Hyperfine Interact (009) 194:15 0 DOI 10.1007/s10751-009-004-7 Precision spectroscopy of antiprotonic helium Vladimir I. Korobov Zhan-Xiang Zhong Published online: 1 August 009 Springer Science + Business Media B.V. 009 Abstract We survey recent progress in the theoretical study of vibrational transitions in the antiprotonic helium atom. Along with the latest experiment they allow to achieve a competitive accuracy in determination of the atomic mass of an electron and thus they have been included into the CODATA06 analysis of the fundamental constants. Improved theoretical calculation of the hyperfine structure in 4 He p atom will be considered as well. We will discuss contributions of order R α 4 to the electron spin-orbit interaction. These corrections are necessary to confirm the latest measurements of the 1.9 MHz intervals of the (n, l) = (37, 35) state in 4 He + p and for precise determination of the antiproton magnetic moment. Keywords Antiprotonic helium Relativistic and radiative corrections Atomic mass of an electron HFS 1 Introduction In past few yearsexperimental [1] and theoretical [] efforts brought the antiprotonic helium spectroscopy to the new level of precision, which allowed to use it in determination of the atomic mass of an electron (http://physics.nist.gov/cuu/constants/ codata.pdf). At present the most precise measurements of m p /m e are the penning trap mass spectroscopy (uncertainty.1 10 9 )[3] andtheg factor of a bound electron in 1 C 5+ (uncertainty 8 10 10 )[4]. Among these experiments only the first one can V. I. Korobov (B) Joint Institute for Nuclear Research, 141980, Dubna, Russia e-mail: korobov@theor.jinr.ru Z.-X. Zhong Wuhan Institute of Physics and Mathematics, Chinese Academy of Science, 430071, Wuhan, People s Republic of China
16 V.I. Korobov, Z.-X. Zhong be considered as direct, the g factor experiment requires sophisticated theoretical efforts to take account of the high order QED corrections. The atomic mass deduced from comparison of the antiprotonic helium theory and experiment (http://physics.nist.gov/cuu/constants/codata.pdf): A r (e) = 0.000 548 579 908 81(91)[1.7 10 9 ]. That means that this system can be competitive in measuring of the fundamental constants. In our contribution we want to report the present status of the theoretical studies, discuss the main sources of uncertainties and the program for future improvements of the theoretical accuracy of the ro-vibrational transition intervals. The second part will be devoted to the R α 4 order corrections to the fine structure in the (37, 35) state of 4 He + p. Ro-vibrational transitions The numerical calculation of the ro-vibrational transitions has been published recently in [], where the contributions up to and including the R α 4 order to the nonrelativistic energies have been taken into account. The radiative corrections of order R α 5 have not been calculated so far but can be approximately evaluated as follows E (5) se = α 5[ 1 ]{ Z 3 He ln (Z He α) δ(r He ) + Z 3 p ln (Z p α) δ(r p ) }, E (5 ) se [ ] δ(rhe = α 5 ZHe 3 A 61 ln (Z He α) + A 60 ), E (5) loop = α5 π Z He [ ] B50 δ(rhe ), (1) where the constants A 61, A 60,andB 50 are taken equal to the constants of the 1s state of the hydrogen atom A 61 = 5.419... [5], A 60 = 30.94... [6], and B 50 = 1.556... [7, 8]. Some examples of the ro-vibrational transitions are shown in Table 1. The theoretical uncertainty is determined by the total contribution of the terms of the last two equations in (1). It is seen that the numerical error is significantly smaller and the major limit for the better theoretical prediction comes from the not