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Vacuum polarization in Coulomb field revisited
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Vacuum polarization in Coulomb field revisited J. Zamastil ∗ , D. Šimsa Department of Chemical Physics and Optics, Charles University, Faculty of Mathematics and Physics, Ke Karlovu 3, 121 16 Prague 2, Czech Republic
article
Article history: Received 16 June 2016 Accepted 14 February 2017 Available online xxxx Keywords: Bound-state QED Vacuum polarization Renormalization
abstract
info
Simplified derivation of Wichmann–Kroll term is presented. The derivation uses two formulas for hypergeometric functions, but otherwise is elementary. It is found that Laplace transform of the vacuum charge density diverges at zero momentum transfer. This divergence has nothing to do with known ultraviolet divergence. The latter is related to the large momentum behavior of the pertinent integral, while the former to the small momentum behavior. When these divergences are removed, the energy shift caused by vacuum polarization for an ordinary hydrogen obtained here is in an exact agreement with the result obtained by Wichmann and Kroll. Also, for muonic hydrogen the result obtained here reasonably agrees with that given in literature. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Evaluation of the influence of one-loop vacuum polarization in Coulomb field on the atomic energy levels beyond the leading, Uehling, term was accomplished long time ago by Wichmann and Kroll [1]. In view of fundamental importance of this result in establishing proton radius puzzle [2] and complexity of original derivation [1], it seems desirable first, to check their result, second, to clarify the assumptions upon which it is based. In fact, Brown et al. [3] checked the 1/r part of the potential created by vacuum polarization, but otherwise, no independent derivation of Wichmann and Kroll result has been made. Wichmann and Kroll gave expression for Laplace transform of the potential created by induced charge density. Their result was transformed to coordinate space by Blomqvist [4]. Recent evaluations of the influence of Wichmann and Kroll term on the energy levels of
∗
Corresponding author. E-mail address: [email protected] (J. Zamastil).
http://dx.doi.org/10.1016/j.aop.2017.02.008 0003-4916/© 2017 Elsevier Inc. All rights reserved.
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muonic hydrogen [5,6] are based on the approximations of the corresponding potential given in [7,8], which are in turn based on Blomqvist’s work. We follow Wichmann and Kroll in reducing the problem to the calculation of Laplace transform. However, instead of using relativistic Coulomb Green functions we use the spectral decomposition of second order Dirac hamiltonian and the two formulas for hypergeometric functions. It is well-known that one-loop vacuum polarization yields a divergent result. Physically meaningful result is obtained only after the regularization and charge renormalization are performed. In this respect the regularization and renormalization performed in [1] (and also in [3]) are somewhat unusual [9] and relation of such a procedure to the standard one is, at least in present authors opinion, somewhat unclear. In view of the discussion of how much of the proton radius puzzle can be ascribed to the evaluation of vacuum polarization, see e.g. [10] and references therein, this relation should be clarified. In this paper we find, in contrast to the previous papers on the subject, that Laplace transform of the vacuum charge density diverges at zero (imaginary) momentum transfer. This divergence does not appear at the leading, Uehling, term. It is related to small momentum behavior of the pertinent integral. While known ultraviolet divergence, that is also present, can be removed by renormalization of the vacuum charge density at any finite value of the momentum transfer, this is not true for the divergence at the zero momentum transfer. We show here that if the renormalization of the vacuum charge density at the zero momentum transfer is made, the results found by Wichmann and Kroll follow. In this paper, we restrict the discussion to the case of light hydrogen-like atoms with low nuclear charge Z . This allows us to consider non-relativistic approximation to the wave function of the reference state and a point nucleus. As we shall see, the relativistic effects are important for intermediate states. However, for the reference state they yield the contribution to the energy shift that is of the order α(Z α)8 probably multiplied by ln(Z α). At the present level of accuracy, this is negligible for both ordinary and muonic hydrogen. The case of atoms with medium and high nuclear charges is discussed for example in [11,12]. The paper is organized as follows. In the following section we show first that the energy shift can be related to the Laplace transform of the vacuum charge density. Second, we calculate this density. Though, admittedly, this Section is somewhat long and technical, once the two formulas for the hypergeometric functions are used, the calculation is quite elementary. Section 3 deals with more conceptual issues of regularization and renormalization. The results of these two Sections are then applied to the calculation of the energy shift in ordinary and muonic hydrogen in Sections 4 and 5, respectively. Finally, some conclusions are drawn in Section 6. 2. The method 2.1. Relation of energy shift and Laplace transform of vacuum charge density The energy shift of the ground state of the hydrogen-like atom caused by vacuum polarization reads
1E =
d3 ⃗ r |ψ0 (⃗ r )|2 V (⃗ r ) = N02
d3 ⃗ r exp{−β r }V (⃗ r ),
(1)
where we substituted for the probability density
|ψ0 (⃗r )|2 = N02 exp{−β r },
N02 =
(mr Z α)3 , β = 2mr Z α. π
(2)
The energy shift of the excited states can be obtained by differentiation with respect to parameter β , see Section 5. The potential created by vacuum charge density ⟨ρ(⃗ r )⟩ satisfies Poisson equation
⃗ 2 V (⃗r ) = e⟨ρ(⃗r )⟩. −∇
(3)
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Following [1] we take Laplace transform of this equation. That is, we multiply this equation by exp{−β r } and integrate over the whole space; on the right hand side we obtain q(β) =
r exp{−β r }e⟨ρ(⃗ r )⟩; d3 ⃗
(4)
on the left hand side (lhs) we obtain
⃗ 2 V (⃗r ) = r exp{−β r }(−)∇ d3 ⃗
=
⃗ 2 exp{−β r } r V (⃗ r )(−)∇ d3 ⃗ β2
d 2 + 2β U (β) = β U (β) , dβ dβ d
where in the second equality we used
⃗ 2 exp{−β r } = − −∇
d2 dr 2
+
2 d
r dr
exp{−β r } =
2
−β +
2β r
exp{−β r }
and we introduced the function U (β) =
d3 ⃗ r
exp{−β r }V (⃗ r ). r Whence the Laplace transform of the Poisson equation (3) leads to the relation d
dβ
β 2 U (β) = q(β) ⇒ U (β) = β −2
β
(5)
dβ ′ q(β ′ ).
0
Since the energy shift (1) can be easily expressed through the function U, Eq. (5), we obtain the sought relation between energy shift and Laplace transform of the charge density
β d d U (β) = N02 − β −2 1E = N02 − dβ ′ q(β ′ ) . dβ dβ 0
(6)
2.2. Second order expression for vacuum charge density
ˆ Starting from the charge symmetric expression for charge density of the quantized Dirac field ψ e + ˆ r) ρ(⃗ ˆ r) = ψˆ (⃗r ), ψ(⃗ 2 one obtains the vacuum charge density in the form ⟨ρ(⃗r )⟩ = ⟨0|ρ(⃗ ˆ r )|0⟩ =
e 2 σ
⟨⃗r | Pˆσ− − Pˆσ+ |⃗r ⟩,
(7)
where Pˆ ± are projectors to positive/negative energy states Pˆ σ± = Xσ± Xσ±
Ď
.
Here, Xσ± are solutions of Dirac equation for particle moving in a static external potential Vˆ
(Vˆ + γ0 γ⃗ · p⃗ˆ + γ0 m)Xσ± = Eσ± Xσ± , where γ are Dirac matrices and obviously Eσ+ > 0, Eσ− < 0.
(8)
By introducing an integration over the energy variable one can rewrite Eq. (7) into the form
⟨ρ(⃗r )⟩ = e
C
dE 2π i σ
⟨⃗r |
Pˆ σ+
E − Eσ+ + iε
+
Pˆ σ−
E − Eσ− − iε
|⃗r ⟩
(9)
where under the integration along the contour C is meant half of the sum of the integration over contours C+ and C− . The contours C+ and C− run along the real axis and they are closed by infinite
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semicircle in the lower and upper half of the complex plane, respectively. By means of the spectral decomposition of the Dirac hamiltonian one further rewrites the last equation into the form
⟨ρ(⃗r )⟩ = e
dE
C
=e C
1
|⃗r ⟩
2π i
⟨⃗r |Tr
dE
1 ⟨⃗r |Tr(E − Vˆ − γ0 γ⃗ · p⃗ˆ + γ0 m) |⃗r ⟩,
2π i
⃗ˆ − γ0 m E − Vˆ − γ0 γ⃗ · p
H
(10)
where in the last equality we introduced the second order Dirac hamiltonian
Hˆ = (E − Vˆ )2 − pˆ 2 − m2 − γ0 γ⃗ · [p⃗ˆ , Vˆ ].
(11)
Straightforward expansion of Eq. (10) in powers of this potential yields
⟨ρ(⃗r )⟩ = ⟨ρ1 (⃗r )⟩ + ⟨ρ3 (⃗r )⟩ + · · · , where
⟨ρ1 (⃗r )⟩ = e
dE C
2π i
C
2π i
⟨⃗r |Tr
1
ˆ0 D
Vˆ
1
ˆ0 D
|⃗r ⟩
and
⟨ρ3 (⃗r )⟩ = e
dE
⟨⃗r |Tr
1
ˆ0 D
Vˆ
1
ˆ0 D
Vˆ
1
ˆ0 D
Vˆ
1
ˆ0 D
|⃗r ⟩.
(12)
1 ˆ− Here D 0 denotes the free electron propagator
ˆ 0 = E − γ0 γ⃗ · p⃗ˆ − γ0 m. D By virtue of Furry theorem the terms with even powers of V vanish. The first term, ⟨ρ1 (⃗ r )⟩, is the Uehling term and its calculation can be found in any textbook on quantum field theory, see e.g. [13]. The second term, ⟨ρ3 (⃗ r )⟩, is the Wichmann–Kroll term. Its evaluation is much more difficult. In fact, to the best of our knowledge, the formula (12) has never been used for actual calculation. The only evaluations of this term, complete, or partial, have been based on the known exact solution of Dirac equation for the electron moving in the Coulomb potential. To this solution we turn next. 2.3. Solution of second order Dirac equation For the Coulomb potential Vˆ = −
Zα rˆ
the second order Dirac Hamiltonian (11) simplifies to the form structurally identical with Schrödinger hamiltonian 2
2
Hˆ = E − m + 2
EZ α rˆ
−
Γˆ (Γˆ − 1) ˆ + rˆ 2
p2r
,
(13)
where the radial momentum operator pˆ r in the coordinate representation reads
pˆ r = −i
d dr
+
1 r
.
Further, following [3,14] we introduced the relativistic angular momentum operator Γˆ
Γˆ = γ0 Kˆ + i(Z α)γ⃗ · n⃗ ,
⃗ · ⃗Lˆ + 1 . Kˆ = γ0 Σ
(14)
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⃗ is the unit vector pointing at an arbitrary direction, ⃗r = r n⃗, Here Kˆ is the relativistic parity operator, n ˆ
and Σ /2 and ⃗L are the usual spin and orbital angular momentum operators, respectively. In contrast to the Dirac hamiltonian on lhs of Eq. (8), the second order Dirac hamiltonian commutes with the relativistic angular momentum operator Γˆ [14]. One of the advantages of the extension (10) is that the second order hamiltonian Hˆ has higher symmetry than the first order hamiltonian, Eq. (8). 2.3.1. Spinor-angular part It follows from the last equation that
Γˆ (Γˆ − 1) = Lˆ 2 − (Z α)2 − i(Z α)γ0 γ⃗ · n⃗.
