Hyperfine Interact (2015) 233:75–82 DOI 10.1007/s10751-015-1149-5
Bound-state QED calculations for antiprotonic helium Vladimir I. Korobov · Laurent Hilico · Jean-Philippe Karr
Published online: 15 April 2015 © Springer International Publishing Switzerland 2015
Abstract We present new theoretical results for the transition energies of the hydrogen isotope molecular ions and antiprotonic helium atoms. Our consideration includes corrections at the mα 7 order in the nonrecoil limit such as the one-loop self-energy, one-loop vacuum polarization, Wichman-Kroll, and complete two-loop contributions. That allowed to get transition energies for the fundamental transition (v = 0, L = 0) → (1, 0) in the hydrogen molecular ion with the relative theoretical uncertainty of ∼ 7 · 10−12 that corresponds to a fractional precision of 1.5 · 10−11 in determination of the electron-to-proton mass ratio, mp /me . Correspondingly, for the two-photon transitions in the antiprotonic helium we have 4.7 · 10−11 as a relative uncertainty for the (33, 32) → (31, 30) transition frequency and a fractional precision of 3.6 · 10−11 for an inferred antiproton-to-electron mass ratio. Keywords Antiprotonic helium · Precision spectroscopy · Nonrelativistic QED
1 Introduction In our recent work [1] we have calculated the relativistic Bethe logarithm contribution at order mα 7 in the two Coulomb center approximation. These results then have been used for improved calculations of the transition energies for the hydrogen isotope molecular
Proceedings of the International Conference on Exotic Atoms and Related Topics (EXA 2014), Vienna, Austria, 15-19 September 2014 V. I. Korobov () Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia e-mail:
[email protected] L. Hilico · J.-P. Karr Laboratoire Kastler Brossel, Universit´e dEvry val dEssonne, UPMC-Sorbonne Universit´es, CNRS, ENS-PSL Research University, Coll`ege de France, 4 place Jussieu, 75005 Paris, France
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+ Table 1 Fundamental vibrational transitions in H+ 2 and HD molecular ions (in MHz)
H+ 2
HD+
ΔEnr
65 687 511.0714
57 349 439.9733
ΔEα 2
1091.0400
958.1514
ΔEα 3
−276.5450
−242.1262
ΔEα 4
−1.9969
−1.7481
ΔEα 5
0.1371(1)
0.1200(1)
ΔEα 6
−0.0010(5)
−0.0009(4)
Etot
65 688 323.7055(5)
57 350 154.693(4)
Table 2 The two-photon transition: (33, 32) → (31, 30), in the antiprotonic helium. (in MHz)
ΔEnr
=
2 145 088 265.34
ΔEα 2
=
-39 349.33
ΔEα 3
=
5 857.84
ΔEα 4
=
92.97
ΔEα 5
=
-8.25(2)
ΔEα 6
=
-0.10(10)
Etotal
=
2 145 054 858.50(10)
ions and antiprotonic helium atoms [2]. The general formula for the one-loop self-energy contribution at the mα 7 order has been obtained in [2, 3]. Since that time few advances have been achieved, which we suppose to discuss below in this work. For convenience we present our ultimate numerical results here in Tables 1 and 2, where the CODATA10 recommended values of constants have been adopted in all our calculations [4]. The error bars in transition frequency (see Table 1) set a limit on the fractional precision in determination of mass ratio to Δμ = 1.5 · 10−11 . μ The proton rms charge radius uncertainty as is defined in the CODATA10 adjustment contributes to the fractional uncertainty at the level of ∼ 4 · 10−12 for the transition frequency. While the muon hydrogen ”charge radius” moves the spectral line blue shifted by 3 kHz that corresponds to a relative shift of 5 · 10−11 . In case if we use the ”muon hydrogen” adjusted Rydberg constant [5] along with the muon charge radius then we get a shift of 1.1 kHz, which is still feasible for detection. For the case of the antiprotonic helium we get 0.10 MHz as a final theoretical uncertainty for the two-photon (33, 32) → (31, 30) transition. Along with the sensitivity of this transition to a change of μ ≡ mp¯ /me , this sets a limit on the fractional precision in determination of mass ratio Δμ = 3.6 · 10−11 . μ
