Finite size effects in DBI and Born-Infeld for screened spherically symmetric objects

We study finite size effects on the linear response of spherically symmetric objects in Born-Infeld (BI) electromagnetism and Dirac-Born-Infeld (DBI) scalar field theories. Previous works show that the linear response coefficients for a point-like source vanish for odd multipoles above the dipole, a feature that resembles the vanishing of Love numbers for black holes. This work goes beyond the point-like idealisation and considers a sphere of finite radius. We find that the vanishing of the linear response coefficients ceases as they acquire a correction due to the finite size of the object. This introduces a hierarchy between the even and odd multipoles of the response coefficients determined by the separation of scales between the radius of the sphere and the screening scale of non-linearities. From a phenomenological viewpoint, the hierarchy between the odd and even multipoles would give access to the screening scale and the object’s radius by measuring the behaviour of the potentials at infinity.

💡 Research Summary

This paper presents a detailed investigation into the finite-size effects on the linear response of spherically symmetric objects within the framework of two renowned non-linear field theories: Born-Infeld (BI) electromagnetism and the Dirac-Born-Infeld (DBI) scalar theory. The core motivation stems from a curious phenomenon observed in prior studies for point-like sources: the linear response coefficients (analogous to tidal Love numbers) vanish for all odd multipoles above the dipole. This work moves beyond the point-source idealization to consider a physically realistic, finite-radius sphere with a uniform charge (or source) density.

The analysis begins by establishing the equivalence between the static, purely electric sector of BI and the static DBI theory, allowing for a unified treatment. The authors then solve for the exact background field profile generated by a sphere of radius r0. The solution reveals a three-zone structure: (i) an outer linear (unscreened) region where the field decays as 1/r, (ii) a screened shell where non-linearities dominate and the field strength saturates near the scale Λ², and (iii) an inner unscreened core within the sphere. This third region arises because the enclosed charge decreases as r³ inside the object, eventually making non-linearities negligible again at small enough radii (r < r0³/rs², where rs is the screening radius). The existence of this core is the key departure from the point-particle model.

The heart of the paper involves studying static, linear perturbations atop this background configuration. The perturbations are decomposed into polar modes (common to both BI’s electric sector and DBI) and axial modes (specific to BI’s magnetic sector). The linearized field equations are solved separately inside and outside the sphere, with appropriate continuity conditions imposed at the surface r = r0. The linear response coefficients k_ℓ are extracted from the asymptotic behavior of the external solutions.

The principal finding is that the vanishing of the odd-multipole response coefficients, a hallmark of the point-source limit, ceases to hold for a finite-sized object. These coefficients acquire non-zero corrections due to the finite radius. Crucially, these corrections are highly suppressed, scaling as k_ℓ (odd) ∝ (r0/rs)^(2ℓ+1). In contrast, the even-multipole coefficients remain relatively large. Consequently, a distinct hierarchy emerges between even and odd multipoles in the linear response, dictated by the separation of scales between the object’s physical size r0 and its screening scale rs (or Λ).

This hierarchy carries significant phenomenological promise. By measuring the multipole structure of the fields (e.g., potentials or forces) at large distances from a compact, screened object, one could in principle disentangle the two fundamental scales: the even multipoles are more sensitive to the object’s size r0, while the odd multipoles are sensitive to the screening scale Λ. This opens a novel avenue for probing the interior properties of exotic compact objects, such as certain dark matter candidates or other dense bodies governed by BI/DBI-like dynamics, through their tidal deformability or response to external fields. The work also revisits the existence of ladder operators and conserved charges in these theories, examining their action on the physical perturbation solutions.

In summary, the study demonstrates that finite-size effects break the delicate symmetry responsible for the vanishing odd Love numbers in point-like BI/DBI systems. However, this breaking is controlled and informative, encoding the object’s intrinsic parameters into a measurable hierarchical pattern in its linear response, thereby bridging a gap between theoretical idealizations and potential observational signatures.