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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
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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 Z 6] and for the two-loop Lamb shift [of order 2 Z 6] 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 nS E 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 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 Z 4 2 10 4 E lnk Z 2 ln Z 2 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 h d r i; (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 E ln 2jH Ej= Z 2 pii is As is well known, the leading one-loop energy shifts 0 2 Z 4 (due to self-energy and vacuum polarization) in hydrogen- the Bethe logarithm, and d r r~ 2V= 4 is a like systems are of order Z 4 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 Z 6 are extremely demanding. For non-S states, 2 d d 1 d=2 V r Z r 2 1 . All matrix elements the Z 6 corrections have been obtained recently [4]. h i 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 Z 4 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 Z 6 correction, for all non-S hydrogenic relativistic correction to the Hamiltonian is 3 states and the normalized difference n n E nS E 1S , including the nonlogarithmic term. Together p~ 4 1 H Z d 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. Z 4 nA60 nS nA60 nS 1 E fA Z 2 A ln Z 2 A g; n3 40 61 60 1 30:924 149 46 1 5 31:455 393 1 2 31:840 465 09 1 6 31:375 130 1 where the indices of the coefficients indicate the power of 3 31:702 501 1 7 31:313 224 1 Z and the power of the logarithm, respectively. We work 4 31:561 922 1 8 31:264 257 1 in D 4 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 Z 6 1 5 Z 6 d E L Z F 2 n~ ij ninj L1 n3 1 3 2" 6 2 n3 4 1 4 i 2 2 jH Ej r~ V ijpir~ Vpj 2H G r~ V ; (3) pi n~ r~ 2 H E 3 ln pj 8 4 R Z 2 1 2 0 i 3 jH Ej j where L Z ln 2 Z , and G 1= E H is the re- p n~ r~ H E ln p n~ r~ : Z 2 duced Green function; 1 is a generalized Bethe logarithm, Z 6 4 jH Ej Here, n~ is a three-dimensional unit vector, and we integrate i i 2 3 1 HRGp H 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 h jp jnihnjHRjmihmjp j i immediately following the operator, i.e., hr~ 2Vp~ 2i 3 Em En n;m ~ 2 2 ~ 2 ~ 2 h r V p~ i, hr VGHRi h r V GHRi, etc. jEn Ej The correction E to the transition current reads E En E ln Em E L3 L3 Z 2 , where Z 6 = n3 contains the gen- D3 F 3 F 3 3 jE Ej 2 eralized Bethe logarithm 3, and ln m hH i Z 2 3 R 2 10 4 r~ 2Vp~ 2 r~ V 2 jH Ej D3 L Z ; pi 1 ln pi : (4) 3" 9 3 4 2 Z 2 2 jH Ej F ji H E ln pi : We temporarily restore the reference state in the notation 3 3 Z 2 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 ln jH 2 2 divergences (in ") in the corrections to the Bethe logarithm Ej= Z , not ln 2jH 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~ V 2 1 103 1 V r~ V ijriVpj; (5) D 2L Z 6 4 2 3 " 60 which in leading order gives rise to the correction h 1 Vi. 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. The sum is 1 1 E h 1 Vi 2h 1 VGH i r~ 4V 2i ijpir~ 2Vpj p~ 2;r~ 2V 2 ijriVpj M R " 192 48 32 11 1 11 1 r~ 4V h r~ V 2i: (6) 240 40" 48 3"
1 The complete one-loop result E EL0 EL1 EL2 EL3 EM reads Z 4 10 4 4 Z 6 1 E ln Z 2 lnk h ijriVpji n3 9 3 l0 3 0 4 n3 1 2 3 5 2 1 779 11 L Z hr~ 2VGH i h ijriVpjGH i L Z hr~ 4Vi 9 3 R 2 R 14 400 120 23 1 589 2 3 1 L Z h2i ijpir~ 2Vpji L Z h r~ V 2i hp~ 2r~ 2Vi hp~ 2 ijriVpji : (7) 576 24 720 3 80 8 180404-2 PHYSICAL REVIEW LETTERS week ending PRL 95, 180404 (2005) 28 OCTOBER 2005
