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.
$$N_k =\frac{\rho_0 \epsilon}{m_T\,m_{\chi}}\int_{E_k}^{E_{k+1}}dE_R\,\varepsilon(E_R) \int_{E^\prime_R}dE^\prime_R\,Gauss(E^\prime_R, E_R) \int_{v_{min}} d \vec v\, v f(\vec v)\, \frac{d\sigma_{{\chi}T}}{dE^\prime_R}$$
Typically the cross-section is given by the either the spin independent and 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,
Due to pure detector effects and also the ability to discriminate background and signal effects the 'efficiency' of a detector, i.e. how much of a dark matter signal will we register as a signal.
To accurately simulate DD experiments one has to take the efficiency curve experimentally determined.
The energy deposited in the experiment is dependent on the relative velocity of DM and the target $E_{\textrm{R}}= 2 \frac{\mu_{T\chi}}{m_T} v^2_{\textrm{esc}}$.
No one expects all DM particles to have a fixed velocity in the galaxy, instead its some kind of distribution, 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 ($\int f(v)/v$).
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
$$ f(v) = N_k^{-1} \left[ e^{-v^2/k v_0^2} - e^{-v_{esc}^2/k v_0^2} \right]^k \Theta(v_{esc}-v) $$
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 $S_{N}$.
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
Now you have the opportunity to be our parameter scanner!
The password is Rapidd_test
Sign into one of these url's, open the notebook up_running.ipynb for the top link and up_running2.ipynb for the bottom link.
On this website you will be able to call the polynomials and try to find the best configuration. Scroll down to your name and use the cells in there.
Depending on the simplified model you want to test you can simply enter
script_SS.widget_show(script_SS.PP)
Referring to our paper arXiv:1802.03174, we did exactly this, but we used a sampling code rather than our eyes.
Setting up a Poissonnian log-likelihood
$$ \mathcal{L}(\mathbf{\Theta}) = \prod_a \mathcal{L}^a(\mathbf{\Theta}) = \prod_a\prod_k{ \frac{N_k^a(\mathbf{\Theta})^{\lambda_k^a} e^{N_k^a(\mathbf{\Theta})}}{\lambda_k^a!}} \,$$
We've developed a new method in calculating DD responses, using a surrogate model we can perform the calculation much faster.
Current work involves using this new tool to learn more about the information we can glean from DD.
We should also do some performance tests against codes on the market.
This technique will make general analysis more readily available.
We're looking to make the first public release of the code but we're keen to improve. What would you like to see?