. where c and d are real coefficients of order one and we The effective interactions between the components of the k k 1 Λ defined the couplings to the real and imaginary components moduli and SM fields, at the first order in = , reads T α ¼ þ β ¼ − 2 of as k ck dk and k ck dk, respectively. We α α LSM ⊃ H j j2 − μ μ2 j j2 can then express generic couplings of the moduli fields to T Λ t DμH Λ 0t H the Standard Model sector as 1 ¯ μ f f þ tfiγ ðα − α γ5ÞDμf þ : : 2Λ V A H c 1We will consider only one modulus field throughout our 1 work. The generalization to several fields is straightforward. ¯ μ f f 2 þ ∂μafγ ðβ − β γ5Þf For simplicity, throughout our work we will consider CP- 2Λ V A conserving Lagrangians and therefore real coefficients in the 1 αG μν βG μνρσ couplings of moduli to matter. The extension to CP-violating − tGμνG þ 2 ∂μaϵ Gν∂ρGσ; ð5Þ couplings is interesting but beyond the goal of our paper. 4 Λ Λ
095028-2 MODULI PORTAL DARK MATTER PHYS. REV. D 99, 095028 (2019) where we have identified the chiral couplings as The interactions of fermions (Ψ, standard or dark) with αf ¼ðα þ α Þ 2 αf ¼ðα − α Þ 2 moduli are essentially different from the scalar and vecto- V L R = and A L R = , with analogous definitions for the couplings of the imaginary part of the rial cases, due to their chirality. The first aspect of this moduli. remark is evident from the amplitudes for their interactions Before we proceed further we can make some remarks with moduli: after having a quick look at Eq. (5): i Ψ Ψ (1) Since the kinetic term of the Higgs needs to be real, M ¯ ¼ − ¯ð Þð − Þðα − α γ Þ ð Þ tΨΨ u p1 p=1 p=2 V A 5 v p2 the Higgs sector only couples with the real part of 2Λ αΨ the modulus field t. VmΨ ¼ −i u¯ðp1Þvðp2Þ; (2) One observes that the Lagrangian in Eq. (5) is Λ invariant under a shift in the imaginary part of the 1 Ψ Ψ M ¯ ¼ u¯ðp1Þðp=1 þ p=2Þðβ − β γ5Þvðp2Þ moduli (a → a + const). This can also be observed aΨΨ 2Λ V A in SUGRA models, for instance, where the Kähler Ψ β mΨ metric depends explicitly on the combination ¼ − A ¯ð Þγ ð Þ ð Þ Λ u p1 5v p2 ; 9 T þ T¯ . In other words, the Lagrangian could be written by imposing the shift symmetry from the where p1 and p2 are the four-momenta of the fermions and 3 beginning. As a consequence, the nature of the mΨ is the mass of the fermion. We notice that if the couplings of a to the Standard Model fields differs fermions are on shell, we have an explicit dependence on from the couplings of t (a naturally develops their mass due to the Dirac equation. As a consequence, derivative-type couplings). above the electroweak scale, standard fermions cannot (3) In Eq. (5) we kept only the leading operators produce any of the dark matter particles. The other aspect which contribute to three-point functions. This is we point out is that the fermionic coupling to the real part of because the 2 → 2 production of dark matter will the moduli is CP even, so that the corresponding rates will 4 dominate over the 3 → 2 production. αΨ depend only on the vector coupling ( V), and that the By analogy, one can write the same type of couplings in fermionic coupling to the imaginary part of the moduli is the dark sector. In our study, we will distinguish three cases CP odd and therefore the corresponding rates will depend χ V of dark mater: scalar (S), fermionic ( ), and vectorial ( ). only on the axionic coupling (βΨ). Their interactions with moduli read A α III. COSMOLOGICAL MODULI PROBLEM S S 2 L ¼ tj∂μSj ; ð6Þ T Λ Because of their gravitational interactions, moduli are long-lived fields. Thus, one has to face two potential χ 1 μ χ χ dangers. If the moduli lifetime is smaller than the age of L ¼ tχ¯iγ ðα − α γ5Þ∂μχ þ H:c: T 2Λ V A our Universe, their decay might have released quite a large amount of entropy in the Universe which would dilute the 1 μ χ χ þ ∂μaχγ¯ ðβ − β γ5Þχ; ð Þ 2Λ V A 7 contents of the Universe. On the other hand, if the moduli lifetime is larger than the age of our Universe, they might still be oscillating around the minimum of their potential. 