Dispersive interaction between two atoms in Proca Quantum Electrodynamics

Neutral but polarizable atoms with no permanent multipoles placed in vacuum will still interact with each other. This can be understood from the Heisenberg uncertainty principle which enforces dipole fluctuation even in states where the expectation value of the dipole operator vanishes. These forces are the so-called dispersive interactions [1, 2, 3, 4, 5]. They are responsible for several phenomena not only in Physics, but also in Chemistry, Biology and many other areas of science. For instance, they explain why noble gases condensate. Besides, since dispersive forces strongly depend on the polarizabilities of the atoms/molecules involved, which in turn scale with the volumes of such atoms/molecules, these forces explain why the temperatures of condensation of noble gases are greater for the larger atoms (with Helium having the smallest and so on). By the same argument, dispersive forces may become dominant in macromolecules interactions. Increase attention are being given to these forces since they are responsible for the cohesion between different layers of the van der Waals heterostructures, materials obtained by stacking different 2d materials and which display fascinating properties[6]. There is still an intense research going also from the quantum field theory perspective[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In Biology, they are responsible for the softness of our skin and explain the remarkable adhesion of geckos on the walls of our houses[19]. We can go even further with more bizarre examples, like the important role of dispersive forces in the generation of the high electrostatic potentials ocurring during the storms, among others [20, 21]. Differently from other kinds of intermolecular forces, dispersion forces are always present due to the ubiquity of Heisenberg uncertainty principle.

We can think of dispersive forces as a consequence of the exchange of virtual photons between the atoms. In Maxwell electrodynamics in vacuum, photons are massless, and this is responsible for some important features of these forces, such as the long range character of the interaction. However, under certain circumstances, photons may acquire an effective mass due to shielding mechanism. One example arises in colloidal systems. When two atoms interact within an electrolyte solution, the potential generated by the atoms ionize the solution and when the potential is not very strong - that is if the atoms are not very close to each other - we may treat the interaction within the linearized Poisson-Boltzmann equation[22]. In this scenario the Maxwell field in the solution behaves as a free Proca effective electromagnetic field, with the photon mass being given by the inverse of the Debye wavelength. Similar mechanisms arise in metals, or inside waveguides. Indeed, in many situations, a gas of photons between conducting plates may be thought of as a massive bi-dimensional photon gas [23]. Such an effective photon mass affects not only real, but also virtual photons, and thus will have consequences for dispersive interactions. In fact, a recent work [24] has also shown that the dispersive interaction between two atoms inside a plane capacitor is screened due to non-additive effects, in a way that can be effectively described by endowing photons with a mass inversely proportional to the separation between capacitor plates. Also, electric as well as magnetic condensates may lead to effectively massive gauge fields and can be used to investigate superconductivity, confinement in QCD, and an holographic duality scenario to explain the metal-insulator transition in condensed matter [25, 26, 27, 28].

Even in vacuum, a tiny, yet non-vanishing, photon mass is also present in scenarios of physics beyond the standard model [29, 30]. Moreover, these Yukawa-type corrections appear in axion physics and modifications of Newtonian gravity. Casimir effect experiments can be used to impose bounds on these Yukawa parameters[31, 32, 33]. However small it may be, a finite mass for the photon changes a lot our physical picture of the world. Indeed, as pointed out in [34], the above bounds may not be correct depending upon the microscopical origin of the mass. For example, if it appears from the Higgs mechanism, it is possible for large-scale fields to be effectively Maxwellian, and in this case astrophysical observations will be insensitive to the mass of the photon.

Many different observations place very stringent bounds for the photon mass. The most rigorous restrictions come from astrophysical observations, which put an upper bound of $10^{-27}$  eV to the mass of the photon [35]. Particle properties, such as the anomalous magnetic moment of the electron, also place strict limits on its mass [36]. Interestingly, some mass bound for photons can be obtained using Schuman ressonances [37]. In Ref. [38] another terrestrial experiment was proposed, in which the longitudinal mode of photons would be measured to restrict the mass of the photon. Two comprehensive reviews, written by the same authors almost 40 years apart, can be found in [39, 40].

Ultimately, dispersive interactions akin to van der Waals forces are to be expected in any theory with Abelian vector bosons, as is the case in some models of nuclear physics. In that context, vector mesons have a considerable mass in vacuum, which are not protected by the global symmetries linked to these particles. In particular, short-range repulsion between nucleons due to the exchange of massive omega vector mesons plays an important role in the structure of both nuclei and neutron stars [41, 42, 43]. In fact, the interplay between attractive interactions mediated by scalar mesons and repulsive interactions, mediated by such vector mesons, are thought to be the mechanism behind nuclear saturation [44, 45]. The omega meson couples to baryon charge, while mesons in general — composed of a valence quark-antiquark pair — should display fluctuations of the dipole moment of baryon number density. Therefore, omega meson exchange should lead to dispersive forces between mesons. Recently, the coupling of omega mesons to pions and sigmas has been proposed as a mechanism to explain the speed of sound peak expected in ultra-dense nuclear matter [46]. Dispersive van der Waals forces within the context of Quantum Chromodynamics have been previously explored in the literature [47, 48].

