# Chapter 11 Systematics RF

In the following sections (11.1-11.6) we report the spread of the different analysis. In the plots the point in red are the one with $$\chi^2_{dof}<1.8$$, red band eq28 while blue band AIC. the fit has the form $\begin{cases} a_{\mu}^{SD}(eq,\ell)=P[0]+a^2P[1]\\ a_{\mu}^{SD}(op,\ell)=P[0]+a^2P[2] \end{cases}$ with some variations:

• log_{n}eq_w{m} mans that we modify the term $$a^2$$ in OS in with $$P a^2/log(w_0^2/m/a^2)^n$$ $$n=-0.2, 1 , 2, 3$$, $$m=1,2$$
• log_{n}op_w{m} same with TM
• +log_{n}eq_w{m} mans that we add a term $$P' a^2/log(w_0^2/m/a^2)^n$$
• a4 means we add $$a^4 P''$$
• Mpi_eq_op that we add a term $$P''' (M_\pi-M_\pi^{phys})$$ in OS and TM with the same coefficient
• for $$a_\mu^W(\ell)$$ there are other two coefficients for the FVE

The fit with log are weighted with a factor $$1/8$$ ($$1/4$$ for the choice of the exponent in the logarithm $$n$$ times $$1/2$$ the choice of the scale $$w_0/m$$)

### 11.0.1 Systematics

To estimate the systematic we use the formula $\overline{x} = \sum_i w_i x_i\\ \overline{\sigma}^2 = \sum_i w_i \sigma_i^2+\sum_i w_i( x_i-\overline{x})^2 \\$ The weights $$w_i$$ are chosen as:

• $$w_i=1/N \,\,\forall\, i$$ with a cut for the analysis with $$\chi^2>1.8$$ (eq28)
• $$w_i=\exp(-\frac{1}{2}(\chi^2+2N_{par}-N_{meas}) )$$ (AIC)
• $$w_i=\exp(-\frac{1}{2}(\chi^2+2N_{par}+ (2N_{npar}^2+2N_{par})/(N_{meas}-N_{par}-1)) )$$ (AICc)
• $$w_i=(1-2|p-0.5|)^2$$ where $$p$$ is the $$p$$-value of the fit (p-value)

### References

Cavanaugh, Joseph E. 1997. “Unifying the Derivations for the Akaike and Corrected Akaike Information Criteria.” Statistics & Probability Letters 33 (2): 201–8. https://doi.org/https://doi.org/10.1016/S0167-7152(96)00128-9.