Ro-vibrational spectroscopy of the hydrogen molecular ion and antiprotonic helium

Transcription

1 of the hydrogen molecular ion and antiprotonic helium V.I. Joint Institute for Nuclear Research , Dubna, Russia Fundamental Constants 2015, February 2015

2 Status of Theory g-factor of a bound electron Status of Theory. 2014

3 H + 2 Present status of theory and atomic mass of electron and HD+ ions Status of Theory g-factor of a bound electron Fundamental transitions in H + 2 and HD + (in MHz). CODATA10 recommended values of constants. H + 2 HD + E nr E α E α E α E α (1) (1) E α (5) (4) E tot (5) (4) The error bars in transition frequency set a limit on the fractional precision in determination of mass ratio to µ µ =

4 RMS radius of proton Status of Theory g-factor of a bound electron The proton rms charge radius uncertainty as is defined in the CODATA10 adjustment contributes to the fractional uncertainty at the level of 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 Antiprotonic helium Status of Theory g-factor of a bound electron E nr = E α 2 = E α 3 = E α 4 = E α 5 = 8.25(2) E α 6 = 0.10(10) E total = (10) Transition (33, 32) (31, 30) (in MHz). CODATA10 recommended values of constants. Along with the sensitivity of this transition to a change of µ m p /m e, this sets a limit on the fractional precision in determination of mass ratio µ µ =

6 Atomic mass of electron A r (e) Status of Theory g-factor of a bound electron At present the most precise measurements of m p /m e are: The penning trap mass spectroscopy (uncertainty ) [D.L. Farnham, et al. Phys. Rev. Lett. 75, 3598 (1995)]; The g factor of a bound electron in 12 C 5+ (uncertainty ) [T. Beier, et al. Phys. Rev. Lett. 88, (2001) and CODATA-10]. The spin-flip energy for a free electron is E = g eµ B B The analogous expression for ions with no nuclear spin E = g e(x )µ B B where the theoretical expression for g e(x ) is written as g e(x ) = g D + g rad + g rec + g ns +... g D is derived from the Dirac equation g D = 2 [1 + 2 ] [ 1 (Zα) 3 2 = ] 3 (Zα) Theoretical uncertainty of the g factor for 12 C 5+ is

7 Status of Theory g-factor of a bound electron Nature, 506, 467 (2014) Here we combine a very precise measurement of the magnetic moment of a single electron bound to a carbon nucleus with a state-of-the-art calculation in the framework of bound-state quantum electrodynamics. The precision of the resulting value for the atomic mass of the electron surpasses the current literature value of the Committee on Data for Science and Technology (CODATA) by a factor of 13. m e = (14)(9)(2) [ ]

8 Other contributions

9 Other contributions One-loop SE corrections in order mα 7

10 Other contributions 1. One-loop SE corrections in order mα 7 Main diagram: Contributions at order mα 7 : + + +

11 Other contributions 1. One-loop SE correction in atomic units We rederived the low-energy part [V.I., J.-P. Karr, and L. Hilico, Phys. Rev, A 89, (2014)], and obtained an expression in atomic units, which may be extended for a general case of two and more external Coulomb sources: { E (7) se = α5 π ( 5 L(Z, n, l) [ ]) 1 3 ln 2 α 2 4πρ Q(E H) 1 Q H B fin au ( +2 H so Q(E H) Q H B [ ]) ln 2 α 2 4 V fin au ( [ ]) 1 24 ln 2 α 2 2iσ ij p i 2 Vp j ( [ ]) 1 3 ln 2 α 2 ( V ) πρ p 2 1 p 2 H so fin au 80 fin au 4 +Z [ 2 ln 2[ α 2] [ ln 2 1 ] ln [ α 2] ] } (1) πρ 4

12 Other contributions 1. Relativistic corrections to BL. Antiprotonic Helium Relativistic Bethe logarithm for the ground electronic state R β (a) 1 β (b) 1 β 2 β Relativistic Bethe logarithm for the ground electronic state R β (a) 1 β (b) 1 β 2 β (8) 650.5(5) 1797.(2) (2) (1) 330.9(2) 177.1(6) (3) (4) (3) (4) (5) (1) (3) (5) (5) (4) (3) (3) (2) (3) (2) (1) (3)

13 Other contributions 1. Relativistic corrections to the Bethe logarithm (R ) / N (R ) R (in a.u.) The relativistic Bethe logarithm L(R) for the ground (1sσ g ) electronic state, for Z 1 = Z 2 = 1 normalized by: N(R) = π ( Z 3 1 δ(r 1) + Z 3 2 δ(r 2) ). [PRA 87, (2013)] Hydrogen molecular ion: E (7) 1loop se = α5 [A 62 ln 2 (α 2 )+A 61 ln(α 2 )+A 60 ] Z 3 1 δ(r 1 )+Z 3 2 δ(r 2 ) 124.9(1) khz,

14 Other contributions Other contributions beyond the self-energy

15 2. One-loop vacuum polarization Other contributions E 1loop vp = α π (Zα) 4 { } n 3 V 40 + (Zα)V 50 + (Zα) 2 V 61 ln(zα) For the hydrogen atom in S-state the coefficients are V 40 (ns) = 4 15 V 50 (ns) = π 5 48 V 61 (ns) = 2 15, V 60 (ns) = 4 15 [ 431 2(n 1) + ψ(n + 1) ψ(1) 105 n n 2 ln n 2 ],

16 2. One-loop vacuum polarization Other contributions Effective potential for the H + 2 ion at mα7 order including higher orders G V P (1 )(R ) R (a.u.) Hydrogen molecular ion: E (7) 1loop vp = α5 [V 61 ln(α 2 ) + V 60 ] Z 3 1 δ(r 1 ) + Z 3 2 δ(r 2 ) 2.9 khz,

