Point prescriptions for theory covariance matrices
The equations below display the different point prescriptions, as they
appear in validphys2.
3 points
Note
theoryids: 163, 180, 173
point_prescription: '3 point'
\[s_{11} = \frac{1}{2}\bigg\{ \Delta_1(+,+)^2 + \Delta_1(-,-)^2 \bigg\}\]
\[s_{12} = \frac{1}{4}\bigg\{\bigg(\Delta_1(+,+) + \Delta_1(-,-) \bigg) \bigg(\Delta_2(+,+) + \Delta_2(-,-) \bigg) \bigg\}\]
3f points
For isolated factorisation scale variation using a 3-point prescription (central, double, half).
Note
theoryids: 163, 177, 176
point_prescription: '3f point'
\[s_{11} = \frac{1}{2}\bigg\{ \Delta_1(+)^2 + \Delta_1(-)^2 \bigg\}\]
\[s_{12} = \frac{1}{2}\bigg\{ \Delta_1(+)\Delta_2(+) + \Delta_1(-)\Delta_2(-) \bigg\}\]
3r points
For isolated renormalisation scale variation using a 3-point prescription (central, double, half).
Note
theoryids: 163, 179, 174
point_prescription: '3r point'
\[s_{11} = \frac{1}{2}\bigg\{ \Delta_1(+)^2 + \Delta_1(-)^2 \bigg\}\]
\[s_{12} = \frac{1}{4}\bigg\{\bigg(\Delta_1(+) + \Delta_1(-) \bigg) \bigg(\Delta_2(+) + \Delta_2(-) \bigg) \bigg\}\]
5 points
Note
theoryids: 163, 177, 176, 179, 174
point_prescription: '5 point'
\[s_{11} = \frac{1}{2}\bigg\{ \Delta_1(+,0)^2 + \Delta_1(-,0)^2 + \Delta_1(0,+)^2 + \Delta_1(0,-)^2 \bigg\}\]
\[\begin{split}\begin{split}
s_{12} = \frac{1}{2}\bigg\{ &\Delta_1(+,0)\Delta_2(+,0) + \Delta_1(-,0)\Delta_2(-,0) \bigg\} \\
+ \frac{1}{4}\bigg\{ &\bigg(\Delta_1(0,+) + \Delta_1(0,-) \bigg)\bigg(\Delta_2(0,+) + \Delta_2(0,-)\bigg)\bigg\}
\end{split}\end{split}\]
Important
This is the sum of 3f points and 3r points.
\(\mathbf{\overline{5}}\) points
Note
theoryids: 163, 180, 173, 175, 178
point_prescription: '5bar point'
\[s_{11} = \frac{1}{2}\bigg\{ \Delta_1(+,+)^2 + \Delta_1(-,-)^2 + \Delta_1(+,-)^2 + \Delta_1(-,+)^2 \bigg\}\]
\[\begin{split}\begin{split}
s_{12} = \frac{1}{4}\bigg\{ &\bigg(\Delta_1(+,+) + \Delta_1(+,-)\bigg) \bigg(\Delta_2(+,+) + \Delta_2(+,-) \bigg) \\
+ &\bigg(\Delta_1(-,+) + \Delta_1(-,-)\bigg) \bigg(\Delta_2(-,+) + \Delta_2(-,-) \bigg) \bigg\}
\end{split}\end{split}\]
7 points - original
Warning
Deprecated prescription!
theoryids: 163, 177, 176, 179, 174, 180, 173
Specify in the runcard seventheories: original
point_prescription: '7 point'
\[\begin{split}\begin{split}
s_{11} = \frac{1}{3}\bigg\{ &\Delta_1(+,0)^2 + \Delta_1(-,0)^2 + \Delta_1(0,+)^2 + \Delta_1(0,-)^2 \\
+ &\Delta_1(+,+)^2 + \Delta_1(-,-)^2 \bigg\}
\end{split}\end{split}\]
\[\begin{split}\begin{split}
s_{12} = \frac{1}{6}\bigg\{ &\bigg(\Delta_1(+,0) + \Delta_1(+,+) \bigg) \bigg(\Delta_2(+,0) + \Delta_2(+,+) \bigg) \\
+ &\bigg(\Delta_1(-,0)+\Delta_1(-,-)\bigg) \bigg(\Delta_2(-,0) + \Delta_2(-,-) \bigg) \\
+ &\bigg(\Delta_1(0,+)+\Delta_1(0,-)\bigg)\bigg(\Delta_2(0,+) + \Delta_2(0,-) \bigg)\bigg\}
\end{split}\end{split}\]
7 points - Gavin (default)
Note
theoryids: 163, 177, 176, 179, 174, 180, 173
point_prescription: '7 point'
\[\begin{split}\begin{split}
s_{11} = \frac{1}{3}\bigg\{ &\Delta_1(+,0)^2 + \Delta_1(-,0)^2 + \Delta_1(0,+)^2 + \Delta_1(0,-)^2 \\
+ &\Delta_1(+,+)^2 + \Delta_1(-,-)^2 \bigg\}
\end{split}\end{split}\]
\[\begin{split}\begin{split}
s_{12} = \frac{1}{6}\bigg\{ &2\bigg(\Delta_1(+,0)\Delta_2(+,0) + \Delta_1(-,0)\Delta_2(-,0) \bigg) \\
+ &\bigg(\Delta_1(0,+)+\Delta_1(0,-)\bigg) \bigg(\Delta_2(0,+) + \Delta_2(0,-) \bigg) \\
+ &\bigg(\Delta_1(+,+)+\Delta_1(-,-)\bigg)\bigg(\Delta_2(+,+) + \Delta_2(-,-) \bigg)\bigg\}
\end{split}\end{split}\]
9 points
Note
theoryids: 163, 177, 176, 179, 174, 180, 173, 175, 178
point_prescription: '9 point'
\[\begin{split}\begin{split}
s_{11} = \frac{1}{4}\bigg\{ &\Delta_1(+,0)^2 + \Delta_1(-,0)^2
+ \Delta_1(0,+)^2 + \Delta_1(0,-)^2 \\
+ &\Delta_1(+,+)^2 + \Delta_1(+,-)^2
+ \Delta_1(-,+)^2 + \Delta_1(-,-)^2 \bigg\}
\end{split}\end{split}\]
\[\begin{split}\begin{split}
s_{12} = \frac{1}{12}\bigg\{&\bigg(\Delta_1(+,0)+\Delta_1(+,+) + \Delta_1(+,-)\bigg) \bigg(\Delta_2(+,0) + \Delta_2(+,+) + \Delta_2(+,-) \bigg) \\
+ &\bigg(\Delta_1(-,0) + \Delta_1(-,+) + \Delta_1(-,-)\bigg)\bigg(\Delta_2(-,0) + \Delta_2(-,+) + \Delta_2(-,-) \bigg) \bigg\}\\
+ \frac{1}{8}&\bigg(\Delta_1(0,+)+ \Delta_1(0,-)\bigg)\bigg(\Delta_2(0,+) + \Delta_2(0,-) \bigg)
\end{split}\end{split}\]