Dynamically emergent gravity from hidden local Lorentz symmetry

Gravity can be regarded as a consequence of local Lorentz (LL) symmetry, which is essential in defining a spinor field in curved spacetime. The gravitational action may admit a zero-field limit of the metric and vierbein at a certain ultraviolet cutoff scale such that the action becomes a linear realization of the LL symmetry. Consequently, only three types of term are allowed in the four-dimensional gravitational action at the cutoff scale: a cosmological constant, a linear term of the LL field strength, and spinor kinetic terms, whose coefficients are in general arbitrary functions of LL and diffeomorphism invariants. In particular, all the kinetic terms are prohibited except for spinor fields, and hence the other fields are auxiliary. Their kinetic terms, including those of the LL gauge field and the vierbein, are induced by spinor loops simultaneously with the LL gauge field mass. The LL symmetry is necessarily broken spontaneously and hence is nothing but a hidden local symmetry, from which gravity is emergent.


Introduction and summary
The local-Lorentz (LL) symmetry is essential in defining a spinor field in curved spacetime [1]. Indeed gravity can be regarded as a LL gauge theory [2,3,4]. We can write down a gauge invariant linear term in the field strength of the LL gauge field, due to the existence of the vierbein, unlike an ordinary Yang-Mills gauge theory. Starting from the leading linear action, we reproduce the Einstein-Hilbert action once we integrate out the LL gauge field. That is, the LL gauge theory is equivalent to the ordinary metric formulation at the level of classical Einstein gravity.
In general, the LL gauge theory is different from the metric theory if we go beyond the Einstein-Hilbert truncation. For example, it makes a physically observable difference whether or not to treat the LL gauge field as an independent variable in a frame with a field-dependent conformal factor [5]. Therefore in writing down quantum gravity, or a low energy effective theory just below an ultraviolet (UV) cutoff scale Λ of the order of the Planck scale, it is important to identify what is the path-integrated off-shell degrees of freedom of the theory. The necessity for the LL gauge symmetry naturally leads to the idea that the LL gauge field is also a dynamical degree of freedom, which acquires a mass of the order of Λ due to spontaneous symmetry breaking.
At Λ, one would expect that the path integral with respect to the gravitational degrees of freedom should include various topology-changing processes, which involve a degenerate configuration of the metric and vierbein that has zero eigenvalues [6,7]. Therefore it is natural to expect that the action should admit such a degenerate limit. It has been argued that this degenerate limit, including a zero-field limit, corresponds to a linear realization of a symmetry in Higgs mechanism [8].
In this Letter, we show that the finiteness of action in the degenerate limit restricts us to write down only three possible LL invariant terms: the cosmological constant, the linear term in the LL field strength, and the spinor kinetic term, whose coefficients are in general a function of LL singlets such as a scalar field and a spinor bilinear. 1 In particular, it is forbidden to have a kinetic term for a scalar field and for an ordinary Yang-Mills gauge field.
We show that all the kinetic terms, including those of the vierbein and of the LL gauge field, are induced by the spinor loop below Λ, and the LL gauge field acquires a mass of the order of Λ at the same time. Consequently, the LL gauge symmetry is spontaneously broken, and hence is nothing but a hidden local symmetry, from which gravity is emergent.

