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1 A Phenomenological Determination of the Pion-Nucleon Scattering Lengths from Pionic Hydrogen T.E.O. Ericson, B. Loiseau, S. Wycech It requires careful analysis of electromagnetic corrections of precise experimental information from hadronic atoms. 1. Introduction Precise knowledge of strong interaction amplitude at zero energy is important : QCD at low energy : Chiral physics Dispersion relations, e.g. GMO sum rule in πN (πNN coupling constant)

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2 (2) and total decay width : Recent experimental results [1] H. Ch. Schröder et al. Eur. Phys. J C21 (2001) 473 “The pion-nucleon scattering lengths from pionic hydrogen and deuterium” [2] D. Gotta et al. hep-ex/0305012, 38th Rencontres de Moriond “Pionic Hydrogen at PSI” (1) For Pionic Hydrogen (Cf. L. Simons’s talk) strong interaction energy shift :

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3 Bohr energy, = 1/137.036, reduced mass: 1s deviation from lowest order B (r) non-relativistic 1s Bohr wave function of a point charge. (4) (3) [3] S. Deser, M.L. Goldberger, K. Baumann, W. Thirring, Phys. Rev. 96 (1954) 774 “Energy Level Displacements in Pi-Mesonic Atoms”, [4] T.L. Trueman, Nucl. Phys. 26 (1961) 57, “Energy level shifts in atomic states of strongly - interacting particles”. - Well known [3], [4] : a : scattering length (elastic threshold scattering amplitude, defined in absence of Coulomb field).

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4 Some determination of 1s as corrections to isospin symmetric world : Coupled channel potentials (numerical resolution) [5] D. Sigg, A. Badertscher, P.F.A. Goudsmit, H.J. Leisi, G.C. Oades Nucl. Phys. A609 (1996) 310, “Electromagnetic corrections to the S-wave scattering lengths in pionic hydrogen”: 1s = - 2.1 ± 0.5 % QCD + QED effective field theory (EFT) approach : Chiral Perturba-tion Theory (ChPT) : ChPT leading order, [6] V.E. Lyuboviitskij, A. Rustsky, Phys. Lett. B494 (2000) 9, “ - p atom in ChPT: strong energy-level shift” : 1s = - 4.3 ± 2.8 % ChPT next to leading, [7] J. Gasser, M.A. Ivanow, E. Lipartia, M. Mojzis, A. Rusetsky, Eur. Phys. J. C26 (2002) 13, “Ground-state energy of pionic hydrogen to one loop” 1s = - 7.2 ± 2.9 % IMPORTANT to understand 1s with accuracy matching high experimental precision

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5 Isospin symmetry not assumed Non-relativistic quantum problem The π and p charge distributions folded to give the Coulomb potential V c (r) Go from a toy model to realistic case Hadronic amplitude low energy expansion : (5) 2. Model for the - p atom

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6 Hadronic interaction at r = 0 EXACT SOLUTION to 2 log Single channel Charge on a spherical shell of radius R V cR (r) r R - /R - /r constant (6) 2.1 Toy model (7). With 2 = 2mE, E total binding :

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7 (8) First term : extended charge wave function at r = 0 in the absence of strong interaction Better e.m. starting function Second term : renormalization from external wave function changed at R by the hadronic scattering by Very insensitive to R Third term : new effect. Use correct interaction energy (or gauge invariance, cf. [8] T.E.O. Ericson, L. Tausher, Phys. Lett. 112B (1982) 425, “A new effect in pionic atoms”, [9] T.E.O. Ericson, B. Loiseau, A.W. Thones, Phys. Rev. C66 (2002) 014005, “Determination of the pion-nucleon coupling constant and scattering lengths”) Matching the logarithmic derivative of the wave function at R : Matching the logarithmic derivative of the wave function at R :

