Abstract
We present the results of a third order calculation of the pion-nucleon scattering amplitude in a chiral effective field theory with pions, nucleons and delta resonances as explicit degrees of freedom. We work in a manifestly Lorentz invariant formulation of baryon chiral perturbation theory using dimensional regularization and the extended on-mass-shell renormalization scheme. In the delta resonance sector, the on mass-shell renormalization is realized as a complex-mass scheme. By fitting the low-energy constants of the effective Lagrangian to the S- and P -partial waves a satisfactory description of the phase shifts from the analysis of the Roy-Steiner equations is obtained. We predict the phase shifts for the D and F waves and compare them with the results of the analysis of the George Washington University group. The threshold parameters are calculated both in the delta-less and delta-full cases. Based on the determined low-energy constants, we discuss the pion-nucleon sigma term. Additionally, in order to determine the strangeness content of the nucleon, we calculate the octet baryon masses in the presence of decuplet resonances up to next-to-next-to-leading order in SU(3) baryon chiral perturbation theory. The octet baryon sigma terms are predicted as a byproduct of this calculation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B.H. Bransden and R.G. Moorhouse, The pion-nucleon system, Princeton University Press, Princeton U.S.A. (1973).
G. Höhler, Landolt-Börnstein, volume 9b2, H. Schopper ed., Springer, Berlin Germany (1983).
G.F. Chew, M.L. Goldberger, F.E. Low and Y. Nambu, Application of dispersion relations to low-energy meson-nucleon scattering, Phys. Rev. 106 (1957) 1337 [INSPIRE].
J. Hamilton and W.S. Woolcock, Determination of pion-nucleon parameters and phase shifts by dispersion relations, Rev. Mod. Phys. 35 (1963) 737 [INSPIRE].
F. Steiner, On the generalized πN potential — a new representation from fixed-t dispersion relations, Fortsch. Phys. 18 (1970) 43 [INSPIRE].
M.G. Olsson and E.T. Osypowski, Systematics of low-energy πN scattering, Nucl. Phys. B 101 (1975) 136 [INSPIRE].
D. Bofinger and W.S. Woolcock, A critical study of effective Lagrangian models for low-energy pion-nucleon scattering, Nuovo Cim. A 104 (1991) 1489 [INSPIRE].
F. Gross and Y. Surya, Unitary, relativistic resonance model for πN scattering, Phys. Rev. C 47 (1993) 703 [INSPIRE].
C. Schutz, J. Haidenbauer, J. Speth and J.W. Durso, Extended coupled channels model for πN scattering and the structure of N ∗(1440) and N ∗(1535), Phys. Rev. C 57 (1998) 1464 [INSPIRE].
O. Krehl, C. Hanhart, S. Krewald and J. Speth, What does ‘ρ exchange’ in πN scattering mean?, Phys. Rev. C 60 (1999) 055206 [nucl-th/9906090] [INSPIRE].
A.M. Gasparyan, J. Haidenbauer, C. Hanhart and J. Speth, Pion nucleon scattering in a meson exchange model, Phys. Rev. C 68 (2003) 045207 [nucl-th/0307072] [INSPIRE].
A. Gasparyan and M.F.M. Lutz, Photon- and pion-nucleon interactions in a unitary and causal effective field theory based on the chiral Lagrangian, Nucl. Phys. A 848 (2010) 126 [arXiv:1003.3426] [INSPIRE].
D. Ronchen et al., Coupled-channel dynamics in the reactions πN → πN , ηN , KΛ, KΣ, Eur. Phys. J. A 49 (2013) 44 [arXiv:1211.6998] [INSPIRE].
C. Ditsche, M. Hoferichter, B. Kubis and U.-G. Meißner, Roy-Steiner equations for pion-nucleon scattering, JHEP 06 (2012) 043 [arXiv:1203.4758] [INSPIRE].
M. Hoferichter, J. Ruiz de Elvira, B. Kubis and U.-G. Meißner, High-precision determination of the pion-nucleon σ term from Roy-Steiner equations, Phys. Rev. Lett. 115 (2015) 092301 [arXiv:1506.04142] [INSPIRE].
M. Hoferichter, J. Ruiz de Elvira, B. Kubis and U.-G. Meißner, Matching pion-nucleon Roy-Steiner equations to chiral perturbation theory, Phys. Rev. Lett. 115 (2015) 192301 [arXiv:1507.07552] [INSPIRE].
