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A confining potential for quarks
595c
Nuclear Physics A497 (1989) 595c-602c North-Holland, Amsterdam
A CONFINING
POTENTIAL
FOR
QUARKS
M. Fabre de la Ripelle Division de Physique Theorique ‘, Institut de Physique NuclBaire, F-91406 Orsay Cedex
We have shown in a previous paper 1 that, with a Schroedinger equation, only a qtj potential in r2i3 can produce hadron masses lying among linear Regge trajectories. Here we intend to analyze in more detail the quality of agreement obtain with this very naive interaction. We assume for the confining qq potential a power law.
v,,= v, + (Y,r,;
, Fij =
4
_ Zj
(1)
function of the distance rij between two quarks q and q (of coordinates Zi and Zj). By using the change of scale z = prij where p = (o,,m/tL2)1/(“+2) the meson masses are given by MgB = V, + m, + mp + ENS(n)
(2)
in terms of the constituants quark masses mp and rns where e is the angular momentum and N the number of nodes of the radial wave UN~(T) eigenfunction of
{Ft[_$ + ----I y 1) + V,,(r) -
Gvt(n)V.JNe(r) =0
(3)
The eigenenergy Ej&)
= (ti2/m)“l(“+2)((Y,)21(n+2)BNL(n)
(4)
is given in terms of the eigenvalues &Neof
Y”NL(z)
=
[e(e+ wz2 + zn-
&N&)]YNL(z)
(5)
which is independent of both the reduced mass m = 2m,ml/(m, ‘Laboratoire
asso&
au C.N.R.S.
0375-9474/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
+ rnq)
(6)
596~
M. Fabre de la Ripelle /A confining potential for quarks
and of the strength (Y, of the potential which operates only as a scale parameter in ENS. The mass spectrum is therefore governed by the eigenvalues sNC(n) independent of the quark masses. According to (2) and (4) the meson masses MNt(n)
=
An
+
&EN&)
(7)
are determined by the parameters A, and B, functions of the two potentials parameters V,,
a,
and the constituant quark masses only, and by the eigenvalues of (5).
The values of n commonly used are n = 0.1 for the Martin potential 2 n = 1 for the Cornell potential 3 and n = 2 for the harmonic oscillator potential used in particular by Isgur and Karl 4. A comparison between the masses provided by (3) for the potential (1) for n = 0.1, 2/3, 1 and 2 is exhibited in table 1 for the meson family. The well known masses of ~(770) and pa(1690) are used as Input to determine the two constants A,, and B, occuring in (7). N
e
P(770)
0
f(m
n = 0.1
n = 213
Exp.
0
770
770
770f3
770
770
0
1
1326
1282
1295f15
1264
1231
0
2
1691
1691
1691f5
1691
1691
(2040)
0
3
1965
2045
2037f25
2079
2151
~~(2350)
0
4
2187
2358
2350* 20
2438
2612
~(1600)
1
0
1540
1587
1590f20
1614
1691
~(2150)
2
0
1988
2192
2100-2200
2305
2612
~~(2250)
1
2
2261
2225f75
+ a2)
p&690) a4
n=l
n=2
Table 1 In the first column the various mesons are identified by their symbol and nominal mass, then the associated Nf? quantum number are exhibited, followed by the theoretical masses
M. Fabre de la Ripelle /A confiningpotenthl
for n = 0.1 and Z/3. The experimental
for quarks
597c
data are given in the next column and finally the
masses for n = 1 and n = 2. For N = 0 and L = 1 the experimental ~~(1320)) which eliminate approximately
average f(ar(1270)
the effect of the spin-orbit
+
force for f? = 1 is
quoted. Everywhere the theoretical and experimental masses are in agreement for n = 2/3, but in disagreement for n # 2/3 (except for Inputs). For solving three quarks systems we assume a proportionality between the quark-antiquark (qq) and the quark-quark
(qq) potentials and we use the hypercentral
approximation, where
only that part of the interaction invariante by rotation in the 6-dimensional space spanned by the two Jacobi coordinates
= ?k and $2 = r7:j where 2 = (Zr+&+&)/3
