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Surface collective motion in the soliton bag model
Volume 199, number 3
PHYSICS LETTERS B
24 December 1987
SURFACE C O L L E C T I V E M O T I O N I N T H E S O L I T O N BAG M O D E L M. IWASAKI and Y. K O N D O
Department of Physics. Kochi University, Kochi 780. Japan Received 23 April 1987
Surface monopole vibration of the baryon is investigated on the basis of the soliton bag model. We derive a relativistic lagrangian for the motion, of which the mass parameter is proportional to the square of the bag radius. With the use of the semi-classical quantization, the energy spectra of the P~ and P33 resonances are reproduced fairly well by our model.
Bag models [ 1 - 3 ] have become useful tools for describing the properties of baryons with light quarks. In these models the surface of the bags is treated as a parameter which is determined so as to minimize the total energy. Recently there have been m a n y attempts [ 4-10 ] to describe the excited states of baryons by considering the surface collective motion o f bags. They are almost studied within the M I T bag model and the surface motion is treated by non-relativistic dynamics, except for ref. [ 10]. However, their results show that the velocity o f the surface is comparable with the velocity o f light. It is the main purpose of this note to investigate the surface motion within the soliton bag model and relativistic dynamics. Because the soliton bag model contains the M I T bag model as a limiting case. The lagrangian in the soliton bag model consists of the scalar a field and the massless quark field ¢/:
L = f d r [~,(y~,0~ +fa)gt+ ~(Oa/Ox~,)-+U(a)] ~ ~ . (1) In the static bag approximation the quarks are confined within a sphere with a radius Ro where ~ ~ 0. On the other hand, the value o f a increases rapidly from zero to a, outside the sphere at which U(~r) has an absolute m i n i m u m ( U ( a v ) = 0 ) . The equations o f motion derived from (1) are i a ' V ~o(r) ~ eoOo(r)
Ago(r)~U'(ao)
(r
(r>Ro) ,
(2) (3)
limit of M I T type. Since the solutions o f (2) and (3) are given in ref. [ 1 ], we do not write them down in this paper. Now let us regard the bag radius R ( t ) as a classical dynamical variable. Our task is to represent the lagrangian (1) as a function o f R and R and then quantize it in the usual manner. One may assume that the quark wave function in the bag take the form o f ¢/= ~ o ( p ) e x p ( - i e o t ) , which is just obtained by a scale transformation, r--,p=-Ror/R in the static wave function. Substituting this into the quark part of L and making use of eq. (2), we get L q ~- ~o -
~/R ,
(4)
where use is made o f eo=~/Ro ( ~ = 2 . 0 4 ) . The remaining part o f L is written as R+AR
L~=-~:gP R3-
f
[½(O~/Ox~)2 +U] 4~rr2dr"
R
(5) The first term is derived from the integration region r < R and called volume energy. The transition region R < r < R + AR ( O~/Or ¢ 0) brings about the second term which is called surface energy. The integration on the right-hand side o f (5) may be carried out in locally inertial coordinates x~ in which the surface element is at rest. Assuming that a(x~,)=~o(Xl, ) and AR<
where ~ u = 0 o ( r ) e x p ( - i E o t ) and we have taken the 437
Volume 199, number 3
PHYSICSLETTERSB
24 December 1987
L~ =-4=pR~ -4~RZ
(a)
(b)
lx/T7~-~2 f [½(&7o/OX/,)2+U]dr', <,¢o
(6)
exp exp
where r in the integrand has been replaced by R and the term ~ is due to the Lorentz transformation d r = ~ d r ' . Furthermore, using the relation U~(dao/ar')2/2 derived from (3) and (V '0"o)2~ (OtTo~Or') 2, the final expression for L is given by
2~"2-W0-
22--'U~ ,,
__
"~%5
exp exp
70~ 21oo
1920
w-f'g4---.I_.~
Ti-iT-TfT6 16o~ .SS.f_.1~X5
(7)
i~85-,,%,.5.5
1"~'0 " 1440 12~"~- "" T ' ~ ' ~
1173
L(R, [~) = - ~ / R - ~npR 3 -4nsR2x/1 _/~2,
2160 2059",
959
where the constant s is defined by
s= f (&ro/Or')2dr.