(15)
Simple calculation yields Kˆ 2 = Jˆ2 +
1 4
and
Γˆ 2 = Kˆ 2 − (Z α)2 ,
ˆ
where ⃗J is the usual total angular momentum operator
⃗Jˆ = ⃗Lˆ + 1 Σ ⃗. 2
The eigenvalues of the operators Kˆ and Γˆ are then obviously K = σ (j + 1/2),
σ = ±1
(16)
and
Γ = ρ (j + 1/2)2 − (Z α)2 ,
ρ = ±1,
(17)
where, as usually, j(j + 1) is the eigenvalue of the operator Jˆ2 . The explicit form of the common eigenvectors |Γ , K , j, m⟩ of the operators Γˆ , Kˆ , Jˆ2 and Jˆz reads
|Γ , K , j, m⟩ =
c1 |j, m⟩σ c2 |j, m⟩−σ
,
(18)
where |j, m⟩σ are well-known spherical spinors. From eigenproblem
Γˆ |Γ , K , j, m⟩ = Γ |Γ , K , j, m⟩ and well-known equations for spherical spinors σ
σ⃗ · n⃗|j, m⟩ = −|j, m⟩
−σ
,
1 ˆ σ ⃗ |j, m⟩σ σ⃗ · L + 1 |j, m⟩ = σ j + 2
(19)
we obtain the relation between the coefficients c c2 =
i Zα
(Γ − K )c1 .
(20)
2.3.2. Radial part Introducing an effective orbital number l l(l + 1) = Γ (Γ − 1)
(21)
we make similarity of hamiltonian (13) with Schrödinger hamiltonian nearly perfect
Hˆ l = E 2 − m2 + 2
EZ α rˆ
p2r
− ˆ +
l(l + 1) rˆ 2
.
(22)
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The eigenfunctions of this operator belonging to the continuous part of the spectrum
Hˆ l Rl (p, r ) = (E 2 − m2 − p2 )Rl (p, r )
(23)
are [15]
Rl (p, r ) =
2 p exp{π a/2}|Γ (l + 1 − ia)|
π
Γ (2l + 2)
× (2pr ) exp{−ipr }F (ia + l + 1, 2l + 2, 2ipr ), l
(24)
where a=
EZ α
(25)
p
and F (α, γ , z ) denotes the confluent hypergeometric function [15,16]. The eigenfunctions belonging to the discrete part of the spectrum are inessential for our purposes. We evaluate the one-loop vacuum polarization in powers of Z α ; though we do not do it directly, as in Eq. (12), we expand around free particle solution. The discrete part of the spectrum does not contribute at any finite order of Z α ; it is a non-perturbative effect. In order for the wave function not to diverge at the origin too strongly l have to be in the limit Z α → 0 nonnegative. Whence physical solutions of Eq. (21) are l = |Γ | − δρ,1 .
(26)
2.4. Spinor-angular integration In Eq. (10) we use the spectral decomposition of the second order hamiltonian 1
Hˆ
=
1 |Γ , K , j, m⟩⟨Γ , K , j, m|γ0 , ˆ ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ ρ,σ ,j,m Hl
where the radial hamiltonian Hˆ l is given by Eq. (22). Somewhat unusual form of this decomposition is given by the fact that the operator Γ is not hermitian, but γ0 Γ is. Further we use the decomposition of the kinetic term
⟨⃗r |γ0 γ⃗ · p⃗ˆ |ψ⟩ = −iγ0 γ⃗ · n⃗
∂ 1 γ0 Kˆ + − ∂r r r
⟨⃗r |ψ⟩
valid for an arbitrary state vector |ψ⟩. By inserting the last two equations into Eq. (10) we easily perform the spinor-angular integration in Eq. (10), see also [3],
⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ = 1, ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ ⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ dΩ Trγ0 ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ σ =±1
dΩ Tr
=
|c1 |2 + |c2 |2 K j + 1/2 = = σ = 0, 2 2 |c | − |c2 | Γ Γ σ =±1 1 σ =±1 σ =±1
⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ c2 c ∗ − c2∗ c1 Zα = −i 12 =− , ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ |c1 | − |c2 |2 Γ ∗ ∗ ⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ c2 c + c2 c1 ⃗γ0 dΩ Triγ0 γ⃗ · n = −i 12 = 0. ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ |c1 | − |c2 |2
⃗ dΩ Triγ0 γ⃗ · n
The first equation follows trivially from the cyclic property of trace. The first equalities in the remaining three equation follow from Eq. (18) and orthonormality relations for spherical spinors. In
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the last two equations, the first property of spherical spinors displayed in Eq. (19) has also been used. The remaining equalities follow from the form of the coefficients c, Eq. (20), and from the explicit form of the eigenvalues K and Γ , Eqs. (16) and (17). To sum up spinor-angular integration in Eq. (10) leads to 1 ⃗ˆ + γ0 m) |⃗r ⟩ dΩ ⟨⃗ r |Tr(E − Vˆ − γ0 γ⃗ · p
H
=
2(2j + 1) E +
Zα
j,ρ
r
−
Zα
1 ∂ 2 ∂r
Γ
+
1
r
⟨r |
1
Hˆ l
|r ⟩.
(27)
The diagonal matrix elements of the inverse of the radial hamiltonian Hˆ l between the eigenstates of the operator rˆ are calculated by means of the spectral decomposition of the radial hamiltonian
⟨r |
∞
1
Hˆ l
|r ⟩ =
dp 0
|Rl (p, r )|2 + ···, − m2 − p2
(28)
E2
where the contribution of the discrete part of the spectrum has been omitted; see remark after Eq. (25). By inserting Eqs. (27) and (28) into Eq. (10) and inserting further that equation into Eq. (4) we get for the Laplace transform of the vacuum charge density
∞ ∞ dE dp k 2 E − p2 − m2 C 2π i 0 k=1 ∂ β ∂ × E − + Zα 1 + Il (p, β), ∂β 2Γ ∂β
q(β) = 8α
(29)
where k=j+
1
(30)
2
and Il (p, β) = 2π
∞
drr exp{−β r }|Rl (p, r )|2 .
(31)
0
When inserting (27) into Eqs. (10) and (4) we integrated by parts 2π
∞
2
drr exp{−β r } 0
1 ∂ 2 ∂r
+
1 r
|Rl (p, r )|2 = −
β ∂ Il (β, p). 2 ∂β
2.5. Radial integration To evaluate the integral Il (p, β), Eq. (31), we use two formulas for the hypergeometric functions. 2.5.1. Two formulas for hypergeometric functions The first formula reads [15]: ∞
dr exp{−λr }r γ −1 F (α, γ , kr )F (α ′ , γ , k′ r ) = Γ (γ )λα+α −γ ′
0 ′
× (λ − k)−α (λ − k′ )−α F (α, α ′ , γ , z ),
z=
kk′
(λ − k)(λ − k′ )
.
(32)
The second formula reads [16]:
1 ′ ′ Γ (γ ) t α −1 (1 − t )γ −α −1 dt , Γ (α ′ )Γ (γ − α ′ ) 0 (1 − tz )α Re(γ ) > Re(α ′ ) > 0, arg(1 − z ) < π .
F (α, α ′ , γ , z ) =
(33)
The proofs of these formulas are not difficult; they can be found in the corresponding references.
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2.5.2. Application to the present case Using Eqs. (24) and (32) we obtain for the integral (31) Il (p, β) =
exp{π a}|Γ (l + 1 − ia)|2 l+1 z Γ (2l + 2)
β − 2ip β + 2ip
−ia
× F (ia + l + 1, −ia + l + 1, 2l + 2, z ),
z=
(2p)2 . β 2 + (2p)2
Using Eq. (33) the last equation can be rewritten as Il (p, β) =
1
0
dt
exp{aA}
t (1 − t )
t (1 − t )z
l+1
1 − tz
,
(34)
where
A = π − 2 arctan
2p
+ i ln
β
1−t t (1 − tz )
.
(35)
2.6. Integration over energy variable The exponential function exp{aA} is an analytic function of the energy E, see Eqs. (25) and (35). The Taylor expansion of this function in E converges for all complex E. Thus, the integration over the energy variable in Eq. (29) is trivial
dE exp{aA} C
2π i E 2 − E02
C 1 2
dE 1 +
=
2π i
C
1 dE 1 + aA + 2 (aA)2 + · · ·
=
2π i
A Z p
α
E 2 − E02
2
E2 + · · ·
dE 1 +
=
E 2 − E02
C
1 2
2π i
A Z p
α
2
E02 + · · ·
E 2 − E02
.
In the last equality we used the identity E2 E2
−
E02
=1+
E02
− E02
E2
and the fact that integral of an analytic function of E, in this case f (E ) = 1, along any closed contour is zero. Proceeding further along these lines one gets
dE exp{aA} C
2π i E 2 − E02
dE cosh{aA}
= C+
2π i E 2 − E02
=−
cosh{a(p)A} 2E0
,
(36)
where E0 =
p2 + m2 ,
a(p) =
E0 p
Zα
(37)
and
dE E exp{aA} C
2π i E 2 − E02
dE E sinh{aA}
= C+
2π i E 2 − E02
=−
sinh{a(p)A} 2
.
(38)
Using Eqs. (31), (36) and (38) in Eq. (29) we obtain for the Laplace transform of the charge density q(β) = −4α
∞
∞
k
k=1
1
dp
0
0
dt t (1 − t )
l+1 ∂ β ∂ cosh{a(p)A} t (1 − t )z × − sinh{a(p)A} + Z α 1 + . ∂β 2Γ ∂β E0 1 − tz
(39)
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For further considerations it is advantageous first to exchange the order of the differentiation with respect to β and the integration with respect to p; second, to make substitution p → β p/2,
(40)
and express the above expression in terms of dimensionless variable
β
. (41) 2m mr After the parametric differentiation with respect to b we set b = m Z α , so b still carries the dependence on Z α . With these changes the Laplace transform of the charge density takes the form then ∞ 1 ∞ dt q(b) = −4α k dp 0 0 t (1 − t ) k =1 b=
×
∂ − ∂b
∂ sinh{a(pb)A(p)} + Z α 1 + 2 2Γ ∂ b
b
b
b cosh{a(pb)A(p)}
(bp)2 + 1
s l +1 ,
(42)
where a(pb) =
(bp)2 + 1 bp
Z α,
A(p) = π − 2 arctan (p) + i ln
1−t
t [1 − tz (p)]
(43)
and z (p) =
p2
1E = mr
,
t (1 − t )z (p)
. (44) 1+ 1 − tz (p) To save the space we write the differentiation with respect to b inside the integration with respect to p, however, it is understood that the integration with respect to p is made before the differentiation with respect to b. The energy shift (6) can be expressed in terms of dimensionless variable b as p2
s=
b m 2 (Z α)3 d r − b −2 db′ q(b′ ) . m m 4π db 0 b= r Z α
(45)
m
2.7. Summation over partial waves Eqs. (42)–(45) are still exact. To obtain closed formula for the sum over all partial waves one is forced to expand the effective orbital number l in powers of (Z α)2 . From Eqs. (17), (26) and (30) we have
ksl+1 =
k,ρ
√ ks
k,ρ
k2 −(Z α)2 −δρ,1 +1
√
= (1 + s)
ks
k2 −(Z α)2
≃ (1 + s)
k
= and
s(1 + s)
(1 − s)
2
ksk −
k
(Z α)2 2
sk ln s
(Z α) 1 + s s ln s 2 1−s 2
−
(46)
√ k k 2 2 sl+1 = s k −(Z α) −δρ,1 +1 2 2 Γ k,ρ k,ρ ρ k − (Z α) √ ks k2 −(Z α)2 (Z α)2 ln s 1 = (1 − s) ≃ ( 1 − s) sk 1 − − 2 2 k k k2 − (Z α)2 k k = s+
(Z α)2 2
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] .
(47)
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Here we used the approximation
√
k2 −(Z α)2
s
≃s
(Z α)2
k
1−
ln s
2
(48)
and elementary summation formulas ∞
, (1 − s)2
k=1
∞ sk k=1
k2
∞
s
ksk =
sk =
k=1
s 1−s
,
∞ k s k=1
k
= − ln(1 − s),
= dilog(1 − s).
Here dilog(1 − s) is the Euler dilogarithm function often denoted by Li2 (s). By inserting Eqs. (46) and (47) into Eq. (42) we obtain q(b) ≃ −4α(Z α)
∞
0
∂ − ∂b
×
1
dp
s(1 + s)
0
b 2Z α
dt t (1 − t )
sinh{a(pb)A(p)} +
b cosh{a(pb)A(p)}
(bp)2 + 1
(Z α)2 s(1 + s) ln(s) b ∂ b cosh{a(pb)A(p)} + (1 − s) 2 1−s 2 ∂b (bp)2 + 1 (Z α)2 × s+ (1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] .
