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High-accuracy calculations in H 2 +

High-accuracy calculations in H 2 +. Jean-Philippe Karr , Laurent Hilico Laboratoire Kastler Brossel (UPMC/ENS) Université d’Evry Val d’Essonne. Vladimir Korobov Joint Institute for Nuclear Research Dubna, Russia. Ro-vibrational transitions:.

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High-accuracy calculations in H 2 +

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  1. High-accuracy calculations in H2+ Jean-Philippe Karr, Laurent Hilico Laboratoire Kastler Brossel (UPMC/ENS) Université d’Evry Val d’Essonne Vladimir Korobov Joint Institute for Nuclear Research Dubna, Russia

  2. Ro-vibrational transitions: Main motivation: improve the determination of mp/me Energy (atomic units) Relative sensitivity Internuclear distance (in units of a0) Required accuracy on Ef - Ei to match CODATA (4.1 10-10) 10 times better would be nice !

  3. Theory : present status H2+ Two-photon transition frequency (v=0, L=2, J=5/2) (v=1, L=2, J=5/2) DEnr 65 412 414.3359 DE(a4) 1 077.303(03) DE(a5) -274.146(02) DE(a6) -1.980 DE(a7) 0.119(23) DEp (leading) -0.041(0.3) DE(5/2→5/2) -2.591(02) DEtot 65 413 213.001(24) n2ph 32 706 606.500(12) MHz } V.I. Korobov PRA 77, 022509 (2008) and ref. therein with recoil QED corrections nonrecoil in progress Proton structure V.I. Korobov, L. Hilico, J.-Ph. Karr PRA 74, 040502(R) (2006) PRA 79, 012501 (2009) Hyperfine sructure Total 8  10-10 Improvement by 2 orders of magnitude

  4. How can the precision be so high ? Enr (u. a.) v=1,L=2,J=5/2 -0.586 Nonrelativistic energies n QED corrections v=0,L=2,J=5/2 -0.596 • Leading terms: corrections to the electronic energy • Weak dependence on v quasi-cancellation • correction to n 1-2% of correction to energy levels • Hyperfine structure depends on (L,J) • two-photon transitions are more favorable • because one can have L=L’ , J=J’

  5. e r1 r2 R p p Theoretical approach • At high orders (ma6 and above) it is sufficient to consider the correction to the electron in the field of the nuclei (nonrecoil limit) Similar to H atom with instead of • Effective Hamiltonian approach: QED corrections are expressed as effective operator mean values • For a grid of values of R, we obtain very precise 1ssg electronic wave functions (DE 10-20 a.u.) Variational expansion: Energy corrections are obtained in a form DEQED(R) Average over ro-vibrational wave functions to get DEQED(v,L) Ts. Tsogbayar and V.I. Korobov J. Chem. Phys. 125, 024308 (2006) Exponents ai, bi are chosen in a quasi-random way.

  6. The one-loop electron self-energy at order (ma)7 • A long-standing problem in hydrogen atom calculations • First high-precision calculation of A60 for 1S and 2S states • K. Pachucki, Ann. Phys. 226,1 (1993) • Derivation of effective operators following NRQED approach; 1S-nS difference • U.D. Jentschura, A. Czarnecki, K. Pachucki, PRA 72, 062102 (2005) • These methods must be adapted to the H2+ case the wave functions are not known analytically numerical calculations NB. The required precision is not too high ( 10-3)

  7. General one-loop result System: electron in an external potential V Valid for l ≠ 0 states and S-state difference: the high-energy part in d(r) drops out Rel. Bethe log. U.D. Jentschura, A. Czarnecki, K. Pachucki, PRA 72, 062102 (2005)

  8. Low energy part: relativistic Bethe logarithm • Leading order a(Za)4 Bethe logarithm • Term in ln(l) cancelled by the high energy part • Term in l cancelled by mass counter-term • Order a(Za)6 : relativistic corrections to the Bethe logarithm HR Relativistic dipole _ HR HR Nonrelativistic quadrupole Relativistic correction to the current

