Yoga Dark Energy: Natural Relaxation and Other Dark Implications of a Supersymmetric Gravity Sector

Abstract

We construct a class of 4D ‘yoga’ (naturally relaxed) models for which the gravitational response of heavy-particle vacuum energies is strongly suppressed. The models contain three ingredients: (i) a relaxation mechanism driven by a scalar field (the ‘relaxon’), (ii) a very supersymmetric gravity sector coupled to the Standard Model in which supersymmetry is non-linearly realised, and (iii) an accidental approximate scale invariance expressed through the presence of a low-energy dilaton supermultiplet. All three are common in higher-dimensional and string constructions and although none suffices on its own, taken together they can dramatically suppress the net vacuum-energy density. The dilaton’s vev τ determines the weak scale M_W of order M_p divided by the square root of τ. We compute the potential for τ and find it can be stabilized in a local de Sitter minimum at sufficiently large field values to explain the size of the electroweak hierarchy, doing so using input parameters no larger than order 60 because the relevant part of the scalar potential arises as a rational function of ln τ. The de Sitter vacuum energy at the minimum is order c M_W^8, proportional to 1/τ^4, with a coefficient c much smaller than M_W^(−4). We discuss ways to achieve c of order 1/M_p^4 as required by observations. Scale invariance implies the dilaton couples to matter like a Brans-Dicke scalar with coupling large enough to be naively ruled out by solar-system tests of gravity. Yet because it comes paired with an axion it can evade fifth-force bounds through the novel screening mechanism described in arXiv:2110.10352. Cosmological axio-dilaton evolution predicts a natural quintessence model for Dark Energy, whose evolution might realize recent proposals to resolve the Hubble tension, and whose axion contributes to Dark Matter. We summarize inflationary implications and some remaining challenges, including the unusual supersymmetry breaking regime used and the potential for UV completions of our approach.

Dedicated to the memory of Steven Weinberg: a physicist’s Standard Model.

1. Introduction

The cosmological constant problem seems hopeless. Despite years of effort and much model-building no technically natural mechanism has been found that reconciles the large vacuum fluctuations associated with known particles with the small gravitational response to the vacuum revealed by the evidence for Dark Energy. Indeed, it is widely believed that a symmetry-based or relaxation-type mechanism does not exist, and this point of view has driven much of the community towards anthropic and/or landscape arguments. Although these might ultimately prove to be the way Nature works, a proper assessment of their likelihood suffers from the absence of compelling-yet-natural alternatives with which to compare.

We here propose a class of models that we hope can provide such a point of comparison. These models are designed to address the low-energy (and so hardest) part of the cosmological constant problem and are built on the interplay of three separate ingredients, all of which seem to play important roles:

• (i) A very supersymmetric gravity sector, for which supermultiplets are split by much less than for Standard Model fields.
• (ii) A relaxation mechanism in which a ‘relaxon’ scalar field dynamically reduces the leading non-gravitational vacuum energy.
• (iii) Accidental approximate scale invariance, including the implied low-energy dilaton, τ, such as is known to be a generic property of low-energy string vacua and higher-dimensional supergravities more generally.

Although each of these ingredients has a plausible UV pedigree, we here avoid unnecessary UV baggage (like extra dimensions) and instead let all three stand on their own within a simple 4D context, with a view to better understanding the underlying mechanisms that could be at work. Indeed, this kind of phenomenological approach lends itself to the cosmological constant problem, which is at heart a low-energy problem rather than a high-energy one. (We do examine UV completions more explicitly in Section 3.3, with a view to understanding the independent new constraints that having a UV provenance for these ingredients can introduce.)

The presence of the dilaton introduces a τ-dependence to particle masses, so we first start with a general EFT at low energies and ask how the τ-dependence associated with the vacuum energy, δV of order m^4(τ), can be dynamically suppressed. We then push the EFT into the UV to see how far it can go, and ask how it extends to energies near the weak scale, but well below the masses of any putative superpartners for Standard Model fields (which therefore do not appear to be supersymmetric at all). We focus on whether our three ingredients suffice to adequately suppress the gravitational response as the Standard Model fields themselves are integrated out. They appear to do so, subject to a few provisos discussed below.

