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)

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/ , 38th Rencontres de Moriond “Pionic Hydrogen at PSI” (1) For Pionic Hydrogen (Cf. L. Simons’s talk) strong interaction energy shift :

3 Bohr energy,  = 1/ , 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).

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 = ± 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 = ± 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 = ± 2.9 %  IMPORTANT to understand  1s with accuracy matching high experimental precision

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

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 :

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) , “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 :

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):

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

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

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

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

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/ , 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]

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) , “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

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

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.

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, …