Effective scalar four-fermion interaction for Ge-phobic exothermic dark matter and the CDMS-II silicon excess

We discuss within the framework of effective four–fermion scalar interaction the phenomenology of a weakly interacting massive particle (WIMP) Dirac dark matter candidate which is exothermic (i.e., is metastable and interacts with nuclear targets down-scattering to a lower-mass state) and Ge-phobic (i.e., whose couplings to quarks violate isospin symmetry leading to a suppression of its cross section off Germanium targets). We discuss the specific example of the CDMS–II Silicon three-candidate effect showing that a region of the parameter space of the model exists where WIMP scatterings can explain the excess in compliance with other experimental constraints, while at the same time the dark matter particle can have a thermal relic density compatible with observation. In this scenario the metastable state $\chi$ and the lowest-mass one ${\chi }^{\prime }$ have approximately the same density in the present Universe and in our Galaxy, but direct detection experiments are only sensitive to the down-scatters of $\chi$ to ${\chi }^{\prime }$. We include a discussion of the recently calculated next-to-leading order corrections to dark matter–nucleus scattering, showing that their impact on the phenomenology is typically small, but can become sizable in the same parameter space where the thermal relic density is compatible to observation.

Figure 1

Mass splitting $\delta =m_{{\chi }^{\prime }}-m_{\chi }$ as a function of $m_{\chi }$. The horizontally (red) hatched area represents the IDM parameter space where the excess measured by CDMS-Si [6] corresponds to a $v_{\min }<v_{\text{esc}}^{\text{Lab}}$ range which is always below the corresponding one probed by LUX and XENON100 (adapted from [23]). As explained in the text, in this case Xenon experiments can constrain the CDMS-Si excess only when some assumptions are made on the galactic velocity distribution. The enclosed region is the result of the combination of four conditions: the thin solid line where $v_{\min }(E_{\min }^{\text{LUX}})=v_{\min }(E_{\max }^{\text{CDMS}-\text{Si}})$; the thick solid line where $v_{\min }(E_{\min }^{\text{LUX}})=v_{\min }(E_{\min }^{\text{CDMS}-\text{Si}})$; the thin long–dashed line where $v_{\min }(E_{\max }^{\text{CDMS}-\text{Si}})=v_{\text{esc}}^{\text{Lab}}$; the thick long–dashed line where $v_{\min }(E_{\min }^{\text{CDMS}-\text{Si}})=v_{\text{esc}}^{\text{Lab}}$ (see text). The corresponding boundaries for XENON100 are less constraining: in particular the thin short-dashed line represents the parameter space where $v_{\min }(E_{\min }^{\text{XENON}100})=v_{\min }(E_{\min }^{\text{CDMS}-\text{Si}})$. The blue shaded strip represents points excluded by the consistency test introduced in Sec. 4.1 of Ref. [23]. The cross represents the benchmark point discussed in detail in Sec. 7.

Figure 2

Isospin–violation parameter space for $m_{\chi }=3\mathrm{GeV}$ and $\delta =-70\mathrm{keV}$ [IDM benchmark point indicated with a (green) cross in Fig. 1]. For each value of ${\overline{\lambda }}_{\theta }$ and ${\overline{\lambda }}_s$ the remaining parameter ${\overline{\lambda }}_d$ is set to the value ${\overline{\lambda }}_{d,\min }$ minimizing the quantity $\mathcal{D}$ defined in Eq. (33). (a) The solid lines show constant values of $r=f_n/f_p=r_{\min }$, i.e., the value of $r$ that minimizes the compatibility ratio $\mathcal{D}$. (b) The solid lines show constant values of $t=t_{\min }$, i.e., the value of the $t$ parameter introduced in Eq. (31) that minimizes the compatibility ratio $\mathcal{D}$. In both figures the NLO corrections of Eq. (26) are included, while the short–dashed and long dashed straight lines represent the “alignment” conditions given in Eqs. (25) and (32), respectively.

Figure 3

Same as in Fig. 2. On the solid lines the upper bound on the suppression scale $M_{*}$ [introduced in Eq. (39)] is fixed to the indicated value when the NLO corrections of Eq. (26) to the expected WIMP rate are included. Dashed lines represent the same for the LO calculation.

