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)


Cavanaugh, Joseph E. 1997. “Unifying the Derivations for the Akaike and Corrected Akaike Information Criteria.” Statistics & Probability Letters 33 (2): 201–8.