Interelectronic-interaction effect on the transition probability in high-Z He-like ions

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Interelectronic-interaction effect on the transition probability in high-Z He-like ions

Paul Indelicato, V. M. Shabaev, A. V. Volotka

To cite this version:

Paul Indelicato, V. M. Shabaev, A. V. Volotka. Interelectronic-interaction effect on the transition probability in high-Z He-like ions. Physical Review A, American Physical Society, 2004, 69, pp.062506- 9. �10.1103/PhysRevA.69.062506�. �hal-00002578�

ccsd-00002578, version 1 - 18 Aug 2004

Interelectronic-interaction effect on the transition probability in high-Z He-like ions

P. Indelicato¹, V. M. Shabaev², and A. V. Volotka²

1Laboratoire Kastler-Brossel,

Ecole Normale Sup´´ erieure et Universit´e P. et M. Curie, Case 74, 4, place Jussieu,

75252 Paris CEDEX 05, France

2Department of Physics,

St.Petersburg State University, Oulianovskaya 1, Petrodvorets, St.Petersburg 198504, Russia

The interelectronic-interaction effect on the transition probabilities in high-Z He-like ions is in- vestigated within a systematic quantum electrodynamic approach. The calculation formulas for the interelectronic-interaction corrections of first order in 1/Z are derived using the two-time Green function method. These formulas are employed for numerical evaluations of the magnetic transition probabilities in heliumlike ions. The results of the calculations are compared with experimental values and previous calculations.

PACS numbers: 32.70.Cs, 31.30.Jv

I. INTRODUCTION

During the last few years, transition probabilities in heliumlike ions were calculated by a number of authors [1, 2, 3, 4]. In these calculations, the interelectronic-interaction effects on the transition probabilities were accounted for by employing the relativistic many-body perturbation theory (RMBPT) [1, 2, 4] and the multiconfiguration Dirac- Fock (MCDF) method [3]. Since these methods are based on using the Coulomb-Breit hamiltonian, they have to deal with a separate treatment of the positive- and negative-energy state contributions. As was first indicated in [3], the contribution from the negative-continuum contribution is very sensitive to the choice of the one-electron model potential, which is used as the starting point of any RMBPT or MCDF calculation. In particular, using a standard Dirac-Fock approximation, in [3] it has been demonstrated that to achieve the agreement between theory and experiment for the magnetic dipole transition 2³S1→1¹S0 it is necessary to take into account both correlation and negative-continuum effects. This statement is closely related to a problem of significant numerical cancellations that may occur in low-Z systems, if an improper one-electron approximation is used. For a rigorous QED approach for low-Z systems and corresponding calculations we refer to [5, 6].

The main goal of the present paper is to perform a complete QED calculation of the interelectronic-interaction correction of first order in 1/Zto the magnetic transition probabilities in high-ZHe-like ions. To derive the calculation formulas for these corrections from the first principles of QED we use the two-time Green function method developed in [7, 8, 9] and described in details in [10]. In Sec. II, we formulate the basic equations of this method for the case of nondegenerate states and apply it for the derivation of the desired formulas. The numerical results for the transitions 2³S1→1¹S0, 2³P2→1¹S0, and 3³S1→2³S1are presented in Sec. III. Both Feynman and Coulomb gauges are used for the photon propagator to demonstrate the gauge independence of the final results. The results of the calculations are compared with previous theoretical results and with experiment.

The relativistic units (¯h=c= 1) and the Heaviside charge unit (α=e²/(4π), e <0 ) are used in the paper.

II. BASIC FORMULAS

We consider the transition of a high-Z two-electron ion from an initial stateato a final stateb with the emission of a photon with momentumk_f and polarizationǫ_f. The transition probability is given as

dW = 2π|τγ_f,b;a|²δ(Eb+k⁰_f−Ea)dkf, (1) whereτγ_f,b;a is the transition amplitude which is connected with theS-matrix element by

hk_f,ǫ_f;b|S|ai= 2πiτγf,b;aδ(εb+k⁰_f−εa), (2) andk⁰_f ≡ |kf|.

