= c_{V,u}^{n} = 1$, because this current counts valence quarks in the nucleon. The axial quark current is proportional to the nucleon spin: N| \^\_5 Q\_i |N &=& 2 s\^Q\_i\^N = Q\_i\^N N| \^\_5 \_N |N where the proportionality constants are measured [@DqN] as $ \Delta u^p = \Delta d^n = 0.84$, $\Delta d^p = \Delta u^n = -0.43$. In the zero-momentum-transfer limit of non-relativistic scattering, the dark matter can have spin-dependent interactions via the axial current, or spin-independent interactions via the first component of the vector current.

The spin-independent scattering amplitude for $\chi$ on a nucleon, is a coherent sum of vector and scalar interactions, for quarks of both chiralities and all flavours. The experimental limit on the cross-section per nucleon is $\sigma_{SI}\lsim 10^{-44}$ cm$^2$ for $m_\chi \sim 100$ GeV [@Xenon]. For the proton ($C_{uR}\leftrightarrow C_{dR}$ for the neutron), with $ C_{qR,V} =\frac{1}{3}( C_{dR,V } + 2C_{uR,V})$, this gives [@BBPS] \_[SI]{} \^2 3 10\^[-17]{} [GeV]{}\^[-2]{} where the $+...$ contains scalar contact interactions neglected in this note. For $\Lambda =$ TeV, this gives \[C\_[qL,V]{} + ( C\_[dR,V ]{} + 2C\_[uR,V]{}) + ...\] 10\^[-2]{}    (SI). \[SIbd\] The spin dependent cross-section per proton is [@BBPS] $$\sigma_{SD} \!\simeq   \!  m_p^2  \left[
 \frac{.42 (C_{qL,A} 