Abstract
Based on the recently proposed Roy-Steiner equations for pion-nucleon (πN) scattering [1], we derive a system of coupled integral equations for the \( \pi \pi \to \overline N N \) and \( \overline K K \to \overline N N \) S-waves. These equations take the form of a two-channel Muskhelishvili-Omnès problem, whose solution in the presence of a finite matching point is discussed. We use these results to update the dispersive analysis of the scalar form factor of the nucleon fully including \( \overline K K \) intermediate states. In particular, we determine the correction \( {\Delta_{\sigma }} = \sigma \left( {2M_{\pi }^2} \right) - {\sigma_{{\pi N}}} \), which is needed for the extraction of the pion-nucleon σ term from πN scattering, as a function of pion-nucleon subthreshold parameters and the πN coupling constant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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].
T. Cheng and R.F. Dashen, Is SU(2) × SU(2) a better symmetry than SU(3)?, Phys. Rev. Lett. 26 (1971) 594 [INSPIRE].
L.S. Brown, W. Pardee and R. Peccei, Adler-Weisberger theorem reexamined, Phys. Rev. D 4 (1971) 2801 [INSPIRE].
V. Bernard, N. Kaiser and U.-G. Meißner, On the analysis of the pion-nucleon sigma term: The Size of the remainder at the Cheng-Dashen point, Phys. Lett. B 389 (1996) 144 [hep-ph/9607245] [INSPIRE].
T. Becher and H. Leutwyler, Low energy analysis of πN → πN, JHEP 06 (2001) 017 [hep-ph/0103263] [INSPIRE].
J. Gasser, H. Leutwyler and M. Sainio, Form-factor of the sigma term, Phys. Lett. B 253 (1991) 260 [INSPIRE].
J.F. Donoghue, J. Gasser and H. Leutwyler, The decay of a light Higgs boson, Nucl. Phys. B 343 (1990) 341 [INSPIRE].
G. Höhler, Pion-Nukleon-Streuung: Methoden und Ergebnisse, in Landolt-Börnstein 9b2, H. Schopper eds., Springer Verlag, Berlin Germany (1983).
J. de Swart, M. Rentmeester and R. Timmermans, The Status of the pion-nucleon coupling constant, PiN Newslett. 13 (1997) 96 [nucl-th/9802084] [INSPIRE].
R. Arndt, W. Briscoe, I. Strakovsky and R. Workman, Extended partial-wave analysis of πN scattering data, Phys. Rev. C 74 (2006) 045205[nucl-th/0605082] [INSPIRE]
V. Baru et al., Precision calculation of the π − d scattering length and its impact on threshold πN scattering, Phys. Lett. B 694 (2011) 473 [arXiv:1003.4444] [INSPIRE].
V. Baru et al., 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].
Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [INSPIRE].
K.M. Watson, Some general relations between the photoproduction and scattering of π mesons, Phys. Rev. 95 (1954) 228[INSPIRE]
B. Moussallam, N (f) dependence of the quark condensate from a chiral sum rule, Eur. Phys. J. C 14 (2000) 111 [hep-ph/9909292] [INSPIRE].
R. Omnès, On the solution of certain singular integral equations of quantum field theory, Nuovo Cim. 8 (1958) 316[INSPIRE]
S. Descotes-Genon, Effet des boucles de quarks lègers sur la structure chirale du vide de QCD, Ph.D. Thesis, Université de Paris–Sud, Paris France (2000).
N.I. Muskhelishvili, Singular Integral Equations, Wolters-Noordhoff Publishing, Groningen The Netherlands (1953) [2nd edition, Dover Publications, New York U.S.A. (2008)].
P. Büttiker, S. Descotes-Genon and B. Moussallam, A new analysis of πK scattering from Roy and Steiner type equations, Eur. Phys. J. C 33 (2004) 409 [hep-ph/0310283] [INSPIRE].
M. Hoferichter, D.R. Phillips and C. Schat, Roy-Steiner equations for γγ → ππ, Eur. Phys. J. C 71 (2011) 1743 [arXiv:1106.4147] [INSPIRE].
I. Caprini, G. Colangelo and H. Leutwyler, Regge analysis of the ππ scattering amplitude, Eur. Phys. J. C 72 (2012) 1860 [arXiv:1111.7160] [INSPIRE].
