ATOMIC PARITY VIOLATION
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1 ATOMIC PARITY VIOLATION OUTLINE Overview of the Atomic Parity Violation Theory: How to calculate APV amplitude? Analysis of Cs experiment and implications for search for physics beyond the Standard Model Neutron skin Nuclear spin-dependent APV effects and weak hadronic interactions Future prospects 1
2 Atomic Parity Violation High energies Instead of search for new processes or particles directly Determine weak charge Q W from atomic parity violation studies and compare the result with Standard Model prediction Low energies Cs experiment, University of Colorado Two sides of the atomic parity violation NUCLEAR SPIN-INDEPENDENT INDEPENDENT : SEARCHES FOR NEW PHYSICS BEYOND THE STANDARD MODEL e q NUCLEAR SPIN-DEPENDENT : STUDY OF IN THE NUCLEUS a e Z 0 q Weak Charge Q W Nuclear anapole moment
3 EXPERIMENTAL STUDIES Cesium: atom with single (valence) electron outside a closed core. Cs Z=55 Need heavy atom for atomic core 1s 5p 6 6s valence electron 7s 6s Cs E1 l=0 to l=0 electric dipole transition is forbidden by parity selection rules 3
4 Atomic Parity Violation r r 7s 6s Cs E1 Non-zero transition amplitude amplitude E e q Both 6s and 7s states acquire an opposite-parity (np 1/ ) admixture Z 0 exchange: Laporte s rule is violated! e Z 0 q Atomic Parity Violation r r 7s 6s Cs E1 Non-zero transition amplitude amplitude E e q Both 6s and 7s states acquire an opposite-parity (np 1/ ) admixture Z 0 exchange: Laporte s rule is violated! e Z 0 q Note: it is really tiny effect!!! E ~10-11 atomic units E1 amplitude for 6s 6p transition is 4.5 atomic units 4
5 The most precise measurement of amplitude ( cesium ) C.S. Wood et al. Science 75, 1759 (1997) 0.35% accuracy Im 7s 6s 1 F=4 F=3 F=4 F=3 mv ( E ) (80) cm = (77) Stark interference scheme to measure ratio of the amplitude and the Stark-induced amplitude β β mv cm 1 Analysis of Cs experiment NUCLEAR SPIN-INDEPENDENT INDEPENDENT 7s 6s Average of 1 & Im si ( E ) β = (56) Weak Charge Q W mv cm NUCLEAR SPIN-DEPENDENT F=4 7s F=3 1 6s F=4 F=3 Difference of 1 & sd ( E ) Im β = (11) mv cm Nuclear anapole moment 5
6 Analysis of Cs experiment NUCLEAR SPIN-INDEPENDENT INDEPENDENT 7s 6s Average of 1 & Im si ( E ) β = (56) mv cm Weak Charge Q W e e How to extract weak charge Q w from Cs experiment? Z 0 q q Electronic sector: Electron-quark parity violating interaction (exchange of virtual Z 0 boson) G µ µ H ( eγ µ γ e){ C uγ u C dγ d} F W = 5 1u + 1 d +... (1) GF H = Q γ 5 ρ( r) W Extraction of weak the charge: Measured value E = E Q theory Neutron density function Theoretical calculation of amplitude inferred W 6
7 Calculation of amplitude 1. Main part Coulomb interactions Electric-dipole matrix elements E theory 1/ np1/ H s 7s H np1/ 1/ 7s d np 6 np d 6s = + E E E E n = 6s np n = 7s np 1/ 1/ Energies matrix elements Sum is separated to main part, n = 6-9 and the tail. Other small corrections: Breit, QED, Neutron skin, e e weak interaction REDUCING THEORY UNCERTAINTY: WHY IS IT SO DIFFICULT? Exact wave function Cs: 55 electrons Ψ v = Ω Ψ (0) v Dirac-Hartree-Fock wave function (lowest order) Many-body operator, describes excitations from lowest-order 55-fold excitations to get exact wave function Even for 100 function basis set Approximate methods: perturbation theory does not converge well, Need to use all-order methods (coupled-cluster method and correlation potential method) 7
8 Development of Theoretical Methods Derive the formulas Write the corresponding computer code debug Optimize the code so it will finish in reasonable time Carry out the numerical calculations Make sure your results (1) make sense () are in fact correct Estimate how accurate are your final numbers Derive the formulas 8
9 Derive the formulas Derive the formulas Contract operators by Wick s theorem H 1 (0) v i j l k d b : c S Ψ > : a a a a : aman ar as a aca aaav 0 > 9
10 TOO MANY TERMS!!! Contract operators by Wick s theorem H 1 (0) v i j l k d b : c S Ψ > : a a a a : aman ar as a aca aaav 0 > 800 TERMS! Solution: develope codes that write formulas, i.e. do all the work for you! Input: ASCII input of terms of the type mnrab ijkl g ρ : a a a a : : a a a a a a : Ψ (0) ijkl mnrvab i j l k m n r b a v v Output: final simplified formula in LATEX you can now code and put in you paper 10
