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List of Tables
1.1
Ground-state baryons with ordinary, strange or charmed flavour. Here q denotes u or d, and qq or qqq stands for a properly symmetrized or antisymmetrized isospin wave function. Masses are in MeV.
2.1
Eigenvalues of the linear potential computed with an increasing number of points in the discretization procedure described in Section 2.5. Note that the parameter
r
0
is optimized for the
n
= 0 state of any given
l
. Another choice could improve the convergence for radial excitations.
2.2
WKB and exact energies for S-wave energy levels in the potential
V
(
r
) =
r
2.3
WKB, exact and variational energies for the lowest Regge trajectory in the potential
V
(
r
) =
r
4.1
Symmetric and scalar states ([56,0
+
]) for quark masses
m
i
= 1 and linear confinement
V
=
1
2
∑
r
ij
using a harmonic-oscillator expansion up to
N
quanta. The oscillator parameter
K
is adjusted by minimizing either the first or second level with an expansion limited to
N
′
quanta (the minimized energy is underlined). The exact values are very close to:
E
0
,
0
= 3
.
8631,
E
1
,
0
= 5
.
3207, and
E
2
,
0
= 6
.
5953.
4.2
Coefficients of the harmonic-oscillator expansion for the ground state and first excitation with
l
P
= 0
+
. It includes up to
N
= 8 quanta, but the oscillator strength is determined by minimizing the ground state energy when the expansion is truncated at the
N
′
= 4 level. Quark masses are
m
i
= 1, and the interquark potential
1
2
∑
r
ij
.
4.3
Ground state bound by
V
=
1
2
∑
r
ij
, using an harmonic-oscillator expansion up to
N
quanta, with the oscillator parameter adjusted at the
N
′
level. The quark masses are (1, 1, 5) for qqQ, and (1, 1, 0.2) for QQq.
4.4
Comparison of the harmonic oscillator expansion and Gaussian expansion for the two-body linear potential. The exact values given by the first zero of the Airy function is 2.33811...
4.5
Empirical variational calculation of the binding energy of the ground state of the potential
V
=
1
2
∑
r
ij
for the quark masses
m
i
= 1 (qqq),
m
i
= 1
,
1
,
5 (qqQ), and
m
i
= 1
,
1
,
0
.
2 (QQq). The exact values are 3.8631, 3.4379, and 4.9392.
4.6
Short-range correlation coefficients
δ
12
and
δ
13
for the ground state bound by a linear potential
V
=
1
2
∑
r
ij
, using either the h.o. expansion or the empirical Gaussian expansion. We consider the symmetric case
m
i
= 1 and the set of constituent masses (1, 1, 5) and (1, 1, 0.2).
4.7
Some energies and short-range correlations of a baryon with quark masses
m
i
= 1, bound by
V
=
1
2
∑
r
ij
0
.
1
, obtained by expansion into generalized coherent states and elaborate minimization
5.1
Ground-state energy
E
0
and correlation coefficient
δ
(3)
(
r
12
) in the potential
V
=
1
2
∑
r
ij
0
.
1
with unit quark masses, as a function of the maximal grand orbital.
N
is the number of coupled equations.
5.2
Energy and correlation coefficient for the ground state (
n
= 0) and its hyper-radial excitation (
n
= 1) for a linear potential
V
=
1
2
∑
r
ij
and masses
m
i
= 1
5.3
Convergence of the hyperspherical expansion for a power-law potential
V
=
1
2
∑
r
ij
β
with quark masses
m
1
=
m
2
= 1 and
m
3
=
m
′
(for
β
= 0
.
1, the correlation coefficients are multiplied by 10
3
)
6.1
Binding energy and short-range correlation coefficient of the two first
J
P
= 0
+
levels in the linear potential
V
=
1
2
∑
r
ij
, with quark masses
m
i
= 1, obtained by solving the Faddeev equations. The substraction constant is
v
0
= 2 and
a
max
denotes the maximal orbital momentum in the Faddeev amplitude.
7.1
Lowest levels and short-range qq correlations for
qqq
′
in the potential
V
=
1
2
∑
r
ij
0
.
1
, calculated either exactly or with two versions of the Born–Oppenheimer method, the extreme and the variational adiabatic approximations
9.1
Three-body ground state en ergy
E
3
(1;
1
2
r
β
) compared with the naïve lower limit
3
_
2
E
2
(1;
r
β
), the improved lower limit
3
2
E
2
(
3
4
;
r
β
) and a simple variational limit approximation
^
E
3
obtained with a Gaussian wave function
9.2
Numerical comparison of the r.m.s. interquark distances
d
3
, in the baryon with quark masses
m
i
= 1 and potential
1
2
ϵ
(
β
)
∑
r
ij
β
and
d
2
, in the two-body system with masses
m
= 3
∕
4 and potential
ϵ
(
β
)
r
β
.
9.3
Ground-state energy of qqQ compared with the lower limits of Eqs. (9.22) and (9.57) for some power-law potentials
ϵ
(
β
)
r
β
and quark mass ratios
M∕m
. The exact result corresponds to a hyperspherical expansion pushed up to a grand orbital momentum
L
= 8.
11.1
Various contributions (in MeV) to the neutron-to-proton mass difference in two non-relativistic models
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