PHYSICAL REVIEW D 99, 055047 (2019)

Dark quark nuggets

Yang Bai,1,2 Andrew J. Long,3,4 and Sida Lu1 1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA 2Theoretical Physics Department, Fermilab, Batavia, Illinois 60510, USA 3Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA 4Leinweber Center for Theoretical Physics, University of Michigan, Ann Arbor, Michigan 48109, USA

(Received 17 October 2018; published 29 March 2019)

“Dark quark nuggets,” a lump of dark quark matter, can be produced in the early universe for a wide range of confining gauge theories and serve as a macroscopic dark matter candidate. The two necessary conditions, a nonzero dark baryon number asymmetry and a first-order phase transition, can easily be satisfied for many asymmetric dark matter models and QCD-like gauge theories with a few massless flavors. For confinement scales from 10 keV to 100 TeV, these dark quark nuggets with a huge dark baryon number have their masses vary from 1023 gto10−7 g and their radii from 108 cm to 10−15 cm. Such macroscopic dark matter candidates can be searched for by a broad scope of experiments and even new detection strategies. Specifically, we have found that the gravitational microlensing experiments can probe heavier dark quark nuggets or smaller confinement scales around 10 keV; collision of dark quark nuggets can generate detectable and transient electromagnetic radiation signals; the stochastic gravitational wave signals from the first-order phase transition can be probed by the pulsar timing array observations and other space-based interferometry experiments; the approximately massless dark mesons can behave as dark radiation to be tested by the next-generation cosmic microwave background experiments; the free dark baryons, as a subcomponent of dark matter, can have direct detection signals for a sufficiently strong interaction strength with the visible sector.

DOI: 10.1103/PhysRevD.99.055047

I. INTRODUCTION Ref. [3]), and therefore quark nugget production is not viable in the SM. The theory of quantum chromodynamics (QCD) is an Nevertheless, the requirements for quark nugget pro- integral part of the Standard Model (SM) of elementary duction are generic, and although SM QCD does not have particles as it successfully explains hadron properties, all the right ingredients, it is not hard to find new physics, nuclear structure, and phenomena. While QCD predicts beyond the Standard Model (BSM), that facilitates the that most matter in the current universe is in the form of formation of these objects. In particular, the formation of hadrons, the theory also admits an exotic phase of “quark matter” at high baryon-number density and low temper- nuggets needs (i) a first-order phase transition to have (at ature [1]. In his seminal work, Witten [2] proposed that least) two phases with different vacuum energies; (ii) a “nuggets” of quark matter could have formed in the early conserved global charge for a small pocket of space to build universe at the epoch of quark confinement, and that these up a large global charge; (iii) a cosmological excess of nuggets could survive in the universe today as a dark matter matter over antimatter, corresponding to a nonzero density candidate. One can understand Witten’s quark nuggets as of a conserved global charge. The SM QCD satisfies the macroscopic nucleons (not nuclei) with a very large baryon last two conditions but not the first one. Regarding the first 30 condition, the literature on BSM physics is replete with number, NB > 10 . Whereas Witten assumed that our QCD confining phase transition was a first-order one, confining gauge theories including the UV completion of a numerical lattice studies later revealed that the transition composite Higgs model [4,5], supersymmetric models – is predicted to be a continuous crossover instead (see, e.g., [6,7], twin Higgs models [8], dark QCD [9 16], and Nnaturalness models [17]. As we will discuss further in Sec. II, the condition of a first-order phase transition is easily satisfied as long as the number of light vectorlike Published by the American Physical Society under the terms of fermions obeys N ≥ 3 for an SUðNÞ gauge theory. (In SM the Creative Commons Attribution 4.0 International license. f Further distribution of this work must maintain attribution to QCD the up and down quarks are light compared to the the author(s) and the published article’s title, journal citation, confinement scale, but the strange quark is marginal, and and DOI. Funded by SCOAP3. consequently the QCD phase transition is not first order).

2470-0010=2019=99(5)=055047(26) 055047-1 Published by the American Physical Society YANG BAI, ANDREW J. LONG, and SIDA LU PHYS. REV. D 99, 055047 (2019)

For the second condition and similar to the U(1) baryon can allow composite objects to form by aggregation. number in the SM, it is natural to have (approximately) Several authors have considered that the dark sector could good symmetry in the new strong-dynamics sector such as undergo a period of dark nucleosynthesis to form com- technibaryon, twin baryon, and dark baryon number posite objects with Oð1Þ constituents [34–37]. The authors symmetries. Finally, for the third condition it is natural of Refs. [38,39] studied a model of asymmetric dark matter to expect that a matter-antimatter asymmetry may be shared in which Oð≫1Þ Dirac fermions become bounded together between the dark and visible sectors [18–26]. through a Yukawa interaction via a light scalar mediator It is interesting to remark here that, based on the and form a nonrelativistic degenerate Fermi gas; they called conditions above, the presence of dark quark nuggets these objects dark matter nuggets. In work by other may be unavoidable in some models of dark baryon dark authors, the properties and production mechanism of these matter [5]. As we will discuss in Sec. II, for models with asymmetric dark matter nuggets was clarified and refined three or greater flavors of light dark quarks, the confining [40,41]. The authors of Ref. [42] considered composite phase transition is expected to be a first-order one, and dark objects, which they called dark blobs, that can be formed quark nuggets can be formed. The dark baryon number from either bosonic or fermionic constituent particles, and could be mainly in the dark quark nugget states, similar to they study the associated detection strategies. the QCD nuggets in Ref. [2]. This observation motivates a The remainder of this article is organized as follows. reevaluation of earlier studies of dark baryon dark matter to In this work we study a class of BSM confining gauge assess whether those models also predict a relic abundance theories, collectively denoted as “dark QCD,” that are of dark quark nuggets. introduced in Sec. II. We discuss the conditions under which In this work we consider a class of BSM confining gauge the confining phase transition is a first-order one, which is a theories, collectively denoted as “dark QCD,” which are necessary condition for the formation of dark quark nuggets. parametrized by the number of colors, the number of In Sec. III we analyze the properties of dark quark matter and flavors of light vectorlike fermions, and the confinement discuss how the Fermi degeneracy pressure provided by the scale. We study the properties of “dark quark matter” and (conserved) dark baryon number supports the dark quark the conditions under which stable “dark quark nuggets” nugget against collapse. Section IV addresses the cosmo- (dQN) can form through a cosmological phase transition in logical production of dark quark nuggets and contains esti- the early universe. Depending on the confinement scale, the mates for their mass, size, and cosmological relic abundance. typical nugget’s mass and radius can reach as large as In Sec. V we discuss various observational signatures ∼ 1023 ∼ 108 MdQN g and RdQN cm. We argue that these including gravitational wave radiation, dark radiation, col- nuggets can survive in the universe today where they liding and merging signatures, and prospects for direct provide a candidate for the dark matter, and we explore detection. We conclude in Sec. VI.InAppendixA,we various observational prospects for their detection. provide a calculation of the phase transition based on the Dark quark nuggets are examples of macroscopic dark effective sigma model for the dark chiral symmetry breaking. matter; for a recent review see Ref. [27]. Given the null results of searching for a weakly interacting massive II. DARK QUANTUM CHOROMODYNAMICS particle with a mass of Oð100 GeVÞ [28], it is natural to In this section we introduce the model being considered explore other well-motivated dark matter models with in the remainder of the article. In particular we are different masses. Since the past several years have seen interested in “dark QCD” with N colors and N flavors renewed interest in these dark matter candidates, let us d f of (approximately massless) vectorlike fermions. In our briefly note some of the recent developments and clarify model, we will assume that there is no dark electroweak their connection to our own work. To our knowledge the gauge group or dark neutrino. More or less, the dark QCD author of Ref. [2] was the first to propose that the dark is anticipated to have a similar asymptotic-free dynamics as matter could consist of macroscopic objects with nuclear our SM QCD. In an ultraviolet energy range, the dark densities, and he called these objects quark nuggets since SUðN Þ QCD has a perturbative gauge coupling and with they were made up of Standard Model quark matter. d the particle content composed of N2 − 1 dark gluons, N Subsequent work introduced a coupling to the QCD axion, d f which led to axion quark nuggets, where quark nuggets are dark quarks, and Nf dark antiquarks. The gauge coupling Λ formed through CP-violating domain walls with modified becomes strong in an infrared scale d, and both confine- properties and enhanced stability [29–31]. Other authors ment and chiral symmetry breaking happen below the dark Λ 2 − 1 proposed that six-flavor quark nuggets could form if the QCD scale d with Nf dark mesons in the low-energy 1 electroweak phase transition were supercooled to the QCD theory. Different from the SM QCD, where the phase scale [32]. The nuggets that are made of techniquarks have 1 also been studied in technicolor models [33]. This counting of dark mesons works for Nd ≥ 3. For Nd ¼ 2, The more recent interest in macro dark matter is the chiral symmetry breaking is SUð2NfÞ → SPð2NfÞ with ’ 2 2 − − 1 motivated by the idea that dark matter s self-interactions Nf Nf dark mesons [43].

