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A modified massive Thirring model and new solitonic coats on the (1 + 1)-dimensional chiral bag
ANNALS
OF PHYSICS
193, 93-101 (1989)
A Modified Massive Thirring Model and New Solitonic Coats on the (1 + 1 )-Dimensional Chiral Bag SHUXI LI. D. B. MITCHELL, Physics Department, Hamilton, Ontario,
AND R. K. BHADURI
McMaster Canada
CJnioersit.v, L8.Y 4Ml
Received October 28, 1988
The bosonisation of a modified massive Thirring solitary-wave solution. This is used as a bosonic (1 + 1 )-dimensional chiral bag. With appropriate insensitive to the bag radius within a certain
model Lagrangian leads to a system with a coating (together with other examples) in a choice of parameters, the baryon energy is range. irrespective of the bosonic coating.
‘0 1989 Academic Press, Inc.
I. INTRoD~CTL~N
This paper is divided into two parts. In the first part, we generalize the (1 + l)dimensional massive Thirring model [l] and obtain an exact bound-state solution in classical field theory. Just as bosonisation of the massive Thirring model [2] leads to the sine-Gordon equation, our modified Thirring Lagrangian suggests a boson system with a new solitary-wave solution. Such a static kink solution is obtained analytically. In the second part of the paper, this new soliton is chirally coupled to a (1 + l)dimensional bag, and the energy of the system obtained as a function of the bag radius. This is a variation of the Zahed model [3], where the sine-Gordon soliton was coupled to the bag. Zahed and Klabucar [4] had investigated the transition of the phenomenological bagged (QCD), vacuum to the sine-Gordon effective Lagrangian in this model. Another, rather different point of interest in these models is to test the so-called Cheshire-cat picture [S]. It is known that in (1 + 1 )-dimensions, this picture is exact (i.e., the physical properties do not depend on the position of the bag wall) provided that equivalent Lagrangians are chosen in the Bose and Fermi sectors according to the rules of bosonisation [6], and with appropriate boundary conditions at the wall surface. In the Zahed model, the confined noninteracting massless fermions inside the bag are not equivalent to the sine-Gordon field outside, and yet the energy of the system is not too sensitive to the bag radius within a reasonable range of the radius. We want to investigate if this insensitivity to the radius is due to the particular choice of the sine-Gordon 93 0003-4916/89 $7.50 Copyright %’ 1989 by Academic Press, Inc All rights of reproduction in any lorm rescrvcd
94
LI,
MITCHELL,
AND
BHADURI
soliton in the Zahed model. To this end, the energy versus radius calculation is repeated for the solitonic coat obtained in Part I, and also with a b4-Lagrangian outside the bag. In Section II, we present the first part of the paper on the modification of the massive Thirring model. The bag model calculations are given in Section III.
II. THE MODIFIED MASSIVE THIRRING
MODEL
We first briefly recapitulate the work of Chang et al. [l] in connection with the exact bound-state solution of the massive Thirring model. They considered the classical Lagrangian density (in (1 + 1)-dimensions)
-%h = iW ar4 + ?j (@,q12- M@7,
(1)
with g > 0. By solving the corresponding Dirac equation, they obtained the boundstate eigenvalue E=Mcos(g/2). (2) Using the representation y”=03,
iyl=ol,
y5 = ~‘9 = a2,
(3)
and writing the eigenvector as
‘OS *tx) e-iEt 4(x,t)= (rltx) rl(x) sin4W> ’
(4)
they found that Ii/(x) = -tan -‘(fi
tanh kx),
2M sin2( g/2) ‘2=q+q=gcosh(2kl)+cos(g,2)’
(5) (6)
In the above equations, B=W-EY(M+E),
k = ,/M-.