×
2
−
2
(49)
2.8. Expansion in Z α Due to approximation (48), Eq. (49) is valid only up to the order α(Z α)3 , so let us expand the hyperbolic functions in Eq. (49) up to the order (Z α)2 : b Zα
sinh{a(pb)A(p)} ≃ A(p)
b cosh{a(pb)A(p)}
(bp) + 1 2
[1 + (bp)2 ]3/2 + (Z α)2 [A(p)]3 , 6b2 p3 (bp)2 + 1 (Z α)2 A(p) 2
1 + (bp)2
p b
≃ + b 2 p (bp )2 + 1 ∂ 1 + (bp)2 (Z α)2 A(p) 2 = + ∂b p2 2 p 1 × 1 + (bp)2 − arctanh 1 + (bp)2
and b
∂ b cosh{a(pb)A(p)} ∂b (bp)2 + 1 ∂ b (bp)2 + 1 (Z α)2 A(p) 2 ≃b + ∂b b 2 p (bp)2 + 1 ∂ 1 (Z α)2 A(p) 2 1 = − + arctanh . ∂b 2 p p2 1 + (bp)2 1 + (bp)2
(50)
J. Zamastil, D. Šimsa / Annals of Physics (
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Inserting these expansion into Eq. (49) we obtain q(b) ≃ −4α(Z α)
d db
Q (b),
Q (b) = Q1 (b) + (Z α)2 Q3 (b),
(51)
where Q1 (b) =
∞
dp
×
s(1 + s)
(1 − s)2 ∞ 1 dp 0
−
1
−
1 + (bp)2
p
+ −[A(p)]3
1 + (bp)2
2
1−s
p2 (52)
− A(p)
s(1 + s) ln(s)
+
1 + (bp)2
s
1
×
p
1
2 p2
1 + (bp)2
2
1
dt
2
1
− A(p)
t (1 − t )
0
×
dt t (1 − t )
0
0
Q3 (b) =
1
1 + (bp)2 − arctanh
+
p2
[1 + (bp)2 ]3/2 12b2 p3
+
1
A(p)
2
2
p
1 1 + (bp)2
× [ln(1 − s) ln(s) + dilog(1 − s)] +
1 + (bp)2
1
4
A(p)
s(1 + s) 1 1−s − ( 1 − s) 2 4 p2 1 + (bp)2 2
1
arctanh
p
1 + (bp)2
s .
(53)
The function q is real; the imaginary part drops out. The powers of function A(p) in above equations can thus be replaced by, see Eq. (43), A(p) → 2 arctan(1/p),
(54)
[A(p)]2 → [2 arctan(1/p)]2 − ln2
1−t t [1 − tz (p)]
[A(p)] → [2 arctan(1/p)] − 3[2 arctan(1/p)] ln 3
3
2
,
1−t t [1 − tz (p)]
.
2.9. Integration over parameter t To integrate over the parameter t we recall definitions of z (p) and s, Eq. (44). Integrals over t that can be evaluated in the closed form are 1
dt
s(1 + s)
= p2 ,
(55)
s = ln(1 + p2 ),
(56)
t (1 − t ) (1 − s)2
0
1
0 1
dt t (1 − t ) dt
s(1 + s)
t (1 − t ) (1 − s)2
0
ln2
1−t t [1 − tz (p)]
=
p2 3
π 2 − 6 dilog
1 p2 + 1
.
For the remaining integrals the resulting expression is either too complex or we did not succeed to evaluate the integrals in terms of the special functions. Nonetheless, the expansion for large p suffices:
0
1
dt
s(1 + s)
t (1 − t ) 1 − s
ln s = −4(ln(p) + 1) +
+
1 5
ln(p) + p4
17 50
π2 3
−
2 3
ln(p) +
+ O(1/p6 ),
14 9
p2 (57)
12
J. Zamastil, D. Šimsa / Annals of Physics ( 1
dt t (1 − t )
0
s ln2
1−t
= 4ζ (3) − 4
t [1 − tz (p)]
−
22 9
)
1 + ln(p) p2
ln(p) +
35 54
p6
–
+3
ln(p) +
1 2
p4
+ O(1/p8 )
(58)
and 1
0
dt t (1 − t ) π2
+
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] = 4ζ (3)
− 6 − 4 ln(p)
3
+
p2
59 36
−
π2 6
+ 53 ln(p) p4
+
197 − 225 +
π2 9
−
47 45
ln(p)
p6
+ O(1/p8 ).
(59)
These are non-trivial results, however, we shall not dwell on their derivation. 3. Observable part of the effect 3.1. Expansion in b—an example In the following we shall need the expansion in b. To illustrate how it is obtained, let us consider the integral I (b) =
∞
dp 1 + (bp)2 f (p),
0
where f (p) is finite for p → 0 and for large p behaves as f (p) =
c1 p2
+
c2 + d2 ln(p) p4
+ ···.
Direct expansion of the integrand of I (b) in powers of b produces divergent integrals in p. The correct expansion of I (b) contains odd powers of b that cannot be obtained by the direct expansion of I (b). The odd powers of b are obtained easily from the expansion of f (p) for large p as follows I (b) =
∞
dp
0
1 + (bp)2 − 1 p2
0
∞
+
dp 0 4
−
∞
dpf (p) + c1
b
8
1 + (bp)2 − 1
dpp 0
b2 2
∞
dpp2 f (p) − 0
c1
p2
c + d ln(p) 2 2
∞
+
4
f (p) −
c1 p2
−
p4 c2 + d2 ln(p) p4
+ ···.
The strategy used in this equation is the following. When the direct expansion in b does not lead to the divergent integral over p we use it. When it does lead to the divergent integral we use the expansion of function f (p) for large p and further use the subtracted function f (p). The above integrals are finite except for the second. The first yields an uninteresting absolute term. After the regularization the second integral yields linear term in b, see the next subsection. The third integral yields quadratic term in b, the fourth integral yields the terms proportional to b3 and b3 ln(b) and the fifth integral yields the quartic term in b. 3.2. Regularization We regularize the integration over electron momentum in Pauli–Villars manner, by subtracting from the pertinent expression the contribution of another pair of heavy fermions with squared
J. Zamastil, D. Šimsa / Annals of Physics (
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13
mass M,
∞
dp
1 + (bp)2 − 1 p2
0
∞
→
dp
1 + (bp)2 − 1 −
M + (bp)2 +
√ M
p2
0
b
= − ln M . 2
We note that the squared mass M is measured in the units of the electron squared mass m2 . This is essentially the same regularization as that used for example in [10]. The only divergences present in Eqs. (52) and (53) are logarithmic. However, it has to be noted that in Eq. (50) the differentiation with respect to b was tacitly moved inside the integration with respect to p and in Eq. (51) it was tacitly moved back outside the integration with respect to p. These operations are allowed for convergent integrals, not for divergent integrals that we actually have. It is clear that these interchanges of parametric differentiation and integration serve as a kind of regularization. Namely, writing in Eq. (50) b
b 1 ∂ ∂ =− , 2 2 ∂ b (bp) + 1 ∂ b p 1 + (bp)2
one changes logarithmically divergent integral in p to convergent integral in p. Question then arises regarding the relation of such a regularization to the standard Pauli–Villars regularization. In a more careful treatment we should first make replacement
1 + (bp)2 →
M + (bp)2
(60)
and differentiate with respect to M:
2 db
b d
1
dp → b 2 db ∞ 1 + (bp)2
b 0
=−
ln(1 + p2 )
∞
b d
b ∂
1
b
∞
0
ln(1 + p2 ) dp M + (bp)2
ln(1 + p2 )
∞
dM
∂ dM ∂M
dp
(M + (bp)2 )3/2 ∂ ln(1 + p2 )[M + 2(bp)2 ] =− dM dp ∂b ∞ 4p2 (M + (bp)2 )3/2 0 ∞ ∂ M ∞ ∂ ln(1 + p2 ) ln(1 + p2 ) dp 2 + lim dp . =− M →∞ ∂ b 2 ∂b 0 2p (1 + (bp)2 )1/2 p2 (M + (bp)2 )1/2 0 4 ∂b
∞
1
0
∞
The first term is exactly the term brought by Eq. (50). The second term is expanded in powers of b: M 2
ln(1 + p2 )
∞
dp 0
p2 (M + (bp)2 )1/2 ln
∞
+b
dp 0
M 1/2 p b
p2
=
M 1/2 2 1
∞
1 + p2
dp 0
ln(1 + p2 ) p2
− 1 + ···.
In the last displayed term we made the substitution p → pM 1/2 /b. The other terms vanish in the limit M → ∞. The first term is independent on b and vanishes upon the differentiation with respect to b. It is seen that different regularization schemes lead to different terms for charge density that do not vanish in the limit M → ∞. However, these are proportional to terms of the order b and b ln(b) and are removed by renormalization. The same analysis could be repeated also for expression (53). In this case more careful treatment does not bring any change, however. When lhs of Eq. (50) is used instead of the right hand side, the pertinent integral over p is now convergent.
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J. Zamastil, D. Šimsa / Annals of Physics (
)
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3.3. Renormalization Charge density (10) is spherically symmetric. Its Fourier transform then depends just on the magnitude of the exchanged momentum f (ω ) = 2
d3 ⃗ k exp{i⃗ k·⃗ r }⟨ρ(⃗ r )⟩,
ω2 = ⃗k · ⃗k.
The usual way of renormalization, presented in the standard textbooks on the subject in the fourdimensional form, see e.g. [13], is to subtract from this the value f (0). This is because the value of the fine structure constant α ≃ 1/137.036 is obtained from measurements, where the exchanged momentum ⃗ k goes to zero. When considering Laplace transform of the charged density q(b) it is natural to choose ‘‘an analytic continuation’’ of the above condition and subtract from q(b) the part that does not vanish for b going to zero. Thus, from Q (b), see Eq. (51), we subtract everything that in the limit b → 0 goes to zero faster than b2 . In fact, since the energy shift is given by Eq. (45) it is obviously independent on the quadratic terms in b. Thus, we can subtract from Q (b) the expansion near b → 0 up to quadratic terms in b. In this paper, we shall stick to this prescription. Note, however, that there are some subtleties involved. After integration over parameter t, described in the previous section is performed, it is not difficult to see that ultraviolet (large p) behavior of the integrand in Eq. (53) is neither worse nor better of that in Eq. (52). On the other hand, it is seen that Eq. (53) contains terms proportional to inverse powers of b. These negative powers of b are connected with infrared (small p) behavior of the integrand. Further, looking at Eq. (42) it is seen, that the absolute value of the negative power of b is increasing with increasing order of the expansion of the charge density in Z α . By renormalizing the Laplace transform of the charge density at zero momentum transfer we are removing first ultraviolet divergence in p proportional to linear term in b and second, negative powers of b that have nothing to do with ultraviolet behavior. Clearly, there seems to be something special about renormalization at zero momentum transfer. Had the charge were renormalized at non-zero value of momentum transfer, one would conclude that for b approaching zero the charge density diverges. 3.4. Renormalized terms The renormalized expressions for integrated Laplace transform, Eqs. (52) and (53), are Q 1 ( b) =
∞
dp
1 + (bp)2 − 1 −
0
−
1 3p2
−
1 2
1
1 + (bp)2
(bp)2
2
−1+
1
1 − p arctan
(bp)
2
p
ln(1 + p ) − 2 ln(p) 2
(61)
p2
2
and
1 − p arctan 1 1 2 p (bp) Q 3 ( b) = 1 + (bp)2 − 1 − dp − 2 2 p2 0 1 1 3 2 dt s(1 + s) ln(s) [1 + (bp)2 ]3/2 3 2 4 × + − 2− p − b p 1−s b2 b 2 8 0 t (1 − t ) 3 1 2 arctan 1p − 2 arctan 1p π 2 − 6 dilog p21+1 + 2 ln(p) × − − 3 4
∞
12p
+
1 + (bp)2 − 1 −
(bp)2 2
p
− arctanh
1
1 + (bp)2
− ln
pb 2
+
(pb)2 4
J. Zamastil, D. Šimsa / Annals of Physics (
2 arctan
×
1 p
1
dt 0 t (1−t )
× −
1 p2 +1
)
–
15
1 − 2 ln(p)
−
6
(bp)2
−1+
1 + (bp)2
1
π 2 − 6 dilog −
2
+
2
p2
2
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] 4p2
+ arctanh
2 arctan
+ ln
1 + (bp)2
1 p
×
1
2
pb
−
2
ln(1 + p ) − 2
(pb)2
+
ζ (3)
p2
4
1
dt s ln2 0 t (1−t )
1−t t [1−tz (p)]
4p2
ζ (3) + 2 . p
(62)
For practical evaluation of the integrals it is advantageous to use for large p the asymptotic expansions following from Eqs. (57) and (58)
1 p
1 1 − p arctan − 2 2
=
2 3
p
0
π2
(ln(p) + 1) −
18
p4
+
dt
s(1 + s) ln(s)
t (1 − t )
1−s
1
19 − 135 −
13 45
2 ln(p) + π30
+
p6
463 6300
−
π2 42
+
p8
13 70
ln(p)
+ O(1/p10 )
(63)
and
2
ln(1 + p2 ) −
1 p
2 arctan
1
dt s ln2 0 t (1−t )
1 −t t [1−tz (p)]
+
4p2
=
1 + 3 ln(p)
5 8
+
p4
−
25 ln 12 6 p
(p)
+
49 30
ln(p) −
217 216
p8
ζ (3) p2
+ O(1/p10 ).
