Mass Insertion Approximation

Prerequisites: Feynman Diagrams. Full path (22 pages)

Intuitive
Concrete
Abstract

Chirality is not well-defined for massive fields. A famous consequence of this fact are pion masses, which can be linked to 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.

https://physics.stackexchange.com/a/298904/37286

Intuitive

A massless fermion has a definite handedness, its chirality: a left-handed one stays left-handed as it propagates. A mass term in the Lagrangian couples the left-handed and the right-handed field, so a massive fermion can be pictured as a particle that keeps switching between its left-handed and right-handed versions as it moves. The mass sets how often this happens.

The mass insertion approximation takes this picture literally. We start from massless fermions, which are simpler to compute with, and treat each switch of handedness as a separate interaction event, drawn as a dot (a "mass insertion") on the fermion line in a Feynman diagram. Each dot costs one power of the mass. If the mass is small compared to the energies in the process, diagrams with few dots dominate, and we can stop after the first one or two.

The picture also gives counting rules for free. A process that needs the handedness of a fermion to change must contain an odd number of insertions, so its amplitude is proportional to the mass and vanishes when the mass goes to zero. A process that keeps the handedness gets corrections only at second order in the mass.

Concrete

Write the Dirac field as $\psi=\psi_L+\psi_R$ with $\psi_{L,R}=P_{L,R}\psi$ , $P_{L,R}=\frac12(1\mp\gamma^5)$ . Since $\bar\psi\gamma^\mu\psi=\bar\psi_L\gamma^\mu\psi_L+\bar\psi_R\gamma^\mu\psi_R$ while $\bar\psi\psi=\bar\psi_L\psi_R+\bar\psi_R\psi_L$ , the Lagrangian splits into $\mathcal{L} = \bar\psi_L i\gamma^\mu\partial_\mu\psi_L + \bar\psi_R i\gamma^\mu\partial_\mu\psi_R \;-\; m\left(\bar\psi_L\psi_R+\bar\psi_R\psi_L\right).$ The kinetic terms preserve chirality and the mass term flips it. In the mass insertion approximation the free propagator is the massless one (with $\not{p}=\gamma^\mu p_\mu$ ), $S_0(p)=\frac{i\not{p}}{p^2}$ , which connects $\psi_L$ with $\bar\psi_L$ (and $\psi_R$ with $\bar\psi_R$ ), and the mass term is a two-point vertex with Feynman rule $-im$ that connects the two chiralities.

Summing all chains of insertions reproduces the massive propagator. With $(\not{p})^2=p^2$ , $S_0+S_0(-im)S_0+S_0(-im)S_0(-im)S_0+\dots = \frac{i}{\not{p}}\sum_{n=0}^\infty\left(\frac{m}{\not{p}}\right)^n = \frac{i}{\not{p} - m} = \frac{i(\not{p}+m)}{p^2-m^2}.$ Splitting into even and odd numbers of insertions, $\frac{i(\not{p}+m)}{p^2-m^2} = \frac{i\not{p}}{p^2}\sum_{k=0}^\infty\left(\frac{m^2}{p^2}\right)^k + \frac{im}{p^2}\sum_{k=0}^\infty\left(\frac{m^2}{p^2}\right)^k.$ The first part keeps the chirality and its first correction is of order $m^2/p^2$ . The second part flips it and starts at order $m$ . When $m^2\ll|p^2|$ for the momenta that matter in a process, the series can be truncated after the first term of each kind.

Example. In the decay $\pi^-\to\ell^-\bar\nu_\ell$ , the weak interaction couples only to the left-handed lepton field. The pion has spin zero, and angular momentum conservation forces the outgoing charged lepton into the helicity state that, for a massless particle, would belong to the right-handed field. The amplitude therefore needs one chirality flip and is proportional to $m_\ell$ , and the decay rate is $\Gamma(\pi^-\to\ell^-\bar\nu_\ell)\propto m_\ell^2\left(1-\frac{m_\ell^2}{m_\pi^2}\right)^2.$ This is why the pion decays almost always to a muon and much more rarely to an electron, even though the electron channel has more phase space.

Abstract

The general setting is a free action that is quadratic in the fields, $S_2=\frac12\Phi^TK\Phi$ , split as $K=K_0-\Delta$ , where $K_0$ is simple (diagonal in some basis, or massless) and $\Delta$ is a small field-independent bilinear, the "insertion". Perturbation theory in $\Delta$ treats it as a two-point vertex. The full propagator obeys the Dyson series $G = G_0 + G_0\Delta G_0 + G_0\Delta G_0\Delta G_0+\dots = (G_0^{-1}-\Delta)^{-1},$ up to factors of $i$ fixed by the conventions for propagators and vertices, so resumming all insertions is exact. Truncating is an expansion in $\Delta$ relative to the scales in $G_0$ : external or loop momenta, or differences between diagonal masses. When loop integrals are dominated by momenta much larger than $\Delta$ , the expansion converges quickly. When the relevant momenta are comparable to the insertion, truncation at low order is not reliable and one uses the resummed propagator.

The selection rules follow from symmetry. For a fermion with Weyl components $\psi_L,\psi_R$ and complex mass term $-m\bar\psi_L\psi_R+\text{h.c.}$ , the massless theory is invariant under independent rephasings $\psi_L\to e^{i\alpha_L}\psi_L$ , $\psi_R\to e^{i\alpha_R}\psi_R$ . The full theory is invariant too if $m$ is treated as a spurion transforming as $m\to e^{i(\alpha_L-\alpha_R)}m$ . Every amplitude must then be a covariant function of $m$ and $m^*$ , which fixes which powers of the mass can multiply a given chirality structure without computing any diagram. The same reasoning in an effective field theory explains why chirality-flipping operators, such as dipole operators for magnetic moments, come with a factor of a fermion mass or another chirality-breaking coupling.

The same method is used for flavour. If the mass matrix of several fields is not diagonal in the basis in which the interactions are simple, one may keep that basis and treat the off-diagonal entries as insertions, instead of diagonalizing the mass matrix and rotating all couplings. In supersymmetric models, the off-diagonal entries of the sfermion mass matrices, normalized to an average sfermion mass squared, are the parameters usually called $\delta_{ij}$ , and flavour-changing processes computed with a single insertion give bounds on them.

Mass Insertion Approximation

Intuitive

Concrete

Abstract

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