  9. Numerical approach • Calculate numerically the integrands using a variational wave function • Numerical integration: • Find the asymptotic behavior of P(k) at k → ∞ - first order perturbation wave function y1 : - approximate form of y1 for k →  : Example: Following terms are evaluated by a fitting procedure. • Analytical integration of the asymptotic form for k >L

  10. Preliminary result L = EL1 + EL2 + EL3 Accuracy: < 10-3, excepted at small R (R < 1 a.u.)

  11. Other contributions • Some of these operator mean values are divergent for S states (in H) • or for the 1Ssg electronic state (in H2+) • Analytical work to extract the divergent part • The obtained finite expression differs from the exact H(1S) result of by the high-energy part i.e. some constant C times (or in H2+). The coefficient C is easily deduced from comparison between the expressions. K. Pachucki, Ann. Phys. 226,1 (1993)

  12. Result A62 = -1

  13. Conclusion Last steps: • Refine the numerical method for low-energy part  accurate values of A60(R) for all R. • Average over ro-vibrational wave functions  correction to ro-vibrational levels. Theoretical accuracy ~ 1 kHz on n OK for significantly improved determination of mp/me  What’s next ? • Two-loop self-energy at order ma2(Za)6 • Vacuum polarization terms • …

  14. And now, for something completely different The muonic hydrogen experiment revisited by U. Jentschura: Ann. Phys. 326, 500-515 and 516-533 (2011). • The observed discrepancy : nexp = ntheor + 0.31 meV might be due to the pm atom forming a 3-body • quasibound state (resonance) with an electron • in the H2 gas target. ? • Order-of magnitude estimate: “In order to assess the validity of the pm-e- atom hypothesis, one would have to calculate its spectrum, its ionization cross sections in collisions with other molecules in the gas target. Furthermore, it would be necessary to study the inner Auger rates of pm-e- as a function of the state of the outer electron, and its production cross sections in the collisions that take place in the molecular hydrogen target used in the experiment.” e- pm (2S)

  15. First check: Schrödinger Hamiltonian (QED effects not included) - Method: Complex Coordinate Rotation 2q Resonances appear as complex poles of the « rotated » Hamiltonian H(reiq). ER = Eres – i G /2 Resonances of ppm and ddm molecules: S. Kilic, J.-Ph. Karr, L. Hilico, PRA 70, 042506 (2004) - Full three-body dynamics; pm atom + particle of charge –e, mass m Lowest 1Se resonance: pmm No resonance for m<25 me ! Binding energy (eV) m/me

  16. …but QED shifts must be included Long-range atom-electron interaction potential: A: dipole moment a: dipole polarizability Charge-dipole Charge-induced dipole Schrödinger Hamiltonian: A≠ 0 (2S-2P degeneracy) V(R) ~ 1/R2 With QED shifts: A = 0 V(R) ~ 1/R4 • How to add QED level shifts to the Schrödinger Hamiltonian ? • 1Po resonances of H- below n=2: E. Lindroth, PRA 57, R685 (1998) • discrete numerical basis set, obtained by discretization of the one-particle • Hamiltonian on a radial mesh. • Add the Uehling potential 2S-2P Lambshift (without FS and HFS): 207.6358 meV One-loop vacuum polarization: 205.1584 meV

  17. The Uëhling potential where r =a mr r x = me/a mr ≈ 0.737… Vvp(r) ~ ln(r)/r at r → 0 Exponential decrease atr →  E.A. Uehling, Phys. Rev. 48, 55 (1935) • Matrix elements of the Uehling potential can be obtained analytically for exponential basis functions • Nonperturbative treatment: Schrödinger equation with Coulomb + Uehling potential • Check: Consistent with published results for muonic systems See also: A.M. Frolov and D.M. Wardlaw, arXiv:1110.3433v1 (15/10/2011) U.D. Jentschura, Ann. Phys. 326, 500-515 (2011). } G.A. Aissing and H.J. Monkhorst, PRA. 42, 7389 (1990) (1st order pert.).

  18. Resonant states with Coulomb+Uehling potential Numerical try: pm atom + particle (-e, m = 100 me) • Conclusions • A nonperturbative treatment of one-loop vacuum polarization in three-body • systems is feasible. • Application to resonant states raises a question: • is the Uehling potential “dilation analytic” ?

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