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
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(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. Right: Thermal equilibrium values of ΓS=H.
In the absence of any other experimental information it is YS Ω M ¼ DM p ≈ 4.2: ð10Þ impossible to provide an estimate of the theoretical YB ΩB MS uncertainty on MS. Lattice computations performed at unphysical values of Thereby the baryon asymmetry before S decoupling must be quark masses find small values for the S binding energy, about 13, 75, 20 MeV [15–17]. Extrapolations to physical YBS ¼ YB þ 2YS ≈ 9.3YB: ð11Þ quark masses suggest that S does not have a sizable binding energy; see, e.g., [18]. Furthermore, the binding energy of One needs to evolve a network of Boltzmann equations for the deuteron is small, indirectly disfavoring a very large the main hadrons: p, n, Λ, Σ0, Σþ, Σ−, Ξ0, Ξ−,andS.Strange binding energy for the (somehow similar) S, which might baryons undergo weak decays with lifetimes τ ∼ 10−10 s, a too be a moleculelike state.1 Despite this, we overoptimisti- few orders of magnitude faster than the Hubble time. This cally treat MS as a free parameter in the following. means that such baryons stay in thermal equilibrium. We We also notice that the S particle could be much larger thereby first compute the thermal equilibrium values taking than what is envisaged in [2–4] and that its coupling to into account the baryon asymmetry. Thermal equilibrium photons, in the case of 2–3 fm size (see the considerations implies that the chemical potentials μðTÞ satisfy on diquark-diquark repulsion at small distances in [19]), could be relevant for momentum transfers k as small as μb ¼ μS=2;b¼fp; n; Λ; …g: ð12Þ k ∼ 60 MeV, compared to k ≈ 0.5 GeV considered by Farrar. Their overall values are determined imposing that the total baryon asymmetry equals B. Cosmological relic density of the hexaquark X eq − eq neq − neq nb nb¯ S S¯ We here compute the relic density of S dark matter, þ 2 ¼ Y S: ð13Þ s s B Ω 2 b studying if it can match the measured value DMh ¼ 0.1186, i.e., Ω ≈ 5.3 Ω [20]. A key ingredient of the DM b The equilibrium values can be analytically computed in computation is the baryon asymmetry. Its value measured in Boltzmann approximation (which becomes exact in the cosmic microwave background (CMB) and Big Bang 0 8 10−10 nonrelativistic limit) nucleosynthesis (BBN) is YB ¼ nB=s ¼ . × .The DM abundance is reproduced (using MS ¼ 1.2 GeV for 2 eq Mi T Mi μ =T n ¼ g K2 e i ; ð14Þ definiteness) for i i 2π2 T
1We thank M. Karliner, A. Francis, and J. Green for where the þ (−) holds for (anti)particles. We then obtain discussions. the abundances in thermal equilibrium plotted in Fig. 1,
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0 assuming MS ¼ 1.2 GeV. We see that the desired where X can be a π or a γ, as preferred by approximate S abundance is reproduced if the interactions that isospin conservation. The Λ can be substituted by the Σ0. S ≈ 25 form/destroy decouple at Tdec MeV. This tem- Defining z ¼ MS=T and Yp ¼ np=s, the Boltzmann perature is so low that baryon antiparticles have equation for the S abundance is negligible abundances, and computations can more 2 2 simply be done neglecting antiparticles. Then, the dYS Y YS sHz ¼ γeqb B − ; ð19Þ desired decoupling temperature is simply estimated dz S eqb2 eqb YB YS eq eq ð2Mp−MS Þ=T imposing nS =np ∼ YBSe ∼ 1, and it decreases if MS is heavier: where the superscript “eqb” denotes thermal equilibrium at fixed baryon asymmetry and YB is summed over all 2M − MS baryons but the dibaryon S. A second equation for Y is not ≈ p ≈ 89 − 0 048 B Tdecjdesired MeV . MS: ð17Þ j ln Y Sj needed, given that the baryon number is conserved: B − 2 − ðYB YB¯ Þþ ðYS YS¯ Þ¼YBS. Furthermore, YS¯ is neg- ligible, and Y ¯ is negligible around decoupling. The S To compute the decoupling temperature, we consider the B production rate is obtained after summing over all bb0 ↔ three different kinds of processes that can lead to the SX processes of Eq. (18). In the nonrelativistic limit the formation of S: interaction rate gets approximated as (1) Strong interactions of two heavier QCD hadrons that contain the needed two s quarks. One example is X T≪MS eqb eqb eqb eqb ΛΛ ↔ S 2γ ≃ σ 0 X, where X denotes pions. These are S nb nb0 h bb vreli : ð20Þ −2 doubly Boltzmann suppressed by e ms=T at temper- b;b0 atures T
2 − YS Mp MS 0 − þ − ∼ σ 2MΛ−MS ΛΛ;nΞ ;pΞ ; Σ Σ ↔ SX; ð18Þ YBðMPlTdec SÞ : ð22Þ YB
2 S Let us consider, e.g., the process Λ þ Λ ↔ S þ X where X The fact that is in thermal equilibrium down to a few tens of denotes other SM particles that do not carry the baryon MeV means that whatever happens at higher temperatures asymmetry, such as pions. Thermal equilibrium of the above gets washed out. Notice the unusual dependence on the cross process implies section for S formation: increasing it delays the decoupling, increasing the S abundance. eq 3=2 nS n gS 2πMS 2 − S ð MΛ MSÞ=T Figure 2 shows the numerical result for the relic S 2 ¼ eq2 ¼ 2 2 e : ð15Þ nΛ nΛ gΛ MΛT abundance, computed by inserting in the Boltzmann 2 σ 0 σ 1 ðM −MΛÞ=T equation a s-wave bb vrel ¼ 0, varied around =GeV . Inserting nΛ ∼ n e p with n ≈ Y Ss gives p p B The cosmological DM abundance is reproduced for 2 MS ≈ 1.2 GeV, while a large MS gives a smaller relic nS n Λ ð2MΛ−MS Þ=T ð2M −MS Þ=T ∼ e ∼ Y Se p : ð16Þ abundance. Bound-state effects at BBN negligibly affect n n B p p the result and, in particular, do not allow one to reproduce the DM abundance with a heavier MS ≈ 1.8 GeV. Namely, nS ≪ np at large T; nS ≫ np at low T.ADM abundance comparable to the baryon abundance is obtained only We conclude this section with some sparse comments. if reactions that form S decouple at the T in Eq. (17). Possible troubles with bounds from direct detection have
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FIG. 3. Sample diagram that dominates nucleon decay into S inside nuclei. The initial state can also be nn or pp.
