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Extension of the logarithmic potential to multiquark systems: Dependence of the energy on depth and range
Volume 83B, number 1
PHYSICS LETTERS
23 April 1979
EXTENSION OF THE LOGARITHMIC POTENTIAL TO MULTIQUARK SYSTEMS:
DEPENDENCE OF THE ENERGY ON DEPTH AND RANGE ~ Richard L. H A L L and C.S. KALMAN Concordia University- Universit~ du Quebec, Elementary Particle Physics Group, Montreal, P.Q. Canada H3G 1M8 Received 2 January 1979 Revised manuscript received 26 February 1979
Quigg and Rosner have recently examined the problem of constructing a quark model consistent with the experimental finding that M(T ') - M(T) -~.M(~ ') - M(xp). They have shown that an energy spectrum with level spacing independent of the mass of the consituent quarks occurs if the potential has a logarithmic shape. Lipkin in a recent letter has attempted to extend the Quigg-Rosner model to multiquark systems by assuming a quasinuclear model with a two-body logarithmic interaction and no additional forces (or bag). In this note we examine some of the features of this two-body interaction and their generalization within the type of quasinuclear model suggested by Lipkin.
Unfortunately, exact solutions to the SchrSdinger equation with the logarithmic potential are presently unavailable, although Bose and Mi:dler-Kirsten [3] have found (high-energy) asymptotic expansions for the wave functions and eigenvalues. In the present article we determine the exact f o r m o f the two-body eigenvalue as functions o f the potential range a and depth V0, and we discuss the extension o f this to the case o f N identical bosons. The SchrSdinger equation for the two-body problem with two-parameter pair potentials is given b y {--(PtZ/m)A r -- Vof(r/a ) }$i(r) = Ei$i(r) ,
(1)
where f i s the shape o f the pair potential, m is twice the reduced mass, and r = Irl. We define the following dimensionless parameters: v
= ml Vola2/h2 ,
e i = --mEia2/fi 2 ,
(2)
and a new variable x = r/a so that eq. (1) becomes (A x + of(x))Oi(x ) = eic~i(x ) ,
(3)
where we use the + if V0 > 0 and - if V0 < 0. For example, in the case o f the hydrogen atom V0 > 0, f ( x ) = ¢r Research supported in part by the National Science and Engineering Research Council of Canada. 80
x - 1 , and we have e n = v2/4n 2, where n = 1 , 2 , 3 ..... For each potential shape f w e get a collection of functions gi defined by ei = g i ( v ) ,
(4)
where i = 1,2, 3, ..., and ei+ 1 <~ e i. The graph L o f g i for the logarithmic potential is shown in fig. 1 and corresponding graphs for some other potentials may be found in refs. [4] and [5]. It appears to be a difficult problem in general to find the gi for a given f; however for the logarithmic potential with V0 < 0 and f ( x ) = In x we prove the following exact result: e i = gi(o) = ~ oln(o/vi) ,
(5)
where the constants oi > 0 are the unique non-trivial zeros of the functions gi. P r o o f o f e q . (5). We replace m in eq. ( 1 ) b y ~,m and consider E i = Ei(X ) for ~ close to 1. With f ( x ) = In x and V0 <: 0 it follows that Ei(~ ) = Ei(1 ) + V0 In ~k1/2 ,
(6)
and consequently we have from eqs. (2) and (4): ~.-lgi(~.O ) = gi(o) + ½ oln X.
(7)
We now differentiate eq. (7) once with respect to o and once with respect to X and obtain
Volume 83B, number 1
PHYSICS LETTERS
g}'(Xv) = (2Xv) -1 ;
(8)
solving eq. (8) we find &(x) = ½(xln x - x ) + A i x + B i ,
(9)
where A i and B i are constants of integration. By substituting eq. (9) in eq. (7) we find B i = 0 and this establishes eq. (5) in which in v i = (1 - 2Ai); this representation for A i is possible since the oi increase with i and, as we shall see, v 1 ~ 7.6. If we use the simple gaussian trial function Co(x) = e x p ( - x 2 /4o2} ,
(10)
then we have from eq. (3) e 1 > (4~a(x), {2x- vlnx}Oo(x))/[I¢oll 2 .