(64)
The remaining asymptotic expansions are obtained trivially either from Eq. (59) or from the expansion of the functions arctan(x) and dilog(x) for small x. 4. Application to ordinary hydrogen In the case of ordinary hydrogen b ≪ 1 and we can expand the above formulas in powers of b. 4.1. Uehling term Expanding the first order renormalized Laplace transform, Eq. (61), in powers of b Q 1 (b) = Q 13 b3 + Q 14
b4
2 we obtain well-known results Q 13 = −
+
1 b3 1 2
∞
dp 0
1 p4
+ ···,
1 5
1 1 + (bp)2
1 + (bp)2 − 1 −
−1+
(bp) 2
2
(bp)2
=−
2 4 15
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J. Zamastil, D. Šimsa / Annals of Physics (
∞
1
dpp4 −
Q 14 = 2
1
1
1
–
− 2+ 4 3p 5p 2 3 ln(1 + p ) ln(p) 1 5π + − + 2 + 4 = . 2 8
0
8
1 − p arctan
)
2p
p
p
2p
48
4.2. Wichmann–Kroll term Likewise, expanding the third order renormalized Laplace transform, Eq. (62), in powers of b Q 3 (b) = Q 33 b3 + Q 34
b4
+ ···,
2
we obtain Q 33 =
∞
1
dp
b3
0
1 p4
(bp)2
1 + (bp)2 − 1 −
2
[1 + (bp)2 ]3/2
7 5 3 2 3 2 4 12 − 3 ln(p) − p − b p 3 18 3 b2 b2 2 8 p2 2 2 pb ( bp ) 1 ( pb ) + − ln 1 + (bp)2 − 1 − − arctanh + 2 2 4 1 + (bp)2 π2 − 6 − 4 ln(p) 1 (bp)2 19 −1+ × − + ln(p) − 3 12 4 2 1 + (bp)2 1 (pb)2 π2 pb 19 + arctanh − − . (65) + ln (1 + 3 ln(p)) = 2 4 45 27 1 + (bp)2
×
2
−
π2
+
2
ln(p)
+
−
1
It is quite remarkable that all the terms b3 ln(b) arising at intermediate stage of the calculation cancel out in the final expression. It looks like some symmetry (gauge invariance?) is ‘‘working behind the scene’’. Thus, although the integrand depends on b this dependence cancels out in the final expression. Further,
∞
dpp4
Q 34 = 2 0
−
× −
1 8
1 1 − p arctan
−
−
p
16 1 3
3
−
1 p
dt
s(1 + s) ln(s)
t (1 − t )
1−s
1
0
2 arctan
− 2 arctan
−
2 3
−
π2 18
+ 23 ln(p)
p4
1 1 2 π − 6 dilog p p2 +1
12p
+ 2 ln(p) p4
2 arctan
×
1 p
p2
2
2
2
−
7 12
− 53 ln(p) p6
2 1 p
−
1 32
π 2 − 6 dilog −
6
1 p2 +1
−
1 − 2 ln(p) p2
− −
19 12
+ ln(p) 4 p
J. Zamastil, D. Šimsa / Annals of Physics (
+
+
3 8
1
dt 0 t (1−t )
−
ζ (3) p2
π2
17
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)]
− 6 − 4 ln(p)
3
+
3
32
–
4p2
−c
)
4p4
2 1 p
2 arctan
ln(1 + p2 ) −
1
dt s ln2 0 t (1−t )
1 −t t [1−tz (p)]
4p2
1 ζ (3) 1 + 3 ln(p) 31π 2 − 2 − − π. = p p4 16 2880 These results are in an agreement with those found in [1]. We obtained the last result numerically only; we reproduced the first nine digits. 5. Applications to muonic hydrogen In case of muonic hydrogen b ≃ 1.36 and Eqs. (61), (62) have to be evaluated exactly. From Eqs. (45) and (51) we get for the energy shift of the ground state caused by Wichmann–Kroll term
(1E )WK (1s) = me
3
mr me
α(Z α)6 d −2 b Q 3 (b) , m π db b= r Z α
mr =
me
mµ 1+
mµ mp
.
The induced charge density is spherically symmetric. Thus, for non-S-states it is sufficient to consider the angular average of the probability density of the reference state. Taking into account the form of the wave functions of the excited states
|ψ2s (r )| = 2
N02 8
exp{−β r } 1 − β r +
β 2r 2
4
and
dΩ 4π
|ψ2p (⃗r )|2 =
N02 96
exp{−β r }β 2 r 2
where
β = mr Z α,
N02 =
(mr Z α)3 , π
and the substitution (41), the energy shift for these states is obtained by the method of differentiation of an integral with respect to a parameter:
(1E )WK (2s) = me
mr
3
me
d b2 d2 d −2 α(Z α)6 1 × 1+b + b Q 3 (b) m π 8 db 4 db2 db b= r
2me
and
(1E )WK (2p) = me
mr me
3
α(Z α)6 1 2 d2 d −2 b b Q ( b ) 3 mr π 96 db2 db b=
Plugging into these formulas the numerical values [17] me α = 1/137.035999074, = 4.83633166 10−3 , mµ me = 5.4461702178 10−4 , me = 0.510998928 106 eV mp
2me
Zα
.
Zα
18
J. Zamastil, D. Šimsa / Annals of Physics (
)
–
and performing numerical integration we obtain
(1E )WK (1s) ≃ me (1E )WK (2s) ≃ me
mr
3
α(Z α)6 0.99287 10−2 ≃ 0.11422 10−4 eV, π
3
α(Z α)6 0.10818 10−2 ≃ 0.12445 10−5 eV π
3
α(Z α)6 0.1988 10−3 ≃ 0.2287 10−6 eV. π
me
mr me
and
(1E )WK (2p) ≃ me
mr me
When integrating over p from 0 to infinity we split the integration into three regions, region I, p ∈ (0, 0.2), region II, p ∈ (0.2, 30) and region III, p ∈ (30, ∞). In the region I, we expanded the integrand of (62) into the series in p and integrated term by term over t and p. In region II we performed two-dimensional numerical integration and in region III we used the asymptotic expansions like (63), (64) valid for large p. For the contribution of Wichmann–Kroll term to the Lamb shift in muonic hydrogen we thus obtain
(1E )WK (2p) − (1E )WK (2s) ≃ −0.10158 10−5 eV which differs by 1% from the result −0.103 10−5 eV given in [5,6]. This is to be expected. The calculation here is for what is to be the exact form of WK potential, while the result obtained in [5,6] is for the approximate form of the WK potential. The error of the approximate form has been estimated to be 1% [8]. 6. Conclusions In this paper we obtained the Laplace transform of the renormalized vacuum charge density, Eqs. (51), (61) and (62). This result is not of the same form as that found by Wichmann and Kroll, Eq. (50) of [1]. Nonetheless, these two forms have the same expansion at small b. The result for the Lamb shift in muonic hydrogen agrees within approximations made in [5,6] with the result given by these authors. So it is very likely that the result found here is equivalent to that found in [1,4]. One thus may conclude that the result found in [1] follows from the charge renormalization at zero momentum transfer described in Section 3.3. However, as pointed out there, this makes zero very special value of the momentum transfer. This is rather alien to the common view of renormalization: it should not depend at which (finite) value is the charge density renormalized. Nonetheless, if the above renormalization prescription is adopted, the contribution of the Wichmann–Kroll term to the Lamb shift in both ordinary and muonic hydrogen is small. Since the terms α(Z α)6 ln(Z α) cancel out, as confirmed here, see comment below Eq. (65), the Wichmann–Kroll term starts to contribute only at the order α(Z α)6 . Acknowledgment The financial support of GAUK (Grant No. 24215) is gratefully acknowledged. References [1] E.H. Wichmann, N.K. Kroll, Phys. Rev. 101 (1956) 843. [2] R. Pohl, et al., Nature 466 (2010) 213; See also A. Antognini, et al., J. Phys.: Conf. Ser. 312 (2011) 032002; R. Pohl, R. Gilman, G.A. Miller, K. Pachucki, Annu. Rev. Nucl. Part. Sci. 63 (2013) 175. [3] L.S. Brown, R.N. Cahn, L.D. McLerran, Phys. Rev. D 12 (1975) 581; L.S. Brown, R.N. Cahn, L.D. McLerran, Phys. Rev. D 12 (1975) 596. [4] J. Blomqvist, Nuclear Phys. B 48 (1972) 95. [5] E. Borie, Phys. Rev. A 71 (2005) 032508; See also E. Borie, Ann. Physics (NY) 327 (2012) 733.
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)
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[6] S.G. Karshenboim, V.G. Ivanov, E.Yu. Korzinin, V.A. Shelyuto, Phys. Rev. A 81 (2010) 060501(R). [7] E. Borie, G.A. Rinker, Rev. Modern Phys. 54 (1982) 67. [8] M.I. Eides, H. Grotch, V.A. Shelyuto, Phys. Rep. 342 (2001) 63; See also P. Vogel, At. Data Nucl. Data Tables 14 (1974) 599; K.-N. Huang, Phys. Rev. A 14 (1976) 1311. [9] See Eqs. (3) and (4) of [3] and text below Eq. (14) of [1]. [10] P. Indelicato, P.J. Mohr, J. Sapirstein, Phys. Rev. A 89 (2014) 042121. [11] G. Soff, P.J. Mohr, Phys. Rev. A 38 (1988) 5066; See also P.J. Mohr, G. Plunien, G. Soff, Phys. Rep. 293 (1998) 227. [12] H. Persson, I. Lindgren, S. Salomonson, P. Sunnergren, Phys. Rev. A 48 (1993) 2772. [13] M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995. [14] L.C. Biedenharn, Phys. Rev. 126 (1962) 845; See also J. Schwinger, Particles, Sources and Fields, Volume II, Addison-Wesley, New York, 1973; P.C. Martin, R.J. Glauber, Phys. Rev. 109 (1958) 1307. [15] L.D. Landau, E.M. Lifschitz, Quantum Mechanics, Nonrelativistic Theory, Pergamon Press, Oxford, 1968. [16] A. Erdélyi (Ed.), Higher Transcendetal Functions Vol. I, Bateman Manuscript Project, McGraw-Hill Book Company, Inc., New York, 1953. [17] P.J. Mohr, B.N. Taylor, D.B. Newell, Rev. Modern Phys. 84 (2012) 1527.