1 α β V V μν V μνρσ Thus, the energy stored in these oscillations may overclose LT ¼ − tVμνV þ 2 ∂μaϵ Vν∂ρVσ: ð8Þ 4 Λ Λ the Universe. One refers to these problems as the cosmo- logical moduli problem [31]. Taken at their face values, We can make some remarks concerning the couplings of such constraints dictate that the moduli fields are either the moduli fields to the dark matter. First of all, similar to superlight or superheavy. We present here a short review on the case for the Higgs sector, the scalar dark matter does not these well-known constraints, following Ref. [31]. couple to the imaginary part of the moduli. An immediate Using simple dimensional analysis the decay width consequence (which will be discussed in what follows) is of a modulus field T of mass mT can be written as Γ ∼ 3 2 that only the real part of moduli contribute to the production T mT =MP. Since the age of the Universe is of order −1 of a scalar dark matter, regardless of the Standard Model H0 , where H0 is the Hubble constant in the present initial states. Universe, we could infer that the modulus will decay at present times if ΓT ∼ H0, or in other words if its mass mT is 3 ð 2 Þ1=3 ≃ 20 This argument is not valid if one includes Yukawa-like around H0MP MeV. couplings in the action. However, their contribution to dark At first, we consider the case when mT < 20 MeV, that matter production in the early Universe is negligible. is, when the modulus has not decayed yet at the present 43 → 2 processes can dominate if 2 → 2 channels are sup- time. As long as the Hubble constant is larger than the pressed, and when strong interactions are generated to compen- _ sate the loss induced by the reduced phase space [30]. modulus mass, the friction term (3HT ) dominates in the
095028-3 CHOWDHURY, DUDAS, DUTRA, and MAMBRINI PHYS. REV. D 99, 095028 (2019) equation of motion of the modulus field and the field T cosmological observations. In the absence of any other remains frozen at its initial value fT . When the Hubble effects, a cosmologically acceptable regime for the modulus −26 constant is of the order ofp theffiffiffiffiffiffiffiffiffiffiffiffiffiffi mass of the modulus or mass is either superlight (mT < 10 eV) or superheavy ∼ ∼ H mT , that is, when TI mT MP (since we know that (mT > 10 TeV). This concludes the short review discussing ∼ 2 T H T =MP), the field starts oscillating around the the constraints on the moduli mass. minimum T 0 of its potential. These coherent oscillations behave like nonrelativistic matter, and thus the energy IV. DARK MATTER PRODUCTION density of the modulus reads 3 3 The evolution of the dark matter number density nDM is T 2 2 T ρT ðTÞ¼ρT ðTIÞ ∼ mT fT : ð10Þ determined by the Boltzmann equation TI TI ρ ð Þ dnDM ¼ −3 ð Þ þ ð Þ ð Þ We know that the radiation energy density γ T scales as H t nDM R T ; 14 4 dt T , and thus ρT =ργ scales as 1=T. As the temperature of the pffiffiffiffiffiffiffiffiffiffiffiffi Universe decreases the energy fraction stored in the moduli ð Þ¼pffiffi1 ρ ð Þ where H t 3 tot t is the Hubble expansion rate, increases. Then, there will be a time when the energy MP ≃ 2 4 1018 density of the modulus oscillations will dominate the total with MP . × GeV being the reduced