The Study On The Nucleus Effects In Dark Matter Direct Detection

With the rapid development of observational method and getting better appara-tus, the human’s understanding for the universe become more profound, at the sametime the relevant theory are gradually improving and comprehensive. As a successfultheory, Big Bang hypothesis has been wildly accepted by human owing to the cosmicmicrowave background (CMB) radiation and thermal relic density of helium and hydro-gen and some other predictions have been confirmed by the experimental observation.However, in the mean time for celebrating the theory success, many more new prob-lems have been put in the front of the scientists. Such as origin of the matter-antimatterasymmetry, the physics of inflation era after the Big Bang and so on are all problemsto be resolved. Among the awkward subjects the dark matter and dark energy becomeour first thing to be understood. contemporary observations has verified that luminousmatter that can be observed directly constitute only4%of the matter in the universe,and most of the matter are’dark’. The2011Nobel Prize in Physics is awarded to threeastrophysicists for their discovery of the accelerating of the universe, the conclusionresulted from the observation for the Supernova. Thus it strike scientists’ enthusiasmto explore the dark energy. Via comprehensive analysis, we have confidence that72%of the whole energy is dark energy,24%is dark matter, and the rest4%of the matteris luminous. The discussion for dark energy is still a obscure stage, we temporarilyhave no efficient method to detect it. Fortunately, the theory study and experimentaldetection has entered a substantial process.Dark matter is the most challenging subject of the20th and21st centuries. Itis also a seemingly different hot topic between particle physics and astrophysics. Thecommon sense of the physicist is astro observations can provide enough foundations forestablishing theoretical models, but can not confirm the correction of property of thosemodels. Until our detectors(no matter in the satellites, mountains, or underground)ob-serve the dark matter, and test if it is consistent with the relevant theory model, we candraw the conclusion. It inevitably involves with interdisciplinary between astronomyand particle physics. In fact the interdisciplinary between the two fields can trace backto the Newton times even earlier(we don’t known whether it is excessive if saying an-cient Greece,ancient China). Undoubtly the research for dark matter surly becomes thefocal point of theoretical research and experimental observations in21centry, both byastrophysics and particle physics. It is not only a challenge, but also providing the most attractive opportunities.As a matter of fact, the conjecture about existence of dark matter was proposedquite a long time ago in1933by Zwicky to explain the anomalously large velocitynear the Coma star clusters astronomically observed at that time. The astronomicalobservation shows that the rotational curves of the test stars in the galaxies did not obeythe gravitational law if only the luminous matter which clusters at the center of thegalaxies existed. Namely, the velocities of a test stars were supposed to be inverselyproportional to square root of their distances from the center of the galaxy, instead, therotational curve turns flat. It means that there must be large amount of matter exist inthe cluster of galaxies that can not be seen, namely dark matter. Besides, the galaxyclusters collision and gravitational lens also confirm the existence of dark matter. Theprimary task for the theorist and experimental physicist is to search for the dark matterdirectly in experiments, meanwhile, identifying the candidate of dark matter becomesthe most important mission.The present theories divide dark matter into three types: Cold dark matter(CDM),Hot dark matter(HDM),Warm dark matter(WDM). On the basis of analysis so far, theCDM occupies the most part of the whole dark matter. It is obviously that dark mat-ter don’t participates strong and electromagnetic interactions. The component of thedark matter my be include one or multiple WIMP(Weakly interacting Massive Parti-cle)that contribute. The most promising candidate of WIMPs should be the lightestsuper-symmetric particle(LSP)neutralino, which is a linear combinationof the SUSYpartners of the photon,Z0, and Higgs bosons. Even there exist small R-parity broken,the neutralino’s lifetime long enough and comparable with the tody’s age of the uni-verse(13.5billon yeas). Of course if it decays, the excess of the positron in the universecan be expained and will be one of the evidence for indirect detection of dark matter.Currently the leading experimental investigation is directly detect the dark matterflux coming from the outer space. This is based on the assumption that dark matterconsist of WIMPs, except the gravitational interaction(it surely exist because it is grav-itation of the dark matter that induced the evidence)between them, dark matter alsoparticipate in weak interaction, actually, we haven’t any support from the experimentsabout this. If the dark matter only take part in gravitational interaction other than weakinteraction, it is impossible for us to ascertain them. It sounds miserable but may likelybe happen to some of extent. Thus we expect dark matter interact weakly with nor-mal materials and neutralino will be our expected candidate. Certainly there are other candidates including darkon, techipion and so on. Our work is based on this assump-tion, dark matter(no matter what kind of is)will interact weakly with general matter ofstandard model in detectors, and produce the observational signals.The direct detection is carried out via collisions between WIMPs and nuclei in de-tectors installed underground. The recoil energy can form electron, photon, or thermalsignals, which is determined by various detections. Besides, physicists hope to producedark matter particles in colliders, but dark matter don’t participate electromagnetic in-teractions nor decay(long lifetime), we have to find other visible particle with explicitenergy and momentum, then deduce and judge the missing particle’s properties, andfinally to determine if it is dark matter. This is long time but full of opportunities andchallenge.The research of our work this thesis centers on the scattered amplitude andform factors between dark matter and nucleus. The velocity of dark matter isabout220～600km/s in the earth frame coordinate. Even the mass of dark matter ar-range from dozens of GeV to hundreds of GeV, is kinetic energy is only a few tens ofKeV, which is too small to cause an inelastic transition for the nucleus. As a result wehave to investigate he dark matter’s property