(8)
It is noted that our lagrangian has a relativistic form, as expected. The kinetic energy of the surface motion originates from the surface potential in our model. On the other hand it originates from the quark kinetic energy in other models based on the MIT bag model. This difference reflects in the mass parameter of the collective motion; M(R) ocR 2 in our model and M(R)ocR-~ in other models. Now we are in a position to quantize our dynamical system (7) in order to investigate the excited states of baryons. From eq. (7), our hamiltonian is
H( R, P) = , , / ~
+
( 4~zsR2) 2
+~/R+ 47rpR3 ,
(9)
where the canonical momentum conjugate to R is given by P = OL/OR. We will use here the simplest quantization prescription, the Bohr-Sommerfeld condition
~PdR=h(n+l)
(n=0,1,2,3,).
(10)
According to ref. [2], we have taken the constant -] on the right-hand side of (10). Let us calculate the energy spectra of baryons. For this aim, we must consider the first-order correction in c~ which is the "fine structure constant" of QCD. This correction replaces the constant in (9) by {--+{- a(I~l --/2/mag) •
(11 )
where # = - 1 for N and # = 1 for A. The constants Ici and Imag are given by 3.409 and 0.363, respectively, 438
O: = 0 4 2 9 S%=0.0
Q: -~0.L65 Sv3=100
Fig. 1. Experimental spectrum of N and A resonances and the calculated masses in MeV for two pairs of the parameters s ~/3in MeV and c~. according to ref. [ 3]. Thus we have three unknown parameters: a, s and p. Actual calculations was carried out for p 1/4= 140 MeV and s 1/3= 0, 100 MeV for convenience. The remaining parameter c~ is determined so as to reproduce the proton mass. The calculated energy spectra of baryons is shown in fig. 1. The cases (a) and (b) correspond to s ~/3= 0 and 100 MeV, respectively. The parameter a for each case is shown in fig. 1. It is seen that our model reproduces the experimental data fairly well. In table 1 we have displayed the minima and maxima of the bag radius R and the maxima of the velocity ,~ for each classical orbit. It means that the surface motion is fully relativistic and the amplitude of the vibration is very large. Moreover, the constant s of the surface energy cannot be determined by the energy spectra only. However, it should be noted that the mean radius of R decreases as the constant s becomes large. Finally, it is concluded that the energy spectra of the P~L and P33 resonances can be described by surface monopole vibration in the soliton bag model. There remains, however, an interesting problem, which is to calculate the static physical quantities of the ground state (nucleon) in our present model. It will be discussed elsewhere as will be the pion effect Jill. The authors would like to express their thanks to
Volume 199, number 3
PHYSICS LETTERS B
24 December 1987
Table 1 Maxima and minima of the bag radius R and maxima of the velocity/~ calculated by our model with the parameter set (a) and (b). Resonance
N(939) N(1440) A(1232)
Set (a)
Set (b)
Rmi.
Rma~
il~max
Rmin
Rm.x
t1~....
0.268 0.184 0.374
1.543 1.793 1.623
1.000 1.000 1.000
0.184 0.124 0.307
1.175 1.419 1.264
0.986 0.995 0.969
P r o f e s s o r S. N i s h i y a m a a n d M r . Y. M i y a u e f o r h e l p ful d i s c u s s i o n s .
References [1] A. Chodos et al., Phys. Rev. D9 (1974) 3471; T. DeGrand et al., Phys. Rev. D 12 (1975) 2060; K. Johnson, Acta Phys. Pol. B 6 (1975) 865. [2] P. Hasenfratz and J. Kuti, Phys. Rep. 40 (1978) 75. [3] R. Friedberg and T.D. Lee, Phys. Rev. D 16 (1977) 1096; D 18 (1978) 2623.
[4] R. Goldflam and L. Wilets, Phys. Rev. D 25 (1982) 1951. [5] T. DeGrand and C. Rebbi, Phys. Rev. D 17 (1978) 2358. [6] P.J. Mulders et al., Phys. Rev. D 27 (1983) 2708; N. Isgur and G. Karl, Phys. Rev. D 19 (1979) 2653. [7] G.E. Brown, J.W. Durso and M.B. Johnson, Nucl. Phys. A 397 (1983) 447. [8] Y. Nogami and g. Tomio, Can. J. Phys. 62 (1984) 260. [9] T. Hatsuda and H. Kuratsuji, Z. Phys. C 27 (1985) 455. [10] H.R. Fiebig, Phys. Rev. D31 (1985) 2902. [ 11 ] J.L. Dethier and L. Wilets, Phys. Rev. D 34 (1986) 207.
