This section describes the effective-Lagrangian technique within the context of a simple toy model, closely following the discussion of [27].

In all branches of theoretical physics a key part of any good prediction is a careful assessment of the theoretical error which the prediction carries. Such an assessment is a precondition for any detailed quantitative comparison with experiment. As is clear from numerous examples throughout physics, this assessment of error usually is reliably determined based on an understanding of the small quantities which control the corrections to the approximations used when making predictions. Perhaps the most famous example of such a small quantity might be the fine-structure constant, , which quantifies the corrections to electromagnetic predictions of elementary particle properties or atomic energy levels.

2.1 The utility of low-energy approximations

2.2 A toy example

2.2.1 Massless-particle scattering

2.3 The toy model revisited

2.3.1 Exhibiting the symmetry

2.3.2 Timely performance the low-energy approximation

2.3.3 Implications for the low-energy limit

2.3.4 Redundant interactions

2.4 Lessons learned

2.4.1 Why are effective Lagrangians not more complicated?

2.5 Predictiveness and power counting

2.5.1 Power-counting low-energy Feynman graphs

2.5.2 Application to the toy model

2.6 The effective Lagrangian logic

2.6.1 The choice of variables

2.6.2 Regularization dependence

2.7 The meaning of renormalizability

2.2 A toy example

2.2.1 Massless-particle scattering

2.3 The toy model revisited

2.3.1 Exhibiting the symmetry

2.3.2 Timely performance the low-energy approximation

2.3.3 Implications for the low-energy limit

2.3.4 Redundant interactions

2.4 Lessons learned

2.4.1 Why are effective Lagrangians not more complicated?

2.5 Predictiveness and power counting

2.5.1 Power-counting low-energy Feynman graphs

2.5.2 Application to the toy model

2.6 The effective Lagrangian logic

2.6.1 The choice of variables

2.6.2 Regularization dependence

2.7 The meaning of renormalizability

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