Dibaryons cannot be the dark matter

The hypothetical $SU(3)$ flavor-singlet dibaryon state $S$ with strangeness $-2$ has been discussed as a dark-matter candidate capable of explaining the curious 5-to-1 ratio of the mass density of dark matter to that of baryons. We study the early-universe production of dibaryons and find that irrespective of the hadron abundances produced by the QCD quark/hadron transition, rapid particle reactions thermalized the $S$ abundance, and it tracked equilibrium until it"froze out"at a tiny value. For the plausible range of dibaryon masses (1860 - 1890 MeV) and generous assumptions about its interaction cross sections, $S$'s account for at most $10^{-11}$ of the baryon number, and thus cannot be the dark matter. Although it is not the dark matter, if the $S$ exists it might be an interesting relic.

Our understanding of QCD and the quark/hadron transition now strongly disfavors this idea (but see Ref. [4]). However, Farrar [5,6] has revived a variant of this idea with a stable (or very long-lived) dibaryon: she argues that the S dibaryon, 2 a six-quark configuration of 2 up quarks, 2 down quarks, and 2 strange quarks, an SU (3) flavor-singlet with baryon number 2, strangeness −2, and spin zero, could explain the ratio of dark matter to baryons.
To date there is no experimental evidence for the S dibaryon, and its existence and properties are a topic of continuing debate in the QCD community. Moreover, there are significant constraints from lattice QCD, hypernuclei, the ALICE experiment at the LHC, and unsuccessful searches [7][8][9][10]. In this paper, we focus on the viability of the dibaryon dark-matter hypothesis, and not the existence of the dibaryon itself. Fortunately, what we know with certainty about the putative S, summarized below, is sufficient to determine its cosmological production: 1. Its mass must be greater than m p + m n − m e − 2BE 1860 MeV (BE ∼ 8 MeV is the binding energy of a nucleon in a nucleus) to guarantee nuclear stability [6]. 3 2. If its mass is greater than 2m p + 2m e 1878 MeV it can decay to two nucleons; for m S < 2055 MeV = m p + m e + m Λ , this is a doubly-weak process with a decay width Γ > G 4 F δm 9 (δm = m S − 1878 MeV); for m S < 1890 MeV the lifetime of the S is greater than the age of the universe.
3. For an S mass between 2055 MeV and 2232 MeV (2232 MeV = 2m Λ ) the dibaryon can decay via a singly weak process into a nucleon and a Λ with a lifetime short compared to the age of the universe (Γ > G 2 F δm 5 ), and for m S > 2232 MeV the dibaryon can decay into two Λ's via the strong interaction with a very short lifetime.
4. In general, the non-strangeness changing interactions of the S, e.g., Λ + Λ ←→ π + π + S, should be strong, although they might be suppressed by factors arising from the wavefunctions of the states involved [13,14].
These facts point to the mass range 1860 MeV to 1890 MeV, where the dibaryon is stable or longlived, and indicate that the reactions that control its abundance are strong and involve lambdas and possibly other strange baryons (e.g., strangeness −1 Σ's and strangeness −2 Ξ's).
2 The S dibaryon is sometimes referred to as the exaquark, or the sexaquark. 3 This constraint ensures that it is energetically impossible for a nucleus to decay into a nucleus with one less proton and one less neutron and an S. We believe there is a much stronger limit based upon the stability of the deuteron.
If mS < mp + mn − me − 2.22 MeV 1875 MeV, the deuteron can decay via a doubly weak process into an S, a positron and a neutrino, with decay width Γ > G 4 F δm 9 . The deuterons in the universe today were produced in the big bang 14 Gyr ago; stability of the deuteron for the age of the universe requires δm < 3 MeV or mS > 1872 MeV.
Stronger constraints likely follow from the formation of neutron stars [11] and observed neutron-star masses [12].
The starting point for our analysis is the early-universe QCD transition from the quark/gluon plasma into hadrons. Lattice calculations imply that the QCD transition is a "crossover" transition at temperature T C = 155 MeV. Using nucleons, lambdas, and S's as the baryonic degrees of freedom, we show that "Baryon Statistical Equilibrium" (BSE) 4 is very rapidly established after the QCD transition, and is maintained only down to a temperature T ∼ 10 MeV, long before a significant fraction of the baryon number is in dibaryons. In particular, our calculations show that the freeze-out abundance of the S's, which determines the present-day relic abundance, is at most 10 −11 that of nucleons, largely independent of the dibaryon mass and the strength of its interactions.

