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) (Cavanaugh 1997)
• $w_i=(1-2|p-0.5|)^2$ 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.