Planck ρ ð Þ energy density of the Universe. Thus, one needs to make mass and tot t the total energy density which changes ρ ð Þ ρ with time, as we will see in what follows. RðTÞ¼ sure that in the present Universe T T0 < c, where T0 is 2 ρ n hσvi → is the production rate (the number the temperature of the present Universe and cpisffiffiffiffiffiffiffiffiffiffiffiffiffiffi the critical SM SMSM DMDM of dark matter particles produced per unit of time and unit density. Using the above equation and T ¼ mT M , one I P of volume; see the Appendix B for details) which, for a can write this condition as 5 process ð1; 2 → 3; 4Þ,isgivenby ρ 2 cMP ∼ 10−26 ð Þ Z Z mT 095028-4 MODULI PORTAL DARK MATTER PHYS. REV. D 99, 095028 (2019) TRH as the temperature in a radiation-dominated Universe dominant over production during the radiation-dominated Γ ¼ ð Þ ¼ 100 after the inflaton decay, ϕ H TRH . era. For instance, for TMAX TRH we find In the present work, we have solved the set of three coupled differential equations above, but it is instructive to 5 2 find analytic solutions for the limiting cases of inflaton and ð10Þ ≈ 1 þ 1 07 1 þ TRH ≃ 3 68 ð Þ BF . × 2 2 . ; 20 radiation domination. In fact, we have checked that this is a TMAX good approximation, since there is a recognizable change of regime in the Hubble rate near the reheating temperature (see, for instance, Ref. [34]). In the radiation-dominated ð12Þ T era, we can use the familiar relations6 B ≈ 1 þ 1.07 × 7 ln MAX ≃ 35.5; ð21Þ F T rffiffiffiffiffi RH 2 d ¼ − ð Þ d ð Þ¼ ge π T ð Þ H T T with H T 90 ; 17 dt dT MP and for n>12, while in the inflaton-dominated era, we have [12] − 5 n−12 sffiffiffiffiffiffiffiffiffiffiffiffiffi ðn>12Þ n TMAX B ≈ 1 þ 1.07 × : ð22Þ 3 5 2 4 F n − 12 T d ¼ − ð Þ d ð Þ¼ gMAXπ T RH 8H T T with H T 72 2 ; dt dT gRH TRHMP ð18Þ The percentage of dark matter production during reheating for n ¼ 10 is therefore ∼73%, whereas for n ¼ 12 ∼97% where gRH and gMAX are, respectively, the relativistic it is . An important point we want to emphasize degrees of freedom at the reheating temperature after here is that if the mediators between the dark and visible inflation and at the maximal temperature reached during sectors are close to the reheating scale and if the production the reheating process, TMAX. rate of dark matter has a high temperature dependence, Ω 2 ≡ ρ The dark matter relic density h mDMnDM= c, we cannot avoid the contribution of the inflaton to the where ρc is the critical density today, may be split into Hubble rate. radiation-dominated (RD) and inflaton-dominated (ID) contributions: V. RESULTS AND DISCUSSION Ωh2 ≅ Ωh2 þΩh2 A. Production rate RD ID Z T ð Þ The squared amplitudes responsible for the model- ∼4 1024 RH R T × mDM dT 6 dependent behavior of the production rates of three differ- T0 T Z ent dark matter candidates are given in Appendix B.7 T ð Þ þ1 07 7 MAX R T Before presenting the exact solution of the rate [see . TRH dT 13 TRH T Eq. (B4)], we recognize from the squared amplitudes three 2 regimes which depend on the relation between the mass of ≡Ωh BF; ð19Þ RD the mediators and the temperature of the thermal bath: where T0 is the temperature of the present Universe. In the (1) The light regime, when the mediator mass is much below the temperature of the thermal bath above equation we have defined a “boost factor” BF Ωh2 T (m ≪ T). ¼ 1 þ ID t;a ( Ω 2 ), which quantifies the fraction of dark matter hRD (2) The pole regime, when the mediator mass is of the produced during the ID era to the amount of dark matter same order as the temperature (mt;a ∼ T), where we produced during the RD era. Notice that