1. Mukesh Kumar Pandey Dept. of Physics, National Taiwan University
Contribution of atomic calculation to constraints on Dark-Matter-electron scattering
Mukesh Kumar Pandey
Dept. of Physics, National Taiwan University
Collaborators:
Jiunn-Wei Chen, Chih-Pang Wu, Chung-Chun Hsieh, (NTU) Chen-Pang Liu, Hsin-Chang Chih (NDHU), Henry T. Wong, Lakhwinder Singh (IOP, Academia Sinica, TEXONO Collaboration)

2. Outline of the talk 2. About Dark matter 3. Why Atomic Physics ?
1. What is the need of this study(Motivation)
2. About Dark matter
3. Why Atomic Physics ?
3. Brief outline about Theoretical approach
4. Result and discussions(preliminary result)
5. Conclusion

3. What is the need of this study?

4. A smaller question
What’s the minimum set of particles and interactions that builds the material world?
This is a problem particle physicists worry about. They are driven to look for “New Physics”.

5. They are “Portals” to New Physics!
Hint of New Physics
Neutrino
Dark matter
Dark energy
They are “Portals” to New Physics!

6. Portals to the Dark Sector
Other dark matter candidates include
MACHOs: MAssive Compact Halo Objects
WIMPs: Weakly Interacting Massive Particles

7. The Solution?
Many exotic particles have been proposed as candidates for dark matter including massive neutrinos, weakly interacting massive particles (WIMP’s), massive compact halo objects (MACHO’s), black holes, and brown dwarfs.
It is unlikely that one explanation will satisfy everyone.

9. Direct Dark Matter Detection
RPP 02/2016

10. Why Atomic Physics?

11. Why Atomic Physics?
Energy scales: Atomic (~ eV) Reactor neutrino (~ MeV) WIMP (~ GeV)
Neutrino: NNM atomic ionization signal larger at lower energy scattering (current Ge detector threshold 0.1 keV)
DM: direct detection, velocity slow (~ 1/1000), max energy 1 keV for mass 1 GeV DM.

12. When atomic structures should be considered (free target approx. fail)?
Incident momentum ~ 100 keV and below
The wavelengths of incident particles are about the same order with the size of the atom.
For Innermost orbital, the related momentum ~ Z me α ~ Z *3 keV (Z = effective nuclear charge)
Energy transfer ~ 10 keV and below
barely overcome the atomic thresholds
For Innermost orbital, binding energy ~ 11 keV (Ge) and 34 keV (Xe)
Phase-space suppression (Ex: WIMP-e scattering)
So here I quickly summarize when the atomic effects should be taken into account.
First, the incident momentum below hundred eV, that means the wavelengths of incident particles are about the same size with target atoms.
Second, the energy transfer below about 10 keV should be careful, which is around the maximum energy level of the target atoms. If the energy transfer is just enough to overcome an atomic threshold, then the behavior will be quite different with free target approximation.
Finally, some cases have strong phase space suppression like WIMP interacted with electrons, because the WIMPs are slow and massive. For given energy transfer T, the minimum momentum transfer is still much larger than the amount which electrons can carry. Thus the most of momentum transfer is carried by the whole atom and results in very strong suppression in transition amplitude.
Opportunity: Applying atomic physics at keV (low for nuclear physics but high for atomic physics)

13. DM Effective Interaction with Electron or Nucleons
short range
long range
spin-indep.
spin-dep.
Leading order (LO):
Effective Field Theory of Dark Matter.
where χ and f denote the non relativistic (NR) DM and fermion fields, respectively, S~ χ, f are their spin operators (scalar DM particles have null S~ χ), the 4-momentum transfer can be reduced to NR limit and replaced by DM 3-momentum transfer |~q| depends on the DM energy transfer T and its scattering angle θ
where µ is the reduced mass of the target with me being the mass of the electron (interacted fermion). The initial state |I denotes the target atom at the ground state, and the Dirac delta function imposes the energy conservation that the energy deposited by DM equals to the recoil energy of the atomic center of mass, Ec.m. , plus the internal excitation energy EF − EI of the atom. Depending on the nature of the final state F|, the scattering processes are classified as
All dynamical information in response functions
where χ and f denote the non relativistic (NR) DM and fermion fields, respectively, S~ χ, f are their spin operators (scalar DM particles have null S~ χ), the 4-momentum transfer can be reduced to NR limit and replaced by DM 3-momentum transfer |~q| depends on the DM energy transfer T and its scattering angle θ

14. The DM-electron interaction can be formulated at the leading order (LO), the spin-independent (SI) part is parametrized by two terms:
For the contact interaction, it is an energy-independent constant and related to c1 by
For the long rang interaction, it is an energy-dependent constant and related to d1 by