for equal masses is taken into account. This approximation is very accurate for the smooth potentials used in the quark potential models 56. As it is exact for both rr = 0 and n = 2 in (1) it is not surprising that it be good for oin52. It has been applied in ref.’ to calculate the Baryon masses with a potential r213. A very good agreement between the theoretical
and experimental
masses have been found
for the spin S = 3/2 Baryons which require only the knowledge of the triplet potential. But it is still possible to simplify our equations by using the property that the sum of one body potentials C$,
V(T,)
and the sum of two-body potentials CiCj V(rij)
have the same
energy spectrum 7. This property is obvious for an harmonic oscillator potential because
where r is the hyperradius. Let us separate the qq potential into two terms :
where the sum is over the cyclic (c) permutations
of 1, 2, 3 and let us perform a hyper-
spherical harmonic expansion of (9). The first term contains potential harmonics PzK(CJij) with even K only while for the second term only odd K whereK 2 3 contribute since there
598~
M. Fabre de la Ripelle /A confining potential for quarks
is no symmetrical harmonic for K = 1. When the expansion is truncated to the first hypercentral term then Vqpp(Tij) and i[Vgp(rij) + V,,(rb)] can be substituted approximation.
for C, V(rij)
are equivalent then c [Vqp( &) + I’,,( (I)]
without any difference in the quality of the hypercentral
It is an identify for an harmonic oscillator according to (8).
This substitution
transform the Schroedinger equation into a separable equation (IO)
which can be trivially solved. The space symmetrical solution is the cyclic permutation
where in At(&) = Ytm(wi)UNt(Ei)/&, uNt(Ei) is an eigenfunction of the radial equations
(12) where the radial wave UNLhas N nodes while E = El + Ez. Actually (10) describes a system where both the pair (ij) and the “Spectator”
particle
(k) interact in a similar way. It might be a better simulation of the qq interaction where three quarks can interact together in the gluon field. The lowest energy state for a given angular momentum e is obtained when the “spectator” particle is in S state (nr = er = 0) and the “interacting”
pair is in a e orbital state
(nZ = O,& = e). For a radial excitation with N nodes nz = N. The Baryon mass is given according to (4) by Msq = ~(VB + m,) + (~)““(~~#~(w
+ EOO)
(13)
for the qq potential Vpq(Tij) = VB + O+ri”j/”
where &N( are
eigeIiVaheS
3(Vn + m,) = -581
Of
(5)
for
R. = 2/3 and for equal mass quarks.
(14)
By setting
MeV and
($Y,,)~/'
=
600 MeV
we obtained the mass spectra of the A and N Baryons exhibited in Table 2.
(15)
M. Fabre de la Ripelle f A confining potential for quarks
N
l
h&q
A and N Baryons
0
0
1232
A (1232 -f 2) c’
0
1
1624
N(1650)f-[1620
- 16801, N(1675);-[1660
A(1620);-[1625
& 25],A(1700);-(1630
A(1905);+[1890
- 19201, A(1910)++[1850
A(1920);+[1860
- 2160]A(1950)~+[1910
0
2
1936
599c
Input - 16901 - 17401 - 19501 - 19601
N(1990)$+, N(2000);+ 0
0
3
4
2207
2447
A(2200);-[2200
f 8O]A(2400);-[2100
- 25001
A(2190);-[2120
- 2230]N(2250);-[2130
- 22701
A(2300);-[2400
f 150]A(2420)y+[2380
- 24501
A( 2390) ; + [2300 - 25001 0
5
2666
N(2600)?-[2580
- 2700]A(2750)?-[2700
0
6
2868
A(2950) y+
1
0
1857
A(1920);+[1850
- 2160)
1
1
2127
A(2150)$-[2150
f lOO],N(22700);-(2100
1
2
2373
A(2390);+[2350
f 1001
f 1501
- 22501
Table 2 The Ne in the first column are followed by the theoretical mass Ms9 and in the last column are listed the Baryons corresponding
to the quantum numbers NL
They are
identified by their symbol, nominal mass, total angular momentum J and parity. Their masses are in square brackets.