the fraction of dark might use the narrow-width approximation (NWA). matter produced during the reheating stage is ðBF − 1Þ=BF. (3) The heavy regime, when the mediator mass is much From this expression we can see that the production of above the temperature (mt;a ≫ T). dark matter during reheating (TRH 095028-5 CHOWDHURY, DUDAS, DUTRA, and MAMBRINI PHYS. REV. D 99, 095028 (2019) 8 > 8 > T ð ≪ Þ > 4 mj T ; > Λ > < m8 j j j T mj ð Þ¼δ K1 ðm ∼ TÞ; ð Þ R0;1 T 0;1 × > Λ4 Γ j 23 > j T > > T12 :> ð ≫ Þ 4Λ4 mj T ; mj 8 > 2 6 > mDMT ð ≪ Þ > 4 mj T ; > Λ > < m2 m6 j j DM j T mj ð Þ¼δ K1 ðm ∼ TÞ; ð Þ R1=2 T 1=2> Λ4 Γ j 24 > j T > 2 10 > m T FIG. 1. Evolution of the production rate of fermionic dark :> DM ðm ≫ TÞ; 4Λ4 j matter as a function of the temperature for different masses of mj dark matter and a real component of the modulus field. δj where the proportionality constants sf are given in Table II. We have numerically computed the total production rates Between those two extremes, we can notice the effects of 2 (B4), where the integration was performed using the CUBA the pole regions once T reaches mt (x ∼ 1) and ma (x ¼ 10 , 5 7 10 13 15 package [35], with a Bose-Einstein distribution function for 10 , and 10 for mt ¼10 , 10 , and 10 GeV, respec- the Higgs and gauge bosons in the initial states. tively). Notice that the production rates for the scalar dark In Fig. 1, we show the exact solutions of the total matter would not have the effect of the poles of ma since it production rate of the fermionic dark matter for a repre- couples only with the real component of the modulus. The sentative set of free parameters, as a function of the variable production rate of a vectorial dark matter would have the x ¼ ms=T which may be regarded as a parametrization of same qualitative features as the fermionic case but with a time. We set the new physics scale Λ to be 1016 GeV (the steeper bend at high temperatures, since the temperature ¼ 1012 12 grand unified theory scale), TMAX GeV, and the dependence in the heavy regime is T in the vector dark mass of the axionic modulus to be 108 GeV. For simplicity, matter case and T10 in the fermionic case. all of the couplings are set to unity. From left to right, the The presence of the pole regions depends on the low- and mass of the real component of the modulus is set to 1010, high-temperature thresholds. It will not appear if the 1013, and 1015 GeV (green, orange, and blue curves, Boltzmann suppression takes place before it (as in the green respectively). The mass of the fermionic dark matter is curve, for x ∼ 100). Since the Universe has a maximal set to be between the mediator masses in the first case temperature (fixed to 1012 GeV in Fig. 1), the production (109 GeV) and to be relatively light in the second and third rate will have maximal values at x ¼ 10−2; 10, and 103.Asa cases (104 GeV). consequence, the pole due to the real component exchange It is easy to understand the mechanism at work in the would not contribute for the cases in orange and blue. dark matter production after taking a look at Fig. 1. First of all, a general feature of the rate is the strong temperature B. Relic abundance dependence: the higher the temperature (small-x region), the more dark matter will be produced. The second generic From the approximate rates given in the last section, feature is the threshold for dark matter production, which is we can get an idea about the parameter space in agreement due to the Boltzmann-suppressed photon distribution hav- with the inferred value of the dark matter relic density ¼ 10 109 Ωh2 ¼ 0 1200 0 0012 ing T 8 7 −4 −4 ð12Þ α2 >BF 2 SM mDM