Graduate Compactification of spatial extra dimensions

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The discussion centers on the compactification of spatial extra dimensions, specifically addressing a term in Equation 65 of a referenced paper. The term 4*pi*n*R in the denominator is explained as necessary for the Torus geometry, which modifies the Coulomb force law to accurately reflect the separation of charges in compactified dimensions. In contrast, a compactification on a circle would indeed involve a factor of 2*pi*n*R. Understanding this distinction is crucial for grasping the implications of the geometry on physical laws. The clarification highlights the importance of geometry in the formulation of theories involving extra dimensions.
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Looking to understand results of a paper.
Hi everyone,
I am looking at a paper on compact dimensions. Equation 65 makes sense except for the term of 4*pi*n*R in the denominator. Why is it 4*pi and not 2*pi? I cannot rationalize this. Please help. Thank you.
https://arxiv.org/ftp/hep-ph/papers/0609/0609260.pdf
 
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The Coulomb force law needs to be modified such that ##R_n## takes into account the true separation of charges. The separation in the compactified extra dimension is given by this ##4\pi n R## term which is specific to the Torus geometry. If the extra dimension was compactified on a circle you may well have a factor of ##2\pi n R## as you expected.
 

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