Helium fine structure theory for determination of α

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1 Journal of Physics: Conference Series Helium fine structure theory for determination of α To cite this article: Krzysztof Pachucki and Vladimir A Yerokhin 0 J. Phys.: Conf. Ser View the article online for updates and enhancements. Recent citations - Precision calculation of fine structure in helium and Li + Zhang Pei-Pei et al - Progress in the accuracy of the fundamental physical constants: 00 CODATA recommended values S G Karshenboim This content was downloaded from IP address on 09/0/08 at 00:57

2 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 Helium fine structure theory for determination of α Krzysztof Pachucki Institute of Theoretical Physics, University of Warsaw, Hoża 69, Warsaw, Poland Vladimir A. Yerokhin Center for Advanced Studies, St. Petersburg State Polytechnical University, Polytekhnicheskaya 9, St. Petersburg 955, Russia Abstract. We present recent progress in the calculation of the helium fine-structure splitting of the P J states, based on the quantum electrodynamic theory. Apart from the complete evaluation of m α 7 and m /M α 6 corrections, we have performed extensive tests by comparison with all experimental results for light helium-like ions and with the known large nuclear charge asymptotics of individual corrections. Our theoretical predictions are still limited by the unknown m α 8 term, which is conservatively estimated to be.7 khz. However, comparison with the latest experimental result for the P 0 P transition [M. Smiciklas and T. Shiner, Phys. Rev. Lett. 05, 00 (00] suggests that the higher-order contribution is in fact much smaller than the theoretical estimate. This means that the spectroscopic determination of α can be significantly improved if another measurement of the P 0 P transition in helium-like Li + or Be + ion is performed.. Introduction The quantum electrodynamic (QED theory of atomic energy levels has achieved a precision level that makes possible the determination of nuclear properties, like the charge radius, the magnetic dipole, or even the nuclear polarizability from measured atomic spectra. If the nuclear structure effects are negligible or can be eliminated, one may obtain fundamental constants from comparison of theoretical predictions with experimental results. The most important examples include the Rydberg constant determined from hydrogen spectroscopy, the electron mass derived from the bound-electron g factor in hydrogen-like ions, and α obtained from the helium fine structure. As first pointed out by Schwartz in 964 [], the splitting of the P J levels in helium can be used for an accurate determination of the fine structure constant α. The attractive features of the fine structure in helium as compared to other atomic transitions are, first, the long lifetime of the metastable P J levels (roughly two orders of magnitude longer than that of the p state in hydrogen and, second, the relative simplicity of the theory. Schwartz s suggestion stimulated a sequence of calculations [ 5], which resulted in a theoretical description of the helium fine structure complete up to order mα 6 (or α 4 Ry and a value of α accurate to 0.9 ppm [6]. The present experimental precision for the fine-structure intervals in helium is sufficient for a determination of α with an accuracy of 4 ppb from Re. [7, 8] and even 5 ppb from Ref. [9]. Published under licence by Ltd

