Relic density of dark matter in the inert doublet model beyond leading order for the low mass region. II. Coannihilation

We examine the relic density of the light mass dark matter region in the inert doublet model (IDM) when the dominant process is due to coannihilation between the lightest neutral scalars of the model. The full one-loop electroweak corrections are computed in an on shell scheme and are found to be well approximated as an effective cross section expressed in terms of $Z$ observables. The electroweak corrections to the subdominant process, which consists of an annihilation into an on shell $W$ and an off shell $W$, that is calculated as a annihilation into a three-body final state, is also performed. In the calculation of the latter in this case of coannihilation, although the SM Higgs exchange is absent, it is induced at one loop, but the cross section does not show a peak behavior despite the crossing of the Higgs resonance. This cross section reveals an important dependence on a parameter that describes the self-interaction of the new scalars (solely within the dark sector), a parameter which is not accessible in tree-level calculations of standard model-IDM interactions.

Figure 1

Tree-level Feynman diagrams (in the nonlinear Feynman gauge [1, 8]) for $AX\to b\overline{b}$ as a representative of $AX\to f\overline{f}$.

Figure 2

Left panel: The tree-level cross section for $\sigma (XX\to \nu \overline{\nu })$ for P58 and P60 as a function of the relative velocity. The dashed curves are the pure $P$-wave ($\propto v^2$) approximation. We see that this pure $v^2$ approximation is only valid for $v<0.4$. Note that the cross sections are quite large; however, remember that a Boltzmann suppression will be applied to these coannihilations cross sections when they are converted to effective velocity/temperature averaged cross sections. Right panel: The $b\overline{b}$ cross section normalised to the $d\overline{d}$ cross section and the $d\overline{d}$ cross section normalized to the $\nu \overline{\nu }$ cross section. See text for more detailed comments on these plots.

Figure 3

Some one-loop diagrams (in the nonlinear Feynman gauge) for $AX\to b\overline{b}$. We only show a subset of the box corrections and the triangles. Note the last two diagrams involve the quartic coupling within the dark sector, ${\lambda }_2$. Unfortunately, the one mediated by the neutral Goldstone is proportional to the fermion mass and is therefore practically not sensitive to ${\lambda }_2$. The one mediated by the $Z$ will also give a contribution proportional to the fermion mass, since the one-loop integration will give, unlike the tree-level structure involving the difference between the two incoming momenta, a new Lorentz structure proportional to the $Z$ four momentum, which ends up giving a contribution proportional to the final fermion mass. Therefore, the ${\lambda }_2$ dependence only affects the $S$ wave.

Figure 4

The tiny ${\lambda }_2$ dependence in the $b\overline{b}$ coannihilation channel. The ${\lambda }_2$ dependence is measured relative to the $d\overline{d}$ cross section, where the ${\lambda }_2$ dependence is vanishing. Observe that the units in the graph indicate a variation which, at its highest, does not cross a couple of per-mille.

Figure 5

A selection of diagrams for the tree-level cross section $XX\to Wf{\overline{f}}^{\prime }$. Since ${\lambda }_L=0$ for the benchmark point P60, Higgs exchange does not take place for P60.

Figure 6

The tree-level cross section $XX\to Wf{\overline{f}}^{\prime }$ for the benchmark points P58 and P60. The arrow at $v\sim 0.56(0.745)$ indicates the position corresponding to the Higgs resonance (if crossed) for $M_X=60(58)\mathrm{GeV}$.

Figure 7

A very small selection of the diagrams that enter the one-loop calculations of $XX\to W{W}^{\star }$. We highlight in particular Higgs exchange through the induced $hXX$ vertex at one-loop. We have also picked configurations where a ${\lambda }_2$ contribution shows up. These are part of what constitutes rescattering in the dark sector before annihilation into SM particles. An example of a pentagon diagram, which can not be considered as a factorized, $XX\to W{W}^{\star }\to Wf{f}^{\prime }$ is also singled out (five-point function).

Figure 8

The relative one-loop contribution to $XX\to Wf{f}^{\prime }$ as a function of the relative velocity $v$ for ${\lambda }_2=0.01$, 1, 2. These corrections are compared to the effect of using an effective tree-level correction with $\alpha (M_Z^2)$.