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It has been recently shown that the discrepancy between the theoretical andexperimental values of the anomalous magnetic moment of the muon can be fullyaccommodated by considering the contribution of few Kaluza-Klein (KK) states ofthe gauged lepton number with masses lighter than the LEP energy, consistentlywith present experimental limits. In this article, we construct the minimalembedding of the Standard Model (SM) into D-brane configurations with a gaugedlepton number. In order to give rise to such KK modes, the lepton number gaugeboson must live on an abelian $U(1)_L$ brane extended along at least one"large" extra dimension in the bulk, with a compactification scale$M_L\sim\mathcal{O}(10-10^2~{\rm GeV})$ for a string scale $M_s\sim 10~{\rmTeV}$. As a consequence, $U(1)_L$ cannot participate to the hypercharge linearcombination. We show that the minimal realisation of this framework containsfive stacks of branes: the SM color $U(3)_c$, weak $U(2)_w$ and abelian $U(1)$stacks extended effectively only in four dimensions, the bulk $U(1)_L$, as wellas a fifth $U(1)^{'}$ brane. With these five abelian factors, one finds besidesthe hypercharge a second anomaly-free linear combination which does not coupleto the SM spectrum, both in the non-supersymmetric case as well as in theminimal supersymmetric extension of the model. It is also shown how theright-handed neutrino can be implemented in the spectrum, and how fermionsarising from the two non-SM branes and coupled to the SM through the $U(1)_L$KK modes can provide Dark Matter candidates. Finally, the possibility ofbreaking Lepton Flavour Universality is studied by replacing $U(1)_L$ with abrane gauging only the muonic lepton number, avoiding most experimentalconstraints and enlarging the parameter space for explaining the discrepancy onthe muon magnetic moment.

Minimal embedding of the Standard Model into intersecting D-brane configurations with a bulk leptonic $U(1)$