439
PHYSICS LETTERS B
24 December 1987
SURFACE C O L L E C T I V E M O T I O N I N T H E S O L I T O N BAG M O D E L M. IWASAKI and Y. K O N D O
Department of Physics. Kochi University, Kochi 780. Japan Received 23 April 1987
Surface monopole vibration of the baryon is investigated on the basis of the soliton bag model. We derive a relativistic lagrangian for the motion, of which the mass parameter is proportional to the square of the bag radius. With the use of the semi-classical quantization, the energy spectra of the P~ and P33 resonances are reproduced fairly well by our model.
Bag models [ 1 - 3 ] have become useful tools for describing the properties of baryons with light quarks. In these models the surface of the bags is treated as a parameter which is determined so as to minimize the total energy. Recently there have been m a n y attempts [ 4-10 ] to describe the excited states of baryons by considering the surface collective motion o f bags. They are almost studied within the M I T bag model and the surface motion is treated by non-relativistic dynamics, except for ref. [ 10]. However, their results show that the velocity o f the surface is comparable with the velocity o f light. It is the main purpose of this note to investigate the surface motion within the soliton bag model and relativistic dynamics. Because the soliton bag model contains the M I T bag model as a limiting case. The lagrangian in the soliton bag model consists of the scalar a field and the massless quark field ¢/:
L = f d r [~,(y~,0~ +fa)gt+ ~(Oa/Ox~,)-+U(a)] ~ ~ . (1) In the static bag approximation the quarks are confined within a sphere with a radius Ro where ~ ~ 0. On the other hand, the value o f a increases rapidly from zero to a, outside the sphere at which U(~r) has an absolute m i n i m u m ( U ( a v ) = 0 ) . The equations o f motion derived from (1) are i a ' V ~o(r) ~ eoOo(r)
Ago(r)~U'(ao)
(r
(r>Ro) ,
(2) (3)
limit of M I T type. Since the solutions o f (2) and (3) are given in ref. [ 1 ], we do not write them down in this paper. Now let us regard the bag radius R ( t ) as a classical dynamical variable. Our task is to represent the lagrangian (1) as a function o f R and R and then quantize it in the usual manner. One may assume that the quark wave function in the bag take the form o f ¢/= ~ o ( p ) e x p ( - i e o t ) , which is just obtained by a scale transformation, r--,p=-Ror/R in the static wave function. Substituting this into the quark part of L and making use of eq. (2), we get L q ~- ~o -
~/R ,
(4)
where use is made o f eo=~/Ro ( ~ = 2 . 0 4 ) . The remaining part o f L is written as R+AR
L~=-~:gP R3-
f
[½(O~/Ox~)2 +U] 4~rr2dr"
R
(5) The first term is derived from the integration region r < R and called volume energy. The transition region R < r < R + AR ( O~/Or ¢ 0) brings about the second term which is called surface energy. The integration on the right-hand side o f (5) may be carried out in locally inertial coordinates x~ in which the surface element is at rest. Assuming that a(x~,)=~o(Xl, ) and AR<
where ~ u = 0 o ( r ) e x p ( - i E o t ) and we have taken the 437
Volume 199, number 3
PHYSICSLETTERSB
24 December 1987
L~ =-4=pR~ -4~RZ
(a)
(b)
lx/T7~-~2 f [½(&7o/OX/,)2+U]dr', <,¢o
(6)
exp exp
where r in the integrand has been replaced by R and the term ~ is due to the Lorentz transformation d r = ~ d r ' . Furthermore, using the relation U~(dao/ar')2/2 derived from (3) and (V '0"o)2~ (OtTo~Or') 2, the final expression for L is given by
2~"2-W0-
22--'U~ ,,
__
"~%5
exp exp
70~ 21oo
1920
w-f'g4---.I_.~
Ti-iT-TfT6 16o~ .SS.f_.1~X5
(7)
i~85-,,%,.5.5
1"~'0 " 1440 12~"~- "" T ' ~ ' ~
1173
L(R, [~) = - ~ / R - ~npR 3 -4nsR2x/1 _/~2,
2160 2059",
959
where the constant s is defined by
s= f (&ro/Or')2dr.
