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The H-dibaryon in hidden gauge theory
Volume 191, number 1,2
PHYSICS LETTERS B
4 June 1987
THE H-DIBARYON IN H I D D E N GAUGE THEORY ~ J. K U N Z and D. M A S A K
Institutfiir TheoretischePhysik, UniversittitGiessen,D-6300 Giessen,Fed. Rep. Germany Received 31 January 1987
Classical solutions in SU (3)v hidden gauge theory are investigated. Unstable SO(3) solutions in the B = 2 and B = 4 sectors converge towards saddle points of a massive SU (3). The H-dibaryon corresponds to the lowest-lyingSO(3) solution. Its properties are calculated and the problem of quantization is discussed.
1. Introduction It is believed that the low-energy behaviour of Q C D can be well described by an effective chiral lagrangian. In such a model baryons emerge as topological solitons, whose winding number is identified with the baryon number B [ 1,2]. In a chiral model with Nf flavours the classical soliton solutions U(r) are constructed by demanding a generalized spherical symmetry [ 3 ] : -i(r×V
),U(r) + [A,, U(r)] - - 0 ,
(1)
where the At generate an SU (2) or SO(3) subgroup o f SU(Nf). Here we consider N f = 3. The classical S U ( 2 ) soliton is embedded in S U ( 3 ) choosing A t = 1).i ( i = 1, 2, 3),
U(r)=(exp[ifoO(r)]
O) 1
(2) "
This Ansatz leads to the (probably) lowest state in the B = 1 sector. After quantization one obtains a reasonable description of the baryon properties [4,5 ]. Note, that the classical energy of solutions with B > 1 rapidly increases with B, e.g. E s = 2 ~ 3 E s = l [1]. Choosing the SO (3) subgroup with A 1=).7, A2 = - 2s and A3 =).2 the classical Ansatz satisfying eq. (1) is [ 3 ]
U(r) =exp{ iPAO+ i[ ( PA) 2 - ~ ]~5}.
(3)
It leads to states with B--2, 6, 10 .... and B = 4, 8, 12 .... and the remarkable relations for the classical energies [3,6-9] E 8 = 2 = 1.92E~=1 and E B _ 4 = 4 E s = l . The SO(3) dibaryon state of the Skyrme model has been identified with the H-dibaryon, predicted in the M I T bag model [ 10 ], which is a six quark b o u n d state with q u a n t u m numbers J " = 0 +, I = 0, Y-- 0 and with mass Mn = 2150 MeV. However, the Skyrme model predictions o f Mn are rather poor and uncertain [8,11 ]. We expect a general improvement o f the Skyrme model predictions, when the low-lying vector mesons are included in the effective lagrangian. We here study the SO (3) soliton solutions obtained in an S U ( 3 ) v hidden gauge model [ 12,13 ]. We obtain similar results for the SO (3) solutions as earlier [ 14 ] for the SU (2) solutions. In the B = 2 sector (and also in the B = 4 sector) a family o f unstable solutions exists, that converges towards Work supported by DFG under contract Ku612/1-1, BMFT and GSI Darmstadt. 174
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 191, number 1,2
PHYSICS LETTERS B
4 June 1987
a saddle point configuration of a massive SU (3) [ 15 ]. Stability of the solutions can be achieved by adding a Skyrme term. 2. Model and Ansgitze
We choose the lagrangian [ 12-14] ~= - ~
( T r [ D , ~ L ~ - - D , ~ R ~ ] 2 + 2 Tr [ D , ~ L ~ + D , ~ R ~ ] 2)
-- ( 1 / 2 g 2) (Tr[F,,] 2) + (1/32e2) ~c~Sky. . . .
(4)
The factor 2 in front of the second term was introduced to insure the KSFR relation for the [ SU (3) v] ~ocalmodel [ 12 ]. The covariant derivatives are defined by Du~L(R) = [0~ --iV#(x)]~L(R)(X) ,
Fuu=OIzVv-GVu-i[Vu, G]
•
(5a,b)
The Wess-Zumino term is omitted, since it does not contribute to the soliton solutions [8]. After gauge fixing,
¢~ =¢R =4, the chiral matrix U is related to the variable ~ via
The SO (3) ansatz for ~ is obtained from eq. (3), = exp(½iq~) + i sin( ½8) A~ exp( - ~iq~) + [cos( ½0) exp( - +iq~) - exp(½iq~)] (A~) 2.
(6a)
The corresponding vector field Ansatz is the Wu-Wu ansatz [ 6] V C -- - ~gp~2~ 1 a
= [ G ( r ) / 2 r ] ( ~ × A ) ¢ + [ H ( r ) / 2 r ] [(/xA)~, (Af)] + ,
V0 = 0 .
