1. 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

2. 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
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3. 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
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EXA2014, Sept.15-19, Vienna

4. 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.
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5. Coulomb Sturmian functions
- range parameter
biorthogonal set

6. 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
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7. 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
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8. partition channels Shorter notation: Jacobi coordinates
Matrix equation by inserting (approximate) unit operators using
a double Sturmian basis:
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9. unknowns
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10. unknowns
equations
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11. unknowns
equations
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12. unknowns
equations
overlap matrix
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13. unknowns equations overlap matrix eigenvalue equation
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14. and can be calculated numerically
unknowns
equations
overlap matrix
eigenvalue equation
and can be calculated numerically
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15. 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:
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16. 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
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17. For an intermediate step is required:
- channel Coulomb interaction
- polarization potential
- channel Green’s function,
again of non-interacting subsystems → convolution integral
- numerical integration
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18. 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
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19. 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:
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20. 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.
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21. 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

22. 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.A (2012) 50-61

23. 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
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24. 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
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