16 0 0 O8 → N SX (where X can be one or two π, γ, e and N 14 14 14 can be O8, N7, C6, depending on the charge of X) 4 FIG. 2. Thermal hexaquark abundance within the SM with the has been performed, but a very conservative limit baryon asymmetry in Eq. (11). 16 0 26 τð O8 → N SXÞ ≳ 10 yr ð23Þ been pointed out in [4,21,22]: a DM velocity somehow is obtained by requiring the rate of such transitions to be smaller than the expected one can avoid such bounds smaller than the rate of triggered background events in SK, reducing the kinetic energy available for direct detection. which is about 10 Hz [27]. A more careful analysis would Using a target made of antimatter (possibly in the upper likely improve this bound by 3 orders of magnitude [2]. atmosphere) would give a sharp annihilation signal, The amplitude for the formation of S is reasonably although with small rates. The magnetic dipole interaction dominated by the sample diagram in Fig. 3: doubly weak of S does not allow one to explain the recent 21 cm production of two virtual strange Λ baryons (e.g., through anomaly along the lines of [23] (an electric dipole would be p → πþΛ and n → π0Λ ; at quark level u → sdu¯ and needed). The interactions of DM with the baryon/photon d → suu¯ Λ Λ → S fluid may alter the evolution of cosmological perturbations ), followed by the strong process , leaving an imprint on the matter power spectrum and the M ≈ M M CMB. However, they are not strong enough to produce NN→SX NN→Λ Λ X × Λ Λ →S: ð24Þ S significant effects. The particle is electrically neutral and 5 has spin zero, such that its coupling to photons is therefore The predicted lifetime is then obtained as [2] suppressed by powers of the QCD scale [3]. So elastic S τðN → N 0SXÞ scattering of with photons is not cosmologically relevant. A light S would affect neutron stars, as they are expected 3 if MS ≲ 1.74 GeV; ≃ yr to contain Λ particles, made stable by the large Fermi 2 × 5 jMjΛ Λ →S 10 if 1.74 GeV ≲ MS ≲ 1.85 GeV; surface energy of neutrons. Then, ΛΛ → S would give a loss of pressure, possibly incompatible with the observed ð25Þ existence of neutron stars with mass 2.0M [1]. However, sun S we cannot exclude S on this basis, because production of Λ where the smaller value holds if is so light that the decay hyperons poses a similar puzzle. S as DM could interact can proceed through real π or ππ emission, while the longer with cosmic ray p giving and photon and other signals [24] lifetime is obtained if instead only lighter eþν or γ can be and would be geometrically captured in the Sun, possibly emitted. affecting helioseismology.3 The key factor is the dimensionless matrix element M Λ Λ → S In the next section we discuss the main problem, which Λ Λ →S for the transition inside a nucleus, seems to exclude S as DM. 4SK searched for dinucleon decays into pions [25] and leptons C. Super-Kamiokande bound on nuclear stability [26] and obtained bounds on the lifetime around ∼1032 years. 16 → N However, these bounds are not directly applicable to O8 Two nucleons N ¼fn; pg inside a nucleus can N 0SX where the invisible S takes away most of the energy make a double weak decay into S, emitting π, γ,ore reducing the energy of the visible pions and charged leptons, in [2]. This is best probed by Super-Kamiokande (SK), which contrast to what is assumed in [25,26]. ∼8 1032 5A numerical factor of 1440 due to spin and flavor effects contains × oxygen nuclei. No dedicated search for 2 has already been factored out from jMjΛ Λ →S here and in the following. Note also that the threshold 2MN − Mπ ¼ 1.74 GeV 3We thank M. Pospelov for suggesting these ideas. neglects the small difference in binding energy between N and N 0.
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ψ ⃗ ⃗ which we now discuss. Following [2], we assume that the function nucðaÞ for the separation a between the center initial state wave function can be factorized into wave of mass of the Λ ’s. The matrix element is given by the functions of the two Λ baryons and a relative wave wave-function overlap
Z M ψ ρ⃗a λ⃗a ρ⃗b λ⃗b ⃗ψ ρ⃗a λ⃗a ψ ρ⃗b λ⃗b ψ ⃗ 3 3ρa 3ρb 3λa 3λb Λ Λ →S ¼ Sð ; ; ; ; aÞ Λ ð ; Þ Λ ð ; Þ nucðaÞd ad d d d : ð26Þ
Here, ρ⃗a;b, λ⃗a;b are center-of-mass coordinates that parametrize the relative positions of the quarks within each Λ . Using the Isgur-Karl (IK) model [28] the wave functions for the quarks inside the Λ and inside the S are approximated by 3 2 ⃗2 ⃗ 1 ρ⃗ þ λ ψ Λ ρ⃗; λ ffiffiffi − ; ð Þ¼ pπ exp 2 2 ð27Þ rN rN
2 2 3 3=4 1 15=2 ρ⃗a2 þ λ⃗a þ ρ⃗b2 þ λ⃗b þ 3 a⃗2 ψ ρ⃗a λ⃗a ρ⃗b λ⃗b ⃗ ffiffiffi − 2 Sð ; ; ; ; aÞ¼ p exp 2 ; ð28Þ 2 rS π 2rS
6 where rN and rS are the radii of the nucleons, respectively, of S. Performing all integrals except the final integral over a ≡ ja⃗j gives Z 3=4 6 3=2 1 3 2rNrS 1 2 −3 2 4 2 M ffiffiffi 4π a = rS ψ j jΛ Λ →S ¼ 2 2 2 2 pπ da a e nucðaÞ: ð29Þ rN þ rS rS
ψ S As shown in Fig. 5 below (and as discussed in [2]), if rS is not following that for a reasonable form of nuc a stable much smaller than rN, the overlap integral is not very much is excluded even if it were very small. 16 0 suppressed and τð O8 → N SXÞ is tens of orders of Numerical computations of the ground-state wave func- 16 magnitude below the experimental limit and is clearly tions of nuclei, including O8, have been performed, e.g., ψ ψ excluded. This conclusion is independent of the form of nuc. in [5]. The quantity that determines nuc is the two-nucleon ρ However, if rS were a few times smaller than rN—a point density NN, defined in Eq. (58) of [5]. We obtain possibility that seems unlikely due to diquark repulsions ρNNðaÞ by interpolating the data given in [5] and adding the (see, e.g., [19]) but cannot firmly be excluded—then constraint ρNNð0Þ¼0, which is a conservative assumption 16 0 ρ 0 ≠ 0 τð O8 → N SXÞ is extremely sensitive to the probability for our purposes since NNð Þ would lead to a larger of the overlap of two nucleons inside the oxygen core at matrix element. There are 28 neutron-neutronR pairs and 64 16 4π 2ρ very small distances (less than, say, 0.5 fm). The wave proton-neutronR pairs in O8 so one has da a nnðaÞ¼ ψ 2 function of nucleon pairs nuc at such small distances has 28 and da4πa ρpnðaÞ¼64. We therefore define the not been probed experimentally. In fact, at such small wave functions distances nucleons are not the appropriate degrees of 7 freedom. Thus, for a very small S one can only make pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τ 16 → N 0S ψ nn ρ 28 ψ pn ρ 64 an educated guess of ð O8 XÞ, since the form of nucðaÞ¼ nnðaÞ= ; nucðaÞ¼ pnðaÞ= : ð30Þ ψ nuc is uncertain. Nevertheless, we will show in the