3 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 In order to match this level of accuracy in the theoretical description of the fine structure, the complete calculation of the next-order, mα 7 contribution and an estimation of the higher-order effects is needed. Work towards this end started in the 990s and extended over two decades [0 9]. In 006 the first complete evaluation of the mα 7 correction to the helium fine structure was reported by one of us (KP [0]. However, the numerical results presented there were in disagreement with the experimental values by more than 0 standard deviations (σ. In our recent investigations [, ], we recalculated, using formulae from Ref. [0], all effects up to order mα 7 to the fine structure of helium and performed calculations for helium-like ions with nuclear charges Z up to 0. The calculations were extensively checked by studying the hydrogenic (Z limit of individual corrections and by comparing them with the results known from the hydrogen theory. We found several problems in previous numerical calculations and, in the meantime, the experimental value of the P P transition was changed by σ [8]. As a result, the present theoretical predictions are in agreement with the latest experimental data for the fine-structure intervals in helium, as well as with most of the experimental data available for light helium-like ions. Our calculation of the mα 7 correction for the fine-structure splitting in light helium-like atoms was reported in Re. [, ]. In this paper, we present a detailed description of all corrections to helium fine structure and a summary of the numerical results.. QED theory of the helium fine structure According to the quantum electrodynamic theory (QED the energy levels of an atomic system are a function of the fine structure constant α and the electron-nucleus mass ratio. We omit possible nuclear structure effects, as their contribution to the helium fine structure is negligible. The fine-structure splitting E (α can be expanded in powers of α, E = E (4 + E (5 + E (6 + E (7 + O(α 8. ( The expansion terms E (n m α n E (n are of order m α n. They implicitly depend on the electron-nucleus mass ratio and may additionally involve powers of ln α. The advantage of this approach is that each of the expansion terms is expressed as the expectation value of some effective Hamiltonian, as presented in the following. For convenience, we first consider the infinite nuclear mass limit, and then account for the finite nuclear mass corrections separately. The dominant contribution to the helium fine structure is induced by the spin-dependent part of the Breit-Pauli Hamiltonian, which is, for an infinitely heavy nucleus, H = ( σ σ 4 r σ r σ r r 5 ( + a e + Z [ 4 r r p σ + ] r r p σ ( + a e { [( ] + + ae 4 r σ + ( + a e σ r p [ } ] ( + a e σ + ( + a e σ r p, ( where r = r r. The above Hamiltonian includes the effect of the anomalous magnetic moment (amm a e, which is given by [4] (neglecting small vacuum-polarization corrections coming from particles heavier than an electron a e = α π ( α π ( α π.94 4(5 ( α π (

4 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 Expanding the amm prefactors in Eq. (, H can be written as a sum of operators contributing to different orders in α H = + α H (5 + α H (6,amm + α H (7,amm (4 Here, and H (5 are the complete effective Hamiltonians to order m α 4 and m α 5, respectively, whereas H (6,amm and H(7,amm are the amm parts of the corresponding higher-order operators. The contributions to the fine structure are E (4 = + O(m/M, (5 E (5 = H (5 + O(m/M, (6 where the expectation values are calculated nonrelativistic Hamiltonian H 0 with the corresponding eigenstate of the H 0 = p + p Z r Z r + r. (7 The finite nuclear mass corrections up to order mα 5 are conveniently divided into three parts, termed the mass scaling, the mass polarization, and the recoil operators. The effect of the mass scaling is accounted for by including the prefactor (m r /m in the operator H, where m r is the reduced mass for the electron-nucleus system. The effect of the mass polarization can be accounted for to all orders by evaluating expectation values of all operators on the eigenfunctions of the Schrödinger Hamiltonian with the mass-polarization operator (m r /M p p included. The third effect is induced by the recoil addition to the Breit-Pauli Hamiltonian H,rec = Z m M [ r r ( p + p σ + r ] r ( p + p σ ( + a e. (8. The spin-dependent m α 6 contribution The m α 6 contribution to the helium fine structure is a sum of the second-order perturbation corrections induced by the Breit-Pauli Hamiltonian and the expectation value of the effective fine-structure Hamiltonian to this order, H (6 E (6 = (E 0 H 0 H(4 +, n (E 0 H 0 H(4 + H (6 + H (6,amm. (9 Here, /(E 0 H 0 is the reduced Green function and n is the spin-independent part of the Breit-Pauli Hamiltonian, n = 8 (p4 + p 4 + Z π [ δ (r + δ (r ] ( δ ij pi r + ri r j r p j, (0 where we have omitted a term with δ (r since it vanishes for the triplet states. H (6 consists of 5 operators first derived by Douglas and Kroll (DK [] in the framework of the Salpeter equation. These operators were later rederived using the much simpler effective field method in Ref. [5]. The result is H (6 5 = B i, ( i=

