Atomic-Physics Tests of QED & the Standard Model

Transcription

1 Atomic-Physics Tests of QED & the Standard Model W.R. Johnson Notre Dame University johnson Abstract A brief review of tests of strong-field QED in many-electron atoms and of atomic paritynonconservation (PNC) induced by Z 0 exchange is given, with emphasis on the role of precise atomic many-body calculations. N D Atomic Physics 11/99

2 Outline (1970 s) Strong-Field Lamb-shift corrections for K-shell electrons in heavy atoms were tested to about 2%. (Dirac Hartree-Fock level atomic theory) (1980 s) QED corrections for valence electrons in highlycharged Li-like, Na-like, and Cu-like ions were tested to about 5%. (Third-order relativistic MBPT) (1990 s) Amplitudes for parity-forbidden transitions in Cs, Tl, Bi, and Pb were measured to better than 1%. Comparisons with precise theory in Cs gave a test of the standard-model at the 1% level and the first measurement of a nuclear anapole moment. (All-order relativistic MBPT) N D Atomic Physics 11/99 1

3 K-shell Binding Energies K-shell Energy (Ry) Experiment (Ry) Dirac-Coulomb (Ry) Nuclear Charge Dirac-Coulomb 1s energy ɛ 1s = 2Z α 2 Z 2 Ry N D Atomic Physics 11/99 2

4 Electron-Electron Interaction Dirac Hartree-Fock Approximation: One-electron orbitals φ a satisfy Dirac equations (H Dirac + V HF )φ a = ɛ a φ a V HF φ a (r) = b b d 3 r r r φ b (r )φ b (r )φ a (r) d 3 r r r φ b (r )φ a (r )φ b (r) Equations solved self-consistently. ɛ a = frozen-core binding energy. N D Atomic Physics 11/99 3

5 Relaxation Energy: Smaller Corrections Breit Interaction: E 1s =E ion E atom = ɛ 1s + 1s b 12 = α 1 α 2 + α 1 ˆr 12 α 2 ˆr 12 2r 12 Hg 1s binding energy (Ry) Coulomb HF correction Relaxation Relaxed Breit Breit Retardation 0.45 Nuclear size Structure Expt Estimated accuracy 0.1 Ry 1eV N D Atomic Physics 11/99 4

6 Theory vs. Experiment Expt. - Theory (Ry) Nuclear Charge Figure 1: Difference between experimental and theoretical K-shell energies as a function of Z. N D Atomic Physics 11/99 5

7 K-electron Self-Energy E 1s = α π (αz)4 F (Zα)mc 2 F (Zα) =A 4 + αza 5 +(αz) 2 A 6 + where, [ln (Zα) ] A 4 = 4 3 A 5 =3π A 6 = [ [ 7ln ] 2 ln 2 ] ln (Zα) ln2 (Zα) F(Zα) A 4 A 4 + (αz) A 5 A 4 + (αz) A 5 + (αz) 2 A Nuclear Charge N D Atomic Physics 11/99 6

8 Non-Perturbative 1s Self-Energy Brown et al Z =80(incorrect numerics) Desiderio 1971 Coulomb (70 90) Mohr 1972 Coulomb (10 100) Cheng 1976 Nuclear Size Soff et al Nuclear Size Blundell & Snyderman 1991 Lindgren et al F(Zα) 3 2 Mohr (Coulomb) Cheng (Coulomb) Cheng (Finite Nucleus) Nuclear Charge N D Atomic Physics 11/99 7

9 Test of Lowest-Order QED QED = Self-Energy + Vacuum Polarization Vacuum-Polarization Uehling Potential Experimental vs. Theoretical QED Atom Expt. Theory Diff. QED Xe (8) (8) W (2) (2) Hg (6) (6) Pb (5) (5) Rn (89) (89) Experimental and Theoretical radiative corrections agree at 2% level of accuracy. N D Atomic Physics 11/99 8

10 Many-Electron Hamiltonian Digression The no-pair Hamiltonian 1 is a many-electron generalization of the Dirac Hamiltomian derived from the field-theoretic Hamiltonian of QED: H = H 0 + V H 0 = i ɛ i a i a i V = 1 2 ijkl v ijkl a i a j a la k ij U ij a i a j 1. the sums are restricted to electron states only, 2. v ijkl is a two-particle matrix element of the sum of the Coulomb and Breit interactions, 3. U ij compensates for including U(r) in h 0. 1 Brown, G.E. & Ravenhall, D.G Proc. R. Soc. London, Ser. A 208, N D Atomic Physics 11/99 9

11 QED Tests for Copper-like Ions Largest atom with a complete isoelectronic sequence. Structure: (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 6 (3d) 10 + valence Outline of MBPT calculations: E (0) = a ɛ a + ɛ v E (1) = E (1) c = 1/2 a (V HF ) aa E (2) = E c (2) + E v (2) E (3) = E (3) c + E (3) v E (2) v E (3) v = ṽ mnav v avmn ṽ abmv v mvab ɛ amn m + ɛ n ɛ v ɛ a ɛ m + ɛ v ɛ a ɛ b abm = ṽ abvm v cdab v mvcd + 17 more terms (ɛ ab ɛ vm )(ɛ cd ɛ mv ) mabcd N D Atomic Physics 11/99 10