These wave functions are plotted in Fig. 4, together with the 6 One should be aware that the IK model has serious short- Miller-Spencer (MS) and the Brueckner-Bethe-Goldstone — comings. One issue is that it is a nonrelativistic model an (BBG) wave function used in [2]. The BBG wave functions assumption that is problematic in particular for small S. Another assume a hard repulsive core between nucleons such that problem is that the value of rN that gives a good fit to the lowest 1þ 3þ ψ 0 5 lying 2 and 2 baryons—rN ¼ 0.49 fm—is smaller than the nuc vanishes at a 063005-6 DARK MATTER IN THE STANDARD MODEL? PHYS. REV. D 98, 063005 (2018) (a) (b) ψ 16 FIG. 4. Wave functions nucðaÞ as a function of the relative distance a between two nucleons in O8. The darker (lighter) green curve ψ nn ψ np shows nuc, obtained from [5], using the AV18 (AV18+UIX) potential; the darker (lighter) red curve shows nuc using the AV18 (AV18 +UIX) potential; the blue curve is the MS wave function used in [2] and the dashed black curve is the BBG wave function with hard core 0 5 radius rcore ¼ . fm. Left: Wave functions over the entire range. Right: Zoom-in plot of the region relevant for our calculation. III. DM AS BLACK HOLES TRIGGERED BY HIGGS FLUCTUATIONS We here present the technical computations relative to the mechanism anticipated in the Introduction. The SM potential is summarized in Sec. III A.InSec.III B we outline the mechanism that generates black holes. Section III C studies the generation of Higgs inhomogeneities. Postinflationary dynamics is studied in Sec. III D. The formation of black holes is considered in Sec. III E. The viability of a critical assumption is discussed in Sec. III F. A. The Higgs effective potential The effective potential of the canonically normalized Higgs field during inflation with Hubble constant H0 is FIG. 5. The dimensionless squared matrix element for nuclear S M 2 S λ ðhÞ 4 2 2 decay into , j jΛ Λ →S, as a function of the radius rS in units ≈ eff − 6ξ VeffðhÞ h HH0h þ V0; ð31Þ of the nucleon radius rN, using different nuclear wave functions. 4 The color coding, defined in Fig. 4, refers to the nuclear wave functions used. The thinner (thicker) curves assume rN ¼ at h ≫ 174 GeV. Here λ is the effective quartic coupling 0 87 0 49 eff . fm (rN ¼ . fm). The Super-Kamiokande bound would computed including quantum corrections. The second mass M 2 ≲ 10−20 10−25 1 74 be evaded for j jΛ Λ →S ð Þ for MS > . GeV V h 1 74 term in eff ð Þ can be generated by various different sources (for MS < . GeV). [8]. We consider the minimal source: a Higgs coupling to 2 2 gravity, Lξ ¼ −ξHRh =2, with Ricci scalar R ¼ −12H0 during inflation. Finally, during inflation the effective poten- The resulting matrix elements from the MS wave tial in Eq. (31) is augmented by thevacuum energy associated 3 ¯ 2 2 ¯ ≃ function qualitatively agree to what is obtained using the with the inflaton sector, V0 ¼ MPlH0, where MPl 18 wave functions extracted from [5]. By contrast, the matrix 2.435 × 10 GeV is the reduced Planck mass. element using the BBG wave function with hard core radius We implement the renormalization group (RG) improve- 0 5 S rcore ¼ . fm is very much suppressed, especially if is ment of the effective potential at next-to-next-to-leading small. The reason is that, according to the assumption of a order (NNLO) precision: running the SM parameters hard core repulsive potential, the nucleons cannot get close at three loops and including two-loop quantum corrections enough to form the small state S. Since we do not consider to the effective potential. We consider fixed values of ψ ≲ 0 5 α 0 1184 125 09 a nucðaÞ that vanishes for a . fm realistic, we con- 3ðMZÞ¼ . and Mh ¼ . GeV, and we vary the clude that a stable S is excluded. main uncertain parameter, the top mass, in the interval Weaker bounds on S production are obtained consider- Mt ¼ 172.5 0.5 GeV [30]. In Fig. 6(a) we show the Λ’ λ ing baryons containing s. resulting effðhÞ as a function of h. 063005-7 GROSS, POLOSA, STRUMIA, URBANO, and XUE PHYS. REV. D 98, 063005 (2018) (a) (b) FIG. 6. Left: RGE running of λeff (solid red lines) obtained by changing the top mass in its 3σ interval defined by Mt ¼ 172.5 0.5 GeV. For comparison, we show the running of λ at three loops (dashed blue lines) without including the Coleman-Weinberg corrections. Right: RGE running of a small ξH. (a) (b) FIG. 7. Left: Number of e-folds Nin at the beginning of the Higgs fall that gives the maximal hend rescued by the reheating temperature. This is computed as a function of the Hubble constant during inflation, for three different values of the uncertain parameter b that −3 approximates the Higgs potential. Continuous (dashed) curves correspond to ξH ¼ 0 (−10 ). Right: SM values of b and of the position hcr of the top of the SM potential as a function of the top mass. The nonminimal coupling to gravity ξH receives SM approximation that encodes the main features of the SM quantum corrections encoded in its renormalization group effective potential in Eq. (31), equation (RGE), which induce ξH ≠ 0 even starting from ξ 0 H ¼ at some energy scale. The RGE running of small 2 4 ξ μ¯ h h values of Hð Þ is shown in Fig. 6(b). As mentioned before, ≈− ffiffiffi − 6ξ 2 2 VeffðhÞ b log 2 p 4 HH0h ; ð32Þ a nonzero ξH can be considered as a proxy for an effective hcr e mass term during inflation. The latter, for instance, can be 2 2 generated by a quartic interaction λ ϕjHj ϕ between the h where h is the position of the maximum of the potential Higgs and the inflaton field or by the inflaton decay into cr with no extra mass term, ξ ¼ 0. The parameters b and h SM particles during inflation. For these reasons, it makes H cr depend on the low-energy SM parameters such as the top sense to include ξ as a free parameter in the analysis of the H mass: they can be computed by matching the numerical Higgs dynamics during inflation, at most with the theo- value of the Higgs effective potential at the gauge-invariant retical bias that its size could be loop suppressed. 