S-matrix element for forward scattering and amputed green fu

[FrancescoS]

I'm studying dispersion relations applied as alternative method to perturbation theory from Weinberg's book (Vol.1)

Let's consider the forward scattering in the lab frame of a massless boson of any spin on an arbitrary target $\alpha$ of mass $m_\alpha>0$ and $\vec{p}_\alpha = 0$.
Weinberg writes "By a repeated use of Eq(10.3.4) or the LSZ theorem, the S-Matrix element here is

$S = \frac{1}{(2\pi)^3\sqrt{4k^0k'^0}|N|^2}lim_{k^2\rightarrow 0}lim_{k'^2\rightarrow 0} \int d^4x d^4 y e^{-ik'y + ikx}(i\Box_y)(i\Box_x)<\alpha | T\left( A^\dagger(y) A(x) \right) |\alpha>,$

where $<VAC| A(x) |k> = (2\pi)^{-3/2} (2\omega)^{-1/2}N e^{ikx}$.I have several questions on this formula:

1. Since he discussed the Green function for a $T$-product of generals operators; why did he choose $A^\dagger$ and $A$ in this case?

2. Where does the factor $|N|^2$ in the denominator come from? Why in the denominator?

3. Why does he talk about LSZ theorem? This S-matrix element seems the usual I get if I take the amputated Green function and I put the in/out coming particles on-shell, expect for the $|N|^2$ factor (I think the answer relies on this renormalization factor, but I don't understand why)

4. Once he choose this renormalization $<VAC| A_l(x)|\vec{q}_1,\sigma> =(2\pi)^{-3/2}N e^{ikx} u_l(q_1,\sigma)$ (eq. 10.3.3). Why now can he change the renormalization adding the $(2\omega)^{-1/2}$ factor?

 

[vanhees71]

The Feynman rules for S-matrix elements regarding the external lines (truncate the external legs from the propagator functions and dress them with asymptotic-free wave functions) originates from the LSZ-reduction theorem.

The normalization factor for the free fields results from the equal-time canonical commutators for the fields. For a scalar field you have
$$[\hat{\phi}(t,\vec{x}),\dot{\hat{\phi}}^{\dagger}(t,\vec{y})]=\mathrm{i} \delta^{(3)}(\vec{x}-\vec{y}).$$
For plane-wave (momentum-eigen) functions the time derivative is where the $1/(2 \omega)$ factor comes from.

Weinberg multiplies this with $N$, which you have to take into account when you do perturbation theory and dress your external legs with self-energy insertions. Depending on your renormalization scheme you might have to compensate the corresponding wave-function renromalization factors in the asymptotic free fields.