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

Main motivation: improve the determination of m p /m e Ro-vibrational transitions: Internuclear distance (in units of a 0 ) Energy (atomic units) Relative sensitivity Required accuracy on E f - E i to match CODATA ( ) 10 times better would be nice ! Düsseldorf HD + v=0 → v=4 Amsterdam HD + v=0 → v=8 Paris H 2 + v=0 → v=1 (  m) (THz) (2 photons) 65.4 s  (kHz)

Two-photon transition frequency ( v=0, L=2, J=5/2 )  ( v=1, L=2, J=5/2 )  E nr  E (  4 ) (03)  E (  5 ) (02)  E (  6 )  E (  7 ) 0.119(23)  E p (leading) (0.3)  E ( 5/2→5/2 ) (02)   E tot (24) 2ph (12) MHz nonrecoil Theory : present status V.I. Korobov PRA 77, (2008) and ref. therein 8  Proton structure QED corrections with recoil Hyperfine sructure Total in progress Improvement by 2 orders of magnitude V.I. Korobov, L. Hilico, J.-Ph. Karr PRA 74, (R) (2006) PRA 79, (2009) } H2+H2+

E nr (u. a.) v=0,L=2,J=5/2 v=1,L=2,J=5/ How can the precision be so high ? Leading terms: corrections to the electronic energy Weak dependence on v quasi-cancellation correction to  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’ Nonrelativistic energies QED corrections

Theoretical approach At high orders (m  6 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 1s  g electronic wave functions (  E  a.u.) Variational expansion: Energy corrections are obtained in a form  E QED (R) Average over ro-vibrational wave functions to get  E QED (v,L) pp e R r2r2 r1r1 Exponents  i,  i are chosen in a quasi-random way. Ts. Tsogbayar and V.I. Korobov J. Chem. Phys. 125, (2006)

The one-loop electron self-energy at order (m  ) 7  A long-standing problem in hydrogen atom calculations - First high-precision calculation of A 60 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, (2005)  These methods must be adapted to the H 2 + case the wave functions are not known analytically numerical calculations NB. The required precision is not too high (  )

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

Low energy part: relativistic Bethe logarithm  Order  (Z  ) 6 : relativistic corrections to the Bethe logarithm  Leading order  (Z  ) 4 Term in ln( ) cancelled by the high energy part Term in cancelled by mass counter-term Bethe logarithm Relativistic dipole Nonrelativistic quadrupole Relativistic correction to the current _ HRHR HRHR HRHR

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  1 : - approximate form of  1 for k →  : Example: Following terms are evaluated by a fitting procedure.  Analytical integration of the asymptotic form for k > 

Preliminary result Accuracy: < 10 -3, excepted at small R (R < 1 a.u.) L = E L1 + E L2 + E L3

Other contributions  Some of these operator mean values are divergent for S states (in H) or for the 1S  g electronic state (in H 2 + )  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 H 2 + ). The coefficient C is easily deduced from comparison between the expressions. K. Pachucki, Ann. Phys. 226,1 (1993)

Result A 62 = -1

Theoretical accuracy ~ 1 kHz on OK for significantly improved determination of m p /m e  Conclusion  Refine the numerical method for low-energy part  accurate values of A 60 (R) for all R.  Average over ro-vibrational wave functions  correction to ro-vibrational levels. Last steps: What’s next ?  Two-loop self-energy at order m  2 (Z  ) 6  Vacuum polarization terms  …

And now, for something completely different The muonic hydrogen experiment revisited by U. Jentschura: Ann. Phys. 326, and (2011). The observed discrepancy : exp = theor meV might be due to the p  atom forming a 3-body  quasibound state (resonance) with an electron  in the H 2 gas target. ? p  (2S) e-e-  Order-of magnitude estimate: “In order to assess the validity of the p  - 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 p  - 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.”

First check: Schrödinger Hamiltonian (QED effects not included) - Method: Complex Coordinate Rotation Resonances of pp  and dd  molecules: S. Kilic, J.-Ph. Karr, L. Hilico, PRA 70, (2004) Resonances appear as complex poles of the « rotated » Hamiltonian H(re i  ). E R = E res – i  /2 22 - Full three-body dynamics; p  atom + particle of charge –e, mass m m/m e Binding energy (eV) No resonance for m<25 m e ! Lowest 1 S e resonance: p 

…but QED shifts must be included Long-range atom-electron interaction potential: A : dipole moment  : dipole polarizability Charge-dipoleCharge-induced dipole Schrödinger Hamiltonian: A ≠ 0 (2S-2P degeneracy) V(R) ~ 1/R 2 With QED shifts: A = 0 V(R) ~ 1/R 4 How to add QED level shifts to the Schrödinger Hamiltonian ?  1 P o 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): meV One-loop vacuum polarization: meV

The Uëhling potential where  =  m r r x = m e /  m r ≈ 0.737… V vp (r) ~ ln(r)/r at r → 0 Exponential decrease at r →  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 E.A. Uehling, Phys. Rev. 48, 55 (1935) Energy shift (meV) p   2S-2P pp  ground state -285 dd  ground state -413 U.D. Jentschura, Ann. Phys. 326, (2011). G.A. Aissing and H.J. Monkhorst, PRA. 42, 7389 (1990) (1 st order pert.). } See also: A.M. Frolov and D.M. Wardlaw, arXiv: v1 (15/10/2011)

Resonant states with Coulomb+Uehling potential Numerical try: p  atom + particle (- e, m = 100 m e ) 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” ?