We assume that in zeroth (one-electron) approximation the initial and final states of the ion are described by one-determinant wave functions

ua(x1,x2) = 1

√2 X

P

(−1)^PψP a1(x1)ψP a2(x2), (3) ub(x1,x2) = 1

√2 X

P

(−1)^PψP b1(x1)ψP b2(x2). (4) To describe the process under consideration we introduce the Green functiongγf,b;a(E^′, E) by

gγf,b;a(E^′, E)δ(E^′+k⁰−E) = 1 2!

Z ∞

−∞

dp⁰₁dp⁰₂dp^′0₁dp^′0₂

×δ(E−p⁰₁−p⁰₂)δ(E^′−p^′0₁ −p^′0₂)

× Z

dx1dx2dx^′₁dx^′₂u^†_b(x^′₁,x^′₂)

×Gγf((p^′0₁,x^′₁),(p^′0₂,x^′₂);k⁰; (p⁰₁,x1),(p⁰₂,x2))γ₁⁰γ₂⁰ua(x1,x2), (5) where

Gγf((p^′0₁,x^′₁),(p^′0₂,x^′₂);k⁰; (p⁰₁,x1),(p⁰₂,x2))

= 2π i

1 (2π)⁵

Z ∞

−∞

dx⁰₁dx⁰₂dx^′0₁dx^′0₂ Z

d⁴y

×exp (ip^′0₁x^′0₁ +ip^′0₂x^′0₂ −ip⁰₁x⁰₁−ip⁰₂x⁰₂+ik⁰y⁰)

×A^ν∗_f (y)h0|T ψ(x^′₁)ψ(x^′₂)jν(y)ψ(x2)ψ(x1)|0i (6) is the Fourier transform of the four-time Green function describing the process, ψ(x) is the electron-positron field operator in the Heisenberg representation, and

A^ν_f(x) = ǫ^ν_fexp (ikf·x) q2k⁰_f(2π)³

(7) is the wave function of the emitted photon. The transition amplitudeSγ_f,b;a ≡ hkf,ǫ_f;b|S|aiis calculated by [7, 8, 10]

Sγf,b;a = Z₃^−1/2δ(Eb+k_f⁰−Ea) I

Γb

dE^′ I

Γa

dEgγf,b;a(E^′, E)

×h 1 2πi

I

Γb

dEgbb(E)i−1/2h 1 2πi

I

Γa

dEgaa(E)i−1/2

. (8)

Heregaa(E) is defined by gaa(E)δ(E^′−E) = 2π

i 1 2!

Z ∞

−∞

dp⁰₁dp⁰₂dp^′0₁dp^′0₂

×δ(E−p⁰₁−p⁰₂)δ(E^′−p^′0₁ −p^′0₂)

× Z

dx₁dx₂dx^′₁dx^′₂×u^†a(x^′₁,x^′₂)G (p^′01,x^′₁),(p^′02,x^′₂); (p⁰1,x₁),(p⁰2,x₂)

γ1⁰γ2⁰ua(x₁,x₂), (9) where

G((p^′0₁,x^′₁),(p^′0₂,x^′₂); (p⁰₁,x1),(p⁰₂,x2))

= 1

(2π)⁴ Z ∞

−∞

dx⁰₁dx⁰₂dx^′0₁dx^′0₂ exp (ip^′0₁x^′0₁ +ip^′0₂x^′0₂ −ip⁰₁x⁰₁−ip⁰₂x⁰₂)

×h0|T ψ(x^′₁)ψ(x^′₂)ψ(x2)ψ(x1)|0i (10) is the Fourier transform of the four-time Green function describing the ion;gbb(E) is defined by a similar equation.

The contours Γa and Γb surround the poles corresponding to the initial and final levels and keep outside all other singularities of the Green functions. It is assumed that they are oriented anticlockwise. The Green functionsGand Gγ_f are constructed by perturbation theory after the transition to the interaction representation and using Wick’s theorem. The Feynman rules forGandGγf are given, e.g., in [10].

Below we consider the transition probability in high-Z He-like ion to zeroth and first order in 1/Z.