055047-2 DARK QUARK NUGGETS PHYS. REV. D 99, 055047 (2019) transition is a crossover one [3], there is a wide range of B. Color confinement model parameter space for the dark QCD phase transition Quantum effects lead to the renormalization group (RG) to be first order. flow of the coupling gd. Let gˆdðμÞ be the running coupling, and let μ be the renormalization scale. The RG flow A. The model equation is ψ ∈ 1 2 … Let iðxÞ for i f ; ; ;Nfg be a collection of ˆ ˆ3 a dgd g 5 Dirac spinor fields or dark quark, and let GμðxÞ for a ∈ μ ¼ β ¼ d b þ Oðgˆ Þ; ð2:3Þ μ gd 16π2 gd d 1 2 … 2 − 1 d f ; ; ;Nd g be the dark gluon fields and a collec- tion of real vector fields that form the connection of an and the leading-order term given by [44,45] SUðNdÞ gauge group under which the ψ i transform in the fundamental representation. The properties of these par- 11 2 b ¼ − N þ N ; ð2:4Þ ticles and their interactions are given by the following gd 3 d 3 f Lagrangian: which can be negative. N Xf 1 We are interested in models with Nf < 11Nd=2 for L ψ¯ γμ ψ − ψ¯ ψ − a μνa ¼ ½ ii Dμ i mi i i GμνG which b < 0, and the theory becomes more strongly 4 gd i¼1 μ ˆ μ coupled in the IR (smaller ). If we take gdð UVÞ¼gUV as 1 θ g2 − d d a ˜ μνa a reference point where the theory is weakly coupled, GμνG ; ð2:1Þ ≪ 4π 4 2π 4π gUV , then by solving the RG flow equation we ˆ μ μ μ observe that gdð Þ diverges at ¼ . As the gauge where coupling becomes larger, the interactions among quarks and gluons become stronger, leading to a color-confining/ a a Dμψ i ¼ ∂μψ i − igdGμT ψ i; chiral-symmetry-breaking phase of the theory. The value of a a a abc b c μ provides a rough (one-loop perturbative) estimate of the Gμν ¼ ∂μGν − ∂νGμ þ g f GμGν; d confinement scale, Λ ≈ μ , which gives 1 d ˜ μνa μνρσ a G ¼ ϵ Gρσ: ð2:2Þ 2 Λ ≈ μ −8π2 2 d UV exp½ =ðjbgd jgUVÞ; ð2:5Þ a The generators of SUðNdÞ are denoted as T , and the assuming that b < 0. abc gd structure constants are denoted by f . Around the confinement scale, the fermion-antifermion ∈ The model parameters are the number of colors Nd operator also develops a nonzero expectation value with 2; 3; 4; … N ∈ 1; 2; 3; … 3 f g, the number of flavors f f g, the hψψ¯ i ∼ Λ , which spontaneously breaks the SUðN Þ ∈ 0 ∞ d f A dark gauge coupling gd ½ ; Þ, the mass parameters flavor symmetry and provides dark mesons as IR degrees mi ∈ ½0; ∞Þ, and the theta parameter θd ∈ ½0; 2πÞ. We will 0 of freedom (d.o.f.). The dark meson decay constant is consider both the case of massless quarks, mi ¼ , and ∼ Λ fπd d, while their masses are related to the dark quark massive quarks, mi ≠ 0. For simplicity, we assume that the 2 2 3 masses by mπ fπ ∼ m Λ . The dark baryon masses have model is CP conserving with θ ¼ 0. There could exist d d i d d m ∼ 4πΛ and are heavier. The temperature of the nonrenormalizable operators for the SM sector interacting Bd d with the dark QCD sector, which will be introduced and confining/chiral-symmetry-breaking phase transition hap- ∼ Λ discussed in a later section. pens at Tc d. Some of our later calculations will be sensitive to some ratios of quantities like m =T , which The fermion mass term in Eq. (2.1) can be written more Bd c requires a nonperturbative tool like lattice QCD to obtain a generally as mijψ¯ iψ j for mij ∈ C, but we have performed a precise value. field redefinition to write it as miψ¯ iψ i with mi being real and non-negative. For mi ¼ 0 the theory respects a chiral 1 1 flavor symmetry, SUðNfÞV ×Uð ÞV ×SUðNfÞA ×Uð ÞA. C. Confining phase transition 1 The symmetry group Uð ÞV has an associated conserved Let us now consider the behavior of this theory in a charge, which is the dark baryon number, Uð1ÞB ; the dark d finite-temperature system, and specifically we are inter- gluons, dark quarks, and dark antiquarks have charges ested in a system whose temperature is close to the critical QB G 0, QB ψ 1=N , and QB ψ¯ −1=N , ∼ d ð aÞ¼ d ð iÞ¼ d d ð iÞ¼ d temperature of the confining phase transition, T Tc. The 1 respectively. The axial Uð ÞA symmetry is anomalous order of magnitude of the critical temperature is set by the under the dark QCD gauge interactions and does not lead confinement scale, Tc ∼ Λd. Suppose that the system is to a light Nambu-Goldstone boson after spontaneous chiral heated to a temperature T>Tc and allowed to cool ≠ 0 symmetry breaking. For mi the subgroup SUðNfÞA × adiabatically to T ∼ Tc. Since the temperature sets the 1 Δ Uð ÞA is explicitly broken. typical momentum transfer j pj of particles in the plasma,

055047-3 YANG BAI, ANDREW J. LONG, and SIDA LU PHYS. REV. D 99, 055047 (2019) the system will be in the unconfined phase for T>Tc ∼ Λd where jΔpj ∼ T>Λd. However, as the temperature reaches close to Λd the system will pass into the confined phase. At the same time a chiral condensate forms, hψψ¯ i ≠ 0, signaling that the chiral symmetry is spontaneously broken.2 We are interested in whether the corresponding phase transition is a first-order one, which is one of the necessary conditions to form the dark quark nuggets. The order of this phase transition has been studied on general grounds by Pisarski and Wilczek (PW) [46] for Nd ≥ 3 (see Ref. [47] for the Nd ¼ 2 case). Using a perturbative ϵ expansion, they argue that the chiral phase transition will be first order if the number of light vectorlike fermion flavors is greater than or equal to three; in our notation, this corresponds to

PW argument∶ Nf ≥ 3 for mi ≪ Λd ⇒ first-order phase transition: ð2:6Þ

The essence of the argument is to write down an effective field theory describing the self-interactions of the chiral condensate, Σij ∼ hψ¯ ið1 þ γ5Þψ ji with i; j ¼ 1; 2; …;Nf. 1 FIG. 1. The nature of the chiral phase transition in dark QCD is Besides the instanton-generated Uð ÞA-breaking term that is suppressed in the large N limit, there are two couplings controlled by the number of colors, Nd, and the number of d massless, vectorlike flavors of fermions, N . Points labeled by 1st, associated with the self-interaction operators, ðTrΣ†ΣÞ2 and f 2nd, and “cross” are known from lattice studies [48–52] to exhibit TrðΣ†ΣÞ2. PW calculate the beta functions for these ≥ 3 a first-order phase transition, a second order phase transition, and couplings and argue that for Nf the RG flow equations a continuous crossover, respectively. Analytical arguments do not have an IR stable fixed point. In the absence of an IR [46,47] imply that points falling into the unshaded (white) region stable fixed point, the theory cannot be smoothly evolved to will exhibit a first-order phase transition. The theory is not arbitrarily low scales (temperatures), but instead some confining in the orange shaded regions: above the dotted line the critical behavior must arise in the form of a first-order beta function remains positive, and between the dotted and dot- phase transition. dashed lines, the theory becomes conformal at low energies. The ’ Whereas the PW argument infers the existence of a first- precise location of the conformal window s boundaries is a matter order phase transition indirectly from RG flow trajectories of active debate [55]. in the chiral effective theory, one can also study the phase transition directly by evaluating the thermal effective chiral phase transition. In Fig. 1 we summarize the results potential for the chiral condensate and calculating the of several lattice studies for different values of Nd and Nf 3 thermal transition rate between coexistent phases. To justify (assuming massless quarks/antiquarks) for Nd ¼ ,4 a perturbative calculation of the effective potential, this [48–52]. We conclude that the PW argument is supported approach is reliable only when the couplings are small, but by numerical lattice simulations, which take all nonper- 2 nevertheless we can infer the behavior at a strong coupling turbative effects into account. For Nd ¼ , more lattice by studying the trending behavior as the coupling is QCD simulations are required to determine the order of increased toward the nonperturbative regime. The results phase transition [53,54]. of this analysis are detailed in Appendix A; in particular, we In Fig. 1, we also indicate the parameter region where the confirm that the chiral effective theory admits a first-order leading-order beta function is positive and the theory is phase transition in the regime consistent with the PW “IR-free” rather than exhibiting confinement or chiral argument. symmetry breaking at low energies. For smaller values Since the PW argument is inherently perturbative in of Nf, the “conformal window” corresponds to a range of nature, one might worry that its conclusions do not apply parameters in which the theory goes to a nontrivial fixed for a strongly coupled system. Thus it is important to “test” point in the IR, and there is neither confinement nor chiral the PW argument against numerical lattice studies of the symmetry breaking. The boundary between the conformal window and models with chiral symmetry breaking (at 2 smaller N ) is an active subject of research for both lattice We assume that both chiral symmetry breaking and color f confinement occur at around the same time during the phase QCD or other semianalytic approaches. In our plot, we take transition at T ¼ Tc ∼ Λd. the point of view based on the review paper in Ref. [55]: the

055047-4 DARK QUARK NUGGETS PHYS. REV. D 99, 055047 (2019) conformal window line is determined by Nd ¼ 2 and A. Modeling dQM as a relativistic Nf ≳ 8 [56,57] and Nd ¼ 3 and Nf ≳ 10 [58,59]. In the degenerate Fermi gas dot-dashed line of Fig. 1, we simply use the information at We suppose that the model from Sec. II is brought to a 2 Nd ¼ , 3 to obtain the conformal window boundary line finite temperature T where the dark gluons, dark quarks, ≈ 2 4 as Nf Nd þ . and dark antiquarks are allowed to reach thermal equilib- Finally let us remark on the range of interest for the rium. We further suppose that the system is prepared with a ≥ 3 model parameters. We will take Nf to ensure a first- nonzero dark baryon number. ≲ 2 order chiral phase transition, and we will take Nf Nd þ Dark QCD mediates interactions among the dark gluons 4 to ensure that confinement occurs. Then the parameter and the dark quarks/antiquarks. If reactions such as ψ iψ¯ i ↔ range of interest is GaGb and ψ iψ¯ i ↔ GaGbGc are in thermal equilibrium, i.e., the thermally averaged rate exceeds the Hubble 3 ≤ ≲ 2 4 ≪ Λ Nf Nd þ with mi d: ð2:7Þ expansion rate at the time of interest, then chemical equilibrium imposes μ ¼ 0 and μψ¯ ¼ −μψ , where μ is We want to stress that there is a wide range of parameter Ga i i the chemical potential of the species. For simplicity, we space in ðN ;N Þ for the dark QCD phase transition to be a d f further suppose that the dark baryon number is shared first-order one. equally by all of the quark and antiquark flavors, which implies that the chemical potentials are equal, μψ ¼ μ, and D. Differential vacuum pressure: B i we also assume that the Nf flavors of dark quarks and During the confining/chiral-symmetry phase transition, antiquarks are degenerate, which lets us write mi ¼ m; the system passes from a phase in which color is uncon- these assumptions do not qualitatively impact our results. fined and the chiral symmetry is unbroken into a second We are interested in this system at a temperature mi ≪ phase in which color is confined and the chiral symmetry is T ≪ μ such that the quarks and antiquarks form a relativ- broken. In general the vacuum energy of these two phases istic degenerate Fermi gas [61]. Let n ¼ nψ − nψ¯ be the will differ, and it is the lower vacuum energy of the ψ-number density, which contains an implicit sum over the confined phase that makes the phase transition energeti- ρ ρ ρ ρ Nf flavors; let ¼ ψ þ ψ¯ þ vacuum be the energy density cally favorable at low temperature. Since the vacuum has an of quarks, antiquarks, and the dark quark matter vacuum; equation of state, ρ ¼ −P, we can equally well talk about and let P ¼ Pψ þ Pψ¯ þ P be the corresponding the differential vacuum pressure between the two phases. vacuum pressure. For a relativistic degenerate Fermi gas, and Following the notation of the MIT bag model of SM neglecting the perturbative interactions among dark quarks nuclear structure [60], we denote this differential vacuum and gluons, these quantities are given by [61] pressure as B, which has a mass dimension equal to 4. In principle B can be expressed in terms of the model μ3 μ4 μ4 Λ ρ − parameters: d, Nd, Nf, and mi. However, a robust n ¼ g 6π2 ; ¼ g 8π2 þ B; and P ¼ g 24π2 B; calculation of B requires nonperturbative methods, such as numerical lattice techniques. Therefore we will generally ð3:1Þ take B as a free parameter, while keeping in mind that it is roughly set by the confinement scale: where B is the differential vacuum pressure from Eq. (2.8) (normalized such that pressure vanishes in the hadronic Δ − ∼ Λ4 phase) and where g 2N N accounts for a sum over B ¼ Pvacuum ¼ Pconfined Punconfined d: ð2:8Þ ¼ d f identically distributed particles that differ in their spin, In Sec. III we will see that B controls the density and energy color, and flavor. The number density of a dark baryon of the dark quark matter that resides inside of dark quark number is given by nuggets. Consequently in Sec. IV C we will find that B also sets the mass scale and radius of cosmologically produced 1 μ3 nB ¼ n ¼ N ; ð3:2Þ dark quark nuggets. d f 2 Nd 3π