Bound-state QED calculations for antiprotonic helium
77
Main diagram:
Contributions at order mα 7 :
+
+
+ Fig. 1 One-loop self-energy NRQED diagrams, which contribute in the mα 7 order. Here, for the middle set of diagrams a cross on a double line means leading order relativistic corrections to the wave function of a bound electron. For the bottom line a cross corresponds to higher order contributions to the vertex function
2 The contributions of order mα 7 2.1 One-loop self-energy In [3] we rederived the low-energy part (Fig. 1) of the one-loop self-energy contribution [6], and obtained a general expression in atomic units, which may be extended for a case of two and more fixed external Coulomb sources: 5 2 α5 1 (7) = L (Z, n, l) + 4πρ Q(E − H )−1 QHB ΔEse + ln α −2 fin π 9 3 2 779 1 11 +2 Hso Q(E − H )−1 QHB + ∇4 V + ln α −2 fin 14400 120 2 1 1 23 + ln α −2 + 2iσ ij p i ∇ 2 Vp j 576 24 2 1 589 2 3 1 + ln α −2 + − p2 Hso + (∇V )2 4πρ p2 fin fin 720 3 2 80 4
16
1 2 2 −2 −2 ln α ln 2 − +Z − ln α + − 0.81971202(1) πρ . (1) 3 4 Here L (Z, n, l) is the relativistic Bethe logarithm. For details and notations see Ref. [3]. In case of the hydrogen molecular ion, (see Table 1) this contribution to the fundamental transition frequency yields:
(7) E1loop−se = α 5 A62 ln2 (α −2 ) + A61 ln(α −2 ) + A60 Z13 δ(r1 ) + Z23 δ(r2 ) ≈ 124.9(1)kHz,
(2)
For the antiprotonic helium, the relativistic Bethe logarithm [L (R; Z1 , Z2 , n) from (1)] for small R 0.5 a.u. is needed, where numerical calculations become unstable. Recently
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Table 3 Relativistic Bethe logarithm L (R) for the ground electronic state of the antiprotonic helium. Here (a) (b) βi are various contributions to L = β1 + β1 + β2 + β3 . See details in Ref. [1, 3] Results of 2013 R
β1(a)
β1(b)
β2
β3
0.1
−137.1
329.2
−102.
−381.08
0.2
−181.5
211.2
−584.1
62.514
0.4
−193.8
160.65
−1382.7
369.822
0.6
−241.21
150.07
−2064.5
590.636
1.0
−304.14
172.37
−2860.8
840.902
β1(a)
β1(b)
β2
β3
0.05
−625.8(8)
650.5(5)
1797.(2)
−1486.18(2)
0.1
−291.5(1)
330.9(2)
177.1(6)
−381.72(3)
0.2
−181.68(4)
208.76(3)
−588.20(4)
63.099(5)
0.4
−194.00(1)
161.76(3)
−1387.92(5)
369.680(5)
0.6
−241.296(4)
151.068(3)
−2069.932(3)
590.555(2)
1.0
−304.531(3)
172.282(2)
−2862.089(1)
840.862(3)
Results of 2014. R
we found how to improve substantially the numerical accuracy. In brief, the relativistic correction to the Bethe logarithm is obtained by taking an integral of a function Pα 2 (k) (see (11) of [3]) over photon momenta k ∈ [0, ∞]. For high k the asymptotic expansion of Pα 2 (k) in terms of 1/k is used, where the unknown coefficients of expansion are extrapolated numerically. In [3] we have managed to extend a set of coefficients, which may be calculated analytically, see Eqs. (A2)–(A5) of this Reference. Along with the extended duodecimal arithmetic precision (about 100 significant decimal digits) that allowed us to improve the calculations. The results are shown in Table 3, they demonstrate that we came closer to R = 0, and that estimates of numerical uncertainty for these new data became available. Thus, it may be concluded that the numerical inaccuracy in the relativistic Bethe logarithm does not contribute to the uncertainty of our present results shown in Tables 1 and 2 above. 2.2 Other contributions. 1.
