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

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 ${\cal O}_i^{p,n}$ and the corresponding long range interaction ${\cal O}_i^{p,n}/q^2$, 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 ${\cal O}_{1}$, ${\cal O}_{3}$, ${\cal O}_{11}$, ${\cal O}_{12}$ and ${\cal O}_{15}$, especially when interferences among contact and long range interactions are considered.