The matrixP elements in this result can be evaluated using standard techniques. In terms of the notation of Ref. [4], we have 3 L i 1 i. Our general result (7), evaluated for hydrogenic states, reproduces the known logarithmic term A61, and is consistent with all formulas reported for the nonlogarithmic term in Eqs. (10) and (12) of Ref. [4]. The evaluation of L is a demanding numerical calculation, and numerical values for non-S states have been presented in Table I of Ref. [4]. Taking advantage of the result [10] for 1S and the validity of Eq. (7) for the nS 1S difference, we can now proceed to indicate results for the nonlogarithmic term A60 for nS states, an evaluation made possible by our generalized NRQED approach (see Table I). A generalization of our NRQED approach leads to the following general result for the 2 Z 6 term of the complete two-loop Lamb shift (including all diagrams with closed fermion loops, see Fig. 1),
2 Z 6 2 E fB ln2 Z 2 B ln Z 2 B g 2n3 62 61 60 2 Z 6 38 4 2 42923 9 5 9 19 b L Z N 2 ln 2 2 3 L Z 2n3 L 4 5 45 3 259200 16 36 64 135 1 2 2179 9 5 9 2 197 3 L2 Z hr~ 2VG r~ 2Vi 2 ln 2 2 3 hr~ 2VG p~ 4i 2 9 10368 16 36 64 1152 8 1 3 2 233 3 1 3 ln 2 2 3 hp~ 4G ijriVpji 2 ln 2 2 3 h ijriVpjG ijriVpji 16 32 576 4 8 16 2 197 3 1 3 2 83 17 2 ln 2 2 3 hfp~ 2;r~ 2V 2 ijriVpjgi 2 ln 2 2304 16 32 64 1152 8 59 17 2 87697 9 2167 9 19 1 2 3 h r~ V 2i 2 ln 2 2 3 L Z L2 Z 72 32 345600 10 9600 40 270 18 2 16841 1 223 1 1 hr~ 4Vi 2 ln 2 2 3 L Z h2i ijpir~ 2Vpji: (8) 207360 5 2880 20 24
2 4 The leading Z term, given by the B40 coefficient, is hydrogenic states. The logarithmic sum 4 is given by 1 ij i j well known and therefore not included here (for a review Eq. (4), with the replacement HR ! 4 r Vp . Finally, see, e.g., Appendix A of Ref. [3]). The above expression is we have valid for the P, D states, and for the normalized difference Z 6 1 jH Ej of S states. The quantity N is defined in terms of the ijrj H E ln pi : (9) n 3 5 2 Z 2 notation adopted in Refs. [6,13], and the two-loop Bethe n logarithm bL is defined in Refs. [7,14]. Although bL has Evaluating the general expression (8) for P states, we 4 n2 1 been determined numerically only for S states (see confirm that B62 nP 27 n2 . Furthermore, we obtain Ref. [14]), it represents a well-defined quantity for all the results 4 n2 1 166 8 ln2 4 n2 1 31 8 ln2 B nP N nP ;B nP N nP : (10) 61 1=2 3 n2 405 27 61 3=2 3 n2 405 27
Numerical values for N nP can be found in Eq. (17) of [13]. Regarding the nonlogarithmic term B60, we fully con- firm results for the fine-structure difference of P states [15]. A further important conclusion to be drawn from Eq. (8) is that all logarithmic two-loop terms of order 2 Z 6 vanish for states with orbital angular momentum l 2. We have also verified that the two-loop result (8) is consistent with the normalized S-state difference n for the logarithmic terms B62 and B61, as derived in Ref. [6] (using a completely different method). Evaluating all matrix elements in Eq. (8), we are now in the position to obtain the n dependence of the nonlogarithmic term, which we write as B60 nS B60 1S bL nS bL 1S A n , where A n is the additional contribution beyond the n dependence of the two-loop Bethe logarithm. The result for A n is 38 4 337043 94261 902609 4 16 4 76 304 76 A n ln 2 N nS N 1S ln2 2 45 3 129600 21600n n2 3 9n n2 45 135n n2 129600 9 135 53 35 419 28003 11 31397 53 35 419 ln 2 2 ln 2 2 3 15 2n 2 10800 2n 2 60 8n 2 30n 10800n 120n 37793 16 304 13 ln2 2 ln 2 8 2 ln 2 2 2 3 n ln n : (11) 10800 9 135 3
