Theoretical transition frequencies beyond 0.1 ppb accuracy in H + 2 , HD + , and antiprotonic helium
TTheoretical transition frequencies beyond 0.1 ppb accuracy in H +2 , HD + , andantiprotonic helium Vladimir I. Korobov, Laurent Hilico,
2, 3 and Jean-Philippe Karr
2, 3 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia Laboratoire Kastler Brossel, UPMC-Paris 6, ENS,CNRS; Case 74, 4 place Jussieu, 75005 Paris, France Universit´e d’Evry-Val d’Essonne, Boulevard Fran¸cois Mitterrand, 91025 Evry Cedex, France
We present improved theoretical calculations of transition frequencies for the fundamental tran-sitions ( L = 0 , v = 1) → ( L (cid:48) = 0 , v (cid:48) = 0) in the hydrogen molecular ions H +2 and HD + with a relativeuncertainty 4 · − and for the two-photon transitions in the antiprotonic helium atom with a rela-tive uncertainty 10 − . To do that, the one-loop self-energy correction of order α ( Zα ) is derived inthe two Coulomb center approximation, and numerically evaluated in the case of the aforementionedtransitions. The final results also include a complete set of other spin-independent corrections oforder mα . The leading order corrections of α ln ( Zα ) − ( Zα ) are also considered that allows toestimate a magnitude of yet uncalculated contributions. PACS numbers: 31.15.A-, 31.30.jf, 31.15.xt
I. INTRODUCTION
The few-body bound-state quantum electrodynamicsis a challenging problem. So far, a complete set of con-tributions up to order mα has been obtained and cal-culated for the two-electron helium-like atoms [1], one-electron molecular ions [2], and antiprotonic helium [3].A contribution of mα order including as well the non-logarithmic part, has been obtained for the fine struc-ture of helium 2 P level in [4]. Four-particle systemswere addressed in [5] (H and its isotopologues) and in[6] (lithium-like atoms). Recently, QED calculations upto mα , and partially mα orders were carried out forthe beryllium atom with four electrons [7].Progress in high-precision spectroscopy of three-bodymolecular or molecule-like systems has opened new pos-sibilities for metrology of nucleus-to-electron mass ra-tios [8]. One-photon ro-vibrational transitions were ob-served in HD + molecular ion with a relative uncertaintyof 1-2 ppb [9, 10]. Spectroscopy of two-photon transitionsin antiprotonic helium at the 2-5 ppb level yielded a newvalue of the antiproton-to-electron mass ratio [11]. Theseexperiments, as well as others [12, 13], are currently beingdeveloped towards higher precision, which motivates theevaluation of higher-order corrections in these systems.The importance of the m p /m e problem is supported byrecent experiments [14] with rubidium atoms, which al-low to deduce a new value of the fine structure constant, α = e / (¯ hc ), with a relative uncertainty 6 . × − . Fur-ther improvement may be hindered by the present limitson the proton-to-electron mass ratio, which is determinedby the latest CODATA adjustment [15] with a relativeuncertainty 4 . × − (see also [16]).Theoretical calculation of the complete set of QED corrections up to order mα has brought the theoreti-cal uncertainty down to 0.3-0.4 ppb in H +2 or HD + [2],and about 1 ppb in antiprotonic helium [3]. Very accu-rate leading order relativistic corrections are also avail-able from [17]. In the present work, we compute thecomplete set of mα order corrections including the one-loop self energy contribution, which represents the mainsource of theoretical uncertainty. This allows us to im-prove the accuracy by about one order of magnitude, thusmaking real the possibility of improving the knowledgeof nucleus-to-electron mass ratios.This paper is organized as follows: In Sec. II theone-loop self-energy contribution at mα order for thehydrogen-like atoms is considered and a general formulafor an arbitrary ( n, l ) state, derived from comparison of[18, 19] and [20, 21] results, is presented. In Sec. III thelow energy part is reconsidered to reformulate the resultof Sec. II in a form which is then suitable to be extendedto the Coulomb two-center problem (Sec. IV A). A list ofother contributions in mα and mα orders, which werealso taken into account in the final results, are consideredin Sec. IV B. Then examples of numerical calculations forthe hydrogen isotope ions as well as for the antiprotonichelium are given in Sec. V. II. THE ONE-LOOP SELF-ENERGYCONTRIBUTION AT ORDER mα .HYDROGEN-LIKE CASE. As a starting point of our consideration we take thegeneral result of Refs. [18, 19] for a bound electron in afield of external Coulomb potential, V ( r ) = − Z/r , writ-ten in the natural relativistic units (¯ h = c = m = 1): a r X i v : . [ phy s i c s . a t o m - ph ] F e b ∆ E (7)se = απ (cid:40) ( Zα ) L H + (cid:18)
59 + 23 ln (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:10) πρ Q ( E − H ) − Q H B (cid:11) + 2 (cid:10) H so Q ( E − H ) − Q H B (cid:11) + (cid:18) (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:10) ∇ V (cid:11) + (cid:18) (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:10) σ ij p i ∇ V p j (cid:11) + (cid:18) (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:68) ( ∇ V ) (cid:69) + 380 (cid:10) πρ p (cid:11) − (cid:10) p H so (cid:11)(cid:41) , (1)where H B = − p π ρ + H so , H so = 14 σ ij ∇ i V p j , πρ = ∆ V, H δso = 2i σ ij p i ( ∇ V ) p j . and σ ij = [ σ i σ j ] / (2 i ) = (cid:15) ijk σ k . Brackets denote averag-ing on the nonrelativistic bound state wave function ψ , E and H = p / V are, respectively, the nonrelativisticenergy of the state and the nonrelativistic Hamiltonian.Here Q is a projector operator on the subspace orthogo-nal to ψ . L H is the low-energy photon contribution orthe relativistic Bethe logarithm for the hydrogen atomstate (assuming Z = 1). The above result is valid forall states with nonzero angular momentum and for thenormalized difference of S states ,∆ n = n ∆ E ( nS ) − ∆ E (1 s ) . A more general expression which would also be validfor individual S states, will differ from (1) only by aterm proportional to the delta function, δ ( r ). To get theunknown contribution for the hydrogen case, we haveto compare with the result of [20, 21] obtained for the1 S state of hydrogen. For this purpose, the expectationvalues in (1) which are divergent for individual S statesshould first be regularized; we will use a regularization bycut-off of a small r spherical domain around the nucleus.Any two such regularizations differ by a term propor-tional to the delta function, so that the result will stilldiffer from an expression valid for all states by a delta-function term.To that end let us introduce two functionals Q and RQ = lim r → (cid:40)(cid:28) πr (cid:29) r + (ln r +ln α + γ E ) (cid:104) δ ( r ) (cid:105) (cid:41) = − ( Zα ) πn (cid:20) −
12 ln Z − + ψ ( n ) − ψ (1) − ln n −
12 + 12 n (cid:21) , (2) R = lim r → (cid:40)(cid:28) πr (cid:29) r − (cid:20) r (cid:104) δ ( r ) (cid:105) + (ln r +ln α + γ E ) (cid:104) δ (cid:48) ( r ) (cid:105) (cid:21)(cid:41) = 2( Zα ) πn (cid:20) −
12 ln Z − + ψ ( n ) − ψ (1) − ln n −
53 + 12 n + 16 n (cid:21) . (3)where (cid:104) φ | δ (cid:48) ( r ) | φ (cid:105) = (cid:68) φ (cid:12)(cid:12)(cid:12) r r ∇ δ ( r ) (cid:12)(cid:12)(cid:12) φ (cid:69) = − (cid:104) ∂ r φ | δ ( r ) | φ (cid:105) − (cid:104) φ | δ ( r ) | ∂ r φ (cid:105) , (cid:104) (cid:105) r denotes integration outside a sphere of radius r . The last line in Eqs. (2-3) contains an expectation value of Q (or R ) for nS states of hydrogen-like atoms. Using these expressions all divergent matrix elements appearing inEq. (1) may be redefined in a finite form: (cid:10) πρ p (cid:11) fin = 8 π ( Zα ) R + 16 π ( Zα ) Q + 4 E (cid:10) V (cid:11) − (cid:10) p V p (cid:11) + 2 E (cid:104) πρ (cid:105) (4a) (cid:10)(cid:2) ∇ V (cid:3)(cid:11) fin = − π ( Zα ) R − π ( Zα ) Q − E (cid:10) V (cid:11) + 4 (cid:10) p V p (cid:11) + 2 (cid:104) p (4 πρ ) p (cid:105) − E (cid:104) πρ (cid:105) (4b) (cid:10) πρ Q ( E − H ) − Q H B (cid:11) fin = (cid:68) H (cid:48) (1) Q ( E − H ) − Q H (cid:48) (2) (cid:69) + 14 (cid:104) π ( Zα ) R + 16 π ( Zα ) Q + 8 E (cid:10) V (cid:11) − E (cid:104) V (cid:105) + (cid:68) H (1) (cid:69) (cid:104) V (cid:105) − (cid:68) H (2) (cid:69) (cid:104) V (cid:105) (cid:105) (4c)In the last expression H (1) = 4 πρ , H (2) = H B , which aretransformed [23, 24] as (cid:40) H (cid:48) (1) = − ( E − H ) U − U ( E − H ) + H (1) H (cid:48) (2) = − ( E − H ) U − U ( E − H ) + H (2) to eliminate the divergent part from the second orderterm, here U = 2 V and U = − V .Thus obtained expression should be compared with thecomplete result for a 1 s state [20, 21]:∆ E (7)se (1 S ) = α ( Zα ) π (cid:26) − ln (cid:2) ( Zα ) − (cid:3) + (cid:20)
