6 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^{phys})+ [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= 2.43586
| P | value |
|---|---|
| P[0] | 0.09142(42) |
| P[1] | 0.079473(46) |
| P[2] | 0.068101(59) |
| P[3] | 0.056821(66) |
| P[4] | 0.048862(66) |
| P[5] | -8.93(72) |
| P[6] | -445(269) |
<r >
6.1 scaling with tau paper
<r >
6.2 max twist corrections
| par | value | no_max_twist |
|---|---|---|
| P[0] | 0.09133(42) | 0.09142(42) |
| P[1] | 0.079478(46) | 0.079473(46) |
| P[2] | 0.068187(68) | 0.068101(59) |
| P[3] | 0.056825(66) | 0.056821(66) |
| P[4] | 0.048899(66) | 0.048862(66) |
| P[5] | -8.67(73) | -8.93(72) |
| P[6] | -534(273) | -445(269) |
<r >
6.3 scaling with tau paper
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6.4 Only phys point ensembles
\chi^2/dof= 0.377803
| P | value |
|---|---|
| P[0] | 0.07941(11) |
| P[1] | 0.068096(87) |
| P[2] | 0.056843(61) |
| P[3] | 0.048881(66) |
| P[4] | -1.3(3.0) |
| P[5] | -3387(2016) |
<r >
<r >
6.5 No C20
\chi^2/dof= 0.854531
| P | value |
|---|---|
| P[0] | 0.09018(63) |
| P[1] | 0.079492(46) |
| P[2] | 0.068095(58) |
| P[3] | 0.056848(66) |
| P[4] | 0.048878(66) |
| P[5] | -5.3(1.7) |
| P[6] | -1703(628) |
<r >
6.6 scaling with tau paper
<r >
6.7 max twist corrections with A12
\chi^2/dof= 2.07209
| x |
|---|
| |P |value | |
| |:—-|:————| |
| |P[0] |0.09128(40) | |
| |P[1] |0.079478(46) | |
| |P[2] |0.068188(68) | |
| |P[3] |0.056826(65) | |
| |P[4] |0.048900(66) | |
| |P[5] |-8.60(71) | |
| |P[6] |-563(265) | |
<r >
6.8 max twist corrections with A12 no C20
\chi^2/dof= 0.802867
| P | value |
|---|---|
| P[0] | 0.09021(58) |
| P[1] | 0.079497(47) |
| P[2] | 0.068176(67) |
| P[3] | 0.056850(66) |
| P[4] | 0.048914(65) |
| P[5] | -5.4(1.6) |
| P[6] | -1689(589) |
<r >
6.9 max twist corrections with A12 no C20 + strange misstuning as error
| par | value | percent |
|---|---|---|
| P[0] | 0.09135(37) | 0.0040684 |
| P[1] | 0.07944(11) | 0.0014365 |
| P[2] | 0.06818(14) | 0.0021009 |
| P[3] | 0.056846(80) | 0.0014004 |
| P[4] | 0.04890(11) | 0.0022075 |
| P[5] | -8.82(21) | -0.0239281 |
| P[6] | -499.98(48) | -0.0009575 |
<r >
6.10 max twist corrections with A12 no C20 + strange misstuning as error, unitary (main)
\chi^2/dof= 0.44991
X1 X2 X33 P[0] 9.025081e-02 7.890561e-04 4 P[1] 7.947758e-02 1.081341e-04 5 P[2] 6.818945e-02 1.429821e-04 6 P[3] 5.685040e-02 9.017466e-05 7 P[4] 4.891721e-02 1.065579e-04 8 P[5] -5.701487e+00 2.227514e+00 9 P[6] -1.592353e+03 7.812041e+02
| par | value | percent |
|---|---|---|
| P[0] | 0.09025(79) | 0.0087429 |
| P[1] | 0.07948(11) | 0.0013606 |
| P[2] | 0.06819(14) | 0.0020968 |
| P[3] | 0.056850(90) | 0.0015862 |
| P[4] | 0.04892(11) | 0.0021783 |
| P[5] | -5.7(2.2) | -0.3906901 |
| P[6] | -1592(781) | -0.4905972 |
<r >
6.11 only phys point + strange misstuning as error
X1 X2 X33 P[0] 7.942148e-02 1.612413e-04 4 P[1] 6.818519e-02 1.503716e-04 5 P[2] 5.684570e-02 7.811211e-05 6 P[3] 4.891968e-02 1.068468e-04 7 P[4] -1.997571e+00 3.772238e+00 8 P[5] -2.839772e+03 2.523184e+03
| par | value | percent |
|---|---|---|
| P[0] | 0.07942(16) | 0.0020302 |
| P[1] | 0.06819(15) | 0.0022053 |
| P[2] | 0.056846(78) | 0.0013741 |
| P[3] | 0.04892(11) | 0.0021841 |
| P[4] | -2.0(3.8) | -1.8884127 |
| P[5] | -2840(2523) | -0.8885165 |
<r >
6.12 only phys + max twist corrections + strange misstuning
\chi^2/dof= 0.343828
| P | value |
|---|---|
| P[0] | 0.07954(11) |
| P[1] | 0.068014(98) |
| P[2] | 0.056890(62) |
| P[3] | 0.048997(67) |
| P[4] | -1.4(2.7) |
| P[5] | -3265(1844) |
<r >