Relic density of dark matter in the inert doublet model beyond leading order for the low mass region. I. Renormalization and constraints

The present paper is the first in a series that addresses the calculation of the full one-loop corrections of dark matter (DM) annihilation cross sections in the low mass region of the inert doublet model (IDM). This series is a sequel to our recent publication concerning these corrections in the high mass region. We first review the renormalization of the model both in a fully on shell (OS) scheme as well as in a mixed scheme that combines on shell (for the masses) and a $\overline{\mathrm{MS}}$ approach when the partial invisible width is closed and does not allow the use of a full OS scheme. The scale dependence introduced by the mixed scheme is shown to be tracked through an analysis of a parametrization of the tree-level cross section and the $\beta$ constant of a specific coupling; this analysis could be followed in theories. We discuss how to minimize the scale dependence. The theoretical uncertainty brought by the scale dependence leads us to introduce a new criterion on the perturbativity of the IDM. This criterion further delimits the allowed parameter space, which we investigate carefully by including a host of constraints, both theoretical and experimental, including in particular, new data from the LHC. We come up with a set of benchmark points that cover three different mechanisms for a viable relic density of DM: (i) a dominance of coannihilation into a fermion pair, (ii) annihilation into two vector bosons of which one is off shell that requires the calculation of a $2\to 3$ process, (iii) annihilation that proceeds through the very narrow standard model Higgs resonance. Since the $2\to 3$ vector boson channel features in all three channels and is essentially a buildup on the simpler annihilation to OS vector bosons, we study the latter in detail in the present paper. We confirm again that the corrected cross sections involve a parameter that can be considered as rescattering in the dark sector, which a tree-level computation is not sensitive to. The setup of the renormalization detailed in the present paper will be the backbone of the accompanying papers where each mechanism requires, calculationally, a specific treatment. One-loop corrections to $2\to 3$ processes for DM annihilation are technically challenging and have not been attempted before. We dedicate one of the accompanying papers to such a computation. The one-loop correction in the presence of a resonance will be presented separately since we need to supplement our general schemes with a complex scheme. For the coannihilation into fermions, our study will show that the annihilation cross sections can be excellently parametrized through simple effective couplings.

Figure 1

The normalized velocity distribution $f(v,x_F=25)$ as a function of $v$.

Figure 2

Born contributions diagrams to $XX\to ZZ$.

Figure 3

The ratio $\sigma v(XX\to W^{+}{W}^{-})/\sigma v(XX\to ZZ)$ as a function of the relative velocity, $v$, shown here for model P1.

Figure 4

A small selection of one-loop contribution diagrams to $XX\to ZZ$. We only picked up some box and triangle contributions. Note the rescattering $XX\to XX$ and $XX\to AA$ diagrams are solely within the dark sector (last two diagrams in the second row and the last diagram in the third row).

Figure 5

The relative (to the tree level) full one-loop corrections to the cross sections $XX\to W^{+}{W}^{-},ZZ$ as a function of the relative velocity for test point P1 with $\mu =M_X/2,M_X,2M_X$ (from left to right). Three values of ${\lambda }_2$ are considered. The percentage deviation due to choosing as input $\alpha (M_Z^2)$ instead of $\alpha (0)$ is also shown. This correction ($\sim 13\%$) is the same for both $ZZ$ and $WW$. It is therefore superimposed for the two cross sections and appears as a common line.