Explicit derivation of the Feynman amplitude of $e^+e^-\rightarrow\mu^+\mu^-$

Question

I'm trying to compute the Feynman amplitude of the process $$ e^+(p_1,s_1)e^-(p_2,s_2)\rightarrow \mu^+(q_1,r_1)\mu^-(q_2,r_2), $$

considering as interaction Lagrangian $$ \mathcal{L}_I=-\lambda_e\phi(x)\bar{\psi}(x)\psi(x)-\lambda_\mu\phi(x)\bar{\chi}(x)\chi(x), $$ where $\psi$ is the field of a scalar particle $H$, $\psi$ of $e$ and $\chi$ of $\mu$.

Using the Wick's theorem I get that the contribution to the transition amplitude is

$$ S=-2\frac{\lambda_e\lambda_\mu}{2}\int d^4x_1 d^4 x_2[\phi(x_1)\phi(x_2)]\bar{\chi}(x_1)\chi(x_1) \bar{\psi}(x_2)\psi(x_2), $$

where the contracted term $[\phi(x_1)\phi(x_2)]=i\Delta_F(x_1-x_2)$.

Then $$ S=-2\frac{\lambda_e\lambda_\mu}{2}\int d^4x_1 d^4 x_2\langle F|\bar{\chi}^-(x_1)\chi^-(x_1)|0\rangle\langle 0 |\bar{\psi}^+(x_2)\psi^+(x_2)| I \rangle i\Delta_F(x_1-x_2). $$

Writing $S=(2\pi)^4\delta^4(p_1+p_2-q_1-q_2)i\eta$, I get that $$ i\eta=-\lambda_e\lambda_\mu \bar{u}'^{r_2}(q_2)v'^{r_1}(q_1)\bar{v}^{s_1}(p_1)u^{s_2}(p_2)\frac{i}{(p_1+p_2)^2-m^2+i\varepsilon}. $$

Now I'd like to do the square of the amplitude in the 16 possible helicity configurations in the center of mass ($\vec{p}_1=-\vec{p}_2$ and $\vec{q}_1=-\vec{q}_2$), but if I try to compute the product of the $u$ and $v$ spinors (even before doing the square), I get that they are all zero.

I have to do (and I did) the calculations using the spinors:

$$ u^\pm(p)=\left( \begin{matrix} \sqrt{E\mp |\vec{p}|} \xi^\pm_p \\ \sqrt{E\pm |\vec{p}|} \xi^\pm_p \end{matrix}\right) $$

$$ v^\pm(p)=\pm\left( \begin{matrix} \sqrt{E\pm |\vec{p}|} \xi^\mp_p \\ -\sqrt{E\mp |\vec{p}|} \xi^\mp_p \end{matrix}\right), $$

where

$$ \xi^+_p=\left( \begin{matrix} cos\frac{\theta}{2} \\ e^{i\phi}sin\frac{\theta}{2} \end{matrix}\right) $$ and $$ \xi^{-}_p=\left( \begin{matrix} -e^{-i\phi}sin\frac{\theta}{2}\\ cos\frac{\theta}{2} \end{matrix}\right). $$

I read on Peskin Schroeder that

$$ u^{r \dagger}(\vec{p}) v^s (-\vec{p})=0 $$

so it may seems that all amplitudes of that form are zero in the center of mass... What am I missing?

Answer 1

Your mistake is hiding in plain sight! In your amplitude you have contributions of the form

$$ \bar v ^{s_1} (\mathbf{p}) u ^ {s_2}(-\mathbf{p}) $$

But you then try to use a formula for $v^\dagger$ rather than $\bar{v}$! Remember that $\bar v$ differs from $v^\dagger$ by a (crucial) factor of $\gamma^0$. This has the effect of making

$$\bar v ^{s_1} (\mathbf{p}) u ^ {s_2}(-\mathbf{p}) \neq 0 $$

In fact the vanishing combination involving barred rather than daggered variables is

$$ \bar v ^{s_1} (\mathbf{p}) u ^ {s_2}(\mathbf{p}) = 0$$

as proved on page 104 of these excellent notes.