Surrogate models for Direct Detection¶

By Andrew Cheek¶

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

Surrogate models for Direct Detection¶

A surrogate model refers to emulators of the true calculations.
We have developed RAPIDD, a working surrogate model for general DD responses.
First I will motivate the need for RAPIDD.
Then I will review the how/if/when RAPIDD works.
Finally I will talk about future analysis that could be done.

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.

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

Halo Integrals¶

The energy deposited in the experiment is dependent on the relative velocity of DM and the target $E_{\textrm{R}}= 2 \frac{\mu_{T\chi}}{m_T} v^2_{\textrm{esc}}$.
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

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

Cross-Section¶

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,
- Anapole (plus other poles)
- Composites
- Vector DM
- Others ...

Parameter Reconstruction¶

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.

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 $S_{N}$.

Introducing Complexity¶

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

Caveat to Complexity¶

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.

We have developed RAPIDD for fast and general analysis¶

Reconstruction Analysis using Polynomials In Direct Detection (RAPIDD) is a surrogate model for DD experiments.
It enables fast evaluation of predicted Direct Detection responses with all operator responses and non-standard halo models.
RAPIDD at worst sees a speed up factor of $\sim$ 20. At best above $200$.

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

Then we took three benchmark points, which are accessible by future detectors.

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

Result for BP3¶

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.

Result for BP3¶

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

Scalar-Vector Profile likelihoods¶

Result for BP2¶

Here is a less illuminating result.

Conclusion¶

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

Operators¶

The list of possible operators is large (18 operators), some examples are $\mathcal{O}_1=\mathbb{1}_{\chi N}\, , \,\mathcal{O}_4 = \overrightarrow{S}_{\chi}\cdot \overrightarrow{S}_{\chi}\, , \, \mathcal{O}_8=\overrightarrow{S}_{\chi}\cdot\overrightarrow{v}^{\perp}\mathbb{1}_N$ and $\mathcal{O}_{10}=\overrightarrow{S}_N\cdot\frac{\overrightarrow{q}}{m_{N}}\mathbb{1}_{\chi}$
It manages to cover in a general way, all elastic interactions with nuclei for DM particles.
However there's a large heirarchy between these operators due to the form factor responses from different nuclei. Coherent effects still play a role.
The interactions to all orders are encoded in this form factor

$$ c_0 + c_2q^2\mathcal{O} + c_4q^4\mathcal{O} + ... \equiv F_{\mathcal{O}}\left(\frac{q^2}{\Lambda^2}\right)\mathcal{O}$$

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.

Velocity as a cut-off¶

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.

A Different EFT and Different Philosophy¶

When learning about the EFT approach for the first time, the general philosophy interested me. In order to tailor your solution to your problem, you should always use an EFT.

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.