PHYSICAL REVIEW D 101, 084022 (2020) Material matter effects in gravitational UV=IR mixing Joseph Bramante and Elizabeth Gould The McDonald Institute and Department of Physics, Engineering Physics, and Astronomy, Queen’s University, Kingston, Ontario K7L 2S8, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Received 12 November 2019; accepted 30 March 2020; published 9 April 2020) We propose a matter effect for the gravitational ultraviolet/infrared (UV=IR) mixing solution to the cosmological constant problem. Previously, the gravitational UV=IR mixing model implied a nonstandard equation of state for dark energy, contradicting observation. In contrast, matter effect gravitational UV=IR mixing accommodates a standard ΛCDM cosmology with constant dark energy. Notably, there are new density-dependent predictions for futuristically precise measurements of fundamental parameters, like the magnetic moments of the muon and electron. DOI: 10.1103/PhysRevD.101.084022 I. INTRODUCTION Nelson [2] (CKN) argues to the contrary, that effective field theory breaks down for any UV cutoff Λ ≳ meV in a The cosmological constant problem can be expressed UV region the size of our universe. as follows. Let us compare the observed vacuum energy in The gravitational UV=IR mixing proposal follows from our universe V4 ∼ ð2 meVÞ4 to the zero-point loop obs the observation that there will be inherently nonlocal vacuum energy contribution from a field with mass m gravitational dynamics in sufficiently large and dense and momentum p, systems. If loop-level zero-point field energy densities Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λ 3 1 Λ4 over a space L are so large that a black hole forms inside δρ ≃ UV dp 2 þ 2 ≃ UV ð Þ Λ 3 p m 2 ; 1 L, we have entered a computational regime for which our Λ ð2πÞ 2 16π IR local effective field theory has broken down, unless the theory properly accounts for the presence of virtual black ≪ Λ where the last expression has assumed m UV and for hole states in loop-level field dynamics. Put another way, Λ ¼ 0 now we neglect the infrared cutoff, setting IR . we should expect that the infrared cutoff of our theory Standard Model field dynamics have been explored up Λ ¼ 1=L, implies a UV cutoff at around the threshold for Λ ∼ IR to an ultraviolet cutoff UV TeV. With no evidence for a black hole formation. It follows that zero-point loop mechanism that cancels these contributions to the cosmo- corrections to vacuum energy in a properly defined logical constant, after probing energies up to a TeV at effective field theory can only be reliably calculated if colliders, the straightforward prediction for vacuum energy 4 from field fluctuations in our universe is at least ∼TeV , 4π ≳ ¼ 2 ðδρ Þ 3 ð Þ which is nearly sixty orders of magnitude larger than L RS G Λ 3 L ; 2 observed. For a review of the cosmological constant problem, see [1]. where 4π L3 is the volume of the space, R ¼ 2GM is the A crucial assumption in the standard cosmological 3 S black hole horizon, δρΛ is the loop contribution to the constant argument is that the local quantum field theory 1 vacuum energy density, and G ¼ 2 is Newton’s constant. underlying the loop calculation in Eq. (1) is validly applied MP Λ up to a cutoff UV across a region the size of our universe. Saturating this inequality leads to a maximum vacuum The gravitational UV=IR mixing solution to the cosmo- energy contribution from loop corrections, logical constant problem proposed by Cohen, Kaplan, and 3 2 MP δρΛ ≲ : ð3Þ 8π L2 Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. In essence, this inequality reflects that Eq. (1) shouldpffiffiffiffiffiffiffiffiffiffiffiffiffi only Further distribution of this work must maintain attribution to Λ ¼ δρ1=4 ≃ the author(s) and the published article’s title, journal citation, be evaluated up to a UV cutoff UV Λ MP=L, and DOI. Funded by SCOAP3. consistent with our ignorance of quantum mechanics 2470-0010=2020=101(8)=084022(6) 084022-1 Published by the American Physical Society JOSEPH BRAMANTE and ELIZABETH GOULD PHYS. REV. D 101, 084022 (2020) around black holes, whose size is determined by the IR to lepton g − 2 measurements at futuristically precise Λ ¼ 1 cutoff IR =L. experiments. If the Hubble parameter is used as an IR cutoff for the Before continuing, for the sake of clarity it is worth Λ ¼ 1 ¼ universe [2], IR =L H, the prediction for loop pointing out that we will exclusively treat the dark energy 4 contributions to vacuum energy is δρΛ ∼ meV , which is that has been observed in our universe as either a constant around the size of the observed vacuum energy. This can be vacuum energy or a (not necessarily static) vacuum energy verified by comparing Eq. (3) to the reduced Friedmann provided by loop corrections in what follows. We include a equation for a background density ρ in flat space, brief discussion of the implications of quintessence fields and other time-varying sources of dark energy in our 3 2 2 ¼ ρ ð Þ conclusions. 