Nearly everywhere the theoretical masses are within the
error bars of the experimental data. This result is not very surprising when one notices that if the quark potential can be reduced to its hypercentral component, justified and the V,,,(ti)
the separable approximation used in (10) is
potential in (12) must produce Baryon masses lying along linear
Regge trajectories like the V&(rij)
potential does for Mesons.
From the values A, =-816.25 MeV and B,
= 784.5 MeV used in (7) for fitting the p
family and those used in (15) to adjust the A family one finds V, = 0.51 V,, and CX~~/CQ= 0.466 where in (1) we set V,,,
= V,, and or/s = ups. Our numbers could easily be adjusted
600~ to give V,
M. Fabre de la Ripelle /A confining potential for quarks
= 0.5
V,,
in agreement with the Lipkin’s rule, but the ratio cx9,/aqp
for the
strength of the confining part is about 10% to small. Up to now we investigated only the long range part of the quark potential to which the light quarks are mainly sensitive except for the spin-spin and spin-orbite short range force which operates only on f! 5 2 orbitals. In order to complete the potential for heavier quarks we have first to introduce the one gluon exchange Coulomb-like potential which provides the modified potential V&(rij)
=
VO
+
CYpqTfr
-
p*p/rij
(16)
By performing the change of scale I = prij already quoted one finds an energy ENL in (2) still given by (4) for n = 2/3 i.e. by tL* r/4 s/4ENL ENC= (,) (a~)
=
(17)
B(m)ENt
where &N( is an eigenvaiue of +
1)/z2 + x”’
Y”Nl
=
[e(e
am)
=
@y&-;R”)‘/4(a,1)8/4
- C(m)/s
-
ENE]YN~(X) (18)
(29”‘”
where m, is the u-quark mass and m is the reduced mass (6). In order to fix the potential parameters we choose to try to fit the p and T families which are mainly sensitive respectively to the long and short range parts of the potential. We used the p(770), ps(l69O),T(9460)
and T(10355) as Inputs and we checked that C(m)(B(m))b/2,
which is independent of mass, is the same for both families. With this choice we obtained the masses of the p and T mesons exhibited in Table 3. Theor.
Theor.
Exp.
Exp.
770
77ozt3
‘Y-(9460)
9460
9460 f 0.2
f (or + a2)
1293
1295f20
T(10023)
10023
10023.4f
ps(l690)
1691
1691f5
T(10355)
10355.5
a4(2040)
2027
2037125
T(10575)
10613
10577 f 4
PS(2350)
2325
2350f
T (10860)
10838
10865 f 8
~(1600)
1567
1590f20
T(11020)
11070
11019 f 9
~(2150)
2143
2145 zt 45
x&895) 9911
99035 1
10265
10263f2
P(770)
20
Xzb(Q915) ps (2250)
2226
2225f75
10355.5 f0.5
xlb(10255) xZb(10270)
Table 3
0.5
M. Fabre de la Ripelle /A confining potential for quarks
601~
For the P states we used the experimental average of the J = 1 and J = 2 states which cancel the spin-orbite effect. The agreement between theory and experiment is of simular quality for the p family than the one obtained in table 1 without one gluon exchange term but the part of the T spectrum above the BB threshold is irregular an cannot be accurately fitted with the potential (16). The theoretical masses in table 3 have been obtained with the potential parameters V, = -1128,
oBI = 1251.75, &
= 76.85
(19)
where energies are in MeV and length in fermi, and with the u and b quarks masses m, = 277.12 MeV,mb = 5168 MeV. These parameters, slightly different from those published in ref.‘, correspond to another fit of the masses. We used h*/m, = 41.47 MeV-fm’
and mpcZ
= 938 MeV. The masses of C$and K* families obtained with the strange-meson mass rns = 530.5 MeV are exhibited in Table 4. N
e
Theor.
