1. Introduction

1. A Phenomenological Determination of the Pion-Nucleon Scattering Lengths from Pionic HydrogenT.E.O. Ericson, B. Loiseau, S. Wycech • 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)  It requires careful analysis of electromagnetic corrections of precise experimental information from hadronic atoms.

2. Recent experimental results For Pionic Hydrogen (Cf. L. Simons’s talk) strong interaction energy shift : (1) (2) and total decay width : [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”

3. - Well known [3], [4]: (3) • Bohr energy,  = 1/137.036, reduced mass: (4) • 1s deviation from lowest order • B(r) non-relativistic 1s Bohr wave function of a point charge. [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”. • a : scattering length (elastic threshold scattering amplitude, defined in absence of Coulomb field).

4.  IMPORTANT to understand 1s with accuracy matching high experimental precision • 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 %

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

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

7. Matching the logarithmic derivative of the wave function at R : (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”)

8. 2.2 Generalization • The true charge distribution gives Vc(r): V(r) Difference quite small R r = Vc(r) - VcR(r) = perturbation -/r VcR(r) Vc(r) NB. Can also be obtained directly • Any interaction with the same near threshold hadronic amplitude and with hadronic range inside R gives the same answer. • 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%  Results in agreement with [5] D. Sigg et al. (1996) for 1-channel

9. 2.3 Coupled channel • Complex Coulomb threshold amplitude : • with (9) • Hadronic K-matrix low energy expansion (10) • K-matrix formalism : charged channel c π-p, neutral channelo  π0n • One expresses a Coulomb K-matrix in terms of a hadronic one.

10. Matching at R • 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”.

11. 3.Numericalresults • : 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 • Folded (π-, p) charge distribution from observed form factors as in [5] D. Sigg et al.

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

13. NN coupling constant in perfect agreement with previous determination from  GMO sum rule [9]+ corrections just mentioned above : 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] [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

14. Previous approaches • 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 • (π0, π-) mass splitting effects - model dependent • approach much used by experimental groups - realistic features - relatively small correction • 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”

15. QCD +QED effective field theory + ChPT, [6], [7] relates AQCD  1s in the order considered  DIFFERENT PROBLEM • Structure of results differ from our approach : • Their expansion even powers of < rn > only • Our expansion < r > and <1/r > 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) • e.m. effects enter both masses and scattering

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 ah. • Within assumptions, ah obtained at 0.6% precision.

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