II. BARYON STATISTICAL EQUILIBRIUM
We consider the thermodynamic system of nucleons, lambdas, and dibaryons at temperatures T < T C . Because other baryons are significantly more massive, only these particles are needed to track the dibaryon abundance. (Later, we will explicitly show that the only other processes of any importance, those that involve Σ and Ξ baryons, can be ignored at temperatures around freezeout of the dibaryon abundance.) Further, we need not differentiate between neutrons and protons, for which we assume a common mass of 939 MeV, and m Λ = 1116 MeV. For the only unknown in the BSE calculation, we consider the range discussed above, 1860 MeV < m S < 1890 MeV, where the S is stable or long-lived.
During the radiation-dominated era the age of the universe and the expansion rate are related by: t = 1/2H = 0.301g positrons, neutrinos, and smaller numbers of other hadrons. Even just after the QCD transition, the age of the universe is much longer than a typical strong-interaction timescale (10 −23 s), and longer than the timescale for weak decays of hadrons (e.g., the Λ lifetime is 2.6 × 10 −10 s), and weak interactions more generally (e.g., e + + e − ←→ ν i +ν i and n + e + ←→ p +ν e ). The same of course holds for electromagnetic interactions. In sum, during the period of interest, T 155 MeV to a few MeV, the constituents of the radiation soup are known and their reactions rapid, which makes equilibrium thermodynamics appropriate.
Recall, there are two types of equilibrium: kinetic equilibrium and chemical equilibrium. If a species i is in kinetic equilibrium its phase-space density is where g i is the number of degrees of freedom, µ i is the chemical potential of species i, and +1 is used for fermions and −1 used for bosons. In the nonrelativistic (NR) limit the number density of species i is independent of spin statistics. As noted above, strong, electromagnetic, and weak interaction rates are so rapid that kinetic equilibrium is maintained throughout.
If a reaction a + b + · · · ←→ · · · + y + z is fast compared to the dynamical timescale for expansion (the age of the universe), chemical equilibrium results, and the sum of chemical potentials in the initial and final states are equal: µ a + µ b + · · · = · · · + µ y + µ z . Chemical equilibrium for the process Λ ←→ N + π + π enforces µ Λ = µ N since µ π = 0 (e.g., π 0 ←→ γ + γ), while chemical equilibrium for the process S + π ←→ Λ + Λ (and other reactions) enforces µ S = 2µ Λ = 2µ N . Using g N = 4 (counting neutrons and protons), g Λ = 2, and g S = 1, and assuming all species are NR, the BSE abundances of the {N, Λ, H} system are Baryon number conservation is expressed as the constancy of baryon number to entropy ratio, where the entropy density is s ≡ 2π 2 g * T 3 /45 and n B ≡ Equilibrium thermodynamics is not the entire story and next we discuss the freeze out of S's, which determines the dibaryon abundance today. S + π + π. The first reaction -a weak process -regulates the number of Λ's through decays and inverse decays. Since the lifetime of the Λ is of the order of 10 −10 s -much shorter than the age of the universe at the times of relevance here -the first reaction ensures that the abundance of Λ's closely tracks its equilibrium value. On the other hand, the second reaction, which regulates the number of S's, cannot keep pace with the expansion at low temperature because of the exponentially decreasing numbers of Λ's. This prevents the dibaryon abundance from increasing and maintaining its BSE value and leads to the small final abundance of S's.
In principle, three rate equations are needed to follow the evolution of the {N, Λ, S} network.
