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Frontiers of Physics

ISSN 2095-0462

ISSN 2095-0470(Online)

CN 11-5994/O4

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2018 Impact Factor: 2.483

Front. Phys.    2024, Vol. 19 Issue (6) : 64203    https://doi.org/10.1007/s11467-024-1430-7
Bayesian method for fitting the low-energy constants in chiral perturbation theory
Hao-Xiang Pan1, De-Kai Kong1, Qiao-Yi Wen2, Shao-Zhou Jiang1()
1. Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, China
2. Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China
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Abstract

The values of the low-energy constants (LECs) are very important in the chiral perturbation theory. This paper adopts a Bayesian method with the truncation errors to globally fit eight next-to-leading order (NLO) LECs Li r and next-to-next-leading order (NNLO) LECs Cir. With the estimation of the truncation errors, the fitting results of Lir in the NLO and NNLO are very close. The posterior distributions of Cir indicate the boundary-dependent relations of these Cir. Ten Cir are weakly dependent on the boundaries and their values are reliable. The other Cir are required more experimental data to constrain their boundaries. Some linear combinations of Ci r are also fitted with more reliable posterior distributions. If one knows some more precise values of Cir, some other Cir can be obtained by these values. With these fitting LECs, most observables provide a good convergence, except for the π K scattering lengths a03 /2 and a0 1/2. An example is also introduced to test the improvement of the method. All the computations indicate that considering the truncation errors can improve the global fit greatly, and more prior information can obtain better fitting results. This fitting method can be extended to the other effective field theories and the perturbation theory.

Keywords chiral perturbation theory      low-energy constants      Bayesian statistics     
Corresponding Author(s): Shao-Zhou Jiang   
Issue Date: 26 August 2024
 Cite this article:   
Hao-Xiang Pan,De-Kai Kong,Qiao-Yi Wen, et al. Bayesian method for fitting the low-energy constants in chiral perturbation theory[J]. Front. Phys. , 2024, 19(6): 64203.
 URL:  
https://academic.hep.com.cn/fop/EN/10.1007/s11467-024-1430-7
https://academic.hep.com.cn/fop/EN/Y2024/V19/I6/64203
Fig.1  The NLO fitting posterior PDFs of aiNLO. The red lines and the light red areas are obtained by Model A. The blue lines and the light blues area are obtained by Model B2. The lines are the distribution curve of aiNLO. The light-colored areas depict the 68% HPDs. The green lines denote the true value.
i 1 2 3 4 5 6 7 8 WAIC LOO
ai,trNLO 0.53 0.80 −3.07 0.3 1.01 0.14 −0.34 0.47
NLO
ai,ANLO 0.541 (35) 0.914 (30) −3.671 (135) 0.366 (18) 1.021 (28) 0.118 (22) −0.413 (14) 0.545 (18) −49.130 −56.310
Pct A 2.1% 14.3% 19.6% 22.0% 1.1% −15.7% 21.5% 16.0%
PctσA 0.3 3.8 −4.5 3.7 0.4 −1.0 −5.2 4.2
ai,B 1 NL O 0.550 (121) 0.842 (79) −3.192 (335) 0.324 (46) 0.972 (68) 0.110 (53) −0.361 (34) 0.503 (43) 14.964 7.889
Pct B1 3.8% 5.2% 4.0% 8.0% −3.8% −21.4% 6.2% 7.0%
PctσB 1 0.2 0.5 −0.4 0.5 −0.6 −0.6 −0.6 0.8
ai,B 2 NL O 0.539 (41) 0.860 (43) −3.252 (175) 0.314 (28) 1.027 (38) 0.149 (28) −0.359 (18) 0.475 (23) 27.307 23.713
Pct B2 1.7% 7.5% 5.9% 4.7% 1.7% 6.4% 5.6% 1.1%
PctσB 2 0.2 1.4 −1.0 0.5 0.4 0.3 −1.1 0.2
NNLO
ai,ANLO 0.924 (574) 0.603 (146) −1.984 (457) 0.416 (383) 0.553 (357) 0.084 (383) −0.243 (53) 0.510 (316) 14.364 7.782
Pct A 74.34% −24.63% −35.37% 38.67% −45.25% −40.00% −28.53% 8.51%
PctσA 0.69 −1.35 2.38 0.30 −1.28 −0.15 1.83 0.13
ai,B 1 NL O 0.534 (116) 0.831 (74) −3.042 (26) 0.316 (43) 0.968 (62) 0.116 (40) −0.346 (27) 0.491 (42) 41.143 32.782
Pct B1 0.75% 3.87% −0.91% 5.33% −4.16% −17.14% 1.76% 4.47%
PctσB 1 0.03 0.42 0.11 0.37 −0.68 −0.60 −0.22 0.50
ai,B 2 NL O 0.525 (81) 0.808 (35) −3.195 (111) 0.319 (18) 0.995 (37) 0.138 (32) −0.354 (12) 0.474 (24) 56.730 53.277
Pct B2 −0.9% 1.0% 4.1% 6.3% −1.5% −1.4% 4.1% 0.9%
PctσB 2 −0.06 0.23 −1.13 1.06 −0.41 −0.06 −1.17 0.17
Tab.1  The NLO and the NNLO fitting results of ai NL O in the example. Row 2 is the true value of aiNLO. Rows 3, 6 and 9 are the NLO fitting results of Model A, B1 and B2, respectively. Rows 12, 15 and 18 are the NNLO fitting results of Model A, B1 and B2, respectively. The percentage PctA, B1,B2 is defined in Eq. (16), and the ratio Pctσ A ,B1,B2 is defined in Eq. (17).
