Elastic Proton-Proton Scattering at tree level

Question

Hello im trying compute the Cross section of an elastic proton-proton scattering and my results are beyond terrrible and i can't figure out where my mistakes lie. I'm going to describe what i tried to do and hopefully someone can help me figure out where i mess up, im not asking you to do these calculations on your own but just see if you can see an obvious mistake in my reasoning or results. Im also aware that the problem could lie entirely on my computer code but i checked it over and over again and can't spot a mistake, so im thinking my error lies in the Physics. I also did not take into account the electromagnetic interaction because it should be negligible compared to the nuclear force. Thank you for reading this in advance

I use the following Lagrangians to Describe the Interactions of the protons with the exchange mesons:

\begin{equation} \begin{array}{c} \mathcal{L}_{\mathrm{NNps}}=-g_{p s} \bar{\psi} i \gamma_{5} \psi \varphi_{\mathrm{ps}} \\ \mathcal{L}_{\mathrm{NNs}}=-g_{s} \bar{\psi} \psi \varphi_{\mathrm{s}} \\ \mathcal{L}_{\mathrm{NNv}}=-g_{v} \bar{\psi} \gamma_{\mu} \psi \varphi_{\mathrm{v}}^{\mu}-\frac{f_{\mathrm{v}}}{4 M_{p}} \bar{\psi} \sigma_{\mu \nu} \psi \cdot\left(\partial^{\mu} \varphi_{\mathrm{v}}^{\nu}-\partial^{\nu} \varphi_{\mathrm{v}}^{\mu}\right) \end{array} \end{equation}

Each proton described by the Dirac Spinor \begin{equation} u\left(\boldsymbol{q}, \lambda\right)=\sqrt{E+m}\left(\begin{array}{c} 1 \\ \frac{2 \lambda q}{E+m} \end{array}\right)\left|\lambda\right\rangle \end{equation}

What i did is to calculate the Matrix Elements for each of these interaction in the C.O.M frame like this (i will only include the calculations on one vertex of the feynman diagrams)

$N=E+m$ for all the calculations that follow

For the Pseudoscalar mesons: \begin{equation} ig_{ps}N\left(\frac{2\lambda q}{N}-\frac{2\lambda' q}{N}\right) \left\langle\lambda'\mid \lambda\right\rangle \end{equation}

For the Scalar mesons: \begin{equation} g_{s}N\left(1-\frac{4\lambda\lambda'q^2}{N^2}\right) \left\langle\lambda'\mid \lambda\right\rangle \end{equation}

For the Vector Mesons: \begin{equation} j^{0}=(f_\mathrm{v}+g_\mathrm{v})N\left(1+\frac{4\lambda\lambda'q^2}{N^2}\right)\left\langle\lambda'\mid \lambda\right\rangle-\frac{f_\mathrm{v}}{2m}(q'^0+q^0)\left(1-\frac{4\lambda\lambda'q^2}{N^2}\right)\left\langle\lambda'\mid \lambda\right\rangle \end{equation} \begin{equation} j^{k}=(f_\mathrm{v}+g_\mathrm{v})N\left(\frac{2\lambda q}{N}+\frac{2\lambda' q}{N}\right)\left\langle\lambda'\mid\sigma^k\mid\lambda\right\rangle-\frac{f_\mathrm{v}}{2m}(q'^k+q^k)\left(1-\frac{4\lambda\lambda'q^2}{N^2}\right)\left\langle\lambda'\mid \lambda\right\rangle \end{equation}

What i did next is to find general expressions for \begin{equation} \left\langle\lambda'\mid \lambda\right\rangle \end{equation} and \begin{equation} \left\langle\lambda'\mid \sigma^k \mid \lambda\right\rangle \end{equation} So my PC could then sum over all posible helicity combinations. I added all 16 different matrix Elements like this: \begin{equation} \begin{aligned} \left\langle\left|\mathcal{M}_{f i}\right|^{2}\right\rangle &=\frac{1}{4}\left(\left|\mathcal{M}_{R R}\right|^{2}+\left|\mathcal{M}_{R L}\right|^{2}+\left|\mathcal{M}_{L R}\right|^{2}+\left|\mathcal{M}_{L L}\right|^{2}\right) \\ &=\frac{1}{4}\left(\left|\mathcal{M}_{R R \rightarrow R R}\right|^{2}+\left|\mathcal{M}_{R R \rightarrow R L}\right|^{2}+\cdots+\left|\mathcal{M}_{R L \rightarrow R R}\right|^{2}+\cdots\right) \end{aligned} \end{equation} Then i took into account both Feynman Diagrams (Picture from David Tong Lectures on Quantum Field Theory)

