11  $w_0$ scale setting

The fit antsaz: $$\left(\frac{w_0}{a}\right)_{lat}=\frac{w_0^{iso}}{a}\left[1+P[5](\xi-\xi^{iso})\right]$$ with $\xi=\frac{M_\pi^2}{(4\pi f_\pi)^2}$, and the chiral LO from (Bär and Golterman 2014, eq (4.8)).

• volume effects:

  • C.06.112 is higher than C.06.80

  • B.072.96 is higher than B.072.64 althought compatible within errors

  • the on on $M_\pi$ and $f_\pi$ in the x-axis are included

• misstuning corrections:

  • $m_0$

  • $m_{\ell,s,c}$

$\chi^2/dof=$ 3.61635

par	a_from_w0	a_from_fpi
a(A)	0.094475(93)	0.09025(79)
a(B)	0.080892(26)	0.07948(11)
a(C)	0.068757(38)	0.06819(14)
a(D)	0.057125(34)	0.056850(90)
a(E)	0.049148(38)	0.04892(11)
P[5]	0.574(24)	0

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<r >

11.1 mu corrections

$\chi^2/dof=$ 1.58581

par	a_from_w0	a_from_fpi
a(A)	0.09420(12)	0.09022(64)
a(B)	0.08117(10)	0.079614(53)
a(C)	0.06890(10)	0.068143(67)
a(D)	0.05751(13)	0.056936(78)
a(E)	0.049078(61)	0.048930(64)
P[5]	0.663(31)	0

<r >

<r >

12 comparing w0 before and after corrections

<r >

12.1 w0 usign a from fpi

P[0] represent $w_0$ at the physical point. BMW value shoul be 0.17236

$\chi^2/dof=$ 3.10512

P	value
P[0]	0.1696(20)
P[1]	0.659(35)
P[2]	8.0(6.4)
P[3]	-1371(720)

<r >

Bär, Oliver, and Maarten Golterman. 2014. “Chiral Perturbation Theory for Gradient Flow Observables.” Physical Review D 89 (3). https://doi.org/10.1103/physrevd.89.034505.