(7)
Note, from Eq. (6), that the “confined” solution exists only for g>O, and from Eq. (2) that for g > rr, E is negative. We generalize Eq. (1) to suggest the modified massive Thirring Lagrangian density
-GMT = W’ apq+$ (@,q12 +i (&I2- M@,
(8)
NEW
SOLITONIC
COATS
ON THE
(‘HIRAL
BAG
9s
where g and h are real but may take positive or negative values. Using the same method as that of Chang et al. [ 11. we find that the bound-state eigenvalue E is given by (provided (h + g) > 0) E=M
cosC(s/2)$-%I
(9)
[ 1 + (h/g) sin’(( g/2) J-)1”” with I+!I(x) the same as in Eq. (5), and YIZ=g+g=2
(M-E)
cosh2(kx)
1 + ,!3tanh’(kx) x g(1 + fl tanh*(kx))* + h( 1 - fl tanh2(kx))2’
(10)
In the above equations, fl and k are defined as before (see Eq. (7)), and it is easy to check that Eqs. (2) and (6) are reproduced when h = 0. Simple solutions are also found when g = 0 and h > 0, and for the special case g = h. Coleman [2] has demonstrated the equivalence of the massive Thirring model [ 1] with the bosonic sine-Gordon Lagrangian
where we are using the notation of Rajaraman [7], with approrpiate sign changes to conform with Section III. The bosonic Lagrangian may be obtained from Eq. (1) by applying the bosonisation rules as given, for example, by Kogut and Susskind [6]. As a result, an interesting equation relates the coupling constants of the fermion system, g, with the coupling parameter (&/wz) of the bosonic Lagrangian ( 11): 2&c Jiqjij=,,.
Jz
(12)
A discussion of the implications of this may be found in Ref. [8]. Using the same rules of Kogut and Susskind [6] on the modified massive Thirring model Lagrangian, Eq. (8), we obtain a form for the new bosonic Lagrangian
The plus sign within the curly brackets is again taken for convenience, and a constant added. The same form would be obtained from Eq. (8) for the special case with g = 0 and a nonzero h. Just as for the classical sine-Gordon equation, it is
96
LI, MITCHELL, AND BHADURI
easy to find, from Eq. (13), the static localized solutions with topological quantum numbers + 1. A little algebra will show that these solutions are d(x) = * 5 tan-‘(mx). (14) d To obtain the topoligical quantum numbers f 1, one should take the dimensionless quantity 19(x) = (d/m) 4(x), which would have the appropriate boundary values as x --+ + co. It is also easy to show that the energy density of the solitary wave
E(X) =-;(gy+g(l+cos(g())
(15)
for the static solution (14) is 4m4 1 E(X) = I (1 +m2x2)2’
&=
+cO 27cm3 I -m E(X) dx = -. A
(17)
For the sine-Gordon system given by Eq. (1 I), other time-dependent solutions are well known [I]. We have not been successful, however, in obtaining any timedependent solutions of PB given by Eq. (13), other than the trivial one by Lorentzboosting the static solution (14).
III. A NEW COATING ON THE (1+ 1)-DIMENSIONAL Zahed [3] has studied a chiral bag with a Lagrangian 9=
[ifjy”iY,q-l?]
CHIRAL
BAG
(with the bag constant B)
I],-$je-i~~o~x’qLl,
+ c=%,,(eqJw(~ -?“I’
(18)
Here qV = 1 inside the bag and zero outside, with 8‘~~ = np A,. The boundary conditions on the bag surface ensure the continuity of the chiral current. For the static bag, np = (0, E(X)), where E(X) =x//xl. In this case, the quark field q is confined within the bag of length 2R, and interacts with the bosonic field e(x) only at the boundary x = f R. Outside the bag, the field 19 is governed by a nonlinear Lagrangian SC,,, , For the latter, Zahed [S] had chosen the sine-Gordon Lagrangian, 2&,,t = L& = ;F,[(a,e)2-
K2(1 + cos e)],
(19)
NEW
which may be reduced F,” = m2/1. In the ground equation is taken, which radius goes to zero. This
SOLITONIC
COATS
to the same form state of the baryon, carries a topological solution, for R = 0,
ON
THE
CHIRAL
BAG
91
as Eq. (11) by putting 9 = &&+$ the kink solution of the sine-Gordon quantum number of one as the bag is
@(X)=&(X){ -rr++tan’[expK~(.~)~]~, where E(X) = s/jxl. The energy of the soliton, for R = 0, is given by E(0) = F,Z1’:
(;)’
d.~= 8KFf.