)
–
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Vacuum polarization in Coulomb field revisited J. Zamastil ∗ , D. Šimsa Department of Chemical Physics and Optics, Charles University, Faculty of Mathematics and Physics, Ke Karlovu 3, 121 16 Prague 2, Czech Republic
article
Article history: Received 16 June 2016 Accepted 14 February 2017 Available online xxxx Keywords: Bound-state QED Vacuum polarization Renormalization
abstract
info
Simplified derivation of Wichmann–Kroll term is presented. The derivation uses two formulas for hypergeometric functions, but otherwise is elementary. It is found that Laplace transform of the vacuum charge density diverges at zero momentum transfer. This divergence has nothing to do with known ultraviolet divergence. The latter is related to the large momentum behavior of the pertinent integral, while the former to the small momentum behavior. When these divergences are removed, the energy shift caused by vacuum polarization for an ordinary hydrogen obtained here is in an exact agreement with the result obtained by Wichmann and Kroll. Also, for muonic hydrogen the result obtained here reasonably agrees with that given in literature. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Evaluation of the influence of one-loop vacuum polarization in Coulomb field on the atomic energy levels beyond the leading, Uehling, term was accomplished long time ago by Wichmann and Kroll [1]. In view of fundamental importance of this result in establishing proton radius puzzle [2] and complexity of original derivation [1], it seems desirable first, to check their result, second, to clarify the assumptions upon which it is based. In fact, Brown et al. [3] checked the 1/r part of the potential created by vacuum polarization, but otherwise, no independent derivation of Wichmann and Kroll result has been made. Wichmann and Kroll gave expression for Laplace transform of the potential created by induced charge density. Their result was transformed to coordinate space by Blomqvist [4]. Recent evaluations of the influence of Wichmann and Kroll term on the energy levels of
∗
Corresponding author. E-mail address: [email protected] (J. Zamastil).
http://dx.doi.org/10.1016/j.aop.2017.02.008 0003-4916/© 2017 Elsevier Inc. All rights reserved.
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muonic hydrogen [5,6] are based on the approximations of the corresponding potential given in [7,8], which are in turn based on Blomqvist’s work. We follow Wichmann and Kroll in reducing the problem to the calculation of Laplace transform. However, instead of using relativistic Coulomb Green functions we use the spectral decomposition of second order Dirac hamiltonian and the two formulas for hypergeometric functions. It is well-known that one-loop vacuum polarization yields a divergent result. Physically meaningful result is obtained only after the regularization and charge renormalization are performed. In this respect the regularization and renormalization performed in [1] (and also in [3]) are somewhat unusual [9] and relation of such a procedure to the standard one is, at least in present authors opinion, somewhat unclear. In view of the discussion of how much of the proton radius puzzle can be ascribed to the evaluation of vacuum polarization, see e.g. [10] and references therein, this relation should be clarified. In this paper we find, in contrast to the previous papers on the subject, that Laplace transform of the vacuum charge density diverges at zero (imaginary) momentum transfer. This divergence does not appear at the leading, Uehling, term. It is related to small momentum behavior of the pertinent integral. While known ultraviolet divergence, that is also present, can be removed by renormalization of the vacuum charge density at any finite value of the momentum transfer, this is not true for the divergence at the zero momentum transfer. We show here that if the renormalization of the vacuum charge density at the zero momentum transfer is made, the results found by Wichmann and Kroll follow. In this paper, we restrict the discussion to the case of light hydrogen-like atoms with low nuclear charge Z . This allows us to consider non-relativistic approximation to the wave function of the reference state and a point nucleus. As we shall see, the relativistic effects are important for intermediate states. However, for the reference state they yield the contribution to the energy shift that is of the order α(Z α)8 probably multiplied by ln(Z α). At the present level of accuracy, this is negligible for both ordinary and muonic hydrogen. The case of atoms with medium and high nuclear charges is discussed for example in [11,12]. The paper is organized as follows. In the following section we show first that the energy shift can be related to the Laplace transform of the vacuum charge density. Second, we calculate this density. Though, admittedly, this Section is somewhat long and technical, once the two formulas for the hypergeometric functions are used, the calculation is quite elementary. Section 3 deals with more conceptual issues of regularization and renormalization. The results of these two Sections are then applied to the calculation of the energy shift in ordinary and muonic hydrogen in Sections 4 and 5, respectively. Finally, some conclusions are drawn in Section 6. 2. The method 2.1. Relation of energy shift and Laplace transform of vacuum charge density The energy shift of the ground state of the hydrogen-like atom caused by vacuum polarization reads
1E =
d3 ⃗ r |ψ0 (⃗ r )|2 V (⃗ r ) = N02
d3 ⃗ r exp{−β r }V (⃗ r ),
(1)
where we substituted for the probability density
|ψ0 (⃗r )|2 = N02 exp{−β r },
N02 =
(mr Z α)3 , β = 2mr Z α. π
(2)
The energy shift of the excited states can be obtained by differentiation with respect to parameter β , see Section 5. The potential created by vacuum charge density ⟨ρ(⃗ r )⟩ satisfies Poisson equation
⃗ 2 V (⃗r ) = e⟨ρ(⃗r )⟩. −∇
(3)
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Following [1] we take Laplace transform of this equation. That is, we multiply this equation by exp{−β r } and integrate over the whole space; on the right hand side we obtain q(β) =
r exp{−β r }e⟨ρ(⃗ r )⟩; d3 ⃗
(4)
on the left hand side (lhs) we obtain
⃗ 2 V (⃗r ) = r exp{−β r }(−)∇ d3 ⃗
=
⃗ 2 exp{−β r } r V (⃗ r )(−)∇ d3 ⃗ β2
d 2 + 2β U (β) = β U (β) , dβ dβ d
where in the second equality we used
⃗ 2 exp{−β r } = − −∇
d2 dr 2
+
2 d
r dr
exp{−β r } =
2
−β +
2β r
exp{−β r }
and we introduced the function U (β) =
d3 ⃗ r
exp{−β r }V (⃗ r ). r Whence the Laplace transform of the Poisson equation (3) leads to the relation d
dβ
β 2 U (β) = q(β) ⇒ U (β) = β −2
β
(5)
dβ ′ q(β ′ ).
0
Since the energy shift (1) can be easily expressed through the function U, Eq. (5), we obtain the sought relation between energy shift and Laplace transform of the charge density
β d d U (β) = N02 − β −2 1E = N02 − dβ ′ q(β ′ ) . dβ dβ 0
(6)
2.2. Second order expression for vacuum charge density
ˆ Starting from the charge symmetric expression for charge density of the quantized Dirac field ψ e + ˆ r) ρ(⃗ ˆ r) = ψˆ (⃗r ), ψ(⃗ 2 one obtains the vacuum charge density in the form ⟨ρ(⃗r )⟩ = ⟨0|ρ(⃗ ˆ r )|0⟩ =
e 2 σ
⟨⃗r | Pˆσ− − Pˆσ+ |⃗r ⟩,
(7)
where Pˆ ± are projectors to positive/negative energy states Pˆ σ± = Xσ± Xσ±
Ď
.
Here, Xσ± are solutions of Dirac equation for particle moving in a static external potential Vˆ
(Vˆ + γ0 γ⃗ · p⃗ˆ + γ0 m)Xσ± = Eσ± Xσ± , where γ are Dirac matrices and obviously Eσ+ > 0, Eσ− < 0.
(8)
By introducing an integration over the energy variable one can rewrite Eq. (7) into the form
⟨ρ(⃗r )⟩ = e
C
dE 2π i σ
⟨⃗r |
Pˆ σ+
E − Eσ+ + iε
+
Pˆ σ−
E − Eσ− − iε
|⃗r ⟩
(9)
where under the integration along the contour C is meant half of the sum of the integration over contours C+ and C− . The contours C+ and C− run along the real axis and they are closed by infinite
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semicircle in the lower and upper half of the complex plane, respectively. By means of the spectral decomposition of the Dirac hamiltonian one further rewrites the last equation into the form
⟨ρ(⃗r )⟩ = e
dE
C
=e C
1
|⃗r ⟩
2π i
⟨⃗r |Tr
dE
1 ⟨⃗r |Tr(E − Vˆ − γ0 γ⃗ · p⃗ˆ + γ0 m) |⃗r ⟩,
2π i
⃗ˆ − γ0 m E − Vˆ − γ0 γ⃗ · p
H
(10)
where in the last equality we introduced the second order Dirac hamiltonian
Hˆ = (E − Vˆ )2 − pˆ 2 − m2 − γ0 γ⃗ · [p⃗ˆ , Vˆ ].
(11)
Straightforward expansion of Eq. (10) in powers of this potential yields
⟨ρ(⃗r )⟩ = ⟨ρ1 (⃗r )⟩ + ⟨ρ3 (⃗r )⟩ + · · · , where
⟨ρ1 (⃗r )⟩ = e
dE C
2π i
C
2π i
⟨⃗r |Tr
1
ˆ0 D
Vˆ
1
ˆ0 D
|⃗r ⟩
and
⟨ρ3 (⃗r )⟩ = e
dE
⟨⃗r |Tr
1
ˆ0 D
Vˆ
1
ˆ0 D
Vˆ
1
ˆ0 D
Vˆ
1
ˆ0 D
|⃗r ⟩.
(12)
1 ˆ− Here D 0 denotes the free electron propagator
ˆ 0 = E − γ0 γ⃗ · p⃗ˆ − γ0 m. D By virtue of Furry theorem the terms with even powers of V vanish. The first term, ⟨ρ1 (⃗ r )⟩, is the Uehling term and its calculation can be found in any textbook on quantum field theory, see e.g. [13]. The second term, ⟨ρ3 (⃗ r )⟩, is the Wichmann–Kroll term. Its evaluation is much more difficult. In fact, to the best of our knowledge, the formula (12) has never been used for actual calculation. The only evaluations of this term, complete, or partial, have been based on the known exact solution of Dirac equation for the electron moving in the Coulomb potential. To this solution we turn next. 2.3. Solution of second order Dirac equation For the Coulomb potential Vˆ = −
Zα rˆ
the second order Dirac Hamiltonian (11) simplifies to the form structurally identical with Schrödinger hamiltonian 2
2
Hˆ = E − m + 2
EZ α rˆ
−
Γˆ (Γˆ − 1) ˆ + rˆ 2
p2r
,
(13)
where the radial momentum operator pˆ r in the coordinate representation reads
pˆ r = −i
d dr
+
1 r
.
Further, following [3,14] we introduced the relativistic angular momentum operator Γˆ
Γˆ = γ0 Kˆ + i(Z α)γ⃗ · n⃗ ,
⃗ · ⃗Lˆ + 1 . Kˆ = γ0 Σ
(14)
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⃗ is the unit vector pointing at an arbitrary direction, ⃗r = r n⃗, Here Kˆ is the relativistic parity operator, n ˆ
and Σ /2 and ⃗L are the usual spin and orbital angular momentum operators, respectively. In contrast to the Dirac hamiltonian on lhs of Eq. (8), the second order Dirac hamiltonian commutes with the relativistic angular momentum operator Γˆ [14]. One of the advantages of the extension (10) is that the second order hamiltonian Hˆ has higher symmetry than the first order hamiltonian, Eq. (8). 2.3.1. Spinor-angular part It follows from the last equation that
Γˆ (Γˆ − 1) = Lˆ 2 − (Z α)2 − i(Z α)γ0 γ⃗ · n⃗.
(15)
Simple calculation yields Kˆ 2 = Jˆ2 +
1 4
and
Γˆ 2 = Kˆ 2 − (Z α)2 ,
ˆ
where ⃗J is the usual total angular momentum operator
⃗Jˆ = ⃗Lˆ + 1 Σ ⃗. 2
The eigenvalues of the operators Kˆ and Γˆ are then obviously K = σ (j + 1/2),
σ = ±1
(16)
and
Γ = ρ (j + 1/2)2 − (Z α)2 ,
ρ = ±1,
(17)
where, as usually, j(j + 1) is the eigenvalue of the operator Jˆ2 . The explicit form of the common eigenvectors |Γ , K , j, m⟩ of the operators Γˆ , Kˆ , Jˆ2 and Jˆz reads
|Γ , K , j, m⟩ =
c1 |j, m⟩σ c2 |j, m⟩−σ
,
(18)
where |j, m⟩σ are well-known spherical spinors. From eigenproblem
Γˆ |Γ , K , j, m⟩ = Γ |Γ , K , j, m⟩ and well-known equations for spherical spinors σ
σ⃗ · n⃗|j, m⟩ = −|j, m⟩
−σ
,
1 ˆ σ ⃗ |j, m⟩σ σ⃗ · L + 1 |j, m⟩ = σ j + 2
(19)
we obtain the relation between the coefficients c c2 =
i Zα
(Γ − K )c1 .