12 Energies of Cu-like Sn (Z=50) (a.u.) Term 4s 1/2 4p 1/2 4p 3/2 E (0) E (2) E (3) B (1) B (2) B (3) Recoil E Tot p 3/2 p 1/2 p 3/2 s 1/2 E Tot (2) (3) E Expt (3) (1) N D Atomic Physics 11/99 11

13 Tests of QED in Cu-like Ions 0.12 Energy (a.u.) Theory Exp. 1st Order QED Z Figure 2: Comparison of experimental and theoretical values of the 4p 3/2 4s 1/2 Lamb-shift in copper-like ions. Conclusion: Calculations of the self-energy and vacuum-polarization corrections carried out in a HF potential agree with experimental Lamb-shift to 5% for high-z Cu-like ions. N D Atomic Physics 11/99 12

14 Atomic Tests of the Standard Model H (1) = G ( ) [ ( ) ψe γ µ γ 5 ψ e c1p ψpi γ µ ψ pi 2 ( )] +c 1n ψni γ µ ψ ni H (2) = G ( ) [ ( ψe γ µ ψ e c2p ψpi γ µ ) γ 5 ψ pi 2 i i +c 2n ( ψni γ µ γ 5 ψ ni )] c 1p = ( sin 2 ) θ W 0.038, c 1n = 1 2, 1 c 2p = 2 g ( A 1 4sin 2 ) θ W 0.047, c 2n = 1 2 g ( A 1 4sin 2 ) θ W g A 1.25 for the partially conserved axial current sin 2 θ W = (24) is Weinberg s angle. N D Atomic Physics 11/99 13

15 Nonrelativistic Nucleon Reduction H (1) eff = G 2 2 γ 5 Q w ρ(r) H (2) eff = G 2 κ 1/2 I(I +1) α I [c 2p ρ pv (r)+c 2n ρ nv (r)] H (a) eff = G 2 K a κ I(I +1) α I ρ v(r) Q w =[2Zc 1p +2Nc 1n ]= N + Z ( 1 4sin 2 θ W ) Nuclear Anapole N D Atomic Physics 11/99 14

16 Single-Double Equations Ψ v =Ψ DHF + δψ { δψ = ρ ma a ma a am abmn ρ mnab a ma na b a a + ρ mv a ma v + ρ mnvb a ma na b a v Ψ DHF m v bmn E C = E DHF C E v = E DHF v + δe C + δe v Later, we will discuss triples N D Atomic Physics 11/99 15

17 Core Excitation Equations (ɛ a ɛ m )ρ ma = bn ṽ mban ρ nb + bnr v mbnr ρ nrab bcn v bcan ρ mnbc (ɛ a + ɛ b ɛ m ɛ n )ρ mnab = v mnab + cd v cdab ρ mncd + rs v mnrs ρ rsab [ + v mnrb ρ ra r c + [ a b m n ] δe C = 1 2 abmn v abmn ρ mnab v cnab ρ mc + rc ṽ cnrb ρ mrac ] 15,000,000 ρ mnab coefficients for Cs (l =6). N D Atomic Physics 11/99 16

18 m a = m a m b n + a a r b n + m c b n + exchange terms m a n b m a n b = a + m r n s b m + a c b d n m a n b + c + a m r n b + m a c n r b + exchange terms Figure 3: equations. Many-body diagrams for the core SD N D Atomic Physics 11/99 17

19 Valence Equations (ɛ v ɛ m + δe v )ρ mv = bn ṽ mbvn ρ nb + bnr v mbnr ρ nrvb bcn v bcvn ρ mnbc (ɛ v + ɛ b ɛ m ɛ n + δe v )ρ mnvb = v mnvb + cd v cdvb ρ mncd + rs v mnrs ρ rsvb [ + v mnrb ρ rv r c + [ v b m n ] v cnvb ρ mc + rc ṽ cnrb ρ mrvc ] δe v = ma ṽ vavm ρ ma + mab v abvm ρ mvab + mna v vbmn ρ mnvb 1,000,000 ρ mnvb coefficients for each state (Cs) N D Atomic Physics 11/99 18

20 Perturbation Expansion of the Energy δe C agrees with MBPT through third order. δe v agrees with MBPT through second order. δe v (3) disagrees with E v (3) from MBPT. Add triple excitations to wave function 1 6 abmnr ρ mnrvab a ma na ra v a b a a Ψ HF E v extra = 1 2 mnab ṽ abmn ρ mnvvab δe (3) v + E (3) v extra gives the entire third-order valence correlation energy N D Atomic Physics 11/99 19