283 ln 2 − (cid:21) ln (cid:2) ( Zα ) − (cid:3) − . (cid:27) (5)which yields (using L (1 S ) = − . E (7)se = απ (cid:40) ( Zα ) L H + (cid:18)
59 + 23 ln (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:10) πρ Q ( E − H ) − Q H B (cid:11) fin + 2 (cid:10) H so Q ( E − H ) − Q H B (cid:11) + (cid:18) (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:10) ∇ V (cid:11) fin + (cid:18) (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:10) H δso (cid:11) + (cid:18) (cid:20)
12 ( Zα ) − (cid:21)(cid:19) (cid:68) ( ∇ V ) (cid:69) fin + 380 (cid:10) πρ p (cid:11) fin − (cid:10) p H so (cid:11) + (cid:20) − ln (cid:0) α − (cid:1) + (cid:18)
163 ln 2 − (cid:19) ln (cid:0) α − (cid:1) + ln Z − + (cid:18)
103 ln 2 + 3715 (cid:19) ln Z − − . (cid:21) ( Zα ) (cid:104) πρ (cid:105) (cid:41) (6)for the hydrogen-like atom. III. THE LOW-ENERGY PART: REDEFININGTHE RELATIVISTIC BETHE LOGARITHM TOATOMIC UNITS
From this point and in what follows we will use atomicunits: m e = ¯ h = e = 1.In expressions (1) and (6), the relativistic Bethe loga-rithm L H is defined using the energy scale Z E h , whichis well suited for the hydrogenic case, but becomes irrel-evant for a system with two Coulomb centers of charges Z , Z . For this reason, we have to redefine the relativis-tic Bethe logarithm L ( Z, n, l ) in a.u.The low-energy part has been considered in more de-tails in [22]. Here we will try to elucidate only the keypoints of the derivation.The relativistic Bethe logarithm is determined in inte-gral form as follows L ( Z, n, l ) = 23 (cid:90) E h kdkP (1) α ( k )+ 23 (cid:90) ∞ E h kdkP (2) α ( k ) , (7)where E h is the Hartree energy.The integrand is a function of energy and is a sum of various contributions:a) relativistic corrections to the wave function P (1) rc ( k ) = 2 (cid:68) H B Q ( E − H ) − Q p ( E − H − k ) − p (cid:69) + (cid:68) p ( E − H − k ) − (cid:16) H B − (cid:104) H B (cid:105) (cid:17) ( E − H − k ) − p (cid:69) ; (8)b) modification of the vertex interactions P (2) rc ( k ) = (cid:28)(cid:18) − p p i − σ ij ∇ j V (cid:19) ( E − H − k ) − p i (cid:29) ; (9)c) nonrelativistic quadrupole contribution P nq ( k ) = 3 k π (cid:90) S d Ω n (cid:0) δ ij − n i n j (cid:1) (cid:26)(cid:68) p i ( n · r ) ( E − H − k ) − ( n · r ) p i (cid:69) − (cid:68) p i ( n · r ) ( E − H − k ) − p i (cid:69)(cid:27) , (10)where k = k n .The complete contribution is P α ( k ) = P (1) rc ( k ) + P (2) rc ( k ) + P nq ( k ). Its asymptotic expansion for large k may bewritten in operator form up to terms of O (1 /k ) (see Appendix for asymptotic expansion of separate contributions): P α ( k ) = − (cid:10) ∇ (cid:11) + 2 k (cid:10) ( H B −(cid:104) H B (cid:105) ) ( E − H ) − ∇ (cid:11) + 45 k (cid:10) ∇ (cid:11) − k (cid:10) ( ∇ V ) (cid:11) + √ k / πZ (cid:104) δ ( r ) (cid:105) − kk πZ (cid:104) δ ( r ) (cid:105) + 1 k (cid:18) (cid:19) πZ (cid:104) δ ( r ) (cid:105) + 1 k (cid:10) ( H B −(cid:104) H B (cid:105) ) ( E − H ) − ( ∇ V ) (cid:11) fin + 1 k (cid:10) ( ∇ V ) (cid:11) fin + 1180 k (cid:10) ( ∇ V ) (cid:11) fin + 116 k (cid:10) H δso (cid:11) + . . . (11)The finite expectation values are defined in a similarway as in the previous section, taking into account thatthe functionals Q and R should be accordingly modified: Q = lim r → (cid:40)(cid:28) πr (cid:29) r + (ln r + γ E ) (cid:104) δ ( r ) (cid:105) (cid:41) , (12) R = lim r → (cid:40)(cid:28) πr (cid:29) r − (cid:20) r (cid:104) δ ( r ) (cid:105) + (ln r + γ E ) (cid:104) δ (cid:48) ( r ) (cid:105) (cid:21)(cid:41) . (13)As is