Bayesian method for fitting the low-energy constants in chiral perturbation theory

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 L_i^r and next-to-next-leading order (NNLO) LECs C_i^r. With the estimation of the truncation errors, the fitting results of L_i^r in the NLO and NNLO are very close. The posterior distributions of C_i^r indicate the boundary-dependent relations of these C_i^r. Ten C_i^r are weakly dependent on the boundaries and their values are reliable. The other C_i^r are required more experimental data to constrain their boundaries. Some linear combinations of C_i^r are also fitted with more reliable posterior distributions. If one knows some more precise values of C_i^r, some other C_i^r can be obtained by these values. With these fitting LECs, most observables provide a good convergence, except for the \pi K scattering lengths a_0^{3/2} and a_0^{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.

Graphical abstract

Keywords

chiral perturbation theory / low-energy constants / Bayesian statistics

References

[1] S. Weinberg, Phenomenological Lagrangians, Physica A 96(1−2), 327 (1979)

[2] J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. 158(1), 142 (1984)

[3] J. Gasser and H. Leutwyler, Chiral perturbation theory: Expansions in the mass of the strange quark, Nucl. Phys. B 250(1−4), 465 (1985)

[4] J. Bijnens, G. Colangelo, and G. Ecker, The mesonic chiral Lagrangian of order p⁶, J. High Energy Phys. 02, 020 (1999)

[5] J. Bijnens, N. Hermansson-Truedsson, and S. Wang, The order p⁸ mesonic chiral Lagrangian, J. High Energy Phys. 01(1), 102 (2019)

[6] J.BijnensI. Jemos, A new global fit of the L^r at next-to-next-to-leading order in chiral perturbation theory, Nucl. Phys. B 854(3), 631 (2012)

[7] J. Bijnens and G. Ecker, Mesonic low-energy constants, Annu. Rev. Nucl. Part. Sci. 64(1), 149 (2014)

[8] Q. H. Yang, W. Guo, F. J. Ge, B. Huang, H. Liu, and S. Z. Jiang, New method for fitting the low-energy constants in chiral perturbation theory, Phys. Rev. D 102(9), 094009 (2020)

[9] K. U. Can, G. Erkol, M. Oka, and T. T. Takahashi, Look inside charmed-strange baryons from lattice QCD, Phys. Rev. D 92(11), 114515 (2015)

[10] K. U. Can, G. Erkol, B. Isildak, M. Oka, and T. T. Takahashi, Electromagnetic structure of charmed baryons in lattice QCD, J. High Energy Phys. 05(5), 125 (2014)

[11] H.BahtiyarK. U. CanG.ErkolM.OkaT.T. Takahashi, Ξ_cγ → Ξ_c^′ transition in lattice QCD, Phys. Lett. B 772, 121 (2017)

[12] T.M. YanH. Y. ChengC.Y. CheungG.L. LinY.C. Lin H.L. Yu, Heavy quark symmetry and chiral dynamics, Phys. Rev. D 46(3), 1148 (1992) [Erratum: Phys. Rev. D 55, 5851 (1997)]

[13] R.J. DowdallC.T. H. DaviesG.P. LepageC.McNeile, V_us 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] A. Bazavov, . (MILC), . Results for light pseudoscalar mesons, PoS LATTICE 2010, 074 (2010)

[15] V. Bernard and E. Passemar, Chiral extrapolation of the strangeness changing Kπ form factor, J. High Energy Phys. 04, 001 (2010)

[16] A.Bazavov. (MILC), ., 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] A. Bazavov, D. Toussaint, C. Bernard, J. Laiho, C. DeTar, L. Levkova, M. B. Oktay, S. Gottlieb, U. M. Heller, J. E. Hetrick, P. B. Mackenzie, R. Sugar, and R. S. Van de Water, Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks, Rev. Mod. Phys. 82(2), 1349 (2010)

[18] M.GoltermanK. MaltmanS.Peris, NNLO low-energy constants from flavor-breaking chiral sum rules based on hadronic τ-decay data, Phys. Rev. D 89(5), 054036 (2014)

