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Fermion Anti-fermion Interaction in a Linearized Quantum Gravity

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Using the linearized gravity in the weak field regime and applying the quantum field theory prescription, the study of the gravitational interactions of massless fermions (neutrinos) in tree-level and through the u- and t-channels has been investigated before, but the gravitational interaction through the s-channel has not taken into account yet. In this paper, we calculate the cross section for the gravitational interaction of a fermion-anti-fermion pair to another pair of fermion-anti-fermion in tree level, that is purely an s-channel interaction, and compare it with the electromagnetic analogous.

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Correspondence to Tahere Olyaei.

Appendix: Yukawa theory

Appendix: Yukawa theory

Fermions’ interaction by exchanging a scalar particle is studied by Yukawa theory [22]

$$ \mathcal{L}_{\text{sf}}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}M^{2}\phi^{2} + \overline{{\Psi} }(i/\!\!\!\partial-m){\Psi} - g\phi\overline{{\Psi} }{\Psi}, $$

where Ψ and ϕ respectively stand for massive fermion and scalar fields, and g is the coupling constant. From this Lagrangian, the scalar propagator and the fermion-fermion-scalar vertex are derived as below

For \(f\bar {f}\to f^{\prime }\bar {f}^{\prime }\) interaction with scalar particle as mediator,

the amplitude is

$$ i\mathcal{M}_{\mathrm{s}}=-ig^{2}\frac{[\bar{v}(\mathbf{p}_{2})u(\mathbf{p}_{1})][\bar{u}(\mathbf{p}_{3})v(\mathbf{p}_{4})]}{q^{2}-M^{2}}. $$


$$\begin{array}{@{}rcl@{}} \overline{|\mathcal{M}_{\mathrm{s}}|^{2}} &=& \frac{g^{4}}{4(q^{2}-M^{2})^{2}}[\bar{v}(\mathbf{p}_{2})u(\mathbf{p}_{1})\bar{u}(\mathbf{p}_{1})v(\mathbf{p}_{2})][\bar{u}(\mathbf{p}_{3})v(\mathbf{p}_{4})\bar{v}(\mathbf{p}_{4})u(\mathbf{p}_{3})]\\ &=&\frac{g^{4}}{4(q^{2}-M^{2})^{2}}\text{Tr}[(/\!\!\!p_{2}-m)(/\!\!\!p_{1}+m)]\text{Tr}[(/\!\!\!p_{3}+m^{\prime})(/\!\!\!p_{4}-m^{\prime})]\\ &=&\frac{4g^{4}}{(q^{2}-M^{2})^{2}}[(p_{1}p_{2})-m^{2}][(p_{3}p_{4})-{m^{\prime}}^{2}], \end{array} $$

where in its computation we have used (21), (25) and (26). Consequently, by using (20), we achieve the differential cross section

$$ \frac{d\sigma_{\mathrm{s}}}{d{\Omega}}=\frac{g^{4}}{16\pi^{2}E^{2}} \frac{|\mathbf{p}_{i}||\mathbf{p}_{f}|^{3}}{(4E^{2} - M^{2})^{2}}, $$

which is independent of the scattering angle.

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Azizi, A., Olyaei, T. Fermion Anti-fermion Interaction in a Linearized Quantum Gravity. Int J Theor Phys 57, 2738–2747 (2018).

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