aa r X i v : . [ phy s i c s . g e n - ph ] F e b Entanglement as a resource for naturalness
Andrei T. Patrascu ELI-NP, Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering,30 Reactorului St, Bucharest-Magurele, 077125, Romania
A novel approach to understanding the hierarchy problem is presented making use of topologicalaspects of the renormalisation group and the ER-EPR interpretation of entanglement. A commondiscussion of the renormalisation group, the black hole horizon and the expected entanglementbetween outgoing Hawking radiation and the interior, the cosmic censorship mechanism and thecosmological constant problem is envisaged.
INTRODUCTION
The discovery of the Higgs boson at the frontier be-tween the regions of stability and metastability of thestandard model vacuum with no other supersymmetricpartners found at reasonable energies has triggered manyquestions regarding naturalness. As it stands, the stan-dard model seems un-natural. Of course this state ofaffairs is only temporary and totally unsatisfactory fromthe perspective of understanding Nature. The Higgs bo-son is the only fundamental scalar of the standard model(known so far) and it is expected that radiative correc-tions in the absence of a protecting supersymmetry wouldbring its mass up to the Planck scale. Understandingwhy the Higgs mass still remains at relatively low ener-gies, comparable to the electroweak scale is regarded asa mystery. One can turn this problem around and realisethat the Higgs scalar may represent a relevant degree offreedom which is strongly dependent on the physics atand beyond the UV cut-off of our theory. Its mass cer-tainly is strongly influenced by such high energy degreesof freedom and hence a protection mechanism should bein place so that its mass becomes natural at the low en-ergies where it has been observed. Overall, the questiontranslates into how we can explain away unnatural phe-nomena considering the idea that otherwise the StandardModel is up to a very high degree of accuracy an effectivefield theory having its underlying degrees of freedom wellhidden. We know of several mechanisms through whichthe high energy degrees of freedom are being hidden froma low energy observer. In gravity, high curvature physicsis being screened via a ”cosmic censorship” principle re-alised in the form of a black hole horizon. In cosmologythe degrees of freedom at the origin of the universe arebeing washed away by the process of inflation, while fi-nally, in effective quantum field theories, the high energydegrees of freedom are being hidden by Universality, aproperty of the renormalisation group which says thatonly a very small number of directions in the parame-ter space would bring us away from a fixed point, whilethe others would pull us back. This basically means thatmany high energy theories would lead to the same macro-scopic low energy theory, which would make it particu-larly hard for us to discern high energy degrees of freedom from physics at our scale. All these mechanisms of hidingaway high energy degrees of freedom however have loop-holes. In the case of black holes, the Hawking radiationis a phenomenon that occurs due to restrictions imposedon the quantum field modes in the spacetime around ablack hole. Such restrictions make the requirement ofan outgoing radiation evident. It is assumed that entan-glement (equivalently wormhole geometry or instantoneffects, basically topologically non-trivial effects) wouldconvey the information behind the horizon to an outsideobserver through various correlations. A similar processoccurs in the case of the Unruh effect and the associatedhorizon together with the particles detected by an ac-celerating observer in vacuum. In the case of cosmology,early-late universe entanglement is considered to be man-ifest and could play an important role in understandingthe problem of the cosmological constant. I showed thattopologically non-trivial effects shifting the mass of theaxion can be interpreted as quantum entanglement. Inthe case of the renormalisation group and for the Higgsmass a similar effect may be at work. While perform-ing a renormalisation group calculation, the basic idea isto integrate (trace) over the high energy modes and toimplement the effects on the low energy physics just byallowing the flows of the coupling constants. This mech-anism is generally possible while not reversible (hencethe renormalisation group is actually just a semi-group).What could be the loophole in this situation? We knowthat at the level of the cut-off scale we must stop trustingout theory as new degrees of freedom may have significanteffects. Restricting ourselves