yet calculated radiative corrections of order R α 5. Especially that is important for the two-photon transitions ( n = ). The numerical uncertainty is primarily determined by the uncertainty in the daughter state decaying via Auger channel and is about the same for one- and two-photon transitions, while the frequency interval of the two-photon transition is doubled. So, the relative error from the numerical uncertainty is about 10 10, or about an order of magnitude smaller than the theoretical error (Fig. 1). As a proposition for the future work we plan to calculate directly the R α 5 order contribution in the nonrecoil limit of the two-center problem. To do that we will use
Precision spectroscopy of antiprotonic helium 17 Table 1 The theoretical ro-vibrational transition intervals (in MHz) Transition Theory 4 He + p (3, 31) (31, 30) 1 13 609 3.5(0.8)(0.) (36, 34) (34, 3) 1 5 107 059.1(.1)(0.3) 3 He + p (35, 33) (33, 31) 1 553 643 100.9(.1)(0.) The state is denoted by (n, l), the principal quantum numbers of the antiprotonic orbital. In parentheses the first error is the theoretical uncertainty due to yet uncalculated higher order corrections. The last error is the numerical uncertainty Fig. 1 Comparison of theory and experiment for some transitions in 4 He + p. Squares are the theoretical data from [, 14], triangles are the data from [15] the analytical expression for the one-loop self-energy correction at this order derived in [9] E (5) se { ( α(zα)6 5 = L + π 9 + [ ]) 1 3 ln (Z α) VQ(E H) 1 QH R + 1 σ ij i Vp j Q(E H) 1 QH R ( 779 + 14400 + 11 [ ]) 1 10 ln (Z α) 4 V ( 3 + 576 + 1 [ ]) 1 iσ 4 ln (Z α) ij p i Vp j ( 589 + 70 + [ ]) 1 ( V) 3 ln (Z α) + 3 p V 1 p σ ij i Vp j }, 80 8 where Q is the projector operator on the subspace orthogonal to Ψ 0, V = Z He Z p, r He r p H R = p 8 + 1 8 V + 1 4 σ ij i Vp j. and L is the relativistic correction to the mean excitation energy (the Bethe logarithm). ()
18 V.I. Korobov, Z.-X. Zhong Concluding the consideration of the ro-vibrational spectroscopy of the antiprotonic helium atoms we may state. The best present theoretical relative accuracy is 7 10 10, that is still large to compete with the g factor experiment [3]. Direct calculation of R α 5 one loop self-energy corrections is strongly desirable and would allow to reach precision of 10 10 in the ro-vibrational transition frequency intervals. In its turn, that would probably makes the antiprotonic helium spectroscopy the best tool for measuring of the electron atomic mass. 3 HFS of the (37, 35) state in 4 He + p atom Another interesting issue related to the high precision spectroscopy of the antiprotonic helium is an accurate measurement of the hyperfine structure of the metastable states. It has two-fold interest. First, it is expected that it may be a way to obtain improved value of the magnetic moment of an antiproton. The other point is that it can be a good benchmark for testing QED theoretical methods for the Coulomb three-body bound states to a high precision. Recently, the ASACUSA experiment has provided new precise values for the two HFS transitions of the (37, 35) state in the 4 He + p atom [10]: τ + = 1.896 6(39) GHz, τ = 1.94 41(39) GHz. Transitions τ + and τ are shown on Fig. That should be compared with the theoretical prediction [11]: τ + = 1.896 35(69) GHz, τ = 1.94 4(69) GHz. It is seen that experimental results are more than an order of magnitude better and have some systematic shift toward larger values. We have to say that so far the theoretical numbers were carried out within the leading order Breit-Pauli approximation taking account of the anomalous magnetic moment of