(11)
By maximizing the expression in (11) with respect to o for each v we find v = 3/2o 2 and it follows that el = gl (v) > ½vln(v/v~U)) ,
(l 2)
v 1 < v ~ u) = ~ e x p ( 3 -
(13)
3,} ~ 8.458,
5
23 April 1979
and "), "- 0.57721 is Euler's constant. Quigg and Rosner [1 ] obtain v 1 ~ 7.63 by numerical integration (the values of E for integer l on their fig. 1 are approximate values for the quantity ½in vi). For higher Sstates the WKB values (ref. [1 ], note 5) on ~ 7r(2n - ½)2 ,
n = 1 ,2, 3,... ,
are within 3% of the corresponding values obtained by numerical integration [1]. The truncated asymptotic expansion which Bose and Miiller-Kirsten [3] (eq. (78)) find for the eigenvalues is consistent with the general form (5) but implies values of v i given by the following formula (q is an odd integer): Oi ~ Oql = e F 2 ( q , I)/4 ,
where F ( q , l ) = 2q + {G +
~
0
Fig. 1. Solution curves for the ground-state energy of the Nboson problem with the logarithmic potential u(rij ) = IVol In (ri'/a)" U, u per b o u n d ' L, lower b o u n d (and exact curve for N / --/ 2)," . v- ~- ~N(m/h 2 IVola ' 2 , e = [ - E / ( N - 1)1 ( m / h2) a 2 .
~ 1/2
q--(35qZ-z1) ~ 108 [2q + G1/2]]
(15) '
and G = 8(l + ~_)2 + q - l ( 6 9 q 2 + 1). Thus from eq. (15) we obtain v 1 ~ Vl0 = 17.93 which by the inequality (13) is too large by a factor greater than 2. It is interesting that we can also discuss the N-body problem with the aid of the above results presented in the form e = g(v). As Lipkin [2] has done, let us consider states with "stretched" angular momenta, namely those with the highest values of spin and J for a given configuration. These are the 1 - , 2 + and 3 - mesons and the 3/2 +, 5 / 2 - and 7/2 + baryons and all have wave functions which are totally symmetric in the spin parameters. Since all such particles are antisymmetric in their color indices, they will have totally symmetric spatial wave functions. Let us then consider Nquarks, each with mass m, interacting by the pair potential V(ri/) = IV01 ln(riJa ) and we define the following parameters (E is the exact energy in the centre-ofmass frame): v = ½N(m/~2)lVola2 ,
0 -2
(14)
e = [-El(N-
1)](m/tt2)a 2 . (16)
These definitions are consistent with eq. (2) which corresponds to the special case N = 2. The theorem of ref. [4] implies the following inequalities for the exact energy of the lowest spatially symmetric N-particle state: ½vln(v/8.458) < e < l v l n ( V / V l ) ,
(17) 81
Volume 83B, number 1
PHYSICS LETTERS
i°e°
bounds on the energy for the higher states of a system of N quarks may then be found using the theorem of Hall [6].
- ~ N ( N - 1)l V0 Iln(v/v 1) < E < - ~ N ( N - 1)1V0 Iln(v/8.458).
23 April 1979
(18) References
The inequalities (17) are shown in fig. 1 in which we have used the value o 1 = 7.63 given by Quigg and Rosner [1]. For the higher states of the N-body system upper bounds can be obtained by the usual Rayleigh-Ritz method. To find lower bounds we require the exact two-body curves e i = gi(o) established for the logarithmic potential by eq. (5) together with the oi calculated by Quigg and Rosner [1]: lower
82
[1] C. Quigg and J.L. Rosner, Phys. Lett. 71B (1977) 153. [2] H.J. Lipkin, Phys. Lett. 74B (1978) 399. [3] S.K. Bose and H.J.W. Miiller-Kirsten, Univ. of Kaiserslautern (Germany) preprint (1978). [4] R.L. Hall and H.R. Post, Proc. Phys. Soc. 90 (1967) 381. [5] R.L. Hall, Proc. Phys. Soc. 91 (1967) 787. [6] R.L. Hall, Phys. Lett. 30B (1969) 320.