However, because strong and weak interaction rates are so much larger than the expansion rate, the nucleons and Λ's will be in chemical equilibrium, and we need only consider the Boltzmann equation for the dibaryons. Using detailed balanced, the equation governing the dibaryon number density can be written as:ṅ (3) In Eq. (3), the term Γ ΛΛ→Sππ is given by where K n (x) is the modified Bessel function of the second kind of order n, and σ(s) is the cross section as a function of s for the process ΛΛ −→ Sππ. 6 This process does not involve a strangenesschanging interaction, so it should proceed via the strong interaction. Absent precise information for σ(s), we assume σ(s) = σ 0 v Møl in the NR limit, where v Møl is the Møller velocity. With this Further, we take σ 0 = 1/m 2 π = 20 mb as the nominal value, and vary σ 0 from 1 b to 1 pb to illustrate the insensitivity of our results to precise knowledge for this cross section. We note that had we included other processes involving Σ's and Ξ's that produce S's, e.g., ΞN → Sπ, ΛΣ → Sπ, or ΣΣ → Sπ, the r.h.s of Eq. (3) would be replaced by the sum i,j Γ ij→SX with Γ ij = σ 0 n EQ i n EQ j /n EQ S . Shown in Fig. 2 is the ratio of Γ ΛΛ→Sππ to the expansion rate for temperatures between 5 MeV and 155 MeV. Around T C = 155 MeV, when Λ's are very abundant, the ratio is greater than 10 19 , ensuring that regardless of its initial abundance, the S's will rapidly come into equilibrium. As the temperature falls, Γ/H drops rapidly, because of the exponentially decreasing numbers of Λ's, and freezes out (achieves a value of unity) around T = 8 MeV.
Shown in the inset of Fig. 2, are the additional S production rates involving Σ's and Ξ's (assuming the same cross section normalization). While these processes can be important around T C , increasing Γ/H by about a factor of 3, the final S abundance is determined by the freeze out of ΛΛ → Sππ, justifying our simple network of N 's, Λ's and S's. We note that σ 0 would have to be less than about 2 × 10 −46 cm 2 (0.2 zeptobarns) for the reactions that regulate S's to be too slow to establish BSE immediately after the QCD transition.
As previously mentioned, the BSE abundance is relatively constant over the range of freeze-out temperatures, at an S-abundance to nucleons in the range 10 −11 to 10 −13 , and so we expect the final abundance of S's to be in this range. Fig. 3 shows the results of integrating Eq. (3) for an for this process (more than 12 orders-of-magnitude), freeze out occurs between T ∼ 6 MeV and 12 MeV, where the BSE abundance of S's is relatively constant, at a value no larger than 10 −11 that of nucleons.
As attractive as the idea of dibaryonic dark matter is, we conclude that dibaryons can be but a tiny fraction of the dark matter. There are two ways to avoid this conclusion; but neither is a easy way out.
The first way out is if the S is light enough (m S 1200 MeV) that the BSE abundance is large at freeze-out. This is the conclusion of Gross et al. [16]. However, a mass this small is strongly excluded by the arguments given in the Introduction.
The second way out is if S's are copiously produced in the QCD transition (e.g., by "entrainment" as proposed in [6]) and their interactions thereafter are too weak to establish the S in BSE.
In our calculations we assumed that the reactions that regulate the S abundance (ΛΛ, ΣΣ, ΣΞ, ΛΣ, ΛΞ and ΞN ↔ S +X) are strong interactions, with σ 0 = 20 mb. 7 As seen from Fig. 2, avoiding 7 We also assumed s-wave reactions. If the reaction is p-wave there is a suppression of 6T /mΛ = 0.8 at T = TC . assumed cross section. Farrar argues that while the S has strong interactions with baryons, a very large suppression in the effective S-baryon-baryon coupling can arise due to wavefunction mismatch [17]. She needs ten orders-of-magnitude suppression to avoid being ruled out by the experimental constraint arising from doubly-strange hypernuclei, and as mentioned here, 20 orders-of-magnitude to avoid BSE and our conclusion that dibaryons are not abundant enough to be the dark matter.
Although we conclude that the dibaryon is unlikely to be dark matter, if a stable or longlived dibaryon does exist they may be an interesting relic, even if the number left over from the early universe is around 10 −11 or so per nucleon. Given that they are neutral and have strong interactions, they should bind to nuclei (however, see Ref. [18] for the argument that the S does not bind to nuclei.) The relevant nuclei in the early universe are p's and 4 He and the formation of S-nuclei should occur at a temperature T ∼ BE/ ln η −1 (BE is the binding energy of an S to a nucleus), which is likely to be of order a fraction of an MeV, i.e., around the time of BBN.
(The tiny S abundance ensures that this will not significantly affect BBN.) Any S's should today masquerade either as an odd, stable form of tritium ( 3 S t) or an odd form of 6 He ( 6 S He). By odd, we mean slightly different mass and possibly different nuclear energy levels. Absent knowledge of their binding energies to p's and 4 He's, we can't estimate the relative abundances of 3 S t and 6 S He. We do not consider their detectability here.