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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 and next-to-next-leading order (NNLO) LECs . With the estimation of the truncation errors, the fitting results of in the NLO and NNLO are very close. The posterior distributions of indicate the boundary-dependent relations of these . Ten are weakly dependent on the boundaries and their values are reliable. The other are required more experimental data to constrain their boundaries. Some linear combinations of are also fitted with more reliable posterior distributions. If one knows some more precise values of , some other can be obtained by these values. With these fitting LECs, most observables provide a good convergence, except for the scattering lengths and . 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.
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Keywords
chiral perturbation theory
low-energy constants
Bayesian statistics
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Corresponding Author(s):
Shao-Zhou Jiang
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Issue Date: 26 August 2024
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1 |
Weinberg S., Phenomenological Lagrangians, Physica A 96(1−2), 327 (1979)
https://doi.org/10.1016/0378-4371(79)90223-1
|
2 |
Gasser J. and Leutwyler H., Chiral perturbation theory to one loop, Ann. Phys. 158(1), 142 (1984)
https://doi.org/10.1016/0003-4916(84)90242-2
|
3 |
Gasser J. and Leutwyler H., Chiral perturbation theory: Expansions in the mass of the strange quark, Nucl. Phys. B 250(1−4), 465 (1985)
https://doi.org/10.1016/0550-3213(85)90492-4
|
4 |
Bijnens J., Colangelo G., and Ecker G., The mesonic chiral Lagrangian of order p6, J. High Energy Phys. 02, 020 (1999)
https://doi.org/10.1088/1126-6708/1999/02/020
|
5 |
Bijnens J., Hermansson-Truedsson N., and Wang S., The order p8 mesonic chiral Lagrangian, J. High Energy Phys. 01(1), 102 (2019)
https://doi.org/10.1007/JHEP01(2019)102
|
6 |
Bijnens J.Jemos I., A new global fit of the Lr at next-to-next-to-leading order in chiral perturbation theory, Nucl. Phys. B 854(3), 631 (2012)
|
7 |
Bijnens J. and Ecker G., Mesonic low-energy constants, Annu. Rev. Nucl. Part. Sci. 64(1), 149 (2014)
https://doi.org/10.1146/annurev-nucl-102313-025528
|
8 |
H. Yang Q., Guo W., J. Ge F., Huang B., Liu H., and Z. Jiang S., New method for fitting the low-energy constants in chiral perturbation theory, Phys. Rev. D 102(9), 094009 (2020)
https://doi.org/10.1103/PhysRevD.102.094009
|
9 |
U. Can K., Erkol G., Oka M., and T. Takahashi T., Look inside charmed-strange baryons from lattice QCD, Phys. Rev. D 92(11), 114515 (2015)
https://doi.org/10.1103/PhysRevD.92.114515
|
10 |
U. Can K., Erkol G., Isildak B., Oka M., and T. Takahashi T., Electromagnetic structure of charmed baryons in lattice QCD, J. High Energy Phys. 05(5), 125 (2014)
https://doi.org/10.1007/JHEP05(2014)125
|
11 |
Bahtiyar H.U. Can K.Erkol G.Oka M.T. Takahashi T., Ξcγ → Ξc′ transition in lattice QCD, Phys. Lett. B 772, 121 (2017)
|
12 |
M. Yan T.Y. Cheng H.Y. Cheung C.L. Lin G.C. Lin Y. L. Yu H., Heavy quark symmetry and chiral dynamics, Phys. Rev. D 46(3), 1148 (1992) [Erratum: Phys. Rev. D 55, 5851 (1997)]
|
13 |