For the hydrogen-like atom the one-loop vacuum polarization, or the Uehling potential contribution, see Fig. 2a, may be written in a form
α (Zα)4 2 −2 + (Zα)V + (Zα) V ln(Zα) + V + . . . (3) ΔE1loop−vp = V 40 50 61 60 π n3 The coefficients are known in an analytical form [4]. For the fundamental transition of the hydrogen molecular ion the V60 coefficient need to be evaluated numerically [7] and corresponding correction to the transition frequency is
(7) E1loop−vp = α 5 V61 ln(α −2 ) + V60 Z13 δ(r1 ) + Z23 δ(r2 ) ≈ 2.4kHz. (4) As in case of the one-loop self-energy contribution we use in (4) atomic units and Z dependence from logarithms is moved to V60 .
Bound-state QED calculations for antiprotonic helium
(a)
79
(b)
Fig. 2 a) The one-loop vacuum polarization diagram, the Uehling potential. b) The Wichman-Kroll diagram
2.
The Wichman-Kroll contribution, Fig. 2b, is expressed by α (Zα)6 {W60 + (Zα)W70 + . . .} π n3 For the hydrogen atom the coefficient of the mα 7 order is known [8]: 19 π 2 δ0l , W60 (nl) = − 45 27 ΔEW K =
and does not depend on n. For the hydrogen molecular ion this contribution may be evaluated as follows: (7) EW K = α 5 W60 Z13 δ(r1 ) + Z23 δ(r2 ) ≈ −0.1 kHz, 3.
4.
(5)
(6)
The complete two-loop contribution in the leading orders is expressed α 2 (Zα)4 (7) ΔE2loop = [B40 + (Zα)B50 + . . .] π n3 Here B50 = −21.55447(13) and state independent (see Fig. 3 for a complete set of diagrams contributing at this order). For the hydrogen molecular ion the two-loop contribution has a form α5 (7) E2loop = (8) [B50 ] Z12 δ(r1 )+Z22 δ(r2 ) ≈ 10.1 kHz, π The three-loop contribution α 3 (Zα)4 (9) ΔE3loop = [0.417504 + . . . ] π n3 is already negligible. For the fundamental transition of the hydrogen molecular ion it gives only α5 (7) E3loop = 2 [0.417504] Z1 δ(r1 )+Z2 δ(r2 ) ≈ 60 Hz. (10) π
3 Evaluation of the leading contributions at the mα 8 order 1.
The one-loop self-energy contribution at the mα 8 order is expressed as
α (Zα)7 (8) −2 ln(Zα) +A . A E1loop = 71 70 π n3
(11)
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Fig. 3 Complete set of diagrams for the two-loop contribution
For the hydrogen-like atom A71 was obtained in an analytical form [9]: 139 − ln 2 . A71 (nS) = π 64
2.
The nonlogarithmic contribution A70 of order mα(Zα)7 was never calculated directly [4]. The one-loop Uehling potential contribution at the mα 8 order may be expressed (8)
EV P =
3.