(20)
2.3.2. Radial part Introducing an effective orbital number l l(l + 1) = Γ (Γ − 1)
(21)
we make similarity of hamiltonian (13) with Schrödinger hamiltonian nearly perfect
Hˆ l = E 2 − m2 + 2
EZ α rˆ
p2r
− ˆ +
l(l + 1) rˆ 2
.
(22)
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The eigenfunctions of this operator belonging to the continuous part of the spectrum
Hˆ l Rl (p, r ) = (E 2 − m2 − p2 )Rl (p, r )
(23)
are [15]
Rl (p, r ) =
2 p exp{π a/2}|Γ (l + 1 − ia)|
π
Γ (2l + 2)
× (2pr ) exp{−ipr }F (ia + l + 1, 2l + 2, 2ipr ), l
(24)
where a=
EZ α
(25)
p
and F (α, γ , z ) denotes the confluent hypergeometric function [15,16]. The eigenfunctions belonging to the discrete part of the spectrum are inessential for our purposes. We evaluate the one-loop vacuum polarization in powers of Z α ; though we do not do it directly, as in Eq. (12), we expand around free particle solution. The discrete part of the spectrum does not contribute at any finite order of Z α ; it is a non-perturbative effect. In order for the wave function not to diverge at the origin too strongly l have to be in the limit Z α → 0 nonnegative. Whence physical solutions of Eq. (21) are l = |Γ | − δρ,1 .
(26)
2.4. Spinor-angular integration In Eq. (10) we use the spectral decomposition of the second order hamiltonian 1
Hˆ
=
1 |Γ , K , j, m⟩⟨Γ , K , j, m|γ0 , ˆ ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ ρ,σ ,j,m Hl
where the radial hamiltonian Hˆ l is given by Eq. (22). Somewhat unusual form of this decomposition is given by the fact that the operator Γ is not hermitian, but γ0 Γ is. Further we use the decomposition of the kinetic term
⟨⃗r |γ0 γ⃗ · p⃗ˆ |ψ⟩ = −iγ0 γ⃗ · n⃗
∂ 1 γ0 Kˆ + − ∂r r r
⟨⃗r |ψ⟩
valid for an arbitrary state vector |ψ⟩. By inserting the last two equations into Eq. (10) we easily perform the spinor-angular integration in Eq. (10), see also [3],
⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ = 1, ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ ⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ dΩ Trγ0 ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ σ =±1
dΩ Tr
=
|c1 |2 + |c2 |2 K j + 1/2 = = σ = 0, 2 2 |c | − |c2 | Γ Γ σ =±1 1 σ =±1 σ =±1
⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ c2 c ∗ − c2∗ c1 Zα = −i 12 =− , ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ |c1 | − |c2 |2 Γ ∗ ∗ ⟨⃗n|Γ , K , j, m⟩⟨Γ , K , j, m|γ0 |⃗n⟩ c2 c + c2 c1 ⃗γ0 dΩ Triγ0 γ⃗ · n = −i 12 = 0. ⟨Γ , K , j, m|γ0 |Γ , K , j, m⟩ |c1 | − |c2 |2
⃗ dΩ Triγ0 γ⃗ · n
The first equation follows trivially from the cyclic property of trace. The first equalities in the remaining three equation follow from Eq. (18) and orthonormality relations for spherical spinors. In
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the last two equations, the first property of spherical spinors displayed in Eq. (19) has also been used. The remaining equalities follow from the form of the coefficients c, Eq. (20), and from the explicit form of the eigenvalues K and Γ , Eqs. (16) and (17). To sum up spinor-angular integration in Eq. (10) leads to 1 ⃗ˆ + γ0 m) |⃗r ⟩ dΩ ⟨⃗ r |Tr(E − Vˆ − γ0 γ⃗ · p
H
=
2(2j + 1) E +
Zα
j,ρ
r
−
Zα
1 ∂ 2 ∂r
Γ
+
1
r
⟨r |
1
Hˆ l
|r ⟩.
(27)
The diagonal matrix elements of the inverse of the radial hamiltonian Hˆ l between the eigenstates of the operator rˆ are calculated by means of the spectral decomposition of the radial hamiltonian
⟨r |
∞
1
Hˆ l
|r ⟩ =
dp 0
|Rl (p, r )|2 + ···, − m2 − p2
(28)
E2
where the contribution of the discrete part of the spectrum has been omitted; see remark after Eq. (25). By inserting Eqs. (27) and (28) into Eq. (10) and inserting further that equation into Eq. (4) we get for the Laplace transform of the vacuum charge density
∞ ∞ dE dp k 2 E − p2 − m2 C 2π i 0 k=1 ∂ β ∂ × E − + Zα 1 + Il (p, β), ∂β 2Γ ∂β
q(β) = 8α
(29)
where k=j+
1
(30)
2
and Il (p, β) = 2π
∞
drr exp{−β r }|Rl (p, r )|2 .
(31)
0
When inserting (27) into Eqs. (10) and (4) we integrated by parts 2π
∞
2
drr exp{−β r } 0
1 ∂ 2 ∂r
+
1 r
|Rl (p, r )|2 = −
β ∂ Il (β, p). 2 ∂β
2.5. Radial integration To evaluate the integral Il (p, β), Eq. (31), we use two formulas for the hypergeometric functions. 2.5.1. Two formulas for hypergeometric functions The first formula reads [15]: ∞
dr exp{−λr }r γ −1 F (α, γ , kr )F (α ′ , γ , k′ r ) = Γ (γ )λα+α −γ ′
0 ′
× (λ − k)−α (λ − k′ )−α F (α, α ′ , γ , z ),
z=
kk′
(λ − k)(λ − k′ )
.
(32)
The second formula reads [16]:
1 ′ ′ Γ (γ ) t α −1 (1 − t )γ −α −1 dt , Γ (α ′ )Γ (γ − α ′ ) 0 (1 − tz )α Re(γ ) > Re(α ′ ) > 0, arg(1 − z ) < π .
F (α, α ′ , γ , z ) =
(33)
The proofs of these formulas are not difficult; they can be found in the corresponding references.
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2.5.2. Application to the present case Using Eqs. (24) and (32) we obtain for the integral (31) Il (p, β) =
exp{π a}|Γ (l + 1 − ia)|2 l+1 z Γ (2l + 2)
β − 2ip β + 2ip
−ia
× F (ia + l + 1, −ia + l + 1, 2l + 2, z ),
z=
(2p)2 . β 2 + (2p)2
Using Eq. (33) the last equation can be rewritten as Il (p, β) =
1
0
dt
exp{aA}
t (1 − t )
t (1 − t )z
l+1
1 − tz
,
(34)
where
A = π − 2 arctan
2p
+ i ln
β
1−t t (1 − tz )
.
(35)
2.6. Integration over energy variable The exponential function exp{aA} is an analytic function of the energy E, see Eqs. (25) and (35). The Taylor expansion of this function in E converges for all complex E. Thus, the integration over the energy variable in Eq. (29) is trivial
dE exp{aA} C
2π i E 2 − E02
C 1 2
dE 1 +
=
2π i
C
1 dE 1 + aA + 2 (aA)2 + · · ·
=
2π i
A Z p
α
E 2 − E02
2
E2 + · · ·
dE 1 +
=
E 2 − E02
C
1 2
2π i
A Z p
α
2
E02 + · · ·
E 2 − E02
.
In the last equality we used the identity E2 E2
−
E02
=1+
E02
− E02
E2
and the fact that integral of an analytic function of E, in this case f (E ) = 1, along any closed contour is zero. Proceeding further along these lines one gets
dE exp{aA} C
2π i E 2 − E02
dE cosh{aA}
= C+
2π i E 2 − E02
=−
cosh{a(p)A} 2E0
,
(36)
where E0 =
p2 + m2 ,
a(p) =
E0 p
Zα
(37)
and
dE E exp{aA} C
2π i E 2 − E02
dE E sinh{aA}
= C+
2π i E 2 − E02
=−
sinh{a(p)A} 2
.
(38)
Using Eqs. (31), (36) and (38) in Eq. (29) we obtain for the Laplace transform of the charge density q(β) = −4α
∞
∞
k
k=1
1
dp
0
0
dt t (1 − t )
l+1 ∂ β ∂ cosh{a(p)A} t (1 − t )z × − sinh{a(p)A} + Z α 1 + . ∂β 2Γ ∂β E0 1 − tz
(39)
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For further considerations it is advantageous first to exchange the order of the differentiation with respect to β and the integration with respect to p; second, to make substitution p → β p/2,
(40)
and express the above expression in terms of dimensionless variable
β
. (41) 2m mr After the parametric differentiation with respect to b we set b = m Z α , so b still carries the dependence on Z α . With these changes the Laplace transform of the charge density takes the form then ∞ 1 ∞ dt q(b) = −4α k dp 0 0 t (1 − t ) k =1 b=
×
∂ − ∂b
∂ sinh{a(pb)A(p)} + Z α 1 + 2 2Γ ∂ b
b
b
b cosh{a(pb)A(p)}
(bp)2 + 1
s l +1 ,
(42)
where a(pb) =
(bp)2 + 1 bp
Z α,
A(p) = π − 2 arctan (p) + i ln
1−t
t [1 − tz (p)]
(43)
and z (p) =
p2
1E = mr
,
t (1 − t )z (p)
. (44) 1+ 1 − tz (p) To save the space we write the differentiation with respect to b inside the integration with respect to p, however, it is understood that the integration with respect to p is made before the differentiation with respect to b. The energy shift (6) can be expressed in terms of dimensionless variable b as p2
s=
b m 2 (Z α)3 d r − b −2 db′ q(b′ ) . m m 4π db 0 b= r Z α
(45)
m
2.7. Summation over partial waves Eqs. (42)–(45) are still exact. To obtain closed formula for the sum over all partial waves one is forced to expand the effective orbital number l in powers of (Z α)2 . From Eqs. (17), (26) and (30) we have
ksl+1 =
k,ρ
√ ks
k,ρ
k2 −(Z α)2 −δρ,1 +1
√
= (1 + s)
ks
k2 −(Z α)2
≃ (1 + s)
k
= and
s(1 + s)
(1 − s)
2
ksk −
k
(Z α)2 2
sk ln s
(Z α) 1 + s s ln s 2 1−s 2
−
(46)
√ k k 2 2 sl+1 = s k −(Z α) −δρ,1 +1 2 2 Γ k,ρ k,ρ ρ k − (Z α) √ ks k2 −(Z α)2 (Z α)2 ln s 1 = (1 − s) ≃ ( 1 − s) sk 1 − − 2 2 k k k2 − (Z α)2 k k = s+
(Z α)2 2
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] .
(47)
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Here we used the approximation
√
k2 −(Z α)2
s
≃s
(Z α)2
k
1−
ln s
2
(48)
and elementary summation formulas ∞
, (1 − s)2
k=1
∞ sk k=1
k2
∞
s
ksk =
sk =
k=1
s 1−s
,
∞ k s k=1
k
= − ln(1 − s),
= dilog(1 − s).
Here dilog(1 − s) is the Euler dilogarithm function often denoted by Li2 (s). By inserting Eqs. (46) and (47) into Eq. (42) we obtain q(b) ≃ −4α(Z α)
∞
0
∂ − ∂b
×
1
dp
s(1 + s)
0
b 2Z α
dt t (1 − t )
sinh{a(pb)A(p)} +
b cosh{a(pb)A(p)}
(bp)2 + 1
(Z α)2 s(1 + s) ln(s) b ∂ b cosh{a(pb)A(p)} + (1 − s) 2 1−s 2 ∂b (bp)2 + 1 (Z α)2 × s+ (1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] .