J. Dowdall R.T. H. Davies C.P. Lepage G.McNeile C., Vus from π and K decay constants in full lattice QCD with physical u, d, s and c quarks, Phys. Rev. D 88, 074504 (2013), arXiv:
|
14 |
Bazavov A., (MILC) ., et al.. Results for light pseudoscalar mesons, PoS LATTICE 2010, 074 (2010)
|
15 |
Bernard V. and Passemar E., Chiral extrapolation of the strangeness changing Kπ form factor, J. High Energy Phys. 04, 001 (2010)
https://doi.org/10.1007/JHEP04%282010%29001
|
16 |
Bazavov A.(MILC) ., et al.., MILC results for light pseudoscalars, in: Proceedings of 6th International Workshop on Chiral dynamics: Bern, Switzerland, July 6–10, 2009, PoS CD09, 007 (2009), arXiv:
|
17 |
Bazavov A., Toussaint D., Bernard C., Laiho J., DeTar C., Levkova L., B. Oktay M., Gottlieb S., M. Heller U., E. Hetrick J., B. Mackenzie P., Sugar R., and S. Van de Water R., Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks, Rev. Mod. Phys. 82(2), 1349 (2010)
https://doi.org/10.1103/RevModPhys.82.1349
|
18 |
Golterman M.Maltman K.Peris S., NNLO low-energy constants from flavor-breaking chiral sum rules based on hadronic τ-decay data, Phys. Rev. D 89(5), 054036 (2014)
|
19 |
Colangelo P., J. Sanz-Cillero J., and Zuo F., Holography, chiral Lagrangian and form factor relations, J. High Energy Phys. 11, 012 (2012)
|
20 |
H. Guo Z., J. Sanz Cillero J., and Q. Zheng H., Partial waves and large NC resonance sum rules, J. High Energy Phys. 06, 030 (2007)
|
21 |
H. Guo Z.J. Sanz-Cillero J.Q. Zheng H., O(p6) extension of the large-NC partial wave dispersion relations, Phys. Lett. B 661, 342 (2008), arXiv:
|
22 |
H. Guo Z.J. Sanz-Cillero J., ππ-scattering lengths at O(p6) revisited, Phys. Rev. D 79, 096006 (2009)
|
23 |
Bijnens J.Colangelo G.Gasser J., Kl4 decays beyond one loop, Nucl. Phys. B 427(3), 427 (1994)
|
24 |
Amorós G.Bijnens J.Talavera P., Kℓ4 form-factors and π‒π scattering, Nucl. Phys. B 585, 293 (2000) [Erratum: Nucl. Phys. B 598, 665(2001)], arXiv:
|
25 |
Colangelo G.Gasser J.Leutwyler H., ππ scattering, Nucl. Phys. B 603(1–2), 125 (2001)
|
26 |
R. Schindler M.R. Phillips D., Bayesian methods for parameter estimation in effective field theories, Ann. Phys. 324, 682 (2009) [Erratum: Ann. Phys. 324, 2051 (2009)], arXiv:
|
27 |
J. Furnstahl R.R. Phillips D.Wesolowski S., A recipe for EFT uncertainty quantification in nuclear physics, J. Phys. G 42(3), 034028 (2015)
|
28 |
Wesolowski S., Klco N., J. Furnstahl R., R. Phillips D., and Thapaliya A., Bayesian parameter estimation for effective field theories, J. Phys. G 43(7), 074001 (2016)
https://doi.org/10.1088/0954-3899/43/7/074001
|
29 |
A. Melendez J., Wesolowski S., and J. Furnstahl R., Bayesian truncation errors in chiral effective field theory: Nucleon‒nucleon observables, Phys. Rev. C 96(2), 024003 (2017)
https://doi.org/10.1103/PhysRevC.96.024003
|
30 |
Svensson I., Ekström A., and Forssén C., Bayesian parameter estimation in chiral effective field theory using the Hamiltonian Monte Carlo method, Phys. Rev. C 105(1), 014004 (2022)
https://doi.org/10.1103/PhysRevC.105.014004
|
31 |
Ekström A., Forssén C., Dimitrakakis C., Dubhashi D., T. Johansson H., S. Muhammad A., Salomonsson H., and Schliep A., Bayesian optimization in ab initio nuclear physics, J. Phys. G 46(9), 095101 (2019)
https://doi.org/10.1088/1361-6471/ab2b14
|
32 |