M. Hoferichter, J. Ruiz de Elvira, B. Kubis and U.-G. Meißner, Roy-Steiner-equation analysis of pion-nucleon scattering, Phys. Rept. 625 (2016) 1 [arXiv:1510.06039] [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Annals Phys. 158 (1984) 142 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral perturbation theory: expansions in the mass of the strange quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
S. Scherer and M.R. Schindler, Quantum chromodynamics and chiral symmetry, Lect. Notes Phys. 830 (2012) 1 [INSPIRE].
E. Epelbaum, J. Gegelia, U.-G. Meißner and D.-L. Yao, Baryon chiral perturbation theory extended beyond the low-energy region, Eur. Phys. J. C 75 (2015) 499 [arXiv:1510.02388] [INSPIRE].
J. Gasser, M.E. Sainio and A. Svarc, Nucleons with chiral loops, Nucl. Phys. B 307 (1988) 779 [INSPIRE].
E.E. Jenkins and A.V. Manohar, Baryon chiral perturbation theory using a heavy fermion Lagrangian, Phys. Lett. B 255 (1991) 558 [INSPIRE].
V. Bernard, N. Kaiser, J. Kambor and U.-G. Meißner, Chiral structure of the nucleon, Nucl. Phys. B 388 (1992) 315 [INSPIRE].
M. Mojzis, Elastic πN scattering to O(p 3) in heavy baryon chiral perturbation theory, Eur. Phys. J. C 2 (1998) 181 [hep-ph/9704415] [INSPIRE].
N. Fettes, U.-G. Meißner and S. Steininger, Pion-nucleon scattering in chiral perturbation theory. 1. Isospin symmetric case, Nucl. Phys. A 640 (1998) 199 [hep-ph/9803266] [INSPIRE].
N. Fettes and U.-G. Meißner, Pion nucleon scattering in chiral perturbation theory. 2. Fourth order calculation, Nucl. Phys. A 676 (2000) 311 [hep-ph/0002162] [INSPIRE].
T. Becher and H. Leutwyler, Baryon chiral perturbation theory in manifestly Lorentz invariant form, Eur. Phys. J. C 9 (1999) 643 [hep-ph/9901384] [INSPIRE].
V. Bernard, N. Kaiser, U.-G. Meißner and A. Schmidt, Aspects of nucleon Compton scattering, Z. Phys. A 348 (1994) 317 [hep-ph/9311354] [INSPIRE].
P.J. Ellis and H.-B. Tang, Pion nucleon scattering in a new approach to chiral perturbation theory, Phys. Rev. C 57 (1998) 3356 [hep-ph/9709354] [INSPIRE].
J.L. Goity, D. Lehmann, G. Prezeau and J. Saez, Regularization for effective field theory with two heavy particles, Phys. Lett. B 504 (2001) 21 [hep-ph/0101011] [INSPIRE].
V. Bernard, T.R. Hemmert and U.-G. Meißner, Infrared regularization with spin 3/2 fields, Phys. Lett. B 565 (2003) 137 [hep-ph/0303198] [INSPIRE].
M.R. Schindler, J. Gegelia and S. Scherer, Infrared regularization of baryon chiral perturbation theory reformulated, Phys. Lett. B 586 (2004) 258 [hep-ph/0309005] [INSPIRE].
P.C. Bruns and U.-G. Meißner, Infrared regularization with vector mesons and baryons, Eur. Phys. J. C 58 (2008) 407 [arXiv:0808.3174] [INSPIRE].
T. Becher and H. Leutwyler, Low energy analysis of πN → πN , JHEP 06 (2001) 017 [hep-ph/0103263] [INSPIRE].
J.M. Alarcon, J. Martin Camalich, J.A. Oller and L. Álvarez-Ruso, πN scattering in relativistic baryon chiral perturbation theory revisited, Phys. Rev. C 83 (2011) 055205 [Erratum ibid. C 87 (2013) 059901] [arXiv:1102.1537] [INSPIRE].
M. Hoferichter, B. Kubis and U.-G. Meißner, Isospin violation in low-energy pion-nucleon scattering revisited, Nucl. Phys. A 833 (2010) 18 [arXiv:0909.4390] [INSPIRE].