Fig.2  The proportions of Oi at each order for the example. The red, green and blue strips in the figure represent the true values, the values obtained by Models B1 and B2, respectively. The lightest and the second lightest colors are the proportions [defined in Eq. (18)] of LO and NLO, respectively. (a) The NLO fit. The darkest color is the proportion of HO. (b) The NNLO fit. The darkest color and the dark gray are the proportions of NNLO and HO, respectively. To avoid layer masking, the colors of the NNLO and the HO true values of O7, O11 and O14 are interchanged. Similarly, the colors of O11, O12 and O13 of Model B2 are also interchanged.
i 102Oi,tr 102Oi,exp 102Oi,B1 102Oi,B2
1 −35.800 −34.637 ± 0.716 −33.778 ± 1.786 −35.023 ± 1.129
2 0.173 0.171 ± 0.003 0.168 ± 0.005 0.172 ± 0.004
3 −0.276 −0.279 ± 0.006 −0.279 ± 0.031 −0.279 ± 0.012
4 0.603 0.590 ± 0.012 0.584 ± 0.013 0.595 ± 0.008
5 27.281 27.753 ± 0.546 27.124 ± 0.756 27.600 ± 0.535
6 −0.524 −0.548 ± 0.010 −0.554 ± 0.020 −0.541 ± 0.014
7 −1.486 −1.434 ± 0.030 −1.403 ± 0.028 −1.459 ± 0.023
8 −0.955 −0.970 ± 0.019 −0.954 ± 0.414 −0.965 ± 0.219
9 −0.227 −0.226 ± 0.005 −0.231 ± 0.008 −0.227 ± 0.004
10 −52.511 −52.773 ± 1.050 −54.107 ± 3.062 −52.493 ± 2.069
11 44.936 46.250 ± 0.899 47.299 ± 2.319 45.728 ± 1.351
12 −4.223 −4.397 ± 0.084 −4.427 ± 0.673 −4.393 ± 0.374
13 −14.674 −14.769 ± 0.293 −14.574 ± 2.295 −14.782 ± 1.325
14 −24.577 −24.765 ± 0.492 −24.351 ± 0.492 −24.755 ± 0.419
15 −15.864 −15.505 ± 0.317 −15.974 ± 1.741 −15.435 ± 0.893
16 3.831 3.746 ± 0.077 3.847 ± 0.145 3.774 ± 0.069
17 −4.193 −4.208 ± 0.084 −4.110 ± 0.146 −4.203 ± 0.086
Tab.2  The comparison of the NLO fitting values for the example. The subscripts tr, exp, B 1 and B2 in the first row represent the true values, the experimental values, the theoretical values from Model B1 and Model B2, respectively. The experimental values in the third column are sampled from the true values. Oi is defined in Eq. (A1).
