advanced_tools:mass_insertion_approximation
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advanced_tools:mass_insertion_approximation [2017/12/04 08:01] |
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| + | ====== Mass Insertion Approximation ====== | ||
| + | <blockquote> | ||
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| + | Chirality is not well-defined for massive fields. A famous consequence of this fact are pion masses, which can be linked to [[advanced_notions:symmetry_breaking:chiral_symmetry_breaking|Chiral Symmetry Breaking]]. | ||
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| + | 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 | ||
| + | $$ P_{L,R} \psi_D \neq \alpha \psi_D. $$ | ||
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| + | There is a computational trick called a "mass insertion", which can be confusing in this regard: | ||
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| + | 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. | ||
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| + | 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. | ||
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| + | Sometimes, the diagram including a mass insertion is computed in order to show that the error induced by neglecting the mass is small indeed. | ||
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| + | <cite>https://physics.stackexchange.com/a/298904/37286</cite></blockquote> | ||
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