Horndeski model in nonlinearly realized supergravity

TL;DR

This paper constructs the Horndeski Lagrangian within non-linearly realized supergravity, demonstrating its bosonic realization and ghost-free gravitino couplings in a super-covariant framework.

Contribution

It introduces a supergravity formulation of the Horndeski Lagrangian with consistent gravitino couplings, extending previous scalar-tensor theories.

Findings

Bosonic part of Horndeski Lagrangian realized in supergravity.

Gravitino couples to Horndeski sector in a super-covariant and ghost-free manner.

Framework ensures consistent supersymmetric extension of scalar-tensor theories.

Abstract

We construct the Horndeski Lagrangian within non-linearly realized supergravity. We will show that the bosonic part of the Horndeski Lagrangian can be realized. Gravitino naturally couples to Horndeski sector in a super-covariant way. Such gravitino couplings are also free from ghosts.

Equations

$$S^2(x,\theta )=0.$$

$$s(x)=\frac{{\chi }_S^{\alpha }{\chi }_{S\alpha }}{2F^S},$$

$$W=W_0+{\mu }^{2}S,$$

$$K=S\bar{S}.$$

$$\mathcal{L}=\int d^2\Theta 2\mathcal{E}\left[\frac{3}{8}\left({\bar{\mathcal{D}}}^2-8\mathcal{R}\right)e^{-\frac{K}{3}}+W\right]+h.c.$$

$$\frac{1}{\sqrt{2}}{\mathcal{D}}^{\alpha }S{|}_{\theta =0}={\chi }_S^{\alpha }=0,$$

$$\mathcal{L}=$$

$$\int d^4\theta E\frac{16S\bar{S}}{{\mathcal{D}}^{\alpha }{\mathcal{D}}_{\alpha }S{\bar{\mathcal{D}}}_{\dot{\alpha }}{\bar{\mathcal{D}}}^{\dot{\alpha }}\bar{S}}\mathcal{F}(\mathcal{C},{\mathcal{D}}_{\alpha }\mathcal{C},{\mathcal{D}}_a\mathcal{C})$$

$$=\mathcal{F}(C,{\zeta }_{\alpha },{\nabla }_{a}C)+\cdots ,$$

$$S(\Phi -\bar{\Phi })=0,$$

$$\Phi (x,\theta )=\varphi (x)\text{(in unitary gauge).}$$

$${\mathcal{F}}_2$$

$${\mathcal{F}}_3$$

$${\mathcal{F}}_4$$

$${\mathcal{F}}_5$$

$$({\mathcal{D}}_a{\mathcal{D}}_b\hat{\varphi }{)}^2$$

$$({\mathcal{D}}_a{\mathcal{D}}_b\hat{\varphi }{)}^3$$

$${\hat{\mathcal{R}}}_s$$

$${\hat{\mathcal{G}}}^{ab}$$

$$\hat{\varphi }|$$

$${\mathcal{D}}_a\hat{\varphi }|$$

$${\mathcal{D}}_a{\mathcal{D}}_b\hat{\varphi }|$$

$$\mathcal{R}|$$

$$G_a|$$

$${\hat{R}}_s|$$

$$\hat{\mathcal{G}}|$$

$${\omega }_m^{ab}$$

$$\mathcal{L}=$$

$$=$$

Full text

††preprint: RESCEU9/19

One of the most important goals of contemporary physics is to construct a theory to describe the entire history of the Universe consistent with observations from its creation to the current state. One may expect the birth of the classical Universe is described by string theory which is the most promissing candidate of quantum gravity. Then physics of inflation, which is an indispensable ingredient of modern cosmology to solve the horizon and the flatness problems as well as to generate curvature perturbations Sato:2015dga , is plausibly described by supergravity as the low energy effective theory of superstring, since observational constraints on the tensor perturbations tell us that the energy scale of inflation is at least several orders smaller than the Planck scale (at that time) Ade:2018gkx .