(6b,c)
These Ans~itze lead to the equations of motion (here without the Skyrme term) 4 r 2 G " = ( G - 2 ) [ 7 H 2 + G ( G - 4 ) ] + 8g2f~ t a [ G - 2 + 2 cos(½0) cos(½#)],
(7a)
4 r 2 H " = H [ H 2 + 7 G ( G - 4 ) +24] + 8gRf~ r 2 [ H - 2 cos(½0) sin(½~b)],
(7b)
r 2 0 " = - - 2 r O ' - 4 H c o s ( ½ 0 ) sin(½#) - 4 G sin(½0) c o s ( ½ ~ ) - 2 sin 0 cos # + 8 sin(½0) cos(½#),
(7c)
r20" = - 2 r # ' - 12H sin(½0) cos(½~a) - 12G cos(½0) sin(½0) - 6 cos 0 sin ~ + 2 4 cos(½0) sin(½0) .
(7d)
The classical energy is given by E = J d3r (~f~ i 2 [8 ,2 + ½0,2 + (4/r2)(1 - c o s 0 cos 0)] + (1/8g2r4){8g2rZ(G '2 + H '2) + G Z ( G - 4 ) 2 + H 2 [ 1 4 ( G - 2 ) 2 + H 2 - 8 ] }
× (2f2~/r2){[H-2 sin(½0) sin(½O)]2+ [ G - 2 + 2 cos(½0) cos(½~)]2}) .
(8)
For the lowest B = 2 state we obtain the boundary conditions
0(oo)=O(oo)=0, 0(0)=0(0) =re,
(9a,b)
G(oo) = H ( o o ) = 0 ,
(9c,d)
G(0) = H ( 0 ) = 2 .
175
Volume 191, number 1,2
PHYSICS LETTERS B
4 June 1987
A degenerate state [ 8 ] is obtained by choosing 0(0) = -z~, H ( 0 ) = - 2 . For the B = 4 state the boundary conditions are 0(~) =0(co)=0,
0(0)-2z¢=0(0) =0,
G(oo) = H ( c o ) = 0 ,
G(0)-4=H(0)
(10a,b)
=0.
(lOc,d)
The baryon density reduces to B ° = - (1/21r2r=)[0'(1 - c o s 0 cos 0) + 0 ' sin 0 sin 0] ,
(11)
which leads to B = 2 and B = 4 , respectively. .
Classical solutions: B = 2. We solved the set of equations (7) choosing the physical parameters f= = 93 MeV and g = 5.85 As in the SU (2) case, we obtained a family of solutions. In addition to the lowest-energy solution, where all the fields are monotonously decreasing, excited states exist, where the two chiral functions 0, 0 develop simultaneously nodes. The solutions converge towards a limiting configuration, where the vector functions G and H become singular at the origin, unless the chiral functions exhibit a spike. The situation is thus analogous to the SU (2) case, where the solutions converged towards the hadroid/sphaleron ( H/S ) configuration [ 14,17,18 ]. We therefore expect a saddle point solution in the SU(3) Yang-Mills-Higgs system [ 15 ]. In figs. 1-3 the fields are shown with no node, one node and in the limiting H/S configuration, the classical energies and the RMS radii are shown in table 1. To address, the question of classical stability of these solutions, we introduced the more general two-parameter Ansatz for the vector field (A,:=A~): V~'~'P)= ( G / 2 r ) { ( 1 - c o s
c~ cos # ) ( i × A ) ~ + i sin ot cosfl [(~×A)o, Ar]_ }
+ [ H sin fl/2r] {2 cos ot ~cA2 - cos o~ [Ac, At] + - sin oL [ ( P × A ) c, Ar] + } , V~'~'m = 0 .
(12)
For o~= ½n = - f l V~",p) reduces to the Ansatz (6b). We calculated the second derivatives of the energy with respect to ot and fl, 02E/O# 2 and OZE/OoL2. The results are shown in table 1. As expected the limiting H/S conI
I
I
I
I
I
I
I
I
I
" < < ~ ..
0
-1 I
I
I
I
0.2
0.4
0.6
0.8
r [fm] Fig. 1. Shown are the vector functions G(r) (solid) and H(r) (dashed) and the chiral functions O(r) (dotted) as well as O(r) (dash-dotted) for the e = ~ case with no nodes. The fields are in dimensionless units, r is in fm.
176
]..
.......;-.> ....
\.
