Bracketing the direct detection exclusion plot for a WIMP of spin one half in non-relativistic effective theory

Abstract

Assuming a standard Maxwellian velocity distribution for the WIMPs in the halo of our Galaxy we use the null results of an exhaustive set of 9 direct detection experiments to calculate the maximal variation of the exclusion plot for each Wilson coefficient of the most general Galilean–invariant effective Hamiltonian for a WIMP of spin one half due to interferences. We consider 56 Wilson coefficients $c_i^{p,n}$ and ${\alpha }_i^{n,p}$ for WIMP–proton and WIMP–neutron contact interactions $O_i^{p,n}$ and the corresponding long range interaction $O_i^{p,n}/q^2$, parameterized by a massless propagator $1/q^2$. For each coupling we provide a different exclusion plot when the following set of operators is allowed to interfere: proton–neutron, i.e. $c_i^p$–$c_i^n$ or ${\alpha }_i^p$–${\alpha }_i^n$; contact–contact or long range–long range, i.e. $c_i^{p,n}$–$c_j^{p,n}$ or ${\alpha }_i^{p,n}$–${\alpha }_j^{p,n}$; contact– long range, i.e. $c_i^{p,n}$–${\alpha }_j^{p,n}$. For each of the 56 Wilson coefficients $c_i^{p,n}$ and ${\alpha }_j^{p,n}$ and for the largest number of interfering operators the exclusion plot variation can reach 3 orders of magnitude and reduces to a factor as small as a few for the Wilson coefficients of the effective interactions where the WIMP couples to the nuclear spin, thanks to the combination of experiments using proton-odd and neutron-odd targets. Some of the conservative bounds require an extremely high level of cancellation, putting into question the reliability of the result. We analyze this issue in a systematic way, showing that it affects some of the couplings driven by the operators $O_1$, $O_3$, $O_{11}$, $O_{12}$ and $O_{15}$, especially when interferences among contact and long range interactions are considered.