III. DARK QUARK MATTER since each dark quark carries a baryon number of 1=Nd and each antiquark has −1=N . Note that nB is independent of The theory discussed in Sec. II admits a state of “dark d d quark matter” (dQM) at zero temperature and finite dark- Nd; raising Nd means that there are more species of dark baryon-number density. In this section we calculate the quarks/antiquarks in the system, but that each one carries a thermodynamic properties of dQM by adapting a similar smaller dark baryon number. calculation from Ref. [2]. The main results of this section appear in Eqs. (3.4) and (3.5), which give energy density B. Dark quark matter inside of nuggets and the dark-baryon-number density of the dark quark Now we suppose that the conserved dark baryon number matter contained within a stable dark quark nugget. is localized in a region of space with finite volume. If the

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FIG. 2. This cartoon illustrates the localized nugget of dark quark matter, which is supported against collapse by the Fermi degeneracy pressure arising from its conserved dark baryon number. 64 1=4 volume is allowed to vary, such as during the formation of a Nf 3=4 nB ¼ B : ð3:5Þ dark quark nugget, then the system will evolve to an d;dQM 3π2 3 Nd equilibrium configuration in which the differential vacuum pressure at the phase boundary is balanced against the Thus the energy per baryon of dark quark matter in dark Δ differential pressure arising from the particles, Pvacuum ¼ quark nuggets is found to be Δ 3 Pparticles. Here we assume that the plasma temperature is ρ 2 3 1=4 3=4 small compared to the phase transition temperature, which dQM 12π N N Δ ≈ ¼ d B1=4 ≃ 3.3 d B1=4: ð3:6Þ lets us write Pvacuum B where B is the differential 1=4 nB ;dQM Nf N vacuum pressure at zero temperature. We also continue d f to assume that T ≪ μ, which lets us neglect the radiation 3 5 7 1=4 pressure that would arise from particles outside of the For instance Nd ¼ Nf ¼ gives . B . nugget and instead write ΔP ≈ gμ4=24π2. A cartoon Looking back over these results, we observe that the particles differential vacuum pressure between the confined and of this situation is illustrated in Fig. 2. Thus the equilibrium unconfined phases, ΔP B from Eq. (2.8),isthe condition is expressed as vacuum ¼ only scale that sets the density and energy of the dark quark μ4 eq matter that resides inside of dark quark nuggets. We will Pjμ μ ¼ g − B ¼ 0; ð3:3Þ ¼ eq 24π2 use Eqs. (3.4) and (3.5) in Sec. IV C to estimate the size and mass of a typical dark quark nugget, and we will use μ ≈ 12π2 1=4 1=4 and its solution is eq ½ =ðNdNfÞ B . For in- Eq. (3.6) in the subsection below to discuss the stability of 3 μ ≃ 1 9 1=4 stance Nd ¼ Nf ¼ gives eq . B . dark quark matter. Now we are equipped to calculate the properties of the dark quark matter that resides inside of a stable dark quark C. Stability of dark quark matter nugget. The energy density of the dark quark matter inside The quantity ρ =nB is used to assess whether the of a dark quark nugget is calculated using ρ from Eq. (3.1) dQM d;dQM μ μ state of dark quark matter is more or less stable than the state and ¼ eq from Eq. (3.3), which gives of dark hadronic matter. Suppose that the lightest stable dark ρ 4 baryons are all degenerate, and let their mass be denoted by dQM ¼ B; ð3:4Þ mBd . In the dark hadronic state and for a volume of V,astate and the density of the dark baryon number is evaluated with with nBd V units of a dark baryon number can have an energy nB from Eq. (3.2), which gives d that is as low as mBd nBd V (if all the dark baryons are at rest with negligible interactions and no additional particles are 3The gravitational pressure is negligible for the range of dQN present). Thus the state of dark quark matter is absolutely ρ masses considered in this paper. stable provided that V

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ρ ≈ 7 for =nBd from Eq. (3.6), the stability of dark quark matter mBd =Tc . This much shorter length compared to the requires neutrino one can lead a dramatical reduction on the energy injection and hence the evaporation rate, and make the dark B1=4 N =N 1=4 N −1=2 quark nugget more stable against the evaporation process. < 0.175 f d d : ð3:7Þ mB 1 3 In addition, it has been argued that the reabsorption effect d will further enhance the stability of nuggets against

Recall that we need Nf=Nd ≳ 1 for a first-order phase evaporation [65]. Therefore, we would ignore the dark transition. Both the differential vacuum energy, B,andthe baryon dissipation from evaporation in the following analysis. dark baryon mass, mBd , are controlled by the confinement 1=4 Since there is no dark neutrino in our model, one may scale of the dark QCD, Λd.InSMQCDwehaveB ≃ 1 4 wonder whether the dark quark nuggets will stay “hot” after 150 MeV and m ≃ 938 MeV to give B = =m ≃ 0.160 Bd Bd their formation below the phase transition. We want to point [62]. For a generic dark QCD model, a nonperturbative tool out that the dark mesons can efficiently thermalize the dark like lattice QCD is needed to estimate this ratio precisely. For quark nuggets with the surrounding medium and make a fixed value of N , there is a critical value of the number of d nuggets cool as the universe cools down. Because dark flavors, N Nc, above which the infrared theory of dark f ¼ f mesons have a short free-streaming length, the cooling of QCD becomes conformal instead of chiral symmetry break- nuggets is mainly through surface evaporation of dark ing. When the number of flavor is close to the critical value, mesons from black-body radiation. To simplify our dis- we anticipate that this ratio is further suppressed and scales cussion, we keep the chemical potential and radius of the B1=4=m ∝ Nc − N =N like Bd ð f fÞ f [32]. So, the dark quark nuggets fixed, which is reasonable within a Hubble time- matter state becomes more stable for a larger value of Nf. scale. We will check the cooling timescales for both an In Eq. (3.7), we have only compared the quark matter earlier time with a tiny chemical potential and a later time state with a free baryon state. In the SM QCD, the most with a large chemical potential. stable state per baryon is the iron nucleus, which has the Using the Stefan-Boltzmann law of blackbody radiation, energy per baryon slightly smaller (≈1%) than a free proton we have the cooling rate given by and neutron. So, if the value of B1=4=m is so close to the Bd π3 upper bound in Eq. (3.7), one may need to check the 2 4 2 − 1 LðTÞ¼ gdπR T ; with gdπ ¼ Nf : ð3:8Þ additional heavy-dark-nuclei evaporation processes, which 30 will depend on more detailed properties of the model like 4 3 The total energy inside has EðTÞ¼3 πR ρðTÞ with ρ ¼ the dark-meson-induced binding energy. For the massless g π2T4=30 when μ ≪ T and ρ ¼ g ðμ4 þ 2π2μ2T2Þ=8π2 dark meson case or the chiral limit, the internucleon dQ dQ for T ≪ μ. Here, we take the d.o.f. as g ¼ð2N N Þ × binding energy is anticipated to be larger by only a factor dQ d f 7=8 for a temperature after the dark quark and dark of around 2 than the SM QCD case [63], so for a wide range antiquark annihilation. Using the energy conservation of model parameters not saturating the bound in Eq. (3.7), dEðTÞ=dt ¼ −LðTÞ, we can derive a differential equation one does not need to worry about evaporation to heavy dark for the temperature change as a function of time and have nuclei. the cooling timescale (the time for the temperature Similar to the SM QCD nugget scenario, the equilibrium decreases from T to T=2) estimated as between the two phases at a temperature below T is c 8 maintained by surface evaporation and emission of light < 16 ln 2 gdQ 3 R; μ ≪ T; particles. The detailed calculation on the establishment of τ gdπ cool ¼ μ2 ð3:9Þ the equilibrium is complicated. Here we would only : 10 gdQ μ ≫ π2 2 R; T: provide simple pictures and argue that the nuggets may T gdπ survive the evaporation and meanwhile stay thermalized When the temperature is high, the nugget radius is smaller with the plasma. In surface evaporation, the nugget emits a ∼ 10−5 1 → 1 than the Hubble scale R dH because there are around dark baryon and undergoes ðNB þ Þ NB þ [64]. 1014 However, such processes require addition energy input nucleation sits within one Hubble volume (see from the environment, as argued above. In SM QCD the Appendix A). So the cooling time is shorter than the energy is dumped into the nuggets by neutrinos, which has Hubble expansion time. When the chemical potential is τ ∝ a long free-streaming length of Oð0.1 mÞ at the QCD scale. high or the temperature is low, one has cool=tH μ2 ∼ 10−8 μ ∼ As we have no dark neutrinos in our model, the energy R=Mpl for the benchmark point with Tc ¼ carrier in the dark quark nugget scenario will be the 100 MeV and R ∼ 0.1 cm from Eq. (4.10). The cooling τ ∼ −1=3 massless dark pions. However, because of the strong time has a mild Tc dependence: cool Tc , so we have a interactions of dark mesons with other hadrons, their sufficiently fast thermalization for the nuggets with the free-streaming length is very short at the order of surrounding medium for the model parameter space in 2 10 =Tc and around 100 fm for Tc ¼ 0.1 GeV and this paper.

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IV. COSMOLOGICAL PRODUCTION growth of the bubbles, while the scattering of the OF DARK QUARK NUGGETS particles in the dark plasma on the bubble wall induces a drag force on the bubble wall. A balance In this section we discuss how dark quark nuggets can between vacuum pressure and thermal pressure is form in the early universe, we calculate their properties, and reached, and the bubble’s radius grows at a non- we estimate their relic abundance. relativistic terminal speed. (3) It is energetically preferable for the dark baryon A. Overview of dark quark nugget production number to remain in the unconfined phase, where Dark quark nuggets may form at a first-order phase dark quarks are light, rather than entering the transition during which dark color is confined and the chiral confined phase, where dark baryons are heavy. Thus, symmetry is spontaneously broken. The production mecha- the dark baryon number accumulates in front of the nism for dark quark nuggets is very similar to the more- advancing bubble walls. familiar QCD quark nugget scenario [2]. Here we briefly (4) The bubbles collide and coalescence with each other. summarize the physical processes that lead to creation of At the end of the phase transition, the dark hadron dark quark nuggets in the early universe. The production phase occupies the majority of the Hubble volume, process is also illustrated in Fig. 3 that shows a schematic with the remaining dark quark-gluon plasma left in phase diagram for dark QCD. isolated regions that form dark quark nuggets. Most (1) The dark sector and the SM sector remain thermal- of the dark baryon number is stored in dQN with the ized with each other until they decouple at a remainder carried by free dark baryons. temperature Tdec. Afterward the temperatures of (5) After the phase transition, the cosmological plasma the two sectors evolve independently, decreasing continues to cool and the remaining regions of dark with the adiabatic expansion of the universe. quark-gluon plasma shrink as the thermal pressure (2) As the temperature of the dark sector cools down to a decreases. When the temperature decreases below temperature T slightly below the critical temperature the chemical potential in these regions, they become Tc, the bubbles of dark hadrons start to nucleate dark quark nuggets, supported by degeneracy Fermi out of the dark quark-gluon plasma. The pressure pressure. difference ΔP ¼ B between the two phases drives the B. Dark baryon number accumulates in the quark nuggets Particles in the plasma scatter from the passing bubble wall, and this causes the dark baryon number to accumulate in the unbroken phase. In front of the wall, the baryon number is carried by the dark quarks and antiquarks, which are approximately massless. However, behind the wall the dark baryon number is carried by dark baryons and anti-

baryons, which acquire a mass mBd .IfmBd is much larger than the temperature of the phase transition, Tc,thenthe amount of baryon number entering the bubble will be Boltzmann suppressed. The authors of Ref. [66] have studied the kinematics of a particle scattering from a bubble wall where the particle’s mass changes. By applying that analysis to the problem of FIG. 3. A schematic phase diagram of the dark QCD sector is dQN formation, we find that the dark baryon number will shown here along with the trajectory through phase space during be kinematically blocked from entering the confined-phase the formation of dark quark nuggets. The entire system is initially bubbles if the dark baryon mass is sufficiently large: in the unconfined phase at high temperature and small chemical 2γ ∼ ≃ 3 6 potential (corresponding to the nonzero dark baryon asymmetry). mBd > wpz with pz prms . Tc: ð4:1Þ The system cools due to cosmological expansion, which triggers a first-order phase transition. Some regions of space enter the The factor of 3.6 in the root-mean-square momentum confined phase where the dark baryon asymmetry is eventually γ followspffiffiffiffiffiffiffiffiffiffiffiffiffi from the Fermi-Dirac distribution. Here w ¼ carried by free dark baryons and antibaryons, but most of the dark 2 1= 1 − vw is the wall’s boost factor, and vw is its speed. baryon asymmetry is collected into pockets of space that cool to ’ form dark quark nuggets. If the chemical potential is large, there It is challenging to calculate the wall s speed from first may be exotic phases, similar to the color superconductivity and principles [67]. (See also Ref. [68], which estimates the the color-flavor-locking phase of QCD [1], but we neglect this maximum deflagration velocity allowed by entropy possibility. increase and argues that vw is nonrelativistic.) However,

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TABLE I. Notation used in this section.

