Vacuum polarization
Uehling potential
Landau pole

Introduction

△φ (r) = -4π ρ (r) (1)
ρ (r) = eUnderscript[∑, σ = 1, 2]   ∫^3k/(2π)^3ψ_k^* (r) ψ_k (r),    e = -| e | <0 (2)

Here the summation runs over all the states in the lower continuum ε < 0. If we consider a potential U(r) as a perturbation then Eq.(2) yeilds

△ δφ (r) = -4π  δρ (r) (3)
δρ (r) = 2 e ReUnderscript[∑, σ = 1, 2] ∫^3k/(2π)^3ψ_k^* (r) δψ_k (r) (4)

Dirac equation

γ_μ (p^μ - eA^μ) ψ = m ψ (5)

γ_μγ_ν (p^μ - eA^μ) (p^ν - eA^ν) ψ = m^2 ψ

γ_μγ_ν = g_μν + σ_μν

γ_μγ_ν (p^μ - eA^μ) (p^ν - eA^ν) ψ =  ( ... =  (p - eA)^2ψ -  /2e σ_μνF^μνψ = m^2 ψ

(p - eA)^2ψ -  /2e σ_μνF^μνψ = m^2 ψ (6)
(( p - eA )^2 - m^2) ψ + e ( Σ · B - i α · E ) ψ = 0 (6)
U = -(Z e^2)/r,   F = -(Z e^2 r)/r^3 (7)

Static electric field:      i∂/∂t-eA_0→ε-U,     e E → F

( (ε - U)^2 - m^2 + △ - i α · F  ) ψ = 0, (8)

Lower continuum     ε = -ε_k

( (ε_k + U)^2 - m^2 + △ - i α · F  ) ψ_k = 0, (9)
(ε_k^2 - m^2 + △) δψ_k (r) = ( -2ε_kU + i α · F  ) ψ_k (r) (10)

The term with U arises due to the electron charge. The sign minus in -2ε_kU modifies the influence of the potential. If the potential produces attraction U<0, then for the states in the lower continuum is represents repulsion.
The term with F arises from the contribution of the spin. We will see that this contribution results in the effective attraction.

ψ_k (r) = u_ke^(i k · r),      u_k^* u_k = 1. (11)

The normalization is conventional for usual quantum mechanical

problems ; it is different from what is often assumed in QED

Overscript[u, -] _ku_k = ± 2m .

(k^2 + △) δψ_k (r) = (-2ε_kU + i α · F ) u_k  e^(i k · r) (12)
δψ_k (r) = 1/(k^2 + △ + i0)[ e^(i k · r) (-2ε_kU + i α · F )] u_k (13)
δρ (r) = 2 e Re  Underscript[∑, σ = 1, 2] ∫^3k/ ... + △ + i0)[ e^(i k · r) (-2ε_kU (u_k^* u_k) + i (u_k^*   α u_k) · F )] (14)
u^* u = 1     Overscript[u, -] γ_0u = 1   ... ;   Overscript[u, -] γ^μu = p^μ/ε, ε = -ε_k (15)
               &n ... sp;   u_k^*   α u_k = Overscript[u, -] _k γ u_k = -k/ε_k (15)
δρ (r) = 2 e Re Underscript[∑, σ = 1, 2] ∫^3k/(2π)^3 e^(-ik · r) 1/(k^2 + △ + i0)[ e^(i k · r) (-2ε_kU - i (k · F)/ε_k) ]  (16)
F = -∇U (16)
δρ (r) = -4 e Re∫^3k/(2π)^3 e^(-ik · r) 1/(k^2 + △ + i0)[ e^(i k · r) ( (2ε_k^2 - i k · ∇ )/ε_k ) U ] (16)
Here Underscript[∑, σ = 1, 2] 2 (17)

Polarization operator

U (r) = e∫U (q) e^(iq · r) ^3q/(2π)^3 , (18)
U (r) = U^* (r) (19)
U (q) = U (-q) = U (q) (19)
U = -Ze^2/r,   U (q) = -(4π Z e^2)/q^2 (20)
∫^3k/(2π)^3 e^(-ik · r) 1/(k^2 + △ + i0)[ e^(i (k + q) · r)  & ... 8;^3k/(2π)^3  1/(k^2 - (k + q)^2 + i0) (2 (k^2 + m^2) + k · q)/(k^2 + m^2)^(1/2) (21)
Definition (22)
-(P (q^2))/(4π) = -4  e^2∫^3k/(2π)^3 1/(k^2 - (k + q)^2 + i0) (2 (k^2 + m^2) + k · q)/(k^2 + m^2)^(1/2) (22)
P (q^2) = P (q^2) (23)
e δρ (r) = Re∫ (-(P (q))/(4π)) U (q) e^(iq · r) ^3q/(2π)^3 (24)
e δφ (r) ≡δU = Re∫ (-(P (q^2))/(4π)) (4π)/q^2 U (q) e^(iq · r) ^3q/(2π)^3 = -Re∫ (P (q^2))/q^2 U (q) e^(iq · r) ^3q/(2π)^3 (25)
δU = 4π Z e^2 Re∫ (P (q^2))/(q^2)^2e^(iq · r) ^3q/(2π)^3 (26)
-(P (q^2))/(4π) = -4  e^2∫^3k/(2π)^3  1/(k^2 - ( ... 63308;^3k/(2π)^3 ((2 (k^2 + m^2) - q^2/2)/(q^2 + 2q · k - i0) + 1/2) ( 1)/(k^2 + m^2)^(1/2) (27)
-(P (q^2))/(4π) = 8  e^2∫^3k/(2π)^3 ((k^2 + m^2 - q^2/4)/( ... = 2  e^2∫^3k/(2π)^3 ( 1)/(k^2 + m^2)^(1/2)      (28)

