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

Main motivation: improve the determination of mp/me

Ro-vibrational transitions:

2/

)/(

pe

pe

mm

mm

cRmm pe /

2/

)/(

pe

pe

mm

mm

Internuclear distance (in units of a0)

Ene

rgy

(ato

mic

uni

ts)

Relativesensitivity

5.0)/(

/

pe

pe

mm

mms

Required accuracyon Ef - Ei

to match CODATA(4.1 10-10)

10 times better would be nice !

Düsseldorf

HD+

v=0 → v=4

Amsterdam

HD+

v=0 → v=8

Paris

H2+

v=0 → v=1

(m)

(THz)

1.395

215

0.782

384

9.166 (2 photons)

65.4

s 0.438 0.38 0.466

(kHz) 38 60 12

Two-photon transition frequency

(v=0, L=2, J=5/2) (v=1, L=2, J=5/2)

Enr 65 412 414.3359

E (4) 1 077.303(03)E (5) -274.146(02)E (6) -1.980E (7) 0.119(23)

Ep (leading) -0.041(0.3)

E (5/2→5/2) -2.591(02)

Etot 65 413 213.001(24) 2ph 32 706 606.500(12) MHz

nonrecoil

Theory : present status

10107.3

ep

ep

mm

mm

/

/

V.I. KorobovPRA 77, 022509 (2008)

and ref. therein

8 10-10

Proton structure

QED correctionswith recoil

Hyperfine sructure

Total

in progress

Improvement by 2 orders of magnitude

V.I. Korobov,L. Hilico, J.-Ph. Karr

PRA 74, 040502(R) (2006)PRA 79, 012501 (2009)

}

H2+

Enr (u. a.)

-0.596v=0,L=2,J=5/2

v=1,L=2,J=5/2-0.586

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 (m6 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 1sg electronic wave functions (E 10-20 a.u.)

Variational expansion:

Energy corrections are obtained in a form EQED(R)

Average over ro-vibrational wave functions to get EQED(v,L)

21

11

rrV

rV

1

pp

e

R

r2r1

N

i

ririririi eeC

1

2121 r

Exponents i, i are chosen in a quasi-random way.

Ts. Tsogbayar and V.I. KorobovJ. Chem. Phys. 125, 024308 (2006)

The one-loop electron self-energy at order (m)7

ZOZAZAAZ

AZZAAZESE

2262

26160

2

502

41404

lnln

ln

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 differenceU.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)

General one-loop result

U.D. Jentschura, A. Czarnecki, K. Pachucki, PRA 72, 062102 (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

00

0

0

3

0

1

3

2

pp

kHEdkkEL

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

_HR

HR HR

01

00

0000

00

R

R

R

HHE

HEE

HHH

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 >

0

)(kPdkk

0

010 kHE

k2

Preliminary result

Accuracy: < 10-3, excepted at small R (R < 1 a.u.)

L = EL1 + EL2 + EL3

Other contributions

Some of these operator mean values are divergent for S states (in H) or for the 1Sg 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.

r 21 rr K. Pachucki, Ann. Phys. 226,1 (1993)

ZOZAZAAZESE 22

622

61606)7( lnln

Result

A62 = -1

Theoretical accuracy ~ 1 kHz on OK for significantly improved determination of mp/me

Conclusion

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.

Last steps:

What’s next ?

Two-loop self-energy at order m2(Z)6

Vacuum polarization terms

…

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 : exp = theor + 0.31 meV

might be due to the p atom forming a 3-bodyquasibound state (resonance) with an electron

in the H2 gas target.?

p (2S)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, 042506 (2004)

Resonances appear as complex polesof the « rotated » Hamiltonian H(rei).

ER = Eres – i /2

2

- Full three-body dynamics; p atom + particle of charge –e, mass m

m/me

Bin

din

g en

ergy

(eV

)

No resonance for m<25 me !

Lowest 1Se resonance:

p

…but QED shifts must be included

Long-range atom-electron interaction potential:

42 2

or RR

ARV

A: dipole moment

: 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-particleHamiltonian on a radial mesh.

Add the Uehling potential2S-2P Lambshift (without FS and HFS): 207.6358 meVOne-loop vacuum polarization: 205.1584 meV

oPlnnl 1

The Uëhling potential

where = mr r x = me/mr ≈ 0.737…

Vvp(r) ~ ln(r)/r at r → 0Exponential 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

1221 rrr iiie

E.A. Uehling, Phys. Rev. 48, 55 (1935)

Energy shift (meV)

p2S-2P -205.1584

pp ground state -285

dd ground state -413

U.D. Jentschura, Ann. Phys. 326, 500-515 (2011).

G.A. Aissing and H.J. Monkhorst,PRA. 42, 7389 (1990) (1st order pert.).}

See also: A.M. Frolov and D.M. Wardlaw,arXiv:1110.3433v1 (15/10/2011)

Resonant states with Coulomb+Uehling potential

Numerical try: p 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” ?