Processing math: 100%

6  scale setting

no bounding has been applied

fjπ(ξ,∞)=fjπ(ξ,L)1−ΔFVE ΔFVE=−2ξℓ ˜g1(λ)˜g1(λ)≃4√π2∞∑n=1m(n)(√nλ)3/2e−√nλξπ≡M2π(4πfπ)2 with m(n) the multiplicities of a three-dimensional vector →n having integer norm n (i.e.~m(n)={6,12,8,6,...}) and λ=MπL. To obtain the above formula we expand K1, the Bessel function of the second kind, by its asymptotic expansion.
Different choices of the expansion variable are possible: one can replace fπ with the LO LEC f and/or replace M2π with 2Bmℓ (and correspondingly MπL with √2BmℓL in the arguments of the functions ˜g1 and ˜g2). At NLO (i.e., for the GL formula) the above changes are equivalent, since any difference represents a NNLO effect.

afjπ(ξ)=afjπ(ξphys){1−2ξlog(ξ/ξphys)+[P+Pdisc(afjπ(ξphys))2](ξ−ξphys)}

P[0]=a(A)fmP[1]=a(B)fmP[2]=a(C)fmP[3]=a(D)fmP[4]=a(E)fm

χ2/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)
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
A-no-max-twB-no-max-twC-no-max-twD-no-max-twE-no-max-twABCDE$\xi$$af_\pi$

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6.1 scaling with tau paper

00.0020.0040.0060.008−0.012−0.01−0.008−0.006−0.004−0.00200.002
tau-MG$a^2$$(a(\tau)-a(MG))/a(\tau)$

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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)
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
ABCDE$\xi$$af_\pi$

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6.3 scaling with tau paper

00.0020.0040.0060.008−0.01−0.008−0.006−0.004−0.00200.002
tau-MG$a^2$$(a(\tau)-a(MG))/a(\tau)$

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6.4 Only phys point ensembles

χ2/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)
0.0050.0060.0070.0080.0090.010.030.0350.040.0450.050.055
BCDE$\xi$$af_\pi$

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00.0020.0040.0060.008−0.000500.00050.0010.00150.0020.0025
tau-MG$a^2$$(a(\tau)-a(MG))/a(\tau)$

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6.5 No C20

χ2/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)
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
ABCDE$\xi$$af_\pi$

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6.6 scaling with tau paper

00.0020.0040.0060.00800.00020.00040.00060.00080.0010.0012
tau-MGno C48$a^2$$a(\tau)-a(MG)$

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6.7 max twist corrections with A12

χ2/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) |
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
ABCDE$\xi$$af_\pi$
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00.0020.0040.0060.008−0.001−0.0008−0.0006−0.0004−0.00020
tau-MGno C48$a^2$$a(\tau)-a(MG)$

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6.8 max twist corrections with A12 no C20

χ2/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)
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
ABCDE$\xi$$af_\pi$
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00.0020.0040.0060.00800.00020.00040.00060.00080.001
tau-MGno C48$a^2$$a(\tau)-a(MG)$

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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
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
ABCDE$\xi$$af_\pi$
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00.0020.0040.0060.008−0.0008−0.0006−0.0004−0.000200.0002
tau-MGno C48$a^2$$a(\tau)-a(MG)$

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6.10 max twist corrections with A12 no C20 + strange misstuning as error, unitary (main)

χ2/dof= 0.44991

X1            X2           X3

3 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
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
ABCDE$\xi$$af_\pi$
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00.0020.0040.0060.008−0.000200.00020.00040.00060.00080.0010.00120.0014
tau-MGno C48$a^2$$a(\tau)-a(MG)$

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6.11 only phys point + strange misstuning as error

X1            X2           X3

3 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
0.0050.010.0150.020.0250.030.0350.030.0350.040.0450.050.0550.060.0650.070.075
BCDE$\xi$$af_\pi$
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00.0010.0020.0030.0040.0050.0060.0070.008−2−10123456
tau-MGno C48$a^2$$a(\tau)-a(MG)$

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6.12 only phys + max twist corrections + strange misstuning

χ2/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)
0.0050.0060.0070.0080.0090.010.030.0350.040.0450.050.055
BCDE$\xi$$af_\pi$
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00.0020.0040.0060.008−1.5e−4−1e−4−0.5e−400.5e−41e−41.5e−42e−4
tau-MGno C48$a^2$$a(\tau)-a(MG)$

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