Because the model we propose has a number of moving parts it is instructive here to summarize the underlying reasons why it works. The first ingredient — the assumption that gravity is described by N = 1 supergravity down to very low energies — is important largely because of the auxiliary fields, F^A, that its linear realization requires to be in the low-energy scalar potential. Although these fields do not propagate, they are required in order to linearly realize supersymmetry. Crucially, their presence changes the way that UV physics can enter into the low-energy potential; because supersymmetry-breaking masses necessarily themselves involve F the contribution of virtual heavy nonsupersymmetric states to the low-energy potential tends to be δV proportional to M^2 F plus its hermitian conjugate (where M is the UV scale) rather than directly as an F-independent term like δV proportional to M^4. Even though M^4 eventually arises once F is integrated out, the form involving F shows that the most UV-sensitive effective couplings have a reduced dimension.

The second important consequence of having low-energy auxiliary fields is the structure that their elimination imposes on the scalar potential, which comes as the usual sum and differences of squares: V = V_F + V_D with

V_F = exp(K/M_p^2) times the quantity K^(ĀB) D_A W times the conjugate of D_B W, minus 3|W|^2/M_p^2, where D_A W is defined as W_A + K_A W/M_p^2, — (1.1)

and

V_D = one half F^(αβ) D_α D_β, — (1.2)

familiar from N = 1 supergravity, where K is the supersymmetric Kähler potential, W is the holomorphic superpotential, F_(αβ) is the inverse of the real part of the holomorphic gauge kinetic function, f_(αβ), and D_α are the ‘moment maps’ for the gauge symmetries (whose detailed form is not needed here). Subscripts on K and W denote differentiation with respect to any complex scalars Z^A. This implies in particular that the dominant ‘globally supersymmetric’ term (the terms unsuppressed by 1/M_p) arise as a square,

V_glob = K^(AB) W_A W_B, — (1.3)

and so can vanish at its minimum very naturally.

The above observation is only useful if (1.1) can also be used when supergravity is coupled to systems like the Standard Model, for which the matter does not come in N = 1 supermultiplets and for which the potential usually need not be positive. The generality of the above form ultimately follows from the generality of the rules for nonlinearly realizing supersymmetry, together with its coupling to supergravity. When supersymmetry is nonlinearly realized (as it must be in such theories) there is always a low-energy superfield X that is nilpotent, X^2 = 0, since this is what is required to represent the goldstino, and it is typically true that W_X is nonzero for this field. For systems where global supersymmetry breaks badly in the UV, for example, the positivity of (1.3) is consistent with the non-supersymmetric low-energy scalar potential U not being positive because W_X is approximately μ^2 + U/(2μ^2) + ⋯ and so |W_X|^2 is approximately μ^4 + U + ⋯, since constant terms in the potential are irrelevant in global supersymmetry. Supergravity complicates things because gravity couples to all sources of energy, but also the gravity sector introduces new auxiliary fields. In what follows we imagine that X is the only supermultiplet to descend from the UV sector with nonzero derivative for W, so that V_glob is proportional to |W_X|^2.

The relaxation mechanism is now built around the structure of the scalar potential described above. A (nonsupersymmetric) relaxon field φ is introduced, whose mass is assumed to be a bit smaller than the electron mass (so that it survives to appear in the low-energy theory below the lightest known dangerous Standard Model field). This scalar appears in particular in W_X, and so long as a configuration exists for which W_X = 0 then this will be a minimum for V_glob. (A very similar mechanism is also commonly at work in supersymmetric gauge theories, where charged scalars automatically seek the zero of the positive D-term potential given in (1.2).) The relaxon field likes in this way to zero out the biggest (order M_p^0) contribution in (1.1), causing W_X to be Planck suppressed once gravitational interactions are included. We return below to why it remains consistent to use the formalism of nonlinearly realized supersymmetry when W_X is suppressed in this way.