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8 Any interaction with the same near threshold hadronic amplitude and with hadronic range inside R gives the same answer. Results in agreement with [5] D. Sigg et al. (1996) for 1-channel Vacuum polarization : long range potential, modifies wave function at r=0, model independent, [10] D. Eiras, J. Soto, Phys. Lett. B491 (2000) 101, “Light fermion mass effects in non-relativistic bound states” : VP = 0.48% 2.2 Generalization V(r) r R - /r V c (r) V cR (r) Difference quite small = V c (r) - V cR (r) = perturbation NB. Can also be obtained directly The true charge distribution gives V c (r):

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9 One expresses a Coulomb K-matrix in terms of a hadronic one. Complex Coulomb threshold amplitude : with (9) Hadronic K-matrix low energy expansion (10) 2.3 Coupled channel K-matrix formalism : charged channel c π - p, neutral channel o π 0 n

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10 Continuity of wave function matrix and its logarithmic derivative at R + true charge distribution (11) (12) Panofsky ratio P = 1.546(9) [11] J. Spuller et al., Phys. Lett. B67 (1977) 479, “A remeasurement of the Panofsky ratio”. Matching at R

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11 3. Numerical results Folded (π -, p) charge distribution from observed form factors as in [5] D. Sigg et al. : empirical, [12] G. Höhler, in “πN scattering”, Lamboldt-Börnstein, New Series, Vol 9b (1983). 1s [1] H. Schröder et al., (2001) + two iterations 1s [1] + sign of

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12 [1] H. Ch. Schröder et al. Main source of uncertainty : empirical range parameters Coulomb corrections in %

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13 NN coupling constant NN coupling constant GMO sum rule [9] + corrections just mentioned above : [13] M. Döring, E. Oset, M.J. Vicente Vacas, nucl-th/0402086, to be published PRC,”S-wave pion nucleon scattering lengths from πN, pionic hydrogen and deuteron data” [14] S.R. Beane, V. Bernard, E. Epelbaum, U.G. Meißner, D.R. Phillips, Nucl. Phys. A720 (2003) 399 in perfect agreement with previous determination from analysed with follow [9] T.E.O. Ericson et al. (2002) + triple scattering correction [13], [14] + additional Fermi motion correction from energy dependence S-wave [14]

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14 Previous approaches Analytical, using potential following basically [4] T.L. Trueman (1961) get log term effective range expansion often considered but extended charge not used several authors incorrectly using binding energy and not potential depth negligible 2 correction, e.g. [15] B.R. Holstein, Phys. Rev. D60 (1999) 114030, “Hadronic atoms and effective interactions” Numerical coupled Klein-Gordon equations, e.g. [5] D. Sigg et al. (1996) : Potential for hadronic part starting from isospin symmetric description extended charge finite size + vacuum polarization low energy expansion of π - p rather poor but tuned (π 0, π - ) mass splitting effects - model dependent approach much used by experimental groups - realistic features - relatively small correction

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15 QCD +QED effective field theory + ChPT, [6], [7] QCD +QED effective field theory + ChPT, [6], [7] Structure of results differ from our approach : Their expansion even powers of only Our expansion and appear Key point, form factor effect : * * EFT : additional short range term proportional to * * HERE : extended charge region essential [6] V.E. Lyuboviitskij et al. (2000), [7] J. Gasser et al. (2002) relates A QCD 1s in the order considered DIFFERENT PROBLEM e.m. effects enter both masses and scattering

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16 4. Some conclusions Coulomb potential for the extended charge plays role of an external field : defined in analogy to a - Isospin symmetry not assumed. Finite charge distribution is a crucial feature. High precision needs an accurate low energy expansion : scattering experiments - QCD constraints from EFT - ChPT. 3 quite understood physical effects in relating 1s to a h. Within assumptions, a h obtained at 0.6% precision.

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17 More remarks and outlook * * Assumption that strong interaction range < e.m. charge radius * * Radiative channel, π - p n to be considered * * Non-relativistic description * * Important to connect our description to that of the EFT approach and to clarify the difference Our approach is general and can be easily applied to other atomic systems : to nuclear system, …

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