(8)
It is noted that our lagrangian has a relativistic form, as expected. The kinetic energy of the surface motion originates from the surface potential in our model. On the other hand it originates from the quark kinetic energy in other models based on the MIT bag model. This difference reflects in the mass parameter of the collective motion; M(R) ocR 2 in our model and M(R)ocR-~ in other models. Now we are in a position to quantize our dynamical system (7) in order to investigate the excited states of baryons. From eq. (7), our hamiltonian is
H( R, P) = , , / ~
+
( 4~zsR2) 2
+~/R+ 47rpR3 ,
(9)
where the canonical momentum conjugate to R is given by P = OL/OR. We will use here the simplest quantization prescription, the Bohr-Sommerfeld condition
~PdR=h(n+l)
(n=0,1,2,3,).
(10)
According to ref. [2], we have taken the constant -] on the right-hand side of (10). Let us calculate the energy spectra of baryons. For this aim, we must consider the first-order correction in c~ which is the "fine structure constant" of QCD. This correction replaces the constant in (9) by {--+{- a(I~l --/2/mag) •
(11 )
where # = - 1 for N and # = 1 for A. The constants Ici and Imag are given by 3.409 and 0.363, respectively, 438
O: = 0 4 2 9 S%=0.0
Q: -~0.L65 Sv3=100
Fig. 1. Experimental spectrum of N and A resonances and the calculated masses in MeV for two pairs of the parameters s ~/3in MeV and c~. according to ref. [ 3]. Thus we have three unknown parameters: a, s and p. Actual calculations was carried out for p 1/4= 140 MeV and s 1/3= 0, 100 MeV for convenience. The remaining parameter c~ is determined so as to reproduce the proton mass. The calculated energy spectra of baryons is shown in fig. 1. The cases (a) and (b) correspond to s ~/3= 0 and 100 MeV, respectively. The parameter a for each case is shown in fig. 1. It is seen that our model reproduces the experimental data fairly well. In table 1 we have displayed the minima and maxima of the bag radius R and the maxima of the velocity ,~ for each classical orbit. It means that the surface motion is fully relativistic and the amplitude of the vibration is very large. Moreover, the constant s of the surface energy cannot be determined by the energy spectra only. However, it should be noted that the mean radius of R decreases as the constant s becomes large. Finally, it is concluded that the energy spectra of the P~L and P33 resonances can be described by surface monopole vibration in the soliton bag model. There remains, however, an interesting problem, which is to calculate the static physical quantities of the ground state (nucleon) in our present model. It will be discussed elsewhere as will be the pion effect Jill. The authors would like to express their thanks to
Volume 199, number 3
PHYSICS LETTERS B
24 December 1987
Table 1 Maxima and minima of the bag radius R and maxima of the velocity/~ calculated by our model with the parameter set (a) and (b). Resonance
N(939) N(1440) A(1232)
Set (a)
Set (b)
Rmi.
Rma~
il~max
Rmin
Rm.x
t1~....
0.268 0.184 0.374
1.543 1.793 1.623
1.000 1.000 1.000
0.184 0.124 0.307
1.175 1.419 1.264
0.986 0.995 0.969
P r o f e s s o r S. N i s h i y a m a a n d M r . Y. M i y a u e f o r h e l p ful d i s c u s s i o n s .
References [1] A. Chodos et al., Phys. Rev. D9 (1974) 3471; T. DeGrand et al., Phys. Rev. D 12 (1975) 2060; K. Johnson, Acta Phys. Pol. B 6 (1975) 865. [2] P. Hasenfratz and J. Kuti, Phys. Rep. 40 (1978) 75. [3] R. Friedberg and T.D. Lee, Phys. Rev. D 16 (1977) 1096; D 18 (1978) 2623.
[4] R. Goldflam and L. Wilets, Phys. Rev. D 25 (1982) 1951. [5] T. DeGrand and C. Rebbi, Phys. Rev. D 17 (1978) 2358. [6] P.J. Mulders et al., Phys. Rev. D 27 (1983) 2708; N. Isgur and G. Karl, Phys. Rev. D 19 (1979) 2653. [7] G.E. Brown, J.W. Durso and M.B. Johnson, Nucl. Phys. A 397 (1983) 447. [8] Y. Nogami and g. Tomio, Can. J. Phys. 62 (1984) 260. [9] T. Hatsuda and H. Kuratsuji, Z. Phys. C 27 (1985) 455. [10] H.R. Fiebig, Phys. Rev. D31 (1985) 2902. [ 11 ] J.L. Dethier and L. Wilets, Phys. Rev. D 34 (1986) 207.
439