i ai,trNNLO ai,ANNLO Pct A PctσA ai,B 1 NN LO Pct B1 PctσB 1 ai,B 2 NN LO Pct B2 PctσB 2
1 0.02 0.176 (298) 780.0% 0.5 0.013 (29) −35.0% −0.2 0.017 (10) −15.0% −0.3
2 0.19 0.060 (293) −68.4% −0.4 0.102 (46) −46.3% −1.9 0.177 (31) −6.8% −0.4
3 −0.72 0.351 (692) −148.8% 1.5 −0.073 (264) −89.9% 2.5 −0.703 (209) −2.4% 0.1
4 0.22 −0.682 (917) −410.0% −1.0 0.917 (735) 316.8% 0.9 0.203 (96) −7.7% −0.2
5 −0.16 0.018 (465) −111.3% 0.4 −0.090 (60) −43.8% 1.2 −0.137 (43) −14.4% 0.5
6 0.26 0.035 (485) −86.5% −0.5 0.189 (71) −27.3% −1.0 0.192 (58) −26.2% −1.2
7 −0.42 0.088 (645) −121.0% 0.8 −0.209 (520) −50.2% 0.4 −0.413 (165) −1.7% 0.0
8 −0.45 0.016 (1005) −103.6% 0.5 −0.136 (188) −69.8% 1.7 −0.472 (118) 4.9% −0.2
9 −0.99 −0.822 (525) −17.0% 0.3 −0.261 (200) −73.6% 3.6 −0.966 (208) −2.4% 0.1
10 −0.06 −0.415 (670) 591.7% −0.5 −0.076 (59) 26.7% −0.3 −0.083 (24) 38.3% −1.0
11 0.24 0.005 (993) −97.9% −0.2 0.163 (646) −32.1% −0.1 0.254 (132) 5.8% 0.1
12 −0.18 −0.182 (605) 1.1% 0.0 −0.194 (85) 7.8% −0.2 −0.219 (51) 21.7% −0.8
13 1.02 0.342 (706) −66.5% −1.0 1.011 (71) −0.9% −0.1 0.997 (57) −2.3% −0.4
14 0.29 −0.226 (181) −177.9% −2.9 0.140 (132) −51.7% −1.1 0.265 (90) −8.6% −0.3
15 −0.11 −0.297 (427) 170.0% −0.4 −0.087 (62) −20.9% 0.4 −0.110 (35) 0.0% 0.0
16 −0.56 0.095 (707) −117.0% 0.9 −0.870 (394) 55.4% −0.8 −0.567 (218) 1.2% 0.0
17 0.19 0.247 (714) 30.0% 0.1 0.188 (112) −1.1% 0.0 0.187 (67) −1.6% 0.0
WAIC 14.364 41.143 56.730
LOO 7.782 32.782 53.277
PM 0.2650 0.0510 0.0177
Tab.3  The NNLO fitting results of the example. Column 2 is the true value of aiNNLO. Columns 3, 6 and 9 are the results of Models A, B1 and B2, respectively. The percentage PctA, B1,B2 is defined in Eq. (16), and the ratio Pctσ A ,B1,B2 is defined in Eq. (17). PM is defined in Eq. (19).
i 102Oi,tr 102Oi,exp 102Oi,B1 102Oi,B2
1 −35.800 −34.637 ± 0.716 −34.219 ± 2.034 −35.189 ± 0.910
2 0.173 0.171 ± 0.003 0.170 ± 0.007 0.172 ± 0.005
3 −0.276 −0.279 ± 0.006 −0.279 ± 0.049 −0.279 ± 0.036
4 0.603 0.590 ± 0.012 0.584 ± 0.032 0.590 ± 0.015
5 27.281 27.753 ± 0.546 27.091 ± 0.843 27.604 ± 0.675
6 −0.524 −0.548 ± 0.010 −0.550 ± 0.028 −0.547 ± 0.023
7 −1.486 −1.434 ± 0.030 −1.434 ± 0.031 −1.469 ± 0.020
8 −0.955 −0.970 ± 0.019 −0.968 ± 0.327 −0.971 ± 0.144
9 −0.227 −0.226 ± 0.005 −0.226 ± 0.008 −0.225 ± 0.010
10 −52.511 −52.773 ± 1.050 −53.136 ± 4.369 −52.212 ± 2.036
11 44.936 46.250 ± 0.899 46.466 ± 2.628 45.904 ± 1.176
12 −4.223 −4.397 ± 0.084 −4.406 ± 0.736 −4.390 ± 0.402
13 −14.674 −14.769 ± 0.293 −14.713 ± 2.670 −14.760 ± 1.834
14 −24.577 −24.765 ± 0.492 −24.304 ± 0.614 −24.692 ± 0.496
15 −15.864 −15.505 ± 0.317 −15.609 ± 1.776 −15.541 ± 1.006
16 3.831 3.746 ± 0.077 3.757 ± 0.154 3.761 ± 0.098
17 −4.193 −4.208 ± 0.084 −4.136 ± 0.134 −4.212 ± 0.093
Tab.4  The comparison of the NNLO fitting values for the example. The subscripts tr, exp, B 1 and B2 in the first row represent the true values, the experimental values, the theoretical values from Model B1 and Model B2, respectively. The experimental values in the third column are sampled from the true values. Oi is defined in Eq. (A1).