4 8 position of the maximum, VeffðhcrÞ¼bhcr= . The result is shown in the right panel of Fig. 7. 1. Analytic approximation Results will be better understood when presented in ξ We will show precise numerical results for the SM case. terms of the dimensionless parameters b, H, hcr=H0, and However, the discussion is clarified by introducing a simple T=H0, where T is the temperature, as they directly control 063005-8 DARK MATTER IN THE STANDARD MODEL? PHYS. REV. D 98, 063005 (2018) the dynamics that we are going to study. The parameter b h2 h4 VT h ≈−b ffiffiffi V h ; controls the flatness of the potential beyond the potential effð Þ log 2 p 4 þ Tð Þ hcr e barrier at hcr, with smaller b corresponding to a flatter 1 2 2 −h=2πT potential. The nonminimal coupling ξH controls the effec- V ðhÞ ≈ M h e : ð36Þ ¯ T 2 T tive Higgs mass during inflation. Finally, MPl=H0 will set the reheating temperature in Eq. (35) and thus the position At h ≲ T we can neglect the exponential suppression in the and size of the thermal barrier. thermal mass, and the maximum of the effective potential in — The position of the potential barrier defined by the Eq. (36) is given by field value where the effective potential has its maximum— strongly depends on the value of the top mass, on the M2 −1=2 nonminimal coupling to gravity, and, after inflation, on the hT ¼ M bW T : ð37Þ max T bh2 temperature of the thermal bath that provides an extra mass cr term. For ξH ≠ 0, the maximum of the Higgs potential gets shifted from h to cr B. Outline of the mechanism During inflation, the Higgs field is subject to quantum 2 −1 2 −12ξ = fluctuations. Depending on the value of H0, these quantum − b W HH0 hmax ¼ H0 12ξ 2 ; ð33Þ fluctuations could lead the Higgs beyond the barrier and H bhcr make it roll toward Planckian values. If TRH is high enough and h is not too far, thermal corrections can “rescue” the where WðzÞ is the product-log function defined by Higgs, bringing it back to the origin [10]. The mechanism z ¼ WeW . The condition relies on a tuning such that the following situation occurs [8]: ∼ 20 (i) At Nin e-folds before the end of inflation, the Higgs background value h is brought by quantum bh2 fluctuation to some h ≠ 0. This configuration must −12ξ H2 > − cr ð34Þ in H 0 e be spatially homogeneous on an inflating local patch large enough to encompass our observable Universe today. We consider the de Sitter metric in flat slicing must be satisfied; otherwise, the effective mass is too coordinates, ds2 ¼ −dt2 þ a2ðtÞdx⃗2. We will dis- negative, and it erases the potential barrier, thus leading cuss later how precisely this assumption must be to a classical instability. satisfied, as well as its plausibility. (ii) When the classical evolution prevails over the 2. The thermal potential quantum corrections, the Higgs field, starting from the initial position h , begins to slow roll down the After the end of inflation, the Higgs effective potential in negative potential. This condition reads receives large thermal corrections from the SM bath at generic temperature T. The initial temperature of the 0 V ðh Þ cH0 eff in > ; thermal bath is fixed by the dynamics of reheating after 3 2 2π ð38Þ inflation. We assume instantaneous reheating, as this is |fflfflfflfflfflffl{zfflfflfflfflfflffl}H0 |{z} most efficient for rescuing the falling Higgs field. The classical quantum reheating temperature is then given by where c is a constant of order 1, fixed to c ¼ 1 in [8]. We will explore what happens choosing c ¼ 0.9 1 4 45 = 1 2 1 2 or c ¼ 1.1. From this starting point t on, the T M = H = ; 35 in RH ¼ 4π3 Pl 0 ð Þ classical evolution of the background Higgs value g is described by 106 75 ̈ 3 _ 0 0 where g ¼ . is the number of SM degrees of hcl þ H0hcl þ VeffðhclÞ¼ ; ð39Þ freedom. After reheating, the Universe becomes radiation dominated, the Ricci scalar vanishes, and so does the where the subscript cl indicates that this is a classical contribution to the effective potential from the nonminimal motion. Dots indicate derivatives with respect to Higgs coupling to gravity. time t. 0 The effective Higgs potential at finite temperature is (iii) At the end of inflation, Nend ¼ , the Higgs is obtained adding an extra thermal contribution VT, which rescued by thermal effects. This happens if the value can be approximated as an effective thermal mass for the of the Higgs field hend at the end of inflation is 2 ≃ 0 12 2 Higgs field, MT . T (see, e.g., [10]) smaller than the position of the thermal potential 063005-9 GROSS, POLOSA, STRUMIA, URBANO, and XUE PHYS. REV. D 98, 063005 (2018) 8 TRH barrier at reheating, hmax. A significant amount of comoving wave number k, the equation for the mode δ PBH arises only if this condition is barely satisfied in hk takes the form all the Universe. This is why the homogeneity 2 assumption in (i) is needed. ̈ _ k 00 δh þ 3H0δh þ δh þ V ðh Þδh ¼ 0; ð43Þ To compute condition (iii) we fix the initial value of the k k a2 k eff cl 1 classical motion hin such that Eq. (38) is satisfied with c ¼ ; where we neglected metric fluctuations. In terms of the next, we maximize the hend obtained solving Eq. (39) by tuning the amount of inflation where the fall happens, as number of e-folds N and of the Mukhanov-Sasaki variable