When R # 0, the matching conditions at the boundary determine 8(R). The singleparticle spectrum for the fermions in the bag is given by [3] c,,=
17I
( 1 n+2
z+G‘
0
where n = 0, + 1, ) 2, etc., and 8 = t[tl(R) - 0( -R)]. This spectrum has spectral asymmetry for nonzero 8, resulting in a fractional fermion number. For example, in the range 0 < 0 7~12,the n = - 1 state continues to be occupied (by n, quarks, where n, is the number of colours) and becomes a positive energy state. Zahed showed that in the range 0 < 6 < I[, the energy of the chiral bag (including the Casimir energy) is n 8’
E Bag=-&+2RB. Furthermore, the continuity demands that
of the boundary F; = (n,.47c).
condition
at x = + R as R -+ 0 (22)
Both Eqs. (21) and (22) remain identical irrespective of the details of the bosonic Lagrangian although B(R) does depend on the latter. Our purpose is to calculate the (bag + soliton) energy versus R when PC,,, in Eq. (18) is of the form (13) obtained in Section II, and compare with Zahed’s model. We also repeat this calculation with another solvable coating, in each case demanding that E(O), the soliton mass in the absence of the bag, remain the same. We find that although the
98
LI,
MITCHELL,
AND
magnitude of E(R) varies from model the range 0.5 fm
BHADURI
to model, it is not a sensitive function of R in this sense the Cheshire-cat picture still holds of the calculation with the Lagrangian given the Zahed model is that we take in Eq. (18)
2&,, = F,’ T [(ape)’ - m’( 1 + cos tI)2].
(23)
This is of the same form as Eq. (13) if we put (24)
the same substitutions as in the sine-Gordon Lagrangian (19). The calculation in the fermion sector yields Eq. (21) as in Zahed [3, 61 and will not be repeated here. In the boson sector, the equation of motion from Eqs. (23) and (18) is (t3,O)’ - m*( 1 + cos 13)sin 0 = 0,
Ix]> R.
(25)
The boundary condition at the bag surface gives E(X) F;g=
1
_- Q-y5e-iey5q, 2
at
1x1= R.
As before, E(X) =x/1x1. To simplify the right-hand side, note that (as in the Zahed model) the fermion sector boundary condition is e -ievsq = -k(x)
at
ylq,
Ixl=R.
Using the relation ysyl = --y,,, we obtain at 1x( = R de(x)
F”‘z=2c
1
q+q=$&,
(26)
sea where 8 = $(0(R) - 0(-R)).
The static solution of Eq. (25) is given by
-dW) = m(i + cos dx
Taking this derivative determines O(R):
to be continuous
F,2m(l +cosB)=g,
e(x)),
(xl> R.
at the surface yields the equation
Ix]= R.
that
(28)
NEW
SOLITONIC
COATS
ON THE
Note that in the hmit of R + 0, the solitary-wave
CHIRAL
BAG
99
solution is (see Eq. (14))
0(x) = 2 tan’(mx). Substituting
this in Eq. (28) as R -+ 0 gives
as stated in Eq. (22). It is also straightforward sector from Eq. (27):
Esoliton = 2 X2 jr
[ (2)’
to calculate the energy in the soliton
+ m’( 1 +
COS
B)‘] dx.
A little algebra yields E sollton= 2mF,Z[n - 8(R) - sin B(R)].