15. DM-Ge & Xe atom ionization differential Cross Sections
Where
are the mass, velocity, initial and final momentum of the DM particle.
The atomic transition amplitudes can be expressed by the generic response functions in the
form of
Therefore, the single differential cross section with respect to energy transfer, dσ/dT, can be compactly
expressed. For elastic scattering or discrete excitation to the final discrete level (n′l′) from

16. WIMP-electron Scattering
The full information of how the detector atom responds to the incident DM particle is encoded in the response function
R(T, θ) is evaluated by well-benchmarked procedure based on an ab-inito method, the (multi-configuration) relativistic random phase approximation, (MC)RRPA.
To expedite the computation, we performed (MC)RRPA calculations only for selected data points, and the full computation is done with an additional approximation: the frozen-core approximation (FCA).
The FCA has a discrepancy less than 20% for all our calculations.

17. Constraints on LDM-electron interactions
The differential event rate, in units of counts per energy per kg detector mass per day, of a direct DM detection experiment, is calculated by
The DM velocity spectrum is assumed to be the standard Maxwell_Boltzmann

18. WIMP-electron Differential Cross section

19. WIMP-electron Short Range (preliminary result)

20. WIMP-electron Long Range

21. WIMP-electron Comparsion
21. Roothaan-Hartree-Fock approach

22. Summary
Dark matter particles are portals to new physics and their properties can be constrained by direct detection.
Ab initio atomic tool indispensable to study DM detector response to light DM signal.
We have investigated the atomic many-body effects in the dark matter scattering amplitudes, which play important roles when the recoil energies of detector go lower to keV scale.
We derive new upper limits on parameter space spanned by dark matter effective coupling strengths and mass using the superCDMSlite, low energy XENON10 and XENON100 ionization-only data.

23. Thanks
The works are supported by the MOST, NCTS, TEXONO, of Taiwan, R.O.C.
Thank you all for your attention.

24. Backup slides

27. Why we study atomic structure ?
The space uncertainty is inversely proportional to its incident momentum:
λ ~ 1/p
LDM with velocity ~10-3
Mass
Energy
Momentum
1 GeV
mχ eV
1 MeV
100 MeV
mχ + 50 eV
100 keV
Neutrino Sources
Reactor ν
~ few MeV
Same as energy
Solar ν (pp)
~ few hundred keV
Atomic Size is inversely proportional to its orbital momentum:
Z me α ~ Z *3.7 keV
Z: effective charge
The physic behavior depends on the energy scales. The atomic size, or the electron orbital radius, are related with their orbital momentum, which is 3.7 keV for hydrogen and for hydrogenlike atoms just multiplied by the effective charge. In such rough estimation, we got a general conclusion that above MeV is the nuclear physic scale and below MeV is the atomic physics scale.
Here I list out some energies and momenta for cold LDM and neutrinos sources. Those numbers are not much larger than the atomic scale, so people must be careful to the atomic structures.

28. Ab initio Theory for Atomic Ionization
MCDF: multiconfiguration Dirac-Fock method
Dirac-Fock method:
is a Slater determinant of one-electron orbitals
and invoke variational principle
to obtain eigenequations for .
multiconfiguration: Approximate the many-body wave function
by a superposition of configuration functions
(for open shell atom)
Ge: 2 e- in 4p ( j = 1/2 or 3/2)
MCRRPA: multiconfiguration relativistic random phase approximation
Here I introduce the methods to calculate electron orbital wave functions. We use Dirac-Fock, or relativistic Hartree-Fock method to get the initial stable orbital wave functions by variational principle.
And for open shell atom like Ge, the valence orbit are not filled up, then we need more configuration states for valence electrons.
And the final perturbed states can be expanded into time-independent orbital wave function states with this exponential i omega t, this is the so-called RRPA methods.
The relativistic random phase approximation (RRPA) is developed from timedependent
Hartree-Fock (TDHF) theory. It has successfully described the closed-shell
system such as heavy noble gas, where the ground state is isolated from the excited
state.
RPA: Expand into time-indep. orbitals in power of external potential

29. JWC et. al. arXiv:
To test the performance of our calculating method, we have done the Photoionization cross sections as a benchmark, for both Ge and Xe targets.
As you can see, above 100 eV, our calculation is good agreement with experimental results. The errors are always under 5%.
The main error are located at 10 to 100 eV for Ge case. It may come from the solid effects but in our calculations where we only consider one Ge atom.
Since 100 eV is an already ambitious threshold for current detectors, we can still set a minimum of T at 100 eV for our calculations, and leave the (T < 100 eV) region for future study.
The main error are located at 10 to 100 eV for Ge case. It may come from the solid effects but in our calculations where we only consider one Ge atom.

31. Comparison of Ge Photoionization cross section of IPM and MCRRPA

32. The full information of how the detector atom responds to the incident DM particle is encoded in the response function