Direct Detection exploits the relative velocity of us and Dark Matter to tell us something about the possible interactions DM has with ordinary matter.
In order to calculate the number of recoils in a given energy bin, one typically needs to evaluate these nested integrals.
$$N_k =\frac{\rho_0 \epsilon}{m_T\,m_{\chi}}\int_{E_k}^{E_{k+1}}dE_R\,\varepsilon(E_R) \int_{E^\prime_R}dE^\prime_R\,Res(E^\prime_R, E_R) \int_{v_{min}} d \vec v\, v f(\vec v)\, \frac{d\sigma_{{\chi}T}}{dE^\prime_R}$$
The energy deposited in the experiment is dependent on the relative velocity of DM and the target $E_{\textrm{R}}= 2 \frac{\mu_{T\chi}}{m_T} v^2_{\textrm{esc}}$.
The velocity distribution of incident DM particles is often assumed to be maxwellian $f(v)=\left(\frac{1}{N}\right)\exp(-v^2/v_0^2)\Theta(v_{esc}-v)$, which can be integrated analytically.
However, to account for uncertainties in halo parameters, and unknowns about the shape of this distribution, one could take results from simulations or data.
One could also use a general form which can recreate the general shape that we expect with
$$ f(v) = N_k^{-1} \left[ e^{-v^2/k v_0^2} - e^{-v_{esc}^2/k v_0^2} \right]^k \Theta(v_{esc}-v) $$
Typically the cross-section is given by either the spin independent or spin dependent cross-sections.
Both of which exhibit similar behaviour with changing energy,
$$ \frac{d\sigma^{WT}}{dE_{\textrm{R}}}=\frac{m_T}{2\mu_{\chi T}^2v^2}\left(\sigma^{SI,T}_0F^2_{SI}(E_R)+\sigma_0^{SD,T}F^2_{SD}(E_R)\right) $$
The behaviour being $\propto 1/v^2$ has important consequences for the halo integral.
Particle theories that are missing from this description,
In [arXiv:1401.3739] it was shown that fitting data to different models will give not only different values for couplings, but also different values for dark matter mass.
This is basically because the nuclear responses are different for different particle models.
General particle interactions are not fully encapsulated by the canonical spin-(in-)dependent parametrisation and misrepresent the physics of DM.
A Non-Relavistic Effective Field Theory has been developed for the 4 point DM-Nucleon interaction,
$$ \mathcal{L}_{\textrm{int}}=\chi\mathcal{O}_{\chi}\chi N \mathcal{O}_{N}N =\sum_{N=n,p}\sum_{i}c_i^{(N)}\mathcal{O}_i\, \chi^+\chi^-N^+N^-$$
Like all EFTs they describe the physics by only using the relevant degrees of freedom.
In Direct Detection these quantities that are relavent are velocity $v$, the tranfer momentum $q$ and the spins of DM and the nucleon $S_{\chi}$ and $S_{N}$.
Just like in the canonical case, $\mathcal{O}_1$, the spin independent response is usually dominant (enhanced by $A^2$).
This enhancement can lead to loop generated $\mathcal{O}_1$ responses being the dominant contribution in DD.
Its been shown for certain simplified models, running from LHC scales to DD scales, operators that aren't present at tree level will be at DD. [arXiv:1605.04917]
In the similar vain, a full UV complete pseudo-scalar dark matter model has been studied at 1-loop in [arXiv:1803.01574]. An $\mathcal{O}_1$ response is generated and becomes dominant.
Instead of using the physics code to produce a result for a given energy bin $N_k^a$ we call a polynomial $\mathcal{P}_k^a$.
To do so we first choose a polynomial order $\mathcal{O}$ appropriate for the physics problem at hand. With $\mathcal{O}$ and the parameter point $\mathbf{\Theta}$ given, the structure of the polynomial is fixed. What remains to be done is to determine the $N_\mathrm{coeffs}$ coefficients, $d^{a}_{k,l}$, that allow to approximate the true behaviour of $N_k^a(\mathbf{\Theta})$ such that
$$ N_k^a(\mathbf{\Theta}) \approx {\cal P}_k^a(\mathbf{\Theta}) = \sum_{l=1}^{N_\mathrm{coeffs}} d^{a}_{k,l} \, \tilde{\Theta}_l \equiv \mathbf{d^{a}_k} \cdot \mathbf{\tilde{\Theta}}$$
$$\vec{N_k^a} =M_{\mathbf{\tilde{\Theta}}}\cdot \mathbf{d^{a}_k}$$
This is simply from the kinematic relations $ v_{\textrm{min}} = \sqrt{m_T E_{R}/(2\mu_T)^2}$
And the fact that the DM distribution has a upper limit in velocity $v_{\textrm{esc}}$.