11 STILL TOO MANY TERMS TO CODE!!! Solution: automated formula and code generation! Codes that write formulas Codes that write codes Input: Output: list of formulas to be programmed final code (need to be put into a main shell) Features: simple input, essentially just type in a formula! Optimize the code so it will finish in reasonable time Less then the lifetime of the Universe 11
12 Optimize the code so it will finish in reasonable time Before your current grant proposal runs out Optimize the code so it will finish in reasonable time Before your patience runs out indieberries.blogspot.com 1
13 Carry out the numerical calculations Make sure your results (1) make sense () are in fact correct Estimate how accurate are your final numbers Codes that write formulas Codes that write codes Codes that analyse data RELATIVISTIC ALL-ORDER METHOD Sum over infinite sets of many-body perturbation theory (MBPT) terms. Scheme: Calculate the atomic wave functions and energies Calculate various matrix elements H Ψ = EΨ v v Calculate derived properties such as amplitudes 13
14 LOWEST-ORDER ORDER ATOMIC WAVE FUNCTION Cs: atom with single (valence) electron outside of a closed core. core 1s 5p 6 6s valence electron Lowest-order wave function Ψ Ψ (0) = v av core Core Creation operator for state v Core wave function Coupled-cluster method ( CCSD ) (0) (0) v v exp( S) v Ψ = Ω Ψ = Ψ DHF wave function exp( S + S ) 1 CCSD single-double coupled-cluster S + S LCCSD coupled-cluster 1 core excitation S1 S valence excitation core excitations core - valence excitations 14
15 LCCSD (ALL-ORDER) ATOMIC WAVE FUNCTION Lowest order Core core valence electron any excited orbital Single-particle excitations Double-particle excitations LCCSD ATOMIC WAVE FUNCTION Lowest order Core (0) Ψ v core valence electron any excited orbital Single-particle excitations (0) (0) ρma amaa Ψv ρmv amav Ψv ma m v Double-particle excitations ρ a (0) 1 mnab m n b a v mnab a a a Ψ mna ρ mnva a a a m n a a v Ψ (0) v 15
16 Calculation of the atomic wave functions reduces to the calculations of excitation coefficients ρ. These are found by iterative solution of a very large system of linear equations. All-order method: Correlation correction to ground state energies of alkali-metal atoms 16
17 RESULTS FOR ALKALI-METAL ATOMS: E1 MATRIX ELEMENTS (A.U.) Na 3p 1/ -3s K 4p 1/ -4s Rb 5p 1/ -5s Cs 6p 1/ -6s Fr 7p 1/ -7s All-order Experiment 3.546(3) 4.10(5) 4.31(3) 4.489(6) 4.77(8) Difference 0.18% 0.1% 0.4% 0.4% 0.5% Experiment Na,K,Rb: U. Volz and H. Schmoranzer, Phys. Scr. T65, 48 (1996), Cs: R.J. Rafac et al., Phys. Rev. A 60, 3648 (1999), Fr: J.E. Simsarian et al., Phys. Rev. A 57, 448 (1998) Theory M.S. Safronova, W.R. Johnson, and A. Derevianko, Phys. Rev. A 60, 4476 (1999) Cs : Comparison with the standard model Standard Model [1] : Q SM W = 73.16(3) Most current result for Cs Expt/Theory: Atomic physics [] : Q inferred W = 73.16(9) (0) expt theory No deviation from the Standard Model [1] C. Amsler et al. (Partical Data Group), Phys. Lett. B 667, 1 (008) [] S. G. Porsev, K. Beloy and A. Derevianko, PRL 10, (009), Phys. Rev. D 8, (010) 17
18 IMPLICATIONS FOR PARTICLE PHYSICS Confirms fundamental running (energy dependence) of the electroweak force over energy span 10 MeV 100 GeV Figure is from Bentz et al. Phys. Lett. B693, 46 (010). 18
19 Probing new physics New physics can be phenomenologically described by weak isospin - conserving S and isospin - breaking T parameters [1]. inferred W SM W Q = Q Q = 0.800S 0.007T Present result []: S < 0.45 Parameter S is important for indirect constraint on the mass of Higgs particle [1]. [1] J.L. Rosner, PRD 65, (00) [] S. G. Porsev, K. Beloy and A. Derevianko, PRL 10, (009), Phys. Rev. D 8, (010) Probing new physics: extra Z bosons e e Z 0 q q + e q e Z q Atomic parity violation is uniquely sensitive to Z Z χ in SO(10) GUT, Marciano & Rosner inferred SM TeV / c Q = QW QW M Z ' x 19
20 Probing new physics: extra Z bosons Z χ in SO(10) GUT, Marciano & Rosner inferred SM TeV / c Q = QW QW M Z ' x Cs result [1] implies M Z ' x > 1.3TeV / c Direct search at Tevatron collider [] M Z ' x > 0.8TeV / c [1] S. G. Porsev, K. Beloy and A. Derevianko, PRL 10, (009) [] T. Aaltonen et al., Phys. Rev. Lett. 99, (007) NEUTRON SKIN AND ATOMIC PARITY VIOLATION 0