Such a mechanism still leaves order M_p^(−2) contributions to (1.1), and because these are not positive definite they cannot as simply be removed using the same kind of relaxon mechanism. Here is where accidental scale invariance finally plays a role. Motivated by the accidental scaling symmetries known to be common in the low-energy limit of higher-dimensional supergravity, we propose that the theory comes to us with an action that is expanded in inverse powers of a large scalar field τ, itself much greater than one,

S = S_0 + S_1 + S_2 + S_3 + ⋯, — (1.4)

with each term in this expansion scaling homogeneously in the sense that S_n goes to λ^(1−ns) S_n when the metric g_(μν) goes to λ g_(μν) and τ goes to λ^s τ for constant λ. This is as would be expected if S_0 goes to λ S_0 and each successive term scales with an additional power of 1/τ relative to the previous one. The scaling of S_0 is chosen to be consistent with the scaling of the 4D Einstein-Hilbert action, which is proportional to M_p^2 times the integral over four-dimensional spacetime of the square root of minus the determinant of the metric times the Ricci scalar R, when written in Einstein frame.

Within a supergravity framework we imagine τ being combined with an axion, a, into a complex axio-dilaton field T = one half of (τ + i a) that, together with a spin-half field ξ, forms a proper supermultiplet, T. Invariance under the axion shift symmetry, a goes to a + c, ensures K depends only on τ = T plus the conjugate of T and that W is T-independent, and the above condition of accidental approximate scale invariance says K admits the expansion

exp of minus K/(3 M_p^2) = τ F − k + h/τ + O(1/τ^2), — (1.5)

where F is possibly a scale-invariant function of other fields, and none of F, k, h and so on can depend on powers of τ. They can be functions of any other fields besides T (and, as it turns out, potentially also on logarithms of τ, as we shall see).

Now comes the final bit of magic. The scale invariance of the leading K = −3 M_p^2 ln(τ F) term in (1.5) suffices to prove that it is automatically of ‘no-scale’ form, for which K satisfies the identity

K^(AB) K_A K_B = 3 M_p^2. — (1.6)

This guarantees the flatness of the potential along the directions in field space that do not appear in W (such as T). But even though the action is not scale invariant in the same way when the first subdominant term in (1.5) is kept, so K = −3 M_p^2 ln(τ F − k), it happens that (1.6) remains true provided only that k does not depend on T. This type of accidental preservation of the no-scale structure beyond leading order in a large field expansion was first noticed in certain string compactifications, where it is called an ‘extended no-scale structure’. The interplay between scale invariance and supersymmetry is more than the sum of its parts: the flat potential for τ gets lifted at one higher order in 1/τ than would naively be expected.

For the present purposes, what is nice about this last observation is that it means that the 1/M_p^2 contributions to the potential also vanish, even after the relaxon has been integrated out, leaving the final dominant result at order V proportional to 1/M_p^4. This is the start of the explanation for why the vacuum energy turns out to be of order V approximately m_vac^4 where m_vac is of order M_TeV^2/M_p and M_TeV is of order the TeV scale.

Because of the underlying scale invariance and the expansion in powers of 1/τ, the powers of 1/M_p in V turn out to go along with powers of 1/τ leading to a result for the potential that has size V of order M^8/(τ M_p)^4, where M is the generic UV scale appearing everywhere in K and W on dimensional grounds. The upshot is that the generic |W_X|^2/τ^2 part of the potential — including in particular any M_TeV^4 contributions due to SM particles with masses M_TeV proportional to one over the square root of τ — is cancelled, leaving a low-energy potential that depends on other parameters. Yet both the weak scale and the vacuum-energy scale are predicted to depend on τ in a manner consistent with V proportional to M_TeV^4.

The next question becomes: why should the field τ be stabilized at such large values? Section 3.2 shows that radiative corrections generically imply the function k can depend on ln τ, and mild assumptions about this dependence give a potential for τ that is stabilized at very large values. Because these functions depend only logarithmically on τ minima can arise at astronomically large values while only dialing in hierarchies amongst the parameters in k that are of order ln τ. Furthermore, standard renormalization-group (RG) methods allow this minimum to be reliably explored without losing control over the underlying radiative corrections.