We take this into account in the determination of polynomials by not using bins which return zero in the training.
When we call the polynomial later, we also evaluate where the discontinuity should be and impose it by hand.
In order to test our code we used RAPIDD and the physics code for some canonical examples.
The first of wich was to test in 2-D, scanning in the $(m_{\chi}, c_1^0)$ plane, which is just the NREFT equivalent to the spin independent case, where there's this weird conversion,
$$\sigma_{\chi N}= \frac{\mu_{\chi N}^2}{\pi\, m_{v}^4}\left(c^0_1\right)^2$$
We also wanted to test RAPIDD in specific cases where finely tuned cancellations were possible.
This inspired us to build the different polynomials contributions seperately
$$ N_k^a(\mathbf{\Theta}) \approx \sum_{ij}\sum_{\tau,\tau^\prime=0,1} {\cal P}_k^{a,i,j,\tau,\tau^\prime}(\mathbf{\Theta}) \, $$
$$ f(v) = N_k^{-1} \left[ e^{-v^2/k v_0^2} - e^{-v_{esc}^2/k v_0^2} \right]^k \Theta(v_{esc}-v) $$
We wanted to provide a case study of how our code could be used in future analysis.
We took the following detector variables
We can use a set of simplified models and try and match them to the data.
If you treat each operator coefficient as a free parameter
$$ \mathcal{L}(\mathbf{\Theta}) = \prod_a \mathcal{L}^a(\mathbf{\Theta}) = \prod_a\prod_k{ \frac{N_k^a(\mathbf{\Theta})^{\lambda_k^a} e^{N_k^a(\mathbf{\Theta})}}{\lambda_k^a!}} \,$$
Rapidd, is a new tool that enables more general analysis in DD experiments.
Tools like this could play a part in both the determination of DM properties and the design of future experiments.
Rapidd is not public yet, we're talking to experimentalists and theorists to try and maximise the utility of our code before release.
If you have any ideas of how to make RAPIDD more useful for you. Let me know!
Thank you
$$ c_0 + c_2q^2\mathcal{O} + c_4q^4\mathcal{O} + ... \equiv F_{\mathcal{O}}\left(\frac{q^2}{\Lambda^2}\right)\mathcal{O}$$
This is very different to the canonical EFT approach where you have operators of a certain mass dimension, and operators with higher mass dimensions are supressed by the cut-off scale.
All the NREFT operators are actually dimensionless, and higher dimensional terms are encoded to arbitrary order by Nuclear Response functions which are calculated using multipole expansions of nuclear electroweak responses.
Using natural units $\overrightarrow{v}^{\perp}$ is a dimensionless parameter which is small for DM in the galaxy $\sim 1 \times 10 ^{-5} $. This gives you a natural reason to truncate the series at higher powers of $v$.
This also helps produce the aforementioned hierarchy of responses.
The operators tend to be cut-off at the $v^2$ scale.
In this way, unlike EFTs at the high energy frontier, the validility of this approach is always true.
You can even encapsulate light (and massless) mediators by making the substitution $c_i\rightarrow \frac{c_i m_N}{q^2 + m_{\phi}^2}$, which technically no longer an Effective Field Theory.
The important difference between HE-EFTs and this NREFT is that we're not only integrating out the interaction mediator (or not in the case of massless fermion), but we've also taken the Non-relativistic description of Nucleons. So we may not be always integrating out the new physics, but we are always integrating out irrelavent SM degrees of freedom.
The essential reasoning behind higher order operators being suppressed is that naturally the couplings should be of order 1. Then from calculating the Wilson coefficients we can infer the scale of the new physics.
This is not always what happens in the NREFT case. As an example, for a pure pseudoscalar DM $c_6$ is determined by the mediator mass and the mass of the DM. This means, if you're uncertain about your DM mass then it propogates through to the cut-off scale.