A. Zeroth order approximation

To zeroth order in 1/Z the transition amplitude is described by the diagrams shown in Fig. 11. Formula (8) gives S_γ⁽⁰⁾_f,b;a=δ(Eb+k_f⁰−Ea)

I

Γ_b

dE^′ I

Γa

dE g_γ⁽⁰⁾_f,b;a(E^′, E), (11) where the superscript indicates the order in 1/Z. According to definition (5) and the Feynman rules forGγ_f [10], we have

g⁽⁰⁾_γf,b;a(E^′, E)δ(E^′+k⁰−E)

=X

P

(−1)^P Z ∞

−∞

dp⁰₁dp⁰₂dp^′0₁dp^′0₂δ(E−p⁰₁−p⁰₂)δ(E^′−p^′0₁ −p^′0₂)

×n hP b1| i

2π X

n1

|n1ihn1| p^′0₁ −εn1(1−i0)

2π

i eαµδ(p^′0₁ +k⁰−p⁰₁)A^µ∗_f

× i 2π

X

n2

|n2ihn2|

p⁰₁−εn2(1−i0)|a1ihP b2| i 2π

X

n3

|n3ihn3|

p⁰₂−εn3(1−i0)|a2iδ(p^′0₂ −p⁰₂) +hP b1| i

2π X

n1

|n1ihn1|

p⁰₁−εn1(1−i0)|a1iδ(p^′0₁ −p⁰₁)

×hP b2| i 2π

X

n2

|n2ihn2| p^′0₂ −εn2(1−i0)

2π

i eαµδ(p^′0₂ +k⁰−p⁰₂)A^µ∗_f i 2π

X

n3

|n3ihn3|

p⁰₂−εn3(1−i0)|a2io

, (12)

whereα^µ=γ⁰γ^µ= (1,α). One obtains g⁽⁰⁾_γf,b;a(E^′, E) = i

2π 2Z ∞

−∞

dp⁰₁ 1

p⁰₁−(E−E^′)−εb1+i0hb1|eαµA^µ∗_f |a1i 1 p⁰₁−εa1+i0

δa2b2

E−p⁰₁−εa2+i0 + i

2π 2Z ∞

−∞

dp⁰₂ 1

p⁰₂−(E−E^′)−εb2+i0hb2|eαµA^µ∗_f |a2i 1 p⁰₂−εa2+i0

δa1b1

E−p⁰₂−εa1+i0 (13)

= i 2π

1

E^′−E_b⁽⁰⁾[hb1|eαµA^µ∗_f |a1iδa2b2+hb2|eαµA^µ∗_f |a2iδa1b1] 1 E−Ea⁽⁰⁾

, (14)

whereEa⁽⁰⁾ =εa1 +εa2 andE_b⁽⁰⁾ =εb1+εb2. Substituting this expression into equation (11) and integrating overE andE^′ we find

S_γ⁽⁰⁾_f,b;a =−2πiδ(Eb+k⁰_f−Ea)[hb1|eαµA^µ∗_f |a1iδa2b2+hb2|eαµA^µ∗_f |a2iδa1b1] (15) or, according to definition (2),

τ_γ⁽⁰⁾_f,b;a=−

hb1|eαµA^µ∗_f |a1iδa2b2+hb2|eαµA^µ∗_f |a2iδa1b1

. (16)

The corresponding transition probability is

dW⁽⁰⁾= 2π|τ_γ⁽⁰⁾_f,b;a|²δ(Eb+k⁰_f−Ea)dkf. (17)

B. Interelectronic-interaction corrections of first order in 1/Z

The interelectronic-interaction corrections to the transition amplitude of first order in 1/Zare defined by diagrams shown in Fig. 2a,b. Formula (8) yields in the order under consideration

S_γ⁽¹⁾_f,b;a = δ(Eb+k⁰_f−Ea)hI

Γb

dE^′ I

Γa

dE g⁽¹⁾_γf,b;a(E^′, E)