[19] P. Colangelo, J. J. Sanz-Cillero, and F. Zuo, Holography, chiral Lagrangian and form factor relations, J. High Energy Phys. 11, 012 (2012)

[20] Z. H. Guo, J. J. Sanz Cillero, and H. Q. Zheng, Partial waves and large N_C resonance sum rules, J. High Energy Phys. 06, 030 (2007)

[21] Z.H. GuoJ. J. Sanz-CilleroH.Q. Zheng, O(p⁶) extension of the large-N_C partial wave dispersion relations, Phys. Lett. B 661, 342 (2008), arXiv:

[22] Z.H. GuoJ. J. Sanz-Cillero, ππ-scattering lengths at O(p⁶) revisited, Phys. Rev. D 79, 096006 (2009)

[23] J.BijnensG. ColangeloJ.Gasser, K_l4 decays beyond one loop, Nucl. Phys. B 427(3), 427 (1994)

[24] G.AmorósJ.BijnensP.Talavera, K_ℓ4 form-factors and π‒π scattering, Nucl. Phys. B 585, 293 (2000) [Erratum: Nucl. Phys. B 598, 665(2001)], arXiv:

[25] G.ColangeloJ. GasserH.Leutwyler, ππ scattering, Nucl. Phys. B 603(1–2), 125 (2001)

[26] M.R. SchindlerD.R. Phillips, Bayesian methods for parameter estimation in effective field theories, Ann. Phys. 324, 682 (2009) [Erratum: Ann. Phys. 324, 2051 (2009)], arXiv:

[27] R.J. FurnstahlD.R. PhillipsS.Wesolowski, A recipe for EFT uncertainty quantification in nuclear physics, J. Phys. G 42(3), 034028 (2015)

[28] S. Wesolowski, N. Klco, R. J. Furnstahl, D. R. Phillips, and A. Thapaliya, Bayesian parameter estimation for effective field theories, J. Phys. G 43(7), 074001 (2016)

[29] J. A. Melendez, S. Wesolowski, and R. J. Furnstahl, Bayesian truncation errors in chiral effective field theory: Nucleon‒nucleon observables, Phys. Rev. C 96(2), 024003 (2017)

[30] I. Svensson, A. Ekström, and C. Forssén, Bayesian parameter estimation in chiral effective field theory using the Hamiltonian Monte Carlo method, Phys. Rev. C 105(1), 014004 (2022)

[31] A. Ekström, C. Forssén, C. Dimitrakakis, D. Dubhashi, H. T. Johansson, A. S. Muhammad, H. Salomonsson, and A. Schliep, Bayesian optimization in ab initio nuclear physics, J. Phys. G 46(9), 095101 (2019)

[32] S. Wesolowski, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, Exploring Bayesian parameter estimation for chiral effective field theory using nucleon–nucleon phase shifts, J. Phys. G 46(4), 045102 (2019)

[33] I. K. Alnamlah, E. A. C. Pérez, and D. R. Phillips, Effective field theory approach to rotational bands in odd-mass nuclei, Phys. Rev. C 104(6), 064311 (2021)

[34] C. J. Yang, A. Ekström, C. Forssén, and G. Hagen, Power counting in chiral effective field theory and nuclear binding, Phys. Rev. C 103(5), 054304 (2021)

[35] A. E. Lovell, F. M. Nunes, M. Catacora-Rios, and G. B. King, Recent advances in the quantification of uncertainties in reaction theory, J. Phys. G 48(1), 014001 (2020)

[36] D. R. Phillips, R. J. Furnstahl, U. Heinz, T. Maiti, W. Nazarewicz, F. M. Nunes, M. Plumlee, M. T. Pratola, S. Pratt, F. G. Viens, and S. M. Wild, Get on the BAND Wagon: A Bayesian framework for quantifying model uncertainties in nuclear dynamics, J. Phys. G 48(7), 072001 (2021)

