Ionization cross sections of neutrino electronagnetic interactions with electrons
Chih-Pan Wu Dept. of Physics, National Taiwan University
P. 1 Constrain ν EM Properties by Ge
Reference: Phys. Lett. B 731, 159, arXiv:1311.5294 (2014). Phys. Rev. D 90, 011301(R), arXiv:1405.7168 (2014). Phys. Rev. D 91, 013005, arXiv:1411.0574 (2015). P. 2 ν-e Non-standard Interactions (NSI)
E E
(T,q) (T,q)
g = −1/ 2 A Weak Magnetic moment g = 2sin +1/ 2 v w Interaction
2 2 1 1 + − 2 me T E An Example for enhanced signals at low T!
P. 3 Solar ν Background in LXe Detectors
99.98% ER rejection
J. Aalbers et. al. (DARWIN collaboration), JCAP 11, 017, arXiv:1606.07001 (2016). J.-W. Chen et. al., Phys. Lett. B 774, 656, arXiv:1610.04177 (2017). P. 4 Outline
• Atomic ionization cross sections of neutrino- electron scattering – Why & when atomic effects become relevant? – MCRRPA: a framework of ab initio method • Discovery potential of ton-scale LXe detectors in neutrino electromagnetic properties – arXiv: 1903.06085 – solar nu v.s. reactor nu + Ge detector
P. 5 Why Atomic Responses Become Important?
The space uncertainty is inversely proportional to its incident momentum: λ ~ 1/p
• Atomic Size is inversely 2 important factors: proportional to its – neutrino momentum orbital momentum: ~ – Energy transfer T Z meα Z *3.7 keV Z: effective charge P. 6 Atomic Ionization Process for ν
| 푀 |2
The weak scattering amplitude: The EM scattering amplitude:
P. 7 Electroweak Currents
Lepton current:
Atomic (axial-)vector current: , 1
, 0 Sys. Error: ~훼 ≈ 1% P. 8 The Form Factors & Related Physical Quantities
neutrino millicharge : : charge form factor charge radius squared :
: anomalous magnetic neutrino magnetic moment : : anapole (P-violating) electric dipole moment : : electric dipole anapole moment : (P, T-violating)
P. 9 “effective” Magnetic Moment
µν and dν interactions are not distinguishable
where f and i are the mass eigenstate indices for the outgoing and incoming neutrinos, Aie(Eν, L) describes how a solar neutrino oscillates to a mass eigenstate νi with distance L from the Sun to the Earth. P. 10 Bound Electron Wave Function
|w >
|u >
ⅇത 훾휇ⅇ < Ψ | 훾휇|Ψ > 푓 푖 < 푤| 훾휇|푢 >
EM interaction Weak interaction Non-standard interactions 휇 휇 휇 5 훾 → 푔푉훾 + 푔퐴훾 훾 → NSI Opⅇrators P. 11 One-Electron Dirac Spinors
Then the radial Dirac equations can be reduced to
P. 12 Atomic Response Functions
Do multipole expansion with J Final continuous wave functions Initial states could be could be obtained by MCRRPA approximated by and expanded in the (J, L) basis bound electron orbital of orbital wave functions wave functions given by MCDF
P. 13 Ab initio Theory for Atomic Ionization MCDF: multiconfiguration Dirac-Fock method Dirac-Fock method: (t) is a Slater determinant of one-electron orbitals ua (r,t) 휕 and invoke variational principle 훿휓ሜ(푡) 푖 − 퐻 − 푉 (푡) 휓(푡) = 0 휕푡 퐼
to obtain eigenequations for 푢푎(푟റ, 푡) . multiconfiguration: Approximate the many-body wave function (t)
by a superposition of configuration functions (t)
2 휓1 = Zn 4푝1/2 (t) = C (t) (t) For Ge: ቐ 2 휓2 = Z푛 4푝3/2 MCRRPA: multiconfiguration relativistic random phase approximation RPA: Expand ua (r,t) into time-indep. orbitals in power of external potential