With this full picture in mind we can return to the question, deferred above, as to why a nonlinearly realized treatment of the Standard Model fields can be consistent even though the relaxon adjusts to ensure that W_X vanishes. These two conditions might normally be thought to contradict one another because it is the auxiliary field, F^X, for the goldstino multiplet X, that is the measure of the size of supersymmetry breaking in the unseen sector that badly breaks supersymmetry (and thereby gives superpartners to the Standard Model large masses). In particular, the formulation of nonlinearly realized supersymmetry assumes F^X is a UV scale and works as an expansion in powers of 1/F^X. But in global supersymmetry the field equations usually predict that F^X is given by

F^X proportional to K^(XX) W_X, — (1.7)

and so large F^X should be inconsistent with small or vanishing W_X.

We argue that there are two reasons why the above framework is nonetheless consistent. First, in supergravity F^X is instead determined by

F^X proportional to K^(XX) D_X W = K^(XX) times the quantity W_X + K_X W/M_p^2 — (1.8)

rather than by (1.7), and so need not vanish even if W_X does. Second, relaxation actually implies that W_X is Planck-suppressed rather than strictly zero. Both of these can be consistent with a large-F^X expansion, if the Planck-suppressed terms in (1.8) are sufficiently big.

Ultimately the suppression of the vacuum energy relative to the weak scale depends on the size of τ, with τ of order 10^26 proving to be consistent with the two observed hierarchies, M_TeV of order M_p over the square root of τ and V of order (M_TeV^2/M_p)^4. However — as discussed in Section 3.3 — additional constraints on how large τ can be arise once its UV origins are more explicit. The same UV frameworks also provide extra sources of suppression (such as warping), making the final solution likely involve a cocktail of suppressions, possibly along the lines described in Section 3.3.

Explicit details of the above construction are given in later sections, but an immediate consequence of the scale invariance and any successful suppression of the cosmological constant is that the dilaton field τ must be very light, with a mass of order the present-day Hubble scale. It follows that it must be cosmologically active up to the current epoch, and so predicts Dark Energy must be described by a specific type of near-scale-invariant quintessence theory, but (remarkably) one for which both the cosmological constant and the quintessence-field mass would be technically natural.

But it gets better than this. A gravitationally coupled scalar as light as the Hubble scale should stick out in tests of gravity like social skills at a physics meeting. Indeed, the low-energy lagrangian relevant to astrophysics is explored in Section 4 where it is shown that the underlying scale invariance forces the dilaton τ to couple to Standard Model matter as does a Brans-Dicke scalar (at leading order in 1/τ — a great approximation when τ is of order 10^26). And it does so with a coupling that is apparently too large to have escaped detection in precision tests of gravity in the solar system and elsewhere. A more careful look, however, shows that its supersymmetric partner (the axion) can save the day, and does so because of the target-space axion-dilaton interactions also automatically predicted by the model. As explored in more detail elsewhere, the axion-dilaton interactions have the effect of making matter-dilaton couplings largely generate external axion fields (rather than dilaton fields), which are much less effective at altering test-particle motions within the solar system and so can escape detection. We call this mechanism ‘axion homeopathy’ because it can work for extremely small direct axion-matter couplings, provided only that these are nonzero.

Because the phenomenology of the axio-dilaton field is so crucial to the viability of such models, Section 5 provides a preliminary discussion of axio-dilaton cosmology and checks that the most basic things work (though without doing justice to the entirety of the constraints that a viable model must ultimately pass — further studies of structure formation and CMB properties within this framework are important to explore). However even if axio-dilaton phenomenology eventually poses challenges to the version of this approach we present here, we regard any such model-building problems within this general framework to be a good trade for progress on the (much harder) cosmological constant problem.

There are also many other ways to test this picture, such as through tests of gravity and the changes predicted in cosmology during well-measured epochs (such as the variations in fundamental masses — in Planck units — that are predicted whenever the dilaton field τ varies in space and time). Intriguingly, some of these may actually help with the Hubble tension by allowing particle masses (in particular the electron mass) all to differ by a common factor at recombination relative to their values today. Such a scaling potentially exploits a mechanism described in the literature, and we briefly check that the basic requirements of this mechanism can be satisfied.

