Quantum Chameleons

We initiate a quantum treatment of chameleon-like particles, deriving classical and quantum forces directly from the path integral. It is found that the quantum force can potentially dominate the classical one by many orders of magnitude. We calculate the quantum chameleon pressure between infinite plates, which is found to interpolate between the Casimir and the integrated Casimir-Polder pressures, respectively in the limits of full screening and no screening. To this end we calculate the chameleon propagator in the presence of an arbitrary number of one-dimensional layers of material. For the E\"ot-Wash experiment, the five-layer propagator is used to take into account the intermediate shielding sheet, and it is found that the presence of the sheet enhances the quantum pressure by two orders of magnitude. As an example of implication, we show that in both the standard chameleon and symmetron models, large and previously unconstrained regions of the parameter space are excluded once the quantum pressure is taken into account.


INTRODUCTION
A wealth of Dark Energy models involve a scalar field with an extremely low mass which plays a role on cosmological scales [1], explaining the accelerated expansion of the Universe. At shorter distances, such as the Solar System scale, some mechanism must take place to suppress the long-range force induced by the new scalar, since such scenario would be otherwise excluded by stringent experimental tests [2,3]. A screening mechanism of the long-range force can naturally occur as a result of the scalar coupling to matter. Indeed, whenever the local matter density is high enough with respect to the other scales of the problem, the properties of the scalar (mass or couplings) tend to change in the local environment and, typically, the scalar tends to get invisible where one could observe it [4,5]. We will refer to any scalar with such property as a chameleon field. For instance in the symmetron model [6][7][8] screening occurs mainly as the coupling to matter decreases with an increasing matter density. The existence of chameleon fields can be tested by laboratory experiments, for instance by neutrons [9] or atomic spectroscopy [10]. The pressure between two parallel plates is also suited to test the potential presence of chameleons [11,12] which could become within reach in the near future [13].
The effects of chameleon fields are typically treated in a classical approximation. However at short enough distances -such as the submicron scale in the Eötwash experiment [14], a quantum treatment of the chameleon mechanism becomes mandatory. In this work we develop the formalism to describe "quantum chameleons" and present some of its consequences. The formalism also sheds new light on the quantum field theory calculation of the Casimir pressure in its various regimes.

