Three-body calculation of the 1s level shift in kaonic deuterium P. Doleschall, J. Révai, Wigner FC, Budapest N. V. Shevchenko INP of the ASCR, Řež

Negative hadron Nucleus EXA2014, Sept.15-19, Vienna A.Deloff in „Fundamentals in hadronic atom theory”: „.. the conventional picture of hadronic atoms (is) based on a two-body model Hamiltonian in which all strong interaction effects have been si- mulated by an absorptive potential representing the complicated inter- action between the hadron and the nucleus...” Negative hadron Nucleus EXA2014, Sept.15-19, Vienna

The simplest case to study deviation from two-body picture: 1/14/2018 The simplest case to study deviation from two-body picture: hadronic (kaonic) deuterium: 3-body problem Jacobi coordinates Hamiltonian EXA2014, Sept.15-19, Vienna EXA2014, Sept.15-19, Vienna

Powerful methods to treat 3-body problem: (a) - Faddeev equations (b) - variational methods (w.f. expansion in coordinate space) But: (a) - everlasting problem with the long range Coulomb force (b) - two very different – and relevant - distance scales in hadronic atoms Some years ago Z. Papp proposed a method for simultaneous treatment of short range and Coulomb forces in 3-body problem. Basic idea: transform the Faddeev integral equations into matrix equations using a special discrete and complete set of the Coulomb Sturmian functions as a basis. Successfully applied for short range + repulsive Coulomb forces (nuclear case) and purely Coulomb systems with attraction and repulsion. EXA2014, Sept.15-19, Vienna

Coulomb Sturmian functions - range parameter biorthogonal set

Infinite tridiagonal set of equations for the matrix elements of the Coulomb Green’s function . Can be solved exactly. The same holds for the matrix elements of the free Green’s function . EXA2014, Sept.15-19, Vienna

Faddeev equations in Noble form: the Coulomb interaction appears in the Green’s functions: – mass coefficients The Coulomb interaction is the same in all 3 equatios, only expressed in different coordinates EXA2014, Sept.15-19, Vienna

partition channels Shorter notation: Jacobi coordinates Matrix equation by inserting (approximate) unit operators using a double Sturmian basis: EXA2014, Sept.15-19, Vienna

unknowns EXA2014, Sept.15-19, Vienna

unknowns equations EXA2014, Sept.15-19, Vienna

unknowns equations EXA2014, Sept.15-19, Vienna

unknowns equations overlap matrix EXA2014, Sept.15-19, Vienna

unknowns equations overlap matrix eigenvalue equation EXA2014, Sept.15-19, Vienna

and can be calculated numerically unknowns equations overlap matrix eigenvalue equation and can be calculated numerically EXA2014, Sept.15-19, Vienna

and can be calculated numerically unknowns equations overlap matrix eigenvalue equation and can be calculated numerically The basic quantities of Papp’s method are the matrix elements of the Green’s functions: EXA2014, Sept.15-19, Vienna

For is the Green’s function of two non- interacting subsystems and can be calculated by a convo- lution integral along a suitable contour in the complex energy plane: known calculated from the matrix equation EXA2014, Sept.15-19, Vienna

For an intermediate step is required: - channel Coulomb interaction - polarization potential - channel Green’s function, again of non-interacting subsystems → convolution integral - numerical integration EXA2014, Sept.15-19, Vienna

There is a “dominant” channel Green’s function: After a brief outline of the formalism a few remarks about the actual calculation (I) There is a “dominant” channel Green’s function: which corresponds to a deuteron and a kaon “feeling” a Coulomb force from the c.m. of the deuteron . Its lowest eigenvalue is the reference point, from which the energy shift is measured At all m.e.-s of are singular, the search for is performed in the vicinity of . EXA2014, Sept.15-19, Vienna

the presence of the polarization potential , causes a certain (real) Even in the absence of the strong interaction of the kaon with the nucleons, the presence of the polarization potential , causes a certain (real) shift of the eigenvalue to from . In principle, the strong shift should be measured from instead of . However, the effect is small, in our case (II) The interaction is isospin conserving and acts in and states, therefore in our “particle” representation the Faddeev compo- nent is a column vector: (III) while the interaction is a 2x2 matrix: EXA2014, Sept.15-19, Vienna

channels, and different for the and variables. For a good choice (IV) A rather heavy numerical work with a lot of small but important technical details.The convergence of the method depends on the good choice of the range parameters of the Sturmian functions. Different in different partition channels, and different for the and variables. For a good choice 30-40 functions for one variable gives an accuracy of ~ .5% (~ 1-2 eV). The dimension of the final matrix for 40 functions in each variable is 4800. EXA2014, Sept.15-19, Vienna

Results Test calculations with 4 models of interactions. All are 1-term separable complex potentials with Yamaguchi form factors, acting in and states. They reproduce different characteristics of the low-energy system, usually described by coupled channel interactions: scattering lengths and/or positions. They all give 1s leve lshifts in kaonic hydrogen within or close to the SIDDHARTA value. Their cross sections are shown below. The choice of the potentials is not the main issue of this calculation

The exact (accurate to 1 eV) results are compared with the commonly used approximations. All of them used results of a proper 3-body (Faddeev) calculation of the system without Coulomb interaction. Deser and corrected Deser – from scattering length Opt. potential - low energy amplitudes (scattering length and eff. radius) N.V. Shevchenko, Nucl. Phys.A890-891 (2012) 50-61

1s level shifts ΔE in kaonic deuterium (eV) Γ=2 Im(ΔE) kaonic hydr. shift Deser corrected Opt.potential from Fadd. exact vI -280 + 268 i -723 + 596 i -675 + 351 i -650 + 434 i -641 + 428 i vII -217 + 292 i -732 + 634 i -694 + 370 i -658 + 460 i -646 + 444 i vIII -219 + 293 i -837 + 744 i -795 + 390 i -747 + 517 i 490 i vIV 266 i -854 + 604 i -750 + 310 i -740 + 422 i -736 + 413 i EXA2014, Sept.15-19, Vienna

Conclusions What remains to do first exact calculation of the level shift in kaonic deuterium Simple Deser order of magnitude estimate Corrected Deser acceptable real part, large error in imaginary part Opt. potential good approximation, as assumed when it was proposed, here is the proof What remains to do Extend the method for energy-dependent interactions in order to - incorporate chirally motivated potentials with inherent energy dependence - properly account for the channel via an energy dependent optical potential EXA2014, Sept.15-19, Vienna