Material Matter Effects in Gravitational UV/IR Mixing

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 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 straightforward 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

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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. κ ¼ 1 black hole, we expect . To understand curvature in gravitational UV=IR mixing cosmology, we first consider the full Friedmann equation III. MATTER EFFECTS AND for a homogeneous, isotropic universe, UV=IR MIXING COSMOLOGY 3 In order to apply the matter effect UV=IR relation to any 2 2 ¼ ρ þ ρ þ ρ þ δρ ¼ ρ ð Þ H MP r m Λ Λ k 11 system, including the observable universe, we need to 8π determine the length scale which defines the IR cutoff of −k where ρ ≡ 2 accounts for the curvature of the universe the system, Λ ¼ 1=L. The Hubble horizon has been k a IR ¼ −1 0 þ1 previously identified as an appropriate IR cutoff for UV=IR and k ; ; for an open, closed, and flat universe, −1 ¼ ¼ Λ respectively. relations in [2], L H IR. It defines the boundary beyond which material recedes from an observer faster than Substituting this full Friedmann equation into Eq. (8) ¼ −1 the speed of light, and so it provides a well-motivated with L H , we find choice for the IR cutoff of the universe, since we are 3 3 interested in defining a boundary within which field 2 2 ¼ κ 2 2 þ ρ ð Þ H MP H MP k 12 densities are restricted by the density of a black hole 8π 8π contained within the same radius. In contrast, [5] proposed As we have already noted setting ρk ¼ 0 and κ ¼ 1, using the proper distance to theR event horizonR of the ∞ dt ∞ da which is consistent with present measurements [3], implies universe as an IR cutoff, L ¼ a ¼ a 2, which 0 a a Ha a flat universe and is compatible with vanishing zero-point resulted in an equation of state for dark energy closer to loop corrections to vacuum energy, and the result of that ≈−0 9 w . . In the following treatment of matter effect computation was the Friedmann equation for a flat UV=IR mixing, we will use the Hubble horizon as the universe. relevant IR cutoff length scale. For a discussion of other IR We have seen that matter effect UV=IR mixing would cutoff choices, see Sec. IV. be consistent with a perfectly flat universe. However, In the case that our length scale is given by the Hubble our starting point in this regard, Eq. (7), really only gave horizon, Eq. (8) becomes an approximate prediction for the scale at which effe- 3 ctive field theory breaks down based on a Schwarzschild ρ þ ρ þ ρ þ δρ ¼ κ 2 2 ð Þ r m Λ Λ 8π MPH : 9 black hole solution. Therefore, we will examine what happens when κ and ρk take on different values, assuming This equation bears a striking resemblance to the the Schwarzschild black hole UV=IR bound, Eq. (7), 3 2 2 8π ¼ ρ þ Friedmann equation in flat space, MPH = 0 applies in curved spacetimes. Using the Friedmann equa- δρΛ, although it has been derived by other means, namely tion and setting L ¼ H−1 we can rephrase Eq. (7) as by placing black hole density restrictions on the validity of effective field theory within a Hubble radius. Note in 3 2 MP κ ¼ 1 δρΛ þ ρ0 ≤ κ ; particular that for a flat universe and , loop correc- d 8π L2 tions to the cosmological constant will vanish, δρΛ → 0,if 3 ð1 − κ Þ 2 2 ≤ ρ ð Þ the vacuum energy in our universe is attributed to a cosmo- d 8π H MP k; 13 ρ ¼ 4 logical constant Λ Vobs. In this case, the Friedmann equation becomes where κd now defines deviations from the inequality given 3 in Eq. (7). We recall that around Eqs. (7) and (8) we argued 2 2 ¼ ρ þ ρ þ ρ ð Þ MPH r m Λ: 10 that our effective field theory is supposed to break down for 8π −1 κd ≳ 1, since this implies a region inside radius H with a Substituting this into Eq. (9) with κ ¼ 1, we find δρΛ ¼ 0. total energy density greater than that required to form a Evidently matter effect gravitational UV=IR mixing black hole. However, we recall that the inequality given in addresses the cosmological constant problem by predicting Eq. (7) used the Schwarzschild horizon as a limit on that effective field theory breaks down for any δρΛ > 0, effective field theories contained in some radius L, but this in the presence of a physical vacuum energy density, would seem to depend on the details of how field theories ρ ¼ 4 Λ Vobs. This altogether implies a constant background are altered at this limit. However, as we will see Eq. (13) 4 vacuum energy Vobs, with effective field theory breaking implies some cosmological restrictions on the range of down if any additional zero-point loop contributions to values κd can take, based on the nonobservation of vacuum energy are introduced. curvature in our universe.