8π H MP ; 4 and noting that the energy density of our universe is II. MATTER EFFECT FOR GRAVITATIONAL presently ρ ∼ meV4. This remarkable result can be under- UV=IR MIXING stood as a consequence of our universe being both flat and Incorporating a matter effect substantially changes the critical, to within current measurement precision [3].In gravitational UV=IR mixing solution to the cosmological other words, the vacuum dominated energy density we have constant problem. While the UV=IR mixing contribution to observed within a Hubble radius is very near to the energy the vacuum energy of the universe proposed by CKN density required to form a black hole. requires that zero-point loop corrections to the vacuum On the other hand, using the Hubble parameter as IR energy density not exceed the threshold where a black hole cutoff and treating Eq. (3) as an equality, would form in a space of size L, we propose that the 3 restriction should be extended to include any matter, δρ ¼ 2 2 ð Þ “ ” Λ 8π H MP: 5 radiation, and bare vacuum energy contained in a space of size L. This proposal aims to incorporate the effect of immediately predicts a problematic equation of state for physical field densities present in a system. vacuum energy in our universe [4]. Our universe was Our contention is that, insofar as a quantum field initially radiation and then matter dominated. During these theoretic description of the volume enclosed in L is expansionary epochs, H will evolve with the expansion breaking down when the field density of the system reaches scale factor a like 1=a2 and 1=a3=2, respectively. This a certain threshold, one should treat “virtual” and “physi- implies that vacuum energy δρΛ in Eq. (5) will also redshift cal” field configurations on equal footing. In other words, approximately like matter and radiation during these the new “matter effect” condition is that both virtual epochs. Reference [4] pointed out that Eq. (3) implies a contributions to the energy density of a system and physical δP “ ” nonstandard equation of state, specifically w ≡ δρ > 0, for or on-shell field configurations, should not exceed the energy density of a black hole filling the space occupied by the vacuum energy density δρΛ. This can be excluded using present measurements of the equation of state for dark the system, ¼ −1 03 Æ 0 03 energy w . [3]. Subsequently, Ref. [5] pro- 4π posed using the future event horizon of the universe as the ≳ ¼ 2 ðδρ þ ρ Þ 3 ð Þ R L RSchw G Λ 0 3 L 6 ¼ ∞ dt IR cutoff in (3), L a t a . However, the most straight- forward version of this future horizon proposal predicted where ρ0 ¼ ρm þ ρr þ ρΛ is the sum of all physical w ≈−0.9. contributions, matter, radiation, and vacuum energy, to In the remainder of this article we will advocate for a the energy density of the system. Note especially that ρ0 different treatment than CKN for setting a black hole bound includes any physical (ρΛ) as opposed to virtual (δρΛ) on the validity of effective field theories. While CKN contribution to the vacuum energy. considered only virtual (i.e., loop-contributions) to vacuum The matter effect gravitational UV=IR inequality indi- energy density, we propose that the correct energy density cating the breakdown of effective field theory for calculat- threshold at which effective field theory breaks down is ing loop contributions to vacuum energy is determined by both physical and virtual energy densities. In δρ ∼ 4 3 2 contrast to CKN, which predicted Λ meV , our pro- MP δρΛ þ ρ0 ≲ : ð7Þ posal leads to the conclusion that loop-contributions to 8π L2 vacuum energy can be consistently set to zero, δρΛ ∼ 0.In the remainder of this paper we detail this “matter effect” Setting this to an equality leads to a vacuum energy variant of the CKN solution to the cosmological constant contribution problem, which is consistent with the observed equation of 3 2 state for dark energy (w ¼ −1), implies certain bounds on MP δρΛ ¼ κ − ρ0: ð8Þ the curvature of the universe, and predicts new corrections 8π L2 084022-2 MATERIAL MATTER EFFECTS IN GRAVITATIONAL UV=IR … PHYS. REV. D 101, 084022 (2020) In the final expression we have introduced an order unity A. Flat and open curvature constant κ, which parametrizes the equality. In the case that in gravitational UV=IR mixing effective field theory breaks down when field energy Matter effect gravitational UV=IR mixing makes some densities imply the formation of an uncharged, spin zero loose predictions regarding the curvature of the universe.