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  In chiral perturbation theory with complex $\phi$-s, would the next lo leading order renormalization $\gamma$-s change?

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The Lagrangian of chiral perturbation theory (with two quark flavors) is written using the following matrix $U$
where $\sigma^i$ are the Pauli matrices, $\phi_i$ are three scalar fields and $f$ is a constant with mass dimension. $U$ is unitary, which makes the $\phi_i$ fields real. 

The Lagrangian at $O(p^2)$ order is 
the Lagrangian at next to leading order $O(p^4)$ is

at next to leading order the coefficients $l_i$ and renormalize with loops with vertices coming only from the $O(p^2)$ Lagrangian like


where the gammas are


Now comes the question. Let's consider the same theory with the same Lagrangian but allowing the $\phi$ fields in $U$ to be complex.

My question is, would the $\gamma_i-s$ change?

asked Aug 28, 2015 in Theoretical Physics by Dmitry hand me the Kalashnikov (735 points) [ revision history ]
edited Aug 30, 2015 by Arnold Neumaier

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