BLED WORKSHOPS
IN PHYSICS
VOL. 1, NO. 1
Proceedins of the Mini-Workshop
Few-Quark Problems (p. 60)
Bled, Slovenia, July 8-15, 2000
Treatment of three-quark problems in Faddeev
theory?
Zoltan Papp1;2??, A. Krassnigg2, and W. Plessas21Institute of Nuclear Research of the Hungarian Academy of Sciences, H–4001 Debrecen,
Hungary2Institute for Theoretical Physics, University of Graz, A–8010 Graz, Austria
We propose a method that allows for the efficient solution of the three-body Fad-
deev equations in the presence of infinitely rising confinement interactions. Such
a method is useful in calculations of nonrelativistic and especially semirelativis-
tic constituent quark models. The convergence of the partial wave series is ac-
celerated and possible spurious contributions in the Faddeev components are
avoided.
We start from the total Hamiltonian of a nonrelativistic or a semirelativistic
three-quark system, which can be written asH = H0 + v + v + v
; (1)
where H0 is the three-body kinetic-energy operator and vÆ = v
Æ + vhfÆ represents
the mutual quark-quark interactions containing both the confinement (v
Æ) and
hyperfine (vhfÆ ) potentials in the subsystems Æ = ;; 

In the nonrelativistic case we may express the kinetic-energy operator by
four equivalent formsH0 = p22 + q22M = p22 + q22M = p2
2
 + q2
2M
 = 3Xi=1 k2i2mi ; (2)
i.e. either through individual particle momenta ki in the center-of-mass system or
in terms of relative momenta pÆ and qÆ conjugate to the usual Jacobi coordinatesxÆ and yÆ, respectively (Æ = ;; 
). In Eq. (2),mi denotes the individual particle
mass, Æ the reduced mass in the two-body subsystem Æ, and MÆ the reduced
mass of this subsystem with the third particle Æ. In the semirelativistic case the
kinetic-energy operator takes the formH0 = 3Xi=1qk2i +m2i ; (3)
where again ki are the individual particle three-momenta in the frame with total
three-momentum P = P3i=1 ki = 0. We note that a Hamiltonian as in Eq. (1) to-
gether with the relativistic kinetic-energy operator (3) represents an allowedmass? Talk delivered by Zoltan Papp?? E-mail: zpapp@physics.csulb.edu
Treatment of three-quark problems in Faddeev theory 61
operator in the point-form formalism of Poincaré-invariant quantum mechanics,
irrespective of the dynamical origin of the interactions.
Strictly speaking the standard Faddeev scheme applies only for potentials
falling of fast enough at large distances. This makes it necessary to modify the
Faddeev formalism. Otherwise one risks unpleasant properties in the Faddeev
components. In particular, for infinitely rising potentials spurious contributions
are picked up and also the partial-wave series becomes slowly convergent.
One can circumvent these difficulties by performing the Faddeev decompo-
sition in such a way that all the long-range potentials are included in a modified
channel Green’s operator. Specifically, in our case at least the long-range parts of
the confinement interactions in all subsystems ;; and 
 should be included in
the modified channel resolvent. One can attain this goal by adopting a different
splitting of the total Hamiltonian intoH = H
 + ṽ + ṽ + ṽ
; (4)
where H
 = H0 + ṽ
 + ṽ
 + ṽ

 (5)
contains, besides the kinetic energy, the long-range parts ṽ
Æ of the confining inter-
actions v
Æ in all subsystems. The potentials ṽÆ are the residual interactions con-
taining the hyperfine potentials and the short-range parts of the confinement.
Based on Eqs. (4) and (5) we now decompose the total wave function intoj	i = j ̃i + j ̃i+ j ̃
i; (6)
where the modified Faddeev components are defined asj ̃i = G
(E)ṽj	i (7)
with G
(E) = (E -H
)-1: (8)
They fulfill the integral equationsj ̃i = G
(E)ṽ(j ̃i + j ̃
i); (9)
with ;; 
 again a cyclic permutation. The new channel resolvent is given byG
(E) = (E -H
 - ṽ)-1 : (10)
It exhibits just the desired property of including the long-range confining inter-
actions in all subsystems ;; 
. Only the short-range potential ṽ remains in
the modified Faddeev equations (9). Specifically, since now G
 contains also the
long-range parts ṽ
 + ṽ

 of the confinement interactions in channels  and 
,
the dependence of the component j ̃i on the Jacobi coordinate yÆ can never be-
come a free motion. Rather the proper confinement-type asymptotic conditions
are imposed on j ̃i. As a result, spurious contributions are avoided in the in-
dividual Faddeev components, and at the same time the partial-wave expansion
converges much faster.
62 Z. Papp
The splitting of the interactions in Eqs. (4) and (5) has to be done with care.
In general, the interaction parts put into H
 must not produce any bound state.
Otherwise the proper behavior of the Faddeev components j ̃i would again be
spoiled. Suppose the potentials contained in H
 would produce bound states.
Then at the corresponding energies, the resolvent G
(E) would become singular.
Consequently, according to Eq. (7), any large Faddeev component je i could be
generated even if the full solution j	i remains infinitesimally small. Therefore,
besides the true physical solutions of the Hamiltonian H, Eqs. (9) would also pro-
duce spurious solutions associated with the discrete eigenstates of the Hamilto-
nianH
. These spurious solutions would occur for any ṽ, thus having no bearing
for the physical spectrum of H. Of course, when adding up the three individual
Faddeev components these spurious solutions would cancel out. However, they
would cause numerical instabilities in the practical calculations. Therefore they
should be avoided by not allowing H
 to produce any bound states.
In the case of confinement interactions the above requirement cannot strictly
be met, since even the longest-range parts of the infinitely rising potential gener-
ate bound states. However, there is a practical way out: one needs to eliminate the
bound states generated by H
 only in the region of physical interest. Outside that
domain, i.e. reasonably far above the physical spectrum, they do not matter. In
practice, upon splitting the interactions in the Hamiltonian (4) an auxiliary short-
range potential is introduced with no effect on the physically interesting states.
It only serves the purpose of cutting off the confinement interaction at short and
intermediate distances thus avoiding low-lying bound states of H
.
We solve Eqs. (9) along the Coulomb-Sturmian (CS) separable expansion ap-
proach. The further details of the method and the demonstration of its power in
the example of the Goldstone-boson-exchange chiral quark model for baryons is
given in Refs. [1℄ and [2℄.
References
1. Z. Papp, Few-Body Systems 26, 99 (1999)
2. Z. Papp, A. Krassnigg, and W. Plessas, Phys. Rev. C 62, 044004 (2000).