Δ a00 Δ a02 Δ l ¯1 Δ l ¯2 Δ l ¯4
Δ a00 2.0× 10 5 3.2× 10 6 1.9× 10 4 1.7 ×105 4.2× 10 4
Δ a02 9.7× 10 7 1.6× 10 4 1.2 ×105 4.2 ×106
Δ ¯1 3.5× 10 1 3.3 ×102 6.7× 10 2
Δ ¯2 1.2× 10 2 7.2 ×103
Δ ¯4 4.8× 10 2
Tab.5  The covariance matrix of a0 0, a02 and l ¯1, l ¯2, l ¯4. This is a symmetric matrix, only the values in the upper right corner of the matrix are given [25].
LECs Free fit Free fit [7] 103L 4r0 103L 4r0 [7] 103L 4r0.3 103L 4r0.3 [7] 103L 4r0.3 103L 4r0.3 [7]
103L1r 1.04(09 ) 1.11(10 ) 0.90(09 ) 0.98(09 ) 0.92(09 ) 1.00(09 ) 0.88(09 ) 0.95(09 )
103L2r 1.00(11 ) 1.05(17 ) 1.49(08 ) 1.56(09 ) 1.41(08 ) 1.48(09 ) 1.57(08 ) 1.64(09 )
103L3r 3.52 (28) 3.82 (30) 3.52 (28) 3.82 (30) 3.52 (28) 3.82 (30) 3.52 (28) 3.82 (30)
103L4r 1.82(25 ) 1.87(53 ) 0 0 0.3 0.3 0.3 0.3
103L5r 1.24(03 ) 1.22(06 ) 1.25(03 ) 1.23(06 ) 1.24(03 ) 1.23(06 ) 1.25(03 ) 1.23(06 )
103L6r 1.46(25 ) 1.46(46 ) 0.12 (05) 0.11 (05) 0.13(06 ) 0.14(06 ) 0.37 (04) 0.36 (05)
103L7r 0.40 (14) 0.39 (08) 0.19 (14) 0.24 (15) 0.23 (14) 0.27 (14) 0.16 (14) 0.21 (17)
103L8r 0.60(12 ) 0.65(07 ) 0.51(12 ) 0.53(13 ) 0.53(12 ) 0.55(12 ) 0.50(12 ) 0.50(14 )
Tab.6  The NLO fit by Model A, of which some different choices of L4r. Columns 2, 4, 6 and 8 are the results from free L4r, L4r0, L4r0.3 and L4r0.3, respectively. Columns 3, 5, 7 and 9 are the results in Ref. [7] for comparison.
Fig.3  The corner plot of the 17-input fitting Lir. The red and blue colors mean the NLO and NNLO fit, respectively. The small and large loops mean the 68% HPD and the 95% HPD, respectively. The light-colored areas are the 68% HPD.
LECs NLO B12 NLO B17 NNLO B17 NLO fit [7] NNLO fit [7] NLO fit 2 [8] NNLO fit 2 [8]
103L1r 0.51(15 ) 0.46(14 ) 0.43(12 ) 1.00(09 ) 0.53(06 ) 0.44(05 ) 0.43(05 )
103L2r 1.08(22 ) 0.88(18 ) 0.83(15 ) 1.48(09 ) 0.81(04 ) 0.84(10 ) 0.74(04 )
103L3r 3.36 (61) 2.94 (49) 2.64 (44) 3.82 (30) 3.07 (20) 2.84 (16) 2.74 (17)
103L4r 0.19(18 ) 0.22(16 ) 0.26(11 ) 0.3 0.3 0.30(33 ) 0.33(08 )
103L5r 1.10(37 ) 1.10(34 ) 1.21(27 ) 1.23(06 ) 1.01(06 ) 0.92(02 ) 0.95(04 )
103L6r 0.05(22 ) 0.08(13 ) 0.12(11 ) 0.14(06 ) 0.14(05 ) 0.22(08 ) 0.20(03 )
103L7r 0.26 (17) 0.34 (18) 0.33 (13) 0.27 (14) 0.34 (09) 0.23 (12) 0.23 (08)
103L8r 0.51(22 ) 0.59(21 ) 0.60(15 ) 0.55(12 ) 0.47(10 ) 0.44(10 ) 0.42(09 )
χ2 (d.o.f.) 1.0(9 ) 4.2(4 ) 4.3(9 )
Tab.7  The fitting results of Lir. The superscripts indicate the input number in the fit. Columns 5 to 8 are the NLO and the NNLO fitting results in Refs. [7, 8], respectively.
