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

The cosmological constant problem [1, 2, 3] seems hopeless. Despite years of effort and much model-building [4] 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 [5, 6]. Indeed, it is widely believed that a symmetry-based or relaxation-type [7, 8] mechanism does not exist, and this point of view has driven much of the community towards anthropic [1, 9] and/or landscape arguments (see for instance [10, 11]). 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-energy¹¹1By this we mean we focus on how the vacuum energies of known particles (e.g. the electron) can avoid gravitating; arguably the hardest part of the problem because its solution involves changing the properties of well-measured particles at experimentally accessible energies. We leave open questions of UV completion (except where these introduce new constraints – see §3.3), but do so knowing that there are multiple ways (including supersymmetry) to suppress the vacuum energy of undiscovered particles above the weak scale. For other recent proposals for a technically natural vacuum energy see for instance [12, 13, 14]. (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:

1. (i)

   A very supersymmetric gravity sector, for which supermultiplets are split by much less than for Standard Model fields (see [15] for a discussion of some other implications and the naturality of this assumption).

2. (ii)

   A relaxation mechanism in which a ‘relaxon’ scalar field²²2There is also no ‘i’ in ‘relaxon’ (as opposed to relaxion [16]) since for us this field need not be an axion. dynamically reduces the leading non-gravitational vacuum energy.

3. (iii)

   Accidental approximate scale invariance, including the implied low-energy dilaton, $\tau$, such as is known to be a generic property of low-energy string vacua [17] and higher-dimensional supergravities more generally [18, 19, 20, 21, 22].

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 (for the reasons given in footnote 1). (We do examine UV completions more explicitly in §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 $\tau$-dependence to particle masses, so we first start with a general EFT at low energies and ask how the $\tau$-dependence associated with the vacuum energy, $\delta V\sim m^{4}(\tau)$, 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 supersymmetric³³3Despite supersymmetry playing an important role one of our first predictions therefore is a successful one: the absence of Standard-Model superpartners at the LHC. 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 ${\cal N}=1$ supergravity down to very low energies – is important largely because of the auxiliary fields, $F^{\scriptscriptstyle A}$, that its linear realization requires to be in the low-energy scalar potential. Although these fields do not propagate,⁴⁴4Concrete evidence for the importance of keeping track of non-propagating fields in low-energy EFTs comes from Quantum Hall systems, although the fields in these examples are usually topological gauge potentials [23, 24]. From this point of view it is suggestive that in known UV completions 4D auxiliary fields like $F$ start life as 4-form fields that often convey topological information from higher dimensions, both in string theory [25] and extra-dimensional gravity more generally [26]. 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 $\delta V\propto M^{2}F+\hbox{h.c.}$ (where $M$ is the UV scale) rather than directly as an $F$-independent term like $\delta V\propto M^{4}$ [15]. 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: $V=V_{\scriptscriptstyle F}+V_{\scriptscriptstyle D}$ with

$$V_{\scriptscriptstyle F}=e^{K/M_{p}^{2}}\left[K^{\bar{A}B}{\overline{D_{\scriptscriptstyle A}W}}D_{\scriptscriptstyle B}W-\frac{3|W|^{2}}{M_{p}^{2}}\right]\quad\hbox{with}\quad D_{\scriptscriptstyle A}W:=W_{\scriptscriptstyle A}+\frac{K_{\scriptscriptstyle A}W}{M_{p}^{2}}\,,$$

(1)

and

$$V_{\scriptscriptstyle D}=\frac{1}{2}\,{\mathfrak{F}}^{\alpha\beta}{\mathfrak{D}}_{\alpha}{\mathfrak{D}}_{\beta}\,,$$