discussed in [18, 22] we have to subtract the lead-ing terms of expansion (11): P (1) α ( k ) = P α ( k ) − F α − A α k − B α k / (14a)and P (2) α ( k ) = P α ( k ) − F α − A α k − B α k / − C α ln kk − D α k . (14b) Constants F , A , B , C , and D are taken by evaluating ex-pectation values of operators appearing in the expansion(11) for the nonrelativistic wave function of a particularstate.The previous definition of the relativistic Bethe loga-rithm L H assumes scaling to ( Zα ) = 1, and thus it maybe expressed in atomic units as L H ( n, l ) = Z − (cid:104) (cid:90) Z E h kdkP (1) α ( k )+ 23 (cid:90) ∞ Z E h kdkP (2) α ( k ) (cid:105) . (15)Comparing (15) with (7) one gets a relation betweenthe two definitions of the relativistic Bethe logarithm: L ( Z, n, l ) = Z L H ( n, l ) + 23 (cid:90) Z E h E h dk (cid:40)(cid:20) kk − k (cid:18) (cid:19)(cid:21) πZ (cid:104) δ ( r ) (cid:105)− k (cid:20)(cid:10) ( H B −(cid:104) H B (cid:105) ) ( E − H ) − ( ∇ V ) (cid:11) fin + (cid:10) ( ∇ V ) (cid:11) fin + 1180 (cid:10) ( ∇ V ) (cid:11) fin + 116 (cid:10) H δso (cid:11)(cid:21)(cid:41) = Z L H ( n, l ) + Z n (cid:20) ln Z − + ln Z − (cid:18)
103 ln 2+ 3715 (cid:19)(cid:21) + ln Z − (cid:20) (cid:10) ( H B −(cid:104) H B (cid:105) ) ( E − H ) − ( ∇ V ) (cid:11) fin + 23 (cid:10) ( ∇ V ) (cid:11) fin + 11120 (cid:10) ( ∇ V ) (cid:11) fin + 124 (cid:10) H δso (cid:11)(cid:21) (16)and now substituting this into expression (6) we immediately get the general expression for the one-loop self-energy A61(R) R ( i n a . u . ) A60(R) R ( i n a . u . ) FIG. 1: The coefficients A ( R ) and A ( R ) for the ground (1 sσ ) electronic state of the two-center problem ( Z = Z = 1, H +2 case) as a function of a bond length R . correction in the mα order in atomic units:∆ E (7)se = α π (cid:40) L ( Z, n, l ) + (cid:18)
59 + 23 ln (cid:20) α − (cid:21)(cid:19) (cid:10) πρ Q ( E − H ) − Q H B (cid:11) fin +2 (cid:10) H so Q ( E − H ) − Q H B (cid:11) + (cid:18) (cid:20) α − (cid:21)(cid:19) (cid:10) ∇ V (cid:11) fin + (cid:18) (cid:20) α − (cid:21)(cid:19) (cid:10) H δso (cid:11) + (cid:18) (cid:20) α − (cid:21)(cid:19) (cid:68) ( ∇ V ) (cid:69) fin + 380 (cid:10) πρ p (cid:11) fin − (cid:10) p H so (cid:11) + Z (cid:20) − ln (cid:2) α − (cid:3) + (cid:20)
163 ln 2 − (cid:21) ln (cid:2) α − (cid:3) − . (cid:21) (cid:104) πρ (cid:105) (cid:41) (17)This formula is quite general and may be extended to thecase of external electric field of two (or more) Coulombsources. One may check that the above expressionmatches the result of Erickson and Yennie for the log-arithmic term for an arbitrary nS state of the hydrogenatom [25]. IV. COULOMB TWO-CENTER PROBLEMA. One-loop self energy
For the case of the two-center Coulomb problem oneneeds to replace the delta-function distribution, Z (cid:104) πρ (cid:105) , in the last line of Eq. (17) by a distribution: V δ = π (cid:104) Z δ ( r ) + Z δ ( r ) (cid:105) . (18)To present our results we will adopt a similar notationas for hydrogen-like ions [26]:∆ E (7)se = α π (cid:104) V δ (cid:105) (cid:104) A ln [ α − ]+ A ln[ α − ]+ A (cid:105) , (19)where A = −
1; expressions for A and A coefficientsare obtained by comparison between Eqs. (17) and (19). A ( R ) = (cid:20) (cid:10) πρ Q ( E − H ) − Q H B (cid:11) fin + 11120 (cid:10) ∇ V (cid:11) fin + 23 (cid:68) ( ∇ V ) (cid:69) fin + (cid:18)
163 ln 2 − (cid:19) (cid:104) V δ (cid:105) (cid:21) / (cid:104) V δ (cid:105) A ( R ) = (cid:20)(cid:18) −
23 ln 2 (cid:19) (cid:10) πρ Q ( E − H ) − Q H B (cid:11) fin + (cid:18) − (cid:19) (cid:10) ∇ V (cid:11) fin + (cid:18) −
23 ln 2 (cid:19) (cid:68) ( ∇ V ) (cid:69) fin + 380 (cid:10) πρ p (cid:11) fin − . (cid:104) V δ (cid:105) + L ( R ) (cid:21) / (cid:104) V δ (cid:105) . (20) A61(R) R ( i n a . u . ) A60(R) R ( i n a . u . ) FIG. 2: The coefficients A ( R ) and A ( R ) for the ground (1 sσ ) electronic state of the two-center problem ( Z = 2, Z = − + ¯ p case) as a function of a bond length R .H +2 HD + ∆ E nr
65 687 511 . . E α . . E α − . − . E α − . − . E α . . E α . . E tot