uk ≡ aδhk, it becomes parametrized by Nin. The left panel of Fig. 7 shows the initial value N obtained following this procedure as a function of in d2u du k2 V00 ðh Þ h in units of H0. Smaller values of b (i.e., smaller values of k k − 2 eff cl 0 cr 2 þ þ 2 2 uk þ 2 uk ¼ : ð44Þ Mt) imply a flattening of the potential, and the classical dN dN a H0 H0 dynamics during inflation is slower. The right side of the curves is limited by the classicality condition in Eq. (38).A It is convenient to consider the evolution of the perturbation making reference to a specific moment before the end of ξH < 0 shifts the position of the potential barrier toward the limiting value hT in Eq. (37)—which does not depend on inflation: at the initial value Nin defined in Sec. III B.We max 0 recall that in our convention Nend ¼ at the end of ξH—above which the rescue mechanism due to thermal effects is no longer effective: its net effect is to reduce the inflation. Equation (44) becomes number of e-folds during which classical motion can happen 2 2 d uk duk k 2 − (for fixed b). − ðN NinÞ − 2 2 þ e uk We anticipate here the feature of PBH formation, which dN dN ainH0 implies the restriction on the parameter space mentioned at 00 VeffðhclÞ 0 point (i): the Higgs fall must start at least N ∼ 20 e-folds þ 2 uk ¼ : ð45Þ in H0 before the end of inflation. The collapse of the mass inside the horizon Ne-folds before the inflation end forms a PBH In this form, the Mukhanov-Sasaki equation is particularly with mass (see also Sec. III E), illustrative. Consider the evolution of the perturbation for a mode of interest k that we fix compared to the reference M¯ 2 ≈ Pl 2N value ainH0 at t ¼ tin. In particular, we consider the case of MPBH e : ð40Þ H0 a mode k that is subhorizon at the beginning of the classical ≫ evolution, that is, k ainH0. From Eq. (45), we see that PBH must be heavy enough to avoid Hawking evaporation. in the subsequent evolution with N 063005-10 DARK MATTER IN THE STANDARD MODEL? PHYS. REV. D 98, 063005 (2018) (a) (b) 2 FIG. 8. Left: Sample evolution of the classical Higgs background (hcl, red solid line) and of a perturbation with k=ainH0 ¼ 10 ζðkÞ (dashed lines). Right: Higgs curvature perturbation h during inflation. We compare the full numerical result with the analytical approximation [last term in Eq. (52), solid horizontal black line]. The vertical dashed gray line marks the instant of horizon exit. We use 12 2 the analytical approximation in Eq. (32) with hcr ¼ 410 GeV, b ¼ 0.09=ð4πÞ (which corresponds to Mt ¼ 172 GeV), and 12 H0 ¼ 10 GeV. 2 2 i j e-folds of inflation, such a mode exits the horizon: ds ¼ a ðtÞ½1 þ 2ζðx;⃗ tÞ δijdx dx ; ð49Þ oscillations stop, and from this point on, further evolution is driven by the time derivative of the classical background. and it is related to the (total) energy density perturbation δρ This is a trivial consequence of the equations of motion on and to the curvature perturbation on a generic slicing Ψ by superhorizon scales. Differentiating Eq. (39) with respect to the gauge invariant formula the cosmic time shows that h_ and δh satisfy the same cl k δρ δρ equation on superhorizon scales, and, therefore, they must ζ ¼ Ψ þ H ¼ Ψ þ H ; ð50Þ δ _ ≪ ρ0 ρ_ be proportional, hk ¼ CðkÞhcl for k aH0 [8]. The proportionality function CðkÞ can be obtained by a match- ing procedure. Deep inside the horizon, in the limit where we introduced the conformal time by means of dt ¼ adτ and H ¼ aH where H ¼ a0=a and prime 0 indicates k ≫ aH0, the Mukhanov-Sasaki variable u ≡ aδh repro- k k the derivative with respect to τ. The virtue of this definition duces the preferred vacuum of a harmonic oscillator in flat ζ Minkowski space, and we have, after introducing the is that, on superhorizon scales, is practically identical to pffiffiffiffiffi R τ τ −ikτ 2 the comoving curvature perturbation defined on hyper- conformal time as dt ¼ ad , uk ¼ e = k. Roughly surfaces of constant comoving time. Furthermore, on matching the absolute value of the solutions at horizon superhorizon scales ζ is conserved provided that the crossing, we determine the absolute value of CðkÞ as pressure perturbation is adiabatic. In conventional single- field inflation models, purely adiabatic perturbations are H0 jδh j¼pffiffiffiffiffiffiffi h_ ðtÞ; ð47Þ generated due to quantum fluctuations of the single field k 2 3 _ cl k hclðtkÞ driving inflation. The considered setup differs from a conventional scenario because of the presence, in addition where we indicate with tk the time of the horizon exit for the to the inflaton, of the Higgs field [8]. During inflation, the — ≡ mode k the time at which k ¼ aðtkÞH0 akH0 (equiv- curvature perturbation on uniform energy density slices is − τ 1 alently, k k ¼ ). The number of e-folds at the horizon given, in a spatially flat gauge, by exit is given by ρ_ δρ ϕ ρ_h k − ζ ¼ H0 ¼ ζϕ þ ζh; ð51Þ ¼ eNin Nk : ð48Þ ρ_ ρ_ ρ_ ainH0 where we separated the inflaton component ζϕ and the Higgs component ζh ≡ H0δρh=ρ_ h [8]. We assume that 1. Primordial curvature perturbations there is no energy transfer between the Higgs and the The primordial curvature perturbation ζðx;⃗ tÞ on uniform inflaton sectors, and that the latter generates, on large energy density slices ρ is defined (at the linear order) by the scales, the perturbations responsible for the CMB anisot- perturbed line element [31,32] ropies and large scale structures. Given these assumptions, 063005-11 GROSS, POLOSA, STRUMIA, URBANO, and XUE PHYS. REV. D 98, 063005 (2018) ζϕ and ζh are separately conserved on superhorizon scales approximation—corresponding to Γ ¼ 0 in Eqs. (54)— [33]. As customary, we can decompose ζh in Fourier Higgs and radiation evolve separately, and the Higgs decay modes. For a given comoving wave number k we have, occurs instantaneously at H ¼ Γ. The system in Eqs. (54) in a spatially flat gauge, can be solved using the following boundary conditions. As far as the classical Higgs field is concerned, we use the field