1  $M_\pi$

Mpi-cA.12.48 = 0.08029(15) , $\chi^2/dof=$ 0.92164

0.08028579 0.0001535833
Mpi-cB.72.64 = 0.056566(43) , $\chi^2/dof=$ 0.0099863

0.05656583 4.271331e-05
Mpi-cB.72.96 = 0.056534(15) , $\chi^2/dof=$ 0.01935

0.05653413 1.460993e-05
Mpi-cC.06.80 = 0.047227(34) , $\chi^2/dof=$ 0.12123

0.04722655 3.352617e-05
Mpi-cD.54.96 = 0.040621(30) , $\chi^2/dof=$ 0.055095

0.04062079 2.978815e-05
Mpi-cE.44.11 = 0.033830(29) , $\chi^2/dof=$ 0.39457

0.0338297 2.881185e-05
Mpi-cC.20.48 = 0.08551(12) , $\chi^2/dof=$ 1.0171

0.0855104 0.0001193227

<r >

1.1 Mpi fit (only as test)

to determine $a\mu_\ell$ we fit $$\frac{M_\pi^2}{f_\pi^2}= 2 \left(\frac{aB}{a^2f_\pi^2}\right) a\mu_\ell \left[1+5\xi\log \xi +P\xi \right]$$ where left hand site is a function of the variable $a\mu_\ell$ and $\left(\frac{aB}{a^2f_\pi^2}\right)$ is a fit parameter for each lattice spacing and $P$ a global parameter for all lattices. We define $\xi=2\left(\frac{aB}{a^2f_\pi^2}\right) a\mu_\ell$. After the fit we solve for $a\mu_\ell$ imposing the physical value of the ratio $\frac{M_\pi^2}{f_\pi^2}\bigg|_{phys}$

$\chi^2/dof=$ 2.16335

P	value
P[0]	892.1(1.5)
P[1]	1018.4(2.3)
P[2]	1203.4(3.5)
P[3]	1384.7(4.9)
P[4]	1.210(81)

<r >

Ens.	$a\mu_\ell$
B	0.0006675(23)
C	0.0005847(23)
D	0.0004948(22)
E	0.0004301(21)

1.2 Mpi fit only physical point (test)

$\chi^2/dof=$ 0.472663

P	value
P[0]	940(19)
P[1]	1067(21)
P[2]	1267(24)
P[3]	1453(26)
P[4]	6.0(1.8)

<r >

Ens.	$a\mu_\ell$	$a\mu_\ell(phys point only)$
B	0.0006675(23)	0.0006650(26)
C	0.0005847(23)	0.0005860(25)
D	0.0004948(22)	0.0004936(22)
E	0.0004301(21)	0.0004303(22)

1.2.1 adding A ensemble (only as test)

$\chi^2/dof=$ 4.64378

P	value
P[0]	812.1(6.2)
P[1]	889.0(1.4)
P[2]	1015.9(2.2)
P[3]	1200.7(3.4)
P[4]	1381.6(4.9)
P[5]	1.448(70)

<r >

Ens.	$a\mu_\ell$
A	0.0007317(57)
B	0.0006684(23)
C	0.0005849(23)
D	0.0004949(22)
E	0.0004301(21)

1.3 a2 term C20.48 no A12.48

adding an extra parameter $$\frac{M_\pi^2}{f_\pi^2}= 2 \left(\frac{aB}{a^2f_\pi^2}\right) a\mu_\ell \left[1+5\xi\log \xi +P\xi +P_1 a^2 \right]$$

$\chi^2/dof=$ 0.753925

P	value
P[0]	713(30)
P[1]	830(25)
P[2]	976(21)
P[3]	1183(18)
P[4]	1382(15)
P[5]	6.55(38)
P[6]	15.0(4.5)

<r >

Ens.	$a\mu_\ell$	$a\mu_\ell$-no-max-tw
A	0.0007483(59)	0.0007499(60)
B	0.0006667(23)	0.0006667(23)
C	0.0005860(25)	0.0005842(23)
D	0.0004940(23)	0.0004944(23)
E	0.0004305(22)	0.0004298(22)

1.4 A12.48 no C20.48

$\chi^2/dof=$ 0.869561

P	value
P[0]	714(30)
P[1]	826(27)
P[2]	968(23)
P[3]	1180(19)
P[4]	1377(16)
P[5]	6.46(44)
P[6]	15.8(4.9)

<r >

1.5 A12.48 no C20.48 + strange misstuning

$\chi^2/dof=$ 10728.9

P	value
P[0]	683.2(5.5)
P[1]	801.9(4.1)
P[2]	946.7(6.0)
P[3]	1163.7(4.4)
P[4]	1363.8(5.7)
P[5]	6.00(25)
P[6]	20.68(98)

<r >

1.6 A12.48 no C20.48 + strange misstuning, unitary (used for $a\mu_\ell$ determination)

$\chi^2/dof=$ 0.407355

P	value
P[0]	686.2(7.7)
P[1]	801.1(5.7)
P[2]	947.3(6.1)
P[3]	1165.3(4.5)
P[4]	1363.1(8.1)
P[5]	6.06(28)
P[6]	20.5(1.3)

<r >

Ens.	$a\mu_\ell$
A	0.0007479(99)
B	0.0006670(27)
C	0.0005862(42)
D	0.0004931(28)
E	0.0004305(31)

1.6.1 reweighting

$\chi^2/dof=$ 0.0950052

P	value
P[0]	0.174(58)
P[1]	0.085(26)
P[2]	0.101(15)
P[3]	0.156(20)
P[4]	2600(105)
P[5]	-1.98(29)e+5

<r >

Ens.	$a\mu_\ell$
A	0.0007461(58)
B	0.0006666(23)
C	0.0005862(25)
D	0.0004940(23)
E	0.0004305(22)