65 688 323 . . v = 0 , L = 0) → ( v (cid:48) = 1 , L (cid:48) = 0) fundamental transition frequency of H +2 andHD + molecular ions (in MHz). Since we are interested in the spin-independent part oftransition frequency we have dropped out the terms fromEq. (17), which correspond to the spin-orbit interaction.They will be considered elsewhere.The coefficients A and A now may be calculatedby averaging of the ”effective” potentials over the vibra-tional wave function of a three-body state (see Sec. Vand Figs. 1–2). B. Other contributions
In addition to the one-loop self-energy correction, wecomputed several other contributions at orders mα and mα , which did not require extensive calculations. Usingthe results from [27] we see that most of the terms areproportional to | Ψ(0) | .To better identify the most relevant terms, we givenumerical values of all the correction terms to the funda-mental vibrational transition frequency ( v = 0 , L = 0) → ( v = 1 , L = 0) in H +2 (see Sec V for details on the nu-merical calculations). For comparison, the one-loop self energy term we have just obtained gives a contribution:∆ E (7) se ≈ ± . (21)The uncertainty here is primarily due to numerical inac-curacy in the calculated data for the relativistic Bethelogarithm [22]. The one-loop vacuum polarization:∆ E (7) vp = α π (cid:20) V ln( Zα ) − + V (cid:21) (cid:104) V δ (cid:105) ≈ . . (22)For S -states in the hydrogen atom these coefficients are V ( nS ) = − , [29] V ( nS ) = 415 (cid:20) − ψ ( n + 1) − ψ (1) − n − n + 128 n − ln n (cid:21) . [30, 31]The coefficient V does not depend on n , the logarithmiccontribution is thus proportional to the delta-function.To estimate the nonlogarithmic contribution in (22), weuse the approximate electronic wave function ψ e ( r e ) ≈ N [ ψ s ( r )+ ψ s ( r )], where ψ s is the ground state wavefunction of the hydrogen atom. The coefficient V forthe 1 S -state is equal to − . The Wichman-Kroll contribution [32]:∆ E (7) W K = α π W (cid:104) V δ (cid:105) ≈ − . . (23)Here W ( nS ) = − π . ∆ E nr . E α −
50 320 . E α . E α . E α − . E α − . E total . , → (34 ,
32) transition frequency of the He + ¯ p atom (in MHz). The complete two-loop contribution [33]:∆ E (7)2 loop = α π [ B ] (cid:10) Z δ ( r )+ Z δ ( r ) (cid:11) ≈ . . (24)Here B = − . The three-loop contribution is already negligible.For the hydrogen molecular ion fundamental transitionit gives [34–36]∆ E (7)3 loop = α π [0 . (cid:104) Z δ ( r )+ Z δ ( r ) (cid:105) ≈ −
60 Hz . (25)The above is the complete set of contributions at mα order in the nonrecoil limit.In the next order ( mα ) we evaluate only the lead-ing ln ( Zα ) − contribution. It represents the second or-der perturbation with two one-loop self-energy operators( mα ( Zα ) ) [37]:∆ E (8)2 loop = α π (cid:104) − (cid:105) ln ( Zα ) − (cid:104) V δ (cid:105) ≈ . (26)Using its value we determine the theoretical uncertaintyof yet uncalculated terms in the mα order and higher. V. NUMERICAL RESULTS
The numerical approach to the two-center problem hasbeen already described in [22, 28]; briefly, the followingexpansion for the electronic wave function is used:Ψ m ( r , r ) = e imϕ r | m | ∞ (cid:88) i =1 C i e − α i r − β i r , (27)where r is the distance from the electron to the z -axisand φ the azimuthal angle. For Z = Z the variationalwave function should be symmetrizedΨ m ( r , r ) = e imϕ r | m | ∞ (cid:88) i =1 C i (cid:16) e − α i r − β i r ± e − β i r − α i r (cid:17) , (28) where (+) is used to get a gerade electronic state and ( − )is for an ungerade state, respectively. Parameters α i and β i are generated in a quasi-random manner.We calculated mean values for all operators appearingin Eq. (20) for the ground (1 sσ ) electronic state of thetwo-center problem, both for Z = Z = 1 (H +2 and HD + case) and Z = 2, Z = − A and A (seeFig. 