to only low energy modes ofour field theory somehow reminds us of the process of re-stricting modes in the case of the Hawking radiation. Onecould imagine a way in which this cut-off could be assim-ilated to a form of ”horizon” for our theory with nothingbeyond it affecting the theory above it, except for an in-teresting flow of the parameters of the theory. Still, therewill be loopholes here too. Topologically non-trivial phe-nomena may have a non-trivial impact on the parametersof the low energy theory in the same way in which entan-glement (seen as a non-trivial spacetime topology via theER-EPR duality) plays a role in the Hawking radiationemitted from the vicinity of the black hole. Topologicaleffects in the Renormalisation group have been studied inthe context of T-duality in ref. [1]. There I showed thatdemanding invariance to changes of topology amounts tocorrections to the Higgs mass that could restore natu-ralness. Here, I expand this idea considering a broaderinterpretation of a horizon triggered entanglement andshow that such topological terms (as entanglement is seenas a topological term, see ref. [2] and [3]) will bring us in-formation beyond the scales to which we would otherwisehave access. This article will have 4 parts. In the nextsection I will briefly present the hierarchy problem fromthe perspective of a coupling between scales showing thatthe lack of naturalness can be related to the idea thatscales are no separated in precisely the way an effectivequantum field theory would tell us. This discussion hasbasically been done also in [4] which I use as a reference.In the next section I will start with a simple approachto a renormalisation group problem and will show whathappens if topological terms are present. I will showwhat effects are to be expected in the case of scalar fieldsand their masses. In the next section I will discuss thismechanism in the context of the analogies with the cos-mic censorship, Hawking radiation, and the cosmologicalconstant, arguing that the solution to all these problemsis of the same type: topologically non-trivial phenomenaconnecting apparently distinct scales and restricting theavailable modes in quantum field theory. The obviousconnection to string-theoretical T-duality and its way ofconnecting all scales is briefly discussed. Finally, somebasic conclusions are being outlined.
NATURALNESS AND RELEVANT DEGREES OFFREEDOM
Nature seems to be hiding things from us, using mech-anisms from the same category. High energy degrees offreedom within a black hole are being hidden by a classi-cal horizon. General relativity seems to tell us that noth-ing passing that horizon can ever come back or transmitany information outside. Still, quantum field theoreti-cal and finally topological phenomena (modes restrictionsvia the Hawking mechanism and entanglement seen as atopological effect by means of ER-EPR) tell us that wemay have some hope of detecting effects of such hiddendegrees of freedom after all. Inflation seems to hide theinitial degrees of freedom of our universe by a similarmechanism which may be overcome by thinking in termsof extraction of entanglement from the vacuum. Finally,the universality of the renormalisation group approachseems to wash away any degrees of freedom remindingus of the high energy physics by means of a ”criticalsurface”. There are only very few types of terms to beadded to the functional form of our action that are rele-vant, hence could bring us out and affect the low energyphysics. These terms can be seen as topologically non-trivial in the parameter space and hence expressible in terms of entanglement via ER-EPR. In ref. [4] a sim-ple model is employed to show how naturalness fails inthe case of renormalisation group flows of unnatural cou-plings. Their example is a simple theory with a scalarfield φ with bare mass m and a massive fermion field ψ with a mass M interacting by means of a Yukawa inter-action with coupling g . The theory can be written as L = 12 ∂ ν φ∂ ν φ − m φ − λ φ + i ¯Ψ γ ν ∂ ν Ψ − M ¯ΨΨ+ gφ ¯ΨΨ(1)We may consider first the scalar mass much higher thanthe fermion mass m >> M and we study the physics atsome low energy scale E << M . We can use perturba-tion theory and calculate the effects of the heavy scalarfield in the full theory and then integrate it out incorpo-rating its effects in the couplings of the light fermion fieldin the low energy theory. The resulting effective theorywill only include the light fermion field with modified(shifted) couplings M ∗ and g ∗ . The resulting effectivetheory will be [4] L = i ¯Ψ γ ν ∂ ν Ψ − M ∗ ¯ΨΨ + g ∗ m ( ¯ΨΨ) (2)In general the effective mass will be equal