Renormalization (see also below) yeilds

const0 (29)
-(P (q^2))/(4π) = 8  e^2∫^3k/(2π)^3 (k^2 + m^2 - q^2/4)/(q^2 + 2q · k - i0) ( 1)/(k^2 + m^2)^(1/2) (30)
-(P (q^2))/(4π) = (8 · 2π)/(2π)^3 e^2Underoverscript[∫, 0, arg3] ᢺ ... (k^2 + m^2)^(1/2) = e^2 /(π^2q) Underoverscript[∫, -∞, arg3] k (31)
1/(2k)   ln ((q + 2 k - i0)/(q - 2 k - i0)) (k^2 + m^2 - q^2/4) k^2/(k^2 + m^2)^(1/2) (31)
= e^2 /(2π^2) Underoverscript[∫, -∞, arg3]   1/qln ((q/2 + k - i0)/(q/2 - k - i0)) (k^2 + m^2 - q^2/4) (kk )/(k^2 + m^2)^(1/2) (31)
-P (q^2) = (2e^2 )/π Underoverscript[∫, -∞, arg3]   1/qln ((q/2 + k - i0)/(q/2 - k - i0)) (k^2 + m^2 - q^2/4) (kk )/(k^2 + m^2)^(1/2) (32)
-P (q^2) = (2e^2 )/π Underoverscript[∫, -∞, arg3]   1/qln ((q/2 + ... ; Underoverscript[∫, -∞, arg3]   1/q (1/(q/2 + k - i0) + 1/(q/2 - k - i0)) (33)
            [((k^2 + m^2)/3 - q^ ... 734;, arg3]   1/(q^2/4 - k^2 - i0) ((k^2 + m^2)/3 - q^2/4) (k^2 + m^2)^(1/2) k (33)

Again throwing away a const (having in mind the renormalization, see below)

P (q^2) = (2e^2 )/π Underoverscript[∫, -∞, arg3]   1/(q^2/4 - k^2 ... 747;, -∞, arg3]   1/(q^2/4 - k^2 - i0) (m^2 - 2k^2 ) (k^2 + m^2)^(1/2) k (34)

[Graphics:HTMLFiles/Uehling-NoColor_54.gif]

Figure 1

Shift the integration counter C from the real axes into the upper semiplane of the complex plane k. The point k = i m on the imaginary axes is a branch point. Make the integration counter C to begin at k=i∞,  going down to k=i m, then circle clockwise and return back to k=i∞. Making a substitution k→i k one finds

P (q^2) = i^24 /(3π) e^2Underoverscript[∫, m, arg3]   1/(q^2/4 + k^2 ) (m^2 + 2k^2 ) (k^2 - m^2)^(1/2)   k (35)

Here an additional coefficient 2 comes from the two branches of the counter.

P (q^2) = -(16 e^2)/(3π) Underoverscript[∫, m, arg3]   1/(q^2 + 4 k^2 ) (m^2 + 2k^2 ) (k^2 - m^2)^(1/2)   k (36)
4k^2 = t '  (37)
k = 1/2t '^(1/2) (37)
dk = 1/4dt '/t '^(1/2) (37)
P (q^2)  -16  /(3π) e^21/41/21/2Underoverscript[∫, 4m^2 ... , arg3]   1/(q^2 + t ' ) (t ' + 2m^2 ) (t ' - 4 m^2)/t '^(1/2)   t ' (38)

Renormalization

P_Reg (q^2) = P (q^2) - P (0) - q^2P ' (0) (39)

The constant was thrown away a couple of times above . This makes no

difference, since the renormalization takes the constant away anyway .

Simplify[1/(q^2 + t ') - Normal[Series[1/(q^2 + t ' ), {q, 0, 2}]]]

q^4/((t^′)^2 (q^2 + t^′))