from nucleus recoil energy and transferredmomentum. The interaction between dark matter and nuclei can also be divided twoparts: spin-independent(SI) interaction and spin-dependent interaction. In this thesiswe have studied:first of all, we studied form factors of spherical nucleus in the case of spin-independent interaction between dark matter and nucleus. In this condition, contri-bution from the nucleus can be reflected with form factor, namely the calculation of theform factors can be transformed to the nucleus density’s fourier transformation.The most simplest nucleus density model is2PF(two-parameter-Fermi)model.Another model is Folding model. This model assumes the uniform density in the nu-clear inner part, and introduce a Gaussian’surface smearing’ density function, convolvethe two together to describe the nucleus density. The result is similar to that of the2PF, but also have some difference. There is no analytic expression for fourier trans-formation of2PF model, but Folding model’s fourier actions can be written, it alsonamed as Helm form factor. The text also give some other density model such asGaussian model,which is proposed by Sick, and Fourier-Bessel-expansion model fordensity and relevant form factor.2PF and Helm model are widely used in the theoryof dark matter detection. As an alternative, we discuss Relativistic Mean field(RMF)of nucleus many-body theory to calculate the nuclear density. RMF theory assumes theinteraction between nucleons happened by exchange bosonic meson such as σ, ω, ρ, π,as a approximation, introducing the expectation value of meson field to replace thefield source terms in the Clein-Gordon equation. Using self-consistent iteration be-tween Dirac equation(nucleon field)and Clein-Gordon equations, we can obtain theground states properties such as nuclear density, binding energy, square root meanand so on. We calculate spherical nucleus density(16O,40Ca,72Ge,132Xe,208Pb), thenperform fourier transformation to the density to get the corresponding form factors.Elements40Ca,72Ge,132Xe are widely used in detectors for dark matter detection. Wecompared2PF, Helm, RMF three types of models. The results from RMF model arelittle different from the other two models, but has the same tendency. Its advantage isRMF theory completely considered nuclear uniqueness, thus more approach the realityand believable.second, we studied effects of nuclear deformation on the form factor in the case ofscalar interaction(spin-independent). Nuclear multiple deformation include: zero orderdeformation(that is spherical shape), quadruple deformation(corresponding ellipsoid),octupole deformation and higher order deformations. For the nucleus in the detectorsin dark matter detections, it is precise enough only consider the quadruple deformation.Therefore we primary address ourselves to the quadruple deformation.2PF model isused to describe the nuclear density distribution and form factor. When the nucleardeformation is considered, this model can be revised. In early period, one revised thehalf-radius to be related with angle in order to adjust to deformed nucleus. Similarly, wemodified the Folding model: beginning from the ellipsoidal equation, we introduce longsemi-axes and short semi-axes, accordingly the surface radius is parameterized. Westill assume the density in nucleus is homogeneous distributed, and convolved it with’smearing surface’ Gaussian function referred aforementioned. The third method tocalculate deformed nuclear density is based on Nilsson mean field, which is introducinga axially symmetric harmonic oscillator, also the considering the spin-orbit couplingterm and, and Dl2which is used to supress the potential well. We start from twoaspects, one is consider both Nilsson mean field and nearest orbital paring interaction ofthe nucleus, another is only Nilsson mean field included. In this paper we calculate thedeformed nucleus densities of73Ge,131Xe. We used three models above to obtain thedensities from different polar angles(10o,30o,45o,60o,90o) and compared with eachmodel. The relevant form factors were carried out by taking fourier transformation to the densities. The results show us that in the detectors for dark matter, the effects ofsmall deformation can be neglected for the nuclear form factors.third, we discuss the scattering amplitude in the spin dependent interaction. Infact most of the dark matter interact with particles of standard model will lead to spin-dependent scattering, which exchange Z or Z’ some other new particles beyond stan-dard model. The WIMP particle interacts with quarks or gluons inside the nucleon.Gluons in the hadron do not participate in the weak interaction at the leading order,so that the fundamental processes concern only the interaction between DM particleand quarks. The cross section of spin-dependent is more complex than that of spin-independent condition. For the spin-independent cross section, the particle-physics andnuclear-physics contributions can be separated, namely the nuclear effects can be fac-tored out and included in a form factor F(q). Actually the calculation process is similarto the nuclear beta decay. J.Engel et al gave scattering amplitude for dark matter andnucleus collision, employing Walecka’s multi-pole expansion method, with the impulseapproximation, expanding the interaction current with vector spherical harmonics, withC-G coefficients including, and get the result related to angles. Final aim is to deal withthe reduced matrix element of the scattering. The amplitude is divided into two situ-ations: momentum transfer q=0and q=0. The detailed expression of reduced matrixelement relies on concrete nuclear models. We obey the quantum field principle andstart from another aspect to deduce renewedly the scattering amplitude. Our advantageis we only consider the valence nucleons outside the full shell,because contributionsfrom the nucleons in the inner part of full shell cancels each other, this make calcula-tion more simplify. We use nucleus structure model derived from Luo, calculate crosssection of spin dependent interaction. The spin-orbital coupling is included, the rele-vant wave-function is written with(j,m,l,s). For the sake of reasonable calculation, thewave-function will be expand with C-G coefficients in the representation which is us-ing spherical harmonic function and spin wave-function as basis, may be some of bittedious in the process but no principle difficulty.The final part in our job discussed the nuclear effect in the theoretical calculationfor the dark matter detection. We hope our scenario can be helpful for the design ofexperiments of dark matter detection, and be some of help to extract any informationabout the fundamental interaction from the experimental data.