[37] P.BedaqueA. BoehnleinM.CromazM.DiefenthalerL.ElouadrhiriT.HornM.KucheraD.Lawrence D.LeeS. LidiaR.McKeownW.MelnitchoukW.NazarewiczK.OrginosY.RoblinM.Scott SmithM.SchramX.N. Wang, A. I. for nuclear physics, Eur. Phys. J. A 57(3), 100 (2021)

[38] S. Wesolowski, I. Svensson, A. Ekström, C. Forssén, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, 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)

[39] M. A. Connell, I. Billig, and D. R. Phillips, Does Bayesian model averaging improve polynomial extrapolations? Two toy problems as tests, J. Phys. G 48(10), 104001 (2021)

[40] Y. H. Lin, H. W. Hammer, and U. G. Meißner, Dispersion-theoretical analysis of the electromagnetic form factors of the nucleon: Past, present and future, Eur. Phys. J. A 57(8), 255 (2021)

[41] T. Djärv, A. Ekström, C. Forssén, and H. T. Johansson, Bayesian predictions for A = 6 nuclei using eigenvector continuation emulators, Phys. Rev. C 105(1), 014005 (2022)

[42] B. Acharya and S. Bacca, Gaussian process error modeling for chiral effective-field-theory calculations of np↔dγ at low energies, Phys. Lett. B 827, 137011 (2022)

[43] D. Odell, C. R. Brune, D. R. Phillips, R. J. deBoer, and S. N. Paneru, Performing Bayesian analyses with AZURE2 using BRICK: An application to the ⁷Be system, Front. Phys. (Lausanne) 10, 888476 (2022)

[44] A. E. Lovell, A. T. Mohan, T. M. Sprouse, and M. R. Mumpower, Nuclear masses learned from a probabilistic neural network, Phys. Rev. C 106(1), 014305 (2022)

[45] G. Hagen, S. J. Novario, Z. H. Sun, T. Papenbrock, G. R. Jansen, J. G. Lietz, T. Duguet, and A. Tichai, Angular-momentum projection in coupled-cluster theory: Structure of ³⁴Mg, Phys. Rev. C 105(6), 064311 (2022)

[46] T. Papenbrock, Effective field theory of pairing rotations, Phys. Rev. C 105(4), 044322 (2022)

[47] S. S. Li Muli, B. Acharya, O. J. Hernandez, and S. Bacca, Bayesian analysis of nuclear polarizability corrections to the Lamb shift of muonic H-atoms and He-ions, J. Phys. G 49(10), 105101 (2022)

[48] Q.Y. ZhaiM. Z. LiuJ.X. LuL.S. Geng, Z_cs(3985) in next-to-leading-order chiral effective field theory: The first truncation uncertainty analysis, Phys. Rev. D 106(3), 034026 (2022)

[49] K. Fraboulet and J. P. Ebran, 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)

[50] W. Jiang and C. Forssén, Bayesian probability updates using sampling/importance resampling: Applications in nuclear theory, Front. Phys. (Lausanne) 10, 1058809 (2022)

[51] A. Ekström, C. Forssén, G. Hagen, G. R. Jansen, W. Jiang, and T. Papenbrock, What is ab initio in nuclear theory, Front. Phys. (Lausanne) 11, 1129094 (2023)

[52] W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103(11), 114502 (2021)

[53] M. Catacora-Rios, G. B. King, A. E. Lovell, and F. M. Nunes, Exploring experimental conditions to reduce uncertainties in the optical potential, Phys. Rev. C 100(6), 064615 (2019)

[54] A. Ekström and G. Hagen, Global sensitivity analysis of bulk properties of an atomic nucleus, Phys. Rev. Lett. 123(25), 252501 (2019)

[55] X.ZhangK. M. NollettD.R. Phillips, S-factor and scattering-parameter extractions from ³He + ⁴He → ⁷Be + γ, J. Phys. G 47, 054002 (2020)

[56] B. K. Luna and T. Papenbrock, Low-energy bound states, resonances, and scattering of light ions, Phys. Rev. C 100(5), 054307 (2019)

[57] E. Epelbaum, J. Golak, K. Hebeler, H. Kamada, H. Krebs, U. G. Meißner, A. Nogga, P. Reinert, R. Skibiński, K. Topolnicki, Y. Volkotrub, and H. Witała, Towards high-order calculations of three-nucleon scattering in chiral effective field theory, Eur. Phys. J. A 56(3), 92 (2020)