Figure 4

Same as in Fig. 2. The shaded regions are bounded by the solid blue lines corresponding to the two conditions ${\mathrm{\Omega }}_{\chi }{h}^2=0.12$ and ${\mathcal{D}}_{\min }=1$ and represent the parameter space where CDMS-Si and SuperCDMS are mutually compatible while at the same time the metastable state $\chi$ can be a thermal relic [i.e., ${\mathrm{\Omega }}_{\chi }{h}^2\le 0.12$ using Eq. (42)]. The dotted curve represents the condition ${\mathrm{\Omega }}_{\chi }{h}^2=0.12$ calculated using the LO scaling law for the expected rate [see Eq. (6)]. Finally, the inner solid (red) line corresponds to $\tau =1/\mathrm{\Gamma }=4\times {10}^{26}$ seconds, as given by Eq. (40). The (red) circle indicates the representative choice of ${\overline{\lambda }}_{\theta }$, ${\overline{\lambda }}_s$ shown in Fig. 5 for which ${\mathrm{\Omega }}_{\chi }{h}^2=0.12$, which corresponds to ${\mathcal{D}}_{\min }=0.7$; the square indicates the choice of ${\overline{\lambda }}_{\theta }$, ${\overline{\lambda }}_s$ shown in Figs. 5, 6, and for which ${\mathcal{D}}_{\min }=1$ (i.e., at the verge of incompatibility between the CDMS–Si result and the SuperCDMS constraint); the (red) triangle indicates the representative choice of ${\overline{\lambda }}_{\theta }$, ${\overline{\lambda }}_s$ shown in Figs. 6, and exemplifies a configuration very close to Eq. (32) where the NLO energy-dependent corrections of Eq. (26) spoil the maximal achievable cancellation in WIMP-Ge scattering.

Figure 5

Measurements and bounds for the function $\tilde{\eta }$ defined in Eq. (15) for ${\sigma }_0={\sigma }_p$ and $m_{DM}=3\mathrm{GeV}$, $\delta =-70\mathrm{keV}$, i.e., for the IDM benchmark point indicated with a (green) cross in Fig. 1. (a) ${\overline{\lambda }}_{\theta }=-0.1$, ${\overline{\lambda }}_s=0.7$ (benchmark point indicated with a circle in Fig. 4, corresponding to ${\mathrm{\Omega }}_{\chi }{h}^2=0.12$). (b) ${\overline{\lambda }}_{\theta }=-0.1$, ${\overline{\lambda }}_s=1$ (benchmark point indicated with a square in Fig. 4 and corresponding to ${\mathcal{D}}_{\min }=1$). In both figures ${\overline{\lambda }}_d={\overline{\lambda }}_{d,\min }$ minimizes the $\mathcal{D}$ function of of Eq. (33), and the thick (blue) horizontal line represents the quantity ${\overline{\tilde{\eta }}}_{\mathrm{fit}}^{\text{CDMS}-\text{Si}}$ introduced in Eq. (35). Note that, consistently with the condition ${\mathcal{D}}_{\min }=1$, in plot (b) the constraint from SuperCDMS “touches” the upper range of the CDMS-Si excess.

Figure 6

The compatibility ratio $\mathcal{D}$ defined in Eq. (33) is plotted as a function of ${\overline{\lambda }}_d$ for $m_{DM}=3\mathrm{GeV}$, $\delta$=-70 keV, i.e., for the IDM benchmark point indicated with a (green) cross in Fig. 1. (a) ${\overline{\lambda }}_{\theta }=-0.1$, ${\overline{\lambda }}_s=1$ (benchmark point indicated with a (red) square in Fig. 4). (b) ${\overline{\lambda }}_{\theta }=-0.1$, ${\overline{\lambda }}_s=1.4$ [benchmark point indicated with a (red) triangle in Fig. 4]. In both figures the solid (blue) line represents the calculation including the NLO corrections of Eq. (26), while the thin dotted (black) line shows the same quantity when the approximate expression for the NLO corrections of Eq. (29) is used, which neglects the terms with explicit energy dependence.

Figure 7

Same as in Fig. 2. The thick solid lines show constant values of the cross section for hadronically-decaying mono $W/Z$ events in fb, while thin solid lines represent constant values of the expected number of $\text{monojet}+\text{missing}$ energy events for the integrated luminosity $\mathcal{L}=19.5{\mathrm{fb}}^{-1}$. In both cases $\sqrt{s}=8\mathrm{TeV}$. The applied kinematic cuts are listed in Sec. 6.