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Dark Matter in the Standard Model? PHYSICAL REVIEW D 98, 063005 (2018) Dark matter in the standard model? Christian Gross,1 Antonello Polosa,2 Alessandro Strumia,1,3 Alfredo Urbano,4,3 and Wei Xue3 1Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Sezione di Pisa, 56127 Pisa, Italy 2Dipartimento di Fisica e INFN, Sapienza Universit`a di Roma, I-00185, Roma, Italy 3Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland 4INFN, Sezione di Trieste, SISSA, via Bonomea 265, 34136 Trieste, Italy (Received 1 June 2018; published 11 September 2018) We critically reexamine two possible dark matter candidates within the Standard Model. First, we consider the uuddss hexaquark. Its QCD binding energy could be large enough to make it (quasi)stable. We show that the cosmological dark matter abundance is reproduced thermally if its mass is 1.2 GeV. However, we also find that such a mass is excluded by the stability of oxygen nuclei. Second, we consider the possibility that the instability in the Higgs potential leads to the formation of primordial black holes while avoiding vacuum decay during inflation. We show that the nonminimal Higgs coupling to gravity must be as small as allowed by quantum corrections, jξHj < 0.01. Even so, one must assume that the Universe survived in e120 independent regions to fluctuations that lead to vacuum decay with probability 1=2 each. DOI: 10.1103/PhysRevD.98.063005 I. INTRODUCTION numerical computations of one key ingredient: the nuclear wave function [5], finding that S seems excluded. In this work we critically reexamine two different intriguing possibilities that challenge the belief that the B. DM as primordial black holes existence of dark matter (DM) implies new physics beyond the standard model (SM). Primordial black holes (PBH) are hypothetical relics that can originate from gravitational collapse of sufficiently large density fluctuations. The formation of PBH is not A. DM as the uuddss hexaquark predicted by standard inflationary cosmology: the primor- The binding energy of the hexaquark dibaryon uuddss is dial inhomogeneities observed on large cosmological expected to be large, given that the presence of the strange scales are too small. PBH can arise in models with large quark s allows it to be a scalar, isospin singlet [1], called H inhomogeneities on small scales, k ≫ Mpc−1. PBH as DM S or , and sometimes named sexaquark. A large binding candidates are subject to various constraints. BH lighter energy might make S light enough that it is stable or long −17 than 6 × 10 M⊙ are excluded because of Hawking lived. All possible decay modes of a free S are kinemat- 5 radiation. BH heavier than 10 M⊙ are safely excluded. ically forbidden if S is lighter than about 1.87 GeV. Then S In the intermediate region, a variety of bounds makes the could be a dark matter candidate within the standard model Ω Ω [2–4]. possibility that PBH ¼ DM problematic but maybe not — In Sec. II A we use the recent theoretical and exper- excluded the issue is presently subject to an intense ∼ imental progress about tetra- and pentaquarks to infer the debate. According to [6] DM as PBH with mass M −15 mass of the S hexaquark. In Sec. II B we present the first 10 M⊙ is not excluded, as previously believed. And the cosmological computation of the relic S abundance, find- HSC/Subaru microlensing constraint on PBH [7] is parti- ing that the desired value is reproduced for MS ≈ 1.2 GeV. ally in the wave optics region. This can invalidate its bound −11 In Sec. II C we revisit the bound from nuclear stability below ∼10 M⊙. (NN → SX production within nuclei) at the light of recent Many ad hoc models that can produce PBH as DM have been proposed. Recently the authors of [8] claimed that a mechanism of this type is present within the Standard Model given that, for present best-fit values of the measured Published by the American Physical Society under the terms of SM parameters, the SM Higgs potential is unstable at the Creative Commons Attribution 4.0 International license. ∼ 1010 Further distribution of this work must maintain attribution to h>hmax GeV [9]. We here critically reexamine the author(s) and the published article’s title, journal citation, the viability of the proposed mechanism, which assumes and DOI. Funded by SCOAP3. that the Higgs, at some point during inflation, has a 2470-0010=2018=98(6)=063005(18) 063005-1 Published by the American Physical Society GROSS, POLOSA, STRUMIA, URBANO, and XUE PHYS. REV. D 98, 063005 (2018) homogeneous vacuum expectation value mildly above the where the κij are