Symbol Definition Equation

MdQN Mass of a typical dark quark nugget Eqs. (4.8), (4.11) RdQN Radius of a typical dark quark nugget Eqs. (4.7), (4.10)

NBd;dQN Amount of dark baryon number in a typical dark quark nugget Eq. (4.6) ndQNðtÞ Cosmological number density of dark quark nuggets at time t Eq. (4.2) Ω 2 dQNh Cosmological relic abundance of dark quark nuggets today Eq. (4.9) Dinit Typical inter-nugget separation distance at the phase transition Eq. (4.3) nBd;dQM Density of dark baryon number of the dQM inside of a dQN Eq. (3.5) ρ dQM Energy density of the dQM inside of a dQN Eq. (3.4)

YBd Cosmological yield of dark baryon number (conserved) nHubðtÞ Cosmological density of dark baryon number at time t Bd Nbary Dimension of the quasi-degenerate dark baryon multiplet Eq. (4.5) mBd Mass of the quasi-degenerate dark baryon multiplet fnug ¼ 1 − ffree Fraction of dark baryon number stored in dark quark nuggets Eq. (4.4)

TdðtÞ & TγðtÞ Temperature of the dark and visible sectors at time t ≈ g;dðtÞ gS;dðtÞ Effective number of relativistic dark-sector species at time t ≈ g;γðtÞ gS;γðtÞ Effective number of relativistic visible-sector species at time t Tc ¼ TdðtcÞ Temperature of the dark sector during the phase transition Tγ;c ¼ TγðtcÞ Temperature of the visible sector during the phase transition

due to the strongly coupled nature of the dark QCD transition, ndQNðtcÞ is the cosmological density of dark interactions, we think it is reasonable to expect that quark nuggets at the time of the phase transition, and fnug is particles in the plasma will induce a large drag force on the fraction of the dark baryon number that gets stored in the wall and lead to a nonrelativistic terminal velocity with the dark quark nuggets (leaving a fraction f ¼ 1 − f γ ≈ 1 free nug w . If that is the case, then Eq. (4.1) imposes a weak to be stored in free dark baryons). constraint, m ≳ 7T Bd c. For the model parameters satisfying The cosmological density of the dark baryon number can Hub this constraint, the dark baryon number is kinematically be written as n ¼ YB s where YB is the cosmological Bd d d preferred to stay in the unbroken phase. Otherwise, if dark baryon number yield, and s is the cosmological γw ≫ 1, effectively all particles in the plasma will have entropy density. We take the yield, YBd , as a free parameter enough energy to enter the bubble, and the dark baryon and note for reference that the cosmological yield of the SM number will hardly remain in the unbroken phase. −10 baryon number is measured to be YB ≃ 10 [69]. The 2π2 45 3 entropy density can be written as s ¼ð = ÞgSTγ;c C. Dark quark nuggets: Mass, size, 3 where gS ¼ gS;γ þ gS;dðTc=Tγ;cÞ counts the effective and relic abundance number of relativistic d.o.f. in the plasma at the phase Let us now estimate the typical mass, size, and relic transition. Here, Tγ;c is the temperature of the visible sector abundance of the dark quark nuggets. The notation used in during the phase transition. this section is summarized in Table I. Already in Sec. III We estimate the density of dark quark nuggets at the phase we have studied the dark quark matter that resides inside of a transition, ndQNðtcÞ, by adopting the results of Appendix A. dark quark nugget, and we have calculated its energy density, In the Appendix we study the dark QCD chiral phase tran- ρ dQM, and number density of the dark baryon number, sition using a chiral effective theory. The main result appears nBd;dQM. Now all that remains is to estimate the typical in Eq. (A19), which gives nnucleations, the average number amount of dark baryon number per nugget, NBd;dQN,and density of chiral-broken-phase bubbles that are nucleated then the nugget’s radius and mass are given by ð4π=3Þ× over the course of the phase transition. We estimate that after 3 4π 3 3 ρ the phase transition is completed, there is roughly one nugget R nB ;dQM ¼ NB ;dQN and ð = ÞR dQM ¼ MdQN. dQN d d dQN produced for each nucleation, i.e., n t ≈ n . We assume that all the nuggets have a comparable dQNð cÞ nucleations amount of the dark baryon number, and that this quantity This lets us infer the density of dark quark nuggets at the is approximately conserved from the time of nugget end of the dark QCD phase transition to be formation until today. Thus we can write NBd;dQN ¼ Hub Hub σ˜ −9=2 fnugnB ðtcÞ=ndQNðtcÞ where nB ðtcÞ is the cosmological ≃ 2 1 1014 3 d d ndQNðtcÞ ð . × Þ HðtcÞ : ð4:2Þ density of the dark baryon number at the time of the phase 0.1

055047-9 YANG BAI, ANDREW J. LONG, and SIDA LU PHYS. REV. D 99, 055047 (2019) 8 2−1 ! 2−2 ! We have defined the dimensionless parameter σ˜ ¼ σ= < ðNfþNd= Þ ðNfþNd= Þ −1 ! −2 ! 2 1 ! 2 ! ;Nd is even 2=3 1=3 ðNf Þ ðNf Þ ðNd= þ Þ ðNd= Þ ðB Tc Þ, and we have introduced σ, which represents N ¼ : bary : 2 N N =2−1=2 ! N N =2−5=2 ! the surface tension of a critical bubble at the time of ð fþ d Þ ð fþ d Þ ;Nis odd ðNf−1Þ!ðNf−2Þ!ðNd=2þ3=2Þ!ðNd=2−1=2Þ! d nucleation; a larger value of σ implies less efficient bubble nucleation, fewer nucleation sites, and more dark baryon ð4:5Þ numbers per nugget. The Hubble parameter is given by 2 2 2 4 Taking N N 3 and m =T 10 gives f ≃ 3M Hðt Þ ¼ðπ =30Þg ðt ÞTγ where g ðt Þ¼g γ þ d ¼ f ¼ Bd c ¼ nug pl c c ;c c ; 99 2% ≃ 0 8% 4 . and ffree . , meaning that most of the dark g;d½TdðtcÞ=Tγ;c . The relation in Eq. (4.2) reveals that 14 −9=2 baryon number is stored in the dark quark nuggets. there are typically ∼10 ðσ˜=0.1Þ dark quark nuggets per Hub By combining the formulas for nB ðtcÞ and ndQNðtcÞ, Hubble volume, regardless of the temperature of the con- d we estimate the amount of dark baryon number inside of a fining phase transition. The typical internugget separation −1=3 dark quark nugget to be distance, Dinit, is then estimated as Dinit ¼ n to obtain dQN f nHubðt Þ nug Bd c 35 fnug YBd   NB ≈ ≃ 2 6 10 −1 2 −2 3 2 d;dQN ð . × Þ −9 g ðt Þ = Tγ σ˜ = n ðt Þ 1 10 D ≃ 77 c ;c ; dQN c init ð cmÞ −3 9 2 10 0.1 GeV 0.1 Tγ σ˜ = × ;c ; ð4:6Þ ð4:3Þ 0.1 GeV 0.1

≈ ≃ 10 where we have used gS g . Here we have taken a and for comparison the Hubble radius is d ≃4.6×106 cm. 1 H fiducial value of fnug ¼ , which corresponds to putting all We estimate fnug as follows. If the bubble wall expands of the dark baryon number into the dark quark nuggets (and sufficiently slowly, then thermal and chemical equilibrium leaving no dark baryon number for free dark baryons), but is maintained at the phase boundary [2]. It is energetically more generally the parameter fnug can be related to the preferable for the dark baryon number to remain in the confinement scale and phase transition temperature unconfined phase where the dark quarks are massless, through Eq. (4.4). rather than enter the confined phase where the dark baryons Using the estimate for NB , it is now straightforward to ≫ d acquire a mass mBd Tc. From these considerations (for 4 estimate the radius and the mass of a typical dark quark more details see Ref. [32]) one can estimate the fraction of nugget. The radius of the dark quark nugget satisfies dark baryon number that goes into the dark quark nuggets 3 ð4π=3ÞR nB ¼ NB where the density of the dark to be dQN d;dQM d baryon number in the dark quark matter state is given by Eq. (3.5). Solving for R gives the typical radius of a pffiffiffiffiffiffi dQN 2π 3=2 dark quark nugget to be NbaryNd mBd −m =T f ¼ 1 − f ≈ 1 − e Bd c : nug free N 3ζ 3 T f ð Þ c 1 4 N = −1=4 ≃ 0 073 d B ð4:4Þ RdQN ð . cmÞ 1 12 4 N = ð0.1 GeVÞ f 1=3 1=3 −1 σ˜ 3=2 fnug YBd Tγ;c Here Nbary represents the number of quasidegenerate × : 1 10−9 0.1 GeV 0.1 baryons with mass mBd in the confined phase (behind the bubble wall) for the lowest-spin and color-singlet state ð4:7Þ as a representation of the unbroken flavor symmetry 5 SUðNfÞV. Using a simple group theory calculation, Similarly the mass of the dark quark nugget satisfies 4π 3 3 ρ one has ð = ÞRdQN dQM ¼ MdQN where the energy density of the dark quark matter is given by Eq. (3.4). This lets us 4Note that there is a typo in Eq. (3.15) of the journal version of estimate the typical nugget mass as Ref. [32]; the value of r is too large by a factor of 8. Upon correcting the error, the quark nugget relic abundance, 3=4 Ω ∼ 1 N B 1=4 f QN =r, is increased by a factor of 8, and Fig. 5 of Ref. [32] ≃ 1 5 1011 d nug MdQN ð . × gÞ 1 4 4 is modified accordingly. N = ð0.1 GeVÞ 1 5These expressions are equal to the dimension of the repre- f −3 9=2 sentation of the baryon multiplet. The dimension is calculated YB Tγ;c σ˜ 2 × d : ð4:8Þ with the aid of a Young tableau having two rows of Nd= boxes 10−9 0.1 GeV 0.1 for even Nd, or two rows with ðNd þ 1Þ=2 and ðNd − 1Þ=2 boxes for odd N [14]. For example, N ¼ 8 for N ¼ N ¼ 3, d bary d f 11 −23 reproducing the SM baryon octet. Recall that 1 × 10 g ≃ 5 × 10 M⊙.