TRH Λ mt > α 14 10 15 13 ðscalarDMÞ; > 35.5 S 5 1.2×10 GeV 10 GeV 10 GeV 10 GeV > 2 < 3 5 −4 −4 −4 Ωh ð10Þ ðαχ Þ2 α2 ðβχ Þ2 BF mDM TRH Λ V SM mt A 2 ma ≈ 10 10 15 13 þ β 12 ðfermionic DMÞ; ð25Þ 0 12 > 3.68 3.2×10 GeV 10 GeV 10 GeV 2 5 10 GeV 2 G 10 GeV . > > > ð12Þ 7 −4 2 2 −4 2 −4 >B Λ α α β 2 : F mDM TRH V SM mt þ V β ma ð Þ 35.5 1.5×1012 GeV 1010 GeV 1015 GeV 2 25 1013 GeV 2 G 1012 GeV vectorialDM : 095028-6 MODULI PORTAL DARK MATTER PHYS. REV. D 99, 095028 (2019) which sets the initial condition for the inflaton energy ρ ≃ 0 175 2 2 density (after the end of inflation) to be end . mϕMP 13 [37], for an inflaton mass of mϕ ¼ 10 GeV and for an −8 inflaton decay width Γϕ ¼ αϕmϕ with αϕ ¼ 10 . We set the reheating scale to be 1010 GeV. In Fig. 2 we set 8 11 ma ¼ 10 GeV and mt ¼ 10 GeV, whereas we explore a scenario with heavier mediators in Fig. 3, with ma ¼ 10 13 10 GeV and mt ¼ 10 GeV. With this set of parameters, it is imperative to consider the presence of the inflaton energy density in the Hubble rate. Indeed, our complete analysis is performed by solving the complete set of Boltzmann equations, and integrating over the whole phase space with Bose-Einstein distribution Ω 2 ¼ 0 12 Λ FIG. 2. Contours respecting h . in the (mDM, ) plane functions for the initial states (since standard fermions do for the real and imaginary parts of the modulus with masses mt ¼ not contribute to dark matter production in our scenario). 11 8 10 GeV and ma ¼ 10 GeV, respectively. Solid (dotted) curves We compute the triple integral of the production rates using are for the exact (approximate) computation of the production a Monte Carlo method (Vegas), as given by the CUBA 10 rates. For illustrative purposes, we set TRH ¼ 10 GeV, package [38]. Depending on the parameter choice, the ¼ 100 TMAX TRH, and all couplings are set to unity. The region Monte Carlo integration outputs close to the pole regions Λ in red is not reliable since 8At a given temperature, the production rate due to the exchange of a mediator j is approximated by RjðTÞ¼ 2 − j j Ω ¼ 0 12 Λ mDM=T ð Þ FIG. 3. Contours respecting h . in the (mDM, ) plane e min Rlight;Rheavy ; in this way, we can account for the for same parameter values as in Fig. 2 but for heavier moduli: six different regimes of production as well as for the Boltzmann 10 13 ma ¼ 10 GeV and mt ¼ 10 GeV. suppression. 095028-7 CHOWDHURY, DUDAS, DUTRA, and MAMBRINI PHYS. REV. D 99, 095028 (2019) whenever the pole can be reached (mt;a 095028-8 MODULI PORTAL DARK MATTER PHYS. REV. D 99, 095028 (2019) PICS No. 06482, PICS MicroDark. This project has and received funding/support from the European Union’s sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Horizon 2020 research and innovation programme under 3 4 2 Γ ¼ ma 1 − mDM the Marie Sklodowska-Curie: RISE InvisiblesPlus (grant a→DMDM 2 2 πΛ ma agreement No. 690575) and the ITN Elusives (Grant 8 > 0 for S; agreement No. 674896). > < 2 1 χ m ðβ β Þ2 DM χ 8 χ 5 2 for ; ð Þ × > ma A5 > β2 4 2 APPENDIX A: WIDTH OF THE MODULI FIELDS : V mDM 4 1 − 2 for V: ma Here we provide the expressions for the decay widths of the moduli fields.10 The real components of the moduli may decay into scalars and vectors of the Standard Model, since APPENDIX B: SQUARED AMPLITUDES the decays into fermions are not allowed above the In what follows, we provide the expressions for the electroweak symmetry breaking scale. Therefore, we have squared amplitudes of s-channel SM annihilations into DM candidates of spin s (jMj2 ): Γ ¼ 4Γ þ 12Γ þ Γ f sf t t→HH t→GG t→DMDM sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2m2 3 ð 2Þ2 4μ2 3 α2 s4ð1 − DMÞ2 X