0
0
C$(1020)
0
1
i(fr(1420)
0
2
&(1850)
0
3
1
0
-f;)
4(1680)
Exp.
Theor.
1020
1019.5
K* (892)
1489
1474f 8
f(Kr(1400)
1836
1853f
1719
10
912
892
1411
1416f 6
K,(1780)
1785
1780f 4
K; (2060)
2102
2060f30
+ K;)
1685:;;
Table 4 In order to test the quality of the potential (16) with the parameters (19) for Baryons we used the Lipkin’s rule to generate the V,, potential between quarks by setting arlP= ore/2 and &
= &/2
and we adjusted the constant V, to obtain A(1232).
The Baryon masses obtained by solving (10) with this potential are shown in table 5. N=O,fZ
0
1
2
3
4
5
6
M39
1232
1646
1964
2234
2473
2687
2886
Table 5
602~
M. Fabre de la Ripelle / A confining potential for quarks
The masses are slightly heavier than those of table 2 obtained without the one gluon exchange component but with cxpp/cxqij # l/2, nevertheless in order to fit the A(1232) mass we have been obliged to use a constant V, = 0.53 Vo which does not follow accurately the Lipkin’s rule. In conclusion we have shown that it is possible to find a simple tr&~& quark potential which, used in a Schroedinger equation, generate hadron masses lying along linear Regge trajectories and can be utilised for both light and heavy quark systems.
References 1. M. Fabre de la Ripelle, Phys. Lett. B205 (1988) 97 2. A. Martin, Phys. Lett. B93 (1980) 338 ; BlOO (1981) 511 3. E. Eichten et al., Phys. Rev. D1'7 (1979) 3090
; D21 (1980)203
4. See contribution to this conference 5. J.M. Richard, Phys. Lett. BlOO
(1981) 515 ; B139 (1984) 408 and references therein
6. J.P. MC Tavish, H. Fiedeldey, M. Fabre de la Ripelle and P. van der Merwe, Few-Body Systems 3 (1988) 99 7. M. Fabre de la Ripelle, Prog. Theor. Phys. 40 (1968) 1454
Nuclear Physics A497 (1989) 595c-602c North-Holland, Amsterdam
A CONFINING
POTENTIAL
FOR
QUARKS
M. Fabre de la Ripelle Division de Physique Theorique ‘, Institut de Physique NuclBaire, F-91406 Orsay Cedex
We have shown in a previous paper 1 that, with a Schroedinger equation, only a qtj potential in r2i3 can produce hadron masses lying among linear Regge trajectories. Here we intend to analyze in more detail the quality of agreement obtain with this very naive interaction. We assume for the confining qq potential a power law.
v,,= v, + (Y,r,;
, Fij =
4
_ Zj
(1)
function of the distance rij between two quarks q and q (of coordinates Zi and Zj). By using the change of scale z = prij where p = (o,,m/tL2)1/(“+2) the meson masses are given by MgB = V, + m, + mp + ENS(n)
(2)
in terms of the constituants quark masses mp and rns where e is the angular momentum and N the number of nodes of the radial wave UN~(T) eigenfunction of
{Ft[_$ + ----I y 1) + V,,(r) -
Gvt(n)V.JNe(r) =0
(3)
The eigenenergy Ej&)
= (ti2/m)“l(“+2)((Y,)21(n+2)BNL(n)
(4)
is given in terms of the eigenvalues &Neof
Y”NL(z)
=
[e(e+ wz2 + zn-
&N&)]YNL(z)
(5)
which is independent of both the reduced mass m = 2m,ml/(m, ‘Laboratoire
asso&
au C.N.R.S.
0375-9474/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
+ rnq)
(6)
596~
M. Fabre de la Ripelle /A confining potential for quarks
and of the strength (Y, of the potential which operates only as a scale parameter in ENS. The mass spectrum is therefore governed by the eigenvalues sNC(n) independent of the quark masses. According to (2) and (4) the meson masses MNt(n)
=
An
+
&EN&)
(7)
are determined by the parameters A, and B, functions of the two potentials parameters V,,
a,
and the constituant quark masses only, and by the eigenvalues of (5).
