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Dark Glueballs and Their Ultralight Axions

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Dark Glueballs and Their Ultralight Axions Dark glueballs and their ultralight axions James Halverson, Brent D. Nelson, and Gustavo Salinas Department of Physics, Northeastern University, Boston, MA 02115-5000 USA Fabian Ruehle Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK Dark gauge sectors and axions are well-motivated in string theory. We demonstrate that if a confining gauge sector gives rise to dark glueballs that are a fraction of the dark matter, and the associated axion has a decay constant near the string scale, then this axion is ultralight and naturally realizes the fuzzy dark matter scenario with a modest tuning of a temperature ratio. Astrophysical observations constrain the size of the glueball component relative to the axionic component, while electric dipole moments constrain mixing with the QCD axion. Extended gauge and axion sectors are ubiquitous in the dark glueball problem is simultaneously eliminated. string theory and provide natural candidates for hidden Constraints on the model arising from astrophysics and sector dark matter. More specifically, gauge groups may electric dipole moment measurements are considered in have high rank, and gauge sectors that interact only via sections III A and III B, respectively. Finally, in sec- gravity, moduli, or axions are common. While not all tion IV, we discuss the scenario in a string theory con- string axions are necessarily associated with gauge sec- text, and consider how many of the assumptions made, tors, each gauge sector must couple via its field strength and conclusion drawn, are supported by broad arguments to some subset of the axions due to the dynamical nature in string theory. of the holomorphic gauge coupling function. It is there- fore natural and interesting to consider cosmologies with both hidden gauge sectors and their associated axions. I. RELIC DENSITIES We study the possibility that a confining dark gauge sector and its axion both contribute to the dark matter relic abundance. Previous studies considered either glue- As in our previous work [11], we consider a dark pure balls or axions that individually satisfied the relic abun- Yang-Mills sector (as these are often present in string dance. In the former case, a single glueball sector may compactifications) with gauge group G and confinement give rise to interesting dark matter candidates [1{10], but scale Λ. In addition, we consider a single axion field a many glueball sectors (as motivated in string theory) lead coupling to this sector with the usual ∼ aF ^F operator, to a dark glueball problem via oversaturation of the relic F being the field strength of the hidden gauge field. abundance [11]. In the latter case, axions of a variety We will assume that the dark sector is reheated to 0 of masses and decay constants may constitute the dark a temperature Trh > Λ, such that the dark sector at −22 that time is a thermal bath of dark gluons. Through- matter, and in the specific case in which ma ' 10 17 out, we will distinguish between the temperature of the eV, with decay constant fa ' 10 GeV, it gives rise to so-called fuzzy dark matter (FDM) [12]. FDM has been observable sector and that of the dark sector by using 0 the subject of many recent studies [13{19], in part to the symbols T and T , respectively. As the dark sec- 0 account for a number of troubling tensions between the tor cools through a transition temperature TΛ ∼ Λ the standard cold dark matter scenario and observed galactic energy density in gluons is converted into glueballs. properties [20, 21]. These confined states are likely to persist to late times We demonstrate that if the dark glueball accounts for provided that (a) any portals to the Standard Model arXiv:1805.06011v2 [hep-ph] 4 Aug 2018 a sizable fraction of the dark matter, it is often accompa- are generated only by Planck-suppressed effects, and (b) nied by ultralight axionic dark matter which is a candi- there are no light matter states (analogues to pions in date for the FDM scenario. This requires a small tuning the Standard Model) into which these glueball states can in the ratio of dark sector to visible sector temperatures, decay. The former is well-motivated by string model- and a solution to the dark glueball problem, but oth- building, while the latter is part of our simplifying as- erwise the scenario arises naturally. Constraints from sumption. Glueballs can, in principle, decay to cosmo- electric dipole moments may be avoided if the mixing logical moduli and/or light