(29)
The total energy of the (bag + soliton), derived from the Lagrangians (18) and (23 ), is given by !@ -- 4xR
E, -~aryon
+ 2RB + 2rnFz[r - H - sin 01,
(30)
where the first two terms come from the fermion sector in the bag and are formally identical to Zahed’s expression. Note, however, that B(R) is now determined from Eq. (28) and is not the same as the sine-Gordon case. In another variation of the calculation, a different coat is taken from the fj”-Lagrangian:
In this case, the kink solution is given by d(x) = (2m,)/&
tanh($
m4x),
We make the transformation (32) such that for the kink 19(x) -+ +rt as x + f co. The Lagrangian (a,O)’ - 4m:
8 COS’ - . 2
(31) then reduces to (33)
100
LI, MITCHELL,
AND BHADURI
For no bag (R = 0), the energy of the kink is
16,,hm:
16fim
F2
E(o)=TT=F
‘$
n.
(34)
For nonzero bag radius R, B(R) is determined by the equation
(35) and the energy of the baryon is given by O2 Emaryon =nc47cR+2RB
+oTF2m4
2-3sini+sin’i
(36)
Note that as R --+0, Eq. (35) again yields the condition that Fz =n,/4~~in accordance with (22). The continuity of the boundary condition as R + 0 therefore fixes the value of I;, in the chiral bag irrespective of the soliton coat. Note that this is in agreement with the bosonisation condition (12) with g = 0, as is the case inside the bag. To make a numerical comparison of the three model calculations (with the coat in the baryon determined from Eqs. (19), (23), and (33)), we have to fix the parameters of the total Lagrangian (18). In each case, F, is taken to be ,/&%, with n, = 3. The remaining parameter in the soliton is fixed by demanding that its energy E(R)IRzo= E(0) be the same in each case. From Eqs. (20), (29), and (34), it then follows that
E(O)=8F,ZK=2mnF,2=716firn 4F2 II. 1.2
I
I
I
I --
1.0 -
XL-----J * 0.0
(37)
I Zahed - modified
1.5 2.0 2.5 3.0 R w FIG. 1. Variation of the ground-state baryon energy with the bag radius R for various solitonic coats. In each case, the parameters have been fixed to yield E(R = 0) = 0.76 GeV. For details, see the text. 0.5
1.0
101
NEW SOLITONIC COATSON THE CHIRAL BAG
Following Ref. [6], we take K = 2.04 fm ~~r, which fixes E(0) = 0.76 GeV, and gives m=2.60fm~ ’ and m,=2.17 fm--‘. In Fig. 1, the energy E(R) of the system is plotted as a function of R for the three cases. Again, following Ref. [6], the bag constant B is taken to be 0.015 GeV’. Note that an appropriate choice of the bag constant is still needed to obtain a relatively flat E(R) versus R curve. Moreover, an uncertain term of the form Z/R in the energy has been dropped. Keeping these limitations in mind, we see from the figure that E(R) is rather insensitive to R in the range 0.5 fm < R < 2 fm, irrespective of the bosonic coat outside. With the appropriate choice of the bag constant B and the solition mass E(O), the Cheshirecat phenomenon [ 5 J thus seems to hold approximately in the ( 1 + 1 )-dimensional bag even though the fermion sector inside the bag is not equivalent to the boson sector outside.
ACKNOWLEDGMENT This research
was supported
by the Natural
Sciences
and Engineering
Research
Council
REFERENCES 1. SHAU-JIN CHANG, S. D. ELLIS. AND B. W. LEE, Phw. Rev. D 11 (1975). 3572. 2. S. COLEMAN, Phys. Rev. D 11 (1975), 2088. 3. I. ZAHED, Phvs. Rev. D 30 (1984), 2221. 4. 1. ZAHED AND D. KLABUCAR, Phys. Rev. D 30 (1984). 2647. 5. S. NADKARNI. H. B. NIELSEN, AND 1. ZAHED, Nucl. Phys. B 253 (1985), 308. 6. J. K~GUT AND L. SUSSKIND, Phys. Reo. D 11 (1975), 3594. 7. R. RAJARAMAN, “Solitons and Instantons,” p. 34, North-Holland, Amsterdam, 1982. 8. S. COLEMAN, “Aspects of Symmetry,” p, 250, Cambridge Univ. Press, Cambridge, 1985.
of Canada.