Observables LO |PctLO NLO |PctNLO HO |PctHO Theory Experiment
ms/m^ |1 25.84 (96.2%) 0.84 (3.1%) 0.19 (0.7%) 26.9± 3.1 27.31.3 +0.7
ms/m^ |2 24.21 (88.4%) 3.23 (11.8%) 0.06 (−0.2%) 27.4± 6.3 27.31.3 +0.7
FK/ Fπ 1.000 (84.1%) 0.184 (15.5%) 0.004 (0.3%) 1.188± 0.036 1.199 ± 0.003
fs 3.782 (66.2%) 1.267 (22.2%) 0.660 (11.6%) 5.709± 0.347 5.712± 0.032
gp 3.782 (77.9%) 0.915 (18.8%) 0.159 (3.3%) 4.856± 0.191 4.958± 0.085
a00 0.159 (72.5%) 0.044 (20.2%) 0.016 (7.4%) 0.2197± 0.005 0.2196± 0.0034
10a02 0.455 (104.0%) 0.019 (−4.4%) 0.002 (0.4%) 0.437 ±0.015 0.444 ±0.012
a01/2mπ 0.142 (63.3%) 0.033 (14.6%) 0.049 (22.1%) 0.224± 0.014 0.224± 0.022
10a03 /2mπ 0.709 (158.3%) 0.084 (−18.8%) 0.177 (−39.5%) 0.448 ±0.090 0.448 ±0.077
103l1r 0 (0.0%) 4.07(98.6%) 0.06 (1.4%) 4.1 ±1.1 4.0 ±0.6
103l2r 0 (0.0%) 3.50 (160.2%) 1.32 (−60.2%) 2.2± 0.9 1.9± 0.2
103l3r 0 (0.0%) 0.18 (104.0%) 0.01 (−4.0%) 0.2 ±3.1 0.3± 1.1
103l4r 0 (0.0%) 6.07 (99.9%) 0.01 (0.1%) 6.1± 2.0 6.2± 1.3
fs 0.531± 0.322 0.868± 0.049
g 0.368± 0.036 0.508± 0.122
?r2?Sπ 0.60± 0.13 0.61± 0.04
cSπ 10± 2 11± 1
Tab.8  The convergences of 17 inputs. The LECs are adopted from the 17-input NLO fitting results obtained by Model B in Tab.7. The second to the fourth columns are the contributions at the LO, NLO and HO, respectively. The percentage Pct LO ,NLO,HO is defined in Eq. (18). The last two columns are the theoretical estimation and the experimental inputs, respectively.
C~i Results Ref. [8] C~i Results Ref. [8]
C~1 0.11(22 ) 0.02(12 ) 10C~10 0.22(32 ) 0.06 (13)
C~2 0.09(58 ) 0.19(34 ) C~11 0.22(07 ) 0.24(02 )
102C ~3 1.29 (77) 0.72 (42) 103C ~12 0.02(05 ) 0.18 (01)
102C ~4 0.05(08 ) 0.22(03 ) 103C ~13 0.13 (35) 1.02(44 )
10C~5 0.08 (04) 0.16 (02) 104C ~14 0.38(27 ) 0.29(06 )
103C ~6 0.76 (151) 0.26(13 ) 103C ~15 0.12 (03) 0.11 (01)
102C ~7 0.70 (63) 0.42 (12) 104C ~16 0.46 (35) 0.56 (06)
10C~8 0.08 (22) 0.45 (09) 104C ~17 0.26 (22) 0.19(16 )
102C ~9 0.46 (43) 0.99 (11)
Tab.9  The values and the errors of C~i, comparing with the results in Ref. [8].