effective couplings determined by the top of the barrier and starts rolling down. When inflation strong interactions at low energies, color factors, quark ends, reheating adds a large thermal mass to the effective masses, and wave functions at the origin. The mij are the Higgs potential, which, under certain conditions, brings the masses of the diquarks in S made of i and j constituent 0 Higgs back to the origin, h ¼ [10]. If falling stops very quarks [13]. Si is the spin of the ith quark. Another close to the disaster, this process generates inhomogeneities important assumption, which is well motivated by studies that lead to the formation of primordial black holes. In on tetraquarks [12], is that spin-spin interactions are Sec. III we extend the computations of [8] adding a non- essentially within diquarks and zero outside, as if they ξ vanishing nonminimal coupling H of the Higgs to gravity, were sufficiently separated in space. which is unavoidably generated by quantum effects [11].We Considering diquark masses to be additive in the ξ find that H must be as small as allowed by quantum effects. constituent quark masses, and taking q and s constituent Under the assumptions made in [8] we reproduce their quark masses from the baryons, one finds results; however, in Sec. III F we also find that such ≃ 0 72 ∼ 0 90 assumptions imply an extreme fine-tuning. m½qq . GeV;m½qs . GeV;q¼fu; dg: The first mechanism is affected by the observed baryon asymmetry, but does not depend on the unknown physics ð4Þ that generates the baryon asymmetry. The second possibil- κ ity depends on inflation, but the mechanism only depends The chromomagnetic couplings ij could be derived as well on the (unknown) value of the Hubble constant during in the constituent quark model using data on baryons inflation. In both cases the DM candidate is part of the SM. κ ≃ 0 10 κ ≃ 0 06 Conclusions are given in Sec. IV. qq . GeV; qs . GeV: ð5Þ II. DM AS THE uuddss HEXAQUARK However, it is known that to reproduce the masses of light scalar mesons, interpreted as tetraquarks, σð500Þ;f0ð980Þ; The hexaquark S ¼ uuddss is stable if all its possible a0ð980Þ; κ [14], we need decay modes are kinematically closed: 8 κqq ≃ 0.33 GeV; κqs ≃ 0.27 GeV: ð6Þ > deν¯ MS <M þ M ¼ 1.8761 GeV; > e d e < Spin-spin couplings in tetraquarks are found to be about a ppeeν¯eν¯e MS < 2ðMp þ MeÞ¼1.8775 GeV; S → factor of 4 larger compared to the spin-spin couplings > peν¯ nMS <M þ M þ M ¼ 1.8783 GeV; :> e p e n among the same pairs of quarks in the baryons, which also 2 1 8791 nn MS < Mn ¼ . GeV: make diquarks. It is difficult to assess if this would change ð1Þ within a hexaquark. At any rate we can attempt a simple mass formula for S, A stable S is a possible DM candidate. A too light S can − 3 2κ 2 − 3 2κ make nuclei unstable. Scanning over all stable nuclei, we MS ¼ðm½qq = qqÞþ ðm½qs = qsÞ; ð7Þ find that none of them gets destabilized by single S 6 which in terms of light tetraquark masses means emission if MS > 1.874 GeV, with Li giving the poten- MS Mσ=2 M . Using the determination of chromo- tially highest sensitivity to MS. ¼ þ f0 magnetic couplings from baryons we would obtain A. Mass of the hexaquark from a diquark model MS ≈ 2.17 GeV; ð8Þ We estimate the mass of the hexaquark S viewing it as a neutral scalar dibaryon constituted by three spin zero whereas keeping the chromomagnetic couplings needed to diquarks fit light tetraquarks gives αβγ S ¼ ϵ ½udα 0½usβ 0½dsγ 0; ð2Þ ;s¼ ;s¼ ;s¼ MS ≈ 1.2 GeV ð9Þ α β γ where , , are color indices. This is possible thanks to the if the same values for the chromomagnetic couplings to fit strange s quark, while spin zero diquarks of the kind light tetraquark masses are taken [or 1.4 GeV using (6)]. ½uu; ½dd; ½ss are forbidden by Fermi statistics because of There is quite a lot of experimental information on antisymmetry in color and spin. We assume the effective tetraquarks [12], whereas hexaquarks, for the moment, Hamiltonian for the hexaquark [12] are purely hypothetical objects. On purely qualitative X grounds we might expect that the mass of S could be 2κ H ¼ ðmij þ ij Si · SjÞ; ð3Þ closer to the heavier value being a dibaryon and not a i≠j¼fu;d;sg dimeson (tetraquark) like light scalar mesons. 063005-2 DARK MATTER IN THE STANDARD MODEL? PHYS. REV. D 98, 063005 (2018) (a) (b) −10 FIG. 1. Left: Thermal equilibrium values of hadron abundances assuming YBS ¼ 7.6 × 10 and MS ¼ 1.2 GeV: the desired abundances are obtained if decoupling of S interactions occurs at T ∼ 25 MeV. We see that in this phase antiparticles are negligible.
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