1 and 2) as well as the other contributions given inSec. IV B. in the form of effective electronic potentialcurves.We then averaged these electronic curves over vibra-tional wave functions in order to obtain energy correc-tions for individual states. Adding these new resultsto previously calculated contributions [2, 3], one obtainsprecise theoretical predictions for the frequencies of ex-perimentally relevant transitions (see Tables I and II).Nonrelativistic energies and leading order correctionswere obtained with the CODATA10 [15] recommendedvalues. It is necessary to note that we used improvedcalculations for the leading order relativistic corrections( mα ) and newly obtained values for the Bethe logarithm[38], which were the major source of inaccuracy in theleading order radiative corrections ( mα ). That allowedto significantly reduce numerical uncertainties in the con-tributions at these orders.In the mα order the uncertainty on the contributionstems from numerical uncertainty in calculation of therelativistic Bethe logarithm [22]. The recoil terms arealready negligible at order α ( m/M ), where they con-tribute about 300 Hz to the fundamental transitions ofthe hydrogen molecular ion.The contribution from the finite charge distributionsof nuclei deserves special discussion. For the fundamen-tal transition in the H +2 ion the CODATA10 uncertaintyresults in 250 Hz uncertainty for the transition energy.If we use instead the charge radius from the muonic hy-drogen measurements [40], the frequency will move by 3kHz; ro-vibrational spectroscopy of H +2 is thus sensitiveto the discrepancy between determinations of the protonradius. The CODATA10 uncertainty due to the deuteronrms charge radius for the HD + fundamental transition is215 Hz and is so far negligible. In the antiprotonic heliumthe value of the rms charge radius of the alpha particleis taken from [41] and results in a frequency uncertaintyof 7 kHz, while the corresponding uncertainty from theantiproton rms charge radius is more than order of mag-nitude less, the antiproton–electron interaction being re-pulsive.At present most accurate experimental results areavailable for the HD + molecular ion and for the antipro-tonic helium. In Table III we compare our new theoret-ical results with the best experimental ones. Agreementis excellent in all cases except for the v = 0 → v = 1transition in HD + where the discrepancy is 2 . σ exp . experiment theoryHD + ( v, L )(0 , → (4 ,
3) [9] 214 978 560.6(5) 214 978 560.948(8)(0 , → (1 ,
1) [10] 58 605 052.00(6) 58 605 052.156(2)(0 , → (0 ,
1) [42] — 1 314 925.7523(1)(0 , → (8 ,
3) [13] — 383 407 177.150(15) He + ¯ p ( n, L ) [11](36 , → (34 ,
32) 1 522 107 062(4) 1 522 107 060.3(2)(33 , → (31 ,
30) 2 145 054 858(5) 2 145 054 858.1(2) He + ¯ p ( n, L )(35 , → (33 ,
31) 1 553 643 100(7) 1 553 643 102.4(3)TABLE III: Comparison with most accurate experimentalmeasurements of transition frequencies for HD + and antipro-tonic helium (in MHz). The two transitions [13, 42] are cur-rently studied experimentally and for convenience of futurecomparison we present our theoretical values for these tran-sitions. VI. CONCLUSION
We have completed the calculation of the α -order one-loop self-energy correction in two-center systems, andused these results to obtain new predictions of experi-mentally relevant ro-vibrational transition frequencies forthe three-body molecular type systems. The theoreticaluncertainty has been improved by about one order ofmagnitude to reach a level of 0.03 ppb in molecular hy-drogen ions (resp. 0.13 ppb in the antiprotonic helium).The achieved accuracy already allows for improved deter-mination of the proton- and antiproton-to-electron massratios [15], and may still be improved further as discussedabove. Particularly, as a first step we intend to improvethe relativistic Bethe logarithm calculations using theasymptotic expansions for P rc ( k ), and P nq ( k ) functionspresented in the Appendix. That may result in reducinguncertainty in the one-loop self energy contribution by a factor of three and reduce theoretical relative uncertaintyfor vibrational transitions to 10 − . VII. ACKNOWLEDGEMENTS
V.I.K. acknowledges support of the Russian Founda-tion for Basic Research under Grant No. 12-02-00417.This work was supported by ´Ecole Normale Sup´erieure,which is gratefully acknowledged.