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To see how these restrictions arise, we first suppose that ρða¼1Þ ρ ρða¼1Þ ρða¼1Þ _ 2 ¼ − k þ 0 ≥ k ¼ k ð Þ κ ≤ 1, which implies that effective field theories are a 2 ; 18 d C C a C0 C ðκ − 1Þ altered a bit before the Eq. (7) limit on field theories is F F F d ρ reached. In this case k will never be negative, which where the inequality on the right-hand side of this expres- implies that the universe is either flat or open. But in fact, sion is obtained from Eq. (16). This in turn implies a bound κ 1 for d < , the universe is necessarily open. Moreover, the on the curvature of a closed UV=IR mixed universe, observation of a nearly flat universe thus requires that κd is not too small, κ ≥ 0.994 at 2σ, using both cosmic micro- ρ0 κ d ≥ d : ð19Þ wave background and baryon acoustic oscillation data [3]. jρkj κd − 1

B. Closed universe limitations IV. ALTERNATIVE IR CUTOFFS It is also possible to fruitfully consider the case that For a cosmological system like our universe, one might κ 1 d > , which implies that the inequality in Eq. (7) is consider choosing one of three seemingly relevant length κ 1 slightly violated. For the case that d > , the matter effect scales to set the IR cutof—the particle horizon which UV=IR mixing formula would predict a closed universe. It determines what could have affected us in the past, the also implies a limit on how large curvature will become in Hubble horizon which is the standard relevant length, and the future. First, we note that the only nonconstant part of the event horizon, which indicates the future causally 2 the left hand side of Eq. (13) is H . This means that we can connected volume. In the limit that the Hubble horizon define a constant is constant, all three scales are the same. See [5–7] for examples of these horizons being utilized. 3 − ≡ ð1 − κ Þ 2 ð Þ In preceding derivations, we have assumed that the IR C0 d 8π MP; 14 1 cutoff length L is given by the Hubble horizon H.Nowwe will examine the case when L is given by two other κ 1 where this definition emphasizes that d > . Equation (13) horizons, the particle horizon becomes Z t dt Lp ¼ a ; ð20Þ − _ 2 −2 ¼ −jρða¼1Þj −2 ð Þ 0 a C0a a k a ; 15 and the event horizon Z where ρ has been defined at a reference scale (a ¼ 1). This ∞ k ¼ dt ð Þ yields a minimum value for ja_j, Le a : 21 t a Neither will prove useful for the purposes of constructing a ρða¼1Þ 2 a_ ≥ k : ð16Þ realistic cosmology based on matter effect gravitational C0 UV=IR mixing. Starting with the matter effect UV=IR inequality (13), _ We see that since a is positive at the present time, it will 3 3 2 always be positive, which implies that a matter effect 2 2 ≤ κ MP ð Þ H MP 2 : 22 UV=IR closed universe cannot have both finite space and 8π 8πL finite time. If we choose κ ¼ 1 as above, it immediately follows that We now obtain a restriction on the curvature of this 1 theory using the Friedmann equation, L ≤ : ð23Þ H 3 _ 2 This implies a Hubble horizon that is larger than or equal to 2 a ¼ −4ρða¼1Þ þ −3ρða¼1Þ M 2 a r a m the particle or event horizon. Neither case is permitted in 8π P a our universe, where it has been observed that L >H−1 þ −2ρða¼1Þ þ ρða¼1Þ þ −3−3wδρða¼1Þ ð Þ p a k Λ a Λ ; 17 −1 and Le >H [8]. where we have defined all energy densities as being fixed at reference scale a ¼ 1. In what follows we will also define V. EXPERIMENTAL CONSEQUENCES ≡ 3 2 FOR MATTER EFFECT UV=IR MIXING CF 8π MP for convenience. Now schematically, we know that our universe is radiation dominated at early times. From Eq. (8), it follows directly that matter effect Therefore, there is a value of a_ at early times where the gravitational UV=IR mixing predicts observable conse- radiation-dominated energy density of the universe is quences for particle experiments with sensitivity to new approximately ρ0 and dynamics at an effective field theory cutoff