I. Caprini, G. Colangelo and H. Leutwyler, in preparation.
D.H. Cohen et al., Amplitude Analysis of the K − K + System Produced in the Reactions π − p → K − K + n and π + n → K − K + p at 6 GeV/c, Phys. Rev. D 22 (1980) 2595 [INSPIRE].
A. Etkin et al., Amplitude Analysis of the \( K_S^0K_S^0 \) System Produced in the Reaction \( {\pi^{ - }}p \to K_S^0K_S^0n \) at 23 GeV/c, Phys. Rev. D 25 (1982) 1786 [INSPIRE].
A. Anisovich et al., Partial wave analysis of \( \overline p p \) → π − π +, π 0 π0, ηη and ηη′, Nucl. Phys. A 662 (2000) 319 [arXiv:1109.1188] [INSPIRE].
R. Arndt, W. Briscoe, I. Strakovsky and R. Workman, Partial-wave analysis and baryon spectroscopy, Eur. Phys. J. A 35 (2008) 311 [INSPIRE].
SAID, http://gwdac.phys.gwu.edu.
F. Huang, A. Sibirtsev, J. Haidenbauer, S. Krewald and U.-G. Meißner, Backward pion-nucleon scattering, Eur. Phys. J. A 44 (2010) 81 [arXiv:0910.4275] [INSPIRE].
J. Hyslop, R. Arndt, L. Roper and R. Workman, Partial wave analysis of K + -nucleon scattering, Phys. Rev. D 46 (1992) 961[INSPIRE]
H. Hellmann, Einführung in die Quantenchemie, Franz Deuticke, Leipzig Germany (1937).
R. Feynman, Forces in Molecules, Phys. Rev. 56 (1939) 340 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
G. Colangelo et al., Review of lattice results concerning low energy particle physics, Eur. Phys. J. C 71 (2011) 1695 [arXiv:1011.4408] [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
B. Ananthanarayan, I. Caprini, G. Colangelo, J. Gasser and H. Leutwyler, Scalar form-factors of light mesons, Phys. Lett. B 602 (2004) 218 [hep-ph/0409222] [INSPIRE].
J.A. Oller and L. Roca, Scalar radius of the pion and zeros in the form factor, Phys. Lett. B 651 (2007) 139 [arXiv:0704.0039] [INSPIRE].
R. García-Martín, R. Kamiński, J. Peláez, J. Ruiz de Elvira and F. Ynduráin, The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV, Phys. Rev. D 83 (2011) 074004 [arXiv:1102.2183] [INSPIRE].
B. Moussallam, Couplings of light I = 0 scalar mesons to simple operators in the complex plane, Eur. Phys. J. C 71 (2011) 1814 [arXiv:1110.6074] [INSPIRE].
G. Colangelo, J. Gasser and H. Leutwyler, ππ scattering, Nucl. Phys. B 603 (2001) 125[hep-ph/0103088] [INSPIRE]
M. Frink, B. Kubis and U.-G. Meißner, Analysis of the pion-kaon sigma term and related topics, Eur. Phys. J. C 25 (2002) 259 [hep-ph/0203193] [INSPIRE].
J. Bijnens and P. Dhonte, Scalar form-factors in SU(3) chiral perturbation theory, JHEP 10 (2003) 061 [hep-ph/0307044] [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]
H. Pagels and W. Pardee, Nonanalytic behavior of the Σ term in π-N scattering, Phys. Rev. D 4 (1971) 3335 [INSPIRE].
J. Gasser and H. Leutwyler, Quark Masses, Phys. Rept. 87 (1982) 77 [INSPIRE].
J. Gasser, M. Sainio and A. Švarc, Nucleons with Chiral Loops, Nucl. Phys. B 307 (1988) 779 [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].
R.A. Arndt, R.L. Workman, I.I. Strakovsky and M.M. Pavan, πN elastic scattering analyses and dispersion relation constraints, nucl-th/9807087 [INSPIRE].
B. Holzenkamp, K. Holinde and J. Speth, A meson exchange model for the hyperon nucleon interaction, Nucl. Phys. A 500 (1989) 485 [INSPIRE].
W.R. Frazer and J.R. Fulco, Partial-Wave Dispersion Relations for ππ → \( N\overline N \), Phys. Rev. 117 (1960) 1603 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1204.6251
Rights and permissions
About this article
Cite this article
Hoferichter, M., Ditsche, C., Kubis, B. et al. Dispersive analysis of the scalar form factor of the nucleon. J. High Energ. Phys. 2012, 63 (2012). https://doi.org/10.1007/JHEP06(2012)063
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2012)063