Appendix A: Asymptotic expansion of P (1) rc ( k ) , P (2) rc ( k ) , and P nq ( k ) Here we present without proof the results forthe asymptotic expansion of the functions defined inEqs. (8)–(10), which appear in the integrand of (7). Ofparticular importance is the term of order 1 /k , whichcontributes to the α ln α part of Eq. (17). It is finiteand the πZ (cid:104) δ ( r ) (cid:105) ”counterterm” is determined by thechoice of regularization of the divergent operators. As in[22] we separate P (1) rc into two parts: P (1 a ) rc ( k ) = (cid:68) p ( E − H − k ) − (cid:16) H B −(cid:104) H B (cid:105) (cid:17) ( E − H − k ) − p (cid:69) (A1a) P (1 b ) rc ( k ) = 2 (cid:68) H B Q ( E − H ) − Q p ( E − H − k ) − p (cid:69) . (A1b)In the following expressions, the second line gives thenumerical values of the asymptotic expansion coefficientsfor the 1S state of the hydrogen atom. P (1 a ) rc ( k ) = − √ k / πZ (cid:104) δ ( r ) (cid:105) + 8 ln 2 − k πZ (cid:104) δ ( r ) (cid:105) + 116 k (cid:10) ( ∇ V ) (cid:11) fin − k (cid:10) ( H B −(cid:104) H B (cid:105) ) ∇ (cid:11) fin + . . . = − √ k / + 8 ln 2 + 12 k + . . . (A2) P (1 b ) rc ( k ) = 2 k (cid:10) ( H B −(cid:104) H B (cid:105) ) ( E − H ) − ∇ (cid:11) + 2 √ k / πZ (cid:104) δ ( r ) (cid:105) + ln kk πZ (cid:104) δ ( r ) (cid:105) + 5 ln 2 − k πZ (cid:104) δ ( r ) (cid:105) + 1 k (cid:10) ( H B −(cid:104) H B (cid:105) ) ( E − H ) − ( ∇ V ) (cid:11) fin + 1 k (cid:10) ( H B −(cid:104) H B (cid:105) ) ∇ (cid:11) fin + . . . = − k + 2 √ k / + ln kk + 3 ln 2 − k + . . . (A3) P (2) rc ( k ) = (cid:10) ∇ (cid:11) k − √ k / πZ (cid:104) δ ( r ) (cid:105) + 4 ln kk πZ (cid:104) δ ( r ) (cid:105)−
12 ln 2 + 4 k πZ (cid:104) δ ( r ) (cid:105) − k (cid:10) ( ∇ V ) p (cid:11) fin − k (cid:10) ( ∇ V ) (cid:11) fin + . . . = 5 k − √ k / + 4 ln kk −
20 ln 2 − k + . . . (A4) P nq ( k ) = − (cid:10) ∇ (cid:11) − k (cid:10) ∇ (cid:11) − k (cid:10) ( ∇ V ) (cid:11) + 8 √ k / πZ (cid:104) δ ( r ) (cid:105) − kk πZ (cid:104) δ ( r ) (cid:105) + 40 ln 2 + 765 k πZ (cid:104) δ ( r ) (cid:105) + 2 k (cid:10) ( ∇ V ) (cid:11) fin + 340 k (cid:10) ( ∇ V ) (cid:11) fin + 12 k (cid:10) ( ∇ V ) p (cid:11) fin = 12 − k + 8 √ k / − kk + 24 ln 2 − k + . . . (A5)The sum of these terms makes up the result given inEq. (11).It is worth noting here that this is the main point wherewe differ from the approach used in [18], where the formalexpansion over 1 /k has been used:1 E − H − k = − k − E − Hk − ( E − H ) k + . . . , (A6)which gives divergent matrix elements for individual S states in the hydrogen-like atom, but still the ”normal-ized difference” ∆ n is finite. This formalism is enough toget a complete result for arbitrary states in the hydrogen atom [see Eq. (3.43) and Table 1 of Ref. [18]], but notsuitable for our generalization. So we took another way,which is to derive an appropriate approximation to the ψ function ψ = ( E − H − k ) − i p ψ , (A7)where ψ is a stationary solution of the Schr¨odinger equa-tion. ψ is a regular function at small r thus providingfinite expectation values for the operators appearing at1 /k order. This procedure is very similar to what wasused to obtain the asymptotic expansion for the nonrel-ativistic Bethe logarithm in [39]. [1] K. Pachucki, Phys. Rev. A , 022512 (2006);V.A. Yerokhin and K. Pachucki, Phys. Rev. A , 022507(2010).