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In softly broken SUSY, the bare mass parameters may be specified at e.g. the GUT scale, and then we can run these down to another scale using RGEs, similar in form to the RGEs for gauge couplings, with a 1-loop and 2-loop differential beta function. Once these parameters have been run to the desired scale, tree-level physical masses are computed via diagonalization of the mass matrices depending on how the mixings were determined. Next, additional accuracy is gained by working out the loop corrections to the physical masses with self-energy graphs.

I am confused by why the physical masses that you determine from diagonalizing the soft parameters are considered tree-level masses if they are determined from 2-loop RGEs. Could you have determined the physical masses at the high scale and computed the self-energy loop corrections there, and then renormalized the physical mass down to the lower scale?

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The physical masses should be independent of the renormalization scale. We, however, only calculate a finite number of loop corrections, resulting in a scale dependence in the physical mass. This scale dependence can be used to estimate the error in the mass calculation from the missing higher orders.

In principle, one could calculate the sparticle mass spectrum at any scale. In practice, though, one wants to minimize the effect of the missing orders. This is typically achieved, in the MSSM, by minimizing the scale dependence of the one-loop EWSB condition $$ \mu^2 + 1/2 m_Z^2 = \frac{1}{\tan^2\beta-1}(m_{H_1}^2 - \tan^2\beta m_{H_2}^2) $$ which is achieved by choosing the scale $\mu=\sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}}$ - the sparticle mass spectrum is typically calculated at this scale. This scale, however, must be found iteratively, because, writing the dependence explicitly, $\mu=\sqrt{m_{\tilde{t}_1}(\mu)m_{\tilde{t}_2}(\mu)}$.

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