PHYSICS LETTERS
23 April 1979
EXTENSION OF THE LOGARITHMIC POTENTIAL TO MULTIQUARK SYSTEMS:
DEPENDENCE OF THE ENERGY ON DEPTH AND RANGE ~ Richard L. H A L L and C.S. KALMAN Concordia University- Universit~ du Quebec, Elementary Particle Physics Group, Montreal, P.Q. Canada H3G 1M8 Received 2 January 1979 Revised manuscript received 26 February 1979
Quigg and Rosner have recently examined the problem of constructing a quark model consistent with the experimental finding that M(T ') - M(T) -~.M(~ ') - M(xp). They have shown that an energy spectrum with level spacing independent of the mass of the consituent quarks occurs if the potential has a logarithmic shape. Lipkin in a recent letter has attempted to extend the Quigg-Rosner model to multiquark systems by assuming a quasinuclear model with a two-body logarithmic interaction and no additional forces (or bag). In this note we examine some of the features of this two-body interaction and their generalization within the type of quasinuclear model suggested by Lipkin.
Unfortunately, exact solutions to the SchrSdinger equation with the logarithmic potential are presently unavailable, although Bose and Mi:dler-Kirsten [3] have found (high-energy) asymptotic expansions for the wave functions and eigenvalues. In the present article we determine the exact f o r m o f the two-body eigenvalue as functions o f the potential range a and depth V0, and we discuss the extension o f this to the case o f N identical bosons. The SchrSdinger equation for the two-body problem with two-parameter pair potentials is given b y {--(PtZ/m)A r -- Vof(r/a ) }$i(r) = Ei$i(r) ,
(1)
where f i s the shape o f the pair potential, m is twice the reduced mass, and r = Irl. We define the following dimensionless parameters: v
= ml Vola2/h2 ,
e i = --mEia2/fi 2 ,
(2)
and a new variable x = r/a so that eq. (1) becomes (A x + of(x))Oi(x ) = eic~i(x ) ,
(3)
where we use the + if V0 > 0 and - if V0 < 0. For example, in the case o f the hydrogen atom V0 > 0, f ( x ) = ¢r Research supported in part by the National Science and Engineering Research Council of Canada. 80
x - 1 , and we have e n = v2/4n 2, where n = 1 , 2 , 3 ..... For each potential shape f w e get a collection of functions gi defined by ei = g i ( v ) ,
(4)
where i = 1,2, 3, ..., and ei+ 1 <~ e i. The graph L o f g i for the logarithmic potential is shown in fig. 1 and corresponding graphs for some other potentials may be found in refs. [4] and [5]. It appears to be a difficult problem in general to find the gi for a given f; however for the logarithmic potential with V0 < 0 and f ( x ) = In x we prove the following exact result: e i = gi(o) = ~ oln(o/vi) ,
(5)
where the constants oi > 0 are the unique non-trivial zeros of the functions gi. P r o o f o f e q . (5). We replace m in eq. ( 1 ) b y ~,m and consider E i = Ei(X ) for ~ close to 1. With f ( x ) = In x and V0 <: 0 it follows that Ei(~ ) = Ei(1 ) + V0 In ~k1/2 ,
(6)
and consequently we have from eqs. (2) and (4): ~.-lgi(~.O ) = gi(o) + ½ oln X.
(7)
We now differentiate eq. (7) once with respect to o and once with respect to X and obtain
Volume 83B, number 1
PHYSICS LETTERS
g}'(Xv) = (2Xv) -1 ;
(8)
solving eq. (8) we find &(x) = ½(xln x - x ) + A i x + B i ,
(9)
where A i and B i are constants of integration. By substituting eq. (9) in eq. (7) we find B i = 0 and this establishes eq. (5) in which in v i = (1 - 2Ai); this representation for A i is possible since the oi increase with i and, as we shall see, v 1 ~ 7.6. If we use the simple gaussian trial function Co(x) = e x p ( - x 2 /4o2} ,
(10)
then we have from eq. (3) e 1 > (4~a(x), {2x- vlnx}Oo(x))/[I¢oll 2 .