the electron. This approximation is limited by the relative order O(α ) and corrections of order R α 4 should be included into consideration in order to comply with the experimental level of accuracy. In terms of the effective Hamiltonian of the hyperfine interaction { H eff = E 1 (s e L) + E (s p L) + E 3 (s e s p ) + E 4 L(L+1)(se s p ) 3 [ (s p L)(s e L) + (s ]} (3) e L)(s p L), one needs to get an improved value for the electron spin-orbit interaction coefficient E 1. It may be done with the use of the NRQED [1]. Some details of the derivation of the R α 6 order contribution may be found in [13]. This contribution may be expressed in terms of the second order term, E A = α 4 He BP Q(E 0 H 0 ) 1 Q (1 + a e )Z a 1 (r m e ra 3 a p e )s e +α 4 He BP Q(E 0 H 0 ) 1 Q (1 + a e)z j 1 (r a P a )s e, m e m (4) a H BP e = p4 e + π [Z 8m 3 e m 1 δ(r 1 ) + Z δ(r )]. e r 3 a
Precision spectroscopy of antiprotonic helium 19 Fig. Schematic diagram of hyperfine sublevels of the (37, 35) state of 4 He + p atom And the effective interaction of this order: H (6) = V 1 + V + V 3. The expressions to the effective interactions V i are obtained via NRQED Feynman rules. For the tree-level diagrams one has ( V 1 = e i 3σ e P[q p e](p e + p e ) ) 1 3m 4 e ( V = e i [σ e P q](p e + p e ) ) 8m 3 e q (Z a) 1 q ( Z i P a M a ), a = 1, ( 4 he, p), here in parentheses are the vertex functions, and 1/q is the photon propagator connecting heavy particles and electron. In addition we have the seagull interaction diagram with one Coulomb and one transverse photon lines: V 3 = e 4 σ e P [ {[ ( 1 4m e q δ ij qi q j q )] ( P Z + P ) } M [ ] ] iq1 (Z 1 ) + (1 ). Radiative corrections (from form factors) have been already included into consideration as contributions from the anomalous magnetic moment. Transforming potentials V i to the coordinate space and atomic units one gets (r a = r e R a, a = 1, ): q 1 V 1 = α 6 c 3Z { a p 16m 4 e, 1 } [r e ra 3 a p e ] s e, { V = α 6 c Z a 4m 3 e M p e, 1 } [r a ra 3 a P a ] s e. (5) (6)
0 V.I. Korobov, Z.-X. Zhong and V 3 = α 6 c Z 1 Z 4m e { [r1 P ] M r 3 1 r + [r P 1 ] M 1 r 1 r 3 [r 1 r ] r 3 1 r3 [ (r1 P 1 ) (r ]} P ) s e. M 1 M The numerical work on evaluation of these contributions is now carried out and results should appear soon. Acknowledgements The Russian Foundation for Basic Research under Grant No. 08-0-00341 is gratefully acknowledged. References 1. Hori, M., Dax, A., Eades, J., Gomikawa, K., Hayano, R.S., Ono, N., Pirkl, W., Widmann, E., Torii, H.A., Juhász, B., Barna, D., Horváth, D.: Phys. Rev. Lett. 96, 43401 (006). Korobov, V.I.: Phys. Rev., A 77, 04506 (008) 3. Farnham, D.L., Van Dyck, Jr., R.S., Schwinberg, P.B.: Phys. Rev. Lett. 75, 3598 (1995) 4. Beier, T., Häffner, H., Hermanspahn, N., Karshenboim, S.G., Kluge, H.-J., Quint, W., Stahl, S., Verdú, J., Werth, G.: Phys. Rev. Lett. 88, 011603 (00) 5. Layzer, A.J.: Phys. Rev. Lett. 4, 580 (1960) 6. Pachucki, K.: Ann. Phys. (N.Y.) 6, 1 (1993) 7. Pachucki, K.: Phys. Rev. Lett. 7, 3154 (1994) 8. Eides, M.I., Shelyuto, V.A.: Phys. Rev., A 5, 954 (1995) 9. Jentschura, U., Czarnecki, A., Pachucki, K.: Phys. Rev., A 7, 0610 (005) 10. Pask, T., Barna, D., Dax, A., Hayano, R.S., Hori, M., Horváth, D., Juhász, B., Malbrunot, C., Marton, J., Ono, N., Suzuki, K., Zmeskal, J., Widmann, E.: J. Phys. B 41, 081008 (008) 11. Korobov, V.I., Bakalov, D.: J. Phys. B 34, L519 L53 (001) 1. Kinoshita, T., Nio, M.: Phys. Rev., D 53, 4909 (1996) 13. Korobov, V.I., Hilico, L., Karr, J.-P.: Phys. Rev., A 79, 01501 (009) 14. Korobov, V.I.: Phys. Rev., A 67, 06501 (003) 15. Kino, Y., Kudo, H., Kamimura, M.: Mod. Phys. Lett. A 18, 388 (003)