The simplest description of massive vector mesons is given by Proca electrodynamics, in which gauge invariance is explicitly lost [49]. While, in some applications, a more fundamental description may be provided by other formulations, which might preserve gauge symmetry, such as the Higgs mechanism or the Stueckelberg action, Proca electrodynamics can be motivated as a simplified version of these formulations, obtained, for instance, by treating extra fields (e.g., the Higgs field) as static, uniform backgrounds. The investigation of dispersive forces mediated by massive photons also raises interesting conceptual questions. For instance, what is the nature of the limit in which the photon mass is taken to zero? The disappearance of the longitudinal polarization could raise the possibility of a discontinuous transition to Maxwell theory, whereas a continuous transition seems to always be found. Yet, the precise dependence of van der Waal forces on the mediator mass near this limit will be revealing of how sensitive these forces are to small photon mass in scenarios beyond the standard model. In the context of the Casimir effect this discussion has revealed surprising subtleties. It was initially predicted that the zero mass limit would be singular due to the longitudinal modes[50], which was subsequently disproved[51, 52] by showing that in the zero mass limit these modes decouple from matter and the plate becomes transparent to them. In this way only transversal modes contribute to the Casimir attraction between plates assuring that this limit is well behaved. Furthermore, understanding dispersive interactions mediated by massive photons may also furnish new insights into Maxwell electrodynamics. For example, a photon mass breaks the degeneracy between the group, phase and signal propagation velocities, present in standard electrodynamics. This will enable us to verify which one plays the dominant role in the physics of dispersive forces.

In this work we employ Proca electrodynamics to investigate the effects of a finite photon mass on the dispersive interaction between two atoms. This subject has recently got renewed interest in the search for signatures of physics beyond standard models[53]. In this paper we employ a Hamiltonian originally proposed by P. Milonni[1] and show that it is possible to analyze the Proca dispersive interaction within first order perturbation theory. Afterwards we analyze the short distance limit, known as the non-retarded regime. We study this limit by two complementary approaches. In section II we evaluate the interaction energy without quantizing the electromagnetic field, while in section III we re-obtain the results of section II by taking the appropriate limit of the full quantum electrodynamics treatment. By doing so, we are able to address the question of when it is necessary to quantize the electromagnetic showing that it leads to a weaker condition for the interatomic distance. In other words, a photonic mass extends the range of validity of the non-retarded regime. We leave section IV for our final remarks.

Let us consider two atoms, $A$ and $B$, held at positions $\mathbf{r_{A}}$ and $\mathbf{r_{B}}$, respectively and for convenience let us define $\bm{R}=\bm{r}_{B}-\bm{r}_{A}$. Throughout this paper we shall assume the magnitude of the distance $R$ to be much larger than the typical size of the atoms, enabling us to employ the electric dipole approximation. Hence, dispersive forces arise from correlations between the fluctuating dipoles in each atom. In the London regime, we consider the field generated by the atomic dipoles as electrostatic. This is a good approximation when the atoms are close to each other, and in the next section we employ a full quantum electrodynamics calculation in order to establish the precise condition $R$ must satify. In the dipole approximation, the interaction Hamiltonian describing the coupling between the atoms is given by

where $\mathbf{d}_{A}$ denotes the electric dipole operator of atom $A$ and $\mathbf{E}_{B}(\mathbf{r_{A}})$ stands for the electrostatic field created by the dipole $B$ at the position of atom $A$. The electrostatic field $\mathbf{E}_{B}$ depends upon the dipole operator of atom $B$, $\mathbf{d}_{B}$, and is given by (see appendix A)

where $\mu\equiv mc\,(\hbar)^{-1}>0$, with $m$ denoting the photon mass in Proca electrodynamics. Substituting the last equation into the Hamiltonian (1) we obtain

Notice that this Hamiltonian is symmetric under the exchange $A\leftrightarrow B$ as it should - the interaction does not depend on which atom we choose as the source of the electric field in Eq. (1). Denoting the ground state of atom $j$ by $|0_{j}\rangle$, the fact that the atom does not possess permanent electric dipole means that $\langle 0_{j}|\mathbf{d}_{j}|0_{j}\rangle=0$. Therefore, perturbation theory applied to the above Hamiltonian vanishes at first order. At second order we have

where we denote the excited states of atoms $A$ and $B$ by $|r\rangle$ and $|s\rangle$, respectively. We assume that the atomic ground state is non-degenerate, as usual. Choosing the $Z$-axis parallel to $\mathbf{R}$ we may rewrite Eq. (7) to obtain

Substituting Eq. (11) into Eq. (8) we obtain

where

and $\Lambda>0$ is a constant depending only on the atomic internal structure, given by

In the last expression we assume an isotropic response of atoms, which enables us to make the replacement $\langle 0|d_{m}|I\rangle\langle I|d_{n}|0\rangle=|\mathbf{d}^{0I}|^{2}\delta_{mn}/3$ for each atom. We emphasize that, in principle, $\Lambda$ depends on $\mu$, since the photon mass also affects the fields inside the atoms. Nonetheless, assuming $\mu a\ll 1$, where $a$ is the atom size, we can consider the atom structure to be independent of $\mu$, and suppose that $\mu$ affects only the intermolecular interaction. We shall assume this to be the case henceforth.