δρðkÞ h_ δ_h V0 h δh values at the end of inflation computed in Sec. III C. The ζðkÞ h − ½ cl k þ effð clÞ k h ¼ H0 ¼ 2 evolution of ρ , in contrast, starts from the temperature in ρ_ 3h_ R h cl Eq. (35).9 Using the solution for ρ , Eq. (53) gives the time 2 R H0 evolution of the temperature. ≃ pffiffiffiffiffiffiffi ; ð52Þ 2 3 _ After instantaneous reheating, the Higgs potential sud- k hclðtkÞ k≪aH0 denly changes from the expression in Eq. (32) to the one in where the last analytical approximation is valid for the Eq. (36), and interactions with the SM bath generate a large T ζðkÞ thermal mass. If hend π2 where radiation perturbations are set to zero. We compute ρ g 4 R ¼ 30 T : ð53Þ ζðkÞ numerically at the time of Higgs decay, H ¼ Γ. After Higgs decay, radiation remains as the only component of The dynamics after reheating is described by the following the Universe and ζðkÞ is, therefore, conserved. system of coupled Higgs-radiation equations: ̈ 3 Γ _ 0 0 E. The power spectrum and the PBH abundance hcl þð H þ Þhcl þ VTðhclÞ¼ ; ð54aÞ The curvature power spectrum is given by ρ_ 4 ρ Γρ R þ H R ¼ h; ð54bÞ 3 k ðkÞ 2 Pζ ¼ jζ j : ð59Þ where the energy density of the Higgs field is given by 2π2 h_ 2 A numerical example is shown in Fig. 9. Its left panel ρ cl VT h ; ξ −10−3 h ¼ 2 þ effð clÞ ð55Þ shows that a small H ¼ has only a minor effect. The cut in the power spectrum at small k arises because of and the Hubble parameter is related to the total energy assumption (i): before that classical rolling starts, the Higgs density by 9More precisely, conservation of total energy determines the ρ þ ρ H2 R h : reheating temperature as ¼ 3 ¯ 2 ð56Þ MPl h_ 2 π2g h_ 2 3M¯ 2 H2 cl V h T4 cl VT h : The damping factor Γ takes into account the Higgs decays Pl 0 þ 2 þ eff ð clÞ ¼ 30 þ 2 þ eff ð clÞ hcl¼hend hcl¼hend at temperature T and represents the energy transportation ð57Þ from the Higgs field to radiation. We use the expression −3 quoted in [8], Γ ≃ 10 T. In the following, we shall adopt The approximation in Eq. (35) is valid because the inflaton the assumption of sudden decay approximation. In this energy density dominates over the Higgs contribution. 063005-12 DARK MATTER IN THE STANDARD MODEL? PHYS. REV. D 98, 063005 (2018) (a) (b) −3 FIG. 9. Left: The power spectrum of Higgs fluctuations produced by rescued Higgs fall for ξH ¼ 0 (red curve) and ξH ¼ −10 (blue curve), which has a minor impact. In the inset we show the number of e-folds at the horizon exit for each mode. Right: How the power spectrum changes when the classicality condition in Eq. (38) is changed by 10%. Z field is away from its minimum and exactly homogeneous. 2 ˜ 2 σ ðRÞ¼ dlnkPδðkÞW ðkRÞ; Relaxing this assumption would increase the power spec- Z trum at small k. 16 4 ˜ 2 The right panel shows the significant effect of a small ¼ dlnk 81 ðkRÞ PζðkÞW ðkRÞ; ð62Þ change in the arbitrary order one parameter c that defines the classicality condition in Eq. (38). Reducing c antici- where W˜ ðkÞ¼exp ð−k2=2Þ is the Fourier transform of the pates the initial moment tin (or equivalently Nin) where the window function. Assuming that PBH are formed when the Higgs starts to roll down the potential. As a consequence, δ ≈ 0 5 density perturbation exceeds th . [37], the fraction of earlier quantum fluctuations get taken into account by our the Universe ending up in PBH is given by the tail of the computation of Sec. III C. assumed Gaussian distribution, Finally, we can now compute the mass and amount of Z PBH generated by Higgs fluctuations. The radius of the ∞ 1 δ2 β γ δ ffiffiffiffiffiffi − Hubble horizon or the wavelength of the modes determines ðMPBHÞ¼ d p exp 2 δ 2πσðM Þ 2σ ðM Þ the typical mass of the PBHs [34], th PBH PBH σ≪δ γσ δ2 ≃ th ffiffiffiffiffiffi − th p exp 2 : ð63Þ γ M2 2πδ 2σ ≈ Pl 2N th MPBH e ; ð60Þ 2 H0 The latter approximation is relevant, given that the obtained −2 power spectra Pζ are of order 10 . We convert σðRÞ to a where N is the number of e-folds when the k mode leaves σ γ ≈ 0 2 function ðMPBHÞ, taking into account that the size R is the horizon; . is a correction factor [35]. For example, ≈ 2 γ 12 −15 related to the mass MPBH as R GMPBH= a. The fraction H0 ¼ 10 GeV and N ¼ 20 gives M ≈ 10 M⊙. PBH of PBH relative to the DM abundance at given mass, To compute the fraction of the Hubble volume collapsing f ðM Þ, is given by [36] to PBHs, we need the variance of the smoothed density PBH PBH 2 2 perturbation over a radius R, σ ðRÞ¼hδ ðx;RÞi, where 1 4 1 2 R 10.75 = γ M⊙ = 3 0 0 0 ≈ 2 7 108 β δðx;RÞ¼ d x δðx ÞWðx − x ;RÞ is the density fluc- fPBH . × : ð64Þ g 0.2 M tuation smoothed by a window function Wðx;RÞ, assumed ;form PBH to be Gaussian, The distribution of PBH as a function of their mass MPBH is strongly peaked at the value that maximizes σðMÞ, in view 1 2 − jxj of the exponential factor in Eq. (63). In terms of the power Wðx;RÞ¼ 3 2 3 exp 2 : ð61Þ ð2πÞ = R 2R spectrum Pζ this means that the PBH abundance roughly scales as e−1=Pζ and is dominated by the top of the peak of The variance can be computed in terms of density power PζðkÞ. The PBH mass distribution is peaked at the value spectrum PδðkÞ, which is related to the curvature pertur- corresponding to the k that maximizes PζðkÞ. The desired −2 bation ζ power spectrum PζðkÞ as [36] abundance is reproduced for max Pζ ∼ 10 , and slightly 063005-13 GROSS, POLOSA, STRUMIA, URBANO, and XUE PHYS. REV. D 98, 063005 (2018) larger (smaller) values produce way too many (too few) the O(4,1) invariance of de Sitter. Thereby we can compute PBH. the evolution of Higgs fluctuations at a fixed point in space This means that, due to the fine-tuned nature of the and study the correlation as a function of time t,or mechanism (as in all models that can generate PBH), the equivalently as a function of the number of e-folds N. amount and mass of PBH depend in an extremely sensitive At a large time separation the correlation is well approxi- way on the uncertain SM and cosmological parameters, mated