α (Zα)7 −2 ln(Zα) +V V . 71 70 π n3
(12)
For the two-center Coulomb problem this contribution has already been calculated numerically in [7] along with the contribution of order mα 7 . The two-loop contribution at the mα 8 order has the following expansion [6] (8) E2loop =
α 2 (Zα)6
3 −2 2 −2 −2 ln (Zα) +B ln (Zα) +B ln(Zα) +B B . 63 62 61 60 π n3 (13)
Bound-state QED calculations for antiprotonic helium
81
which for the ground state of a hydrogen atom provides: ΔE(1S) ≈
α 2 (Zα)6 [−282 − 62 + 476 − 61] . π2
(14)
The coefficients B6k may be calculated using the following expressions with expectation values regularized as in [2, 3]: 8 3 Z π δ(r) , 27 1 2 1 4 = + ∇ V Q(E0 − H )−1 Q ∇ 2 V ∇ V fin fin 9 18 16 31 3 + 2 ln 2 Z π δ(r) . + 9 15
Z 6 B63 = − Z 6 B62
(15)
The numerical data for expectation values in (15) are already available from previous calculations performed for the one-loop self energy at the mα 7 order [3]. As is seen from (14) the largest contribution is to be 1 2 1 4 Z 6 B61 = −2 ln 2 + ∇ V Q(E0 − H )−1 Q∇ 2 V ∇ V fin fin 9 18 19 2 4 + N (n, l) + ∇ V Q(E0 − H )−1 Q ∇ 2 V fin 3 135 19 4 1 ij i 2 j + + ∇ V 2iσ p ∇ Vp fin 24 270 2π 2 48781 2027π 2 56 2 + + ln 2− ln 2+8 ln 2+ζ (3) Z 3 π δ(r) .(16) + 64800 864 27 3 The only quantity that requires additional numerical efforts is the low energy contribution N (n, l), and it is defined by the integral 2Z k dk δπδ(r) p(E0 − H − k)−1 p , (17) N= 3 0 where
δπδ(r) p(E0 − H − k)−1 p ≡ p(E0 −H −k)−1 (π δ(r)−π δ(r)) (E0 −H −k)−1 p +2 π δ(r)Q(E0 − H )Qp(E0 − H − k)−1 p .
4.
This integrand is of the same type as were considered in [1] for calculation of the relativistic Bethe logarithm and thus may be obtained in a similar way. Pure relativistic correction of order m(Zα)8 may be evaluated approximately using the expansion of the Dirac energy for the hydrogen atom. It contributes at a level of 1 Hz and may be neglected.
4 Summary –
A new limit of precision for theoretical predictions is achieved. Relative uncertainty is + −11 for the now 7 · 10−12 for the hydrogen molecular ions H+ 2 and HD , and 4.7 · 10 antiprotonic helium.
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The proton rms charge radius uncertainty as is defined in the CODATA10 adjustment contributes to the fractional uncertainty at the level of ∼ 4 · 10−12 for the transition frequency. While the muon hydrogen ”charge radius” moves the spectral line blue shifted by 3 kHz that corresponds to a relative shift of 5 · 10−11 . A use of the ”muon hydrogen” Rydberg constant along with the ”muon” charge radius results in the still detectable 1.1 kHz shift. The leading corrections at the mα 8 order may be evaluated to at least 10 % precision. That will allow to infer a value of the rms charge radius from the ro-vibrational spectroscopy of the hydrogen molecular ions. For the antiprotonic helium that will make possible to get antiproton-to-electron mass ratio to a level of few ppt (parts per trillion).
Acknowledgments This work has been partially supported by the Russian Foundation for Basic Research ´ under a grant No. 12-02-00417-a, by the Heisenberg-Landau program of BLTP JINR and by Ecole Normale Sup´erieure, which is gratefully acknowledged. Conflict of interests
The authors declare that they have no conflict of interest.
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
Korobov, V.I., Hilico, L., Karr, J.-Ph.: Phys. Rev. A 87, 062506 (2013) Korobov, V.I., Hilico, L., Karr, J.-Ph.: Phys. Rev. Lett. 112, 103003 (2014) Korobov, V.I., Hilico, L., Karr, J.-Ph.: Phys. Rev. A 89, 032511 (2014) Mohr, P.J., Taylor, B.N., Newell, D.B.: Rev. Mod. Phys. 84, 1527 (2012) Nez, F.: R∞ c = 3 289 841 960.2509 MHz; private communication Jentschura, U., Czarnecki, A., Pachucki, K.: Phys. Rev. A 72, 062102 (2005) Karr, J.-Ph., Hilico, L., Korobov, V.I.: Phys. Rev. A 90, 062516 (2014) Wichmann, E.H., Kroll, N.M.: Phys. Rev. 101, 843 (1956) Karshenboim, S.G.: Zh. Eksp. Teor. Fiz. 106, 414 (1994). [JETP 79, 230 (1994)]