CHAMELEON FORCES FROM THE PATH INTEGRAL
We start with the general chameleon Lagrangian where V (φ) is the interaction potential, J is a source and A(φ)J describes how the chameleon couples to the source [15]. In this letter J is always considered as a matter density i.e. J(x) = ρ(x) and is assumed to be static. Both V and A can contain higher dimensional operators suppressed by a scale M , in which case the Lagrangian describes a low-energy effective field theory (EFT). In the following we define the effective potential V J = V + AJ. In general J, φ, A(φ) depend on space, this will often be left implicit. The source term is assumed to depend on an external parameter L, to be understood, for instance, as a measure of the distance between two objects. Our aim is to understand how the quantum system reacts when changing L. To this end we consider the generating functional of connected chameleon correlators When the source is static the vacuum energy is given by T where T = dt. We will work with T = 1 conventionally and refer to the generating functional directly as the vacuum energy.
All the information about the force (or pressure) that one source induces to another in the presence of the chameleon field is contained in the variation of the vacuum energy with respect to L. This variation is given by This seems a simple result however the calculation of the quantum average A(φ) is highly non-trivial and can in general be evaluated only in the expansion, i.e. the loop expansion of quantum field theory. Writing the chameleon field as φ = φ cl + η where η represents the quantum fluctuations around the classical field φ cl , we find the first two terms in the expansion to be where ∆ J (x, x, ) is the Feynman propagator of the fluctuation, which satisfies the equation of motion where V = δ 2 V /δ 2 φ cl , A = δ 2 A/δ 2 φ cl . The convention adopted here is that F > 0 for an attractive force. The first term in Eq. (4) is the classical force, pictured in Fig. 1i. This term does not involve relativistic retardation and only requires to solve the background equation of motion. When for instance A(φ) = yφ, V (φ) = 1 2 m 2 φ 2 , one recovers exactly the Yukawa force (note one can use φ (x) = y d 4 x i∆(x, x )J(x )). This method of calculation gives the same result as those starting from the classical stress-energy tensor used in [16].
The second term in Eq. (4) is the quantum force, pictured in Fig. 1ii. It can be obtained by taking the derivative ∂ L of the functional determinant obtained in the explicit evaluation of E[J], then recognizing the geometric series of insertions corresponding to the Green function satisfying Eq. (5).
and realise that η 2 (x) has to be the connected correlator for η in presence of the source term. The vacuum energy (in)famously contains infinities which usually have to be subtracted by hand (see e.g. [17,18]). In our approach all divergences automatically vanish thanks to the ∂ L since they are L-independent, as should be the case as ∂ L E[J] is an observable. Inspecting the derivative with respect to L of the functional determinant one can see that ∂ L removes all diagrams which do not link a source to the other, i.e. the "tadpole" diagrams of the extended sources, pictured in Fig. 1iii. Thus in Eq. (4) the infinite part of ∆(x, x, ) (which is L-independent) does not contribute and one can readily use its finite part ∆ fin (x, x). [19] THE CHAMELEON QUANTUM FORCE Computing the quantum force (Eq. (4)) requires the knowledge of the ∆ J propagator. However we can readily deduce some important general properties prior to any calculation. Whenever the source term A J is large with respect to other scales involved in the interaction potential, the Green function should vanish (i.e. be "screened") inside the source and vanish at its surface, as illustrated in Fig. 1iv. These are precisely the conditions of the standard Casimir effect.
In the opposite limit, when the coupling to the source A J can be treated perturbatively, the functional determinant Eq. (6) can be truncated at quadratic order, in which case it is the limit of no screening where the force is This corresponds to the bubble diagram shown in Fig. 1v, which is precisely the Casimir-Polder force integrated over extended sources.
This is a generalisation of the Casimir-Polder potential in presence of an unscreened scalar, which matches results of Refs. [20,21] The chameleon models are often effective theories whose predictions are valid below a cutoff scale, better determined in a given experimental situation. When self-interactions such as L ⊃ φ n /Λ n−4 are present, the cutoff is expected to be ∼ 4πΛ since higher order diagrams are expected to produce fast-growing 1/(ΛL) n contributions to the force which cannot be neglected when L ∼ 1/4πΛ. The cutoff resulting from the matter interactions is slightly more subtle because of screening. Consider the contributions to the force from a leading interaction M −2 φ 2 J and the next-to-leading interaction M −4 φ 4 J (shown in Fig. 1vi ), which contributes at twoloop as M −2 d 4 x∂ L J(∆(x, x)) 2 . We obtain that the two loop contribution is negligible for Therefore since ∆ fin J (x bd , x bd ) → 0 in the presence of screening, the cutoff tends to infinity. This is not surprising as a Casimir pressure should not depend on the coupling to the plates, only on the mass and degrees of freedom of the field living between the plates.
Our conclusions about the validity of the chameleon EFT differ from those drawn in Ref. [22] for the following reason. The reasoning of Ref. [22] would hold if the source occupied the whole space. However one should take into account that whenever an empty region exists, the fluctuation gets confined there when the effective mass induced by the source becomes large (as pictured in Fig. 1iv ). As a consequence the contributions to the 1−loop potential in the source region are suppressed by the vanishing wave function of the fluctuation, and the chameleon EFT is not violated -even when the effective mass induced by the source becomes infinite.

QUANTUM FORCE BETWEEN PLATES
We now study the case of a chameleon in an environment whose constant density changes piece-wise along the direction z, which means the mass of the chameleon is a piecewise constant along z. This case is important as it is a sensible approximation whenever the profile of φ cl near the interfaces is irrelevant compared to the distance L. This approximation is especially accurate for symmetron models [23].
Let us consider the case of 3 regions, for which the effective mass V J ≡ m 2 (z) takes the form This can be readily used to calculate the chameleon pressure between plates of homogeneous mass density ρ, in which case m 2 2 is seen as the intrinsic mass and the sources in regions 1, 3 are identified with M −2 J 1,3 = (M ) −2 ρ 1,3 = m 2 1,3 − m 2 2 . Defining ω(z) = |p 0...2 | 2 + i − m 2 (z), the equation of motion is (∂ 2 z + ω 2 (z))∆(z, z ) = 0. The solution in regions i = 1, 2, 3 is simply (φ + i , φ − i ) = (e iωiz , e −iωiz ). The solution everywhere can be found by continuity of the solution and its derivative at each of the interfaces, defining momentum-dependent transfer matrices of the form The Feynman propagator solves (∂ 2 z + ω 2 (z))∆(z, z ) = In order to obtain the quantum force induced by the fluctuation between regions 1 and 3, one needs to vary E[J] with respect to L. The variation of the source term will simply give a Dirac delta function at z = L. Hence we only need the propagator at the interface between regions 2 and 3, which is .
The variation of the source is more precisely A ∂ L J = (m 2 2 − m 2 3 )δ(z − L) = (ω 2 3 − ω 2 2 )δ(z − L) and the final expression for the pressure between regions 1 and 3 is where one has performed a Wick rotation and introduced ω i = iγ i = i ρ 2 + m 2 i . Let us consider some limiting cases. For m 1,3 → ∞, the expression gives the Casimir pressure from a massive scalar, At the opposite end, weak coupling is defined by (m 2 1,3 − m 2 2 )/m 2 2 1 in which case a perturbative expansion is possible. The leading order in the expansion is quadratic and gives We have checked this corresponds exactly to the Casimir-Polder force integrated over regions 1 and 3.
Although the limits taken above are conceptually simple, the transition between both as a function of L is non trivial, as shown in Fig. 2. We see that the transition occurs over 3 orders of magnitude in L and takes place near L ∼ 1/m 1,3 . Qualitatively, this is the typical distance for which the chameleon fluctuation has high enough momentum to start travelling in the 1, 3 regions. This behaviour can be seen as a validity cutoff on the Casimir pressure, in the sense that at close enough distance the pressure becomes constant instead of keeping growing.
The validity cutoff of the prediction in the presence of a higher-dimensional coupling to matter is shown in Fig. 2. The minimum value allowed for M L, defined as (M L) val ≡ L ∆ fin J (x bd , x bd ) following Eq. (9), reaches a maximal value of ∼ 0.02, which is similar to ∼ 1/4π. In the screened regime, however, the validity cutoff tends to be much higher and goes to infinity when m 1,3 /m 2 → ∞.