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FIG. 4. The typical mass (left panel) and radius (right panel) of a dark quark nugget are shown here as functions of the critical temperature of the confining phase transition. We assume TγðtcÞ¼TdðtcÞ ≡ Tc, but if the dark sector is colder, then the mass and radius 2=3 1=3 are reduced according to Eqs. (4.10) and (4.11). The dimensionless parameter σ˜ ≡ σ=ðB Tc Þ measures the surface tension of the confined-phase bubbles at the time of formation, which affects the initial dQN density through Eq. (4.2). The dark quark nuggets are assumed to occupy the majority of dark matter energy density. If the scale of the confining phase transition is larger than ∼10 TeV, then free dark baryons overclose the universe; see the discussion in Sec. VB. Also shown is the Subaru-HSC microlensing constraint after taking the wave effects into account [71,72].

Finally we estimate the relic abundance of dark quark [5,70] where the dark matter’s mass and asymmetry are Ω ρ ’ nuggets in the universe today. Let dQN ¼ dQNðt0Þ= comparable to the baryon s mass and asymmetry. 2 2 3 ρ Solving Eq. (4.9) for YB lets us write Eqs. (4.7) and ð MplH0Þ where dQNðt0Þ is the cosmological energy d density of dark quark nuggets in the universe today and (4.8) as H0 ¼ 100h km=sec=Mpc with h ≃ 0.674 [69]. Since the   Ω h2 1=3 B −1=3 dark quark nuggets are nonrelativistic, we can write R ≃ ð0.081 cmÞ dQN ρ dQN 0.12 ð0.1 GeVÞ4 dQNðt0Þ¼MdQNndQNðt0Þ where ndQNðt0Þ is their cosmo- −1 3=2 logical number density today. If the nuggets do not merge Tγ;c σ˜ or evaporate (see Sec. VD), then their comoving number × 0 1 0 1 ; ð4:10Þ 3 . GeV . density, ndQNðtÞaðtÞ , is conserved; here aðtÞ is the 2 −3 9 2 Friedmann-Robertson-Walker (FRW) scale factor at Ω h Tγ σ˜ = M ≃ ð2.1 × 1011 gÞ dQN ;c time t. While the universe expands adiabatically, the dQN 0.12 0.1 GeV 0.1 comoving entropy density, sðtÞaðtÞ3, is conserved. Combining these formulas gives the relic abundance of ð4:11Þ dark quark nuggets today to be In Fig. 4 we show the dark quark nugget’s mass and radius for the interesting range of phase transition temperatures 3 1 M n ðt Þ g ðt0ÞTγðt0Þ from Tγ;c ¼ keV to 1 PeV. Ω h2 dQN dQN c S dQN ¼ 3 2 2 3 MplðH0=hÞ gSðtcÞTγ;c 3 4 V. SIGNATURES AND TESTABLE N = B 1=4 ≃ 0 090 d PREDICTIONS ð . Þ 1 4 4 N = ð0.1 GeVÞ f In this section we discuss various observational signa- tures of the theory that we have presented above. Some of fnug YBd × 1 10−9 : ð4:9Þ these observables directly test for the presence of dark quark nuggets in our universe while other indirectly probe the dark QCD model. For reference, the relic abundance of dark matter is Ω 2 ≃ 0 12 measured to be DMh . [69]. Thus the nuggets A. Dark radiation Ω 2 ≃ 0 12 can make up all of the dark matter ( dQNh . )if In addition to a dark matter candidate, the dark QCD the differential vacuum pressure is at the nuclear energy model also admits a dark radiation candidate. The presence 4 scale, B ≃ ð0.1 GeVÞ , and if the dark baryon asymmetry is of dark radiation in the universe is felt through its ≃ 10−9 “ around YBd . This result illustrates the same coinci- gravitational influence, particularly during the formation dence” that comes up in models of asymmetric dark matter of the cosmic microwave background (CMB). In this

055047-11 YANG BAI, ANDREW J. LONG, and SIDA LU PHYS. REV. D 99, 055047 (2019) 11 4=3 4 4 section we discuss how CMB observations lead to con- Δ TdðtcmbÞ Neff ¼ 4 7 g;dðtcmbÞ 4 : ð5:4Þ straints on the dark QCD model and its dark radiation. TγðtcmbÞ In general we can write the energy density of particles in the dark sector as In general the dark and visible sectors may have different Δ temperatures. The parameter Neff is already strongly constrained [69], due to the absence of evidence for dark ρ ¼ ρ þ ρ ; ð5:1Þ d d;rad d;mat radiation at the CMB epoch, and next-generation observa- ρ tions [74] are projected to improve the sensitivity by an where d;rad is the energy density of (relativistic) dark ρ order of magnitude: radiation and d;mat is the energy density of (nonrelativistic) Δ 0 2 95% 2018 dark matter. The various particle species in the dark Neff < . at C:L:; current limit-Planck ; — sector quark and gluons in the unconfined phase and σðΔN Þ¼0.03; projected sensitivity-CMB-S4: mesons and baryons in the confined phase—are distributed eff between radiation and matter. ð5:5Þ In the following discussion we consider the model with The presence of dark radiation at the epoch of nucleo- 0 Δ 1 mi ¼ in Eq. (2.1), which corresponds to massless dark synthesis is more weakly constrained, Neff < at quarks in the unconfined phase and massless dark mesons 95% C.L. [75]. 6 Δ (Goldstone bosons) in the confined phase. If all species of To make a prediction for Neff we must estimate Td=Tγ, particles in the dark sector are in thermal equilibrium at a but this ratio depends on the history of interactions between common temperature Td, then the energy densities in the the dark and visible sectors. Without loss of generality, we dark sector are given by7 identify three scenarios. (1) The dark and visible sectors are thermalized at the π2 CMB epoch. If the dark sector remains in thermal ρ T4; d;rad ¼ 30g;d d equilibrium with the visible sector at the CMB 8 >2 2 −1 7 4 epoch, then we take Td ¼ Tγ in Eq. (5.4) to evaluate > ðNd Þþ8ð NdNfÞ; unconfined phase Δ < Neff. We can distinguish two cases, either (1a) the  2 −1 ≥ 3 g ¼ ðN Þ for Nd ; confined phase; dark sector is still in the unconfined phase at tcmb or ;d > f : 2 2 − −1 2 (1b) it is in the confined phase. For case (1a) we find ð Nf Nf Þ for Nd ¼ ; confined phase Δ ≫ 1 ≥ 2 ≥ 1 Neff for any Nd and Nf . For case Δ ≫ 1 ≥ 2 ≥ 2 ð5:2Þ (1b) we have Neff for any Nd and Nf , Δ 0 1 but Neff ¼ if Nf ¼ , because there is no Gold-  1 0; unconfined phase stone boson. Nevertheless, a model with Nf ¼ is ρ d;mat ¼ ρ ρ ρ : ð5:3Þ not expected to have a first-order phase transition B þ B þ dQN; confined phase d d [46] or allow for the formation of dQNs. In light of Δ the constraints on Neff in Eq. (5.5), this first The first equality also defines the effective number of scenario is not viable. relativistic species in the dark sector, denoted by g;d. The (2) The dark and visible sectors decouple prior to the ρ Δ terms in d;mat count the energy density of nonrelativistic CMB epoch. The Neff constraints are relaxed if species carrying a dark baryon number, which includes the dark sector decoupled from the Standard Model dark baryons, dark antibaryons, and dark quark nuggets. at a time tdec

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Δ FIG. 5. The predicted dark radiation, parametrized by Neff, is shown for the three cases depending on whether the dark sector is in the unconfined or the confined phase at the time when it thermally decouples from the visible sector (tdec) and the time when the CMB is generated (tcmb). Observational constraints (5.5) strongly prefer case (2c) in which the confining phase transition occurs while the dark 106 75 and visible sectors are still in thermal equilibrium. We assume that decoupling occurs before the electroweak epoch with g;γ ¼ . , Δ and otherwise Neff is larger according to Eq. (5.8). We also assume massless dark mesons, but if the dark mesons are instead allowed to Δ 1 decay to SM particles before tcmb, then the predicted Neff is smaller. For Nf ¼ there is no dark radiation for cases (2b) and (2c).

Here g;dðtÞ denotes the effective number of relativistic remains in the confined phase at the CMB epoch. These species in the dark sector at time t, and it is given by cases are illustrated in Fig. 5. For each of these three cases, Δ Eq. (5.2). Similarly g;γðtÞ denotes the effective number of the predicted Neff is given by relativistic species in the visible sector (Standard Model 8 > 0 027 2 2 − 1 7 4 2 d.o.f.). Assuming no new light d.o.f. beyond the Standard > . ½ ðNd Þþ8 ð NdNfÞ; ð aÞ > 8 Model and the dark QCD, then this factor is as large as > > 2 N2 −1 7 4N N 4=3 > > ½ ð d Þþ8ð d fÞ 106 75 ≳ 160 > < 0.027 2 1 3 ;N≥ 3 g;γ ¼ . for Tγ GeV before electroweak sym- > ½N −1 = d 3 91 ≲ < f ; ð2bÞ metry breaking, and it decreases to g;γ ¼ . for Tγ > 2 2 −1 7 4 4=3 Δ > ½ ðNd Þþ8ð NdNfÞ 0 2 Neff ¼ : 0.027 2 1 3 ;N 2 : . MeV after neutrino scattering and electron-positron > ½2N −N −1 = d ¼ > f f annihilations have frozen out. At the time of decoupling > ( > 0 027 2 − 1 ≥ 3 T ðt Þ¼Tγðt Þ, but as particle species go out of > . ½Nf ;Nd d dec dec > ; ð2cÞ equilibrium the temperatures will begin to differ. Solving : 0 027 2 2 − − 1 2 . ½ Nf Nf ;Nd ¼ Eq. (5.6) for tdec