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ized difference n of the nS 1S Lamb shift. Elucidating discussions regarding the latter point can be found near Eqs. (2) and (3) of Ref. [16], and in Appendix A of Ref. [3]. Accurate theoretical values for n can be inferred from the results reported here and are compiled in Table II. The Rydberg constant is currently known to a relative accuracy of 6:6 10 12, limited essentially by the experimental accuracy of the 2S 8D and 2S 12D measurements (see Table V of [3]). Using the improved theory as pre- sented in this Letter, it will become possible to determine the Rydberg constant to an accuracy on the level of 10 14, provided the ongoing experiments concerning the hydro- FIG. 1. Two-loop Feynman diagrams for the Lamb shift. The gen 1S 3S transition [17,18] reach a sub-kHz level of bound-electron propagator is denoted by a double line. accuracy. The authors acknowledge helpful conversations with P.J. Mohr and R. Bonciani. This work was supported by Numerically, A n is found to be much smaller than EU grant No. HPRI-CT-2001-50034. A. C. acknowledges bL nS bL 1S , which implies that the main contribution support by the Natural Sciences and Engineering Research to B60 nS B60 1S is exclusively due to the two-loop Council of Canada. U. D. J. acknowledges support from the Bethe logarithm. As an example, we consider A 5 Deutsche Forschungsgemeinschaft (Heisenberg program). 0:370 042 and B60 5S B60 1S 21:2 1:1 , where the error is due to the numerical uncertainty of the two-loop Bethe logarithm bL 5S (see Ref. [14]). The test of standard model theories and the determina- [1] M. Fischer et al., Phys. Rev. Lett. 92, 230802 (2004). tion of fundamental constants (specifically, of the Rydberg [2] B. de Beauvoir et al., Phys. Rev. Lett. 78, 440 (1997). constant and of the electron mass) provide the main moti- [3] P.J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 (2005). vations for carrying out the QED calculations in ever [4] U. D. Jentschura, E.-O. Le Bigot, P.J. Mohr, P. Indelicato, higher orders of approximation. Recently, our knowledge and G. Soff, Phys. Rev. Lett. 90, 163001 (2003). [5] T. Appelquist and S. J. Brodsky, Phys. Rev. Lett. 24, 562 of the ground-state Lamb shift has been improved by a (1970); Phys. Rev. A 2, 2293 (1970); R. Barbieri, J. A. fully numerical calculation of the two-loop self-energy [8]. Mignaco, and E. Remiddi, Lett. Nuovo Cimento 3, 588 However, because of the structure of the hydrogen spec- (1970); B. E. Lautrup, A. Peterman, and E. de Rafael, trum, the decisive quantity for the determination of the Phys. Lett. B 31, 577 (1970). Rydberg constant from spectroscopic data is the normal- [6] K. Pachucki, Phys. Rev. A 63, 042503 (2001). [7] K. Pachucki and U. D. Jentschura, Phys. Rev. Lett. 91, 113005 (2003). [8] V.A. Yerokhin, P. Indelicato, and V.M. Shabaev, Phys. TABLE II. Theoretical values of the normalized Lamb-shift Rev. A 71, R040101 (2005). 3 difference n n E nS E 1S , based on the results re- [9] H. A. Bethe, Phys. Rev. 72, 339 (1947). ported in this Letter [see Eq. (11)]. Units are kHz. [10] K. Pachucki, Ann. Phys. (N.Y.) 226, 1 (1993). [11] U. Jentschura and K. Pachucki, Phys. Rev. A 54, 1853 n n n n (1996). 2 187 225.70(5) 12 279 988.60(10) [12] C. Itzykson and J. B. Zuber, Quantum Field Theory 3 235 070.90(7) 13 280 529.77(10) (McGraw-Hill, New York, 1980). 4 254 419.32(8) 14 280 962.77(10) [13] U. D. Jentschura, J. Phys. A 36, L229 (2003). 5 264 154.03(9) 15 281 314.61(10) [14] U. D. Jentschura, Phys. Rev. A 70, 052108 (2004). 6 269 738.49(9) 16 281 604.34(11) [15] U. D. Jentschura and K. Pachucki, J. Phys. A 35, 1927 7 273 237.83(9) 17 281 845.77(11) (2002). 8 275 574.90(10) 18 282 049.05(11) [16] Th. Udem, A. Huber, B. Gross, J. Reichert, M. Prevedelli, 9 277 212.89(10) 19 282 221.81(11) M. Weitz, and T. W. Ha¨nsch, Phys. Rev. Lett. 79, 2646 10 278 405.21(10) 20 282 369.85(11) (1997). 11 279 300.01(10) 21 282 497.67(11) [17] Th. Udem (private communication). [18] O. Arnoult (private communication).
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