QUANTUM BOUNDS ON SYMMETRON DARK ENERGY
The symmetron mechanism, which efficiently relaxes experimental constraints on a light scalar, has triggered a lot of activity in the modified gravity literature, see for instance [24,25]. The mechanism relies on a Z 2 symmetry restoration in the presence of matter and is usually realised as The classical symmetron force between plates is suppressed and is approximated by F cl = µ 4 4λ e −2µL [16]. As made clear in Fig. 2, the classical force is suppressed with respect to the quantum one by ∼ (µL) 4 /λ which is small at distances L < 1/µ, for which the forces become active.
A simple bound on symmetron dark energy comes from molecular spectroscopy [20,21]. Indeed in this case the Casimir-Polder force between nuclei is unscreened and is constrained by high precision measurements of certain transitions in simple molecules and ions like H 2 , HD + , p 4 He + and on the ground state of muonic molecular deuterium ion ddµ + . Observables like bouncing neutrons [26], neutron scattering [27] require more theoretical developments and are not discussed here.
At masses below the meV (see Fig. 2 in [21]), the main bound on the symmetron comes from the torsion pendulum Eötwash experiment. In this experiment an electrostatic shielding sheet is placed between the plates, thus the particle propagates in 5 different regions. The 5-layer propagator is obtained using the method described in this Letter, and the subsequent change in force is shown in Fig. 3 for a massless particle. We see that when the sheet is dense enough, it screens the propagation and the force is enhanced by a factor 16, because the pressure is now between the plate and the sheet, which is twice closer than the opposite plate. In the screening limit of the sheet the Casimir pressure is found to be π 2 /(30(L + W sheet ) 4 ) (L = 55 µm, W sheet = 10 µm for Eötwash [14]).
Interestingly, without the sheet, the L = 55 µm measurement is close to be sensitive to a Casimir pressure between plates induced by the symmetron -or more generally any chameleon-like particle. Once the effect of the sheet is taken into account, the pressure is enhanced and Eötwash becomes sensitive to the symmetron Casimir pressure. The L = 55µm measurement excludes a large part of the symmetron parameter space as shown in Fig. 3. A sensitivity up to M ∼ 1 TeV and to µ ∼ 58 meV is obtained. The exclusion region from the classical symmetron force [16,28] is also shown for comparison. The exclusion region is finite, depends on λ and vanishes for λ 0.4. The exclusion region near the transition requires a treatment of the VEV profile at the interface which is beyond the piecewise constant mass approximation used here. CONCLUSION We have started the study of forces induced by chameleon-like particles in a fully-fledged quantum approach. The formalism clarifies the role taken by screening in the quantum picture and provides a natural interpolation between Casimir and Casimir-Polder forces in extended sources. We have computed propagators with piecewise constant masses in an arbitrary number of 1D regions and analyzed in details the quantum chameleon pressure between plates. Our conclusions relative to the validity of the chameleon EFT differ from [22] and are less restrictive. We find that symmetron dark energy is strongly bounded by its quantum force, for instance from molecular spectroscopy and the Eötwash experiment. For the latter, taking into account the presence of the intermediate sheet is crucial as it increases the Eötwash sensitivity to Casimir pressure for chameleons. The Eötwash bound from classical force is mostly overwhelmed by the quantum one. Other developments can be considered using similar techniques for other kind of observables and experiments including bouncing neutrons, small scale Casimir experiments, neutron scattering. In presence of a UV completion for the chameleon EFT, our formalism provides the basis to do reliable chameleon particle physics.