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2 2 4 MP L Λ ≃ − ρ0; ð24Þ δðg − 2Þ − δðg − 2Þ ≃ ρ0 UV 2L2 ME UV=IR M2 P 2 ρ0 L ρ ≃ 10−13 ; where we reiterate that 0 is the energy density in the region g=cm3 107 cm that the experimental measurements take place and L is the ð Þ length scale of the region, which is related to the infrared 26 Λ ∼ 1 cutoff as IR L. Note that unlike Eqs. (8) and (24) has not 3 assumed a spherical geometry for the region bounded by L. where we have adopted a terrestrial density of gram=cm ¼ 107 In the original gravitational UV=IR proposal, CKN and a plausible collider length scale of L cm. We noted that electron g − 2 measurements might some day see that the precision measurements of lepton magnetic reach the precision required to observe nonlocal gravita- moment required to test matter effect gravitational UV=IR tional corrections. CKN calculated the minimum possible mixing lie beyond present experimental capabilities. In correction to the electron g − 2 by simultaneously varying particular, the precision of the Brookhaven E821 muon g-2 both UV and IR cutoffs [2]. On the other hand, in a measurement would need to be improved by over ten orders companion paper [9], we have found that a straightforward of magnitude for the matter effect to become relevant. Λ However, this matter effect would be sought after first choice for the infrared cutoff IR, leads to interesting and measurable consequences for the muon g − 2. Indeed, we finding a larger vacuum gravitational UV=IR mixing effect, have found that this could serve as an explanation of the which could be found at current experiments [9]. anomalous muon g − 2 anomaly observed in the last decade. Thus far, the muon g-2 has been measured to VI. CONCLUSIONS match Standard Model predictions at parts-per-billion precision. The Brookhaven Muon E821 measurement We have introduced a matter effect variant of the exp −10 gravitational UV=IR mixing prescription for the cosmo- [10] of aμ ≡ ðg − 2Þ =2 ¼ 11659208.0ð6.2Þ × 10 can μ logical constant problem. It appears that nonlocal gravita- be compared to the Standard Model prediction [11], tional effects could indeed resolve the cosmological SM ¼ 11659182 05ð3 56Þ 10−10 aμ . . × . constant problem, so long as physical field densities are However, for the purposes of calculating the relative properly incorporated when determining the UV cutoff for impact of a matter effect UV=IR mixing correction to which field theory breaks down in a space of size L.Aswe − 2 δð − 2Þ lepton g measurements, g ME, as compared to the have discussed, UV=IR mixing appears to mildly prefer a δð − 2Þ original UV=IR mixing correction g UV=IR, here we flat or open universe, since in these cases the black hole will not need to extensively address the definition of the IR density bound is not saturated. However, slightly beyond Λ cutoff IR. the naive limits of the bound, a closed universe with a With the UV cutoff for new dynamics defined as in restricted curvature is also possible. The matter effect Eq. (24), we would like to consider whether terrestrial UV=IR theory predicts local-density-dependent corrections experiments could reach the precision necessary to test to extremely precise measurements of fundamental con- matter effect UV=IR mixing. The leading order muon g-2 stants, like the magnetic moment of electrons and muons. correction from UV=IR mixing is [12] Many related topics remain open for further exploration. While we have shown that zero-point loop corrections to 2 2 α mμ α mμ vacuum energy can vanish for a universe with a cosmo- δðg − 2Þ ≃ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi logical constant, which in this case means a constant π Λ π M2 UV P − ρ 2 2 0 background vacuum energy density, it would be interesting L pffiffiffi 2 2 to determine how this proposal changes in the presence of a αmμL L ¼ 2 1 þ ρ þ quintessence field, or some other source of varying vacuum π 2 0 ; MP MP energy density. Along similar lines, future work might also ð25Þ consider whether zero-point loop corrections could account for a subdominant portion of the observed vacuum energy, so that the left-hand side of Eq. (8) is nonvanishing. Such a where α is the fine structure constant and we have expanded proposal could be constrained by bounds on the equation of in the limit that the matter effect correction is small, 2 state for dark energy, which similarly restricted the original M ρ ≪ P 0 2L2, which we believe is satisfied by any presently CKN proposal [2], which attributed all of the observed conceivable terrestrial experiment. Therefore, the matter vacuum energy to zero-point loop corrections [4,5]. effect correction to UV=IR mixing in the laboratory, We have found the implications of a gravitational relative to the nonmatter effect UV=IR mixing correction UV=IR mixing matter effect for particle experiments, but [i.e., the first expansion term relative to the second in we have assumed a constant background matter density. Eq. (25)], is approximately A time-varying background matter density should also be

084022-5 JOSEPH BRAMANTE and ELIZABETH GOULD PHYS. REV. D 101, 084022 (2020) considered. We look forward to exploring these and other vacuum energy could be sourced by zero-point loop aspects of gravitational UV=IR mixing in future work. corrections. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada. ACKNOWLEDGMENTS Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by We wish to thank the anonymous referees for many helpful comments and suggestions. We wish to particularly the Province of Ontario through the Ministry of Economic thank one anonymous referee for a discussion of the Development & Innovation. possibility that a subdominant portion of the observed

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