Degenerate gravity at ultraviolet cutoff
We take the vierbein e a µ and the LL gauge field ω a bµ as fundamental degrees of freedom to describe gravity at a certain ultraviolet cutoff scale Λ, after integrating out possible quantum gravitational/stringy modes. Here and hereafter, the bold roman letters a, b, . . . and the greek ones µ, ν, . . . denote the tangent space basis and the spacetime coordinates in a given chart, respectively. Metric field is defined as a composite of vierbein: g µν = η ab e a µ e b ν , where η = diag(−1, 1, 1, 1) is the flat spacetime metric.
We assume that the action at Λ is invariant under the diffeomorphisms (diff) and the LL transformation: where we employ the short-hand notation We assume that the "bare" action at Λ must admit any degenerate limit |e| → 0 so that it becomes a linear realization of the LL symmetry, where |e| := det a,µ e a µ . In particular, the inverse e a µ and g µν are forbidden, just as we do not add a term with an inverse power of the Higgs field in the Standard Model action. Under this assumption, we find that only the following three terms are compatible with the linear realization at Λ: where 2 ω abµ σ ab +iA µ is the covariant derivative associated with the LL and ordinary gauge symmetries, in which is the Lorentz generator; and V B and M 2 B are the potential and the Planck mass-squared parameter, respectively, which are in general arbitrary functions of singlets under both diff and the LL transformation such as φ, ψψ, etc. The following comments are in order: • The apparent existence of the inverse vierbein e a µ in the action (3)  • The linear realization of the LL symmetry forbids the scalar and ordinary gauge kinetic terms − 1 2 |e| g µν tr (D µ φ) † D ν φ and − 1 2g 2 |e| g µρ g νσ tr (G µν G ρσ ), where g in the denominator is a gauge coupling, • It is also forbidden to put the Levi-Civita connection and the Levi-Civita spin connection, Γ µ ρσ = g µν 2 (−∂ ν g ρσ + ∂ ρ g σν + ∂ σ g νρ ) and Ω a bµ = e a λ ∂ µ e b λ + Γ λ σµ e b σ , respectively.
• In principle, we can also add the so-called Immirzi term ∝ |e| [abcd] e a µ e b ν F cdµν . This becomes a total derivative term if we put the ordinary vacuum solution ω = Ω, and hence we omit it here for simplicity. 2 We see that the linear realization severely restricts the possible form of terms at Λ.
At this level of the action (3), there are no apparent kinetic terms for both the vierbein and the LL gauge field, while there is a mixing term such as ee∂ω as well as the interaction term with the spinor field. One may regard the vierbein and the LL gauge field as auxiliary fields. The kinetic terms will be generated at the loop level by integrating out the spinor fields, as we will see later. In this sense, these auxiliary fields might be interpreted as composite fields of spinor fields, which will become dynamical below Λ at the loop level. We may also recall the compositeness condition [11]. Namely, the action (3) at the cutoff scale Λ is a boundary condition of the low energy effective theory for this system.

Generation of LL gauge kinetic term
We demonstrate that a spinor loop generates a kinetic term for the LL gauge field, starting from the (bare) action (3) with only taking into account the spinor mass term mψψ in V B and with regarding M 2 B as a constant, for simplicity. In this Letter, we assume a flat spacetime background e a µ = δ a µ and ω a bµ = 0, although in general they should be determined dynamically from the self-consistency condition using a general form of the effective potential V eff , including the vierbein and the LL gauge field, to be induced at the loop level. As we do not deal with a general form of V eff , we do not treat generation of mass-terms either. We also do not discuss the kinetic-term generation for the scalars and the ordinary gauge bosons, which can be trivially done as in the ordinary Nambu-Jona-Lasino model; see e.g. Ref. [12] for a review.
We calculate the kinetic term for the LL gauge field Zω 2 g µν g ρσ F a bµρ F b aνσ , which contains a term where Z ω is the field-renormalization factor and the square brackets for indices denote antisymmetrization. Generation of a finite value of Z ω indicates that the LL gauge field has become dynamical. The kinetic operator (4) is induced from the two-point function of the LL gauge field: After some computation, we obtain a result containing the following tensor structure: with the form factors where we have cut off the momentum integral by Λ; the dots denote higher powers of p 2 Λ 2 +m 2 ; and m 2 is a quadratically divergent mass-renormalization constant. Now we can read off Z ω = f (0) for m 2 Λ 2 1: It is remarkable that both the logarithmic and quadratic divergences have canceled out in Z ω .
To conclude, we have found that the LL gauge field acquires the kinetic term and becomes a dynamical field. We have also computed the same vacuum polarization diagram for the vierbein. We have found that the (dimensionless) vierbein acquires a quartically divergent mass operator around the symmetric phase e a µ = 0, as well as a logarithmically divergent kinetic term for its trace mode.