The values of n commonly used are n = 0.1 for the Martin potential 2 n = 1 for the Cornell potential 3 and n = 2 for the harmonic oscillator potential used in particular by Isgur and Karl 4. A comparison between the masses provided by (3) for the potential (1) for n = 0.1, 2/3, 1 and 2 is exhibited in table 1 for the meson family. The well known masses of ~(770) and pa(1690) are used as Input to determine the two constants A,, and B, occuring in (7). N
e
P(770)
0
f(m
n = 0.1
n = 213
Exp.
0
770
770
770f3
770
770
0
1
1326
1282
1295f15
1264
1231
0
2
1691
1691
1691f5
1691
1691
(2040)
0
3
1965
2045
2037f25
2079
2151
~~(2350)
0
4
2187
2358
2350* 20
2438
2612
~(1600)
1
0
1540
1587
1590f20
1614
1691
~(2150)
2
0
1988
2192
2100-2200
2305
2612
~~(2250)
1
2
2261
2225f75
+ a2)
p&690) a4
n=l
n=2
Table 1 In the first column the various mesons are identified by their symbol and nominal mass, then the associated Nf? quantum number are exhibited, followed by the theoretical masses
M. Fabre de la Ripelle /A confiningpotenthl
for n = 0.1 and Z/3. The experimental
for quarks
597c
data are given in the next column and finally the
masses for n = 1 and n = 2. For N = 0 and L = 1 the experimental ~~(1320)) which eliminate approximately
average f(ar(1270)
the effect of the spin-orbit
+
force for f? = 1 is
quoted. Everywhere the theoretical and experimental masses are in agreement for n = 2/3, but in disagreement for n # 2/3 (except for Inputs). For solving three quarks systems we assume a proportionality between the quark-antiquark (qq) and the quark-quark
(qq) potentials and we use the hypercentral
approximation, where
only that part of the interaction invariante by rotation in the 6-dimensional space spanned by the two Jacobi coordinates
= ?k and $2 = r7:j where 2 = (Zr+&+&)/3
for equal masses is taken into account. This approximation is very accurate for the smooth potentials used in the quark potential models 56. As it is exact for both rr = 0 and n = 2 in (1) it is not surprising that it be good for oin52. It has been applied in ref.’ to calculate the Baryon masses with a potential r213. A very good agreement between the theoretical
and experimental
masses have been found
for the spin S = 3/2 Baryons which require only the knowledge of the triplet potential. But it is still possible to simplify our equations by using the property that the sum of one body potentials C$,
V(T,)
and the sum of two-body potentials CiCj V(rij)
have the same
energy spectrum 7. This property is obvious for an harmonic oscillator potential because
where r is the hyperradius. Let us separate the qq potential into two terms :
where the sum is over the cyclic (c) permutations
of 1, 2, 3 and let us perform a hyper-
spherical harmonic expansion of (9). The first term contains potential harmonics PzK(CJij) with even K only while for the second term only odd K whereK 2 3 contribute since there
598~
M. Fabre de la Ripelle /A confining potential for quarks
is no symmetrical harmonic for K = 1. When the expansion is truncated to the first hypercentral term then Vqpp(Tij) and i[Vgp(rij) + V,,(rb)] can be substituted approximation.
for C, V(rij)
are equivalent then c [Vqp( &) + I’,,( (I)]
without any difference in the quality of the hypercentral
It is an identify for an harmonic oscillator according to (8).