M. Mai, P.C. Bruns, B. Kubis and U.-G. Meißner, Aspects of meson-baryon scattering in three and two-flavor chiral perturbation theory, Phys. Rev. D 80 (2009) 094006 [arXiv:0905.2810] [INSPIRE].
J. Gegelia and G. Japaridze, Matching heavy particle approach to relativistic theory, Phys. Rev. D 60 (1999) 114038 [hep-ph/9908377] [INSPIRE].
J. Gegelia, G. Japaridze and X.Q. Wang, Is heavy baryon approach necessary?, J. Phys. G 29 (2003) 2303 [hep-ph/9910260] [INSPIRE].
T. Fuchs, J. Gegelia, G. Japaridze and S. Scherer, Renormalization of relativistic baryon chiral perturbation theory and power counting, Phys. Rev. D 68 (2003) 056005 [hep-ph/0302117] [INSPIRE].
J.M. Alarcon, J. Martin Camalich and J.A. Oller, Improved description of the πN -scattering phenomenology in covariant baryon chiral perturbation theory, Annals Phys. 336 (2013) 413 [arXiv:1210.4450] [INSPIRE].
Y.-H. Chen, D.-L. Yao and H.Q. Zheng, Analyses of pion-nucleon elastic scattering amplitudes up to O(p 4) in extended-on-mass-shell subtraction scheme, Phys. Rev. D 87 (2013) 054019 [arXiv:1212.1893] [INSPIRE].
R.A. Arndt, W.J. Briscoe, I.I. Strakovsky and R.L. Workman, Extended partial-wave analysis of πN scattering data, Phys. Rev. C 74 (2006) 045205 [nucl-th/0605082] [INSPIRE].
S. Bellucci, J. Gasser and M.E. Sainio, Low-energy photon-photon collisions to two loop order, Nucl. Phys. B 423 (1994) 80 [Erratum ibid. B 431 (1994) 413] [hep-ph/9401206] [INSPIRE].
W. Rarita and J. Schwinger, On a theory of particles with half integral spin, Phys. Rev. 60 (1941) 61 [INSPIRE].
T.R. Hemmert, B.R. Holstein and J. Kambor, Chiral Lagrangians and Δ(1232) interactions: formalism, J. Phys. G 24 (1998) 1831 [hep-ph/9712496] [INSPIRE].
S. Weinberg, Effective chiral Lagrangians for nucleon-pion interactions and nuclear forces, Nucl. Phys. B 363 (1991) 3 [INSPIRE].
G. Ecker, Chiral perturbation theory, Prog. Part. Nucl. Phys. 35 (1995) 1 [hep-ph/9501357] [INSPIRE].
T.R. Hemmert, B.R. Holstein and J. Kambor, Systematic 1/M expansion for spin 3/2 particles in baryon chiral perturbation theory, Phys. Lett. B 395 (1997) 89 [hep-ph/9606456] [INSPIRE].
V. Pascalutsa and D.R. Phillips, Effective theory of the Δ(1232) in Compton scattering off the nucleon, Phys. Rev. C 67 (2003) 055202 [nucl-th/0212024] [INSPIRE].
F. Hagelstein, R. Miskimen and V. Pascalutsa, Nucleon polarizabilities: from Compton scattering to hydrogen atom, Prog. Part. Nucl. Phys. 88 (2016) 29 [arXiv:1512.03765] [INSPIRE].
H.-B. Tang and P.J. Ellis, Redundance of Δ isobar parameters in effective field theories, Phys. Lett. B 387 (1996) 9 [hep-ph/9606432] [INSPIRE].
V. Pascalutsa, Correspondence of consistent and inconsistent spin-3/2 couplings via the equivalence theorem, Phys. Lett. B 503 (2001) 85 [hep-ph/0008026] [INSPIRE].
H. Krebs, E. Epelbaum and U.-G. Meißner, Redundancy of the off-shell parameters in chiral effective field theory with explicit spin-3/2 degrees of freedom, Phys. Lett. B 683 (2010) 222 [arXiv:0905.2744] [INSPIRE].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
M.R. Schindler, T. Fuchs, J. Gegelia and S. Scherer, Axial, induced pseudoscalar and pion-nucleon form-factors in manifestly Lorentz-invariant chiral perturbation theory, Phys. Rev. C 75 (2007) 025202 [nucl-th/0611083] [INSPIRE].