5 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 Table. Effective operators contributing to H (6 (left column and H H (right column Operator m α 6 Operator m α 7 /π B = Z 8 p r p r σ H = Z 4 p r r B = Z r r σ r ( r p H = Z 4 r r r σ r r r r r r σ r r p σ B = Z σ H = Z 4 B 4 = r p r 4 σ H 4 = r p r 4 σ r r σ ( r p σ B 5 = r 6 r σ r σ H 5 = 4 r 6 r σ r σ B 6 = 5 8 p r p r σ B 7 = 4 p r p r σ B 8 = i 4 p r σ ( p p H 6 = 4 p r p r σ H 7 = 4 p r p r σ H 8 = Z r 4 r p r σ B 9 = i 4 p r p r r p σ H 9 = i p r p r r p σ B 0 = i r ( r p 8 r 5 p σ H 0 = i r ( r p 4 r 5 p σ B = r ( r p 6 r 5 σ p σ H = r ( r p 8 r 5 σ p σ B = p 6 r σ p σ H = p 8 r σ p σ B = p r σ r 5 r σ H = 6 p r σ r 5 r σ B 4 = i 4 p r r σ p σ H 4 = i 8 p r r σ p σ B 5 = i 8 p r r σ p σ H 5 = i 8 p r ( r σ p σ + r σ p σ r r σ r σ r p H 6 = 4 p σ p r r p H 7 = 8 p σ ( p σ r + p r r r 5 σ where the B i are given in Table. The finite nuclear mass corrections to the m α 6 contribution can be divided into the mass scaling, the mass polarization, and the operator parts. The mass scaling prefactor is (m r /M 4 for the B, B, B 4, and B 5, (m r /M 5 for the other B i operators, (m r /M 6 for the secondorder corrections involving the first term in Eq. (0, and (m r /M 5 for all other second-order corrections. The mass polarization effect is most easily accounted for by including the mass polarization operator in the zeroth-order Hamiltonian. The operator part comes from recoil corrections to, H(4 n, and H(6. The recoil part of H(4 is given by Eq. (8. The spinindependent recoil part of the Breit-Pauli Hamiltonian is n,rec = Z m M a=, p i a ( δ ij r a + ri ar j a r a (p j + pj. ( Recoil corrections to the DK operators were studied by Zhang [4] and by Pachucki and 4

6 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 Sapirstein [9]. The result is given by the effective Hamiltonian H (6,rec = m [ iz M 4 p σ ( p p iz r r 4 p ( σ r p ( p + p Z 4 p σ r r ( p + p + Z σ r r r r ( p + p + Z σ r r r r ( r ( p + p +Z σ r r r r ( r p Z σ r r 4 ( p + p Z 4 σ r r σ r r ]. ( 4. The spin-dependent mα 7 correction The mα 7 correction to the helium fine structure can be conveniently separated into four parts E (7 = E (7 log + E(7 first + E(7 sec + E (7 L. (4 The first term above combines all terms with ln Z and ln α [,5,0], [ E (7 Z log = ln[(z α ] i p δ (r p σ 4 ( σ ( σ δ (r i p δ (r p σ + 8Z [ δ (E 0 H 0 (r + δ (r ] ]. (5 The second part of E (7 is induced by effective Hamiltonians to order mα 7. They were derived by one of us (K.P. in Re. [0, ]. (The previous derivation of this correction by Zhang [, ] turned out to be not entirely consistent. The result is E (7 first = H Q + H H + H (7,amm. (6 The Hamiltonian H Q is induced by the two-photon exchange between the electrons, the electron self-energy and the vacuum polarization. It is given by [0] H Q = Z 9 80 i p δ (r p σ ( σ ( σ δ (r [ ] + i p δ (r p σ 0 ln Z 5 8 π [ ] ln Z r 7 ( σ r ( σ r 4 π i p r p σ. (7 Here, the terms with ln Z compensate the logarithmic dependence implicitly present in the expectation values of the singular operators /r and /r 5, so that matrix elements of H Q do not have any logarithms in their /Z expansion. The singular operators are defined through their integrals with the arbitrary smooth function f and d r r 7 ( r i r j δij lim ɛ 0 d r f( r lim r ɛ 0 d r r [ r 7 [ ] d r r θ(r ɛ + 4 π δ (r (γ + ln ɛ f( r (8 f( r (9 (r i r j δij r θ(r ɛ + 4 π ] 5 δ (r (γ + ln ɛ ( i j δij f( r, 5