This substitution
transform the Schroedinger equation into a separable equation (IO)
which can be trivially solved. The space symmetrical solution is the cyclic permutation
where in At(&) = Ytm(wi)UNt(Ei)/&, uNt(Ei) is an eigenfunction of the radial equations
(12) where the radial wave UNLhas N nodes while E = El + Ez. Actually (10) describes a system where both the pair (ij) and the “Spectator”
particle
(k) interact in a similar way. It might be a better simulation of the qq interaction where three quarks can interact together in the gluon field. The lowest energy state for a given angular momentum e is obtained when the “spectator” particle is in S state (nr = er = 0) and the “interacting”
pair is in a e orbital state
(nZ = O,& = e). For a radial excitation with N nodes nz = N. The Baryon mass is given according to (4) by Msq = ~(VB + m,) + (~)““(~~#~(w
+ EOO)
(13)
for the qq potential Vpq(Tij) = VB + O+ri”j/”
where &N( are
eigeIiVaheS
3(Vn + m,) = -581
Of
(5)
for
R. = 2/3 and for equal mass quarks.
(14)
By setting
MeV and
($Y,,)~/'
=
600 MeV
we obtained the mass spectra of the A and N Baryons exhibited in Table 2.
(15)
M. Fabre de la Ripelle f A confining potential for quarks
N
l
h&q
A and N Baryons
0
0
1232
A (1232 -f 2) c’
0
1
1624
N(1650)f-[1620
- 16801, N(1675);-[1660
A(1620);-[1625
& 25],A(1700);-(1630
A(1905);+[1890
- 19201, A(1910)++[1850
A(1920);+[1860
- 2160]A(1950)~+[1910
0
2
1936
599c
Input - 16901 - 17401 - 19501 - 19601
N(1990)$+, N(2000);+ 0
0
3
4
2207
2447
A(2200);-[2200
f 8O]A(2400);-[2100
- 25001
A(2190);-[2120
- 2230]N(2250);-[2130
- 22701
A(2300);-[2400
f 150]A(2420)y+[2380
- 24501
A( 2390) ; + [2300 - 25001 0
5
2666
N(2600)?-[2580
- 2700]A(2750)?-[2700
0
6
2868
A(2950) y+
1
0
1857
A(1920);+[1850
- 2160)
1
1
2127
A(2150)$-[2150
f lOO],N(22700);-(2100
1
2
2373
A(2390);+[2350
f 1001
f 1501
- 22501
Table 2 The Ne in the first column are followed by the theoretical mass Ms9 and in the last column are listed the Baryons corresponding
to the quantum numbers NL
They are
identified by their symbol, nominal mass, total angular momentum J and parity. Their masses are in square brackets.
Nearly everywhere the theoretical masses are within the
error bars of the experimental data. This result is not very surprising when one notices that if the quark potential can be reduced to its hypercentral component, justified and the V,,,(ti)
the separable approximation used in (10) is
potential in (12) must produce Baryon masses lying along linear
Regge trajectories like the V&(rij)
potential does for Mesons.
From the values A, =-816.25 MeV and B,
= 784.5 MeV used in (7) for fitting the p
family and those used in (15) to adjust the A family one finds V, = 0.51 V,, and CX~~/CQ= 0.466 where in (1) we set V,,,
= V,, and or/s = ups. Our numbers could easily be adjusted
600~ to give V,
M. Fabre de la Ripelle /A confining potential for quarks
= 0.5
V,,
in agreement with the Lipkin’s rule, but the ratio cx9,/aqp
for the
strength of the confining part is about 10% to small. Up to now we investigated only the long range part of the quark potential to which the light quarks are mainly sensitive except for the spin-spin and spin-orbite short range force which operates only on f! 5 2 orbitals. In order to complete the potential for heavier quarks we have first to introduce the one gluon exchange Coulomb-like potential which provides the modified potential V&(rij)
=
VO
+
CYpqTfr
-
p*p/rij
(16)
By performing the change of scale I = prij already quoted one finds an energy ENL in (2) still given by (4) for n = 2/3 i.e. by tL* r/4 s/4ENL ENC= (,) (a~)
=
(17)
B(m)ENt
where &N( is an eigenvaiue of +
1)/z2 + x”’
Y”Nl
=
[e(e
am)
=
@y&-;R”)‘/4(a,1)8/4
- C(m)/s
-
ENE]YN~(X) (18)
(29”‘”
where m, is the u-quark mass and m is the reduced mass (6). In order to fix the potential parameters we choose to try to fit the p and T families which are mainly sensitive respectively to the long and short range parts of the potential. We used the p(770), ps(l69O),T(9460)
and T(10355) as Inputs and we checked that C(m)(B(m))b/2,
which is independent of mass, is the same for both families. With this choice we obtained the masses of the p and T mesons exhibited in Table 3. Theor.