Observables LO |PctLO NLO |PctNLO NNLO |PctNNLO HO |PctHO Theory Experiment
ms/m^ |1 25.84 (95.1%) 1.45 (5.3%) 0.12 (−0.5%) 0.003 (0.01%) 27.2± 4.1 27.31.3 +0.7
ms/m^ |2 24.21 (88.4%) 3.60 (13.1%) 0.38 (−1.4%) 0.020 (−0.07%) 27.4± 10.7 27.31.3 +0.7
FK/ Fπ 1.000 (83.2%) 0.197 (16.4%) 0.007 (0.6%) 0.002 (−0.12%) 1.202± 0.050 1.199± 0.003
fs 3.782 (66.7%) 1.342 (23.7%) 0.494 (8.7%) 0.050 (0.88%) 5.668± 0.351 5.712± 0.032
gp 3.782 (76.9%) 0.834 (17.0%) 0.284 (5.8%) 0.018 (0.37%) 4.918± 0.080 4.958± 0.085
a00 0.159 (72.2%) 0.045 (20.6%) 0.015 (6.8%) 0.001 (0.34%) 0.2204± 0.004 0.2196± 0.0034
10a02 0.455 (104.1%) 0.020 (4.6%) 0.003 (−0.7%) 0.005 (1.20%) 0.437 ±0.017 0.444 ±0.012
a01/2mπ 0.142 (63.2%) 0.034 (15.2%) 0.049 (21.6%) 0.000 (0.00%) 0.225± 0.013 0.224± 0.022
10a03 /2mπ 0.709 (161.9%) 0.093 (−21.2%) 0.179 (−40.8%) 0.000 (0.04%) 0.438 ±0.061 0.448 ±0.077
103l1r 0 (0.0%) 3.57 (90.4%) 0.38 (9.7%) 0.004 (−0.11%) 4.0 ±0.94 4.0 ±0.6
103l2r 0 (0.0%) 3.32 (174.8%) 1.36 (−71.8%) 0.057 (−3.01%) 1.9± 0.82 1.9± 0.2
103l3r 0 (0.0%) 0.25 (−115.1%) 0.46 (214.6%) 0.001 (0.48%) 0.2± 3.01 0.3± 1.1
103l4r 0 (0.0%) 6.85 (104.8%) 0.27 (−4.1%) 0.044 (−0.67%) 6.5± 1.91 6.2± 1.3
fs 0.472± 0.461 0.868± 0.049
g 0.508± 0.029 0.508± 0.122
?r2?Sπ 0.59± 0.07 0.61± 0.04
cSπ 11± 1 11± 1
Tab.10  Same as Tab.8, except for the NNLO fit.
LECs Results Ref. [8] Ref. [7] Ref. [71] LECs Results Ref. [8] Ref. [7] Ref. [71]
C1r 14.82 (41.49) 14[37 ] 12 25.33 1.11+0.60 C21r −0.41 (0.82) 0.28 (0.56) −0.48 0.51 0.09+ 0.09
C2r 3.48 (8.98] 16(1 ] 3.0 0 C22r 5.88 [15.71] 14(13 ] 9.0 2.98 2.21+ 1.70
C3r 1.70 [6.05] 2.9[6.0 ] 4.0 0.43 0.09+ 0.09 C23r 0.92 [3.52] 5.6(0.9 ] −1.0 0
C4r 18.54 (29.94) 26 [16) 15 18.11 0.85+0.51 C25r −21.17 (58.67) 34(33 ) −11 25.76+5.02 3.49
C5r −3.62 (19.23) 31 [7) −4.0 10.88 1.11+ 0.85 C26r −4.30 [42.04) 31(36 ] 10 23.04 4.59+2.98
C6r −3.43 [4.22) 7.9 [1.8) −4.0 0 C28r 0.45 [3.95] 4.9 [0.9) −2.0 1.530.09 +0.00
C7r 1.00 [6.26] 2.4[6.1 ] 5.0 0 C29r −23.20 [24.09) 49 [11) −20 8.42+2.04 1.79
C8r 10.52 [16.84] 15[16 ] 19 17.85 +1.36 1.28 C30r 3.44 [4.61] 9.0(1.9 ] 3.0 3.150.17 +0.09
C10r 3.51 [13.75] 13(6 ] −0.25 5.53 0.51+ 0.43 C31r 2.15 (9.56) 0.71 (6.70) 2.0 3.91 1.11+ 0.60
C11r −2.90 (4.17) 2.6 (1.8) −4.0 0 C32r 1.79 [3.63] 5.6(1.9 ] 1.7 1.45+0.260.17
C12r −6.02 (5.57) 18(2 ) −2.8 2.89 0.09+ 0.09 C33r −0.01 [3.58] 0.69 [3.12) 0.82 0.43+0.43 0.17
C13r 1.74 (2.06) 2.2(0.9 ) 1.5 0 C34r 9.27 (10.48] 0.68(4.67 ) 7.0 5.61+2.471.53
C14r −3.32 (3.15) 4.2 (1.2) −1.0 7.40 1.79+ 1.19 C36r 1.27 [5.12] 4.1(4.3 ] 2.0 0
C15r 1.30 (1.13) 1.2(1.0 ) −3.0 0 C63r 11.09 [22.82] 6.6 [16.8) 21.08 +2.13 1.79
C16r 1.10 [4.20] 0.81 (1.34) 3.2 0 C66r 3.90 [26.83] 6.5 [25.4] 6.800.60 +0.34
C17r 0.27 [2.62] 3.6(1.6 ] −1.0 1.450.34 +0.09 C69r −1.16 [19.86] 4.6[19.0 ] 4.420.09 +0.00
C18r −1.46 [5.45] 1.1 [5.4] 0.63 5.10 0.77+ 0.60 C83r −1.11 [20.31] 14(16 ] 14.79 1.87+ 1.45
C19r −5.42 (5.22) 5.3(2.8 ) −4.0 2.30 1.11+ 0.77 C88r −24.09 [63.17] 38 [59] 14.37+7.91 5.78
C20r 0.53 [3.45] 2.9 [2.3) 1.0 1.45+0.260.17 C90r 20.83 [62.90] 35 [44) 19.72 +4.68 3.74
Tab.11  The values of Cir are in units of 106. The brackets “[” and “]” represent strong dependence on the lower and the upper boundaries, respectively. “(” and “)” represent weak dependence on the lower and the upper boundaries, respectively. The results with an asterisk mean the input boundaries on the website [68] are very close to those in Ref. [7] (less than 10 10). The symbol “ 0” for the results in Ref. [71] means these values are zeros in the large- NC limits.