So what is the catch? For the long-distance physics (below the eV scale) relevant to astrophysics, we do not see a fundamental one yet and not for want of looking. The main provisos about which we worry are described in Sections 3 and 6 below. They start with the observation that the large value for τ required to explain the hierarchies also implies a breakdown of EFT methods well below electroweak scales. It does so because τ of order 10^26 implies the axion decay constant is f_a of order M_p/τ, about 10 eV. This need not in itself be a problem because supersymmetric extra dimensions could provide a plausible UV completion at these scales, while remaining consistent with SM degrees of freedom being four-dimensional as assumed here. The worries come once the low-energy picture is embedded into such a UV completion, because new constraints on the value of τ can arise depending on precisely how this is done. For instance a natural choice in extra-dimensional models identifies τ with the extra-dimensional volume modulus, but this seems to require τ no larger than about 10^20. Either τ must arise differently in the UV completion or there must be additional sources of hierarchical suppression (or both). We explore some of the options in Section 3.3. Other worries include ensuring the naturalness of having the relaxon be so light; exploring the detailed stability of the nonlinearly realized supergravity form in regions for which W_X is Planck-suppressed; and so on.

Our presentation is organized as follows. Section 2 describes the main mechanism in some detail, starting by explicitly writing out the EFT applicable at energies just below the electron mass. Since all of the dangerous Standard Model particles are integrated out at this point the EFT at these scales illuminates most clearly the interplay between scale-invariance and the relaxation mechanism. Standard Model fields are then reintroduced, allowing their couplings to low-energy states to be made more precise.

Knowledge of Standard Model couplings allows a more explicit assessment of naturalness issues, such as why low-energy gravitational physics is relatively insensitive to the loops of Standard Model fields. This is the topic of Section 3. Since most of the hierarchies of scale in the model are set by the background value of the dilaton field, this section also shows how this field can be stabilized, along the lines described above.

Section 4 makes a down payment on the most pressing phenomenological challenges, including a derivation of the low-energy EFT relevant to astronomical and cosmological tests. These give the field equations used to evade the constraints on dilaton-matter couplings coming from tests of General Relativity (GR) in the solar system (whose results are merely quoted here). Section 5 provides a preliminary evaluation of the cosmological evolution of the axio-dilaton fields for comparison with some features of late-universe cosmology. This includes a brief discussion of potential relevance to the Hubble tension, mentioned above.

Finally Section 6 summarizes some of the implications of our proposal; contrasts our approach with other discussions that build on the role of scale invariance. Along the way this section outlines several topics for future investigation — such as possible implications for dark matter, inflation, baryogenesis and neutrino physics — that we do not study here.

(Sections 2 to 5 — the model, naturalness issues, phenomenological issues and axio-dilaton cosmology — are omitted for length; the complete text is at the source.)

6. Concluding Remarks

To summarize: we propose here a framework for understanding the Dark Energy density in a technically natural way that concentrates on the cancelation of the contributions to the vacuum energy coming from Standard Model particles. The framework relies on the following core ingredients:

• (i) A very supersymmetric gravity sector coupled to a Standard Model sector in which supersymmetry is badly broken and so non-linearly realised using the formalism of constrained superfields. The low-energy presence of supergravity and goldstino auxiliary fields plays an important role by imposing a supersymmetric form on the low-energy scalar potential.
• (ii) A relaxation mechanism wherein a relaxon field φ adjusts to suppress the leading vacuum energy. Here φ is a new light scalar field that also realizes supersymmetry nonlinearly (as does the Standard Model sector).
• (iii) Approximate accidental scale invariance under which a dilaton field τ and the metric scale by constant factors, whose breaking is captured by an expansion of the action in powers of 1/τ. The field τ belongs to the gravitationally coupled supersymmetric sector and so comes with axion and dilatino partners.

In the explicit realisation presented here, a minor tuning of lagrangian parameters of order 1/60 allows the dilaton field to be stabilised at an exponentially large value τ of order 10^26, and this large value explains the size of the electroweak hierarchy inasmuch as Standard Model particles acquire masses of order M_TeV, that is M_p divided by the square root of τ. This exponentially large value for τ is obtained naturally because the potential arises as a rational function of ln τ due to the generic presence of logarithms of mass ratios amongst the UV particles that are integrated out above the weak scale (along the lines proposed in a different context some time ago).