21 Neutron skin correction to amplitude The relative neutron skin correction to the amplitude is independent from electronic structure and is given by: E E ns 3 = 7 ( αz ) R R np p Difference between rms radii of neutron and proton distributions rms radius of proton distribution Neutron skin correction is 0.% for Cs, but 0.6% for Fr and Ra + Therefore, neutron skin can be extracted from APV studies if their uncertainties are smaller than these values. Pollock et al., PRC 46, 587 (199), Brown et al., PRC 79, (009) The neutron skin calculations, Skxs0 Different isotopes for a given element are connected by lines. The filled circles show nuclei of interest for atomic PV. Alex Brown, 008 PREX workshop, Brown et al., PRC 79, (009) 1
22 Isotopic chain APV experiments (1) APV isotopic chain experiments should allow to largely cancel dependence on atomic theory. () Errors due to neutron skin in two isotopes are correlated. This should allows to extract new physics from such experiments (3) The sensitivity to neutron skin effect is the largest for the lightest and heaviest pair of isotopes in the chain. It may be possible to extract neutron skin from such experiments. R np Atom A ns R np E E Yb (Z=70) (35) (8) (67) (14) Fr (Z=87) (36) (11) (53) (16) Brown et al., PRC 79, (009) NUCLEAR SPIN-DEPENDENT PARITY VIOLATION EFFECTS
23 Parity violation in atoms The other part of the story NUCLEAR SPIN-DEPENDENT : STUDY OF IN THE NUCLEUS a Nuclear anapole moment Spin-dependent parity violation: Nuclear anapole moment a Parity-violating nuclear moment Valence nucleon density 7s 6s 1 F=4 F=3 F=4 F=3 H (a) G F = κ α I a Anapole moment ρ ( r) v Nuclear anapole moment is parity-odd, time-reversal-even E1 moment of the electromagnetic current operator. 3
24 Constraints on nuclear weak coupling contants W. C. Haxton and C. E. Wieman, Ann. Rev. Nucl. Part. Sci. 51, 61 (001) Nuclear anapole moment? The constraints obtained from the Cs experiment were found to be inconsistent with constraints from other nuclear measurements, which favor a smaller value of the 133 Cs anapole moment. κ = 0.117(16) Possible atomic calculation solution? Incomplete correlation calculation of spin-dependent amplitude? 4
25 More spin-dependent effects κ = κ + κ + κ a hf (V e,a N ) interaction Same Hamiltonian as anapole moment term with κa κ Weak-hyperfine interference term This term does not reduce to the same interaction but effective constant κ hf can be calculated. W.R. Johnson, M.S. Safronova and U.I. Safronova, Phys. Rev. A 67, (003) New all-order (CCSD) calculation of spin-dependent Electric-dipole matrix elements E (, a ) 7s d j j H 6s 7s H j j D 6s = A + E (, a) (, a) 1 A j v E6s E j j w E7s j matrix elements First four terms in the sums are replaced by all-order matrix elements 1% accuracy is expected 5
26 Nuclear anapole moment: Test of hadronic weak interations The constraints obtained from the Cs experiment were found to be inconsistent with constraints from other nuclear measurements, which favor a smaller value of the 133 Cs anapole moment. All-order (LCCSD) calculation of spin-dependent amplitude: k = 0.107(16)* [ 1% theory accuracy ] No significant difference with previous value k = 0.11(16) is found. NEED NEW EXPERIMENTS!!! Fr, Yb, Ra + *M.S. Safronova, Rupsi Pal, Dansha Jiang, M.G. Kozlov, W.R. Johnson, and U.I. Safronova, Nuclear Physics A 87 (009) 411c NEED NEW EXPERIMENTAL STUDIES 6
27 EXPERIMENTAL STUDIES All-order Correlation potential CI+MBPT 7
28 Conclusion A: New analysis of atomic experiment is Cs: Nuclear spin-independent part: (1) Provided most accurate to-date test of the low-energy electroweak sector of the SM. () Confirmed fundamental running (energy dependence) of the electroweak force. (3) Placed constraints are on a variety of new physics scenarios beyond the SM. B: New analysis of atomic experiment is Cs: Nuclear spin-independent part (anapole moment) (1) New calculations, accurate to 1% - essentially the same result. () Constraints on nuclear weak coupling constants are still inconsistent with nuclear physics experiments. 8
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