I
-I
,
\ I
~
/ ~r
0.2
I
I
I
0.4
0.6
0.8
r [fm] Fig. 2. Same as fig. 1 for the solution with one node in both chiral functions (e = oo).
Volume 191, number 1,2
PHYSICS LETTERSB
4 June 1987
2
1
22
0
-I
0
0.2
0.4
0.6
0.8
[fm]
Fig. 3. Same as fig. 1 for the hadroid/sphaleron solution. Note that the chiral functions take the value of n at r - 0 and 0 elsewhere.
figuration is unstable. The B = 2 solutions with no and one node are also unstable with respect to this variation. one can reduce the energy c o n t r i b u t i o n from the vector field to zero, by changing the parameters from a = - f l - ½ ~ z to a = f l = 0 . The chiral fields will then collapse to a spike, Adding the fourth-order Skyrme term leads to a lower topological b o u n d for the energy:
E>~6~2(f~/e)B . For small enough e the solutions become probably globally stable. The influence of the Skyrme term on the classical solutions is also demonstrated in table 1. Comparing the energies of these solutions with the SU (2) solutions [ 14 ], we observe the approximate relation EB=2,~ 1.98EB= l, Classical solutions." B = 4. The classical solutions in the B = 4 sector have O ( r ) - H ( r ) = 0 ; the equations for O(r) and G(r) thus reduce to the equations for the S U ( 2 ) soliton, if the replacement O(r)--20su(z)(r), G(r) = 2G sU(2) (r) is made. Thus all earlier results [ 14], correspondingly modified, hold. We also obtain the energy relation Es= 4 = 4E8= ~, consequently these solutions can decay into B = 2 solutions [ 8 ].
4. Quantization The standard way to quantize the classical solutions of a chiral lagrangian is the collective coordinate method applied by Adkins, Nappi a n d Witten [ 19] to the S U ( 2 ) baryons. For S O ( 3 ) c S U ( 3 ) the classical solutions to the quantized energy become [ 8] Equantized= Ed,ssical + J( J + 1 )/21+ 2 [p2 + q2 + 3 (p + q) +pq]/3I',
(13)
Table 1 Shown here are the energiesE and the second derivatives E~, Ea~ of the energieswith respect to the parameters in eq. (12), all are given in MeV. Also the quantal corrections due to vibrations are listed as AEq.Additionally given are the RMS radii in fm. The hadroid/sphaleron is labelled H/S and the other solutions by S~ where e refers to the Skyrme parameter and n to the number of nodes in the chiral functions.
H/S S~ S~ S~0
Ed
AEq
E~
Eaa
RMS
3361 2068 2931 2496
72 97 76
-831 -272 -514 119
-627 -154 -370 82
0.51 0.15 0.58
177
Volume 191, number 1,2
PHYSICS LETTERSB
4 June 1987
where (p, q) labels the appropriate irreducible representation of SU (3). For the flavour singlet (p, q) = (0, 0) we get a J = 0 ground state, such that Equantize d = Eclassica I •
This corresponds to the H-dibaryon. By considering vibrational excitations of the type [20] ~(r)--,~(r e~) a n d V~,(r)~e ~ V~,(r ez) we obtain small q u a n t u m corrections to the energy shown in table 1.
5. Conclusions In the SO(3) subgroup of S U ( 3 ) we have constructed finite-energy solutions for a chiral lagrangian with vector mesons. The non-trivial b o u n d a r y conditions of the vector field functions in the h i d d e n gauge model at the origin lead to solutions even without stabilization term. But these solutions do not correspond to m i n i m a . They converge towards saddle point solutions of a massive S U ( 3 ) theory. Global m i n i m a can be obtained by adding a Skyrme term. To obtain reliable predictions for the properties of the constructed H-dibaryon, an SU (3) calculation of the baryon spectrum with symmetry breaking mass terms in the presence of vector mesons is necessary first.