7 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 where γ is the Euler constant. The effective Hamiltonian H H represents the anomalous magnetic moment (amm correction to the Douglas-Kroll mα 6 operators and is given by [0] 7 H H = H i, (0 i= where the H i are presented in Table. The last term of E (7 first in Eq. (6, the Hamiltonian H (7,amm is the mα7 amm correction to the Breit-Pauli Hamiltonian, see Eq. (. The third part of E (7 is given by the second-order matrix elements of the form [0] E (7 sec = (E 0 H 0 H(5 nlog + [ + n ] (E 0 H 0 H(5, ( where H (5 nlog is the effective Hamiltonian responsible for the nonlogarithmic mα5 correction to the energy n H (5 nlog = 7 6 π r + 8Z 45 [ δ (r + δ (r ]. ( is the spin-independent part of the Breit-Pauli Hamiltonian given by Eq. (0, and H(5 is the mα 5 amm correction to, see Eq. (. The fourth part of E (7 is the contribution induced by the emission and reabsorption of virtual photons of low energy. It is denoted as E (7 L Bethe logarithm. The expression for E (7 L reads [6] E (7 L = π δ ( p + p (H 0 E 0 ln ( r + i Z π r + r r σ + σ ln and interpreted as the relativistic correction to the [ (H0 E 0 Z [ ] ( (H0 E 0 r Z ] ( p + p r + r r, ( where δ... denotes the first-order perturbation of the matrix element... by, implying perturbations of the reference-state wave function, the reference-state energy, and the electron Hamiltonian. 5. Results for helium fine-structure Summary of the individual contributions to the fine-structure intervals of helium is given in Table. Numerical results are presented for the large ν 0 and the small ν intervals, defined by ν 0 = [ E( P 0 E( P ] /h (4 ν = [ E( P E( P ] /h. (5 We note that the style of breaking the total result into separate entries used in Table differs from that used in the summary tables of the previous papers by Pachucki et al. [0, ]. In particular, the lower-order terms listed in Table III of Ref. [0] and in Table II of Ref. [] contained contributions of higher orders, whereas in the present work the entries in Table contain only the contributions of the order specified. A term-by-term comparison with the independent calculation by Drake [8] was performed in Ref. []. We observe good agreement between the two calculations for the lower-order terms, 6

8 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 Table. Summary of individual contributions to the fine-structure intervals in helium, in khz. The parameters [5] are α = (94, cr = ( khz, and m/m = The label (+m/m indicates that the corresponding entry comprises both the non-recoil and recoil contributions of the specified order in α. Term ν 0 ν ν 0 mα 4 (+m/m mα 5 (+m/m mα ( mα 6 m/m mα 7 log(zα mα 7, nlog mα 8 ±.7 ±.7 Total theory ± ± ±.7 Experiment (70 a (5 d 908.5(0 f (0 b (5 a (94 a (9 c (0 e a Ref. [7], b Ref. [6], c Ref. [7], d Ref. [8], e Ref. [8], f Ref. [9]. namely, for the mα 4, mα 5, and mα 6 corrections. However, for the recoil correction to order mα 6, our results differ from those of Drake by about 0.5 khz for both intervals. The reason for this disagreement seems to be different for the large and the small intervals. For the large interval, the deviation is due to the recoil operator part, whereas for the small interval, it is mainly due to the mass polarization part (see discussion in Ref. []. Our present estimates of the uncalculated higher-order effects for helium are larger than those in the previous studies [7, 8]. The previous estimates were significantly less than khz. They were based on logarithmic contributions to order mα 8 corresponding to the hydrogen fine structure. However, a larger contribution might originate from the nonlogarithmic relativistic corrections. So our present estimate is obtained by multiplying the mα 6 contribution for the ν 0 = ν 0 + ν interval by the factor of (Zα, which yields a conservative estimate of ±.7 khz for all ν 0, ν, and ν 0 intervals. All nuclear structure effects are completely negligible at the current precision level. The finite nuclear size correction is estimated to yield 8 Hz for ν 0 and 6 Hz for ν. Our result for the ν 0 interval of helium agrees well with all recent experimental values [7, 6, 7]. For the ν interval, theoretical result is by about σ larger than the values obtained in Re. [7, 8] but in agreement with the latest measurement by Hessels and coworkers [8]. Our theoretical prediction for the ν 0 interval is in excellent agreement with the very recent measurement of Smiciklas and Shiner [9]. Comparison with this experimental result suggests that the higher-order contribution might in fact be much smaller than our conservative estimate. This means that, if an independent measurement on Li + or Be + confirms the smallness of the mα 8 terms, the helium determination of α will be significantly improved. The measurement should be performed for the P 0 P transition, since it is not affected by the singlet-triplet mixing effects, which strongly depend on Z. In summary, the theory of the fine structure of helium and light helium-like ions is now complete up to orders mα 7 and α 6 m /M. The theoretical predictions agree with the latest 7