The low-energy scalar potential is calculable as a series in 1/τ of the form

V = V_2/τ^2 + V_3/τ^3 + V_4/τ^4 + ⋯ — (6.1)

and because Standard Model particle masses are m proportional to τ^(−1/2) their loops contribute δV of order m^4, proportional to 1/τ^2, and so contribute to V_2. Strictly speaking these contributions actually arise as order m^2 corrections to w_X.

The supersymmetry of the gravity sector implies V_2 proportional to |w_X|^2 is positive since it must turn off in the hypothetical limit where global supersymmetry is unbroken. The relaxation field appears in V_2 and it prefers to minimize w_X towards zero in the large-τ limit. In global supersymmetry it would do so by seeking V_2 = w_X = 0, but in supergravity this combination instead becomes Planck suppressed. This suppression does not mean superpartners cannot remain heavy, however, because supersymmetry-breaking fields like F^X, proportional to W_0/M_p, remain at the weak scale despite being Planck suppressed.

The interplay between scale invariance and supersymmetry (as manifested in ‘extended no-scale structure’) then leads to V_3 also vanishing, leaving in V_4/τ^4 a naturally small, positive, cosmological constant that is order (M_TeV^2/M_p)^4 (and so is the right size). The fact that supersymmetry is so mildly broken in the gravity sector allows it to protect the series form of V given in (6.1), as well as the cancellation of the V_3 terms.

The large value of τ determines both the small cosmological-constant scale, m_vac proportional to 1/τ, and the electroweak scale, m_TeV proportional to M_p over the square root of τ, fixing their ratio to be m_vac/m_TeV proportional to one over the square root of τ. This contains the seeds of the oft-made observation that the weak scale is the geometric mean of the Planck and cosmological-constant scales.

Weinberg’s no-go argument applies (as it must since the underlying mechanism relies on scale invariance) in the sense that quantum corrections to the scalar potential (such as subdominant powers of 1/τ) are present. But Weinberg’s argument does not say how large these corrections must be and supersymmetry — through the cancellation of V_3 — is what keeps them small.

The presence of auxiliary fields in the scalar potential required by supersymmetry in the gravity sector allows supersymmetry also to have implications for the naturalness of the relaxon φ. This is because its mass term arises from a dimension-three operator g Φ^2 X in W rather than from a dimension-two operator in V; involving X because it is a strong supersymmetry-breaking effect. Having its mass come from W_X both ensures that the φ mass is proportional to τ^(−1/2) (and so at most lies at TeV energies) and makes radiative corrections enter through g, which is dimensionless.

Similar arguments may also apply to the Higgs boson itself if its scalar potential also arises as a contribution λ H X, involving the combination of the Higgs doublet H with its conjugate minus the square of the electroweak vev, in W, although in the Higgs case having a mass proportional to τ^(−1/2) is more important (and more generic). The same physics underlies both the electroweak and cosmological constant scales at a fundamental level (without need for anthropic arguments).

There are several prices we pay (plus a few opportunities we reap) for this suppression. First, the small size of the vacuum energy in the low-energy theory requires τ of at least 10^26 and then this drives the axion decay constant so low that the low-energy EFT must fail at eV energies. Although plausible UV physics (such as supersymmetric large extra dimensions) could plausibly intervene at these scales, it is not yet known how it does so. In particular, the UV pedigree for τ is not known and so we do not know whether such large values are allowed. This is not an empty worry because if τ is of order the two-thirds power of the extra-dimensional volume, as suggested by the simplest string models, then extra-dimensional constraints preclude it from being larger than of order 10^20.

A second price we pay is the large Brans-Dicke-like coupling of the dilaton to ordinary matter, which flirts with inconsistency with current tests of gravity. Although the model seems to bring its own evasion mechanism — wherein the SL(2, R)-invariant axion-dilaton self-couplings divert dilaton couplings into generating harder-to-detect axionic response — there are also numerous potential signals for their presence in tests of gravity and cosmology. Some of the low-energy axio-dilaton signals might even solve problems, such as the current puzzle over apparent inconsistencies in measurements of the Hubble scale H_0. Whether these eventually kill or verify the model, we regard it as progress to trade the cosmological constant problem for exercises in late-epoch model-building, and leave a more detailed treatment of this mechanism for future research.

The remainder of this section puts our scenario into the context of earlier approaches using similar ingredients and briefly summarizes several open issues.