axions. In [11], it was demon- between the dark axion and the QCD axion is minimal. strated that glueball lifetimes will exceed the lifetime of This paper is organized as follows. In section I we com- the universe for all realistic UV parameter combinations > 16 pute the relic abundances of dark axions and glueballs, when fa ∼ 10 GeV. Furthermore, glueball lifetimes discuss their dependence on the confinement scale, and from decay into light moduli will be similarly long unless compare these results to the observed dark matter abun- the confinement scale satisfies Λ 1 GeV. As will be dance. In section II we discuss the region in parameter shown in the next section, the confinement scales neces- space in which the FDM scenario can be realized, while sary to generate a FDM axion are of the order Λ ∼ 100 eV. Thus, we should expect dark glueballs to be present 2 cosmologically for the parameter regime of interest. In The multiplicative factor !(Λ) is simply unity in the case addition, glueball decays to SM particles are highly sup- that 3 ! 2 interactions are already decoupled at the time 0 pressed by large powers of MGUT, in which case the en- of confinement, where Td ' Λ. More typically, we expect 0 ergy exchange between the visible and dark sectors is Td < Λ, and 3 ! 2 interactions have time to affect the minimal and large preferential reheating can be main- final relic abundance. In this case, assuming radiation tained [22]." domination at the time glueballs are formed1, we find 0 0 After confinement (T < TΛ), the number density of " !#−1 relic glueballs is set by the rate of 3 ! 2 interactions, 4 f 6=5g0 4=5 3:6eV 3=5 as was studied by Carlson et al. [1]. Freezeout occurs !(Λ) = W 7:45 × 1012 ; 5 ξ2=5 Λ when these interactions cease to be effective, leaving a 0 (7) dark glueball relic, at a decoupling temperature Td ≤ Λ. We will assume that throughout these processes, the where W (x) is the Lambert W-function and f is the cou- comoving entropy density is conserved in each sector due pling constant of the effective operator that governs glue- to thermal equilibrium and minimal interactions between ball 3 ! 2 interactions. the sectors, so that the ratio For the axion relic density, we will follow the standard treatment, as contained in [24, 25]. We assume the axion s ξ := (1) mass and scalar potential arise from non-perturbative ef- s0 fects at a time when the universe is radiation dominated. The equation of motion for the background field a(t) is is a constant. We have assumed T 0 > Λ, and thus it is rh given by reasonable to assume both sectors are relativistic at high enough Trh. This allows us to identify the parameter ξ 2 d 3 d 2 a(t) as 2 + a(t) + ma(t)fa sin = 0 ; (8) dt 2t dt fa 3 gST −1 ξ := 0 03 ; (2) where we have used 2t = H in a radiation-dominated gST epoch. Let t1 be the time at which the axion field begins with g and g0 being the number of relativistic degrees to oscillate about its background value. Since the first S S −2 of freedom in entropy in the visible and dark sectors right term scales with t , this occurs when the second term after reheating. in (8) grows comparable to the first at a time As the dark sector is composed of dark gluons at the time of BBN, the number of degrees of freedom in ma(t1)t1 ∼ 1 : the dark sector has to satisfy the current experimental −22 For axion masses of the order ma ∼ 10 eV, we ex- bounds, given by [23] 6 pect this time to be t1 ∼ O(10 sec), consistent with the 4 assumption of a radiation-dominated epoch. 0 Trh g?jBBN ≤ 148:6 0 : (3) After the time t1, the axion field evolves adiabatically, Trh with the total number of axions serving as the adiabatic This bound can be expressed in terms of the entropy invariant. At this point, the number density for the axion ratio, using (2), to give field is m (t )a2 a2 4 n = a 1 i = i ; 0 10 a g ≥ 0:083 ; (4) 2 2t1 ξ where ai is the initial amplitude of the field oscillation, if we assume g0 j = g0 ≡ g0. Notice that mild prefer- −1 ? BBN S and we have taken ma(t1) = t1 for definiteness. Since ential reheating into the visible sector, ξ ∼ O(1 − 10), is the value of ai is arbitrary, we can parameterize it as ai = needed to allow for smaller gauge groups. faθi, where −π ≤ θi ≤ +π is the initial misalignment Since ξ is assumed to be constant, and comparing to angle. Therefore, the relic density today would be the visible sector entropy density today, the relic abun- dance of glueballs is [1] 2 2 3 0 0 0 0 fa θi R1 ρa = mana = ma ; 0 2t1 R0 2 Td Ωg(ξ)h = : (5) 3:6 eV ξ 0 with R1 being the scale factor at t1 and ma, R0, the axion mass and scale factor today.
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