OF PHYSICS
193, 93-101 (1989)
A Modified Massive Thirring Model and New Solitonic Coats on the (1 + 1 )-Dimensional Chiral Bag SHUXI LI. D. B. MITCHELL, Physics Department, Hamilton, Ontario,
AND R. K. BHADURI
McMaster Canada
CJnioersit.v, L8.Y 4Ml
Received October 28, 1988
The bosonisation of a modified massive Thirring solitary-wave solution. This is used as a bosonic (1 + 1 )-dimensional chiral bag. With appropriate insensitive to the bag radius within a certain
model Lagrangian leads to a system with a coating (together with other examples) in a choice of parameters, the baryon energy is range. irrespective of the bosonic coating.
‘0 1989 Academic Press, Inc.
I. INTRoD~CTL~N
This paper is divided into two parts. In the first part, we generalize the (1 + l)dimensional massive Thirring model [l] and obtain an exact bound-state solution in classical field theory. Just as bosonisation of the massive Thirring model [2] leads to the sine-Gordon equation, our modified Thirring Lagrangian suggests a boson system with a new solitary-wave solution. Such a static kink solution is obtained analytically. In the second part of the paper, this new soliton is chirally coupled to a (1 + l)dimensional bag, and the energy of the system obtained as a function of the bag radius. This is a variation of the Zahed model [3], where the sine-Gordon soliton was coupled to the bag. Zahed and Klabucar [4] had investigated the transition of the phenomenological bagged (QCD), vacuum to the sine-Gordon effective Lagrangian in this model. Another, rather different point of interest in these models is to test the so-called Cheshire-cat picture [S]. It is known that in (1 + 1 )-dimensions, this picture is exact (i.e., the physical properties do not depend on the position of the bag wall) provided that equivalent Lagrangians are chosen in the Bose and Fermi sectors according to the rules of bosonisation [6], and with appropriate boundary conditions at the wall surface. In the Zahed model, the confined noninteracting massless fermions inside the bag are not equivalent to the sine-Gordon field outside, and yet the energy of the system is not too sensitive to the bag radius within a reasonable range of the radius. We want to investigate if this insensitivity to the radius is due to the particular choice of the sine-Gordon 93 0003-4916/89 $7.50 Copyright %’ 1989 by Academic Press, Inc All rights of reproduction in any lorm rescrvcd
94
LI,
MITCHELL,
AND
BHADURI
soliton in the Zahed model. To this end, the energy versus radius calculation is repeated for the solitonic coat obtained in Part I, and also with a b4-Lagrangian outside the bag. In Section II, we present the first part of the paper on the modification of the massive Thirring model. The bag model calculations are given in Section III.
II. THE MODIFIED MASSIVE THIRRING
MODEL
We first briefly recapitulate the work of Chang et al. [l] in connection with the exact bound-state solution of the massive Thirring model. They considered the classical Lagrangian density (in (1 + 1)-dimensions)
-%h = iW ar4 + ?j (@,q12- M@7,
(1)
with g > 0. By solving the corresponding Dirac equation, they obtained the boundstate eigenvalue E=Mcos(g/2). (2) Using the representation y”=03,
iyl=ol,
y5 = ~‘9 = a2,
(3)
and writing the eigenvector as
‘OS *tx) e-iEt 4(x,t)= (rltx) rl(x) sin4W> ’
(4)
they found that Ii/(x) = -tan -‘(fi
tanh kx),
2M sin2( g/2) ‘2=q+q=gcosh(2kl)+cos(g,2)’
(5) (6)
In the above equations, B=W-EY(M+E),
k = ,/M-.