6.1 Relation to other approaches

All three of our ingredients have been separately used previously in related contexts, so it is useful to clarify what differs from these in our particular framework.

Low-energy supergravity. Approaches to quintessence and Dark Energy that use supergravity equations of motion go back more than 20 years. Even if supersymmetry is valid at high energies, there is a basic question that these constructions do not address: why should it survive down to the extremely low energies required to be relevant to quintessence, given the apparent absence of supersymmetry at the intervening energies containing the observed Standard Model particles?

This question is a special case of a larger problem faced even by nonsupersymmetric quintessence models: why is it legitimate to compute and use a carefully designed quintessence potential entirely within the classical approximation? This is often phrased as the statement that vacuum energies and very small scalar masses are not technically natural: quantum effects cause scalar masses and potentials to change dramatically once heavier particles are integrated out, making them particularly sensitive to a system’s UV sector. This sensitivity can also undermine the usual low-energy arguments that justify using the classical approximation in gravitating systems.

By contrast, addressing these issues is the main motivation of the model presented here. Our tool for doing so is the explicit coupling of low-energy supergravity to a non-supersymmetric Standard Model permitted by the nonlinear realization and its coupling to supergravity, since this allows one to trace the low-energy effects of integrating out non-supersymmetric sectors.

Relaxation mechanisms. Relaxation mechanisms that use a field’s dynamical evolution to suppress the apparent cosmological constant also have a long history, and more recently have been applied to the electroweak hierarchy problem. We feel that none have been entirely convincing as solutions to these problems on their own, and the same would have been true for us if we had not combined relaxation with the other two ingredients.

Scale invariance. Scale invariance has an equally long history, with both early applications to the cosmological constant problem and elsewhere. Early proposals assumed a scale-invariant action with scaling broken only by quantum anomalies, leaving open whether such examples exist. They very early ran up against no-go results that identified why even completely unbroken scale invariance cannot prevent the lifting of classically flat dilaton directions.

Our approach evades some of these issues because scale invariance is only approximate, even at the classical level. The arguments of the no-go theorems broadly apply, inasmuch as the central issue is to quantify the lifting of flat directions by quantum effects. But the no-go arguments do not forbid the use of supersymmetry to suppress (though not completely eliminate) this lifting, as we do here using the low-energy distillation of the extended no-scale structure found in string models.

Approximate scale invariance has also been conjectured to be related to small cosmological constants and the existence of cosmologically light dilatons and within an anthropic context within string constructions. These papers (and we) both use the robust link between potentials that predict a small cosmological constant and Hubble-scale dilaton masses described in the literature, though in the present paper we provide an explicit mechanism for building a potential with a naturally small vacuum energy that exploits this connection (without the need to resort to anthropic arguments).

Indeed, scale invariance is more broadly suggestive as an ingredient for solving naturalness problems because the essence of these problems is that Nature seems to be closer to scale invariance (that is, some masses or energies are smaller) than we would normally expect. This has led to its exploration for other purposes, such as to higher-derivative theories of gravity (which are more broadly scale invariant in the UV, but bring associated difficulties with ghosts). Scale invariance has also been applied to inflationary models or motivating the choices required to obtain inflation using only the Standard Model Higgs as the inflaton.

This history teaches that scale invariance, though attractive, seems to come with undesirable extras (such as ghosts in higher-derivative gravity or dangerous dilatons in the models described here). We regard the dilaton to be amongst the more benign of these options, in that its existence need not point towards a fundamental instability, though viability of the model requires threading a minefield of potential observational tests.

6.2 General implications and future directions

In the picture we paint the small observed size of the dark-energy density points towards a very rich low-energy ‘dark’ sector consisting of supersymmetric gravity, a cosmologically active axio-dilaton multiplet, a somewhat heavier gravitationally coupled relaxon plus possibly other dark ingredients. This picture leaves open a great many interesting directions worth further exploring.