by its dominant exponential and parametrized in τ mainly the top mass and the Hubble constant H0. terms of a correlation time corr or equivalently in terms of τ Furthermore, uncertainties in the computation of black number of e-folds Ncorr ¼ H0 corr as [38] δ hole formation (such as the value of th) imply uncertainties 2 −jt1−t2j=τ 2 −jN1−N2j=N of many orders of magnitude in the PBH density. hhðt1; x⃗Þhðt2; x⃗Þi ≃ hh ie corr ¼hh ie corr : For example, a 10% variation in the order one arbitrary ð66Þ constant c that parametrizes the classicality condition has an order one impact on the power spectrum (Fig. 9), and O(4,1) invariance implies that the spatial correlation length consequently a huge impact on the PBH abundance. is exponentially large eHNcorr , namely that space is expo- In addition to SM parameters and to H0, the PBH nentially inflated. We demand that N is larger than about abundance also depends on two extra parameters that corr 40 in order to produce a smooth region as large as our characterize the assumed initial condition: constant hðxÞ¼ Universe at N ≈ 20 e-folds before the end of inflation. h at N e-folds before the end of inflation, when the in in in Before computing numerically N for the SM potential, it approximated classicality condition is satisfied. The next corr is useful to consider some simple limits: section reconsiders if this assumption is justified. (0) A massless free scalar h is a simple but special case, because it does not “thermalize” to the equilibrium F. Is a homogeneous Higgs background distribution. Rather, it undergoes random walk diffu- a sensible assumption? sion. Starting from h 0, after Ne-folds one has pffiffiffiffi ¼ ≈ 2 1=2 The computation was based on assumption (i): at Nin hh i ¼ NH0=2π and the correlation length in a 20 2π e-folds before the end of inflation, the Higgs field must region with given hhi is Ncorr ¼ hhi=H.Imposing 40 0 16 be away from its minimum and constant within the Ncorr > implies H0 < . hhi. 2 presently observable horizon. (m) A free scalarp withffiffiffiffiffiffiffiffi squared mass m > 0 fluctuates as hh2i1=2 ¼ 3=2H2=2πm with correlation length 3 2 2 2 −12ξ 2 1. Approximate Higgs homogeneity? Ncorr ¼ H =m [38].Form ¼ HH one gets −1 4ξ 40 −0 006 ξ 0 We here show that approximate homogeneity is a natural Ncorr ¼ = H > for . < H < . Roughly, product of inflation, provided that the SM Higgs potential this will be our final result. satisfies certain conditions. Quantum corrections in infla- tionary (de Sitter) spacetime have been studied in [38], which showed that long wavelength fluctuations can be described by a Fokker-Planck equation for ρðh; NÞ, the probability of finding the Higgs field at the value h at Ne-folds of inflation, ∂ρ H2 ∂2ρ ∂ V0 ρ − 0 eff ¼ 2 2 þ 2 : ð65Þ ∂N 8π ∂h ∂h 3H0 The first term on the right-hand side is a diffusion term due to quantum fluctuations while the second term is a drift (or transport) term due to the potential. After some e-folds, the ρ ∝ distribution converges to its equilibrium value eqðhÞ −8π2 3 4 expð VeffðhÞ= H Þ [38]. We are interested in the probability of having a roughly constant Higgs away from its minimum h ¼ 0 within the presently observable horizon. This configuration is a natural outcome of inflationary dynamics provided that FIG. 10. Contour plot of the correlation length N of Higgs the correlation length of Higgs fluctuations is larger than corr inflationary fluctuations as a function of the main parameters, ξH the present horizon. Following [38], the computation of the and hcr=H0. The needed homogeneous patch, Ncorr > 40, can be correlation length is simplified, observing that correlation obtained in the green region, which corresponds to a small functions depend on space and time separations respecting negative ξH. 063005-14 DARK MATTER IN THE STANDARD MODEL? PHYS. REV. D 98, 063005 (2018) (a) (b) 12 2 12 FIG. 11. We fix the SM to Mt ¼ 172 GeV [which corresponds to hcr ¼ 4 × 10 GeV, b ¼ 0.09=ð4πÞ ] and H0 ¼ 10 GeV and explore the dependence on the parameters that define the assumed initial condition. Left: Final outcome as a function of the homogeneous initial value of the Higgs field hin at Nin e-folds before the end of inflation and of Nin. The desired PBH abundance is produced inside the white band. Inflationary fluctuations in hin are much larger than the needed tuning. Right: We fix Nin and introduce dhin=dN as an additional parameter. (λ) A massless scalar with an interaction λh4=4 and λ > 0 abundance is obtained within the narrow strip in Fig. 11(a). fluctuates as h2 1=2 0.36H=λ1=4 with N Its boundaries have been computed as follows: pffiffiffi h i ¼ corr ¼ 7 62 λ 40 λ 0 036 (i) A slightly smaller h leads to a negligible PBH . = .SoNcorr > for < . . This is satisfied in in the SM at large energy. amount. In Fig. 11(a) we show the peak value of the The above two results agree, up to order one factors, power spectrum and shade in gray regions where it is −2 approximating a generic VðhÞ as a massive field with below 10 . 2 m ¼ V00. The precise value of the correlation length can be (ii) A slightly larger hin leads to a too large hend not obtained as proposed in [38]: the dominant exponential that rescued by reheating. In Fig. 11(a) we shade in red solves Eq. (65) can be computed as the eigenvalue of a regions where hend is above the maximal value that λ can be rescued by instantaneous reheating. Schrödinger-like equation. In the SM effðhÞ runs to 00 _ negative values, making the curvature V negative at large In Fig. 11(b) we show that a variation in hin has a smaller field values; the patch with a large correlation length is effect as it gets redshifted away. created while h is climbing the potential in its region with Within the assumption that hin is homogeneous, its tuned positive curvature. The result is shown in Fig. 10, and value can be justified on an anthropic basis provided that basically agrees with the result anticipated at point (m): ξH PBH make all DM [8,39]. However, one single fluctuation δ ≳ 10−3 must be negative and small. Such a small ξH is roughly hin hin away from the assumed perfect homo- compatible with the size of quantum corrections to ξH. The geneity can lead to one vacuum decay bubble that, after running Higgs quartic coupling is small enough that it does inflation, expands engulfing the observable Universe. not spoil the mechanism, as qualitatively understood at As discussed previously, inflation can produce an point (λ). approximate homogeneity within a large patch, but up to δ ∼ 2π 10 In conclusion, a roughly homogeneous Higgs field (up to fluctuations of order hin H0= . Additional fluctua- fluctuation of order H0) encompassing our whole horizon is tions in h_ are less significant and we ignore them. The ξ in a natural outcome of inflation, provided that H is small Hubble constant H0 cannot be significantly reduced, for enough. In a multiverse context, its fine-tuned value that ≳ two reasons: First, H0 hcr is needed to keep the Higgs leads to DM as PBH can be justified on an anthropic basis. fluctuating around the top of its potential barrier, until it starts to classically roll down. Second, it would suppress δ 2. Exact Higgs homogeneity? the unwanted prefall fluctuations hin at the price of Figure 11(a) shows that the initial homogeneous Higgs fields hðxÞ¼h at N N e-folds before the end of in ¼ in 10 inflation must be tuned to a part in about 10−3 in order to Technically, such fluctuations appear in our equations when (motivated, e.g., by the arbitrariness in the classicality condition) produce a final hend close to the maximal value that can be we start from a different initial time: this shifts the position and rescued by reheating, as needed to produce a substantial the height of the presumed peak of the power spectrum, and amount of primordial black holes. An interesting PBH thereby the abundance and mass of PBHs. 063005-15 GROSS, POLOSA, STRUMIA, URBANO, and XUE PHYS. REV. D 98, 063005 (2018) suppressing postfall fluctuations that generate PBHs; seems worrying enough that one wonders if it can be see Eq. (52). avoided or alleviated. Thereby we must take into account the effect of One possibility is devising a mechanism that suppresses fluctuations in the initial hðxÞ. Pictorially, the status of vacuum decay by rescuing the Higgs more efficiently the Universe should now be represented in Fig. 10 not by a than the thermal barrier considered in [8,10]. For example, point (which can lie in the desired region), but by a dot, a nonthermal distribution11 or an inflation that initially much thicker than the desired region. decays to the SM particles more coupled to h increases its ≈ 20 This is a problem because, at Nin e-folds before the thermal mass. The green dashed line in Fig. 11 shows the end of inflation (which needs at least 60 e-folds), the present extra rescued region imposing what we believe is the most 120 max max 0 Universe is composed by about e causally disconnected optimistic possibility: hend 063005-16 DARK MATTER IN THE STANDARD MODEL? PHYS. REV. D 98, 063005 (2018) 120 smaller than the DM abundance. Among sparse comments, ð1=2Þe , that trying to justify it through anthropic con- we mention that direct detection of S on an antimatter target siderations leads to the issues discussed in Sec. III F. gives an annihilation signal. We conclude that tentative searches and interpretations In the second part of the paper, we considered the of dark matter as a phenomenon beyond the Standard proposal of [8]: given that the SM Higgs potential can be Model remains a justified field. unstable at large field values, during inflation the Higgs might fall from an assumed homogeneous vacuum expect- ACKNOWLEDGMENTS ation value h beyond the potential barrier toward the Planck We thank G. Ballesteros, I. Bombaci, A. Bonaccorso, A. scale. If the fall is tuned such that h almost reaches the Francis, A. Gnech, J. Green, M. Karliner, D. Racco, and Y. maximal value that can be rescued by thermal effects, this Wang for discussions and M. Redi for participating in an process generates small-scale inhomogeneities that form early stage of this project. We especially thank G. Farrar for primordial black holes. While DM as PBH seems excluded discussions about S breakup cross sections and A. Riotto in the proposed mass range (just above the bound on for many useful suggestions at all stages of this work, and Hawking radiation), the proposal is interesting enough to in particular for clarifications about nonadiabaticity of ζ deserve further scrutiny. We confirmed the computations of h ξ because of interactions with the plasma. A. U. thanks all [8] and extended them, adding a nonminimal coupling H participants in the Dark Matter Coffee Break at SISSA for of the Higgs to gravity. We find that inflationary fluctua- ξ many stimulating discussions about the physics of primor- tions can generate a quasihomogenous h only if H is as dial black holes. This work was supported by the ERC small as allowed by quantum corrections that unavoidably Grant No. 669668 NEO-NAT. generate it. Furthermore, we find in Sec. III F that the amount and mass of PBH depend on an extremely sensitive Note added.—G. Farrar in arXiv:1805.03723 proposed that a way on the uncertain SM and cosmological parameters, co-stable S dark matter with 1.86 GeV [1] R. L. Jaffe, Strangelets, Nucl. Phys. B, Proc. Suppl. 24B,8 [8] J. R. Espinosa, D. Racco, and A. Riotto, A Cosmological (1991). Signature of the Standard Model Higgs Vacuum Instability: [2] G. R. Farrar and G. Zaharijas, Nuclear and nucleon tran- Primordial Black Holes as Dark Matter, Phys. Rev. Lett. sitions of the H dibaryon, Phys. Rev. D 70, 014008 (2004). 120, 121301 (2018). [3] G. R. Farrar, Stable sexaquark, arXiv:1708.08951. [9] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, [4] G. R. Farrar, 6-quark dark matter, Proc. Sci., ICRC2017 G. F. Giudice, G. Isidori, and A. 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