S. Scherer, Introduction to chiral perturbation theory, Adv. Nucl. Phys. 27 (2003) 277 [hep-ph/0210398] [INSPIRE].
R.G. Stuart, Pitfalls of radiative corrections near a resonance, in Z0 physics, J. Tran Thanh Van ed., Editions Frontieres, Gif-sur-Yvette France (1990), pg. 41 [INSPIRE].
A. Denner, S. Dittmaier, M. Roth and D. Wackeroth, Predictions for all processes e + e − → 4 fermions + γ, Nucl. Phys. B 560 (1999) 33 [hep-ph/9904472] [INSPIRE].
T. Bauer, J. Gegelia, G. Japaridze and S. Scherer, Complex-mass scheme and perturbative unitarity, Int. J. Mod. Phys. A 27 (2012) 1250178 [arXiv:1211.1684] [INSPIRE].
A. Denner and J.-N. Lang, The complex-mass scheme and unitarity in perturbative quantum field theory, Eur. Phys. J. C 75 (2015) 377 [arXiv:1406.6280] [INSPIRE].
C. Hacker, N. Wies, J. Gegelia and S. Scherer, Including the Δ(1232) resonance in baryon chiral perturbation theory, Phys. Rev. C 72 (2005) 055203 [hep-ph/0505043] [INSPIRE].
J. Gegelia and S. Scherer, How to define physical properties of unstable particles, Eur. Phys. J. A 44 (2010) 425 [arXiv:0910.4280] [INSPIRE].
R. Koch and E. Pietarinen, Low-energy πN partial wave analysis, Nucl. Phys. A 336 (1980) 331 [INSPIRE].
R. Koch, A calculation of low-energy πN partial waves based on fixed t analyticity, Nucl. Phys. A 448 (1986) 707 [INSPIRE].
E. Matsinos, W.S. Woolcock, G.C. Oades, G. Rasche and A. Gashi, Phase-shift analysis of low-energy π ± p elastic-scattering data, Nucl. Phys. A 778 (2006) 95 [hep-ph/0607080] [INSPIRE].
V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga and D.R. Phillips, Precision calculation of the π − deuteron scattering length and its impact on threshold πN scattering, Phys. Lett. B 694 (2011) 473 [arXiv:1003.4444] [INSPIRE].
V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga and D.R. Phillips, Precision calculation of threshold π − d scattering, πN scattering lengths and the GMO sum rule, Nucl. Phys. A 872 (2011) 69 [arXiv:1107.5509] [INSPIRE].
V. Pascalutsa, M. Vanderhaeghen and S.N. Yang, Electromagnetic excitation of the Δ(1232)-resonance, Phys. Rept. 437 (2007) 125 [hep-ph/0609004] [INSPIRE].
Particle Data Group collaboration, K.A. Olive et al., Review of particle physics, Chin. Phys. C 38 (2014) 090001 [INSPIRE].
A. van Hameren, OneLOop: for the evaluation of one-loop scalar functions, Comput. Phys. Commun. 182 (2011) 2427 [arXiv:1007.4716] [INSPIRE].
T. Hahn and M. Pérez-Victoria, Automatized one loop calculations in four-dimensions and D-dimensions, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565] [INSPIRE].
F. James and M. Roos, Minuit: a system for function minimization and analysis of the parameter errors and correlations, Comput. Phys. Commun. 10 (1975) 343 [INSPIRE].
E. Epelbaum, H. Krebs and U.-G. Meißner, Improved chiral nucleon-nucleon potential up to next-to-next-to-next-to-leading order, Eur. Phys. J. A 51 (2015) 53 [arXiv:1412.0142] [INSPIRE].
D. Siemens, V. Bernard, E. Epelbaum, A. Gasparyan, H. Krebs and U.-G. Meißner, Elastic pion-nucleon scattering in chiral perturbation theory: a fresh look, arXiv:1602.02640 [INSPIRE].
N. Fettes and U.-G. Meißner, Pion-nucleon scattering in an effective chiral field theory with explicit spin 3/2 fields, Nucl. Phys. A 679 (2001) 629 [hep-ph/0006299] [INSPIRE].
S. Dürr, Validity of ChPT — is M π = 135 MeV small enough?, PoS(LATTICE2014)006 [arXiv:1412.6434] [INSPIRE].
V. Bernard, N. Kaiser and U.-G. Meißner, Critical analysis of baryon masses and sigma terms in heavy baryon chiral perturbation theory, Z. Phys. C 60 (1993) 111 [hep-ph/9303311] [INSPIRE].
