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
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)
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” ?