Thus, regularization means (40)
1/(q^2 + t ') q^4/((t^′)^2 (q^2 + t^′)) (40)
P_Reg (q^2) = -( e^2)/(3π) Underoverscript[∫, 4m^2, arg3] q^4/((t^′)^2 (q^2 + t^′)) (t ' + 2m^2 ) (t ' - 4 m^2)/t '^(1/2)   t ' (41)
P (q^2) = -( α)/(3π) q^4Underoverscript[∫, 4m^2, arg3]   1/(t ' + q^2 - i0 ) (t ' + 2m^2)/t '^2 (t ' - 4 m^2)/t '^(1/2)   t ' (42)
t = -q^2 (43)
 (t) =  (-q^2) = P (q^2) (43)
t = -q^2,  (t) = -( α)/(3π) t^2Underoverscript[∫, 4m^2, arg3]   1/(t ' - t - i0 ) (t ' + 2m^2)/t '^2 (t ' - 4 m^2)/t '^(1/2)   t ' (44)
t ' = 4m^2ζ^2 (45)
P (q^2) = -( e^2)/(3π) q^4Underoverscript[∫, 1, arg3]   1/(4m^2ζ^2 ... /(4m^2ζ^2 + q^2 ) (ζ^2 + 1/2)/ζ^4 (ζ^2 - 1)^(1/2)   ζ (45)
P (q^2) = -(2 α)/(3π) q^4Underoverscript[∫, 1, arg3]   1/(4m^2ζ^2 + q^2 ) (1 + 1/(2ζ^2)) ((ζ^2 - 1)^(1/2)   ζ)/ζ^2 (46)
P (q^2) ≃ -( α)/(3π) q^2ln (q^2/m^2) ,   q^2≫m^2 (47)

Scalar particles

For spinor particles it was found that

P (q^2) = -( α)/(3π) q^4Underoverscript[∫, 4m^2, arg3]   1/(t ' + ... /(t ' + q^2 - i0 ) (2m^2 - t '/2 + 3t '/2)/t '^2 (t ' - 4 m^2)/t '^(1/2)   t ' (48)

Here m^2-t'/2 comes from the charge contribution, while 3t'/2 originates from the spin. These two terms have different signs. The charge tends to make P(q) positive, while the spin results in a opposite, negative contribution, which dominates.

Calculating the polarization for scalar particles one needs to make three amendments:
1. To change the total sign, which is related to statistics (i.e. to eliminate the negative energy)
2. To eliminate the spin contribution, i.e. the term 3t'/2.
3.To reduce the result by a factor of 2, which exists for fermions due to their spin 1/2, but is absent for scalars.
Then the polarization P_0 created by scalar particles reads

P_0 (q^2) = -(-1/2) ( α)/(3π) q^4Underoverscript[∫, 4m^2, arg3]    ... bsp; 1/(t ' + q^2 - i0 ) (t ' - 4m^2)/t '^2 (t ' - 4 m^2)/t '^(1/2)   t ' (49)
P_0 (q^2) ≃ -( α)/(12π) q^2ln (q^2/m^2),    q^2 = q^2≫m^2 (50)

The sign of the vacuum polarization is same for fermions  ( Landau, Abrikosov, Khalatnikov, 1954) )  and scalars ( Fradkin, 1955? ).

Uehling potential

δU (r) = -4π Z e^2 Re∫ (P (q^2))/(q^2)^2e^(iq · r) ^3q/(2π)^3 (51)
P (q^2) = -(2 α)/(3π) q^4Underoverscript[∫, 1, arg3]   1/(4m^2ζ^2 + q^2 ) (1 + 1/(2ζ^2)) (ζ^2 - 1)^(1/2)/ζ^2ζ (51)
∫1/(q^2 + μ^2) e^(iq · r) ^3q/(2π)^3 = (exp (-μr))/(4π r) (52)
δU = 4π Z e^2 Re∫ (P (q^2))/(q^2)^2e^(iq · r) ^3q/(2π)^3 (53)
δU (r) = 4π Z e^2 (-(2 α)/(3π)) Underoverscript[∫, 1, arg3] ᢺ ... ;ζ   (1 + 1/(2ζ^2)) (ζ^2 - 1)^(1/2)   /ζ^2e^(-2mrζ) (54)
U (r) = -(Z e^2)/r (55)
U (r) + δU = -Ze^2/r (1 + (2 α)/(3π) Underoverscript[∫, 1, arg3] (1 + 1/(2ζ^2)) (ζ^2 - 1)^(1/2)   /ζ^2e^(-2mrζ)   ζ ) (55)
Consider  r≫1/m (56)
δU≃ -Ze^2/r (2 α)/(3π) Underoverscript[∫, 1, arg3] ζ ... ) Underscript[|, λ = 2mr] =  -Ze^2/r ( α)/(2π)^(1/2) e^(-2mr)/(2mr)^(3/2) (56)
δU≃ -Ze^2/r ( α)/(2π)^(1/2) e^(-2mr)/(2mr)^(3/2), r≫1/m (57)
δU≃ -Ze^2/r (2 α)/(3π) ln (1/mr),        r≪1/m (58)

Landau (Moscow) pole

e^2 (r) ≃e^2 ( 1 + (2 α)/(3π) ln (1 + 1/mr)    ) (59)

More accurate account of polarization can be obtained by iterating the vacuum polarization, which gives

e^2 (r) ≃e^2/(1 - (2 α)/(3π) ln (1 + 1/mr)    ) (60)

In the momentum representation (for spinors)

e^2 (q) ≃e^2/(1 - ( α)/(3π) ln (q^2/m^2)    ), | q^2 | ≫m^2 (61)

For scalars

e^2 (q) ≃e^2/(1 - ( α)/(12π) ln (q^2/m^2)    ), | q^2 | ≫m^2 (62)
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