×
2
−
2
(49)
2.8. Expansion in Z α Due to approximation (48), Eq. (49) is valid only up to the order α(Z α)3 , so let us expand the hyperbolic functions in Eq. (49) up to the order (Z α)2 : b Zα
sinh{a(pb)A(p)} ≃ A(p)
b cosh{a(pb)A(p)}
(bp) + 1 2
[1 + (bp)2 ]3/2 + (Z α)2 [A(p)]3 , 6b2 p3 (bp)2 + 1 (Z α)2 A(p) 2
1 + (bp)2
p b
≃ + b 2 p (bp )2 + 1 ∂ 1 + (bp)2 (Z α)2 A(p) 2 = + ∂b p2 2 p 1 × 1 + (bp)2 − arctanh 1 + (bp)2
and b
∂ b cosh{a(pb)A(p)} ∂b (bp)2 + 1 ∂ b (bp)2 + 1 (Z α)2 A(p) 2 ≃b + ∂b b 2 p (bp)2 + 1 ∂ 1 (Z α)2 A(p) 2 1 = − + arctanh . ∂b 2 p p2 1 + (bp)2 1 + (bp)2
(50)
J. Zamastil, D. Šimsa / Annals of Physics (
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Inserting these expansion into Eq. (49) we obtain q(b) ≃ −4α(Z α)
d db
Q (b),
Q (b) = Q1 (b) + (Z α)2 Q3 (b),
(51)
where Q1 (b) =
∞
dp
×
s(1 + s)
(1 − s)2 ∞ 1 dp 0
−
1
−
1 + (bp)2
p
+ −[A(p)]3
1 + (bp)2
2
1−s
p2 (52)
− A(p)
s(1 + s) ln(s)
+
1 + (bp)2
s
1
×
p
1
2 p2
1 + (bp)2
2
1
dt
2
1
− A(p)
t (1 − t )
0
×
dt t (1 − t )
0
0
Q3 (b) =
1
1 + (bp)2 − arctanh
+
p2
[1 + (bp)2 ]3/2 12b2 p3
+
1
A(p)
2
2
p
1 1 + (bp)2
× [ln(1 − s) ln(s) + dilog(1 − s)] +
1 + (bp)2
1
4
A(p)
s(1 + s) 1 1−s − ( 1 − s) 2 4 p2 1 + (bp)2 2
1
arctanh
p
1 + (bp)2
s .
(53)
The function q is real; the imaginary part drops out. The powers of function A(p) in above equations can thus be replaced by, see Eq. (43), A(p) → 2 arctan(1/p),
(54)
[A(p)]2 → [2 arctan(1/p)]2 − ln2
1−t t [1 − tz (p)]
[A(p)] → [2 arctan(1/p)] − 3[2 arctan(1/p)] ln 3
3
2
,
1−t t [1 − tz (p)]
.
2.9. Integration over parameter t To integrate over the parameter t we recall definitions of z (p) and s, Eq. (44). Integrals over t that can be evaluated in the closed form are 1
dt
s(1 + s)
= p2 ,
(55)
s = ln(1 + p2 ),
(56)
t (1 − t ) (1 − s)2
0
1
0 1
dt t (1 − t ) dt
s(1 + s)
t (1 − t ) (1 − s)2
0
ln2
1−t t [1 − tz (p)]
=
p2 3
π 2 − 6 dilog
1 p2 + 1
.
For the remaining integrals the resulting expression is either too complex or we did not succeed to evaluate the integrals in terms of the special functions. Nonetheless, the expansion for large p suffices:
0
1
dt
s(1 + s)
t (1 − t ) 1 − s
ln s = −4(ln(p) + 1) +
+
1 5
ln(p) + p4
17 50
π2 3
−
2 3
ln(p) +
+ O(1/p6 ),
14 9
p2 (57)
12
J. Zamastil, D. Šimsa / Annals of Physics ( 1
dt t (1 − t )
0
s ln2
1−t
= 4ζ (3) − 4
t [1 − tz (p)]
−
22 9
)
1 + ln(p) p2
ln(p) +
35 54
p6
–
+3
ln(p) +
1 2
p4
+ O(1/p8 )
(58)
and 1
0
dt t (1 − t ) π2
+
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] = 4ζ (3)
− 6 − 4 ln(p)
3
+
p2
59 36
−
π2 6
+ 53 ln(p) p4
+
197 − 225 +
π2 9
−
47 45
ln(p)
p6
+ O(1/p8 ).
(59)
These are non-trivial results, however, we shall not dwell on their derivation. 3. Observable part of the effect 3.1. Expansion in b—an example In the following we shall need the expansion in b. To illustrate how it is obtained, let us consider the integral I (b) =
∞
dp 1 + (bp)2 f (p),
0
where f (p) is finite for p → 0 and for large p behaves as f (p) =
c1 p2
+
c2 + d2 ln(p) p4
+ ···.
Direct expansion of the integrand of I (b) in powers of b produces divergent integrals in p. The correct expansion of I (b) contains odd powers of b that cannot be obtained by the direct expansion of I (b). The odd powers of b are obtained easily from the expansion of f (p) for large p as follows I (b) =
∞
dp
0
1 + (bp)2 − 1 p2
0
∞
+
dp 0 4
−
∞
dpf (p) + c1
b
8
1 + (bp)2 − 1
dpp 0
b2 2
∞
dpp2 f (p) − 0
c1
p2
c + d ln(p) 2 2
∞
+
4
f (p) −
c1 p2
−
p4 c2 + d2 ln(p) p4
+ ···.
The strategy used in this equation is the following. When the direct expansion in b does not lead to the divergent integral over p we use it. When it does lead to the divergent integral we use the expansion of function f (p) for large p and further use the subtracted function f (p). The above integrals are finite except for the second. The first yields an uninteresting absolute term. After the regularization the second integral yields linear term in b, see the next subsection. The third integral yields quadratic term in b, the fourth integral yields the terms proportional to b3 and b3 ln(b) and the fifth integral yields the quartic term in b. 3.2. Regularization We regularize the integration over electron momentum in Pauli–Villars manner, by subtracting from the pertinent expression the contribution of another pair of heavy fermions with squared
J. Zamastil, D. Šimsa / Annals of Physics (
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13
mass M,
∞
dp
1 + (bp)2 − 1 p2
0
∞
→
dp
1 + (bp)2 − 1 −
M + (bp)2 +
√ M
p2
0
b
= − ln M . 2
We note that the squared mass M is measured in the units of the electron squared mass m2 . This is essentially the same regularization as that used for example in [10]. The only divergences present in Eqs. (52) and (53) are logarithmic. However, it has to be noted that in Eq. (50) the differentiation with respect to b was tacitly moved inside the integration with respect to p and in Eq. (51) it was tacitly moved back outside the integration with respect to p. These operations are allowed for convergent integrals, not for divergent integrals that we actually have. It is clear that these interchanges of parametric differentiation and integration serve as a kind of regularization. Namely, writing in Eq. (50) b
b 1 ∂ ∂ =− , 2 2 ∂ b (bp) + 1 ∂ b p 1 + (bp)2
one changes logarithmically divergent integral in p to convergent integral in p. Question then arises regarding the relation of such a regularization to the standard Pauli–Villars regularization. In a more careful treatment we should first make replacement
1 + (bp)2 →
M + (bp)2
(60)
and differentiate with respect to M:
2 db
b d
1
dp → b 2 db ∞ 1 + (bp)2
b 0
=−
ln(1 + p2 )
∞
b d
b ∂
1
b
∞
0
ln(1 + p2 ) dp M + (bp)2
ln(1 + p2 )
∞
dM
∂ dM ∂M
dp
(M + (bp)2 )3/2 ∂ ln(1 + p2 )[M + 2(bp)2 ] =− dM dp ∂b ∞ 4p2 (M + (bp)2 )3/2 0 ∞ ∂ M ∞ ∂ ln(1 + p2 ) ln(1 + p2 ) dp 2 + lim dp . =− M →∞ ∂ b 2 ∂b 0 2p (1 + (bp)2 )1/2 p2 (M + (bp)2 )1/2 0 4 ∂b
∞
1
0
∞
The first term is exactly the term brought by Eq. (50). The second term is expanded in powers of b: M 2
ln(1 + p2 )
∞
dp 0
p2 (M + (bp)2 )1/2 ln
∞
+b
dp 0
M 1/2 p b
p2
=
M 1/2 2 1
∞
1 + p2
dp 0
ln(1 + p2 ) p2
− 1 + ···.
In the last displayed term we made the substitution p → pM 1/2 /b. The other terms vanish in the limit M → ∞. The first term is independent on b and vanishes upon the differentiation with respect to b. It is seen that different regularization schemes lead to different terms for charge density that do not vanish in the limit M → ∞. However, these are proportional to terms of the order b and b ln(b) and are removed by renormalization. The same analysis could be repeated also for expression (53). In this case more careful treatment does not bring any change, however. When lhs of Eq. (50) is used instead of the right hand side, the pertinent integral over p is now convergent.
14
J. Zamastil, D. Šimsa / Annals of Physics (
)
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3.3. Renormalization Charge density (10) is spherically symmetric. Its Fourier transform then depends just on the magnitude of the exchanged momentum f (ω ) = 2
d3 ⃗ k exp{i⃗ k·⃗ r }⟨ρ(⃗ r )⟩,
ω2 = ⃗k · ⃗k.
The usual way of renormalization, presented in the standard textbooks on the subject in the fourdimensional form, see e.g. [13], is to subtract from this the value f (0). This is because the value of the fine structure constant α ≃ 1/137.036 is obtained from measurements, where the exchanged momentum ⃗ k goes to zero. When considering Laplace transform of the charged density q(b) it is natural to choose ‘‘an analytic continuation’’ of the above condition and subtract from q(b) the part that does not vanish for b going to zero. Thus, from Q (b), see Eq. (51), we subtract everything that in the limit b → 0 goes to zero faster than b2 . In fact, since the energy shift is given by Eq. (45) it is obviously independent on the quadratic terms in b. Thus, we can subtract from Q (b) the expansion near b → 0 up to quadratic terms in b. In this paper, we shall stick to this prescription. Note, however, that there are some subtleties involved. After integration over parameter t, described in the previous section is performed, it is not difficult to see that ultraviolet (large p) behavior of the integrand in Eq. (53) is neither worse nor better of that in Eq. (52). On the other hand, it is seen that Eq. (53) contains terms proportional to inverse powers of b. These negative powers of b are connected with infrared (small p) behavior of the integrand. Further, looking at Eq. (42) it is seen, that the absolute value of the negative power of b is increasing with increasing order of the expansion of the charge density in Z α . By renormalizing the Laplace transform of the charge density at zero momentum transfer we are removing first ultraviolet divergence in p proportional to linear term in b and second, negative powers of b that have nothing to do with ultraviolet behavior. Clearly, there seems to be something special about renormalization at zero momentum transfer. Had the charge were renormalized at non-zero value of momentum transfer, one would conclude that for b approaching zero the charge density diverges. 3.4. Renormalized terms The renormalized expressions for integrated Laplace transform, Eqs. (52) and (53), are Q 1 ( b) =
∞
dp
1 + (bp)2 − 1 −
0
−
1 3p2
−
1 2
1
1 + (bp)2
(bp)2
2
−1+
1
1 − p arctan
(bp)
2
p
ln(1 + p ) − 2 ln(p) 2
(61)
p2
2
and
1 − p arctan 1 1 2 p (bp) Q 3 ( b) = 1 + (bp)2 − 1 − dp − 2 2 p2 0 1 1 3 2 dt s(1 + s) ln(s) [1 + (bp)2 ]3/2 3 2 4 × + − 2− p − b p 1−s b2 b 2 8 0 t (1 − t ) 3 1 2 arctan 1p − 2 arctan 1p π 2 − 6 dilog p21+1 + 2 ln(p) × − − 3 4
∞
12p
+
1 + (bp)2 − 1 −
(bp)2 2
p
− arctanh
1
1 + (bp)2
− ln
pb 2
+
(pb)2 4
J. Zamastil, D. Šimsa / Annals of Physics (
2 arctan
×
1 p
1
dt 0 t (1−t )
× −
1 p2 +1
)
–
15
1 − 2 ln(p)
−
6
(bp)2
−1+
1 + (bp)2
1
π 2 − 6 dilog −
2
+
2
p2
2
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)] 4p2
+ arctanh
2 arctan
+ ln
1 + (bp)2
1 p
×
1
2
pb
−
2
ln(1 + p ) − 2
(pb)2
+
ζ (3)
p2
4
1
dt s ln2 0 t (1−t )
1−t t [1−tz (p)]
4p2
ζ (3) + 2 . p
(62)
For practical evaluation of the integrals it is advantageous to use for large p the asymptotic expansions following from Eqs. (57) and (58)
1 p
1 1 − p arctan − 2 2
=
2 3
p
0
π2
(ln(p) + 1) −
18
p4
+
dt
s(1 + s) ln(s)
t (1 − t )
1−s
1
19 − 135 −
13 45
2 ln(p) + π30
+
p6
463 6300
−
π2 42
+
p8
13 70
ln(p)
+ O(1/p10 )
(63)
and
2
ln(1 + p2 ) −
1 p
2 arctan
1
dt s ln2 0 t (1−t )
1 −t t [1−tz (p)]
+
4p2
=
1 + 3 ln(p)
5 8
+
p4
−
25 ln 12 6 p
(p)
+
49 30
ln(p) −
217 216
p8
ζ (3) p2
+ O(1/p10 ).