Discussion
The LL gauge symmetry is spontaneously broken once the vierbein background is determined to be any non-zero value such as the flat spacetime in the above example, whereas the zerofield limit e a µ → 0 corresponds to the symmetric phase in the Higgs mechanism. The vierbein field plays the role of the Higgs field, being a fundamental representation of the LL gauge symmetry. A difference here is that the "Higgs" field is a vector field, unlike in an ordinary Higgs mechanism in which it is a scalar. Once vierbein kinetic terms are generated from loop effects, its LL covariant derivative in the broken phase e a µ = 0 will also contribute to the mass of the LL gauge field through the Higgs mechanism, to be added to m 2 ω above. The vierbein background induces, via the spinor loop, not only the transverse term with f p 2 but also directly the non-vanishing longitudinal term with g p 2 . This is the very characteristic of the LL gauge symmetry, in contrast to the usual Higgs mechanism, where the gauge symmetry is only broken by the mass term of the gauge field.
Our result is consistent with the Weinberg-Witten theorem [13]: The above-mentioned longitudinal term results in the violation of the naive current conservation, p µ I abcdµν (p) = 0. That is, the LL-current conservation is spontaneously broken as soon as the LL gauge field becomes dynamical no matter whether the LL gauge field becomes massive or not. This implies that the LL gauge symmetry is necessarily spontaneously broken, that is, the LL symmetry is nothing but the hidden local symmetry; see e.g. Ref. [14] for a review. In other words, the theory at Λ retains the LL gauge invariance even when we integrate out the auxiliary LL gauge field, where the LL gauge invariance is "hidden" in the UV scale physics and is dynamically emergent in the infrared-scale physics, analogously to the hidden local symmetry carried by rho meson in the low-energy QCD.
In the original action at Λ, there is a local diff × SO(1, 3) symmetry, which has 4 + 6 = 10 degrees of freedom. Within 16 degrees of freedom of vierbein fluctuations, the 6 modes of the SO(1, 3) Nambu-Goldstone direction are eaten as the longitudinal modes of the LL gauge field, while the remains correspond to the 10 classical degrees of freedom of the graviton. Among them, 4 modes of vierbein fluctuations are reduced by the transverse condition for the diff. At the quantum level, remaining 6 degrees of freedom are further subtracted by the loop of 4 diff ghost fields, resulting in the 2 degrees of freedom of quantum graviton fluctuation. 3 As we lower the scale below Λ, the LL gauge field appears to become dynamical at the lower loop level than the vierbein (except for its trace mode). It may be intriguing that the LL gauge field seems to become dynamical earlier than the vierbein, which should be compared to the common classical procedure to integrate out the LL gauge field first as an auxiliary field to make the vierbein dynamical.
In this Letter we have taken the action (3) as a low energy effective action: The fluctuating mode of vierbein is charged under the spontaneously broken (hidden) LL gauge symmetry, and the situation is analogous to the hidden local symmetry below the QCD scale. One might see it as a hint for a quantum Einstein-Hilbert theory that uses the renormalization procedure of the hidden local symmetry at the loop level, on the same footing as the chiral perturbation theory; see e.g. Ref. [16] for a review.
We have assumed that the flat vierbein background becomes a vacuum solution. On physical ground, we expect that the vierbein fluctuation will become massless around a vacuum e a µ = 0 in the end. To confirm this expectation, we need to compute the full effective potential. In this Letter, we have computed the spinor loop correction with the naive momentum cutoff. It is important to improve it by a non-perturbative method such as the functional renormalization group. These points will be pursued in a separate publication.