So the Matrix Element is \begin{equation} \mathcal{M}=\mathcal{M}_1-\mathcal{M}_2 \end{equation} Where the Matrix Elements Correspond to each Feynman diagrams and the minus sign is used because of Fermi-Dirac Statistics. For each of the Feynman diagrams i summed for the six exchange Bosons listed below with the following properties and i used their respective propagator in computing the Matrix Elements: \begin{equation} \begin{array}{lllllllc} \hline \hline \text { Meson } & J^{P} & I^{G} & g_{\alpha}^{2} / 4 \pi & f_{\alpha} / g_{\alpha} & m_{\alpha}(\mathrm{MeV}) & \Lambda_{\alpha}(\mathrm{GeV}) & 2 n_{\alpha} \\ \hline \pi & 0^{-} & 1^{-} & 14.4 & & 138.03 & 1.7 & 2 \\ \eta & 0^{-} & 0^{+} & 3 & & 548.8 & 1.5 & 2 \\ \rho & 1^{-} & 1^{+} & 0.9 & 6.1 & 769 & 1.85 & 4 \\ \omega & 1^{-} & 0^{-} & 24.5 & 0 & 782.6 & 1.85 & 4 \\ \delta & 0^{+} & 1^{-} & 2.488 & & 983 & 2.0 & 2 \\ \sigma^{a} & 0^{+} & 0^{+} & 8.9437 & & 550 & 1.9 & 2 \\ & & & (18.3773) & & (720) & (2.0) & (2) \\ \hline \end{array} \end{equation} At each vertex is used a form factor like this: \begin{equation} \mathcal{F}_{\alpha}\left[\left(\boldsymbol{q}^{\prime}-\boldsymbol{q}\right)^{2}\right]=\left[\frac{\Lambda_{\alpha}^{2}-m_{\alpha}^{2}}{\Lambda_{\alpha}^{2}+\left(\boldsymbol{q}^{\prime}-\boldsymbol{q}\right)^{2}}\right]^{n_{\alpha}} \end{equation}

Lastly i use the following equation to compute the Cross section \begin{equation} \sigma=\frac{1}{64 \pi^{2} s} \frac{\mathrm{p}_{f}}{\mathrm{p}_{i}} \int\left|\mathcal{M}_{f i}\right|^{2} \mathrm{~d} \Omega \end{equation}

Where the cross section must include a factor of $1/2$ because each final state may only be counted once in integrating over the angles

To my mind all of this should yield reasonable results. Not exact since this is a tree level treatment of the scattering but my results as i mentioned before do not even come close to the experimental results. I know that for Moller Scattering (Elastic electron scattering) tree level treatment is not enough since the integral is diverging. Is this one case where tree level treatment also doesn't work? Again i thank you for reading all this. The Graph below is my attempt where you can see my results where the Green points are the experimental findings and the blue curve is the cross section i calculated.

Answer 1

I'm not sure where the coupling constants and form factors come from, but they may well have been derived from a fit to elastic nucleon-nucleon scattering phase shifts (such as the Bonn meson exchange potential).

However, the nucleon-nucleon is sufficiently strong to generate bound states (such as the deuteron) and resonances, so that the tree level amplitude has to be iterated. Typically the tree level amplitude is converted into a potential, and used in a non-relativistic Schroedinger equation, or iterated using a Bethe-Salpeter equation. For high quality fits you also need certain higher order corrections, such as two-pion exchange. This means that a tree level approximation is at best qualitatively useful (and you can see that near threshold your result is off by a factor 2-3).

At low energy, using $\sqrt{s}$ is a little misleading, because $s$ is dominated by the rest masses. More relevant is the relative three-momentum in the CM frame. If that exceed a few hundred MeV, then relativistic effects and in-elastic channels become important. Also, for momenta bigger than a GeV or so you start to resolve the quark-gluon substructure, and a meson-exchange description is not appropriate.

You may think that at high energies, $\sqrt{s}>10$ GeV or so, perturbative QCD is reliable. Unfortunately, this is not the case. Both total and elastic nucleon-nucleon cross sections are dominated by small angle scattering, and pQCD is not applicable. There are semi-phenomenological models based on pomerons and Regge trajectories, but as theories of total cross sections these are not grounded in QCD.

Answer 2

turns out my method is simply wrong. More matrix elements need to be taken into account which means multiple meson exchanges. This is because perturbation theory doesn't work in the strong interaction. To get an accurate result one would have to take into account infinte Matrix Elements but since this is not possible Ladder approximations have to be used.