(64)
The remaining asymptotic expansions are obtained trivially either from Eq. (59) or from the expansion of the functions arctan(x) and dilog(x) for small x. 4. Application to ordinary hydrogen In the case of ordinary hydrogen b ≪ 1 and we can expand the above formulas in powers of b. 4.1. Uehling term Expanding the first order renormalized Laplace transform, Eq. (61), in powers of b Q 1 (b) = Q 13 b3 + Q 14
b4
2 we obtain well-known results Q 13 = −
+
1 b3 1 2
∞
dp 0
1 p4
+ ···,
1 5
1 1 + (bp)2
1 + (bp)2 − 1 −
−1+
(bp) 2
2
(bp)2
=−
2 4 15
16
J. Zamastil, D. Šimsa / Annals of Physics (
∞
1
dpp4 −
Q 14 = 2
1
1
1
–
− 2+ 4 3p 5p 2 3 ln(1 + p ) ln(p) 1 5π + − + 2 + 4 = . 2 8
0
8
1 − p arctan
)
2p
p
p
2p
48
4.2. Wichmann–Kroll term Likewise, expanding the third order renormalized Laplace transform, Eq. (62), in powers of b Q 3 (b) = Q 33 b3 + Q 34
b4
+ ···,
2
we obtain Q 33 =
∞
1
dp
b3
0
1 p4
(bp)2
1 + (bp)2 − 1 −
2
[1 + (bp)2 ]3/2
7 5 3 2 3 2 4 12 − 3 ln(p) − p − b p 3 18 3 b2 b2 2 8 p2 2 2 pb ( bp ) 1 ( pb ) + − ln 1 + (bp)2 − 1 − − arctanh + 2 2 4 1 + (bp)2 π2 − 6 − 4 ln(p) 1 (bp)2 19 −1+ × − + ln(p) − 3 12 4 2 1 + (bp)2 1 (pb)2 π2 pb 19 + arctanh − − . (65) + ln (1 + 3 ln(p)) = 2 4 45 27 1 + (bp)2
×
2
−
π2
+
2
ln(p)
+
−
1
It is quite remarkable that all the terms b3 ln(b) arising at intermediate stage of the calculation cancel out in the final expression. It looks like some symmetry (gauge invariance?) is ‘‘working behind the scene’’. Thus, although the integrand depends on b this dependence cancels out in the final expression. Further,
∞
dpp4
Q 34 = 2 0
−
× −
1 8
1 1 − p arctan
−
−
p
16 1 3
3
−
1 p
dt
s(1 + s) ln(s)
t (1 − t )
1−s
1
0
2 arctan
− 2 arctan
−
2 3
−
π2 18
+ 23 ln(p)
p4
1 1 2 π − 6 dilog p p2 +1
12p
+ 2 ln(p) p4
2 arctan
×
1 p
p2
2
2
2
−
7 12
− 53 ln(p) p6
2 1 p
−
1 32
π 2 − 6 dilog −
6
1 p2 +1
−
1 − 2 ln(p) p2
− −
19 12
+ ln(p) 4 p
J. Zamastil, D. Šimsa / Annals of Physics (
+
+
3 8
1
dt 0 t (1−t )
−
ζ (3) p2
π2
17
(1 − s) [ln(1 − s) ln(s) + dilog(1 − s)]
− 6 − 4 ln(p)
3
+
3
32
–
4p2
−c
)
4p4
2 1 p
2 arctan
ln(1 + p2 ) −
1
dt s ln2 0 t (1−t )
1 −t t [1−tz (p)]
4p2
1 ζ (3) 1 + 3 ln(p) 31π 2 − 2 − − π. = p p4 16 2880 These results are in an agreement with those found in [1]. We obtained the last result numerically only; we reproduced the first nine digits. 5. Applications to muonic hydrogen In case of muonic hydrogen b ≃ 1.36 and Eqs. (61), (62) have to be evaluated exactly. From Eqs. (45) and (51) we get for the energy shift of the ground state caused by Wichmann–Kroll term
(1E )WK (1s) = me
3
mr me
α(Z α)6 d −2 b Q 3 (b) , m π db b= r Z α
mr =
me
mµ 1+
mµ mp
.
The induced charge density is spherically symmetric. Thus, for non-S-states it is sufficient to consider the angular average of the probability density of the reference state. Taking into account the form of the wave functions of the excited states
|ψ2s (r )| = 2
N02 8
exp{−β r } 1 − β r +
β 2r 2
4
and
dΩ 4π
|ψ2p (⃗r )|2 =
N02 96
exp{−β r }β 2 r 2
where
β = mr Z α,
N02 =
(mr Z α)3 , π
and the substitution (41), the energy shift for these states is obtained by the method of differentiation of an integral with respect to a parameter:
(1E )WK (2s) = me
mr
3
me
d b2 d2 d −2 α(Z α)6 1 × 1+b + b Q 3 (b) m π 8 db 4 db2 db b= r
2me
and
(1E )WK (2p) = me
mr me
3
α(Z α)6 1 2 d2 d −2 b b Q ( b ) 3 mr π 96 db2 db b=
Plugging into these formulas the numerical values [17] me α = 1/137.035999074, = 4.83633166 10−3 , mµ me = 5.4461702178 10−4 , me = 0.510998928 106 eV mp
2me
Zα
.
Zα
18
J. Zamastil, D. Šimsa / Annals of Physics (
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–
and performing numerical integration we obtain
(1E )WK (1s) ≃ me (1E )WK (2s) ≃ me
mr
3
α(Z α)6 0.99287 10−2 ≃ 0.11422 10−4 eV, π
3
α(Z α)6 0.10818 10−2 ≃ 0.12445 10−5 eV π
3
α(Z α)6 0.1988 10−3 ≃ 0.2287 10−6 eV. π
me
mr me
and
(1E )WK (2p) ≃ me
mr me
When integrating over p from 0 to infinity we split the integration into three regions, region I, p ∈ (0, 0.2), region II, p ∈ (0.2, 30) and region III, p ∈ (30, ∞). In the region I, we expanded the integrand of (62) into the series in p and integrated term by term over t and p. In region II we performed two-dimensional numerical integration and in region III we used the asymptotic expansions like (63), (64) valid for large p. For the contribution of Wichmann–Kroll term to the Lamb shift in muonic hydrogen we thus obtain
(1E )WK (2p) − (1E )WK (2s) ≃ −0.10158 10−5 eV which differs by 1% from the result −0.103 10−5 eV given in [5,6]. This is to be expected. The calculation here is for what is to be the exact form of WK potential, while the result obtained in [5,6] is for the approximate form of the WK potential. The error of the approximate form has been estimated to be 1% [8]. 6. Conclusions In this paper we obtained the Laplace transform of the renormalized vacuum charge density, Eqs. (51), (61) and (62). This result is not of the same form as that found by Wichmann and Kroll, Eq. (50) of [1]. Nonetheless, these two forms have the same expansion at small b. The result for the Lamb shift in muonic hydrogen agrees within approximations made in [5,6] with the result given by these authors. So it is very likely that the result found here is equivalent to that found in [1,4]. One thus may conclude that the result found in [1] follows from the charge renormalization at zero momentum transfer described in Section 3.3. However, as pointed out there, this makes zero very special value of the momentum transfer. This is rather alien to the common view of renormalization: it should not depend at which (finite) value is the charge density renormalized. Nonetheless, if the above renormalization prescription is adopted, the contribution of the Wichmann–Kroll term to the Lamb shift in both ordinary and muonic hydrogen is small. Since the terms α(Z α)6 ln(Z α) cancel out, as confirmed here, see comment below Eq. (65), the Wichmann–Kroll term starts to contribute only at the order α(Z α)6 . Acknowledgment The financial support of GAUK (Grant No. 24215) is gratefully acknowledged. References [1] E.H. Wichmann, N.K. Kroll, Phys. Rev. 101 (1956) 843. [2] R. Pohl, et al., Nature 466 (2010) 213; See also A. Antognini, et al., J. Phys.: Conf. Ser. 312 (2011) 032002; R. Pohl, R. Gilman, G.A. Miller, K. Pachucki, Annu. Rev. Nucl. Part. Sci. 63 (2013) 175. [3] L.S. Brown, R.N. Cahn, L.D. McLerran, Phys. Rev. D 12 (1975) 581; L.S. Brown, R.N. Cahn, L.D. McLerran, Phys. Rev. D 12 (1975) 596. [4] J. Blomqvist, Nuclear Phys. B 48 (1972) 95. [5] E. Borie, Phys. Rev. A 71 (2005) 032508; See also E. Borie, Ann. Physics (NY) 327 (2012) 733.
J. Zamastil, D. Šimsa / Annals of Physics (
)
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[6] S.G. Karshenboim, V.G. Ivanov, E.Yu. Korzinin, V.A. Shelyuto, Phys. Rev. A 81 (2010) 060501(R). [7] E. Borie, G.A. Rinker, Rev. Modern Phys. 54 (1982) 67. [8] M.I. Eides, H. Grotch, V.A. Shelyuto, Phys. Rep. 342 (2001) 63; See also P. Vogel, At. Data Nucl. Data Tables 14 (1974) 599; K.-N. Huang, Phys. Rev. A 14 (1976) 1311. [9] See Eqs. (3) and (4) of [3] and text below Eq. (14) of [1]. [10] P. Indelicato, P.J. Mohr, J. Sapirstein, Phys. Rev. A 89 (2014) 042121. [11] G. Soff, P.J. Mohr, Phys. Rev. A 38 (1988) 5066; See also P.J. Mohr, G. Plunien, G. Soff, Phys. Rep. 293 (1998) 227. [12] H. Persson, I. Lindgren, S. Salomonson, P. Sunnergren, Phys. Rev. A 48 (1993) 2772. [13] M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995. [14] L.C. Biedenharn, Phys. Rev. 126 (1962) 845; See also J. Schwinger, Particles, Sources and Fields, Volume II, Addison-Wesley, New York, 1973; P.C. Martin, R.J. Glauber, Phys. Rev. 109 (1958) 1307. [15] L.D. Landau, E.M. Lifschitz, Quantum Mechanics, Nonrelativistic Theory, Pergamon Press, Oxford, 1968. [16] A. Erdélyi (Ed.), Higher Transcendetal Functions Vol. I, Bateman Manuscript Project, McGraw-Hill Book Company, Inc., New York, 1953. [17] P.J. Mohr, B.N. Taylor, D.B. Newell, Rev. Modern Phys. 84 (2012) 1527.