Maxwell Lagrangian at weak fields

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The discussion centers on the requirement for the constant C in the equation related to the Maxwell Lagrangian to vanish when considering weak fields. The author of the referenced paper suggests that for the Lagrangian to align with the standard Maxwell Lagrangian under weak field conditions, C must equal zero. This is based on the premise that in the absence of any field, the equation simplifies to zero, necessitating that the constant also be zero to maintain consistency. The confusion arises from the interpretation of "weak fields," which implies a scenario where the field strength is negligible. Ultimately, the conclusion is that the constant must vanish to ensure the Lagrangian behaves correctly in weak field limits.
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In http://arxiv.org/abs/hep-th/9506035 the author said after writing this equation:

$$\frac{1}{4}\eta^{\mu\nu\lambda\rho} F_{\mu\nu}F_{\lambda\rho} = \eta_{\sigma\tau\alpha\beta}\frac{\partial L}{\partial F_{\sigma\tau}} \frac{\partial L}{\partial F_{\alpha\beta} } + 2C$$

where C was arbitrary constant of integration." In fact, if L is to agree with the usual Maxwell Lagrangian at weak fields the constant must vanish". Why? I mean why should the constant vanish. It seems that I don't understand what he meant by Maxwell Lagrangian at "weak fields".
 
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Imagine the case of no field at all. Everything apart from the constant is zero, so the constant has to be zero as well.
 
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