102b1 102b2 102b3 102b4 102b5 102b6 102b7 102b8 102ai L O
O1 −50.00000 −50.00000 −50.00000 −50.00000
O2 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000
O3 −0.27574 −81.82470 −20.69689 −95.19898 −130.30649
O4 −0.15389 55.22236 −22.53571 10.47685 14.87959 10.84243 85.28317 164.01310
O5 32.80394 52.16045 52.16045 42.66749 12.43968 77.33250 52.35691
O6 −10.25934 −10.95043 −11.71888 −28.67356 −26.84030 −32.24206
O7 −2.39804 2.37745 −9.62388 9.39379 9.39265 0.57071 42.90526
O8 −24.83947 69.61634 −2.52600 −10.10231 7.19485 99.63693
O9 0.51431 99.49478 30.47334 32.08646 32.08646 113.26784 96.25052 32.08646 77.78096
O10 −69.04271 −69.23904 38.65428 20.36290 10.78947 10.06250 82.44726
O11 −62.98961 −66.28358 42.37512 −8.89931 13.20056 6.67356 91.91503
O12 50.00000 10.00000 −135.00000 10.00000 10.00000 −100.00000
O13 74.39589 103.26600 103.32687 156.81033 95.54550 83.85802 100.61745 105.09613 107.97725
O14 −33.59097 −54.18416 −54.20951 −41.78655 −28.19762 −28.53647 −48.27921
O15 28.77264 26.35989 40.37381 49.81383 40.01344 63.80493 40.01344 45.46315
O16 −9.19703 3.17944 1.51912 5.66408 −5.66251 −8.10996 10.10760 58.11734
O17 −29.07789 −62.10099 −44.02866 −48.43771 −31.06314 −34.52566 −40.99873 −12.31774
  Table A1 The values of parameters bi and aiLO in Eq. (A1). Because the values are exact, more significant digits are given.
i μe,NLO σe,NLO pNLO μaiNLO σaiNLO μe,NNLO σe,NNLO pNNLO μaiNNLO σaiNNLO
1 0.140 0.050 1 0.616 0.308 0.040 0.020 1 0.023 0.012
2 0.100 0.050 1 0.751 0.376 0.040 0.020 0 0.178 0.089
3 0.020 0.050 0 3.232 1.616 0.100 0.030 1 0.758 0.379
4 0.040 0.050 1 0.268 0.134 0.120 0.036 1 0.196 0.098
5 0.140 0.050 1 1.097 0.549 0.060 0.020 1 0.146 0.073
6 0.110 0.050 0 0.108 0.054 0.020 0.020 0 0.200 0.100
7 0.110 0.050 1 0.281 0.140 0.060 0.020 1 0.347 0.173
8 0.140 0.050 1 0.434 0.217 0.010 0.020 0 0.484 0.242
9 0.070 0.050 0 0.130 0.039 0 0.958 0.479
10 0.120 0.050 0 0.050 0.020 0 0.061 0.031
11 0.110 0.050 0 0.060 0.020 0 0.275 0.138
12 0.010 0.050 1 0.040 0.020 1 0.217 0.109
13 0.140 0.050 1 0.060 0.020 1 0.987 0.494
14 0.140 0.050 1 0.070 0.021 1 0.279 0.139
15 0.020 0.050 0 0.030 0.020 1 0.098 0.049
16 0.060 0.050 0 0.050 0.020 1 0.622 0.311
17 0.140 0.050 1 0.050 0.020 1 0.187 0.093
  Table A2 The fitting parameters and the priors in Model B2. The subscripts NLO and NNLO represent the NLO and NNLO fit, respectively. The definitions of these parameters are in Eqs. (10)−(12) and the text below them.
i μaiNLO σaiNLO μaiNNLO σaiNNLO
1 0.616 0.308 0.023 0.012
2 0.751 0.376 0.178 0.089
3 3.232 1.616 0.758 0.379
4 0.268 0.134 0.196 0.098
5 1.097 0.549 0.146 0.073
6 0.108 0.054 0.200 0.100
7 0.281 0.140 0.347 0.173
8 0.434 0.217 0.484 0.242
9 0.958 0.479
10 0.061 0.031
11 0.275 0.138
12 0.217 0.109
13 0.987 0.494
14 0.279 0.139
15 0.098 0.049
16 0.622 0.311
17 0.187 0.093
  Table A3 The prior of the LECs in Model B2. The subscripts NLO and NNLO represent the NLO and NNLO fit, respectively.
