1. Electron Recoil & Dark Matter Direct Detection
Test
Electron Recoil & Dark Matter Direct Detection
Just now Henry introduced to you the status of Jingping Lab in China Sichuan Province and CDEX experiment. My talk is also based on the needs of CDEX experiment.
Qing Wang Tsinghua Univ. Beijing

2. Background and Status of ER
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Background and Status of ER
Our present understanding
My talk include two parts: one is the background and status of electron recoil, the other is our present understanding of it.

3. Cosmology, Astrophysics
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Cosmology, Astrophysics
Search DM
particle physics
We know that there are mainly too fields in searching dark matter. One is in cosmology and astrophysics, another is in particle physics.

4. Cosmology & Astrophysics： Found ！
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Cosmology & Astrophysics： Found ！
In cosmology and astrophysics, we know that there are many strong evidences to show the existence of dark matter.

5. Particle Physics： Not Found ！
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Particle Physics： Not Found ！
No unambiguous evidence has been obtained to date
Collider Experiment
Direct Detection
Indirect Detection
But in particle physics, up to now, there is still no ambiguous evidence to show the existence of dark matter. Among the typical three ways of detection of dark matter, I am interested in this talk in direct detection.

6. Nuclear Recoil & Electron Recoil
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Within direct detection, people search dark matter by detecting nucleon recoil or electron recoil colliding from dark matter .
Nuclear Recoil & Electron Recoil

7. Electron Recoil not considered at very beginning !
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Electron Recoil not considered at very beginning !
This is the first theoretical paper of direct detection written by Goodman and Witten in 1985, only nucleon recoil is investigated, electron recoil is ignored. So electron recoil is not considered at very beginning.

8. Estimation of Recoil Energy
Test
m，v0, vˊ
Estimation of Recoil Energy
M, v，v0ˊ
Target recoil v target initial v0’ DM initial v DM final v’
For M>>m rest nucleon ER~2v02m2/M~ 2keV ×(m/10GeV)2×100GeV/M
v0~10-3； for electron m>>M ER~2v02M~1eV Chemistry & Biology
Theory & Exp not prefer electron recoil
Angle between incident particle & target
CM frame
traditional threshold
Lower detection bound
The reason that people ignore electron recoil is due to the fact that, if you use energy and momentum conservation to estimate the recoil energy, for the case that incident dark matter particle is much lighter than target nucleon but heavier than electron, the typical nucleon recoil energy is at order of keV, while the electron recoil energy is at order of eV. People thought that electron recoil energy is too small for our particle physics detectors, and then just ignore it. Therefore at very beginning theory and experiment all not prefer the electron recoil. But notice that nucleon recoil energy is quadratically depend dark matter mass, when dark matter mass decrease, nucleon recoil energy decrease very fast, while electron recoil energy keeps almost same. So when dark matter mass becomes low enough, usually in sub GeV region, nucleon recoil energy will be even smaller than electron recoil energy.
m reduce from 10GeVto 1GeV，NR reduce from 2keV to 20eV，ER keep 1eV ！
Too small for electron recoil energy

9. 1st rise up ER energy~keV
Test
1st rise up
ER energy~keV
The 1st rise up of electron recoil starts in 2008, DAMA people began to investigate electron recoil in this PRD paper.

10. Test
Bounded electron of NaI(Tl) has 10-4 probability v2>1/2（p>0.5MeV）
They found that traditional estimation of electron recoil energy is based on assumption that electron is at rest before the collision. In fact, for Sodiumiodide (NaI) crystal, bounded electron has 10-4 probability to have large velocity close to speed of light. Or electron in the shell of atom has some momentum distribution. It is this large momentum tail which may make electron recoil has relatively large energy. Consider that the small probability of this large momentum tail, DAMA people assume dark matter donot couple to quark then donot have nucleon recoil at very begining, only interact with lepton, just like the case professor Liao discussed yestoday. Then they show electron recoil do can produce some measurable effect.
Sodiumiodide

11. Theory & Exp not prefer ER again !
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Theory & Exp not prefer ER again !
Phys.Rev. D80 (2009)
DAMA signal was explained as ERavoid contradiction with other exps
Assume DM only interact with leptons leptophilic
But Leptons loop induce enough NR
And electroncannot seen as free particle
While CDMS analysis on ER spectrum gives no signal
With this result, DAMA people give an explanation of DAMA experiment. That is positive signal of DAMA is due to electron recoil not from nucleon recoil. While negative results from other experiments are for nucleon recoil. Then there is not contradiction among DAMA positive and other experiment negative results. The assumption is the dark matter interaction is leptophillic or donot interact with quark. After this first result published, other people soon show in this paper that even in this letophillic case, these kind loops may still induce big enough nuclear recoil. Further, the calculation was criticized that it take electron as free particle. Later, CDMS analysis their collected electron recoil data, they not find any dark matter signal. This is DAMA result, and this is CDMS result. So, theory and experiment do not prefer electron recoil again.
Phys.Rev. D81 (2010)
with some velocity distribution

12. Typical values： mX = 0.1–1 GeV, mχ = 0.1–1 TeV ， αDM = αem
Test
A new gauge boson X, couples to SM particles and the WIMP through kinetic vector boson mixing with properties: me ≤ mX ≤ mχ β ≤ mχ αDM
Typical values： mX = 0.1–1 GeV, mχ = 0.1–1 TeV ， αDM = αem
mχ = 10 GeV
mχ = 100GeV
2nd rise up
mχ = 1000 GeV
The 2nd rise up of electron recoil starts in 2009 from this nuclear physics paper. Instead of discussing relatively higher recoil energy, they change to search secluded dark matter by detecting low energy recoil electrons. They show that for hydrogenic atom and massless mediator, in 10eV energy region, there are sizable event rate for electron recoil.
pick back small recoil energy~10eV
hydrogenic atom；massless mediator

13. Test
Concerning detection ability, they also propose theoretical mechanism to realize detection of this very low recoil energy either for gas-based and for semiconductor detectors.

