d-dimensional Oscillating Scalar Field Lumps and the Dimensionality of Space

Extremely long-lived, time-dependent, spatially-bound scalar field configurations are shown to exist in $d$ spatial dimensions for a wide class of polynomial interactions parameterized as $V(\phi) = \sum_{n=1}^h\frac{g_n}{n!}\phi^n$. Assuming spherical symmetry and if $V''<0$ for a range of values of $\phi(t,r)$, such configurations exist if: i) spatial dimensionality is below an upper-critical dimension $d_c$; ii) their radii are above a certain value $R_{\rm min}$. Both $d_c$ and $R_{\rm min}$ are uniquely determined by $V(\phi)$. For example, symmetric double-well potentials only sustain such configurations if $d\leq 6$ and $R^2\geq d[3(2^{3/2}/3)^d-2]^{-1/2}$. Asymmetries may modify the value of $d_c$. All main analytical results are confirmed numerically. Such objects may offer novel ways to probe the dimensionality of space.