17 Other contributions 3. The Wichman-Kroll contribution E WK = α π (Zα) 6 { } n 3 W 60 + (Zα)W For the hydrogen atom in S-state the coefficients are W 60 (ns) = π2 27, W 70 (ns) = π 16 31π Hydrogen molecular ion: E (7) WK = α5 W 60 Z 3 1 δ(r 1 ) + Z 3 2 δ(r 2 ) 0.1 khz,

18 Other contributions 4. Complete two-loop contribution

19 Other contributions 4. Complete two-loop contribution E 2loop = Here B 50 = (13). ( α ) 2 (Zα) 4 [ ] π n 3 B 40 + (Zα)B N.B. Insertion of two radiative photons in the electron line contributes to B 50 Hydrogen molecular ion: E (7) 2loop = α5 π [B 50] Z 2 1 δ(r 1 )+Z 2 2 δ(r 2 ) 10.1 khz,

20 5. Three-loop contribution Other contributions Three-loop contribution E 3loop = is already negligible. Hydrogen molecular ion: ( α ) 3 (Zα) 4 [ ] π n E (7) 3loop = α5 π 2 [ ] Z 1δ(r 1 )+Z 2 δ(r 2 ) 60 Hz,

21 Vacuum polarization Two-loop self-energy Prospects for the future

22 Vacuum polarization Two-loop self-energy The one-loop contribution at mα 8 order is expressed E (8) 1loop = α (Zα) 7 [ A71 π n 3 ln(zα) 2 ] +A 70 Here [ ] 139 A 71 (ns) = π 64 ln 2 The nonlogarithmic contribution A 70 of order mα(zα) 7 was never calculated directly. Hydrogen molecular ion: E (8) 1loop se = α6 [ A71 ln(zα) 2 ] + A 70 Z 4 π 1 δ(r 1 )+Z2 4 δ(r 2 ) 0.5 khz,

23 Uehling potential Vacuum polarization Two-loop self-energy The one-loop contribution at mα 8 order is expressed E (8) VP = α (Zα) 7 [ V71 π n 3 ln(zα) 2 ] +V 70 Here V 71 (ns) = π 5 96 V 70 (ns) = π 5 48 ( ψ(n+1) ψ(1) ln n ln n ) 48n 2 Hydrogen molecular ion: E (8) 1loop vp = α6 [ V71 ln(zα) 2 ] + V 70 Z 4 π 1 δ(r 1 )+Z2 4 δ(r 2 ) 17 Hz,

24 Wichman-Kroll contribution Vacuum polarization Two-loop self-energy The Wichman-Kroll contribution at mα 8 order E (8) WK = α (Zα) 7 [ ] π π n π3 2880

25 Two-loop self-energy Vacuum polarization Two-loop self-energy The two-loop contribution at mα 8 order is expressed ( E (8) α ) 2 (Zα) 6 2loop = [ B63 π n 3 ln 3 (Zα) 2 +B 62 ln 2 (Zα) 2 +B 61 ln(zα) 2 ] +B 60 E(1S) α2 (Zα) 6 Hydrogen molecular ion: π 2 [ ] [ E (8) 2loop vp = α6 B 73 ln 3 (Zα) 2 +B 72 ln 2 (Zα) 2 π ] Z +B 71 ln(zα) 2 3 +B 70 1 δ(r 1 )+Z2 3 δ(r 2 ) 0.5 khz,

26 Vacuum polarization Two-loop self-energy The coefficients B 6k may be calculated using expressions Z 6 B 63 = 8 27 Z 3 πδ(r) Z 6 B 62 = 1 2 V Q(E 0 H) 1 Q 2 V 9 fin V 18 fin + 16 [ ] ln 2 Z 3 πδ(r) The largest contribution is [ 1 Z 6 B 61 = 2 2 V Q(E 0 H) 1 Q 2 V 9 fin V ] ln 2 18 fin N(n, l) + 2 V Q(E 0 H) 1 Q 2 V fin V 270 fin + 1 2iσ ij p i 2 Vp j 24 [ π ] 2π2 ln ln 2+8 ln2 2+ζ(3) Z 3 πδ(r)

27 Low energy part N(n, l) Vacuum polarization Two-loop self-energy The only quantity that needs numerical computations is N(n, l), and it is defined by and N = 2Z 3 Λ 0 k dk δ πδ(r) p(e 0 H k) 1 p δ πδ(r) p(e 0 H k) 1 p p(e 0 H k) 1( πδ(r) πδ(r) ) (E 0 H k) 1 p +2 πδ(r) Q(E 0 H)Q p(e 0 H k) 1 p

28 Summary Present status of theory and atomic mass of electron Vacuum polarization Two-loop self-energy A new limit of precision for theoretical predictions is achieved. Relative uncertainty is now for the hydrogen molecular ions H + 2 and HD+, and about for the antiprotonic helium. The proton rms charge radius uncertainty as is defined in the CODATA10 adjustment contributes to the fractional uncertainty at the level of 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 The two-loop correction at the mα 8 order become now the major uncertainty in the theory. The two-loop SE correction at the mα 8 order is now under consideration and we hope to get the results by the end of this year. That should allow to improve theoretical uncertainty by a factor of 3-5.

29 Acknowledgements Vacuum polarization Two-loop self-energy This work was done in collaboration with D. Bakalov, INRNE, Sofia, Bulgaria. L. Hilico, J.-P. Karr, LKB, Paris, France Zhen-Xiang Zhong, WIPM CAS, Wuhan, China Special thanks to Krzysztof Pachucki and Masaki Hori!

30 Vacuum polarization Two-loop self-energy Thank you for your attention!