i a t −i t i t ua (r,t) = e ua (r) + wa+ (r)e + wa− (r)e +... −i t i t Ca (t) = Ca +[Ca ]+ e +[Ca ]− e +... P. 14 Here use square brackets with subscripts to designate the coefficients in powers of e ±iωt in the expansion of various matrix elements:
훾훼훽: Lagrange multipliers MCDF Equations: † 훿훼: functional derivatives EC+= C H 0 † a b ab 0 with respect to 푢 b 훼 C* Cδ† H−= u 0 a b ab 0 ab The zero-order equations are MCDF equations for unperturbed
orbitals ua and unperturbed weight coefficients Ca. MCRRPA Equations: ECHCHCVC − + = ( ) a( ab0 b ab b) ab b bb C*** C iδ† − H − C C + C Cδ† H a b( ab ab ) ( am b a b ) ab 0 ab ab −w +u = C* Cδ† V ( ) a b ab ab The first-order equations are the MCRRPA equations describing the linear response of atom to the external perturbation v . ± P. 15 Atomic Structure of Ge
Multiconfiguration of Ge Ground State (Coupled to total J=0) : Selection Rules for J=1, λ=1:
Angular Momentum Selection Rule:
Parity Selection Rule:
P. 16 Multipole Expansion
Transition matrix elements of atomic ionization by nu-EM interactions:
P. 17 Benchmark: Ge & Xe Photoionization
Exp. data: Ge solid Theory: Ge atom (gas)
Above 100 eV error under 5%.
B. L. Henke, E. M. Gullikson, and J. C. Davis, Atomic Data and Nuclear Data Tables 54, 181-342 (1993). J. Samson and W. Stolte, J. Electron Spectrosc. Relat. Phenom. 123, 265 (2002). I. H. Suzuki and N. Saito, J. Electron Spectrosc. Relat. Phenom. 129, 71 (2003). L. Zheng et al., J. Electron Spectrosc. Relat. Phenom. 152, 143 (2006). P. 18 Approximation Schemes
Longitudinal Photon Approx. (LPA) : VT = 0 2 Equivalent Photon Approx. (EPA) : VL = 0, q = 0 Strong q2-dependence in the denominator : long-range interaction Real photon limit q2 ～0 : relativistic beam or soft photons qμ～0
2 Free Electron Approx. (FEA) : q = -2 meT
Main contribution comes from the phase space region similar with 2-body scattering Atomic effects can be negligible :
Eν >> Z meα T ≠ Bi (binding energy) P. 19 Numerical Results: Weak Interaction
E = 1 MeV Ev = 10 keV ve e
(1) short range interaction 2E 2 cutoff : T = ve 0.38 keV (2) neutrino mass is tiny Max 2E + m ve e (3) Eν >> Z meα Kinematic forbidden by the inequality: p 2m T q 2E −T FEA works well away from r e Max e (backward scattering, m → 0) the ionization thresholds. νe P. 20 Numerical Results: NMM
E = 1 MeV ve E = 10 keV ve
Similar with WI cases. FEA still faces a cutoff with lower Eν . 2 For right plot, EPA becomes better when T approaches to Eν (q -> 0). Consistent with analytic Hydrogen results.
P. 21 Numerical Results: Millicharge
E = 1 MeV E = 10 keV ve ve
EPA worked well due to q2 dependence in the denominator of scattering formulas of F1 form factor (a strong weight at small scattering angles).
P. 22 Double Check on Our Simulation
• We perform ab initio many-body calculations for atomic initial & final states WF in ionization processes, and test by – Comparing with photo-absorption experimental data, for typical E1 transition, the difference is <5%. – In general, we have confidence to report a 5~10% theoretical errors. – It agrees with some common approximations under the crucial condition as we know in physical picture
P. 23 Solar ν As Signals in LXe Detectors
J.-W. Chen et. al., arXiv:1903.06085 (2019). P. 24 Expected Experimental Limits
Assuming an energy resolution from the XENON100 experiment
J.-W. Chen et. al., arXiv:1903.06085 (2019). P. 25 What Else?
