Abstract
Starting from hyperbolic dispersion relations, we derive a closed system of Roy-Steiner equations for pion-nucleon scattering that respects analyticity, unitarity, and crossing symmetry. We work out analytically all kernel functions and unitarity relations required for the lowest partial waves. In order to suppress the dependence on the high energy regime we also consider once- and twice-subtracted versions of the equations, where we identify the subtraction constants with subthreshold parameters. Assuming Mandelstam analyticity we determine the maximal range of validity of these equations. As a first step towards the solution of the full system we cast the equations for the \(\pi \pi \to \overline N N\) partial waves into the form of a Muskhelishvili-Omnès problem with finite matching point, which we solve numerically in the single-channel approximation. We investigate in detail the role of individual contributions to our solutions and discuss some consequences for the spectral functions of the nucleon electromagnetic form factors.
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D. Gotta et al., Pionic hydrogen, in Precision Physics of Simple Atoms and Molecules, Lect. Notes Phys. 745 (2008) 165.
T. Strauch et al., Pionic deuterium, Eur. Phys. J. A 47 (2011) 88 [arXiv:1011.2415] [INSPIRE].
V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga and D.R. Phillips, 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, 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].
J. Gasser, Hadron Masses and Sigma Commutator in the Light of Chiral Perturbation Theory, Annals Phys. 136 (1981) 62 [INSPIRE].
A. Bottino, F. Donato, N. Fornengo and S. Scopel, Size of the neutralino-nucleon cross-section in the light of a new determination of the pion-nucleon sigma term, Astropart. Phys. 18 (2002) 205 [hep-ph/0111229] [INSPIRE].
J.R. Ellis, K.A. Olive and C. Savage, Hadronic Uncertainties in the Elastic Scattering of Supersymmetric Dark Matter, Phys. Rev. D 77 (2008) 065026 [arXiv:0801.3656] [INSPIRE].
K.A. Olive, The impact of XENON100 and the LHC on Supersymmetric Dark Matter, arXiv:1202.2324 [INSPIRE].
A. Walker-Loud, Evidence for non-analytic light quark mass dependence in the baryon spectrum, arXiv:1112.2658 [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].
S. Roy, Exact integral equation for pion-pion scattering involving only physical region partial waves, Phys. Lett. B 36 (1971) 353 [INSPIRE].
B. Ananthanarayan, G. Colangelo, J. Gasser and H. Leutwyler, Roy equation analysis of ππ scattering, Phys. Rept. 353 (2001) 207 [hep-ph/0005297] [INSPIRE].
S. Descotes-Genon, N. Fuchs, L. Girlanda and J. Stern, Analysis and interpretation of new low-energy ππ scattering data, Eur. Phys. J. C 24 (2002) 469 [hep-ph/0112088] [INSPIRE].
R. García-Martín, R. Kaminski, 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].
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.
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].
B. Ananthanarayan and P. Büttiker, Comparison of pion-kaon scattering in SU(3) chiral perturbation theory and dispersion relations, Eur. Phys. J. C 19 (2001) 517 [hep-ph/0012023] [INSPIRE].
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].
T. Becher and H. Leutwyler, Low energy analysis of πN → πN, JHEP 06 (2001) 017 [hep-ph/0103263] [INSPIRE].
G. Hite and F. Steiner, New dispersion relations and their application to partial-wave amplitudes, Nuovo Cim. A 18 (1973) 237 [INSPIRE].
R. Koch, A New Determination of the πN Sigma Term Using Hyperbolic Dispersion Relations in the (ν 2 , t) Plane, Z. Phys. C 15 (1982) 161 [INSPIRE].
G. Höhler, Determinations of the πN Sigma term, PiN Newslett. 15 (1999) 123.
J. Stahov, The subthreshold expansion of the πN invariant amplitudes in dispersion theory, PiN Newslett. 15 (1999) 13.
J. Stahov, Calculation of πN partial waves from hyperbolic dispersion relations, PiN Newslett. 16 (2002) 116.
N.I. Muskhelishvili, Singular Integral Equations, Wolters-Noordhoff Publishing, Groningen (1953) [Dover Publications, 2nd edition (2008)].