(7)
Note, from Eq. (6), that the “confined” solution exists only for g>O, and from Eq. (2) that for g > rr, E is negative. We generalize Eq. (1) to suggest the modified massive Thirring Lagrangian density
-GMT = W’ apq+$ (@,q12 +i (&I2- M@,
(8)
NEW
SOLITONIC
COATS
ON THE
(‘HIRAL
BAG
9s
where g and h are real but may take positive or negative values. Using the same method as that of Chang et al. [ 11. we find that the bound-state eigenvalue E is given by (provided (h + g) > 0) E=M
cosC(s/2)$-%I
(9)
[ 1 + (h/g) sin’(( g/2) J-)1”” with I+!I(x) the same as in Eq. (5), and YIZ=g+g=2
(M-E)
cosh2(kx)
1 + ,!3tanh’(kx) x g(1 + fl tanh*(kx))* + h( 1 - fl tanh2(kx))2’
(10)
In the above equations, fl and k are defined as before (see Eq. (7)), and it is easy to check that Eqs. (2) and (6) are reproduced when h = 0. Simple solutions are also found when g = 0 and h > 0, and for the special case g = h. Coleman [2] has demonstrated the equivalence of the massive Thirring model [ 1] with the bosonic sine-Gordon Lagrangian
where we are using the notation of Rajaraman [7], with approrpiate sign changes to conform with Section III. The bosonic Lagrangian may be obtained from Eq. (1) by applying the bosonisation rules as given, for example, by Kogut and Susskind [6]. As a result, an interesting equation relates the coupling constants of the fermion system, g, with the coupling parameter (&/wz) of the bosonic Lagrangian ( 11): 2&c Jiqjij=,,.
Jz
(12)
A discussion of the implications of this may be found in Ref. [8]. Using the same rules of Kogut and Susskind [6] on the modified massive Thirring model Lagrangian, Eq. (8), we obtain a form for the new bosonic Lagrangian
The plus sign within the curly brackets is again taken for convenience, and a constant added. The same form would be obtained from Eq. (8) for the special case with g = 0 and a nonzero h. Just as for the classical sine-Gordon equation, it is
96
LI, MITCHELL, AND BHADURI
easy to find, from Eq. (13), the static localized solutions with topological quantum numbers + 1. A little algebra will show that these solutions are d(x) = * 5 tan-‘(mx). (14) d To obtain the topoligical quantum numbers f 1, one should take the dimensionless quantity 19(x) = (d/m) 4(x), which would have the appropriate boundary values as x --+ + co. It is also easy to show that the energy density of the solitary wave
E(X) =-;(gy+g(l+cos(g())
(15)
for the static solution (14) is 4m4 1 E(X) = I (1 +m2x2)2’
&=
+cO 27cm3 I -m E(X) dx = -. A
(17)
For the sine-Gordon system given by Eq. (1 I), other time-dependent solutions are well known [I]. We have not been successful, however, in obtaining any timedependent solutions of PB given by Eq. (13), other than the trivial one by Lorentzboosting the static solution (14).
III. A NEW COATING ON THE (1+ 1)-DIMENSIONAL Zahed [3] has studied a chiral bag with a Lagrangian 9=
[ifjy”iY,q-l?]
CHIRAL
BAG
(with the bag constant B)
I],-$je-i~~o~x’qLl,
+ c=%,,(eqJw(~ -?“I’
(18)
Here qV = 1 inside the bag and zero outside, with 8‘~~ = np A,. The boundary conditions on the bag surface ensure the continuity of the chiral current. For the static bag, np = (0, E(X)), where E(X) =x//xl. In this case, the quark field q is confined within the bag of length 2R, and interacts with the bosonic field e(x) only at the boundary x = f R. Outside the bag, the field 19 is governed by a nonlinear Lagrangian SC,,, , For the latter, Zahed [S] had chosen the sine-Gordon Lagrangian, 2&,,t = L& = ;F,[(a,e)2-
K2(1 + cos e)],
(19)
NEW
which may be reduced F,” = m2/1. In the ground equation is taken, which radius goes to zero. This
SOLITONIC
COATS
to the same form state of the baryon, carries a topological solution, for R = 0,
ON
THE
CHIRAL
BAG
91
as Eq. (11) by putting 9 = &&+$ the kink solution of the sine-Gordon quantum number of one as the bag is
@(X)=&(X){ -rr++tan’[expK~(.~)~]~, where E(X) = s/jxl. The energy of the soliton, for R = 0, is given by E(0) = F,Z1’:
(;)’
d.~= 8KFf.
