Abstract
We apply the nuclear model with explicit mesons to photoproduction of neutral pions off protons at the threshold. In this model the nucleons do not interact with each other via a potential but rather emit and absorb mesons that are treated explicitly on equal footing with the nucleons. We calculate the total cross section of the reaction for energies close to threshold and compare the calculations with available experimental data. We show that the model is able to reproduce the experimental data and determine the range of the parameters where the model is compatible with the experiment.
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Notes
The action of \(W^\dagger \) includes integration \(\int _V d^3r\) to get rid of the coordinate of the annihilated pion.
For this one-dimensional problem the ground state wave-function needs around four to five gaussians to converge.
First the matrix element with the operator \(r_ir_j\) is calculated as
$$\begin{aligned} \langle e^{i{\vec {s}}{\vec {r}}}| r_i r_j|e^{-\alpha r^2}\rangle =\frac{-\partial ^2}{\partial s_i\partial s_j} \langle e^{i{\vec {s}}{\vec {r}}}|e^{-\alpha r^2}\rangle =\frac{-\partial ^2}{\partial s_i\partial s_j} e^{-\frac{1}{4\alpha }s^2}\left( \frac{\pi }{\alpha }\right) ^{3/2} =\left( \frac{1}{2\alpha }\delta _{ij}-\frac{1}{4\alpha ^2}s_is_j\right) e^{-\frac{1}{4\alpha }s^2}\left( \frac{\pi }{\alpha }\right) ^{3/2} \,.\end{aligned}$$(57)And now the sought matrix element evaluates as
$$\begin{aligned}{} & {} \langle e^{i{\vec {s}}{\vec {r}}}| ({\vec {e}}\frac{\partial }{\partial {\vec {r}}}) ({\vec {\sigma }}{\vec {r}}) |e^{-\alpha r^2}\rangle =({\vec {e}}{\vec {\sigma }}) \langle e^{i{\vec {s}}{\vec {r}}}| e^{-\alpha r^2}\rangle -2\alpha \langle e^{i{\vec {s}}{\vec {r}}} |({\vec {e}}{\vec {r}})({\vec {\sigma }}{\vec {r}})|e^{-\alpha r^2}\rangle \nonumber \\{} & {} \quad = ({\vec {e}}{\vec {\sigma }})e^{-\frac{1}{4\alpha }s^2} \left( \frac{\pi }{\alpha }\right) ^{3/2} -2\alpha \left( \frac{1}{2\alpha }({\vec {e}}{\vec {\sigma }}) -\frac{1}{4\alpha ^2}({\vec {e}}{\vec {s}})({\vec {\sigma }}{\vec {s}}) \right) e^{-\frac{1}{4\alpha }s^2}\left( \frac{\pi }{\alpha }\right) ^{3/2} \nonumber \\{} & {} \quad = \frac{1}{2\alpha }({\vec {e}}{\vec {s}})({\vec {\sigma }}{\vec {s}}) e^{-\frac{1}{4\alpha }s^2}\left( \frac{\pi }{\alpha }\right) ^{3/2} \end{aligned}$$(58)- $$\begin{aligned} \langle \uparrow | {\vec {\sigma }}{\vec {s}} |\uparrow \rangle = s_z \,,\, \langle \downarrow | {\vec {\sigma }}{\vec {s}} |\uparrow \rangle = s_x+is_y \,,\, s_z^2+(s_x-is_y)(s_x+is_y)=s^2 \,.\end{aligned}$$(63)
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Numerous discussions with Hans Fynbo and Karsten Riisager are gratefully acknowledged.
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Fedorov, D.V., Mikkelsen, M. Threshold Photoproduction of Neutral Pions Off Protons in Nuclear Model with Explicit Mesons. Few-Body Syst 64, 3 (2023). https://doi.org/10.1007/s00601-022-01783-9
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DOI: https://doi.org/10.1007/s00601-022-01783-9