Theor.
Exp.
Exp.
770
77ozt3
‘Y-(9460)
9460
9460 f 0.2
f (or + a2)
1293
1295f20
T(10023)
10023
10023.4f
ps(l690)
1691
1691f5
T(10355)
10355.5
a4(2040)
2027
2037125
T(10575)
10613
10577 f 4
PS(2350)
2325
2350f
T (10860)
10838
10865 f 8
~(1600)
1567
1590f20
T(11020)
11070
11019 f 9
~(2150)
2143
2145 zt 45
x&895) 9911
99035 1
10265
10263f2
P(770)
20
Xzb(Q915) ps (2250)
2226
2225f75
10355.5 f0.5
xlb(10255) xZb(10270)
Table 3
0.5
M. Fabre de la Ripelle /A confining potential for quarks
601~
For the P states we used the experimental average of the J = 1 and J = 2 states which cancel the spin-orbite effect. The agreement between theory and experiment is of simular quality for the p family than the one obtained in table 1 without one gluon exchange term but the part of the T spectrum above the BB threshold is irregular an cannot be accurately fitted with the potential (16). The theoretical masses in table 3 have been obtained with the potential parameters V, = -1128,
oBI = 1251.75, &
= 76.85
(19)
where energies are in MeV and length in fermi, and with the u and b quarks masses m, = 277.12 MeV,mb = 5168 MeV. These parameters, slightly different from those published in ref.‘, correspond to another fit of the masses. We used h*/m, = 41.47 MeV-fm’
and mpcZ
= 938 MeV. The masses of C$and K* families obtained with the strange-meson mass rns = 530.5 MeV are exhibited in Table 4. N
e
Theor.
0
0
C$(1020)
0
1
i(fr(1420)
0
2
&(1850)
0
3
1
0
-f;)
4(1680)
Exp.
Theor.
1020
1019.5
K* (892)
1489
1474f 8
f(Kr(1400)
1836
1853f
1719
10
912
892
1411
1416f 6
K,(1780)
1785
1780f 4
K; (2060)
2102
2060f30
+ K;)
1685:;;
Table 4 In order to test the quality of the potential (16) with the parameters (19) for Baryons we used the Lipkin’s rule to generate the V,, potential between quarks by setting arlP= ore/2 and &
= &/2
and we adjusted the constant V, to obtain A(1232).
The Baryon masses obtained by solving (10) with this potential are shown in table 5. N=O,fZ
0
1
2
3
4
5
6
M39
1232
1646
1964
2234
2473
2687
2886
Table 5
602~
M. Fabre de la Ripelle / A confining potential for quarks
The masses are slightly heavier than those of table 2 obtained without the one gluon exchange component but with cxpp/cxqij # l/2, nevertheless in order to fit the A(1232) mass we have been obliged to use a constant V, = 0.53 Vo which does not follow accurately the Lipkin’s rule. In conclusion we have shown that it is possible to find a simple tr&~& quark potential which, used in a Schroedinger equation, generate hadron masses lying along linear Regge trajectories and can be utilised for both light and heavy quark systems.
References 1. M. Fabre de la Ripelle, Phys. Lett. B205 (1988) 97 2. A. Martin, Phys. Lett. B93 (1980) 338 ; BlOO (1981) 511 3. E. Eichten et al., Phys. Rev. D1'7 (1979) 3090
; D21 (1980)203
4. See contribution to this conference 5. J.M. Richard, Phys. Lett. BlOO
(1981) 515 ; B139 (1984) 408 and references therein
6. J.P. MC Tavish, H. Fiedeldey, M. Fabre de la Ripelle and P. van der Merwe, Few-Body Systems 3 (1988) 99 7. M. Fabre de la Ripelle, Prog. Theor. Phys. 40 (1968) 1454