Wesolowski S., J. Furnstahl R., A. Melendez J., and R. Phillips D., Exploring Bayesian parameter estimation for chiral effective field theory using nucleon–nucleon phase shifts, J. Phys. G 46(4), 045102 (2019)
https://doi.org/10.1088/1361-6471/aaf5fc
|
33 |
K. Alnamlah I., A. C. Pérez E., and R. Phillips D., Effective field theory approach to rotational bands in odd-mass nuclei, Phys. Rev. C 104(6), 064311 (2021)
https://doi.org/10.1103/PhysRevC.104.064311
|
34 |
J. Yang C., Ekström A., Forssén C., and Hagen G., Power counting in chiral effective field theory and nuclear binding, Phys. Rev. C 103(5), 054304 (2021)
https://doi.org/10.1103/PhysRevC.103.054304
|
35 |
E. Lovell A., M. Nunes F., Catacora-Rios M., and B. King G., Recent advances in the quantification of uncertainties in reaction theory, J. Phys. G 48(1), 014001 (2020)
https://doi.org/10.1088/1361-6471/abba72
|
36 |
R. Phillips D., J. Furnstahl R., Heinz U., Maiti T., Nazarewicz W., M. Nunes F., Plumlee M., T. Pratola M., Pratt S., G. Viens F., and M. Wild S., Get on the BAND Wagon: A Bayesian framework for quantifying model uncertainties in nuclear dynamics, J. Phys. G 48(7), 072001 (2021)
https://doi.org/10.1088/1361-6471/abf1df
|
37 |
Bedaque P.Boehnlein A.Cromaz M.Diefenthaler M.Elouadrhiri L.Horn T.Kuchera M.Lawrence D. Lee D.Lidia S.McKeown R.Melnitchouk W.Nazarewicz W.Orginos K.Roblin Y.Scott Smith M.Schram M.N. Wang X., A. I. for nuclear physics, Eur. Phys. J. A 57(3), 100 (2021)
|
38 |
Wesolowski S., Svensson I., Ekström A., Forssén C., J. Furnstahl R., A. Melendez J., and R. Phillips D., Rigorous constraints on three-nucleon forces in chiral effective field theory from fast and accurate calculations of few-body observables, Phys. Rev. C 104(6), 064001 (2021)
https://doi.org/10.1103/PhysRevC.104.064001
|
39 |
A. Connell M., Billig I., and R. Phillips D., Does Bayesian model averaging improve polynomial extrapolations? Two toy problems as tests, J. Phys. G 48(10), 104001 (2021)
https://doi.org/10.1088/1361-6471/ac215a
|
40 |
H. Lin Y., W. Hammer H., and G. Meißner U., Dispersion-theoretical analysis of the electromagnetic form factors of the nucleon: Past, present and future, Eur. Phys. J. A 57(8), 255 (2021)
https://doi.org/10.1140/epja/s10050-021-00562-0
|
41 |
Djärv T., Ekström A., Forssén C., and T. Johansson H., Bayesian predictions for A = 6 nuclei using eigenvector continuation emulators, Phys. Rev. C 105(1), 014005 (2022)
https://doi.org/10.1103/PhysRevC.105.014005
|
42 |
Acharya B. and Bacca S., Gaussian process error modeling for chiral effective-field-theory calculations of np↔dγ at low energies, Phys. Lett. B 827, 137011 (2022)
https://doi.org/10.1016/j.physletb.2022.137011
|
43 |
Odell D., R. Brune C., R. Phillips D., J. deBoer R., and N. Paneru S., Performing Bayesian analyses with AZURE2 using BRICK: An application to the 7Be system, Front. Phys. (Lausanne) 10, 888476 (2022)
https://doi.org/10.3389/fphy.2022.888476
|
44 |
E. Lovell A., T. Mohan A., M. Sprouse T., and R. Mumpower M., Nuclear masses learned from a probabilistic neural network, Phys. Rev. C 106(1), 014305 (2022)
https://doi.org/10.1103/PhysRevC.106.014305
|
45 |
Hagen G., J. Novario S., H. Sun Z., Papenbrock T., R. Jansen G., G. Lietz J., Duguet T., and Tichai A., Angular-momentum projection in coupled-cluster theory: Structure of 34Mg, Phys. Rev. C 105(6), 064311 (2022)