R. Omnès, On the Solution of certain singular integral equations of quantum field theory, Nuovo Cim. 8 (1958) 316 [INSPIRE].
R. Koch and E. Pietarinen, Low-Energy πN Partial Wave Analysis, Nucl. Phys. A 336 (1980) 331 [INSPIRE].
G. Höhler, Pion-Nukleon-Streuung: Methoden und Ergebnisse, in Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology — New Series / Elementary Particles, Nuclei and Atoms 9b2, H. Schopper ed., Springer Verlag, Berlin (1983).
J. Stahov, Determination of πN low-energy parameters from forward dispersion relations, PiN Newslett. 13 (1997) 174.
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].
J. Gasser and G. Wanders, One-channel Roy equations revisited, Eur. Phys. J. C 10 (1999) 159 [hep-ph/9903443] [INSPIRE].
G. Wanders, The Role of the input in Roy’s equations for ππ scattering, Eur. Phys. J. C 17 (2000) 323 [hep-ph/0005042] [INSPIRE].
Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [INSPIRE].
T.W.B. Kibble, Kinematics of General Scattering Processes and the Mandelstam Representation, Phys. Rev. 117 (1960) 1159 [INSPIRE].
M. Döring, C. Hanhart, F. Huang, S. Krewald and U.-G. Meißner, Analytic properties of the scattering amplitude and resonances parameters in a meson exchange model, Nucl. Phys. A 829 (2009)170 [arXiv:0903.4337] [INSPIRE].
W.B. Kaufmann and G.E. Hite, Tests of current algebra and partially conserved axial-vector current in the subthreshold region of the pion-nucleon system, Phys. Rev. C 60 (1999) 055204 [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].
G.E. Hite, W.B. Kaufmann and R.J. Jacob, New evaluation of the πN Sigma term, Phys. Rev. C 71 (2005) 065201 [INSPIRE].
D. Bugg, A. Carter and J. Carter, New values of pion-nucleon scattering lengths and F 2, Phys. Lett. B 44 (1973) 278 [INSPIRE].
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].
W.R. Frazer and J.R. Fulco, Partial-Wave Dispersion Relations for Pion-Nucleon Scattering, Phys. Rev. 119 (1960) 1420 [INSPIRE].
S.W. MacDowell, Analytic Properties of Partial Amplitudes in Meson-Nucleon Scattering, Phys. Rev. 116 (1959) 774 [INSPIRE].
J. Baacke and F. Steiner, πN partial wave relations from fixed-t dispersion relations, Fortsch. Phys. 18 (1970) 67 [INSPIRE].
F. Steiner, On the generalized πN potential — a new representation from fixed-t dispersion relations, Fortsch. Phys. 18 (1970) 43 [INSPIRE].
F. Steiner, Partial wave crossing relations for meson-baryon scattering, Fortsch. Phys. 19 (1971) 115 [INSPIRE].
W.R. Frazer and J.R. Fulco, Partial-Wave Dispersion Relations for \(\pi \pi \to N\overline N\), Phys. Rev. 117 (1960) 1603 [INSPIRE].
M. Jacob and G.C. Wick, On the general theory of collisions for particles with spin, Annals Phys. 7 (1959) 404 [Annals Phys. 281 (2000) 774] [INSPIRE].
D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, Quantum Theory of Angular Momentum, World-Scientific Publishing, Singapore (1988).
K.M. Watson, Some general relations between the photoproduction and scattering of π mesons, Phys. Rev. 95 (1954) 228 [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].
M.J. Musolf, H.-W. Hammer and D. Drechsel, Nucleon strangeness and unitarity, Phys. Rev. D 55 (1997) 2741 [Erratum ibid. D 62 (2000) 079901] [hep-ph/9610402] [INSPIRE].
W.R. Frazer and J.R. Fulco, Effect of a Pion-Pion Scattering Resonance on Nucleon Structure. II, Phys. Rev. 117 (1960) 1609 [INSPIRE].
E. Pietarinen, A calculation of \(\pi \pi \to N\overline N\) amplitudes in the pseudophysical region, Preprint Series in Theoretical Physics HU-TFT-17-77, Helsinki University, unpublished.