(2)

familiar from ${\cal N}=1$ supergravity, where $K$ is the supersymmetric Kähler potential, $W$ is the holomorphic superpotential, ${\mathfrak{F}}^{\alpha\beta}$ is the inverse of the real part of the holomorphic gauge kinetic function, ${\mathfrak{f}}_{\alpha\beta}$, and ${\mathfrak{D}}_{\alpha}$ are the ‘moment maps’ for the gauge symmetries [46] (whose detailed form is not needed here). Subscripts on $K$ and $W$ denote differentiation with respect to any complex scalars $Z^{\scriptscriptstyle A}$. This implies in particular that the dominant ‘globally supersymmetric’ term (the terms unsuppressed by $1/M_{p}$) arise as a square,

$$V_{\rm glob}=K^{{\overline{{\scriptscriptstyle A}}}{\scriptscriptstyle B}}{\overline{W_{\scriptscriptstyle A}}}W_{\scriptscriptstyle B}\,,$$

(3)

and so can vanish at its minimum very naturally.

The above observation is only useful if (1) can also be used when supergravity is coupled to systems like the Standard Model, for which the matter does not come in ${\cal N}=1$ 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 described in [27], together with its coupling to supergravity [28, 29, 30]. Ref. [15] explores more concretely why generic potentials are consistent with the above supergravity form, using this general framework. 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 [27], and it is typically true that $W_{\scriptscriptstyle X}\neq 0$ for this field. For systems where global supersymmetry breaks badly in the UV, for example, the positivity of (3) is consistent with the non-supersymmetric low-energy scalar potential $U$ not being positive because $W_{\scriptscriptstyle X}\simeq\mu^{2}+U/(2\mu^{2})+\cdots$ and so $|W_{\scriptscriptstyle X}|^{2}\simeq\mu^{4}+U+\cdots$, 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 sector⁵⁵5The generality of the emergence of $X$ in the far infrared carrying the main supersymmetry-breaking order parameter (even if supersymmetry should be partly broken in a more complicated way, including by $D$-terms in the UV, say) is argued in [27] with nonzero derivative for $W$, so that $V_{\rm glob}\propto|W_{\scriptscriptstyle X}|^{2}$.

The relaxation mechanism is now built around the structure of the scalar potential described above. A (nonsupersymmetric) relaxon field $\phi$ 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 $W_{\scriptscriptstyle X}$, and so long as a configuration exists for which $W_{\scriptscriptstyle X}=0$ then this will be a minimum for $V_{\rm glob}$. (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 (2).) The relaxon field likes in this way to zero out the biggest (order $M_{p}^{0}$) contribution in (1), causing $W_{\scriptscriptstyle X}$ 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 $W_{\scriptscriptstyle X}$ is suppressed in this way.

Such a mechanism still leaves order $M_{p}^{-2}$ contributions to (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 $\tau\gg 1$,

$$S=S_{0}+S_{1}+S_{2}+S_{3}+\cdots\,,$$

(4)

with each term in this expansion scaling homogeneously in the sense that $S_{n}\to\lambda^{1-ns}S_{n}$ when $g_{\mu\nu}\to\lambda g_{\mu\nu}$ and $\tau\to\lambda^{s}\tau$ for constant $\lambda$. This is as would be expected if $S_{0}\to\lambda S_{0}$ and each successive term scales with an additional power of $1/\tau$ relative to the previous one. The scaling of $S_{0}$ is chosen to be consistent with the scaling of the 4D Einstein-Hilbert action, $S_{\scriptscriptstyle EH}\propto M_{p}^{2}\int{\rm d}^{4}x\,\sqrt{-g}\;{\cal R}$, when written in Einstein frame.

Within a supergravity framework we imagine $\tau$ being combined with an axion, ${\mathfrak{a}}$, into a complex axio-dilaton field ${\cal T}=\frac{1}{2}(\tau+i{\mathfrak{a}})$ that, together with a spin-half field $\xi$, forms a proper⁶⁶6By so doing we assume that the mass splittings in this dilaton supermultiplet are small enough that all members remain in the low-energy theory, unlike for the Standard Model sector. We verify below that the gravitational coupling of these fields do keep their splittings to be similar to those in the gravity sector. supermultiplet, $T$. Invariance under the axion shift symmetry ${\mathfrak{a}}\to{\mathfrak{a}}+c$ ensures $K$ depends only on $\tau=T+{\overline{T}}$ and that $W$ is $T$-independent, and the above condition of accidental approximate scale invariance says $K$ admits the expansion

$$e^{-K/(3M_{p}^{2})}=\tau F-k+\frac{h}{\tau}+{\cal O}(1/\tau^{2})$$

(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 $\tau$. They can be functions of any other fields besides $T$ (and, as it turns out [31], potentially also on logarithms of $\tau$, as we shall see).