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¯ 9 antibaryons (Bd), and the dark quark nuggets (dQN). In this section we estimate the relic abundances of the dark baryons and antibaryons. We assume that the dark baryon number is conserved, which forbids the dark baryons/ antibaryons from decaying, and instead they contribute to the dark matter. The dark baryons and antibaryons are kept in thermal equilibrium with the dark mesons, such as the dark pions ¯ πd, through annihilation reactions such as Bd þ Bd ↔ πd þ πd and multimeson final states. Let hσvi denote the thermally averaged cross section for this annihilation reaction. At temperatures below the mass of the dark ≪ baryon/antibaryon, T mBd , the thermally averaged cross section is well approximated by FIG. 6. Here we show the predicted dark radiation, parame- Δ trized by Neff, for case (3) in which the dark sector is never 1 GeV 2 thermalized with the SM and the temperature ratio, Td=Tγ,is hσvi ≈ ð50 mb · cÞ ; ð5:11Þ determined by initial conditions. Several values of N and N are d f mBd shown, and we consider two cases depending on whether the dark ≲ 3 sector is confined at the CMB epoch. Provided that Td Tγ= the where we have used the low-βpp¯ annihilation rates [80]. dark radiation is small enough to evade existing limits, and if σ ≃ 130 2 This is roughly h vi =mB . T ≳ Tγ=10, then the next-generation CMB-S4 program may d d If the dark baryon asymmetry is negligibly small, then uncover evidence for dark radiation. Ω the relic abundances of dark baryons and antibaryons, Bd Ω ¯ and Bd , are controlled by thermal freeze-out, which occurs in order to generate quark nuggets while avoiding constraints when the plasma temperature in the dark sector is approx- imately T ðt Þ ≃ m =20. The standard freeze-out calcu- from dark radiation. Alternatively, it may be possible to open d fo Bd up the parameter space by lifting the dark meson mass and lation [73] gives the relic abundances to be allowing it to decay to Standard Model particles before the −1 2 2 2 hσvi mB CMB epoch. Ω h ¼ Ω ¯ h ≃ ð0.052Þ d 3. The dark and visible sectors never thermalize. If the Bd Bd 130m−2 200 TeV Bd dark sector never reaches thermal equilibrium with the −1=2 mBd =TdðtfoÞ TdðtfoÞ g Standard Model, and if the freeze-in population is negli- × 20 100 : ð5:12Þ TγðtfoÞ gible (see also Ref. [76]), then the ratio Td=Tγ is controlled by the physics that populated the dark and visible sectors ≤ 1 initially. For instance, if both sectors are populated directly The factor of TdðtfoÞ=TγðfoÞ arises because the dark from decay of the inflaton field ϕ after cosmological and visible sectors may be thermally decoupled at the time inflation has evacuated the observable universe [77–79], of dark baryon freeze-out. However, as we have already discussed in Sec. IV C, a dark-baryon-number asymmetry then Td=Tγ is proportional to a ratio of branching fractions is required for the formation of dark quark nuggets, and this BFðϕ → darkÞ=BFðϕ → SMÞ. The ratio Td=Tγ can be made arbitrarily small in a model in which the inflaton asymmetry may affect the relic abundance of free dark decays predominantly to the visible sector, and the con- baryons and antibaryons as well (as we encounter in models Δ of asymmetric dark matter [5,70]). Recall from Eq. (4.4) straints from Neff can be avoided. In Fig. 6, we show the predicted dark radiation as a function of the temperature that the fraction of the dark baryon number carried by the 1 − ≪ 1 free dark baryons is ffreeYB where ffree ¼ fnug is ratio Td=Tγ. Even a small splitting, Td=Tγ ∼ 1=3, is enough d to evade existing constraints, but still provide a target for desirable for the formation of nuggets. If the dark baryon next-generation surveys. However, if the two sectors do not asymmetry is large enough, then the relic abundances are thermalize, then the dark and visible baryon asymmetries given by may either arise directly from the inflaton decay (if it is CP- and baryon-number violating), or baryogenesis may occur 9The free dark baryons may undergo dark nucleosynthesis to separately in the two sectors. form dark nuclei, and this idea has been explored recently by several authors [34,36,38,39,41]. Since the dark baryons typically make up a subdominant population of the dark matter, the total B. Free dark baryons and antibaryons dark matter relic abundance is approximately not affected. Also, the dark baryon number for the dark nucleus coagulation is After the confining phase transition occurs, the dark dramatically smaller than the one in nuggets, and their detection baryon number is carried by the dark baryons (Bd), the dark potential could be dramatically different from nuggets.

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1=4 free dark baryons ΩB þ Ω ¯ N ∶ d Bd ≃ ð0.031Þ f Ω 3=4 dark quark nuggets dQN N d m =B1=4 f =f × Bd free nug : 10 0.01 ð5:15Þ Note that the free dark baryons are a subdominant pop- ulation of the dark matter provided that  3 4 −1 N = B1=4 1 − 1 3 3 d fnug ¼ ffree > þ . 1=4 ; ð5:16Þ Nf mBd 0 636 3 which evaluates to fnug > . for Nd ¼ Nf ¼ and 10 1=4 FIG. 7. The relic abundances of free dark baryons and anti- mBd ¼ B . An expression for fnug appears in Eq. (4.4), baryons are shown here in comparison with the relic abundance and by comparing with the limit above, we find that free of dark quark nuggets. Note that the curve for free dark baryons dark baryons typically make up a subdominant component 1 − (5.13) scales as ffree ¼ fnug, whereas the curve for dark quark of the dark matter, which is predominantly composed of 0 99 nuggets scales as fnug; we have taken fnug ¼ . for illustration, dark quark nuggets. but this value may vary greatly across models. For the free dark Since the free dark baryons and antibaryons are very baryon thermal relic abundance, we have used TdðtfoÞ¼TγðtfoÞ. abundant, it may be possible to detect their presence with direct detection experiments on Earth. Their gravitational m 1 − f YB influence is expected to be exceedingly weak, and therefore Ω h2 ≃ ð0.14Þ Bd nug d Bd 50 GeV 0.01 10−9 an additional, direct coupling between the dark sector and 2 the SM is required. The nature of this interaction depends and Ω ¯ h ≈ 0; 5:13 Bd ð Þ on (as yet unspecified) UV physics. As an example we will ψ¯ γ ψ ¯ γμ Λ2 use the vector-vector interactions, d;L μ d;LdR dR= UV, which could be generated by integrating out a heavy scalar which is insensitive to hσvi.IfYB < 0, then the expres- d coupling to both a dark quark and an ordinary quark and sions for Ω and Ω ¯ are exchanged. For sure, since dark Bd Bd using the Fierz transformation. Then the matrix element quark nuggets have the energy density with a factor of 1 − for spin-independent (SI) scattering of a dark baryon around fnug=ð fnugÞ larger than that from free dark M 0 0 off a proton or neutron is written as p;n ¼ Jψd Jp;n= baryons, the specific parameter choice of m ¼ 50 GeV 2 0 0 0 Bd 4Λ ψ¯ γ ψ ≈ −9 ð UVÞ where Jψd ¼hBdj d djBdi Nd and Jp;n ¼ and YB ¼ 10 will have dark matter overclose the ¯ 0 d hp; njdγ djp; ni ≈ 1; 2. For a Fermionic dark baryon, the universe. SI scattering cross section for a neutron is The relic abundance of free dark baryons is shown in 2 2 −4 2 Fig. 7 as a function of the dark baryon mass scale and N μ − Λ N σSI d Bd n ≃ 2 5 10−44 2 UV d − ¼ 4 ð . × cm Þ ; the dark baryon asymmetry. Requiring the relic abundance Bd n 4πΛ 10 TeV 3 of dark baryons to be smaller than the observed density of UV Ω 2 ≃ 0 12 5:17 dark matter, DMh . , yields an upper bound [81] ð Þ of m ≲ 200 TeV. Recall from Eq. (4.1) that we need Bd where μ − ¼ m m =ðm þ m Þ ≈ m is the reduced T ≲ m =7 Bd n Bd n Bd n n c Bd to ensure that nuggets are able to form, and ≫ mass for mBd mn. Recent null results from the one therefore the overclosure condition implies an upper bound tonne-year exposure of XENON1T [28] imply an upper on the dark-sector temperature at the phase transition: bound on the dark baryon scattering cross section at the level of σSI ≲ ð4.1 × 10−47 cm2Þðm =30 GeVÞ½Ω = Bd−n Bd dQN Ω Ω ¯ < Ω ⇒ T ≲ 30 : : Ω Ω ¯ Ω Bd þ Bd DM c TeV ð5 14Þ ð Bd þ Bd Þ, where the factor arises because dark baryons are only a subdominant component of the dark matter. Thus the nonobservation of free dark baryons by However, the temperature in the dark sector may be smaller ≤ XENON1T imposes than the temperature in the visible sector, Tc Tγ;c, which affects the corresponding lower bounds on the dQN mass N 5=16 N 1=16 B −1=16 Λ ≳ ð42 TeVÞ d f and radius through Eqs. (4.10) and (4.11). UV 3 3 ð0.1 GeVÞ4 For comparison Fig. 7 also shows the relic abundance of 1=4 ≲ 200 ffree=fnug dark quark nuggets (4.9).FormBd TeV the relative × : ð5:18Þ abundances are given by 0.01

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This limit also means that if the cutoff scale is not too far from 40 TeV, the future results from direct detection experiments could have a chance to discover the dark baryon.

C. Stochastic gravitational wave background It is well known that cosmological phase transitions can generate a stochastic background of gravitational waves (GW) if the transition is first order [82]. First-order phase transitions in dark sectors have also been studied specifi- cally; see, e.g., Refs. [83–90]. In general three processes contribute to the stochastic GW background during a first- order phase transition: the collision of the scalar field bubbles, sound waves in the plasma, and the magneto- FIG. 8. We show the GW spectrum that is predicted to arise hydrodynamic (MHD) turbulence. The total GW spectrum from a first-order confining phase transition in dQCD along with is then well approximated by the linear sum of these three the projected sensitivities of various future GW interferometer contributions: and pulsar timing array experiments [92–100]. We vary the phase transition temperature from Tγ;c ¼ 10 keV to 100 TeV, and we Ω 2 ≈ Ω 2 Ω 2 Ω 2 4 3 gwh ϕh þ swh þ turbh : ð5:19Þ show ðα; β=HÞ¼ð0.1; 10 Þ (solid lines) and ð1; 10 Þ (dashed lines). The interferometer sensitivities are calculated using 2 3 2 1=2 The spectra of these three sources are determined by several Ωgw ¼ð2π f =3H0ÞSn where Sn is the noise amplitude spec- key parameters from the bubble nucleation process. The tral density; often the power-law integrated sensitivity is shown parameter β−1 measures the duration of the phase tran- instead, which can be 1 or 2 orders of magnitude stronger. sition, and it is customary to write the dimensionless ratio β =H where H is the Hubble parameter at the time when 1 6 1 β z Tγ g = GWs are generated; see also Eq. (A12). We assume that the 8 9μ p ;c fsw ¼ð . HzÞ 10 100 100 ; universe is radiation dominated during the phase transition vw H GeV with the dominant energy component having a temperature ð5:21Þ ≈ α T Tγ;c. The dimensionless parameter measures the ≃ 10 released vacuum energy as compared to the radiation where zp is a simulation-derived factor and g is the energy of the plasma after the phase transition is completed; effective number of relativistic species. Using Eqs. (5.3) α 4 see also Eq. (A20). The parameter also controls the and (5.7) we can write g ¼ g;γ þ g;dðTd=TγÞ . The efficiency with which energy is transferred into the bulk κ efficiency coefficient f is in general a function of vw motion of the fluid; this efficiency is parametrized by κ , α κ α f and , and a numerical fit of fðvw; Þ is done in Ref. [67] and an explicit expression appears below. The parameter vw for four different scenarios of wall velocity. In our measures the speed of the bubble wall in the rest frame of calculation we use the plasma. For bubbles that reach a terminal velocity (rather than α2=5 “ ” κ ¼ ; ð5:22Þ running away ), the contribution to gravitational waves f 0.017 þð0.997 þ αÞ2=5 from the bubble collisions themselves has been shown by recent numeric study to be negligible [91]. The GW signal which corresponds to a subsonic wall velocity. from MHD turbulence also turns out to be negligible for the Using the formulas above we have calculated the parameter range we are considering. Therefore we only predicted spectrum of gravitational wave radiation, and present the formula for the sound wave contribution, which we present our results in Fig. 8. For comparison we also fits to [91] show the projected sensitivities of various GW interferom- eter observatories and several pulsar timing array experi-ffiffiffi −1=3 β −1 3 2 p 2 −6 g 2 ¯ 4 f Ω 1 3 Ω h ¼ð8.5 × 10 Þ Γ U v ments. In calculating gwh we fix vw ¼ cs ¼ = ,we sw 100 f H w f ≡ ≡ sw assume Tγ;c TγðtcÞ¼TdðtcÞ Tc, we vary Tc from 7 7=2 10 keV to 100 TeV (corresponding to the different colors), × : ð5:20Þ 4 3 f=f 2 and we choose two combinations of α and β: ðα; β=HÞ¼ þ ð swÞ 4 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0.1; 10 Þ (solid lines) and ð1; 10 Þ (dashed lines). We also Γ ≈ 4 3 ¯ ≈ 3 4 κ α 3 Here = is the adiabatic index, and Uf ð = Þ f choose Nd ¼ Nf ¼ , which determines g ¼ g;γ þ g;d 3 8 is the root-mean-squared fluid velocity. The peak fre- through Eq. (5.2) to be g ¼ . , 13.0, 154.25, and 154.25 10 quency, fsw,isgivenby for Tc ¼ keV, 100 MeV, 100 GeV, and 100 TeV.