https://doi.org/10.1103/PhysRevC.105.064311
|
46 |
Papenbrock T., Effective field theory of pairing rotations, Phys. Rev. C 105(4), 044322 (2022)
https://doi.org/10.1103/PhysRevC.105.044322
|
47 |
S. Li Muli S., Acharya B., J. Hernandez O., and Bacca S., Bayesian analysis of nuclear polarizability corrections to the Lamb shift of muonic H-atoms and He-ions, J. Phys. G 49(10), 105101 (2022)
https://doi.org/10.1088/1361-6471/ac81e0
|
48 |
Y. Zhai Q.Z. Liu M.X. Lu J.S. Geng L., Zcs(3985) in next-to-leading-order chiral effective field theory: The first truncation uncertainty analysis, Phys. Rev. D 106(3), 034026 (2022)
|
49 |
Fraboulet K. and P. Ebran J., Addressing energy density functionals in the language of path-integrals I: Comparative study of diagrammatic techniques applied to the (0+0)D O(N)-symmetric φ4-theory, Eur. Phys. J. A 59(4), 91 (2023)
https://doi.org/10.1140/epja/s10050-023-00933-9
|
50 |
Jiang W. and Forssén C., Bayesian probability updates using sampling/importance resampling: Applications in nuclear theory, Front. Phys. (Lausanne) 10, 1058809 (2022)
https://doi.org/10.3389/fphy.2022.1058809
|
51 |
Ekström A., Forssén C., Hagen G., R. Jansen G., Jiang W., and Papenbrock T., What is ab initio in nuclear theory, Front. Phys. (Lausanne) 11, 1129094 (2023)
https://doi.org/10.3389/fphy.2023.1129094
|
52 |
I. Jay W. and T. Neil E., Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103(11), 114502 (2021)
https://doi.org/10.1103/PhysRevD.103.114502
|
53 |
Catacora-Rios M., B. King G., E. Lovell A., and M. Nunes F., Exploring experimental conditions to reduce uncertainties in the optical potential, Phys. Rev. C 100(6), 064615 (2019)
https://doi.org/10.1103/PhysRevC.100.064615
|
54 |
Ekström A. and Hagen G., Global sensitivity analysis of bulk properties of an atomic nucleus, Phys. Rev. Lett. 123(25), 252501 (2019)
https://doi.org/10.1103/PhysRevLett.123.252501
|
55 |
Zhang X.M. Nollett K.R. Phillips D., S-factor and scattering-parameter extractions from 3He + 4He → 7Be + γ, J. Phys. G 47, 054002 (2020)
|
56 |
K. Luna B. and Papenbrock T., Low-energy bound states, resonances, and scattering of light ions, Phys. Rev. C 100(5), 054307 (2019)
https://doi.org/10.1103/PhysRevC.100.054307
|
57 |
Epelbaum E., Golak J., Hebeler K., Kamada H., Krebs H., G. Meißner U., Nogga A., Reinert P., Skibiński R., Topolnicki K., Volkotrub Y., and Witała H., Towards high-order calculations of three-nucleon scattering in chiral effective field theory, Eur. Phys. J. A 56(3), 92 (2020)
https://doi.org/10.1140/epja/s10050-020-00102-2
|
58 |
Metropolis N., W. Rosenbluth A., N. Rosenbluth M., H. Teller A., and Teller E., Equation of state calculations by fast computing machines, J. Chem. Phys. 21(6), 1087 (1953)
https://doi.org/10.1063/1.1699114
|
59 |
K. Hastings W., Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57(1), 97 (1970)
https://doi.org/10.1093/biomet/57.1.97
|
60 |
Duane S., Kennedy A., J. Pendleton B., and Roweth D., Hybrid Monte Carlo, Phys. Lett. B 195(2), 216 (1987)
https://doi.org/10.1016/0370-2693(87)91197-X
|
61 |
D. Homan M. and Gelman A., The No-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo, J. Mach. Learn. Res. 15, 1593 (2014)