−1 2

I

Γb

dE^′ I

Γa

dE g⁽⁰⁾_γf,b;a(E^′, E) 1 2πi

I

Γa

dE g_aa⁽¹⁾(E) + 1 2πi

I

Γb

dE g⁽¹⁾_bb (E)i

, (18)

wheregaa⁽¹⁾(E) andg⁽¹⁾_bb (E) are defined by the first order interelectronic-interaction diagram (Fig. 3). Let us consider first the contribution of the diagrams shown in Fig. 2a. According to the definition (5) and the Feynman rules for Gγf [10], we have

g_γ^(1a)_f,b;a(E^′, E)δ(E^′+k⁰−E)

=X

P

(−1)^P Z ∞

−∞

dp⁰₁dp⁰₂dp^′0₁dp^′0₂δ(E−p⁰₁−p⁰₂)δ(E^′−p^′0₁ −p^′0₂)

× i 2π

3Z ∞

−∞

dq⁰dωn 1 p^′0₁ −εP b1+i0

X

n

hP b1|eαµA^µ∗_f |ni 1 q⁰−εn(1−i0)

×hnP b2|I(ω)|a1a2i 1 p⁰₁−εa1+i0

1 p^′0₂ −εP b2+i0

1 p⁰₂−εa2+i0

×δ(p⁰₁−ω−q⁰)δ(q⁰−k⁰−p^′0₁)δ(p⁰₂+ω−p^′0₂)

+ 1

p^′0₂ −εP b2+i0 X

n

hP b2|eαµA^µ∗_f |ni 1 q⁰−εn(1−i0)

×hP b1n|I(ω)|a1a2i 1 p⁰₂−εa2+i0

1 p^′0₁ −εP b1+i0

1 p⁰₁−εa1+i0

×δ(p⁰₂−ω−q⁰)δ(q⁰−k⁰−p^′0₂)δ(p⁰₁+ω−p^′0₁)o

, (19)

whereI(ω) =e²α^µα^νDµν(ω) and

Dρσ(ω,x−y) =−gρσ

Z dk (2π)³

exp (ik·(x−y))

ω²−k²+i0 (20)

is the photon propagator in the Feynman gauge. One finds g_γ^(1a)_f,b;a(E^′, E) = i

2π 3X

P

(−1)^PX

n

Z ∞

−∞

dp⁰₂dp^′0₂ 1 p^′0₂ −εP b2+i0

1

E^′−p^′0₂ −εP b1+i0

× 1

p⁰₂−εa2+i0

1

E−p⁰₂−εa1+i0hP b1|eαµA^µ∗_f |ni

× 1

E−p^′0₂ −εn(1−i0)hnP b2|I(p^′0₂ −p⁰₂)|a1a2i + i

2π 3X

P

(−1)^PX

n

Z ∞

−∞

dp⁰₁dp^′0₁ 1 p^′0₁ −εP b1+i0

1

E^′−p^′0₁ −εP b2+i0

× 1

p⁰₁−εa1+i0

1

E−p⁰₁−εa2+i0hP b2|eαµA^µ∗_f |ni

× 1

E−p^′0₁ −εn(1−i0)hP b1n|I(p^′0₁ −p⁰₁)|a1a2i. (21) The expression (21) is conveniently divided into irreducible and reducible parts. The reducible part is the one with εP b2+εn =Ea⁽⁰⁾ in first term and withεP b1+εn =Ea⁽⁰⁾ in second term. The irredicible part is the reminder. Using the identities

1 p⁰₁−εa1+i0

1

E−p⁰₁−εa2+i0 = 1 E−Ea⁽⁰⁾

1

p⁰₁−εa1+i0+ 1

E−p⁰₁−εa2+i0

, (22) 1

p^′0₁ −εP b1+i0

1

E^′−p^′0₁ −εP b2+i0 = 1 E^′−E_b⁽⁰⁾

1

p^′0₁ −εP b1+i0+ 1

E^′−p^′0₁ −εP b2+i0

, (23)

we obtain for the irreducible part τ_γ^(1a,irred)_f,b;a = 1

2πi I

Γ_b

dE^′ I

Γa

dE g_γ^(1a,irred)_f,b;a (E^′, E)