to the baremass plus corrections coming from the high energyphysics. If we calculate them up to one loop they looklike M ∗ = [ M + M g π ln ( Λ M )] = ( M + ∆ M ) (3)The correction to the bare mass is proportional to thebare mass itself which, considering it was small in theoriginal theory, will remain small in the effective theoryas well. If however the situation changes and the fermionmass is much heavier than the scalar mass we will havea very different result( m ∗ ) = m + g π [Λ + M + m ln ( Λ M )+ O ( M Λ )] (4)This shows that starting with a small scalar mass pertur-bative corrections can produce corrections of the magni-tude of the cut-off scale Λ where out theory would breakdown anyways. The procedure of integrating away thehigh energy fields did not work as planned. The resultingtheory would contain a field with a mass proportional tothe cut-off. One could imagine the scalar to be the Higgsfield and the heavy fermion ot be the top quark. We ex-pect the cut-off scale of the standard model to be at mostthe Planck scale and at least 1 TeV (following [4]). How-ever, even a correction of the order of (1 TeV) wouldpush the mass of the Higgs up by six orders of magni-tude. From the perspective of the renormalisation groupthis would show how relevant and irrelevant parametersat low energy depend on their initial high energy val-ues at or beyond the cut-off scale. Indeed, in the caseof the Higgs the coupling and mass at the low energyend would depend strongly on extremely small changesin the cut-off couplings. Such a strong sensitivity is abenchmark for relevant degrees of freedom and for theHierarchy problem. Therefore our cut-off region, asideof looking analogue to a horizon could also be regardedas the equivalent of a ”fast scrambler” using quantumgravity terminology. If we look at the trajectories of theRG flows associated to the couplings/masses of the Higgscompared to other quantities, we see that them beingrelevant is translated into a topologically non-trivial linkbetween the region beyond the cut-off and the region be-low. Such a topological feature reminds us of wormholesolutions and entanglement. Let us see in a simple ex-ample how a topologically non-trivial parameter spacestructure can play a significant role TOPOLOGY, ENTANGLEMENT, AND THEHIERARCHY PROBLEM
The renormalisation group approach depends upon thefact that integrating over high momentum modes in thetheory and shifting the scale of the theory results in anaction that resembles the structure of the original actionwith the only distinction arising in the functional formof the coupling parameters (which start flowing). How-ever, what would happen if additional topological struc-ture is being added by the process of integrating over themodes? The high energy modes may have topologicallynon-trivial connections with the low energy ones. Afterall, if we think in terms of fixed points and scale inde-pendence at such fixed points, we come to the conclusionthat fluctuations that we wash away by integration arebeing recovered from the next scale we consider in ourcalculations. Such fluctuations can be strongly entan- gled. It would be interesting to analyse this situationin the context of entanglement extraction from an in-termediate vacuum state with early and late conformalbackgrounds. That such states can be used for entangle-ment farming has been shown in [5] and [6]. Demandinginvariance to such new topological changes would implynew terms to be visible at the lower energy domains.Such terms would offer non-trivial quantum correlationwith the high energy physics and would appear as ”worm-hole corrections” in the low energy physics. Indeed, such”wormhole corrections” have recently been shown to shiftthe mass of the axion [7]. But let us see how this works.We can write our partition function as Z = Z D φe − S [ φ ] (5)let us establish a cut-off Lambda such that the Fouriermodes of our fields vanish far above this cut-off φ k = 0 , k > Λ (6)As the renormalisation group procedure goes, we are onlyinterested about physics at long length scales L so we donot care about the modes φ k >> /L . We may writeour theory using a lower cut-off Λ ′ = Λ ζ which is valid forΛ ′ >> /L . Let me now write the Fourier modes as φ k = φ − k + φ + k (7)where + describes the high energy modes. Our actioncan be decomposed in terms of these modes as S [ φ k ] = S [ φ − k ] + S [ φ + k ] + S I [ φ − k , φ + k ] (8)Here the term S [ φ − k , φ + k ] involves interactions betweenthe high and low energy modes and hence a form of scalemixing. We can re-write our partition function as Z = Z Y k< Λ dφ k e − S = Z Y k< Λ ′ dφ − k e − S [ φ − k ] Z Y Λ ′