Atomic many body calculations for dark matter scattering on electrons: Ge & Xe Detectors 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)
Outline of the talk 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 5. Conclusion
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” .
Hint of New Physics • Neutrino • Dark matter • Dark energy They a re “ Portals ” to New Physics!
Why Atomic Physics?
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.
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 m e 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) Opportunity: Applying atomic physics at keV (low for nuclear physics but high for atomic physics)
Brief outline about Theoretical approach
Ab initio Theory for Atomic Ionization MCDF: multiconfiguration Dirac-Fock method ( , ) ( t ) u a r t Dirac-Fock method: is a Slater determinant of one-electron orbitals and invoke variational principle ( t ) i H V ( t ) ( t ) 0 I t u a ( r , t ) . to obtain eigenequations for ( t ) multiconfiguration: Approximate the many-body wave function ( t ) (for open shell atom) by a superposition of configuration functions And for open shell atom like Ge, the valence orbit are not filled up, then we need more configuration states for valence electrons. MCRRPA: multiconfiguration relativistic random phase approximation The relativistic random phase approximation (RRPA) is developed from time dependent 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.
Atomic ionization: ab initio MCRRPA Theory • MCRRPA: multiconfiguration relativistic random phase approx. Hartree-Fock : Reducing the N-body problem to a 1-body problem by solving the 1-body effective potential ↓ self consistently. RPA: Including 2 particle 2 hole excitations ↓ RRPA: Correcting the relativistic effect ↓ MCRRPA: More than one configurations in Hartree-Fock; Important for open shell system like Ge where the energy gap is smaller than the closed shell case 10
DM Effective Interaction with Electron or Nucleons Leading order (LO): short range spin-indep. spin-dep. long range where χ and f denote the DM and fermion fields, respectively, S~ χ, f are their spin operators (scalar DM particles have null S~ χ), the DM 3-momentum transfer |~q| depends on the DM energy transfer T and its scattering angle θ
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 c 1 by For the long rang interaction, it is an energy-dependent constant and related to d 1 by
DM-Ge & Xe atom ionization differential Cross Sectio ns Where are the mass, velocity, initial and final momentum of the DM particle.
Re Respons sponse e Function nction The full information of how the detector atom responds to the incident DM particle is encoded in the response function denote the many-body initial (bound) and final (ionized) state. M and µ the total and reduced mass of the ion plus free electron system, respectively, with µ ≈ m e . The summation is over all electrons, and the ith electron has its binding energy E Bi , relative coordinate ~r i , and relative momentum ~p i . 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 To ex expedit pedite th the co comput mputati ation on, we we perf perform ormed ed (M (MC)R )RRPA RPA calcula culatio tions ns on only ly for or se sele lected ted data poi dat point nts, s, and and the the full full comput omputati ation on is is done done wit with an an addit dditional ional app pprox oximat mation on: the the froz ozen-core ore app pproxima oximation tion (FCA). The FCA has a discre repancy ancy less s than 20% f for all our calcu lculati lations. s. FCA approximation, In this scheme, the final-state continuum wave function of the ionized electron is solved by the Dirac equation which has an electromagnetic mean field determined from the ionic state given by (MC)DF.
Cons onstra raints nts on on WI WIMP MP-electro ron n int ntera raction ions 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
Result and discussions
WIMP-electro ectron n Diffe ffere rential ntial Cros oss section on In these figure, we show some results of Averaged velocity- weighted differential cross sections for ionization of Ge and Xe atoms by LDM of various masses with the effective short-range (up) and long-range (down) interactions Important observations First, the sharp edges correspond to ionization thresholds of specific atomic shells. They clearly indicate the effect of atomic structure, and the peak values sensitively depend on atomic calculations. If direct DM detectors have good enough energy resolution, these peaks can serve as powerful statistical hot spots. Second, away from these edges, the comparison between Ge and Xe cases do point out that the latter has a larger cross section, but the enhancement is not as strong as Z 2 for coherent scattering nor Z for incoherent sum of free electrons. Third, the long-range interaction has a larger inverse energy dependence than the short-range one. As a result, lowering threshold can effectively boost a detector’s sensitivity to the long-range DM-electron interaction .
Comparison of Velocity-averaged differential cross section for a Xe atom Plots of the velocity averaged differential cross-sections for 10 GeV WIMP masses for short range(upper) and long range(down) types are presented in these figures. We find very good agreement with similar calculations by B. M. Roberts and V. V. Flambaum for Xe atoms. arXiv:1904.07127, B. M. Roberts, V. V. Flambaum
WIMP MP-electro electron n S Sho hort Ra Rang nge e Important observations First, the lowest reach of a direct search experiment in LDM mass is determined by its energy threshold. According to what we set for CDMSlite, XENON100, and XENON10: 80, 56, and 13.8 eV, the lightest DM masses can be probed are ∼ 50, 30, and 10 MeV, respectively. For the contact interaction, where the DM-Xe differential cross section is universally bigger than the DM-Ge one, XENON100 gives a better limit than CDMSlite is mainly due to its larger exposure mass-time. On the other hand, the very low threshold of XENON10 The exclusion limits on DM-electron interaction not only makes it able to constrain the lower-mass strengths depend on several factors: region where XENON100 has no observable electron Experimentally, detector species, energy recoil signals resolution, background, and exposure mass-time. Also results in a slightly better limit than XENON100, despite a smaller exposure mass-time by almost three orders of magnitude.
WIMP MP-electro electron n L Lon ong g Ra Rang nge Important observations Differential cross section has a sharper energy dependence and weights more at low T, this explains why XENON10 ’s constraint is much better than others in the entire plot Also for the low-energy weighting, the finer energy resolution and lower background of CDMSlite makes its power to constrain d1 better than XENON100. In above two Figure, the exclusion limits derived in Ref., using the same XENON10 and XENON100 data, are compared. The differences in the overall exclusion curves are obvious and most likely of theoretical origins.
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