Question

Hello, this is for quantum field theory. Please write neatly. Nice clear steps would be great. Please find the question below. It has two parts. Consider a theory with a single fermion species $\psi$ coupled to a single real scalar field $\phi$, with an interaction vertex given by $\mathcal{L}_{int} = -g \bar{\psi} \psi \phi$. The Lagrangian is expressed as: $\mathcal{L} = \frac{1}{2} \left[ (\partial \phi)^2 - m^2 \phi^2 \right] + \bar{\psi} (i \partial \! - M + g \phi^2) \psi$, where $g$, $M$ and $m$ are real constants. (a) Draw all fully connected (and amputated) Feynman diagrams, up to order $g^2$, contributing to the scattering process $\psi \bar{\psi} \to \psi \bar{\psi}$. Take the interaction term to be normal ordered. [2] (b) Translate these diagrams into an expression describing the scattering amplitude to this order. [6] (c) Comment on the ultraviolet behaviour of the loop integral in part (b) and whether or not this can be renormalised by the fundamental parameters of the given La- grangian. [2]

          Hello, this is for quantum field theory. Please write neatly. Nice clear steps would be great.
Please find the question below. It has two parts.
Consider a theory with a single fermion species $\psi$ coupled to a single real scalar field $\phi$, with an interaction vertex given by $\mathcal{L}_{int} = -g \bar{\psi} \psi \phi$. The Lagrangian is expressed
as:
$\mathcal{L} = \frac{1}{2} \left[ (\partial \phi)^2 - m^2 \phi^2 \right] + \bar{\psi} (i \partial \! - M + g \phi^2) \psi$,
where $g$, $M$ and $m$ are real constants.
(a) Draw all fully connected (and amputated) Feynman diagrams, up to order $g^2$,
contributing to the scattering process $\psi \bar{\psi} \to \psi \bar{\psi}$. Take the interaction term to be
normal ordered.
[2]
(b) Translate these diagrams into an expression describing the scattering amplitude
to this order.
[6]
(c) Comment on the ultraviolet behaviour of the loop integral in part (b) and whether
or not this can be renormalised by the fundamental parameters of the given La-
grangian.
[2]

Hello, this is for quantum field theory. Please write neatly. Nice clear steps would be great.
Please find the question below. It has two parts.
Consider a theory with a single fermion species ψ coupled to a single real scalar field ϕ, with an interaction vertex given by ℒint = -g ψ̅ψϕ. The Lagrangian is expressed
as:
ℒ = (1)/(2)[ (∂ϕ)^2 - m^2 ϕ^2 ] + ψ̅ (i ∂ - M + g ϕ^2) ψ,
where g, M and m are real constants.
(a) Draw all fully connected (and amputated) Feynman diagrams, up to order g^2,
contributing to the scattering process ψψ̅→ψψ̅. Take the interaction term to be
normal ordered.
[2]
(b) Translate these diagrams into an expression describing the scattering amplitude
to this order.
[6]
(c) Comment on the ultraviolet behaviour of the loop integral in part (b) and whether
or not this can be renormalised by the fundamental parameters of the given La-
grangian.
[2]

Added by William S.

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