In order to realize inflation in supergravity, first of all, we must realize a (quasi) de Sitter universe which inevitably breaks supersymmetry. Recently the simplest supergravity model of de Sitter space, dubbed pure de Sitter supergravity, was proposed Bergshoeff:2015tra ; Hasegawa:2015bza , in which the physical degrees of freedom are graviton and massive gravitino. In this model, spontaneous supersymmetry breaking is realized with a constrained chiral superfield $S(x,\theta)$ satisfying the nilpotent condition $S^{2}(x,\theta)=0$ Rocek:1978nb ; Ivanov:1978mx ; Lindstrom:1979kq , which describes the Goldstino first found by Volkov and Akulov Volkov:1973ix . Such a superfield is also known to be the low energy effective description of anti-D3 brane in string theory Kallosh:2014wsa ; Bergshoeff:2015jxa ; Kallosh:2015nia ; Garcia-Etxebarria:2015lif ; Cribiori:2019hod . The nilpotent superfield has also been applied to supergravity cosmology models e.g. in Antoniadis:2014oya ; Ferrara:2014kva ; DallAgata:2014qsj ; McDonough:2016der ; Kallosh:2017wnt .

Turning our eyes to the bottom-up approach to inflationary cosmology, the cosmic microwave background (CMB) observations strongly favor single-field inflation as we do not find any trace of isocurvature modes Akrami:2018odb . Then the generalized G-inflation Kobayashi:2011nu is the theory we should consider because it is the most general single-field inflation model containing practically all the known models of single-field inflation from potential driven models including the Starobinsky model Starobinsky:1980te to kinetically driven ones ArmendarizPicon:1999rj ; Kobayashi:2010cm with or without various nonminimal couplings to gravity Spokoiny:1984bd ; Germani:2010gm . This model is based on the Horndeski theory Horndeski:1974wa or the generalized galileon Deffayet:2011gz which are the most general covariant scalar-tensor theory whose field equations are of second-order so that ghost instability is absent.

It is unanimously agreed that sensible theories should not contain fundamental ghost fields Carroll:2003st ; Woodard:2006nt ; Sbisa:2014pzo , but ghost-free nature is important even from low energy effective field theory (LEFT) viewpoint. In general, LEFT contains infinite number of derivative interactions originating from integrating heavy fields out, which would be schematically given by $f(\frac{\partial}{M_{H}},\phi_{L})$ where $M_{H}$ denotes heavy field mass and $\phi_{L}$ being light fields. For instance, the propagator of a heavy scalar field $M_{H}^{-2}(1+\partial^{2}/M_{H}^{2})^{-1}$ gives rise to such interaction. All the terms are suppressed by heavy field mass and $M_{H}$ becomes the cut-off scale of LEFT. On the other hand, ghost-free type higher-derivative terms does not have any pole of heavy fields. Therefore, even if they dominate the dynamics, any heavy fields would not be excited, and ghost-free higher derivative terms can be the dominant part of dynamics as long as strong coupling regime is avoided. In other words, among various higher-derivative terms, only ghost-free ones can be leading order terms. Also, the ghost-free type interactions may have hidden symmetry Aoki:2018lwx ; Aoki:2019rvi , which becomes manifest in the affine-metric formulation.

The purpose of this work is to construct the Horndeski model coupled to pure de Sitter supergravity, namely, to embed the Horndeski Lagrangian within nonlinearly realized supergravity to make the generalized G-inflation possible in supergravity111One might wonder if this is a marriage of losers, as LHC does not discover supersymmetry yet, and trivial but strong constraints have been obtained against the Horndeski theory from GW170817 and GRB 170817A in terms of the propagation speed of gravitational waves Baker:2017hug ; Creminelli:2017sry ; Sakstein:2017xjx ; Ezquiaga:2017ekz using the formula derived in Kobayashi:2011nu . (See also Copeland:2018yuh for nontrivial analysis.) These arguments only apply in the low energy regime well below TeV scale and we should consider the full theories to describe physics of inflation Kobayashi:2019hrl .. Supergravity realization of ghost free higher-derivative interaction has been a difficult issue, since supersymmetry predicts not only the desired term but also additional derivative couplings, which lead to ghost instabilities. Ghost-free higher derivatives in supersymmetric models are discussed e.g. in Khoury:2010gb ; Khoury:2011da ; Farakos:2012je ; Sasaki:2012ka ; Koehn:2012ar ; Koehn:2012np ; Farakos:2012qu ; Farakos:2013zya ; Gwyn:2014wna ; Aoki:2014pna ; Aoki:2015eba ; Fujimori:2016udq ; Fujimori:2017kyi ; Fujimori:2017rcc ; Nitta:2018yzb ; Nitta:2018vyc . As we will show, all the issues of ghost instabilities are circumvented once supersymmetry is nonlinearly realized, and the Horndeski Lagrangian is consistently embedded within supergravity. In particular, the resultant supersymmetric Horndeski model has couplings to gravitino without ghost instabilities.