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α β A robust calculation of and in dQCD is challenging, Similarly we can define a distance dyr such that Γ 1 −1 since the theory becomes strongly coupled at the phase collide ¼ yr , which gives transition. Using a low-energy chiral effective descrip- 2=9 −4=3 2 tion of the phase transition in Appendix, we find that B Tγ;c σ˜ 4 d ≃ ð4.0 pcÞ : ðα; β=HÞ¼ð0.1; 10 Þ may be typical values; see Fig. 12. yr ð0.1 GeVÞ4 0.1 GeV 0.1 We also present the GW spectrum for ðα; β=HÞ¼ð1; 103Þ, ð5:24Þ which is more favorable for detection, to allow for the possibility that the transition is more strongly first order We estimate the amount of energy liberated during a than the chiral effective theory would suggest. If the collision as 2 × M v2 =2, which is just the kinetic confinement scale is on the lower end, corresponding to dQN dQN energy of the two incident dQNs. Suppose that a fraction Tc ∼ 10 keV, then the GW signal will be probed by pulsar timing array observations such as EPTA [92], IPTA [93], frad of this energy goes into visible, SM radiation. If Δ and SKA [94]. Alternatively if T ∼ 100 MeV to 100 GeV, the collision takes a time t to complete, then the c ≈ then the GW signal could be accessible to future space- corresponding power output is estimated as Pcollide 2 Δ based gravitational wave interferometer experiments such fradMdQNvdQN= t, which evaluates to as LISA [95], Taiji [96,97], DECIGO [98], BBO [98], Δ −1 −3 and ET [99]. −11 frad t Tγ;c P ≃ ð4.8 × 10 L⊙Þ collide 0.01 10 s 0.1 GeV D. Cosmic rays from colliding and σ˜ 9=2 × ; ð5:25Þ merging dark quark nuggets 0.1 Let us now turn our attention to astroparticle probes of ≃ 3 8 1026 dark quark nuggets in the universe today. If a pair of dark where L⊙ . × W is the luminosity of the Sun. To quark nuggets were to collide today, some fraction of the assess whether a telescope on Earth could detect this δΩ 1 initial energy would be liberated as dark radiation (mostly radiation, we assume an angular resolution of ¼ °× 1 π 180 2 dark mesons), and a new dQN would be formed from the ° ¼ð = Þ sr. Then the frequency-weighted spectral ν 2 δΩ merger. If the dark sector has a direct coupling to the density is estimated as Iν ¼ Pcollide=ðdyr Þ, which Standard Model, the dark mesons may decay into ultrahigh evaluates to energy SM particles, and the observation of these cosmic −4=9 rays thereby provides a new channel for the indirect −15 W frad B νIν ≃ 4.1 × 10 detection of dark quark nuggets. m2 sr 0.01 ð0.1 GeVÞ4 −1 3 1 2 Tγ = σ˜ = × ;c : ð5:26Þ 1. Collisions of dark quark nuggets near the Sun 0.1 GeV 0.1 Let us begin by estimating the rate of dQN collisions nearby to the Sun. Here we assume that dark quark For comparison, the observed cosmic backgrounds of ν 10−10 −2 −1 nuggets make up all of the dark matter, ρ ≈ ρ ≃ x rays and gamma rays run from Iν ¼ Wm sr dQN DM 10 ν 10−13 −2 −1 0.3 GeV=cm3, and that all nuggets have the same mass and at Eγ ¼ keV down to Iν ¼ Wm sr at Eγ ¼ 10 GeV [101]. If a dQN collision produces photons with radius: MdQN given by Eq. (4.11) and RdQN given by Eq. (4.10). The rate of dQN collisions per unit volume is energies in this range, then the signal could be detectable 1=4 γ ≈ 2 for B ∼ Tγ;c ≲ 10 MeV. This is represented in Fig. 9 estimated as collide ndQNvdQNAdQN where ndQN ¼ ρ where we plot νIν for different phase transition temper- DM=MdQN is the number density of dQNs near the Sun, ≃ 10−3 atures. The radiation energy is related to the Fermi vdQN ¼ vDM is the typical speed of a dQN in the π 2 momentum of the dark quark matter or the phase transition Milky Way, and AdQN ¼ RdQN is the geometrical cross temperature, Tc. This is similar to a neutron-star merge section of a dark quark nugget. (The gravitational enhance- event, where semirelativistic neutrons collide with each ment to AdQN is negligible.) Now consider a spherical other to generate energetic photons up to the neutron’s region of radius d centered at the Sun. The rate of dQN kinetic energy. For T ≳ 10 keV, dQN collisions will Γ ≈ c collisions within this region is roughly collideðdÞ produce energetic x rays and gamma rays, which provide γ 4π 3 3 collide d = , which evaluates to transient signals that telescopes can seek out. −2 3 4 B = Tγ Γ ≃ ð16 yr−1Þ ;c 2. Visible radiation from dQN collisions collide ð0.1 GeVÞ4 0.1 GeV We expect that the collisions of dark quark nuggets will σ˜ −6 d 3 × : ð5:23Þ release an enormous number of dark mesons, which may 0.1 10 pc decay into SM-sector particles that could be detected from

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gravitationally bound system, which allows them to merge after radiating away excess kinetic energy. A pair of dark quark nuggets can form a gravitationally bound binary system if their relative speed is smaller than their escape speed, vrel ð . MeVÞð UV=TeVÞ ; here we have taken Λ 10 Λ Λ4 Λ a charged particle moving in a circle of radius r is mπd ¼ d= , fπd ¼ d, B ¼ d, and Tγ;c ¼ d. Similar considerations can yield an estimate of frad in Eq. (5.26). 10 More generally, the merger time for a pair of masses m1 and 5 256 −3 4 1 − 2 7=2 m2 is given by tGW ¼ð = ÞGN D ð e Þ ½m1m2ðm1 þ 3. Mergers of gravitationally bound dQN systems −1 m2Þ if the orbital radius and eccentricity are D and e, 0 The preceding calculation only accounts for head-on respectively. To obtain Eq. (5.31) we take e ¼ , D ¼ Dinit, collisions, but a pair of dark quark nuggets may also form a and m1 ¼ m2 ¼ MdQN.

055047-18 DARK QUARK NUGGETS PHYS. REV. D 99, 055047 (2019) 4 2 3 −9=2 P ¼ 2αγ =ð3r Þ where γ is the boost factor. This Tγ;c σ˜ em F L2Δt ≃ ð2.5 × 10−15Þ motivates us to estimate the two-dark-gluon-radiation dQN 0.1 GeV 0.1 ∼ α2 2 power as P2dg d=r for nonrelativistic motion. Using L 2 Δt this expression in Eq. (5.30) gives dr=dt ∼ α2= G M2 , × : ð5:34Þ d ð N dQNÞ 10 m 1 yr τ ∼ and we estimate the merger timescale as merge 2 α2 1 F 2Δ DinitGNMdQN= d, which gives Imposing < dQNL t leads to a lower bound on the confinement temperature, 21 2 −8 −2 σ˜ = Tγ α τ ≃ 2 3 1023 ;c d merge ð . × sÞ : σ˜ 3=2 −2=3 Δ −1=3 0.1 0.1 GeV 1 7 4 L t Tγ;c > ð . TeVÞ 0 1 10 1 : ð5:32Þ . m yr ð5:35Þ This merger timescale is longer than the age of the universe today for From Eq. (4.11) we recall that Tγ;c > 10 TeV implies 1 1020 ≃ 2 10−4 10 ≲ MdQN < × GeV × g. For TeV Tγ;c σ˜ 21=16 α −1=4 merge d the flux of dQNs through a terrestrial detector can be Tγ;c

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Regarding directions for future work, there are several places at which our analysis could be extended and our calculations could be refined. (1) We have taken the dark baryon asymmetry to be a free parameter, which may differ from the baryon asymmetry in the visible sector, and it would be useful to investigate how these asymmetries are generated initially in the early universe. (2) While the dark and visible sectors may be thermalized in the early universe, this scenario is becoming tightly constrained by CMB limits on dark radiation. We also consider a scenario in which the two sectors are thermally decoupled, and it would be interesting to study how the two sectors are populated and what interactions control their relative temperatures, which we have taken as a free parameter. (3) We have argued that dQN mergers may be frequent for an intermediate mass range, and it would be very interest- ing to study the effect of these mergers on the dQN mass distribution and the associated observables. (4) Our analy- sis of the observational prospects for colliding dQNs in the Milky Way halo compares the predicted luminosity against the observed diffuse background, but one would like to explore how these transient signals could appear in a FIG. 10. The predictions and signatures of dark quark nugget specific detector. (5) Finally, the QCD-like gauge theory dark matter are summarized here. The predicted dQN mass and studied in this paper provides just one example in which radius fall within a couple decades of the brown line (depending macroscopic dark matter can arise from a first-order phase on specic choices for model parameters; see also Fig. 4). Very transition in the early universe. It is worthwhile to explore high-mass nuggets are excluded by searches for microlensing, similar early-universe relics that could be produced in other and we do not expect very low-mass nuggets, because the dQCD (supersymmetric) gauge theories or even nongauge theo- model in which they arise also predicts a population of free dark baryons in excess of the dark matter relic abundance. The rst ries. Overall, we trust that the theory and phenomenology order phase transition, which gives rise to the dQNs, creates a of dark quark nuggets will provide a rich research program stochastic background of gravitational waves that can be probed in the era of macroscopic dark matter. by GW interferometry and pulsar timing array observations. If the theory also admits a direct coupling between the dark and visible ACKNOWLEDGMENTS sectors, then low-mass nuggets can be probed in laboratories on Earth, while the collisions of high-mass nuggets in the Milky Way We thank Thomas Appelquist, Jonathan Feng, Patrick halo could be probed through cosmic ray observations. Fox, David Weir, and Thomas DeGrand for discussions. The work of Y.B. and S. L. is supported by the U. S. Department Δ ≃ 0 21 of Energy under Contract No. DE-SC0017647. A. J. L. is dark radiation is at the level of Neff . , which runs Δ ≲ 0 2 supported at the University of Chicago by the Kavli Institute into CMB constraints that impose Neff . at 95% con- fidence level, and which can be tested definitively with for Cosmological Physics through Grant No. NSF PHY- next-generation CMB-S4 instruments. Whereas the dark 1125897 and an endowment from the Kavli Foundation and radiation constraints only rely upon the dark mesons’ its founder Fred Kavli. A. J. L. is supported at the University gravitational influence, a direct coupling between the dark of Michigan by the U.S. Department of Energy under Grant and visible sectors opens the possibility to find evidence for No. DE-SC0007859. This work was performed at the Aspen free dark baryons and antibaryons at direct detection Center for Physics, which is supported by National Science experiments on Earth. Assuming a vector-vector interaction Foundation Grant No. PHY-1066293. Y.B. also thanks the between dark quarks and SM quarks, we estimate the particle theory group of the University of Chicago and the SI Center for Future High Energy Physics at the Institute of interaction cross section in Eq. (5.17), and we find σ − ∼ Bd n High Energy Physics of the Chinese Academy of Sciences 10−44 cm2 if the scale of new physics is 10 TeV. The for the hospitality. sensitivities of current dark matter direct detection experi- ments like XENON1T are more than adequate to probe APPENDIX: LOW-ENERGY DESCRIPTION OF these interactions, even if the dark baryons are only a THE PHASE TRANSITION subdominant population of the dark matter. Thus the detections of dark radiation and free dark baryons may We can study the phase transition from the low-energy provide the first clues for the physics of dark QCD and dark perspective by using a chiral effective theory. To describe quark nugget dark matter. the phase of broken chiral symmetry, the appropriate