mt fH mt 0 2 jMj2 ¼ S s λt ð Þ ð Þ ¼ 1 − þ α þ Γ → : 0 4 2 2 2 2 0 s ; B1 πΛ2 8 2 16 G t DMDM Λ ðs − m Þ þ m Γ si; mt t t t si 2 ð Þ 2 2 3 4m X A1 ðαχÞ m s ð1 − DMÞ jMj2 ¼ DM s λt ðsÞ 1=2 4 2 2 2 2 si;1=2 Λ ðs − mt Þ þ mt Γt The distinct ways of writing the mass parameter of the si 11 χ 2 2 3 X Higgs lead us to define the function ðβχβ Þ m s þ 5 DM λa ð Þ ð Þ 4 2 2 2 2 1 2 s ; B2 8 si; = Λ ðs −maÞ þ maΓa < si αH ½case of Eq:ð3Þ ; 2 4 ð Þ ≡ 2μ2 ð Þ 4 4m 6m fH x 0 2Λ hFi A2 2 ð1 − DM þ DMÞ X :α 1 − þ ½ ð4Þ αV s 2 H x x M case of Eq: . jMj2 ¼ s s λt ð Þ P 1 4 2 2 2 2 1 s Λ ðs − m Þ þ m Γ si; t t t si 2 In the case of the imaginary component, decay into 2 4 4m β s ð1 − DMÞ X þ V s λa ð Þ ð Þ scalars is prohibited, and we have simply 4 2 2 2 2 1 s : B3 Λ ðs − m Þ þ m Γ si; a a a si Γ ¼ 12Γ þ Γ a a→GG a→DMDM We parametrize the contribution of the Ni SM initial states m3 ¼ a 3β2 þ Γ ð Þ with spin si for the production of DM of spin sf through the 2 G a→DMDM: A3 πΛ λj exchange of a field j by si;sf, which may be functions of The partial decay widths of the real and imaginary parts the Mandelstam variable s and the masses and couplings of the modulus into dark matter read, respectively, involved in the processes. They are given in Table I. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 t 4 TABLE I. Coefficients of the squared amplitudes: λs ;s (left) mt mDM i f Γ → ¼ 1 − λa – t DMDM 2 2 and si;sf (right) [Eqs. (B1) (B3)]. πΛ mt 8 α2 2m2 2 DM > S 1 − DM > 32 m2 for S; <> t SM spin-0 spin-1=2 spin-1 ðαχ Þ2 2 4 2 V mDM mDM 2 2 2 × 2 1 − 2 for χ; ðA4Þ ð Þ 2 ð Þ 1 2 ð Þ > 8 m m spin-0 fH s fH s = fH s > t t 1 2 > 2 2 4 spin- = 00 0 > β 4m 6m : V 1 − DM þ DM V 1 2α2 α2 1 4α2 64 2 4 for ; spin-1 = G G = G mt mt DM 10Finite-temperature effects on the decay width of moduli (as SM spin-0 spin-1=2 spin-1 discussed, for instance, in Ref. [39]) will not be relevant for us since we will not consider the moduli as ultrarelativistic species in spin-0 0 0 0 our analysis. 1 2 11 spin- = 000 In what follows, we assume that both couplings of moduli to 16β2 64β2 spin-1 0 G G the Higgs are equal, so that αμ ¼ αH. 095028-9 CHOWDHURY, DUDAS, DUTRA, and MAMBRINI PHYS. REV. D 99, 095028 (2019) The total production rate of the dark matter candidate TABLE II. Coefficients appearing in the approximate rates pffiffiffi t a ≪ [Eqs. (23) and (24)]: δs (left) and δs (right). We have defined with spin sf from Ni ultrarelativistic (mi T; s) thermal 2 2 2 f 2f α ≡ 2α þ 3α , since we assume μ0 ≪ s. Except for the NWA particles of spin si in terms of the Mandelstam variable s is SM H G given by cases, we have used Bose-Einstein statistics for the initial-state distribution functions. X RðTÞ ¼ RðTÞ sf si→sf DM si Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi regimes of t spin-0 spin-1=2 spin-1 4π ∞ 2 S 4m 3 2 3 ¼ f ds 1 − DM Light π α2α2 ζð3Þ ðαχ Þ2α2 π α2 α2 6 108000 S SM 8π5 V SM 21600 V SM 2048π 4m2 s DM Z Z NWA 1 α2α2 1 ðαχ Þ2α2 1 α2 α2 X ∞ ∞ 1024π4 S SM 256π4 V SM 2048π4 V SM jMj2 i i i i Heavy 64π7 α2α2 72ζð5Þ2 χ 2 2 32π7 α2 α2 × NiSi s →s dp1f1 dp2f2; 19845 5 ðα Þ α 19845 V i f s S SM π V SM SM 0 4 si pi1 ðB4Þ DM regimes of a spin-0 spin-1=2 spin-1 where the symmetrization factors Si;f ¼ 1=ni;f! account for Light 0 6ζð3Þ2 χ 2 2 8π3 β2 β2 i ¼ 5 ðβ Þ β 225 V G ni;f identical particles in the initial/final state and f1;2 π A G 3 χ 2 2 3 2 2 −pi =T −1 NWA 0 ðβ Þ β β β ðe 1;2 1Þ are the distribution functions of the initial- 16π4 A G 8π4 V G Heavy 0 3456ζð5Þ2 ðβχ Þ2β2 8192π7 β2 β2 state particles. π5 A G 6615 V G The integration over the initial momenta in Eq. 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