Let us start with a review the construction of the pure de Sitter supergravity model Bergshoeff:2015tra ; Hasegawa:2015bza . Throughout the manuscript, we use the notation of Wess:1992cp .

The pure de Sitter supergravity model consists of the standard supergravity multiplet and a nilpotent chiral superfield $S(x,\theta)$ Rocek:1978nb ; Ivanov:1978mx ; Lindstrom:1979kq satisfying

[TABLE]

The nontrivial solution to this constraint is that the lowest component of $S(x,\theta)|_{\theta=0}=s(x)$ is no longer an independent degree of freedom, but becomes the fermion bilinear

[TABLE]

where $\chi_{S}$ is the left handed Weyl spinor and $F^{S}$ the auxiliary component in $S(x,\theta)$. Since the auxiliary field $F^{S}$ is in the denominator in (2), the expectation value of $F^{S}$ should be non-vanishing everywhere. In other words, this constrained superfield can be defined only when $\langle F^{S}\rangle\neq 0$, namely supersymmetry is spontaneously broken. This is not a problem as the supersymmetry is broken in de Sitter space anyway. One may think of this constrained superfield as an effective description of sGoldstino $s(x)$ decoupling Komargodski:2009rz .

Due to the constraint on $S(x,\theta)$, the most general superpotential is

[TABLE]

where $W_{0}$ and $\mu$ are real constants. Besides that, the form of the Kähler potential is restricted to

[TABLE]

The action of the pure de Sitter supergravity is given by

[TABLE]

where $W$ and $K$ are defined in (3) and (4). One needs to integrate out the auxiliary complex scalars $F^{S}$, $M$ and vector $b_{a}$, which requires straightforward but tedious calculations. The complete component action including fermions is shown in Bergshoeff:2015tra ; Hasegawa:2015bza . Here we will focus on the system in the unitary gauge of supersymmetry:

[TABLE]

which significantly simplifies the Lagrangian. Under the unitary gauge condition, the system is simply given by the standard supergravity action with $\chi_{S}=S=0$,

[TABLE]

Here $R$ is the Ricci scalar and $\psi_{\mu}$ is the gravitino. Here, the physical degrees of freedom are graviton $e_{\mu}^{a}$ and massive gravitino $\psi_{\mu}$ and nothing else. The mass of gravitino $m_{3/2}$ is given by $m_{3/2}=W_{0}$. The constant part $\Lambda^{4}\equiv\mu^{2}-3W_{0}^{2}$ corresponds to vacuum energy density. Note that if $\mu=0$ the cosmological constant can never be positive, which was the standard lore in supergravity: the pure supergravity system with graviton and gravitino can only lead to negative cosmological constant, namely, anti-de Sitter spacetime. However, with a nilpotent superfield, we can realize the system having graviton, massive gravitino and positive cosmological constant, which is the pure de Sitter supergravity.

We now show the Horndeski model Horndeski:1974wa ; Deffayet:2011gz ; Kobayashi:2011nu coupled to pure de Sitter supergravity. In the case with linearly realized supersymmetry, the couplings having derivatives more than two lead to ghosts or dynamically propagating auxiliary fields, which are obstacles to realize Horndeski type interactions. So far, the known ghost free higher-derivative interactions with linearly realized supergravity are only of the kind of $\mathcal{L}_{2}=P(X,\phi)$ Khoury:2010gb ; Koehn:2012ar and $\mathcal{L}_{5}=G^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ Farakos:2012je where $X=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$ and $G_{\mu\nu}$ is the Einstein tensor. The latter is only known in the new minimal supergravity, and more general coupling ${\tilde{G}}_{5}(X,\phi)G^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ without ghosts has not been found yet.

