Analytical Green's Functions for Continuum Spectra

Green's functions with continuum spectra are a way of avoiding the strong bounds on new physics from the absence of new narrow resonances in experimental data. We model such a situation with a five-dimensional model with two branes along the extra dimension $z$, the ultraviolet (UV) and the infrared (IR) one, such that the metric between the UV and the IR brane is AdS$_5$, thus solving the hierarchy problem, and beyond the IR brane the metric is that of a linear dilaton model, which extends to $z\to\infty$. This simplified metric, which can be considered as an approximation of a more complicated (and smooth) one, leads to analytical Green's functions (with a mass gap $m_g \sim \textrm{TeV}$ and a continuum for $s > m_g^2$) which could then be easily incorporated in the experimental codes. The theory contains Standard Model gauge bosons in the bulk with Neumann boundary conditions in the UV brane. To cope with electroweak observables the theory is also endowed with an extra custodial gauge symmetry in the bulk, with gauge bosons with Dirichlet boundary conditions in the UV brane, and without zero (massless) modes. All Green's functions have analytical expressions and exhibit poles in the second Riemann sheet of the complex plane at $s=M_n^2-i M_n\Gamma_n$, denoting a discrete (infinite) set of broad resonances with masses ($M_n$) and widths ($\Gamma_n$). For gauge bosons with Neumann or Dirichlet boundary conditions, the masses and widths of resonances satisfy the (approximate) equation $s= - 4 m_g^2 \mathcal W_n^2[\pm (1+i)/4]$, where $\mathcal W_n$ is the $n$-th branch of the Lambert function.