(11)
By maximizing the expression in (11) with respect to o for each v we find v = 3/2o 2 and it follows that el = gl (v) > ½vln(v/v~U)) ,
(l 2)
v 1 < v ~ u) = ~ e x p ( 3 -
(13)
3,} ~ 8.458,
5
23 April 1979
and "), "- 0.57721 is Euler's constant. Quigg and Rosner [1 ] obtain v 1 ~ 7.63 by numerical integration (the values of E for integer l on their fig. 1 are approximate values for the quantity ½in vi). For higher Sstates the WKB values (ref. [1 ], note 5) on ~ 7r(2n - ½)2 ,
n = 1 ,2, 3,... ,
are within 3% of the corresponding values obtained by numerical integration [1]. The truncated asymptotic expansion which Bose and Miiller-Kirsten [3] (eq. (78)) find for the eigenvalues is consistent with the general form (5) but implies values of v i given by the following formula (q is an odd integer): Oi ~ Oql = e F 2 ( q , I)/4 ,
where F ( q , l ) = 2q + {G +
~
0
Fig. 1. Solution curves for the ground-state energy of the Nboson problem with the logarithmic potential u(rij ) = IVol In (ri'/a)" U, u per b o u n d ' L, lower b o u n d (and exact curve for N / --/ 2)," . v- ~- ~N(m/h 2 IVola ' 2 , e = [ - E / ( N - 1)1 ( m / h2) a 2 .
~ 1/2
q--(35qZ-z1) ~ 108 [2q + G1/2]]
(15) '
and G = 8(l + ~_)2 + q - l ( 6 9 q 2 + 1). Thus from eq. (15) we obtain v 1 ~ Vl0 = 17.93 which by the inequality (13) is too large by a factor greater than 2. It is interesting that we can also discuss the N-body problem with the aid of the above results presented in the form e = g(v). As Lipkin [2] has done, let us consider states with "stretched" angular momenta, namely those with the highest values of spin and J for a given configuration. These are the 1 - , 2 + and 3 - mesons and the 3/2 +, 5 / 2 - and 7/2 + baryons and all have wave functions which are totally symmetric in the spin parameters. Since all such particles are antisymmetric in their color indices, they will have totally symmetric spatial wave functions. Let us then consider Nquarks, each with mass m, interacting by the pair potential V(ri/) = IV01 ln(riJa ) and we define the following parameters (E is the exact energy in the centre-ofmass frame): v = ½N(m/~2)lVola2 ,
0 -2
(14)
e = [-El(N-
1)](m/tt2)a 2 . (16)
These definitions are consistent with eq. (2) which corresponds to the special case N = 2. The theorem of ref. [4] implies the following inequalities for the exact energy of the lowest spatially symmetric N-particle state: ½vln(v/8.458) < e < l v l n ( V / V l ) ,
(17) 81
Volume 83B, number 1
PHYSICS LETTERS
i°e°
bounds on the energy for the higher states of a system of N quarks may then be found using the theorem of Hall [6].
- ~ N ( N - 1)l V0 Iln(v/v 1) < E < - ~ N ( N - 1)1V0 Iln(v/8.458).
23 April 1979
(18) References
The inequalities (17) are shown in fig. 1 in which we have used the value o 1 = 7.63 given by Quigg and Rosner [1]. For the higher states of the N-body system upper bounds can be obtained by the usual Rayleigh-Ritz method. To find lower bounds we require the exact two-body curves e i = gi(o) established for the logarithmic potential by eq. (5) together with the oi calculated by Quigg and Rosner [1]: lower
82
[1] C. Quigg and J.L. Rosner, Phys. Lett. 71B (1977) 153. [2] H.J. Lipkin, Phys. Lett. 74B (1978) 399. [3] S.K. Bose and H.J.W. Miiller-Kirsten, Univ. of Kaiserslautern (Germany) preprint (1978). [4] R.L. Hall and H.R. Post, Proc. Phys. Soc. 90 (1967) 381. [5] R.L. Hall, Proc. Phys. Soc. 91 (1967) 787. [6] R.L. Hall, Phys. Lett. 30B (1969) 320.