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--- advanced_tools:mass_insertion_approximation [2017/09/29 09:42]
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+++ advanced_tools:mass_insertion_approximation [2017/12/04 08:01]
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-====== Mass Insertion Approximation ======
-<blockquote>
-Chirality is not well-defined for massive fields. A famous consequence of this fact are pion masses, which can be linked to [[symmetry_breaking:chiral_symmetry_breaking|Chiral Symmetry Breaking]].
-In the Lagrangian, you can define left- and right-handed Weyl fermions independently. A mass term will mix these, giving a massive Dirac fermion. Weyl fermions fulfill either
-$$ P_{L} \psi_L = \psi_L, \quad \text{or} \quad  P_R \psi_R = \psi_R$$
-but a Dirac fermion is not an eigenstate of the projection operators
-$$ P_{L,R} \psi_D \neq \alpha \psi_D. $$
-There is a computational trick called a "mass insertion", which can be confusing in this regard:
-A Dirac fermion can be considered as a coupled system of two Weyl fermions, where the mass is the coupling parameter.
-If a fermion's mass is small compared to the energy of a given process, one can approximate the Dirac fermion by its two (massless) Weyl components.
-The advantage is that for massless fields, loop integrals usually take much simpler forms.
-Corrections to the massless case can then be included by adding a Feynman rule for the mass term in the Lagrangian, which is a bilinear coupling between the left- and right-handed Weyl fermions.
-If you were to resum all possible mass insertions, the result is the same as if you had started with the massive Dirac fermion from the start.
-Since the underlying assumption of the approximation is that the mass is small compared to other energy scales in the theory, the corrections are usually small, though.
-Sometimes, the diagram including a mass insertion is computed in order to show that the error induced by neglecting the mass is small indeed.
-<cite>https://physics.stackexchange.com/a/298904/37286</cite></blockquote>

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