[2] V.I. Korobov, Phys. Rev. A , 022509 (2008).[3] V.I. Korobov, Phys. Rev. A , 042506 (2008).[4] K. Pachucki and V.A. Yerokhin, Phys. Rev. Lett. ,070403 (2010); K. Pachucki and V.A. Yerokhin, Phys.Rev. A , 062516 (2009).[5] J. Komasa, K. Piszczatowski, G. (cid:32)Lach, M. Przybytek,B. Jeziorski, and K. Pachucki, J. Chem. Theory Comput. , 3105 (2011).[6] Z.-C. Yan, W. N¨ortersh¨auser, and G.W.F. Drake, Phys.Rev. Lett. , 243002 (2008); , 249903(E) (2009).[7] M. Puchalski, J. Komasa, and K. Pachucki, Phys. Rev. A , 030502(R) (2013).[8] B. Roth, J. Koelemeij, S. Schiller, L. Hilico, J.-Ph. Karr,V.I. Korobov, and D. Bakalov, Precision Physics of Sim-ple Atoms and Molecules , ed. S. Karshenboim, LectureNotes in Physics , 205. Springer-Verlag, Berlin, Hei-delberg (2008).[9] J.C.J. Koelemeij, B. Roth, A. Wicht, I. Ernsting, andS. Schiller, Phys. Rev. Lett. , 173002 (2007).[10] U. Bressel, A. Borodin, J. Shen, M. Hansen, I. Ernsting,and S. Schiller, Phys. Rev. Lett. , 183003 (2012). [11] M. Hori, A. S´ot´er, D. Barna, A. Dax, R. Hayano,S. Friedreich, B. Juh´asz, T. Pask, E. Widmann,D. Horv´ath, L. Venturelli, and N.Zurlo, Nature ,484 (2011).[12] J.-Ph. Karr, A. Douillet, and L. Hilico, Appl. Phys. B , 1043 (2012).[13] J.C.J. Koelemeij, D.W.E. Noom , D. de Jong, M.A. Had-dad, and W. Ubachs, Appl. Phys. B , 1075 (2012).[14] R. Bouchendira, P. Clad´e, S. Guellati-Kh´elifa, F. Nez,and F. Biraben, Phys. Rev. Lett. 106, 080801 (2011).[15] P.J. Mohr, B.N. Taylor, and D.B. Newell, Rev. Mod.Phys. , 1527 (2012).[16] J. Verd´u J, S. Djeki´c, S. Stahl, T. Valenzuela, M. Vo-gel, G. Werth, T. Beier, H.J. Kluge, and W. Quint,Phys. Rev. Lett. , 093002 (2004); G. Werth, J. Alonso,T. Beier, K. Blaum, S. Djeki´c, H. H¨affner, N. Her-manspahn, W. Quint, S. Stahl, J. Verd´u, T. Valenzuela,and M. Vogel, Int. J. Mass Spectrom. , 152 )2006).[17] Z.-X. Zhong, Z.-C. Yan, and T.-Y. Shi, Phys. Rev. A ,064502 (2009).[18] U. Jentschura, A. Czarnecki, and K. Pachucki, Phys.Rev. A , 062102 (2005).[19] A. Czarnecki, U.D. Jentschura, and K. Pachucki, Phys.Rev. Lett. , 180404 (2005). [20] K. Pachucki, Phys. Rev. A , 648 (1992).[21] K. Pachucki, Ann. Phys. (N.Y.) , 1 (1993).[22] V.I. Korobov, L. Hilico, and J.-Ph. Karr, Phys. Rev. A , 062506 (2013).[23] K. Pachucki, Phys. Rev. Lett. , 4561 (2000).[24] V.I. Korobov and Ts. Tsogbayar, J. Phys. B, , 2661,(2007).[25] G.W. Erickson and D.R. Yennie, Ann. Phys. (NY) ,271 (1965); ibid. , 447 (1965).[26] J.R. Sapirstein, D.R. Yennie, in: T. Kinoshita (Ed.), Quantum Electrodynamics , World Scientific, Singapore(1990).[27] M.I. Eides, H. Grotch, and V.A. Shelyuto,
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