M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].
A. Martin, Unitarity and high-energy behavior of scattering amplitudes, Phys. Rev. 129 (1963) 1432 [INSPIRE].
P. Büttiker and U.-G. Meißner, Pion-nucleon scattering inside the Mandelstam triangle, Nucl. Phys. A 668 (2000) 97 [hep-ph/9908247] [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].
J.L. Basdevant, J.C. Le Guillou and H. Navelet, Crossing and physical partial-wave amplitudes, Nuovo Cim. A 7 (1972) 363 [INSPIRE].
A. Schenk, Absorption and dispersion of pions at finite temperature, Nucl. Phys. B 363 (1991) 97 [INSPIRE].
C.D. Froggatt and J.L. Petersen, Phase-shift analysis of π+ π− scattering between 1.0 GeV and 1.8 GeV based on fixed momentum transfer analyticity. 2., Nucl. Phys. B 129 (1977) 89 [INSPIRE].
E. Pietarinen, Dispersion relations and experimental data, Nuovo Cim. A 12 (1972) 522 [INSPIRE].
R. Koch, Improved πN Partial Waves, Consistent With Analyticity And Unitarity, Z. Phys. C 29 (1985) 597 [INSPIRE].
R. Koch, A Calculation of Low-Energy πN Partial Waves Based on Fixed-t Analyticity, Nucl. Phys. A 448 (1986) 707 [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].
R.A. Arndt, W.J. Briscoe, I.I. Strakovsky and R.L. Workman, Partial-wave analysis and baryon spectroscopy, Eur. Phys. J. A 35 (2008) 311 [INSPIRE].
SAID, http://gwdac.phys.gwu.edu.
A. Anisovich et al., Partial-wave analysis of \(\overline p p \to {\pi^{-} }{\pi^{+} },{\pi^0}{\pi^0},\eta \eta\) and ηη′, Nucl. Phys. A 662 (2000)319 [arXiv:1109.1188] [INSPIRE].
M.E. Sainio, Analyticity constrained pion-nucleon analysis, PoS(CD09)013.
P. Metsä, Forward analysis of πN scattering with an expansion method, Eur. Phys. J. A 33 (2007) 349 [arXiv:0705.4528] [INSPIRE].
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].
T.N. Pham and T.N. Truong, Muskhelishvili-Omnès Integral Equation with Inelastic Unitarity: Single- and Coupled-Channel Equations, Phys. Rev. D 16 (1977) 896 [INSPIRE].
I. Caprini, Omnès representations with inelastic effects for hadronic form factors, Rom. J. Phys. 50 (2005) 7.
S.M. Flatté, Coupled-Channel Analysis of the πη and \(K\overline K\) Systems Near \(K\overline K\) Threshold, Phys. Lett. B 63 (1976) 224 [INSPIRE].
R. García-Martín, R. Kaminski, J.R. Peláez and J. Ruiz de Elvira, Precise determination of the f 0(600) and f 0(980) pole parameters from a dispersive data analysis, Phys. Rev. Lett. 107 (2011) 072001 [arXiv:1107.1635] [INSPIRE].
M.M. Nagels, T.A. Rijken and J.J. de Swart, A Low-Energy Nucleon-Nucleon Potential from Regge Pole Theory, Phys. Rev. D 17 (1978) 768 [INSPIRE].
P.M.M. Maessen, T.A. Rijken and J.J. de Swart, Soft Core Baryon Baryon One Boson Exchange Models. 2. Hyperon-Nucleon Potential, Phys. Rev. C 40 (1989) 2226 [INSPIRE].
V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen and J.J. de Swart, Construction of high quality N N potential models, Phys. Rev. C 49 (1994) 2950 [nucl-th/9406039] [INSPIRE].
T.A. Rijken, H. Polinder and J. Nagata, Extended-soft-core NN potentials in momentum space. 2. Meson-pair exchange potentials, Phys. Rev. C 66 (2002) 044009 [nucl-th/0201020] [INSPIRE].