As expected, if $\mu=0$ we reobtain London’s formula, hereafter denoted by $U_{\rm London}$. For $\mu R\ll 1$, the leading corrections to the London potential are given by

and hence we see that a small $\mu$ weakens the interaction, as could be expected, since the field produced by each fluctuating dipole becomes exponentially damped with distance. Notice that the linear term is not present, which could be anticipated by Eq.(4), since the Proca correction for the dipole electric field is already of order $\mu^{2}$, as the reader may verify. In the regime $\mu R\gg 1$, on the other hand, the interaction between atoms is exponentially suppressed. In Fig.1 we plot the interaction energy normalized by the usual London interaction as a function of the dimensionless parameter $\mu R$.

Equation (12) reproduces exactly the interaction energy of two atoms interacting inside a colloidal system[22], justifying the use of Proca electrodyamics as an effective theory for these systems. Within this effective scenario the intraatomic field is Maxwellian and thus $\Lambda$ is not dependent on $\mu$. This means that the ratio $U_{\rm NR}/U_{\rm London}$ depends only on the colloidal properties which defines $\mu$ and not on the atoms partaking in the interaction.

Before closing this section, we follow [54] (see section 7.6) and rewrite $\Lambda$ in the form

where $\alpha_{j}$ denotes the dynamical polarizability of atom $j$, defined by

Since the intermolecular interaction is due to the correlation between induced dipoles, the polarizability is a key concept to the physical understanding of the intermolecular interaction. Equation (17) illustrates the dispersive character of the interaction, expressed by the dependence of the atomic response $\alpha_{j}$ on the frequency of the external perturbation. As we shall see in the next section, the polarizability is the most convenient starting point for analyzing the interatomic interaction within quantum electrodynamics.

In this paper we have derived the dispersive interaction energy between two polarizable atoms, with no permanent dipole moments, within Proca electrodynamics for any distance regime. We emphasize that we employed a very convenient Hamiltonian, introduced in Ref.[1], which allows for a first order perturbative calculation.

We have found that, despite the inclusion of an extra energy scale corresponding to the photon mass, dispersive interactions within Proca electrodynamics are still marked by the existence of no more than two asymptotic regimes, corresponding to the London and Casimir-Polder limits of Maxwellian electrodynamics. A timescale consisting of interatomic distance over phase velocity may be used to distinguish between these two regimes, but, in contrast to Maxwell’s theory, the nonlinear relationship between frequency and wavenumber in the massive case prevents its unambiguous physical interpretation as a retardation time. This is due to the fact that the phase velocity is both frequency-dependent and superluminal in the massive theory. We have computed closed-form analytical expressions that approximate the dispersive interaction energy in the retarded and non-retarded regimes and presented a detailed discussion on their differences with respect to the case of a massless photon. Our results reveal an enhancement of the non-retarded regime, which extends over a broader range of interatomic distances in the presence of a photon mass.

We have also shown that the effects of the photon mass weaken the interaction energy. More strikingly, we have shown that, for a given interatomic distance, this weakening is very similar for both the retarded and the non-retarded regimes. This means that the mass correction is nearly independent of the atomic transition frequency. As a perspective, it would be interesting to explore if this still holds beyond the atomic scenario, analyzing for instance the Casimir-Proca force between dielectric media.

Our results should be relevant not only to beyond the standard model physics scenarios, but also for any theories or effective descriptions featuring massive vector bosons or an effective photon mass. An effective photon mass may emerge, for instance, inside electrolyte solutions, metals or waveguides [23, 24]. Other massive vector bosons can be found in effective descriptions of nuclear matter, such as relativistic mean-field models of nuclear interactions [44, 45, 41, 42, 43]. In the nuclear context, the possible polarization of mesons into a baryon-numer dipole could lead to new couplings with vector mesons, with potentially interesting phenomenological consequences [46].

We hope our findings will inform future investigations on potential observable effects of a finite, albeit small, photon mass [53]. Because bounds on the mass of the photon are rather small, the detection of a photon mass via van der Waals interactions would be far from trivial. Nonetheless, possible implications of the finite-mass modifications in the condensation of gases and other physical phenomena could potentially prove more revealing of the photon mass then the van der Waals interaction energy in itself. It would be interesting to explore this possibility in future work.