When R # 0, the matching conditions at the boundary determine 8(R). The singleparticle spectrum for the fermions in the bag is given by [3] c,,=
17I
( 1 n+2
z+G‘
0
where n = 0, + 1, ) 2, etc., and 8 = t[tl(R) - 0( -R)]. This spectrum has spectral asymmetry for nonzero 8, resulting in a fractional fermion number. For example, in the range 0 < 0
E Bag=-&+2RB. Furthermore, the continuity demands that
of the boundary F; = (n,.47c).
condition
at x = + R as R -+ 0 (22)
Both Eqs. (21) and (22) remain identical irrespective of the details of the bosonic Lagrangian although B(R) does depend on the latter. Our purpose is to calculate the (bag + soliton) energy versus R when PC,,, in Eq. (18) is of the form (13) obtained in Section II, and compare with Zahed’s model. We also repeat this calculation with another solvable coating, in each case demanding that E(O), the soliton mass in the absence of the bag, remain the same. We find that although the
98
LI,
MITCHELL,
AND
magnitude of E(R) varies from model the range 0.5 fm
BHADURI
to model, it is not a sensitive function of R in this sense the Cheshire-cat picture still holds of the calculation with the Lagrangian given the Zahed model is that we take in Eq. (18)
2&,, = F,’ T [(ape)’ - m’( 1 + cos tI)2].
(23)
This is of the same form as Eq. (13) if we put (24)
the same substitutions as in the sine-Gordon Lagrangian (19). The calculation in the fermion sector yields Eq. (21) as in Zahed [3, 61 and will not be repeated here. In the boson sector, the equation of motion from Eqs. (23) and (18) is (t3,O)’ - m*( 1 + cos 13)sin 0 = 0,
Ix]> R.
(25)
The boundary condition at the bag surface gives E(X) F;g=
1
_- Q-y5e-iey5q, 2
at
1x1= R.
As before, E(X) =x/1x1. To simplify the right-hand side, note that (as in the Zahed model) the fermion sector boundary condition is e -ievsq = -k(x)
at
ylq,
Ixl=R.
Using the relation ysyl = --y,,, we obtain at 1x( = R de(x)
F”‘z=2c
1
q+q=$&,
(26)
sea where 8 = $(0(R) - 0(-R)).
The static solution of Eq. (25) is given by
-dW) = m(i + cos dx
Taking this derivative determines O(R):
to be continuous
F,2m(l +cosB)=g,
e(x)),
(xl> R.
at the surface yields the equation
Ix]= R.
that
(28)
NEW
SOLITONIC
COATS
ON THE
Note that in the hmit of R + 0, the solitary-wave
CHIRAL
BAG
99
solution is (see Eq. (14))
0(x) = 2 tan’(mx). Substituting
this in Eq. (28) as R -+ 0 gives
as stated in Eq. (22). It is also straightforward sector from Eq. (27):
Esoliton = 2 X2 jr
[ (2)’
to calculate the energy in the soliton
+ m’( 1 +
COS
B)‘] dx.
A little algebra yields E sollton= 2mF,Z[n - 8(R) - sin B(R)].