M. Hoferichter, C. Ditsche, B. Kubis and U.-G. Meißner, Dispersive analysis of the scalar form factor of the nucleon, arXiv:12046251, accepted for publication in JHEP.
G.C. Oades, Finite contour dispersion relations and the subthreshold expansion coefficients of the πN invariant amplitudes, PiN Newslett. 15 (1999) 307.
B.R. Martin and G.C. Oades, Threshold and subthreshold πN scattering amplitudes: Comparison with chiral perturbation theory predictions, PiN Newslett. 16 (2002) 133.
N. Fettes, Pion-nucleon physics in Chiral Perturbation Theory, Thesis, University of Bonn (2000).
M.M. Pavan, R.A. Arndt, I.I. Strakovsky and R.L. Workman, Determination of the πNN coupling constant in the VPI/GW πNN partial wave and dispersion relation analysis, PiN Newslett. 15 (1999) 171 [Phys. Scripta 87 (2000) 65 ] [nucl-th/9910040] [INSPIRE].
G. Höhler, Some results on πN phenomenology, PiN Newslett. 15 (1999) 7.
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].
G. Höhler and E. Pietarinen, Electromagnetic Radii of Nucleon and Pion, Phys. Lett. B 53 (1975) 471 [INSPIRE].
M.A. Belushkin, H.-W. Hammer and U.-G. Meißner, Dispersion analysis of the nucleon form-factors including meson continua, Phys. Rev. C 75 (2007) 035202 [hep-ph/0608337] [INSPIRE].
J. Gasser, H. Leutwyler and M.E. 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. Colangelo, Hadronic contributions to a μ below one GeV, Nucl. Phys. Proc. Suppl. 131 (2004) 185 [hep-ph/0312017] [INSPIRE].
F.-K. Guo, C. Hanhart, F.J. Llanes-Estrada and U.-G. Meißner, Quark mass dependence of the pion vector form factor, Phys. Lett. B 678 (2009) 90 [arXiv:0812.3270] [INSPIRE].
Bateman Manuscript Project, Higher Transcendental Functions 1, A. Erdélyi ed., McGraw-Hill, New York (1953).
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].
S. Descotes-Genon and B. Moussallam, The \(K_0^*\) (800) scalar resonance from Roy-Steiner representations of πK scattering, Eur. Phys. J. C 48 (2006) 553 [hep-ph/0607133] [INSPIRE].
S. Mandelstam, Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. General theory, Phys. Rev. 112 (1958) 1344 [INSPIRE].
S. Mandelstam, Analytic properties of transition amplitudes in perturbation theory, Phys. Rev. 115 (1959) 1741 [INSPIRE].
S. Mandelstam, Construction of the perturbation series for transition amplitudes from their analyticity and unitarity properties, Phys. Rev. 115 (1959) 1752.
A. Martin, Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity - I., Nuovo Cim. A 42 (1965) 930 [INSPIRE].
A. Martin, Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity - II., Nuovo Cim. A 44 (1966) 1219 .
S.W. MacDowell, Analytic continuation of reduced pion-nucleon partial-wave amplitudes, Phys. Rev. D 6 (1972) 3512 [INSPIRE].
F.F.K. Cheung and F.S. Chen-Cheung, Uniqueness of amplitudes satisfying the Mandelstam representation, Phys. Rev. D 5 (1972) 970 [INSPIRE].
H. Lehmann, Analytic properties of scattering amplitudes as functions of momentum transfer, Nuovo Cim. 10 (1958) 579 .
J. Stahov, Dispersion relations on hyperbolas and higher pion-nucleon partial waves (in Croatian), Thesis, University of Zagreb (1983).
T. Regge, Introduction to complex orbital momenta, Nuovo Cim. 14 (1959) 951 [INSPIRE].
P.D.B. Collins, An introduction to Regge theory and high energy physics, Cambridge University Press, Cambridge (1977).
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Ditsche, C., Hoferichter, M., Kubis, B. et al. Roy-Steiner equations for pion-nucleon scattering. J. High Energ. Phys. 2012, 43 (2012). https://doi.org/10.1007/JHEP06(2012)043
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DOI: https://doi.org/10.1007/JHEP06(2012)043