Portals to the Dark Sector:
1. Remain a large region for the possibility of LDM (Ex: Dark Sectors) 2. Other interactions, or interacted with electrons
K. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014). R. Essig, J. A. Jaros, W. Wester, P. H. Adrian, S. Andreas et al., arXiv:1311.0029. P. 26 Spin-Indep. DM-e Scattering in Ge & Xe
J.-W. Chen et. al., arXiv:1812.11759 (2018). P. 27 Sterile Neutrino Direct Constraint
q2 < 0 q2 > 0
• Non-relativistic massive sterile neutrinos decay into SM neutrino. -14 • At ms = 7.1 keV, the upper limit of μνsa < 2.5*10 μB at 90% C.L. • The recent X-ray observations of a 7.1 keV sterile neutrino with decay -28 -1 -21 lifetime 1.74*10 s can be converted to μνsa = 2.9*10 μB , much tighter because its much larger collecting volume.
J.-W. Chen et al., Phys. Rev. D 93, 093012, arXiv:1601.07257 (2016). P. 28 Constraints on millicharged DM
L. Singh et. al. (TEXONO Collaboration), arXiv:1808.02719 (2018). P. 29 Summary
• Low energies nu-e Scattering can be the signal or important background in direct detection experiments, but the atomic effects should be taken into consideration now. • Ab initio atomic many-body calculations of ionization processes in Ge and Xe detectors performed with ~5% estimated error. That can be applied for 1. Constraining neutrino EM properties, 2. Study on solar neutrino backgrounds in DM detection, 3. Calculating DM atomic ionization cross sections.
P. 30 THANKS FOR YOUR ATTENTION! Scattering Diagrams and Detector Response Detected Signals 3 1 2
• elastic scattering, excitation, ionization • electron recoils (ER) or nucleus recoils (NR) 1. The particle-detector interaction 2. dσ/dT for the primary scattering process 3. The following energy loss mechanism P. 32 DM Effective Interaction with Electron or Nucleons Leading order (LO): short range
spin-indep. long range spin-dep. Differential cross section for spin-independent contact (e) interaction with electron (c1 ) : Initial & final states of detector material
P. 33 Toy Model: Analytic Hydrogen WFs
exp.-decay with the rate ∝ orbital momentum ~ 3.7 keV
Oscillated like sin/cos function with frequency ∝ electron momentum 1/2 ~ (2meT) • The initial state of the hydrogen atom at the ground state, the
spatial part |I>spat = |1s>
1. elastic scattering: 2. discrete excitation (ex): Phase space is fixed in 2-body scattering Energy and momentum transfer can → 4-momentum transfer is fixed be shared by nucleus and electrons → scattering angle is fixed → Inelastic scattering → Maximum energy transfer is limited (energy loss in atomic energy level) → Phase space suppression 4 minc mtar by a factor r = 2 (minc + mtar ) P. 35 Reduce Mass System for Atom Two particles can reduce to one system at their center of mass, with internal motion: (m1 , p1 ) p2 p2 p2 p2 1 + 2 = tot + rel = T − B 2m1 2m2 2M 2 106 eV M = m1 + m2 C.M. A * 109 eV ptot = p1 + p2 m m , = 1 2 prel = (v1 − v2 ) m1 + m2 (m , p ) 2 2 If the system received a 4-momentum transfer (T , q) , then the relative momentum would be: q (hit m ) m 1 p = 1 2 (T − B) (for M ) rel q (hit m2 ) m2 P. 36 Comparison of DM-H Cross Sections with the Electron and Proton ela (p) ion (p) ex (p) ion (e) Spin-indep. contact interaction ela (e) ex (e) J.-W. Chen et al., Phys. Rev. D 92, 096013 (2015). P. 37 Multipole Expansion & Operators MCRRPA Transition Amplitude: P. 38 Neutrino-Impact Ionization Cross Sections neutrino weak scattering : neutrino magnetic moment scattering : neutrino millicharge scattering: P. 39 ν-N Coherent Scattering through ν EM properties NMM: Millicharge: P. 40