(29)
The total energy of the (bag + soliton), derived from the Lagrangians (18) and (23 ), is given by !@ -- 4xR
E, -~aryon
+ 2RB + 2rnFz[r - H - sin 01,
(30)
where the first two terms come from the fermion sector in the bag and are formally identical to Zahed’s expression. Note, however, that B(R) is now determined from Eq. (28) and is not the same as the sine-Gordon case. In another variation of the calculation, a different coat is taken from the fj”-Lagrangian:
In this case, the kink solution is given by d(x) = (2m,)/&
tanh($
m4x),
We make the transformation (32) such that for the kink 19(x) -+ +rt as x + f co. The Lagrangian (a,O)’ - 4m:
8 COS’ - . 2
(31) then reduces to (33)
100
LI, MITCHELL,
AND BHADURI
For no bag (R = 0), the energy of the kink is
16,,hm:
16fim
F2
E(o)=TT=F
‘$
n.
(34)
For nonzero bag radius R, B(R) is determined by the equation
(35) and the energy of the baryon is given by O2 Emaryon =nc47cR+2RB
+oTF2m4
2-3sini+sin’i
(36)
Note that as R --+0, Eq. (35) again yields the condition that Fz =n,/4~~in accordance with (22). The continuity of the boundary condition as R + 0 therefore fixes the value of I;, in the chiral bag irrespective of the soliton coat. Note that this is in agreement with the bosonisation condition (12) with g = 0, as is the case inside the bag. To make a numerical comparison of the three model calculations (with the coat in the baryon determined from Eqs. (19), (23), and (33)), we have to fix the parameters of the total Lagrangian (18). In each case, F, is taken to be ,/&%, with n, = 3. The remaining parameter in the soliton is fixed by demanding that its energy E(R)IRzo= E(0) be the same in each case. From Eqs. (20), (29), and (34), it then follows that
E(O)=8F,ZK=2mnF,2=716firn 4F2 II. 1.2
I
I
I
I --
1.0 -
XL-----J * 0.0
(37)
I Zahed - modified
1.5 2.0 2.5 3.0 R w FIG. 1. Variation of the ground-state baryon energy with the bag radius R for various solitonic coats. In each case, the parameters have been fixed to yield E(R = 0) = 0.76 GeV. For details, see the text. 0.5
1.0
101
NEW SOLITONIC COATSON THE CHIRAL BAG
Following Ref. [6], we take K = 2.04 fm ~~r, which fixes E(0) = 0.76 GeV, and gives m=2.60fm~ ’ and m,=2.17 fm--‘. In Fig. 1, the energy E(R) of the system is plotted as a function of R for the three cases. Again, following Ref. [6], the bag constant B is taken to be 0.015 GeV’. Note that an appropriate choice of the bag constant is still needed to obtain a relatively flat E(R) versus R curve. Moreover, an uncertain term of the form Z/R in the energy has been dropped. Keeping these limitations in mind, we see from the figure that E(R) is rather insensitive to R in the range 0.5 fm < R < 2 fm, irrespective of the bosonic coat outside. With the appropriate choice of the bag constant B and the solition mass E(O), the Cheshirecat phenomenon [ 5 J thus seems to hold approximately in the ( 1 + 1 )-dimensional bag even though the fermion sector inside the bag is not equivalent to the boson sector outside.
ACKNOWLEDGMENT This research
was supported
by the Natural
Sciences
and Engineering
Research
Council
REFERENCES 1. SHAU-JIN CHANG, S. D. ELLIS. AND B. W. LEE, Phw. Rev. D 11 (1975). 3572. 2. S. COLEMAN, Phys. Rev. D 11 (1975), 2088. 3. I. ZAHED, Phvs. Rev. D 30 (1984), 2221. 4. 1. ZAHED AND D. KLABUCAR, Phys. Rev. D 30 (1984). 2647. 5. S. NADKARNI. H. B. NIELSEN, AND 1. ZAHED, Nucl. Phys. B 253 (1985), 308. 6. J. K~GUT AND L. SUSSKIND, Phys. Reo. D 11 (1975), 3594. 7. R. RAJARAMAN, “Solitons and Instantons,” p. 34, North-Holland, Amsterdam, 1982. 8. S. COLEMAN, “Aspects of Symmetry,” p, 250, Cambridge Univ. Press, Cambridge, 1985.
of Canada.