Self-annihilation of the neutralino dark matter into two photons or a $Z$ and a photon in the minimal supersymmetric standard model

We revisit the one-loop calculation of the annihilation of a pair of the lightest neutralinos into a pair of photons, a pair of gluons and also a $Z$ photon final state. For the latter we have identified a new contribution that may not always be negligible. For all three processes we have conducted a tuned comparison with previous calculations for some characteristic scenarios. The approach to the very heavy Higgsino and $W$-ino is studied and we argue how the full one-loop calculation should be matched into a more complete treatment that was presented recently for these extreme regimes. We also give a short description of the code that we exploited for the automatic calculation of one-loop cross sections in the minimal supersymmetric model that could apply, both for observables at the colliders and for astrophysics or relic density calculations. In particular, the automatic treatment of zero Gram determinants which appear in the latter applications is outlined. We also point out how generalized nonlinear gauge-fixing constraints can be exploited.

Figure 1

Typical classes of diagrams common to both ${\tilde{\chi }}_1^0{\tilde{\chi }}_1^0\to \gamma \gamma$ and ${\tilde{\chi }}_1^0{\tilde{\chi }}_1^0\to Z\gamma$. For the former, the same “flavor” circulates in the loops and we do not have any mixing as in (a) and (d). Diagrams with Goldstone bosons are not shown. In the heavy $W$-ino and Higgsino limit, (a) is the dominant diagram. Diagrams (d) and (f) with $H,h$ exchange do not contribute for $v=0$.

Figure 2

An additional class of diagrams describing the ${\tilde{\chi }}_i^0\to {\tilde{\chi }}_1^0\gamma$ transition that only appear in the case of ${\tilde{\chi }}_1^0{\tilde{\chi }}_1^0\to Z\gamma$. A representative of the blob (represented by the first graph) is the virtual correction shown in the second diagram. The last graph shows the counterterm contribution.

Figure 3

Dependence of the $\gamma \gamma$ and $Z\gamma$ cross sections as a function of the LSP neutralino mass $M_{{\tilde{\chi }}_1^0}$. For the Higgsino case we take $M_2=2M_1=4\times {10}^5\mathrm{TeV}$ and for the $W$-ino $\mu =M_1=2\times {10}^5\mathrm{TeV}$. In both cases $\tan \beta =10$, $M_A=100\mathrm{TeV}$ and all sfermions $4\times {10}^5\mathrm{TeV}$. We use $M_W=M_Z{c}_W$, with $s_W=0.473$, $M_Z=91.1884\mathrm{GeV}$, ${\alpha }^{-1}=127.9$.

Figure 4

The first figure shows the dependence on $M_1$ for $\mu (M_2)$ fixed in the Higgsino ($W$-ino) limit for $\gamma \gamma$ and $Z\gamma$. The values for the SUSY parameters are: $\tan \beta =10$, $m_{\tilde{f}}=4\times {10}^8\mathrm{GeV}$, $A=0$, $M_A=100\mathrm{TeV}$. $\mu =50\mathrm{TeV}$, $M_2=2M_1$ in the Higgsino case. $M_2=10\mathrm{TeV}$ and $\mu =M_1$ in the $W$-ino case. In the figure at the bottom, the only parameters that are changed are $\mu =10\mathrm{TeV}$ in the Higgsino case and $\tan \beta =2$, $M_2=500\mathrm{GeV}$ in the $W$-ino case. The SM parameters are as in Fig. 3.

Figure 5

Comparison, for ${\tilde{\chi }}_1^0{\tilde{\chi }}_1^0\to \gamma \gamma$ as a function $M_{{\tilde{\chi }}_1^0}$, between our prediction for the full one-loop calculation (continuous line, black), analytical one-loop expressions based on the approximation of Eq. (2) (dotted, blue) and the nonperturbative prediction based on the fitting functions including resonances [14] (dashed, red). The input SUSY parameters are $M_2=2M_1=50\mathrm{TeV}$, $\tan \beta =10$, $A=0$, $m_{\tilde{f}}=M_A=100\mathrm{TeV}$. The SM parameters are as in Fig. 3. The inset is a close-up for small $M_{{\tilde{\chi }}_1^0}$.