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 $V(\phi) = \sum_{n=1}^h\frac{g_n}{n!}\phi^n$\mathrm{.}Assumingsphericalsymmetryandif$V’'<0$forarangeofvaluesof$\phi(t,r)$, such configurations exist if: i) spatial dimensionality is below an upper-critical dimension $d_c$; ii) their radii are above a certain value $R_{\mathrm{m}\mathrm{i}\mathrm{n}}$. Both $d_c$ and $R_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are uniquely determined by $V(\varphi )$. For example, symmetric double-well potentials only sustain such configurations if $d\le 6$ and $R^2\geq d[3(2^{3/2}/3)^d-2]^{-1/2}$\mathrm{.}Asymmetriesmaymodifythevalueof$d_c$. All main analytical results are confirmed numerically. Such objects may offer novel ways to probe the dimensionality of space.