21 Example E (3) v = δe (3) v + E (3) v extra Table 1: Na 3s and 3p states (a.u.) Term 3s 1/2 3p 1/2 3p 3/2 δe v (3) [-4] [-4] [-4] E (3) v extra [-4] [-4] [-4] E v (3) [-4] [-4] [-4] N D Atomic Physics 11/99 20

22 Correlation Energy Correlation energy (cm -1 ) Na K Rb Expt. E (2) E (2) +E (3) δe υ Nuclear charge Z Cs Fr Figure 4: Energy comparisons for alkali-metal atoms. N D Atomic Physics 11/99 21

23 Higher-order terms E (4)+ (cm -1 ) s 1/2 3p 1/2 3p 3/ Nuclear Charge Z Figure 5: Contributions beyond 3rd-order for Na-like ions. E (4)+ < 1% of correlation energy for Z>20 N D Atomic Physics 11/99 22

24 All-order energies Table 2: Summary for sodium 3s and 3p states (cm 1 ) Term 3s 3p 1/2 3p 3/2 DHF δe (4) E (3) extra Breit RM+MP Theory (4) Expt N D Atomic Physics 11/99 23

25 Results for Heavy Atoms Table 3: Energies for cesium and francium cm 1 Cs 6s 7s 8s 9s Theory Expt Cs 6p 1/2 7p 1/2 8p 1/2 9p 1/2 Theory Expt Fr 7s 8s 9s 10s Theory Expt Fr 7p 1/2 8p 1/2 9p 1/2 10p 1/2 Theory Expt N D Atomic Physics 11/99 24

26 Fine-Structure Intervals Table 4: Fine-structure (cm 1 ). This work Expt. K 4p 3/2 4p 1/ p 3/2 5p 1/ p 3/2 6p 1/ p 3/2 7p 1/ Rb 5p 3/2 5p 1/ p 3/2 6p 1/ p 3/2 7p 1/ p 3/2 8p 1/ Cs 6p 3/2 6p 1/ p 3/2 7p 1/ p 3/2 8p 1/ p 3/2 9p 1/ Fr 7p 3/2 7p 1/ p 3/2 8p 1/ p 3/2 9p 1/ p 3/2 10p 1/ N D Atomic Physics 11/99 25

27 Dipole Transitions Table 5: Matrix Elements for heavy alkali metals (a.u.) K Rb Cs Fr p 1/2 s n =4 n =5 n =6 n =7 SD Expt (5) 4.231(3) 4.489(7) 4.277(8) p 3/2 s n =4 n =5 n =6 n =7 SD Expt (8) 5.977(4) 6.324(7) 5.898(15) N D Atomic Physics 11/99 26

28 Application to PNC F ez I = n + n F ez n n H W I E n E I F H W n n ez I E n E F, where H W = H (1) eff + H(2) eff + H(a) eff F ez I = E PNC [1 + A(F F,F I )] For 133 Cs (I =7/2), we obtain: E PNC =0.906(9) i e a 0 Q w /N A(F F,F I )= [ ] N D Atomic Physics 11/99 27

29 Status of PNC Experiments (a) Optical rotation: n + n φ = E PNC /M 1 6p 1/2 6p 3/2 transition Element Group 10 8 φ Thallium Oxford -15.7(5) Thallium Seattle -14.7(2) Lead Oxford -9.8(1) Lead Seattle -9.9(1) Bismuth Oxford -10.1(20) (b) Stark interference: Add E(t) =A cos ωt and detect the hetrodyning signal R = E PNC /β 6s 1/2 7s 1/2 (mv/cm) Element Group R 4 3 R 3 4 Cesium Paris (1984) -1.5(2) -1.5(2) Cesium Boulder (1988) -1.64(5) -1.51(5) Cesium Boulder (1997) (8) (8) N D Atomic Physics 11/99 28

30 Cesium: Theory vs. Experiment β =27.02(8) a 3 0 (1999) (eliminating axial vector + anapole contribution) I(E PNC )= 0.836(4) e a 0 (1997) (dividing by theoretical matrix element) Q w = 71.9(4) K =0.50(7) (Expt.+ Atomic Theory) (Expt.+ Atomic Theory) Marciano & Rosner (with radiative corrections) Q w = 73.20(13) (Standard Model) Expt. - Theory = 1.3(4) Principal source of error is atomic theory. N D Atomic Physics 11/99 29

31 Conclusions I Strong-field QED corrections inner-shell electrons in heavy atoms are tested to 2%. II Comparing theoretical and experimental values of the Lamb-shift for valence electrons in highlycharged ions provide stringent tests of higher-order MBPT calculations. III Tests of the standard-model by comparing experiment and theory show a 3σ effect in Q w. IV Modern atomic PNC experiments are underway at Stonybrook, Berkeley, and Seattle. Calculations are needed to aid in extracting quantitive information from such measurements. N D Atomic Physics 11/99 30