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Σ ∼ dynamical variable is the quark condensate, ij To study the chiral symmetry breaking phase transition ψ¯ 1 γ ψ h ið þ 5Þ ji, which transforms as a bifundamental in this model, we calculate the thermal effective potential, φ under the flavor symmetry group, UðNfÞL ×UðNfÞR. Veffð ;TÞ, which is the Helmholtz free energy or equiv- We also now specify to Nf ¼ 3 flavors for which det Σ alently the negative pressure of the system. In total ΣðxÞ 2 2 18 is cubic in the field and a renormalizable operator. The represents Nf ¼ d.o.f., which is made transparent by effective theory can be written as [105] the following parametrization: φ ϕ L μν ∂ Σ∂ Σ† þ i I eff ¼ g Trð μ ν Þ Σ ¼ pffiffiffi δ þ Θa ðTaÞ þ iΘaðTaÞ : ðA5Þ  ij 6 ij R ij I ij 2 † † − B − mΣTrðΣΣ Þ − ðμΣ det Σ þ μΣ det Σ Þ The matrices denoted by Ta are the N2 − 1 ¼ 8 generators  f λ κ of SUðN Þ¼SUð3Þ. The fields ϕ ðxÞ, Θa ðxÞ, and ΘaðxÞ ΣΣ† 2 ΣΣ†ΣΣ† f I R I þ 2 ½Trð Þ þ 2 Trð Þ ; ðA1Þ couple to the field φðxÞ and contribute to the effective potential. The one-loop thermal effective potential can be where gμν is the inverse of the metric. The five model calculated using standard techniques [106], and by doing so parameters are the vacuum energy density B, the squared we find 2 mass parameter mΣ, the dimensionless couplings λ and κ, 1 2 2 μΣ 3 1 λ κ 4 and the complex mass parameter μΣ. Without loss of V ðφ;TÞ¼B − mΣφ − pffiffiffi φ þ þ φ eff 2 3 6 4 2 6 generality, it is possible to perform a field redefinition X 4 (global phase rotation) that makes μΣ real and non-negative. T 2 2 þ ν J ½m ðφ;TÞ=T : ðA6Þ The symmetry structure of this theory is discussed at i 2π2 B i i¼φ;ϕI ;ΘR;ΘI length in Refs. [46,105]. In the vacuum where hΣiji¼0, the symmetry group is SUðN Þ ×SUðN Þ × Uð1Þ .In We have neglected the (zero-temperature, one-loop) f Lpffiffiffi f R V the vacuum where hΣiji¼ðfΣ= 6Þδij, the symmetry is Coleman-Weinberg correction [107], which primarily 1 serves to renormalize the tree-level couplings. The thermal spontaneously broken to SUðNfÞV ×Uð ÞV, and the spec- 2 − 1 correction is expressed as a sum over species that couple to trum contains Nf massless Goldstone bosons corre- φ; the multiplicities are νφ ¼ νϕ ¼ 1 and νΘ ¼ νΘ ¼ 8; sponding to the broken symmetry generators of SUðN Þ . I R I f A the background-dependent masses are To study the phase transition between the symmetricpffiffiffi and broken phases, it is convenient to write Σ φ= 6 δ . ij ¼ð Þ ij 2 5λ κ 2 2 2 λ κ 2 mφ ¼ þ T − m − pffiffiffi μΣφ þ 3 þ φ ; Thus the effective Lagrangian reduces to 6 2 Σ 6 2 6  5λ κ 2 λ κ 1 1 μΣ 2 2 2 ffiffiffi 2 L ∂ φ 2 − − 2 φ2 − ffiffiffi φ3 mϕ ¼ þ T − mΣ þ p μΣφ þ þ φ ; eff ¼ ð μ Þ B mΣ p I 6 2 6 2 6 2 2 3 6  5λ κ 1 λ κ 1 λ κ 2 2 2 ffiffiffi 2 φ4 mΘ ¼ þ T − mΣ þ p μΣφ þ þ φ ; þ þ : ðA2Þ R 6 2 6 2 2 4 2 6 2 5λ κ 2 2 1 λ κ 2 2 mΘ ¼ þ T − mΣ − pffiffiffi μΣφ þ þ φ ; For models with mΣ > 0, μΣ > 0, λ > 0, and κ > 0, the I 6 2 6 2 6 scalar potential has its global minimum at φ ¼ fΣ where ðA7Þ the vacuum expectation value is given by and the bosonic thermal function is defined by the integral sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 2 ∞ 2 − x þy mΣ 2 JBðyÞ¼ 0 dxx logð1−e Þ. In the dark QCD model fΣ ¼ γ þ γ þ 1 ; ðA3Þ λ=2 þ κ=6 under consideration here, we only keep the contribution to φ from light d.o.f. and ignore the heavy field (e.g., dark and thep dimensionlessffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi parameter γ > 0 is defined by baryons) contributions, which are Boltzmann suppressed. 2 Around the temperature of the chiral phase transition, the γ ≡ μΣ= 24ðλ=2 þ κ=6ÞmΣ. We choose thermal effective potential admits a pair of local minima at pffiffiffiffiffiffiffiffiffiffiffiffiffi φ ¼ 0 and φ ¼ vφðTÞ, which correspond to the phases of γ γ2 1 2 2 ð þ þ Þ þ 2 2 unbroken and broken chiral symmetry, respectively. The B ¼ 12 mΣfΣ; ðA4Þ degeneracy condition, such that the potential vanishes at φ ¼ fΣ, and therefore B 0 Veffð ;TcÞ¼Veff½vφðTcÞ;Tcðcritical temperatureÞ; corresponds to the differential vacuum energy (or pressure) A8 between the phases at φ ¼ 0 and φ ¼ fΣ. ð Þ

055047-21 YANG BAI, ANDREW J. LONG, and SIDA LU PHYS. REV. D 99, 055047 (2019) defines the critical temperature Tc at which the two phases have equal pressure. For T>Tc the system is completely in the chiral-unbroken phase, and for T

055047-22 DARK QUARK NUGGETS PHYS. REV. D 99, 055047 (2019) pffiffiffi σ 1 3 where L is the latent heat of the phase transition and is the For the numerical estimate we have fixed vsh ¼ = , ’ ω 1 10 η ≪ 1 bubble s surface tension at the time of its nucleation. ¼ , and gðtnÞ¼ . A value of n implies that the Parametrically the latent heat is set by the differential phase transition occurs after little supercooling, and T is ≈ 4 n vacuum pressure, B, and Ref. [109] estimates L B, just slightly below Tc. The parameter β is given by which we will now adopt as a fiducial reference point. Eq. (A12), which evaluates to Using Eq. (A15) we evaluate the bubble nucleation rate, given by Eq. (A9), and the parameter β, defined in 2π 1 − η σ˜ −3=2 β=H ¼ n σ˜ 3 ≃ ð1.0 × 105Þ Eq. (A12). Then by solving Eq. (A13) we obtain the n 3 η3 0.1 fiducial bubble nucleation temperature, T , and we n n η −3 calculate the dimensionless supercooling parameter, n : ðA18Þ η ≡ − 0 0028 σ˜ 3=2 n ðTc TnÞ=Tc, which is found to be . ð0.1Þ rffiffiffi  pffiffiffi  π 9 3 ω 4 3 −1=2 3 2 Tcv 9 The density of bubble nucleation sites is given by η ≈ σ˜ = log pffiffiffi sh η : ðA16Þ n 3 3 4 15=2 n 4 2π Hnσ˜ Eq. (A14), which evaluates to Here we have introduced the dimensionless tension param- −9=2 −9 −3 14 σ˜ ηn σ˜ ≡ σ 2=3 1=3 n H ¼ð2.1×10 Þ : eter, =ðB Tc Þ, which affects the rate of bubble nucleations n 0.1 0.0028ð σ˜ Þ3=2 nucleation through Eq. (A15) and controls the amount of 0.1 supercooling through Eq. (A16). The Hubble parameter at ðA19Þ the fiducial bubble nucleation time, Hn ¼ HðtnÞ, depends on the dominant energy component of the universe at this This corresponds to roughly 1014 nucleation sites per ∼ −3 time. To be general, we allow that the temperature of the Hubble volume, VH Hn . Note that nnucleations is very plasma in the (dark) sector undergoing the phase transition sensitive to the amount of supercooling, ηn, and to the may be different from the temperature in the (visible) model parameters through σ˜. If the dark sector radiation sector. By writing the energy densities of radiation in the energy density is subdominant to the visible sector radi- ρ π2 30 4 dark and visible sectors as dðtÞ¼ð = Þg;dðtÞTdðtÞ ation, then nnucleations is insensitive to the temperature in the ρ π2 30 4 ∼ 3 ∼ 6 3 and γðtÞ¼ð = Þg;γðtÞTγðtÞ , the Hubble parameter is dark sector, but instead nnucleations Hn Tγ =Mpl. 3 2 2 ρ ρ π2 30 4 given by MplHn ¼ dðtnÞþ γðtnÞ¼ð = ÞgðtnÞTγðtnÞ Finally it is useful to define a dimensionless parameter, 4 where gðtÞ¼g;γðtÞþg;dðtÞðTd=TγÞ . Using this V φ 0;T 0 − V vφ T ;T 0 expression, the supercooling factor in Eq. (A16) becomes α ≡ effð ¼ ¼ Þ eff½ ð nÞ ¼ 2 4 ; ðA20Þ ðπ =30Þg ðt ÞTγðt Þ σ˜ 3=2 σ˜ n n ηn ≃ 0.0028 1 þ 0.027 log 0.1 0.1 that measures the vacuum energy released during the phase T Tγðt Þ transition and controls the strength of the resulting sto- þ 0.014 log c þ 0.029 log n 0.1 GeV T chastic gravitational wave background. The numerator of c η Eq. (A20) is the difference in the vacuum energies between n φ 0 φ − 0.033 log : ðA17Þ the symmetric ( ¼ ) and broken phases [ ¼ vφðTnÞ]at 0.0028 the fiducial bubble nucleation temperature, Tn; its value is

FIG. 12. The parameters α and β=H that feed into the gravitational wave spectrum are shown here as a function of the dimensionless couplings λ ¼ κ.

055047-23 YANG BAI, ANDREW J. LONG, and SIDA LU PHYS. REV. D 99, 055047 (2019) bounded from above by B, the differential vacuum pressure and the results are shown in Fig. 12. In evaluating α we at zero temperature. Using the thermal effective potential assume that the dark and visible sectors are at the same described above, we have numerically evaluated α and β, temperature, TγðtnÞ¼TdðtnÞ.

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