[58] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21(6), 1087 (1953)

[59] W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57(1), 97 (1970)

[60] S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195(2), 216 (1987)

[61] M. D. Homan and A. Gelman, The No-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo, J. Mach. Learn. Res. 15, 1593 (2014)

[62] J. Salvatier, T. V. Wiecki, and C. Fonnesbeck, Probabilistic programming in python using PyMC3, PeerJ Comput. Sci. 2, e55 (2016)

[63] P.Gregory, Bayesian Logical Data Analysis for the Physical Sciences, Cambridge: Cambridge University Press, 2005

[64] J. Bijnens, G. Colangelo, and G. Ecker, Renormalization of chiral perturbation theory to order p⁶, Ann. Phys. 280(1), 100 (2000)

[65] A.GelmanJ. B. CarlinH.S. SternD.B. DunsonA.Vehtari D.B. Rubin, Bayesian Data Analysis, 3rd Ed., Boca Raton: CPC Press, 2013

[66] A. Vehtari, A. Gelman, and J. Gabry, Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC, Stat. Comput. 27(5), 1413 (2016)

[67] G. Amoroós, J. Bijnens, and P. Talavera, Two-point functions at two loops in three flavor chiral perturbation theory, Nucl. Phys. B 568(1−2), 319 (2000)

[68] J.Bijnens, Chiral perturbation theory, URL: home.thep.lu.se/~bijnens/chpt/ (2019)

[69] J. Bijnens and P. Dhonte, Scalar form-factors in SU(3) chiral perturbation theory, J. High Energy Phys. 10, 061 (2003)

[70] J. Gasser, C. Haefeli, M. A. Ivanov, and M. Schmid, Integrating out strange quarks in ChPT, Phys. Lett. B 652(1), 21 (2007)

[71] S. Z. Jiang, Z. L. Wei, Q. S. Chen, and Q. Wang, Computation of the O(p⁶) order low-energy constants: An update, Phys. Rev. D 92(2), 025014 (2015)

[72] S. Z. Jiang, Y. Zhang, C. Li, and Q. Wang, Computation of the p⁶ order chiral Lagrangian coefficients, Phys. Rev. D 81(1), 014001 (2010)

[73] K. Kampf and B. Moussallam, Tests of the naturalness of the coupling constants in ChPT at order p⁶, Eur. Phys. J. C 47(3), 723 (2006)

[74] M. Jamin, J. A. Oller, and A. Pich, Order p⁶ chiral couplings from the scalar Kπ form-factor, J. High Energy Phys. 02, 047 (2004)

[75] J.BijnensP. Talavera, K_ℓ3 decays in chiral perturbation theory, Nucl. Phys. B 669(1–2), 341 (2003)

[76] V. Cirigliano, G. Ecker, M. Eidemuüller, R. Kaiser, A. Pich, and J. Portolés, The ⟨ SPP⟩ Green function and SU(3) breaking in K_ℓ3 decays, J. High Energy Phys. 04, 006 (2005)

[77] R. Unterdorfer and H. Pichl, On the radiative pion decay, Eur. Phys. J. C 55(2), 273 (2008)

[78] V. Cirigliano, G. Ecker, M. Eidemüller, R. Kaiser, A. Pich, and J. Portolés, Towards a consistent estimate of the chiral low-energy constants, Nucl. Phys. B 753(1-2), 139 (2006)

[79] V. Bernard and E. Passemar, Matching chiral perturbation theory and the dispersive representation of the scalar Kπ form-factor, Phys. Lett. B 661(2−3), 95 (2008)

[80] B. Moussallam, Flavor stability of the chiral vacuum and scalar meson dynamics, J. High Energy Phys. 08, 005 (2000)

Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

H. X. Pan thanks Qin-He Yang for providing the original program. This project was supported by the Guangxi Science Foundation under Grants No. 2022GXNSFAA035489.

RIGHTS & PERMISSIONS

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