SUSY QCD-corrected and Sommerfeld-enhanced stau annihilation into heavy quarks with scheme and scale uncertainties

We investigate stau-antistau annihilation into heavy quarks in the phenomenological minimal supersymmetric Standard Model within the DM@NLO project. We present the calculation of the corresponding cross section including corrections up to $\mathcal{O}({\alpha }_s)$ and QED Sommerfeld enhancement. The numerical impact of these corrections is discussed for the cross section and the dark matter relic density, where we focus on top-quark final states and consider either neutralino or gravitino dark matter. Similarly to previous work, we find that the presented corrections should be included when calculating the relic density or extracting parameters from cosmological observations. Considering scheme and scale variations, we estimate the theoretical uncertainty that affects the prediction of the annihilation cross section and thus the prediction of the relic density.

Figure 1

Parameter regions in the $M_1–M_{{\tilde{\tau }}_L}$ plane that are compatible with the Planck limits given in Eq. (1.1), where the relic density has been computed using micromegas. All other parameters are fixed to those given in Table 1. The red dot indicates scenarios I defined in Table 1. The green contours correspond to the contribution of the process ${\tilde{\tau }}_1{\tilde{\tau }}_1^{*}\to t\overline{t}$. The black contour lines indicate the difference $m_{{\tilde{\tau }}_1}-m_{{\tilde{\chi }}_1^0}$ in GeV between the physical masses of the lighter stau and the lightest neutralino.

Figure 2

Tree-level Feynman diagrams for stau-antistau annihilation into top-antitop pairs via Higgs ($h^0$ or $H^0$) and vector boson ($\gamma$ or $Z^0$) exchange.

Figure 3

QCD one-loop corrections to the Higgs-top-top and vector-top-top vertices appearing in the processes of Fig. 2.

Figure 4

Real gluon emission to the processes of Fig. 2.

Figure 5

Multiple photon exchange in the initial state leading to the Sommerfeld enhancement.

Figure 6

Annihilation cross section of the process ${\tilde{\tau }}_1{\tilde{\tau }}_1^{*}\to t\overline{t}$ as a function of the center-of-mass momentum $p_{\mathrm{cm}}$ for scenario I of Table 1 using the standard DM@NLO renormalization scheme (left) and the alternative scheme (right). The upper panels show tree-level results and different levels of corrections as discussed in Sec. 3. The lower panels show the corresponding relative corrections. The gray areas indicate the thermal distribution in arbitrary units.

Figure 7

Ratios of the ${\tilde{\tau }}_1{\tilde{\tau }}_1^{*}\to t\overline{t}$ cross section for renormalization scale $\mu$ varied around the central scale ($\mu =1\mathrm{TeV}$) at leading (dashed line) and next-to-leading order (solid line) for the DM@NLO renormalization scheme (left) and the alternative scheme (right).

Figure 8

Ratios of the ${\tilde{\tau }}_1{\tilde{\tau }}_1^{*}\to t\overline{t}$ cross section calculated in the DM@NLO and the alternative renormalization schemes at leading (orange) and at next-to-leading order (green).

Figure 9

Comparison of experimental and theoretical uncertainties in the $M_1–M_{{\tilde{\tau }}_L}$ plane around reference scenario I (indicated by the red dot). The yellow band shows the experimental uncertainties given in Eq. (1.1) as measured by the Planck satellite at the $1\sigma$ confidence level. The leading (next-to-leading) order relic density from both our renormalization schemes is denoted by blue (black) lines. The predictions in the DM@NLO (alternative) renormalization scheme are shown using the solid (dashed) lines. As in Fig. 1, the green contours indicate the relative contribution of the process ${\tilde{\tau }}_1{\tilde{\tau }}_1^{*}\to t\overline{t}$ to the total annihilation cross section, based on the micromegas calculation.

Figure 10

Parameter regions in the $M_1–M_{{\tilde{\tau }}_L}$ plane (left) and $m_{{\tilde{\chi }}_1^0}–m_{{\tilde{\tau }}_1}$ plane (right) that are compatible with the Planck limits given in Eq. (1.1), where the stau relic density has been computed using micromegas (orange) and our full NLO and Sommerfeld corrected cross section (blue). All other parameters are fixed to those given for scenario I in Table 1. The red dot correspond to scenario I. The green contours correspond to the relative contribution of the process ${\tilde{\tau }}_1{\tilde{\tau }}_1^{*}\to t\overline{t}$ to the total annihilation cross section.

Figure 11

Parameter regions in the $m_{\tilde{G}}–T_R$ plane which are compatible with the Planck limits given in Eq. (1.1) for the case of gravitino dark matter, where the stau relic density has been computed using micromegas (yellow) and our full NLO and Sommerfeld-corrected cross section in the DM@NLO scheme (orange). All other parameters are fixed to those given for scenario II in Table 1. The blue contours correspond to the gravitino relic density based on the micromegas calculation. The black lines indicate the relative nonthermal contribution in percent according to Eq. (2.3) to the total gravitino relic density, again based on the micromegas calculation.