J.M. Alarcon, J. Martin Camalich and J.A. Oller, The chiral representation of the πN scattering amplitude and the pion-nucleon sigma term, Phys. Rev. D 85 (2012) 051503 [arXiv:1110.3797] [INSPIRE].
H. Leutwyler, Theoretical aspects of chiral dynamics, arXiv:1510.07511 [INSPIRE].
J. Gasser, Hadron masses and sigma commutator in the light of chiral perturbation theory, Annals Phys. 136 (1981) 62 [INSPIRE].
B. Borasoy and U.-G. Meißner, Chiral expansion of baryon masses and σ-terms, Annals Phys. 254 (1997) 192 [hep-ph/9607432] [INSPIRE].
J.M. Alarcon, L.S. Geng, J. Martin Camalich and J.A. Oller, The strangeness content of the nucleon from effective field theory and phenomenology, Phys. Lett. B 730 (2014) 342 [arXiv:1209.2870] [INSPIRE].
X.-L. Ren, L.-S. Geng and J. Meng, Scalar strangeness content of the nucleon and baryon sigma terms, Phys. Rev. D 91 (2015) 051502 [arXiv:1404.4799] [INSPIRE].
S. Dürr et al., Sigma term and strangeness content of octet baryons, Phys. Rev. D 85 (2012) 014509 [Erratum ibid. D 93 (2016) 039905] [arXiv:1109.4265] [INSPIRE].
Y.-B. Yang, A. Alexandru, T. Draper, J. Liang and K.-F. Liu, πN and strangeness sigma terms at the physical point with chiral fermions, arXiv:1511.09089 [INSPIRE].
S. Dürr et al., Lattice computation of the nucleon scalar quark contents at the physical point, arXiv:1510.08013 [INSPIRE].
QCDSF-UKQCD collaboration, R. Horsley et al., Hyperon sigma terms for 2 + 1 quark flavours, Phys. Rev. D 85 (2012) 034506 [arXiv:1110.4971] [INSPIRE].
MILC collaboration, W. Freeman and D. Toussaint, Intrinsic strangeness and charm of the nucleon using improved staggered fermions, Phys. Rev. D 88 (2013) 054503 [arXiv:1204.3866] [INSPIRE].
P. Junnarkar and A. Walker-Loud, Scalar strange content of the nucleon from lattice QCD, Phys. Rev. D 87 (2013) 114510 [arXiv:1301.1114] [INSPIRE].
JLQCD collaboration, H. Ohki et al., Nucleon strange quark content from N f = 2 + 1 lattice QCD with exact chiral symmetry, Phys. Rev. D 87 (2013) 034509 [arXiv:1208.4185] [INSPIRE].
M. Engelhardt, Strange quark contributions to nucleon mass and spin from lattice QCD, Phys. Rev. D 86 (2012) 114510 [arXiv:1210.0025] [INSPIRE].
XQCD collaboration, M. Gong et al., Strangeness and charmness content of the nucleon from overlap fermions on 2 + 1-flavor domain-wall fermion configurations, Phys. Rev. D 88 (2013) 014503 [arXiv:1304.1194] [INSPIRE].
ETM collaboration, A. Abdel-Rehim et al., Direct evaluation of the quark content of the nucleon from lattice QCD at the physical point, arXiv:1601.01624 [INSPIRE].
G.S. Bali, S. Collins, D. Richtmann, A. Schäfer, W. Söldner and A. Sternbeck, Direct determinations of the nucleon and pion σ terms at nearly physical quark masses, arXiv:1603.00827 [INSPIRE].
A. Denner and S. Dittmaier, Reduction schemes for one-loop tensor integrals, Nucl. Phys. B 734 (2006) 62 [hep-ph/0509141] [INSPIRE].
B.C. Lehnhart, J. Gegelia and S. Scherer, Baryon masses and nucleon sigma terms in manifestly Lorentz-invariant baryon chiral perturbation theory, J. Phys. G 31 (2005) 89 [hep-ph/0412092] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1603.03638
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Yao, DL., Siemens, D., Bernard, V. et al. Pion-nucleon scattering in covariant baryon chiral perturbation theory with explicit Delta resonances. J. High Energ. Phys. 2016, 38 (2016). https://doi.org/10.1007/JHEP05(2016)038
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2016)038