14. ★ larger contribution from initial larger momentum state phase space
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Recoil energy~1-10eV
Later a more thorough and realistic discussion was made in this PRD paper this year. They even give exclusion lines. They revised original free electron calculation to bounded electron calculation and find original effect of large momentum tail change to two compete effects, one increase the rate and another decrease the rate.
Free electron with momentum distribution change to bound state wave function，
effect of large momentum tail change to：
★ larger contribution from initial larger momentum state phase space
★ smaller contribution due to overcome ionization energy

15. Test
More later, Based these teoretical discussions, Xenon10 acheaved low energy detection and analyzed their electron recoil data. This is their expected rate and this is exclusion lines for contact interaction.

16. This is the result consider different form factor corrections.
Test
This is the result consider different form factor corrections.

17. keV region: NR dominant ！ eV region: ER dominant !
Test
Recoil energy:
keV region: NR dominant ！
eV region: ER dominant !
10~several hundred eV region: ?
That’s CDEX most interested region !
Now we already know that in keV recoil energy region, nuclear recoil is dominant, while in ev recoil energy region, electron recoil is dominant, then how about this intermediate recoil energy region, that’s what we are interested, and CDEX is also plan to make its contribution in this region.

18. If R0 Universal Static target： 0.245, 0.433,0.571 Event rate
Test
Static free electron
If R0 Universal
0.245, 0.433,0.571
ER spectrum
Phase B：>
NR spectrum
NR spectrum
Phase A：>
ER spectrum
Event rate
Static target：
To qualitatively estimate the event rate of direct detection experiment, we use this typical nucleon calculation result. The differential event rate is exponentially depend on recoil energy if the target particle is at rest before the collision. For fixed incident energy and universal total event rate, the rate is controlled by reduced mass of dark matter and target. If we plot this reduced mass in terms of dark matter mass, we find that if dark matter particle is heavy there is a region we call phase A where nuclear spectrum is above electron recoil spectrum. In contrast, if dark matter particle is light, we enter into another phase where electron spectrum is heavy than nucleon spectrum. This result does not include effect of initial electron momentum distribution.
Recoil energy
Incident kinetic energy
Incident particle mass
Needs initial momentum distribution and bounded states effects!

19. Impact of electron Initial velocity
Test
Free electron with some momentum distribution
Impact of electron
Initial velocity
To see the effect of electron initial velocity, I show you differential rates for different initial momentum distribution of electrons. This red line distribtuion has relatively larger momentum tail, then it produce higher event rate.

20. Effect of initial velocity
Test
Free electron with some momentum distribution
Energy<0
Phase B
Too big energy ？
Phase A
Effect of initial velocity
Phase B
In general, if we plot rate in logarithmic coordinate, the exponential behavior of nucleon recoil will be an straight line, just like this blue line. While if we consider the initial velocity for electron, the line for electron bended as this red line. Then the most of region present experiment locate is in this phase A region where nucleon rate is higher than electron rate. While in general there exist another regions of phase B where electron recoil rate is above nucleon recoil. Unfortunately, in many cases, this region of phase B often leads too small event rate, and this region of phase B often is unphysical.

21. Test
Free electron with some momentum distribution
4.3, 7×10-7
We have checked case of Sodiumiodide(NaI) for different dark matter mass. Most of realistic situation is in phase A. Here the scattering amplitude of electron with dark matter is assumed to be cm^2 or fp which is the order DAMA people used in 2008 paper. And we further assume it is equal to that of Sodiumiodide.
4.7, 5×10-8

22. While if dark matter mass is small, there appear case B.
Test
Free electron with some momentum distribution
0.37, 0.33
2.8, 3×10-5
While if dark matter mass is small, there appear case B.
4.3, 7×10-7

23. Test
Up to now, we only consider the momentum distribution effect. That’s not enough, because electron inside an atom is bounded not a free particle. In this diagram, we have calculated event rate for helium with dark matter particle mass of 10GeV. The black line is for the helium nucleon, red line is for free electron with some momentum distribution, and blue line is for bounded electron in a free Helium atom. We see below 100MeV, bounded electron do can produce significant event rate. Here cross section of electron scattering amplitude is taken to be cm^2 which is just at edge of eclusion line of 2009 paper, and nuclear scattering amplitude is also assumed to be the same with electron.

24. More steep Reduce little bit increase
Test
When dark matter mass become small
More steep
Reduce little bit
increase
When dark matter mass become smaller,nucleon spectrum become more steep, bounded electron of free atom spectrum reduce a little bit, and free electron spectrum increase.

25. At 10~several hundred eV region:
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At 10~several hundred eV region:
Competation result of ER and NR is still not clear
May or may not produce measurable event rate
More nuclear & atomic
calculation is needed !
To finish my talk, my conclusion is that: at ten to several hundred eV region, competation result of ER and NR is still not clear
They may or may not produce big event rate, More nuclear & atomic calculation is needed! Here I stress that for Germanian detector used in CDEX and many other experiments, just bound electron of free atom calculation is not enough, since as a semiconductor, germanian has such complex band structure, one must consider this band effect into the calculation to make reliable result.