Squark-pair annihilation into quarks at next-to-leading order

The Minimal Supersymmetric Standard Model (MSSM) is under intense scrutiny at the LHC and in dark matter searches. Interestingly, scenarios with light squarks of the third generation remain not only viable, but also well motivated by the observed Standard-Model-like Higgs boson mass and dark matter relic density. The latter often requires important contributions from squark-pair annihilation. Following up on previous work, we present in this paper a precision analysis of squark-pair annihilation into quarks at next-to-leading order of QCD including Sommerfeld enhancement effects. We discuss all technical details of our one-loop, real emission and resummation calculations, their implementation in the precision tool DM@NLO, as well as the numerical impact on the annihilation cross section and cosmological relic density in phenomenological MSSM scenarios respecting, in particular, current LHC constraints. We demonstrate that including these radiative corrections leads to substantial shifts in the preferred parameter regions by up to 20 GeV.

Figure 1

Contribution of selected processes to the total annihilation cross section ${\sigma }_{\mathrm{ann}}$ in the $M_1-M_{{\tilde{t}}_R}$ or $M_1-M_{{\tilde{q}}_L}$ plane around reference scenarios I or II, respectively. The orange band indicates the parameter region in agreement with the Planck limit given in Eq. (1.1) at the $2\sigma$ confidence level. The green levels indicate the relative importance of the processes that can be corrected by DM@NLO (first and fifth plots) and of selected individual processes (remaining plots). The grey region corresponds to $m_{{\tilde{t}}_1}<m_{{\tilde{\chi }}_1^0}$. The red dots indicate scenarios I and II of Table 1.

Figure 2

Tree-level Feynman diagrams associated with the squark-pair annihilation into quark pairs for the case of squarks of identical (upper row) or different (lower row) types.

Figure 3

Upper part: Leading-order cross section $\sigma v$ as a function of the center-of-mass momentum $p_{\mathrm{cm}}$ into different subchannels according to Fig. 2 in the two pMSSM scenarios I and II of Table 1. In the legend, ${\sigma }_{\text{Tree}}$ denotes the total tree-level cross section, the subscripts $g$, $\chi$ and $g\chi$ correspond to gluino exchange squared, gaugino exchange squared, and gluino-gaugino interference. The superscripts indicate the squared $t$-channel (T), the squared $u$-channel (U), the sum of both ($\mathrm{T}+\mathrm{U}$), and the $t-u$ interference contributions (TU). For the gaugino exchange, the superscript “Int” refers to the sum of all involved diagrams. Lower part: Contributions relative to the total tree-level result ${\sigma }_{\text{Tree}}$.

Figure 4

Upper part: Decomposition of the leading-order cross section into the color basis in typical pMSSM scenarios for identical (left) and nonidentical (right) incoming particles. The superscripts $\overline{\mathbf{3}}$ and 6 refer to the respective color representation. Lower part: Contributions relative to the total tree-level cross section ${\sigma }_{\text{Tree}}$.

Figure 5

Gluino self-energy diagrams relevant for the gluino mass and wave-function renormalization.

Figure 6

Vertex corrections diagrams associated with the squark-pair annihilation into quark pairs depicted in Fig. 2.

Figure 7

Box diagrams associated with the squark-pair annihilation into quark pairs depicted in Fig. 2.

Figure 8

Real gluon radiation diagrams associated with squark-pair annihilation into quark pairs depicted in Fig. 2.

Figure 9

Ladder diagram for a LO Coulomb potential.

Figure 10

Annihilation cross section $\sigma v$ for the stop annihilation into top quarks (first panel) and neutralino-stop co-annihilation (three remaining panels) for scenario I, computed using the micrOMEGAs tree-level calculation (MO), our leading-order calculation (Tree), our fixed-order NLO calculation (NLO, only first panel), our fixed-order NLO calculation without the velocity-enhanced part of the box contributions (${\mathrm{NLO}}_{\mathrm{B}}$, only first panel), and our full NLO calculation including resummation (F). The lower part shows various relative cross sections according to the second part of the legend.

Figure 11

Same as Fig. 10 for the three quark-annihilation processes relevant in scenario II. In the last panel, we show, in addition, the cross section for chargino $u$-channel exchange with $\beta$-enhanced Yukawa corrections (${\sigma }_{\mathrm{\Delta }{M}_b}$).

Figure 12

Parameter regions compatible with the Planck limits given in Eq. (1.1) presented in the $M_1-M_{{\tilde{t}}_R}$ plane around scenario I and in the $M_1-M_{{\tilde{q}}_L}$ plane around scenario II. The orange band corresponds to the micrOMEGAs calculation, while the blue band stems from the full DM@NLO one-loop calculation. The right panel corresponds to a zoom into the left panel around scenario I (or scenario II), which is indicated by the red dot. Grey regions are excluded due to stop LSP.

Figure 13

Same as Fig. 12, but projected into the plane of the physical neutralino and stop masses.

Figure 14

Upper part: Neutralino relic density ${\mathrm{\Omega }}_{{\tilde{\chi }}_1^0}{h}^2$ along the parameter region satisfying the experimental constraints on the relic abundance. Both the bino mass parameter $M_1$ and the squark mass parameter $M_{{\tilde{t}}_R}$ (or $M_{{\tilde{q}}_L}$) are shown around scenario I (or scenario II). In both plots we show the values obtained using micrOMEGAs (${\mathrm{\Omega }}_{\mathrm{MO}}{h}^2$), our tree-level calculation (${\mathrm{\Omega }}_{\text{Tree}}{h}^2$), and our full one-loop calculation including the resummation (${\mathrm{\Omega }}_{\mathrm{F}}{h}^2$). For scenario I, we also show the value obtained by correcting only neutralino-stop co-annihilation (${\mathrm{\Omega }}_{{\tilde{\chi }}_1^0{\tilde{t}}_1}{h}^2$), and the value obtained by correcting only stop-pair annihilation (${\mathrm{\Omega }}_{{\tilde{t}}_1{\tilde{t}}_1}{h}^2$). Similarly, for scenario II, we show the effect of correcting only the stop and sbottom-pair annihilations (${\mathrm{\Omega }}_{{\tilde{q}}_1{\tilde{q}}_1}{h}^2$) and the stop-sbottom annihilations (${\mathrm{\Omega }}_{{\tilde{t}}_1{\tilde{b}}_1}{h}^2$). Lower part: Impact of the different contributions relative to the relic density obtained by using micrOMEGAs.

Figure 15

Box diagram corresponding to the gluon exchange.