Now comes the final bit of magic. The scale invariance of the leading $K=-3M_{p}^{2}\ln(\tau F)$ term in (5) suffices to prove [32] that it is automatically of ‘no-scale’ form [33], for which $K$ satisfies the identity

$$K^{{\overline{{\scriptscriptstyle A}}}{\scriptscriptstyle B}}K_{{\overline{{\scriptscriptstyle A}}}}K_{\scriptscriptstyle B}=3M_{p}^{2}\,.$$

(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 (5) is kept, so $K=-3M_{p}^{2}\ln(\tau F-k)$, it happens that (6) remains true [17] 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 [34, 35], where it is called an ‘extended no-scale structure’ and fits within the general approach described in [36]. The interplay between scale invariance and supersymmetry is more than the sum of its parts [17]: the flat potential for $\tau$ gets lifted at one higher order in $1/\tau$ than would naively be expected.⁷⁷7For aficionados: this is how we evade (really, co-exist with) Weinberg’s no-go theorem [1]. Although the theorem says flat directions built on scale invariance must be lifted, it does not say by how much and so does not preclude supersymmetry making the lifting smaller than would otherwise be generic.

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 $V\propto 1/M_{p}^{4}$. This is the start of the explanation for why the vacuum energy turns out to be of order $V\simeq m_{\rm vac}^{4}$ where $m_{\rm vac}\sim M_{\scriptscriptstyle TeV}^{2}/M_{p}$ and $M_{\scriptscriptstyle TeV}$ is of order the TeV scale.

Because of the underlying scale invariance and the expansion in powers of $1/\tau$, the powers of $1/M_{p}$ in $V$ turn out to go along with powers of $1/\tau$ leading to a result for the potential that has size $V\sim M^{8}/(\tau M_{p})^{4}$, where $M$ is the generic UV scale appearing everywhere in $K$ and $W$ on dimensional grounds. The upshot is that the generic $|W_{\scriptscriptstyle X}|^{2}/\tau^{2}$ part of the potential – including in particular any $M_{\scriptscriptstyle TeV}^{4}$ contributions due to SM particles with masses $M_{\scriptscriptstyle TeV}\propto 1/\sqrt{\tau}$ – 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 $\tau$ in a manner consistent with $V\propto M_{\scriptscriptstyle TeV}^{4}$.

The next question becomes: why should the field $\tau$ be stabilized at such large values? §3.2 shows that radiative corrections generically imply the function $k$ can depend on $\ln\tau$ [31], and mild assumptions about this dependence give a potential for $\tau$ that is stabilized at very large values. Because these functions depend only logarithmically on $\tau$ minima can arise at astronomically large values while only dialing in hierarchies amongst the parameters in $k$ that are of order $\ln\tau$. 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 $W_{\scriptscriptstyle X}$ vanishes. These two conditions might normally be thought to contradict one another because it is the auxiliary field, $F^{\scriptscriptstyle X}$, 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 $F^{\scriptscriptstyle X}$ is a UV scale and works as an expansion in powers of $1/F^{\scriptscriptstyle X}$. But in global supersymmetry the field equations usually predict that $F^{\scriptscriptstyle X}$ is given by

$$F^{\scriptscriptstyle X}\propto K^{{\overline{{\scriptscriptstyle X}}}{\scriptscriptstyle X}}{\overline{W_{\scriptscriptstyle X}}}\,,$$

(7)

and so large $F^{\scriptscriptstyle X}$ should be inconsistent with small or vanishing $W_{\scriptscriptstyle X}$.