• Naturalness: The main conceptual issue is to better verify through explicit calculations that the general arguments about nonlinear realization of supersymmetry do indeed preserve the supergravity structure of the scalar potential, but more explicitly in the regime of small W_X.
• Super-eV completions: τ of at least 10^26 plays an important role suppressing the vacuum energy in the low-energy 4D theory and this drives a breakdown of EFT methods at eV energies (due to the small axion decay constant). Can extra dimensions provide the UV completion that unitarizes the theory up to the electroweak scale? If so, how does τ arise in this completion and can it take the large values that are needed?
• Axio-dilaton phenomenology: The biggest phenomenological issue is to see whether the required dilaton properties could have evaded contemporary precision tests of gravity, and can be consistent with what we know about cosmology. How robust is the screening mechanism to dynamical issues and how sensitive is it to the detailed axion couplings? How does it stand up to more detailed studies of cosmology, including issues of structure formation? Can the baryonic wind-up mechanism of Section 5.2 allow the axion a to contribute to (or to replace) Dark Matter? Are the predicted variations of masses over cosmological times consistent with observations? Can they help solve the current Hubble tension?
• Dark matter: Our discussion leaves open what plays the role of dark matter, which could be included in the simplest scenarios by supplementing the Standard Model sector by another non-supersymmetric particle whose mass varies as m proportional to τ^(−1/2). More economical options might also be worth exploring, however, including those for which the dark matter mass varies differently with τ or the option where the axion a itself plays the role of dark matter, as sketched in Section 5.2. Could the relaxon be the dark matter? (Although its coupling to the Higgs bilinear resembles scalar-portal dark-matter models its small coupling strength is too small to allow the φ field to be thermally produced, unlike in the minimal case.)
• Neutrino physics: Our picture potentially populates the low-energy world with a rich spectrum of weakly coupled very light fermions, and explains why they are there (some are superpartners to known light bosons, like the graviton). Their presence is a required consequence of the supersymmetry of the gravity sector. Should these mix with Standard Model neutrinos they would provide natural candidates for light sterile neutrinos, and could open up new ways to express lepton-number violation at low energies, with possible implications for lepto- and baryogenesis. If super-heavy sterile fermions mix with SM neutrinos (as in the see-saw mechanism) with a Dirac mass m_D proportional to τ^(−1/2) that scales like other SM masses then physical neutrino masses are order m_ν of order 1/τ, and so would explain the coincidence between neutrino masses and the cosmological constant scale, where the minimum of the potential is of order the fourth power of the neutrino mass. What observable features do the resulting neutrino/dark-sector interactions imply?
• Pre-BBN cosmology: The picture also likely modifies pre-nucleosynthesis cosmology in a variety of ways that are worth exploring. Among these are the interplay between φ and the Higgs H in the scalar potential, since if the potential for these both lie within |w_X|^2 then only the vev of a linear combination gets fixed by the single condition w_X = 0. What would this mean for late-time cosmology and/or the epoch of the electroweak phase transition?

The field φ also seems designed to be a good inflaton candidate. After all, nonlinearly realized supersymmetry naturally provides large positive (de Sitter-like) vacuum energies (the potential’s |w_X|^2 term) for any value of φ away from its minimum and φ has been designed as a field whose evolution parameterizes changes between large nonzero w_X and vanishing w_X. The early-universe evolution of φ while positive energies dominate therefore provides an attractive picture of inflation in which the inflaton is not completely divorced from Standard Model physics, and changes to τ become correlated with changes to the size of the observable universe. The inflationary models found in this way come with scale invariance baked in (much like for the possible UV embeddings described in Section 3.3), in a way that is known to help such models agree with observations. We report the details of this scenario, and other ‘yoga breathing’ exercises (relaxed inflation) in a future publication.

Yoga Dark Energy ties together many of the scales of physics and so its implications are legion; further investigations are underway into several of these directions.

Acknowledgements

We thank Aizhan Akhmetzhanova, Clare Burrage, Michele Cicoli, Ed Copeland, Shanta de Alwis, Emilian Dudas, Nemanja Kaloper, Justin Khoury, Lloyd Knox, Francesco Muia, José de Jesús Padua Argüelles and Henry Tye for many helpful conversations. CB’s research was partially supported by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. The work of FQ has been partially supported by STFC consolidated grants ST/P000681/1, ST/T000694/1.

(Appendices A and B, and the reference list, are omitted for length; the complete text is at the source.)