21  scratch area

We have the identity $$\begin{gather} c(t)=\int_{E_0}^{\infty}dE \rho(E) e^{-Et} =\cancel{\rho\frac{e^{-Et}}{-t}\bigg|_{E_0}^{\infty}} -\int_{E_0}^{\infty}dE \rho'(E)\frac{e^{-Et}}{-t} \end{gather}$$ thus using as a base $\tilde b_t(E)=\frac{e^{-Et}}{t}$ to approximate $\Delta$ using the HLT we can compute the convolution $\int\rho'\Delta$ $$\begin{gather} \sum_t g_tc(t)=\int_{E_0}^{\infty}dE \rho'(E) \sum_t g_t\frac{e^{-Et}}{t}= \int_{E_0}^{\infty}dE \rho'(E) \tilde \Delta(E) \\=\cancel{\rho \Delta\bigg|_{E_0}^{\infty} }-\int_{E_0}^{\infty}dE \rho(E) \tilde \Delta'(E)\,. \end{gather}$$ So if we use as base $\tilde b_t(E)=\frac{e^{-Et}}{t}$ to approximate $\Delta$ with the HLT we can compute the convolution $-\int\rho\Delta'$. For $Z_0$ the kernel function is $$\begin{gather} \theta_0(\omega-\omega_0)=\frac{c}{1+e^{\frac{\omega-\omega_0}{\sigma}}}\,,\\ -\int \theta_0(\omega-\omega_0)= c\left[\sigma \log\left(e^{\frac{\omega_0}{\sigma}}+e^{\frac{\omega}{\sigma}}\,. \right)-\omega\right] \end{gather}$$

• One could also try $-\int \theta_0(\omega-\omega_0)\approx-(\omega-\omega_0)\theta_0(\omega-\omega_0)$

• We could iterate the procedure and use as base $b_t''=\frac{e^{-Et}}{t^2}$ to approximate $\int \int\Delta$ $$\int\int\Delta=\int\int \theta_0(\omega-\omega_0)\approx \frac{1}{2} (\omega-\omega_0)^2\theta_0(\omega-\omega_0)$$

• We try a different function that approach the $\theta$, the algebraic function $$\theta_0^{algebraic}=\frac{c}{2}\left( \frac{x}{\sqrt{1+x^2}}+1\right)\,, \quad x=\frac{\omega_0-\omega}{\sigma}$$

21.1 Stability

21.2 Z0

$\theta_0(\omega-\omega_0)$ = 0.0593(19) , $\chi^2/dof=$ 0.15622

$-\int \theta_0(\omega-\omega_0)$ = 0.0583(43) , $\chi^2/dof=$ 0.18837

$-(\omega-\omega_0)\theta_0(\omega-\omega_0)$ = 0.0583(43) , $\chi^2/dof=$ 0.18837

$0.5(\omega-\omega_0)^2\theta_0(\omega-\omega_0)$ = 0.0475(80) , $\chi^2/dof=$ 0.20377

$\theta_0^{algebraic}(\omega-\omega_0)$ = 0.0593(19) , $\chi^2/dof=$ 0.1562

<r >

21.3 Z0

<r >

21.4 Z0

<r >

21.5 Z0

<r >

21.6 Z0

<r >

21.7 $\sigma$ extrapolation

21.8 Z0

<r >