9 Journal of Physics: Conference Series 64 ( doi:0.088/ /64//0007 experimental results for helium, as well as with most of the experimental data for light heliumlike ions. A combination of the theoretical and experimental results [9] for the P 0 P interval in helium yields an independent determination of the fine structure constant α α = (64(4(68, (6 where the first error is the experimental uncertainty, the second one is the numerical uncertainty, and the third comes from the estimate of the m α 8 term (±.7 khz. The result (6 is accurate to 7 ppb and in agreement with the recent value obtained from the electron g factor [9]. Acknowledgments Support by NIST through Precision Measurement Grant PMG 60NANB7D65 is gratefully acknowledged. References [] Schwartz C 964 Phys. Rev. 4 A8 [] Douglas M and Kroll N 974 Ann. Phys. (NY 8 89 [] Hambro L 97 Phys. Rev. A 5 07 [4] Hambro L 97 Phys. Rev. A [5] Hambro L 97 Phys. Rev. A [6] Lewis M L and Serafino P H 978 Phys. Rev. A [7] Zelevinsky T, Farkas D and Gabrielse G 005 Phys. Rev. Lett [8] Borbely J S, George M C, Lombardi L D, Weel M, Fitzakerley D W and Hessels E A 009 Phys. Rev. A (R [9] Smiciklas M and Shiner D 00 Phys. Rev. Lett [0] Yan Z -C and Drake G W F 995 Phys. Rev. Lett [] Zhang T 996 Phys. Rev. A 54 5 [] Zhang T 996 Phys. Rev. A [] Zhang T, Yan Z -C and Drake G W F 996 Phys. Rev. Lett [4] Zhang T 997 Phys. Rev. A [5] Pachucki K 999 J. Phys. B 7 [6] Pachucki K and Sapirstein J 000 J. Phys. B 597 [7] Pachucki K and Sapirstein J 00 J. Phys. B 5 78 [8] Drake G W F 00 Can. J. Phys [9] Pachucki K and Sapirstein J 00 J. Phys. B 6 80 [0] Pachucki K 006 Phys. Rev. Lett [] Pachucki K and Yerokhin V A 009 Phys. Rev. A ; ibid (E; ibid (E [] Pachucki K and Yerokhin V A 00 Phys. Rev. Lett [] Pachucki K and Yerokhin V A 00 Can. J. Phys. in print [4] Aoyama T, Hayakawa M, Kinoshita T and Nio M 007 Phys. Rev. Lett [5] Mohr P J, Taylor B N and Newell D B 008 Rev. Mod. Phys [6] Giusfredi G, Pastor P C, Natale P D, Mazzotti D, de Mauro C, Fallani L, Hagel G, Krachmalnicoff V and Inguscio M 005 Can. J. Phys. 8 0 [7] George M C, Lombardi L D and Hessels E A 00 Phys. Rev. Lett [8] Castillega J, Livingston D, Sanders A and Shiner D 000 Phys. Rev. Lett [9] Hanneke D, Fogwell S and Gabrielse G 008 Phys. Rev. Lett