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

$$F^{\scriptscriptstyle X}\propto K^{{\overline{{\scriptscriptstyle X}}}{\scriptscriptstyle X}}{\overline{D_{\scriptscriptstyle X}W}}=K^{{\overline{{\scriptscriptstyle X}}}{\scriptscriptstyle X}}\left[{\overline{W_{\scriptscriptstyle X}}}+\frac{K_{{\overline{{\scriptscriptstyle X}}}}{\overline{W}}}{M_{p}^{2}}\right]$$

(8)

rather than by (7), and so need not vanish even if $W_{\scriptscriptstyle X}$ does. Second, relaxation actually implies that $W_{\scriptscriptstyle X}$ is Planck-suppressed rather than strictly zero. Both of these can be consistent with a large-$F^{\scriptscriptstyle X}$ expansion, if the Planck-suppressed terms in (8) are sufficiently big.

Ultimately the suppression of the vacuum energy relative to the weak scale depends on the size of $\tau$, with $\tau\sim 10^{26}$ proving to be consistent with the two observed hierarchies $M_{\scriptscriptstyle TeV}\sim M_{p}/\sqrt{\tau}$ and $V\sim(M_{\scriptscriptstyle TeV}^{2}/M_{p})^{4}$. However – as discussed in §3.3 – additional constraints on how large $\tau$ 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 §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 $\tau$ must be very light, with a mass of order the present-day Hubble scale.⁸⁸8The size of the dilaton mass in this model is a special case of a general result [37] that a gravitationally coupled scalar field, ${\cal L}\sim M_{p}^{2}(\partial\theta)^{2}+V(\theta)$, whose potential at its minimum successfully gives the observed dark energy density, generically predicts a mass for $\theta$ that is of order the present-day Hubble scale, $H$. It does so because the generic condition $V^{\prime\prime}(\theta)\sim V(\theta)\sim{\cal O}(1)$ at the minimum implies a mass $m^{2}\sim V/M_{p}^{2}$, which is order $H^{2}$ whenever the scalar potential dominates the universal energy density. 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 [38, 39], 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 §4 where it is shown that the underlying scale invariance forces the dilaton $\tau$ to couple to Standard Model matter as does a Brans Dicke scalar [40] (at leading order in $1/\tau$ – a great approximation when $\tau\sim 10^{26}$). 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 [41]. 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 in [42], 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, §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 $\tau$ varies in space and time). Intriguingly, some of these may actually help with the Hubble tension [43] by allowing particle masses (in particular $m_{e}$) all to differ by a common factor at recombination relative to their values today. Such a scaling potentially exploits the mechanism described in [44], 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 §3 and §6 below. They start with the observation that the large value for $\tau$ required to explain the hierarchies also implies a breakdown of EFT methods well below electroweak scales. It does so because $\tau\sim 10^{26}$ implies the axion decay constant is $f_{a}\sim M_{p}/\tau\sim 10$ eV. This need not in itself be a problem because supersymmetric extra dimensions [45] 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 $\tau$ can arise depending on precisely how this is done. For instance a natural choice in extra-dimensional models identifies $\tau$ with the extra-dimensional volume modulus, but this seems to require $\tau\mathrel{\raise 1.29167pt\hbox{$<$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}10^{20}$. Either $\tau$ 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 §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 $W_{\scriptscriptstyle X}$ is Planck-suppressed; and so on.

Our presentation is organized as follows. §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 §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.

§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 in [42] 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). §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 §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.

The goal in this section is to set up the lagrangian for a low-energy nonsupersymmetric world coupled to supergravity, for which there is a relaxation mechanism that dynamically removes the gravitational response of any vacuum energy obtained by integrating out Standard Model fields. When doing so we lean heavily on the description given in [15] of the form this lagrangian must take [27, 28, 29, 30].

The idea is two-fold: first combine supersymmetry and scale invariance with a relaxation mechanism to make the leading part of the scalar potential small and then use the ‘extended no-scale’ mechanism to minimize the damage to the scalar potential that inevitably must come once subdominant scale-invariance breaking effects are included.