References [ 1] T.H.R. Skyrme, Proc. R. Soc. (London) A 260 (1961 ) 127;Nucl. Phys. 31 (1962) 556. [2] A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stern, Phys. Rev. Lett. 49 (1982) 1124;Phys. Rev. D 27 (1983) 1153. [3] A.P. Balachandran, V.P. Nair, N. Panchapakesan and S.G. Rajeev, Phys. Rev. D 28 (1983) 2830. [4] G.S. Adkins and C.R. Nappi, Nucl. Phys. B 249 (1985) 507. [5] M. Chemtob, Nucl. Phys. B 256 (1985) 600. [6] A.P. Balachandran, A. Barducci, F. Lizzi, V.G.J. Rodgers and A. Stern, Phys. Rev. Lett. 52 (1984) 887. [7] H. Gomm, F. Lizzi and G. Sparano, Phys. Rev. D 31 (1985) 226. [8] A.P. Balachandran, F. Lizzi, V.G.J. Rodgers and A. Stern, Nucl. Phys. B 256 (1985) 525. [9 ] R.L. Jaffe and C.L. Korpa, Nucl. Phys. B 258 (1985) 468. [ I0] R.L. Jaffe, Phys. Rev. Lett. 38 (1977) 195. [ 11] S.A.Yost and C.R. Nappi, Phys. Rev. D 32 (1985) 816. [ 12] M. Bando, T. Kugo, S. Uehara, K. Yamawakiand T. Yanagida,Phys. Rev. Lett. 54 (1985) 1215. [ 13] T. Fujiwara, T. Kugo, H. Terao, S. Uehara and K. Yamawaki, Progr. Theor. Phys. 73 (1985 ) 926. [ 14] J. Kunz and D. Masak, Phys. Len. B 179 (1986) 146. [ 15] J. Kunz and D. Masak, in preparation. [ 16] A.C.T. Wu and T.T. Wu, J. Math. Phys. 15 (1974) 53. [ 17] J. Boguta, Phys. Rev. Lett. 50 (1983) 148. [I 8] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212. [ 19] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [20] L.C. Biedenharn, Y. Dothan and M. Tarlini, Phys. Rev. D 31 (1985) 649.
178
PHYSICS LETTERS B
4 June 1987
THE H-DIBARYON IN H I D D E N GAUGE THEORY ~ J. K U N Z and D. M A S A K
Institutfiir TheoretischePhysik, UniversittitGiessen,D-6300 Giessen,Fed. Rep. Germany Received 31 January 1987
Classical solutions in SU (3)v hidden gauge theory are investigated. Unstable SO(3) solutions in the B = 2 and B = 4 sectors converge towards saddle points of a massive SU (3). The H-dibaryon corresponds to the lowest-lyingSO(3) solution. Its properties are calculated and the problem of quantization is discussed.
1. Introduction It is believed that the low-energy behaviour of Q C D can be well described by an effective chiral lagrangian. In such a model baryons emerge as topological solitons, whose winding number is identified with the baryon number B [ 1,2]. In a chiral model with Nf flavours the classical soliton solutions U(r) are constructed by demanding a generalized spherical symmetry [ 3 ] : -i(r×V
),U(r) + [A,, U(r)] - - 0 ,
(1)
where the At generate an SU (2) or SO(3) subgroup o f SU(Nf). Here we consider N f = 3. The classical S U ( 2 ) soliton is embedded in S U ( 3 ) choosing A t = 1).i ( i = 1, 2, 3),
U(r)=(exp[ifoO(r)]
O) 1
(2) "
This Ansatz leads to the (probably) lowest state in the B = 1 sector. After quantization one obtains a reasonable description of the baryon properties [4,5 ]. Note, that the classical energy of solutions with B > 1 rapidly increases with B, e.g. E s = 2 ~ 3 E s = l [1]. Choosing the SO (3) subgroup with A 1=).7, A2 = - 2s and A3 =).2 the classical Ansatz satisfying eq. (1) is [ 3 ]
U(r) =exp{ iPAO+ i[ ( PA) 2 - ~ ]~5}.
(3)
It leads to states with B--2, 6, 10 .... and B = 4, 8, 12 .... and the remarkable relations for the classical energies [3,6-9] E 8 = 2 = 1.92E~=1 and E B _ 4 = 4 E s = l . The SO(3) dibaryon state of the Skyrme model has been identified with the H-dibaryon, predicted in the M I T bag model [ 10 ], which is a six quark b o u n d state with q u a n t u m numbers J " = 0 +, I = 0, Y-- 0 and with mass Mn = 2150 MeV. However, the Skyrme model predictions o f Mn are rather poor and uncertain [8,11 ]. We expect a general improvement o f the Skyrme model predictions, when the low-lying vector mesons are included in the effective lagrangian. We here study the SO (3) soliton solutions obtained in an S U ( 3 ) v hidden gauge model [ 12,13 ]. We obtain similar results for the SO (3) solutions as earlier [ 14 ] for the SU (2) solutions. In the B = 2 sector (and also in the B = 4 sector) a family o f unstable solutions exists, that converges towards Work supported by DFG under contract Ku612/1-1, BMFT and GSI Darmstadt. 174
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 191, number 1,2
PHYSICS LETTERS B
4 June 1987
a saddle point configuration of a massive SU (3) [ 15 ]. Stability of the solutions can be achieved by adding a Skyrme term. 2. Model and Ansgitze
We choose the lagrangian [ 12-14] ~= - ~
( T r [ D , ~ L ~ - - D , ~ R ~ ] 2 + 2 Tr [ D , ~ L ~ + D , ~ R ~ ] 2)
-- ( 1 / 2 g 2) (Tr[F,,] 2) + (1/32e2) ~c~Sky. . . .
