Rapidd Tutorial: A Quick Look at a New Tool for Direct Detection

By Andrew Cheek

Based on arXiv:1802.03174 with D. Cerdeño, E. Reid and H. Schulz

What is RAPIDD?

  • Reconstruction Analysis using Polynomials In Direct Detection, and its a surrogate model for DD experiments.
  • A surrogate model is an engineering term which basically refers to emulators of the true calculations.
  • RAPIDD at worst sees a speed up factor of $\sim$ 20. At best above $200$.
  • This talk will explain why calculational speed up is desired, and how its acheived.
  • You will also get the opportunity to play with some polynomials we built.

Dark Matter Direct Detection Calculation

  • 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.

$$N_k =\frac{\rho_0 \epsilon}{m_T\,m_{\chi}}\int_{E_k}^{E_{k+1}}dE_R\,\varepsilon(E_R) \int_{E^\prime_R}dE^\prime_R\,Gauss(E^\prime_R, E_R) \int_{v_{min}} d \vec v\, v f(\vec v)\, \frac{d\sigma_{{\chi}T}}{dE^\prime_R}$$

Cross-Section

  • 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,

    • Anapole (plus other poles)
    • Composites
    • Vector DM
    • Others ...

Experimental Effects

  • 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.

  • Resolution effects are also very important, and they are essentially the effect of gaussian smearing from bin to bin.

Halo Integrals

  • 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

$$ f(v) = N_k^{-1} \left[ e^{-v^2/k v_0^2} - e^{-v_{esc}^2/k v_0^2} \right]^k \Theta(v_{esc}-v) $$

NREFT

  • 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 $S_{N}$.

Simplicity vs. Complexity

  • Simplicity allows quick understanding and calculation, but there are always caveats.
  • The more complex NREFT basis will allow analysis to be more general and model independent.
  • By widening the parameter space, we can test what DD experiments could tell us thing about the particle nature of Dark Matter in general.
  • A computational drawback to the NREFT is that we're going from $\sigma_{0}^{\textrm{SI}}$ and $\sigma_{0}^{\textrm{SD}}$ to

Professor + DD = RAPIDD

  • We have succeeded in speeding up these calculations by using a 'surrogate' model to mimmick the direct detection calculation.

How does RAPIDD work?

  • 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

$$ N_k^a(\mathbf{\Theta}) \approx {\cal P}_k^a(\mathbf{\Theta}) = \sum_{l=1}^{N_\mathrm{coeffs}} d^{a}_{k,l} \, \tilde{\Theta}_l \equiv \mathbf{d^{a}_k} \cdot \mathbf{\tilde{\Theta}}$$

  • For example, for a quadratic polynomial in a two dimensional parameter space $\mathbf{\Theta}=(m_{\chi},\,c_1)=(x, y)$, these would take on the form $\mathbf{d^{a}_k} = (\alpha, \beta_x,\beta_y, \gamma_{xx}, \gamma_{xy}, \gamma_{yy})$

How does RAPIDD work?

  • This is done by collecting each $N_k^a(\mathbf{\Theta})$ for the set of sample points and solving this matrix equation

$$\vec{N_k^a} =M_{\mathbf{\tilde{\Theta}}}\cdot \mathbf{d^{a}_k}$$

  • Where $M_{\mathbf{\tilde{\Theta}}}$ is a quantity similar to a Vandermorde matrix where each row contains the values of $\mathbf{\tilde{\Theta}}$ for each sampled point, and $\vec{N_k^a}$ is a vector of the resulting number of events. This allows us to solve for $\mathbf{d^{a}_k}$ using the (pseudo-) inverse of $M_{\mathbf{\tilde{\Theta}}}$, which in the PROFESSOR program is evaluated by means of a singular value decomposition.

Instant issue for low masses

  • In the situation where we model a DM with low mass (below 30 GeV) we have a discontinuous spectrum.
  • 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.

Tests

  • 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,

$$\sigma_{\chi N}= \frac{\mu_{\chi N}^2}{\pi\, m_{v}^4}\left(c^0_1\right)^2$$

3-D Test 2 (Cancellation)

  • 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

$$ N_k^a(\mathbf{\Theta}) \approx \sum_{ij}\sum_{\tau,\tau^\prime=0,1} {\cal P}_k^{a,i,j,\tau,\tau^\prime}(\mathbf{\Theta}) \, $$

  • For example when isoscalar and isovector couplings are free (would cause problems with quark universality).

6-D Test (with Halo)

  • Finally we tested how RAPIDD works with the general halo function

$$ f(v) = N_k^{-1} \left[ e^{-v^2/k v_0^2} - e^{-v_{esc}^2/k v_0^2} \right]^k \Theta(v_{esc}-v) $$

Using RAPIDD to Constrain Models

  • We wanted to provide a case study of how our code could be used in future analysis.

  • We took the following detector variables

  • Say there's a positive DM signal in a Xenon experiment, possibly 1 in a Germanium experiment, and with a high certainty none in Argon.

Analysing with simplified models

  • 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

Playing with our code

  • Now you have the opportunity to be our parameter scanner!

  • http://6440e1f6.ngrok.io

  • http://dacfe6ee.ngrok.io

  • 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)

Tutorial Problem result

  • 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

$$ \mathcal{L}(\mathbf{\Theta}) = \prod_a \mathcal{L}^a(\mathbf{\Theta}) = \prod_a\prod_k{ \frac{N_k^a(\mathbf{\Theta})^{\lambda_k^a} e^{N_k^a(\mathbf{\Theta})}}{\lambda_k^a!}} \,$$

  • where a runs over experiments and k runs over bins. $N_k^a(\Theta)$ is the counts given by the theoretical expectation at a parameter point $\Theta$ and $\lambda_k^a$ is the data result.

Tutorial Problem result

  • We can see how the unbinned spectra shape up here, and can also see which contributions are dominant.

Scalar-Scalar Profile likelihoods

Scalar-Vector Profile likelihoods

Fermion-Scalar Profile likelihoods

Fermion-Vector Profile likelihoods

Outlook

  • 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?