Quantity μe,NLO12 σ e,NLO12 pNLO12 μe,NLO17 σ e,NLO17 pNLO17 μe,NNLO17 σ e,NNLO17 pNNLO 17
ms/m^ |1 0.050 0.050 0.5 0.050 0.050 0.5 0.020 0.020 0.5
ms/m^ |2 0.050 0.050 0.5 0.050 0.050 0.5 0.020 0.020 0.5
FK/ Fπ 0.050 0.050 0.5 0.050 0.050 0.5 0.020 0.020 0.5
fs 0.150 0.050 1 0.150 0.050 1 0.020 0.020 0.5
gp 0.050 0.050 0.5 0.050 0.050 0.5 0.020 0.020 0.5
fs 0.050 0.050 0.5 0.050 0.050 0.5 0.020 0.020 0.5
g 0.050 0.050 0.5 0.050 0.050 0.5 0.020 0.020 0.5
a00 0.100 0.050 1 0.100 0.050 1 0.020 0.020 0.5
10a02 0.050 0.050 0.5 0.050 0.050 0.5 0.020 0.020 0.5
a01/2mπ 0.350 0.105 1 0.350 0.105 1 0.020 0.020 0.5
10a03 /2mπ 0.250 0.075 0 0.250 0.075 0 0.020 0.020 0.5
?r2?Sπ 0.200 0.060 0.5 0.200 0.060 0.5 0.020 0.020 0.5
cSπ 0.200 0.060 0.5 0.020 0.020 0.5
l ¯1 0.200 0.060 0.5 0.050 0.050 0.5
l ¯2 0.200 0.060 0.5 0.050 0.050 0.5
l ¯3 0.200 0.060 0.5 0.050 0.050 0.5
l ¯4 0.200 0.060 0.5 0.050 0.050 0.5
  Table A4 The parameters for fitting Lir and C~i. The superscripts 12 and 17 represent the fit with 12 and 17 inputs, respectively. The subscripts NLO and NNLO represent the NLO and NNLO fitting, respectively. The definitions of these parameters are in Eqs. (10)−(12) and the text below them.
i μLir, NL O12 σ Li r, NL O12 μLir, NL O17 σ Li r, NL O17 μ C~i,NNLO17 σ C~i,NNLO17
1 0.500 0.300 0.500 0.300 0.077 0.989
2 1.000 0.500 1.000 0.500 0.190 3.102
3 3.000 1.000 3.000 1.000 1.073 0.007
4 0.200 0.200 0.200 0.200 0.068 0.095
5 1.000 0.500 1.000 0.500 0.040 0.071
6 0.000 0.300 0.000 0.300 0.806 1.772
7 0.300 0.300 0.300 0.300 0.561 1.581
8 0.500 0.500 0.500 0.500 0.043 0.250
9 0.173 0.497
10 0.405 1.819
11 0.066 0.227
12 0.013 0.053
13 0.244 0.388
14 0.007 0.693
15 0.022 0.072
16 0.214 1.122
17 0.207 0.222
  Table A5 The parameters of the priors of Lir and C~i. Their definition is above Eq. (20). The superscripts 12 and 17 represent the fit with 12 and 17 inputs, respectively. The subscripts NLO and NNLO represent the NLO and the NNLO fits, respectively.
  Fig.A1 The posterior distributions of the NNLO fitting C~i. The vertical coordinate is the posterior PDF and the horizontal coordinate is the value of C~i. The pink shaded area depicts the 68% HPD. The blue line is the distribution curve of Lir.
  Fig. A2 The posterior distributions of Cir. The horizontal axis represents the value of Cir, and the upper and the lower boundaries are given in Eq. (38) in Ref. [8]. The vertical coordinate is the posterior PDF. The pink shaded area depicts the 68% HPD. The blue line is the distribution curve of Cir.
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[1] Jin-Man Chen, Ze-Rui Liang, De-Liang Yao. Low-energy elastic (anti)neutrino−nucleon scattering in covariant baryon chiral perturbation theory[J]. Front. Phys. , 2024, 19(6): 64202-.
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