|
62 |
Salvatier J., V. Wiecki T., and Fonnesbeck C., Probabilistic programming in python using PyMC3, PeerJ Comput. Sci. 2, e55 (2016)
https://doi.org/10.7717/peerj-cs.55
|
63 |
Gregory P., Bayesian Logical Data Analysis for the Physical Sciences, Cambridge: Cambridge University Press, 2005
|
64 |
Bijnens J., Colangelo G., and Ecker G., Renormalization of chiral perturbation theory to order p6, Ann. Phys. 280(1), 100 (2000)
https://doi.org/10.1006/aphy.1999.5982
|
65 |
Gelman A.B. Carlin J.S. Stern H.B. Dunson D.Vehtari A. B. Rubin D., Bayesian Data Analysis, 3rd Ed., Boca Raton: CPC Press, 2013
|
66 |
Vehtari A., Gelman A., and Gabry J., Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC, Stat. Comput. 27(5), 1413 (2016)
https://doi.org/10.1007/s11222-016-9696-4
|
67 |
Amoroós G., Bijnens J., and Talavera P., Two-point functions at two loops in three flavor chiral perturbation theory, Nucl. Phys. B 568(1−2), 319 (2000)
https://doi.org/10.1016/S0550-3213(99)00674-4
|
68 |
Bijnens J., Chiral perturbation theory, URL: home.thep.lu.se/~bijnens/chpt/ (2019)
|
69 |
Bijnens J. and Dhonte P., Scalar form-factors in SU(3) chiral perturbation theory, J. High Energy Phys. 10, 061 (2003)
https://doi.org/10.1088/1126-6708/2003/10/061
|
70 |
Gasser J., Haefeli C., A. Ivanov M., and Schmid M., Integrating out strange quarks in ChPT, Phys. Lett. B 652(1), 21 (2007)
https://doi.org/10.1016/j.physletb.2007.06.058
|
71 |
Z. Jiang S., L. Wei Z., S. Chen Q., and Wang Q., Computation of the O(p6) order low-energy constants: An update, Phys. Rev. D 92(2), 025014 (2015)
https://doi.org/10.1103/PhysRevD.92.025014
|
72 |
Z. Jiang S., Zhang Y., Li C., and Wang Q., Computation of the p6 order chiral Lagrangian coefficients, Phys. Rev. D 81(1), 014001 (2010)
https://doi.org/10.1103/PhysRevD.81.014001
|
73 |
Kampf K. and Moussallam B., Tests of the naturalness of the coupling constants in ChPT at order p6, Eur. Phys. J. C 47(3), 723 (2006)
https://doi.org/10.1140/epjc/s2006-02606-7
|
74 |
Jamin M., A. Oller J., and Pich A., Order p6 chiral couplings from the scalar Kπ form-factor, J. High Energy Phys. 02, 047 (2004)
https://doi.org/10.1088/1126-6708/2004/02/047
|
75 |
Bijnens J.Talavera P., Kℓ3 decays in chiral perturbation theory, Nucl. Phys. B 669(1–2), 341 (2003)
|
76 |
Cirigliano V., Ecker G., Eidemuüller M., Kaiser R., Pich A., and Portolés J., The ⟨ SPP⟩ Green function and SU(3) breaking in Kℓ3 decays, J. High Energy Phys. 04, 006 (2005)
|
77 |
Unterdorfer R. and Pichl H., On the radiative pion decay, Eur. Phys. J. C 55(2), 273 (2008)
https://doi.org/10.1140/epjc/s10052-008-0584-8
|
78 |
Cirigliano V., Ecker G., Eidemüller M., Kaiser R., Pich A., and Portolés J., Towards a consistent estimate of the chiral low-energy constants, Nucl. Phys. B 753(1-2), 139 (2006)
https://doi.org/10.1016/j.nuclphysb.2006.07.010
|
79 |
Bernard V. and Passemar E., Matching chiral perturbation theory and the dispersive representation of the scalar Kπ form-factor, Phys. Lett. B 661(2−3), 95 (2008)
https://doi.org/10.1016/j.physletb.2008.02.004
|
80 |
Moussallam B., Flavor stability of the chiral vacuum and scalar meson dynamics, J. High Energy Phys. 08, 005 (2000)
https://doi.org/10.1088/1126-6708/2000/08/005
|
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