The situation is drastically different in the nonlinearly realized supersymmetry as we will show. It seems rather natural to expect that Horndeski type interaction can be realized if supersymmetry is spontaneously broken: When supersymmetry is broken at some scale, its very low energy effective theory, where the heavy superparticles decouple, would become effectively non-supersymmetric system, and Horndeski type Lagrangian can be realized in such case. As pointed out in Farakos:2017mwd , the following expression would realize (almost) arbitrary couplings within supergravity:

[TABLE]

where ${\cal C}$ is a general superfield, which may also have spinor or vector indices in general, and $C={\cal C}|_{\theta=0}$. We have also defined $\zeta_{\alpha}\equiv{\cal D}_{\alpha}{\cal C}|_{\theta=0}$. The derivative ${\cal D}_{a}$ is vector derivative in superspace and $\nabla_{a}=e_{a}^{m}\nabla_{m}$ denotes the spacetime derivative covariant under diffeomorphism as well as supersymmetry. The ellipses part denotes the terms including the Goldstino $\chi_{S}$, which vanish in the unitary gauge (6).

The coupling (8) is crucial to realize the Horndeski model. From the above observation, it would be enough to find the superfield function $\mathcal{F}$ which has the Horndeski couplings in the lowest component. Since the original Horndeski model is described by a single real scalar field, we consider a chiral superfield $\Phi$ satisfying the following constraint Komargodski:2009rz ; Farakos:2017mwd ; Ferrara:2015tyn ; Carrasco:2015iij ; DallAgata:2015zxp ; Ferrara:2016een

[TABLE]

where $S$ is the nilpotent superfield. All, but ${\rm Re}\Phi|$, of components in $\Phi$ become dependent fields of $\chi_{S}^{\alpha}$ and vanish in unitary gauge, namely

[TABLE]

This constrained superfield can be realized as the effective theory of broken supersymmetry where all components but $\phi$ in $\Phi$ acquire mass due to soft supersymmetry breaking.

Let us show the function ${\cal F}_{i}$ which corresponds to $\mathcal{L}_{i}$ ($i=2,3,4,5$).

[TABLE]

where $\hat{\phi}=\frac{1}{2}(\Phi+\bar{\Phi})$, ${\cal X}=-\frac{1}{2}{\cal D}_{a}\hat{\phi}{\cal D}^{a}\hat{\phi}$, and we have defined

[TABLE]

The superfields $\hat{\cal R}_{s}$ and ${\cal G}^{ab}$ are given by (see Farakos:2017mwd )

[TABLE]

The lowest components of the superfields are as follows (in unitary gauge).

[TABLE]

Here we emphasize that $\hat{R}$ and $\hat{G}^{mn}$ are Ricci scalar and Einstein tensor consisting of super-covariantized spin-connection $\omega_{m}^{ab}$

[TABLE]

where $A_{[a}B_{b]}=\frac{1}{2}(A_{a}B_{b}-A_{b}B_{a})$. The second order derivative term ${\cal D}_{a}{\cal D}_{b}\hat{\phi}|$ also contains the spin-connection. Thus, we find that these functions ${\cal F}_{i}$ give the Horndeski Lagrangian for $\phi$ nontrivially coupled to gravitino $\psi_{m}^{\alpha}$. Let us show the Horndeski Lagrangian in supergravity:

[TABLE]

where we have defined $X=-\frac{1}{2}\nabla_{m}\phi\nabla^{m}\phi$, $(\nabla_{m}\nabla_{n}\phi)^{2}=\nabla_{m}\nabla_{n}\phi\nabla^{m}\nabla^{n}\phi$ and $(\nabla_{m}\nabla_{n}\phi)^{3}=\nabla_{m}\nabla_{n}\phi\nabla^{n}\nabla^{k}\phi\nabla_{k}\nabla^{m}\phi.$ The purely bosonic part of this action is precisely the same as Horndeski model.

