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And Two-Loop Lamb Shift for Arbitrary Excited Hydrogenic States Missouri University of Science and Technology Scholars' Mine Physics Faculty Research & Creative Works Physics 01 Oct 2005 Calculation of the One- and Two-loop Lamb Shift for Arbitrary Excited Hydrogenic States Andrzej Czarnecki Ulrich D. Jentschura Missouri University of Science and Technology, [email protected] Krzysztof Pachucki Follow this and additional works at: https://scholarsmine.mst.edu/phys_facwork Part of the Physics Commons Recommended Citation A. Czarnecki et al., "Calculation of the One- and Two-loop Lamb Shift for Arbitrary Excited Hydrogenic States," Physical Review Letters, vol. 95, no. 18, pp. 180404-1-180404-4, American Physical Society (APS), Oct 2005. The definitive version is available at https://doi.org/10.1103/PhysRevLett.95.180404 This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Physics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. PHYSICAL REVIEW LETTERS week ending PRL 95, 180404 (2005) 28 OCTOBER 2005 Calculation of the One- and Two-Loop Lamb Shift for Arbitrary Excited Hydrogenic States Andrzej Czarnecki,1 Ulrich D. Jentschura,2 and Krzysztof Pachucki3 1Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2J1 2Max-Planck-Institut fu¨r Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 3Institute of Theoretical Physics, Warsaw University, Hoz˙a 69, 00-681 Warsaw, Poland (Received 8 August 2005; published 27 October 2005; corrected 28 October 2005) General expressions for quantum electrodynamic corrections to the one-loop self-energy [of order Z6] and for the two-loop Lamb shift [of order 2 Z6] are derived. The latter includes all diagrams with closed fermion loops. The general results are valid for arbitrary excited non-S hydrogenic states and 3 for the normalized Lamb shift difference of S states, defined as n n E nSÿE 1S. We present numerical results for one-loop and two-loop corrections for excited S, P, and D states. In particular, the normalized Lamb shift difference of S states is calculated with an uncertainty of order 0.1 kHz. DOI: 10.1103/PhysRevLett.95.180404 PACS numbers: 12.20.Ds, 06.20.Jr, 31.15.2p, 31.30.Jv The theory of quantum electrodynamics, when applied of space is d 3 ÿ 2. Units are chosen so that @ c to the hydrogen atom and combined with accurate mea- 0 1, and the electron mass is unity. A nonrelativistic, surements [1,2], leads to the most accurately determined ‘‘Bethe-style’’ [9] calculation of the contribution due to physical constants today [3] and to accurate predictions for ultrasoft photons, in the dipole approximation, leads to a transition frequencies. Of crucial importance are higher- dimensionally regularized energy shift EL0, order corrections to the bound-state energies, which in- volve both purely relativistic atomic-physics effects and 4 Z4 2 10 4 E ÿ lnk Z2 ln Zÿ2 are mixed with the quantum electrodynamic (QED) cor- L0 3 n3 0 3" 9 3 rections. In general, this leads to a double expansion for the hd ri; (1) energy shifts, both in terms of the QED coupling (the fine-structure constant) and the nuclear charge number Z. where lnk n3 hpi H ÿ E ln2jH ÿ Ej= Z2pii is As is well known, the leading one-loop energy shifts 0 2 Z4 (due to self-energy and vacuum polarization) in hydrogen- the Bethe logarithm, and d rr~ 2V= 4 is a like systems are of order Z4 in units of the electron d-dimensional Dirac delta function obtained via the action mass. Analytic calculations for higher excited states in the of the Laplacian on the d-dimensional Coulomb potential order Z6 are extremely demanding. For non-S states, 2ÿd d 1ÿd=2 V rÿZr ÿ 2 ÿ 1 . All matrix elements the Z6 corrections have been obtained recently [4]. hi are to be evaluated with regard to the reference state, However, excited S states are very important for spectros- as given by a nonrelativistic (Schro¨dinger-Pauli) wave copy, and the corresponding gap in our knowledge is filled function, and the summation convention is used throughout in the current Letter (see Table I). Regarding the two-loop this Letter. correction, complete results for the 2 Z4 effect were Following [4,10,11], we now consider corrections due to obtained in 1970 (see Ref. [5]). Here, we derive general the relativistic Hamiltonian, the quadrupole term, and the expressions which allow the determination of the entire relativistic and retardation corrections to the current. The two-loop 2 Z6 correction, for all non-S hydrogenic relativistic correction to the Hamiltonian is 3 states and the normalized difference n n E nSÿ E 1S, including the nonlogarithmic term. Together p~ 4 1 H ÿ Zd r ijriVpj: (2) with other available analytic [6,7] and numerical calcula- R 8 2 4 tions for the 1S state [8], our results allow for a much improved understanding of the higher-order two-loop cor- ij 1 i j Here, 2i ; . The resulting, dimensionally regu- rections for general excited hydrogenic states, and pave the larized, correction to the Bethe logarithm is way for an improved determination of fundamental con- stants from hydrogen spectroscopy. The one-loop bound-state self-energy, for the states TABLE I. Values of the nonlogarithmic self-energy correction under investigation here, can be written as A60 (‘‘relativistic Bethe logarithm’’) for higher excited S states. Z4 nA60 nS nA60 nS 1E fA Z2A ln Zÿ2A g; n3 40 61 60 1 ÿ30:924 149 46 1 5 ÿ31:455 393 1 2 ÿ31:840 465 09 1 6 ÿ31:375 130 1 where the indices of the coefficients indicate the power of 3 ÿ31:702 501 1 7 ÿ31:313 224 1 Z and the power of the logarithm, respectively. We work 4 ÿ31:561 922 1 8 ÿ31:264 257 1 in D 4 ÿ 2 spacetime dimensions, and the dimension 0031-9007=05=95(18)=180404(4)$23.00 180404-1 © 2005 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 95, 180404 (2005) 28 OCTOBER 2005 Z Z6 1 5 Z6 d E L Z F 2 n~ ij ÿ ninj L1 n3 1 3 2" 6 2 n3 4 1 4 i 2 2 jH ÿ Ej r~ V ijpir~ Vpj 2H Gr~ V ; (3) pi n~ r~2 H ÿ E3 ln pj 8 4 R Z2 1 ÿ2 0 i 3 jH ÿ Ej j where L Zln2 Z , and G 1= EÿH is the re- ÿ p n~ r~ H ÿ E ln p n~ r~ : Z2 duced Green function; 1 is a generalized Bethe logarithm, Z6 4 jH ÿ Ej Here, n~ is a three-dimensional unit vector, and we integrate i i 2 3 1 ÿ HRGp H ÿ E ln 2 p ~ n 3 Z over the entire solid angle n~ . Throughout this Letter, r 4 X i i and r~ are understood to exclusively act on the quantity 2 hjp jnihnjHRjmihmjp ji immediately following the operator, i.e., hr~ 2Vp~ 2i 3 Em ÿ En n;m ~ 2 2 ~ 2 ~ 2 hr Vp~ i, hr VGHRih r VGHRi, etc. jEn ÿ Ej The correction E to the transition current reads E En ÿ E ln ÿ Em ÿ E L3 L3 Z2 , where Z6 =n3 contains the gen- D3 F 3 F 3 3 jE ÿ Ej 2 eralized Bethe logarithm 3, and ln m hH i Z2 3 R 2 10 4 r~ 2Vp~ 2 r~ V2 jH ÿ Ej D3 ÿ L Z ; pi 1 ln pi : (4) 3" 9 3 4 2 Z2 2 jH ÿ Ej F ji H ÿ E ln pi : We temporarily restore the reference state in the notation 3 3 Z2 of the matrix element, and the sums over n and m include i i 2 1 ijrj ri i both the discrete as well as the continuous part of the Here, j p p~ 2 V, and @=@r denotes the derivative with respect to the ith Cartesian coordinate. The spectrum. The argument of the logarithm in 1 is lnjH ÿ 2 2 divergences (in ") in the corrections to the Bethe logarithm Ej= Z , not ln2jH ÿ Ej= Z as in lnk0, and this fact are compensated by high-energy virtual photons, which in is important for the precise definition of 1, and of all other generalized Bethe logarithms in the following. nonrelativistic QED (NRQED) are given by effective op- In the dimensional scheme, the quadrupole correction erators. From a generalized Dirac equation (see Chap. 7 of Ref. [12]), one easily obtains the effective one-loop poten- EL2, which was denoted as Fnq in former work [10,11], is tial found to be expressible as EL2 D2 F 2, where 1 2 2 r~ V2 1 103 1V ÿ r~ V ijriVpj; (5) D 2L Z 6 4 2 3 " 60 which in leading order gives rise to the correction h 1Vi. r~ 4V 1 12 2L Z This correction is a contribution to the middle-energy part 40 " 5 E , which originates from high-energy virtual photons, M r~ 2Vp~ 2 1 34 with electron momenta of order Z. The corrections of 2L Z ; 2 1 6 " 15 relative order Z to h Vi involve relativistic correc- tions to the wave function and to the operators, and a two- and F 2 contains the generalized Bethe logarithm 2, Coulomb-vertex scattering amplitude.
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