1. Theoretical uncertainties in predictions for direct detection experiments
Felix Kahlhoefer
Dark LHC
27 September 2014
Oxford
Rudolf Peierls Centre for Theoretical Physics
© Banksy

2. Introduction
We have discussed the uncertainties of LHC searches for DM:
PDF uncertainties
QCD corrections & scale ambiguities
Systematics in background estimates
Let us assume that all of these are under control and we can really constrain the fundamental interactions of dark matter with quarks.
How do we compare these results
with dark matter direct detection?
Felix Kahlhoefer
Dark LHC 2014, Oxford

3. Outlook How reliable is this procedure?
Identify relevant operators in the non-relativistic limit
Calculate nuclear matrix elements
Include nuclear form factors
Calculate recoil spectra
Predict experimental event rates
Felix Kahlhoefer
Dark LHC 2014, Oxford

4. From the LHC to the non-relativistic limit
Step 1: We need to work out which effective operators give the dominant contribution in the non-relativistic limit.
March-Russell et al. (2012)
Vector operator (e.g. Z’ exchange)
Axialvector operator (Majorana DM)
Careful: These operators induce spin-independent interactions at the 1-loop level.
Haisch & FK, arXiv:
Scalar operator (Higgs)
Felix Kahlhoefer
Dark LHC 2014, Oxford

5. From quarks to nucleons
Step 2: We need to work out the effective couplings of dark matter to protons and neutrons (fp, fn or ap, an).
Easy for vector current: fp = 2 gu + gd, fn = 2 gd + gu
Slightly more complicated for the axial current:
Typically gu = -gd = -gs
 ap = (1.36 ± 0.05) gu
Felix Kahlhoefer
Dark LHC 2014, Oxford

6. From quarks to nucleons
Difficult for the scalar current:
Young,
Large uncertainties in the strange contents of the nucleus:
fTs = σs / mN could be anywhere between zero and 0.15.
The resulting fp then varies between 0.27 and 0.38.
Felix Kahlhoefer
Dark LHC 2014, Oxford

7. From nucleons to nuclei
Step 3: We need to include the loss of coherence for finite momentum transfer
Spin dependent form factor
XENON100 Collaboration,
Neutron
(10-30%)
The necessary form factors depend on the nuclear shell models.
Proton
( %)
Spin-independent form factor
Typical momentum transfer in direct detection experiments
Zheng et al.
Felix Kahlhoefer
Dark LHC 2014, Oxford

8. From nucleons to nuclei
Even for spin-independent interactions, there are small differences between the distribution of protons and neutrons (neutron skin thickness), leading to different form factors for the two contributions.
Zheng et al
Zheng et al
Only really matters for models with destructive interference.
Felix Kahlhoefer
Dark LHC 2014, Oxford

9. From matrix elements to recoil spectra
Step 4: Calculate differential event rates
with
The local DM density can be measured in essentially two different ways:
Locally from the vertical motion of stars
Globally from the rotation curve of the Milky Way
Since dσ/dER ~ 1/v2, direct detection experiments probe the velocity integral
Felix Kahlhoefer
Dark LHC 2014, Oxford

10. Local dark matter density
Global measurements need to assume certain properties of the DM halo, such as a certain density profile or spherical symmetry.
For sufficiently strong assumptions, the (averaged) local DM density can be determined with very high precision:
Catena & Ullio, : ± GeV / cm3
McMillan, : 0.4 ± 0.04 GeV / cm3
Relaxing these assumptions, however, a much wider range becomes allowed:
Iocco et al : 0.2 – 0.56 GeV / cm3
Weber & de Boer, : 0.2 – 0.4 GeV / cm3
Felix Kahlhoefer
Dark LHC 2014, Oxford

11. Local dark matter density
What if the halo is not spherical?
What if there are additional structures (e.g. a dark disk)?
Read,
Typical results: 0.35 ± 0.15 GeV / cm3
Felix Kahlhoefer
Dark LHC 2014, Oxford

12. The Standard Halo Model
v0: Velocity dispersion (= circular velocity)
vesc: Escape velocity
Frandsen et al
McCabe,
Felix Kahlhoefer
Dark LHC 2014, Oxford

13. The dark matter velocity distribution
Moreover, the dark matter velocity distribution may deviate significantly from the simple Maxwell-Boltzmann form:
Anisotropic halos
Debris flows
A dark disk contribution
Vogelsberger et al.
Frandsen et al
Fairbairn et al
Felix Kahlhoefer
Dark LHC 2014, Oxford

14. Extracting particle physics
Given all these astrophysical uncertainties, can we hope to extract particle physics properties from direct detection experiments?
No, if we only have data from a single experiment.
Yes, if we combine the data from several different targets.
Basic idea: Map differential event rates into vmin-space in order to compare them independently of astrophysical uncertainties.
Fox et al
Felix Kahlhoefer
Dark LHC 2014, Oxford

15. Dark Matter @ LHC 2014, Oxford
A few simple steps
Introducing the rescaled velocity integral We can write and solve for g(vmin):
Feldstein & FK,
Repeating this procedure for different values of mDM, we can find the one(s) that give the best fit to the data (lowest χ2).
Best-fit velocity integral
Felix Kahlhoefer
Dark LHC 2014, Oxford

16. Integrating the velocity integral
If we ‘know’ g(vmin) and mχ, we can reconstruct ρ σp.
Only problem: Low-energy threshold of direct detection. ?
We can never probe the lowest velocity dark matter particles.
Felix Kahlhoefer
Dark LHC 2014, Oxford

17. A lower bound on the cross section
But: Since the velocity integral g(vmin) must be monotonically decreasing, we can always construct a lower bound.
Conservative lower bound
Feldstein & FK,
If we have an upper bound on ρ, we obtain a lower bound on σ!
Felix Kahlhoefer
Dark LHC 2014, Oxford

18. More uncertainties beyond this talk!
Frandsen et al
Collar,
Sorensen,
Experimental uncertainties:
Quenching factors
detector resolution
Felix Kahlhoefer
Dark LHC 2014, Oxford

19. Dark Matter @ LHC 2014, Oxford
Conclusions
Various uncertainties have to be taken into account when comparing LHC searches to direct detection experiments:
Contributions from loop-induced spin-independent interactions.
Nuclear matrix elements (in particular for scalar mediators)
Form factors (in particular for spin-dependent interactions and isospin- dependent scattering)
Local dark matter density (uncertainties are large if we relax our assumptions on the global structure of the halo)
Velocity distribution (it is not enough to just vary the parameters of the Standard Halo Model)
Experimental uncertainties (quenching factors, detector resolution)
New tools are in development to deal with astrophysical uncertainties by combining data from several direct detection experiments.
Felix Kahlhoefer
Dark LHC 2014, Oxford