Although the original Horndeski action is known to be free from ghosts, the absence of the ghosts in our supersymmetric system is still nontrivial. Since the gravitino couples to Horndeski Lagrangian through $\hat{R}$ and $\hat{G}$ as well as second-order derivative terms of $\phi$, such as $\nabla^{2}\phi$, one may wonder if the higher-derivative terms for gravitinos are produced. However, we can prove the absence of ghosts as follows. For the equations of motion of $\phi$ and graviton, it is proven by a simple observation that the difference appears only in the explicit form of the spin-connection. Therefore, the equations of motion of our model are derived from that of the original Horndeski system by simply replacing the non-supersymmetric spin-connection by (26). One finds that there is no second order derivative term of spin-connection, which implies that the equation of motion of the gravitino is the first order. Also, our action does not lead to more-than second-order derivatives for bosons. The gravitino variation in our action is always associated with the spin-connection. As the third-order derivative terms are canceled in the equations of motion of $\phi$ and graviton $e_{m}^{a}$, one can check that the coefficients of the variation $\delta\omega_{m}{}^{ab}$ is up to the second order in derivatives. Since there is no derivative terms of the gravitino $\psi_{m}^{\alpha}$ in the definition of the spin-connection (26), $\frac{\delta\omega_{m}{}^{ab}}{\delta\psi_{n}^{\alpha}}$ would not increase the number of derivatives. Equivalently, the gravitino equation of motion is up to the first order differential equation of fermion and up to the second order differential equation of bosons. Thus, we find that our supersymmetric Horndeski action (Horndeski model in nonlinearly realized supergravity) is free from ghosts.

Besides the pure de Sitter supergravity action (5), one may add the Kähler and superpotential terms of $\Phi$, but then one should take into account the fact that $\Phi$ is a constrained superfield.

We note that the nonlinear realization is supposed to be the LEFT of spontaneously broken supersymmetry, and the theory breaks down at sufficiently high-energy scale. In our case, the cut-off scale is given by the unitarity bound $E_{\rm cut}=\sqrt{6\sqrt{2\pi}m_{3/2}M_{pl}}$, at which four-fermi coupling becomes strong Casalbuoni:1988sx . Also, if gravitino mass is much below the cut-off scale, huge gravitino production may take place Hasegawa:2017hgd ; Hasegawa:2017nks . If we consider inflationary models based on our supersymmetric Horndeski Lagrangian, the Hubble scale during inflation should be below the cut-off scale. Such constraint gives the inequality $m_{3/2}>2.8\times 10^{3}{\rm TeV}\left(\frac{H}{10^{13}{\rm GeV}}\right)$. Therefore, light superparticles would not be expected in our construction, which is very consistent with the collider experimental result so far.

In this work, we have shown the embedding of Horndeski Lagrangian within (nonlinearly realized) 4D ${\cal N}=1$ supergravity. Supersymmetric higher-derivative couplings often have the issues of ghosts or dynamical auxiliary fields, which have been an obstacle for the construction of general inflationary models with derivative interactions. As we have shown, such issues could be circumvented once supersymmetry is spontaneously broken, and indeed, we have constructed the Horndeski Lagrangian with constrained superfields, which describe the low energy effective theory of broken supersymmetry. In pure de Sitter supergravity model, the physical degrees of freedom are graviton, massive gravitino, and we have added a single real scalar field $\phi$ in the Horndeski Lagrangian. The supergravity realization of Horndeski Lagrangian is achieved in (Horndeski model in nonlinearly realized supergravity) with a constrained superfield $\Phi$. It is remarkable that gravitino couples to the Horndeski scalar $\phi$ in super-covariant way. As we discussed, additional gravitino couplings do not lead to higher-derivative terms, and do not spoil the ghost-free property of the original Horndeski Lagrangian, although we naively expect fermionic ghosts at first glance. This might be understood as the consequence of supersymmetry of the gravity multiplet: Since graviton is coupled to a real scalar without ghosts, its fermionic partner, gravitino, is also free from ghosts.

We are now able to construct supergravity inflation model with Horndeski Lagrangian, where the inflaton $\phi$ is super-gravitationally coupled to gravitino. In such a case, one needs to consider e.g. the gravitino production through (non-)perturbative processes, which would be calculable on the bases of the Lagrangian we have shown.

It would also be possible to realize larger class of the ghost free scalar-tensor system Langlois:2015cwa within our framework. However, as in our case, additional gravitino couplings would show up and one needs to check whether or not such couplings lead to fermionic ghost instabilities.