5  scale setting

no bounding has been applied

f_\pi^j(\xi, \infty) =\frac{f_\pi^j(\xi, L) }{1-\Delta_{FVE}} \Delta_{FVE} = - 2 \xi_\ell ~ \widetilde{g}_1(\lambda) \\ \widetilde{g}_1(\lambda) \simeq 4 \sqrt{\frac{\pi}{2}} \sum_{n=1}^\infty \frac{m(n)}{(\sqrt{n} \lambda)^{3/2}} e^{- \sqrt{n} \lambda} \\ \xi_\pi \equiv \frac{M_\pi^2}{(4 \pi f_\pi)^2} with m(n) the multiplicities of a three-dimensional vector \vec{n} having integer norm n (i.e.~m(n) = \{6, 12, 8, 6, ...\}) and \lambda=M_\pi L. To obtain the above formula we expand K_1, the Bessel function of the second kind, by its asymptotic expansion.
Different choices of the expansion variable are possible: one can replace f_\pi with the LO LEC f and/or replace M_\pi^2 with 2 B m_\ell (and correspondingly M_\pi L with \sqrt{2 B m_\ell} L in the arguments of the functions \widetilde{g}_1 and \widetilde{g}_2). At NLO (i.e., for the GL formula) the above changes are equivalent, since any difference represents a NNLO effect.

af_\pi^j(\xi) = af_\pi^j(\xi^{phys}) \left\{1-2\xi\log(\xi/\xi)+ [P+P_{disc} (af_\pi^j(\xi^{phys}))^2] (\xi-\xi^{phys})\right\}

P[0]=a(A) \text{fm}\\ P[1]=a(B) \text{fm}\\ P[2]=a(C) \text{fm}\\ P[3]=a(D) \text{fm}\\ P[4]=a(E) \text{fm}

\chi^2/dof= 1.48522

P	value
P[0]	0.09124(46)
P[1]	0.079388(47)
P[2]	0.068089(55)
P[3]	0.056824(59)
P[4]	0.048845(65)
P[5]	-8.87(85)
P[6]	-502(312)

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5.1 scaling with tau paper

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5.2 Extra FVE

here we add an residual volume dependece a f_\pi(L)=a f_\pi(L=\infty)\left\{\ 1+P_{FVE}\xi \frac{e^{M_\pi L}}{(M_\pi L)^{3/2}} \right\}

\chi^2/dof= 1.59947

P	value
P[0]	0.09117(46)
P[1]	0.079376(49)
P[2]	0.068070(66)
P[3]	0.056799(86)
P[4]	0.048823(78)
P[5]	-8.93(86)
P[6]	-477(318)
P[7]	21(39)

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5.3 No C20

\chi^2/dof= 0.393088

P	value
P[0]	0.09026(66)
P[1]	0.079410(48)
P[2]	0.068084(55)
P[3]	0.056846(58)
P[4]	0.048859(66)
P[5]	-6.0(1.8)
P[6]	-1496(655)

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5.4 scaling with tau paper

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