= 1

2πi I

Γb

dE^′ I

Γa

dE 1

E^′−E_b⁽⁰⁾ 1 E−Ea⁽⁰⁾

×nX

P

(−1)^P i 2π

3Z ∞

−∞

dp⁰₂dp^′0₂ 1

p^′0₂ −εP b2+i0 + 1

E^′−p^′0₂ −εP b1+i0

× 1

p⁰₂−εa2+i0 + 1

E−p⁰₂−εa1+i0

εP b2+εn6=E_a⁽⁰⁾

X

n

hP b1|eαµA^µ∗_f |ni

× 1

E−p^′0₂ −εn(1−i0)hnP b2|I(p^′0₂ −p⁰₂)|a1a2i

+X

P

(−1)^P i 2π

3Z ∞

−∞

dp⁰₁dp^′0₁ 1

p^′0₁ −εP b1+i0+ 1

E^′−p^′0₁ −εP b2+i0

× 1

p⁰₁−εa1+i0 + 1

E−p⁰₁−εa2+i0

ε_{P b1}+εn6=E_a⁽⁰⁾

X

n

hP b2|eαµA^µ∗_f |ni

× 1

E−p^′0₁ −εn(1−i0)hP b1n|I(p^′0₁ −p⁰₁)|a1a2io

. (24)

The expression in the curly braces of equation (24) is a regular function ofE or E^′ when E ≈Ea⁽⁰⁾ andE^′ ≈E_b⁽⁰⁾ (see [10] for details). Calculating the residues and taking into account the identity

i 2π

1

x+i0+ 1

−x+i0

=δ(x), (25) we find

τ_γ^(1a,irred)_f,b;a = −X

P

(−1)^Pn

εP b2+εn6=E_a⁽⁰⁾

X

n

hP b1|eαµA^µ∗_f |ni 1 Ea⁽⁰⁾−εP b2−εn

hnP b2|I(εP b2−εa2)|a1a2i

+

εP b1+εn6=E⁽⁰⁾_a

X

n

hP b2|eαµA^µ∗_f |ni 1 E⁽⁰⁾a −εP b1−εn

hP b1n|I(εP b1−εa1)|a1a2io

. (26)

A similar calculation of the irreducible part of the diagrams shown in Fig. 2b yields

τ_γ^(1b,irred)_f,b;a = −X

P

(−1)^Pn

εa2+εn6=E_b⁽⁰⁾

X

n

hP b1P b2|I(εP b2−εa2)|na2i 1 E_b⁽⁰⁾−εa2−εn

hn|eαµA^µ∗_f |a1i

+

εa1+εn6=E⁽⁰⁾_b

X

n

hP b1P b2|I(εP b1−εa1)|a1ni 1 E_b⁽⁰⁾−εa1−εn

hn|eαµA^µ∗_f |a2io

. (27)

For the reducible part of the diagrams shown in Fig. 2a we have τ_γ^(1a,red)_f,b;a = 1

2πi I

Γb

dE^′ I

Γa

dE g^(1a,red)_γf,b;a (E^′, E)

= 1 2πi

I

Γ_b

dE^′ I

Γa

dE 1

E^′−E_b⁽⁰⁾ 1 E−Ea⁽⁰⁾

nX

P

(−1)^P i 2π

3Z ∞

−∞

dp⁰₂dp^′0₂

×

εP b2+εn=E⁽⁰⁾_a

X

n

h 1 E−Ea⁽⁰⁾

1

p^′0₂ −εP b2+i0+ 1

E−p^′0₂ −εn+i0

+ 1

E^′−p^′0₂ −εP b1+i0

1

E−p^′0₂ −εn+i0

i 1

p⁰₂−εa2+i0 + 1

E−p⁰₂−εa1+i0

×hP b1|eαµA^µ∗_f |nihnP b2|I(p^′0₂ −p⁰₂)|a1a2i

+X

P

(−1)^P i 2π

3Z ∞

−∞

dp⁰₁dp^′0₁

εP b1+εn=E⁽⁰⁾_a

X

n