In short, how to reproduce the derivation in the footnote on page 588 of Weinberg's QFT book? The derivation there are necessary for lamb-shift calculation.

Complete version of my question:

Weinberg's QFT book describes how to calculate lamb shift of the hydrogen atom in chapter 14. During the course of calculation, there are spurious shift terms that persist in the limit of zero coulomb-field strength. These spurious zeroth-order shift terms should cancel among themselves. (Page 588, "By the same argument we can anticipate ... cancels the second and third terms ..., as well as the footnote on this page.)

The book described the justification of this cancellation, during which he employed the following intermediate result (hereafter referred to as result A):

$$

\sum_{M} \left | \tilde{\Gamma}_{MN}^{0}( \mathbf{k}) \right |^2

=\left ( \frac{\sqrt{\mathbf{k}^2+m_e^2}-m_e}{2\sqrt{\mathbf{k}^2+m_e^2}} \right )

$$

(page 588 footnote.) I tried to reproduce this result following Weinberg's description but failed so far. Actually, it seems that, if an incorrect approximation is used, one get exactly the above result A.

My attemped derivation is as follows:

The derivation starts from

$$ \tilde{\Gamma}_{MN}^{\rho}(\mathbf{k}) \equiv \int d^3 y \,

e^{-i \mathbf{k} \cdot \mathbf{y}} \bar{v}_M(\mathbf{y}) \gamma^{\rho} u_N(\mathbf{y}) $$

where $v_M$, $u_N$ are four-component electronic and positronic "wave functions" defined in chapter 5. They are just solutions of the Dirac equation for hydrogen. According to the description of Weinberg, a relativistic approximation for the positronic wave function is used, $$ v_{\vec{p},\sigma}(\vec{x})= \frac{1}{(2 \pi)^{3/2}}v(\vec{p},\sigma)e^{-i \vec{p} \cdot \vec{x}}, $$ where $v(\vec{p},\sigma)$ is positron spinor introduced in section 5.5 of the book. It satisfy $$ \sum_{\sigma} v(\vec{p},\sigma) v^{\dagger}(\vec{p},\sigma)= \frac{1}{2 p^0}[ -i p^\mu \gamma_\mu -m_e]\beta $$ where $p^0=\sqrt{\vec{p}^2 +m_e^2}$.

Note that to maintain a consistent notation of $v_{\vec{p},\sigma}$ between chapter 5 and chapter 14, $v_{\vec{p},\sigma}(\vec{x})$ 's $\vec{x}$ dependence has been changed to $e^{-i \vec{p}\cdot\vec{x}}$ instead of $e^{i \vec{p}\cdot\vec{x}}$ as given in footnote of page 588. This change is not obligatory but convenient. For $u_N$ one use the extreme nonrelativistic approximation $\beta u_N(\vec{x}) =u_N(\vec{x})$ (page 588, footnote).

Using the above formulas one get

$$

\sum_{M} \left | \tilde{\Gamma}_{MN}^{0}( \mathbf{k}) \right |^2

= \int \tilde{u}_N^\dagger (\vec{k}-\vec{p}) \frac{\sqrt{\vec{p}^2+m_e^2}-m_e}{2 \sqrt{\vec{p}^2+m_e^2}} \tilde{u}_N (\vec{k}-\vec{p}),

$$

where

$$

\tilde{u}_N(\vec{q}) \equiv \int d^3 y \frac{1}{(2\pi)^{3/2}} e^{-i \vec{k} \cdot \vec{y}} u_N(\vec{y}).

$$

If one can treat $\tilde{u}^{\dagger}(\vec{q}) \tilde{u} (\vec{q})$ as a delta function $\delta^3(\vec{q})$, one get immediately result A.

If the typical momentum inside a hydrogen atom is much smaller than the $|\vec{k}|$ values, this delta-function approximation can be justified. In a Coulomb field this criteria amounts to

$$

Z \alpha m_e \ll |\mathbf{k}|.

$$

However it turns out the opposite is true: the intermediate result A is used to get the left-hand side of the following result:

$$

- \frac{e^2}{2 (2 \pi)^3} \int d^3 k \left( \frac{1}{\mathbf{k}^2} -

\frac{1}{\mathbf{k}^2 +\mu^2} \right) \left ( \frac{\sqrt{\mathbf{k}^2+m_e^2}-m_e}{2 \sqrt{\mathbf{k}^2+m_e^2}} \right ) \simeq - \frac{\alpha \mu^2}{ \pi m_e}

$$

(page 588 footnote). As can be seen, the significant values of $|\mathbf{k}|$ are those that are smaller than $\mu$. i.e. , $|\mathbf{k}|< \mu$.

However, in the first paragraph of page 579, it says $\mu$ is "chosen to be much larger than typical electron kinetic energies, but much less than typical electron momenta. " And the imediately following equation (14.3.1) states unambiguously that：

$$ (Z \alpha)^2 m_e \ll \mu \ll Z \alpha m_e. $$

Thus $|\mathbf{k}|< \mu \ll Z \alpha m_e $, which is contradictory to the above mentioned justification criteria for the delta function approximation $Z \alpha m_e \ll |\mathbf{k}|$.

So, how do one understand this? Is Weinberg correct here? If so, how? And if not, how do one eventually justify the cancelation of the spurious zeroth-order shifts, which is of course physically required?