(4)
The factor 2 in front of the second term was introduced to insure the KSFR relation for the [ SU (3) v] ~ocalmodel [ 12 ]. The covariant derivatives are defined by Du~L(R) = [0~ --iV#(x)]~L(R)(X) ,
Fuu=OIzVv-GVu-i[Vu, G]
•
(5a,b)
The Wess-Zumino term is omitted, since it does not contribute to the soliton solutions [8]. After gauge fixing,
¢~ =¢R =4, the chiral matrix U is related to the variable ~ via
The SO (3) ansatz for ~ is obtained from eq. (3), = exp(½iq~) + i sin( ½8) A~ exp( - ~iq~) + [cos( ½0) exp( - +iq~) - exp(½iq~)] (A~) 2.
(6a)
The corresponding vector field Ansatz is the Wu-Wu ansatz [ 6] V C -- - ~gp~2~ 1 a
= [ G ( r ) / 2 r ] ( ~ × A ) ¢ + [ H ( r ) / 2 r ] [(/xA)~, (Af)] + ,
V0 = 0 .
(6b,c)
These Ans~itze lead to the equations of motion (here without the Skyrme term) 4 r 2 G " = ( G - 2 ) [ 7 H 2 + G ( G - 4 ) ] + 8g2f~ t a [ G - 2 + 2 cos(½0) cos(½#)],
(7a)
4 r 2 H " = H [ H 2 + 7 G ( G - 4 ) +24] + 8gRf~ r 2 [ H - 2 cos(½0) sin(½~b)],
(7b)
r 2 0 " = - - 2 r O ' - 4 H c o s ( ½ 0 ) sin(½#) - 4 G sin(½0) c o s ( ½ ~ ) - 2 sin 0 cos # + 8 sin(½0) cos(½#),
(7c)
r20" = - 2 r # ' - 12H sin(½0) cos(½~a) - 12G cos(½0) sin(½0) - 6 cos 0 sin ~ + 2 4 cos(½0) sin(½0) .
(7d)
The classical energy is given by E = J d3r (~f~ i 2 [8 ,2 + ½0,2 + (4/r2)(1 - c o s 0 cos 0)] + (1/8g2r4){8g2rZ(G '2 + H '2) + G Z ( G - 4 ) 2 + H 2 [ 1 4 ( G - 2 ) 2 + H 2 - 8 ] }
× (2f2~/r2){[H-2 sin(½0) sin(½O)]2+ [ G - 2 + 2 cos(½0) cos(½~)]2}) .
(8)
For the lowest B = 2 state we obtain the boundary conditions
0(oo)=O(oo)=0, 0(0)=0(0) =re,
(9a,b)
G(oo) = H ( o o ) = 0 ,
(9c,d)
G(0) = H ( 0 ) = 2 .
175
Volume 191, number 1,2
PHYSICS LETTERS B
4 June 1987
A degenerate state [ 8 ] is obtained by choosing 0(0) = -z~, H ( 0 ) = - 2 . For the B = 4 state the boundary conditions are 0(~) =0(co)=0,
0(0)-2z¢=0(0) =0,
G(oo) = H ( c o ) = 0 ,
G(0)-4=H(0)
(10a,b)
=0.
(lOc,d)
The baryon density reduces to B ° = - (1/21r2r=)[0'(1 - c o s 0 cos 0) + 0 ' sin 0 sin 0] ,
(11)
which leads to B = 2 and B = 4 , respectively. .
Classical solutions: B = 2. We solved the set of equations (7) choosing the physical parameters f= = 93 MeV and g = 5.85 As in the SU (2) case, we obtained a family of solutions. In addition to the lowest-energy solution, where all the fields are monotonously decreasing, excited states exist, where the two chiral functions 0, 0 develop simultaneously nodes. The solutions converge towards a limiting configuration, where the vector functions G and H become singular at the origin, unless the chiral functions exhibit a spike. The situation is thus analogous to the SU (2) case, where the solutions converged towards the hadroid/sphaleron ( H/S ) configuration [ 14,17,18 ]. We therefore expect a saddle point solution in the SU(3) Yang-Mills-Higgs system [ 15 ]. In figs. 1-3 the fields are shown with no node, one node and in the limiting H/S configuration, the classical energies and the RMS radii are shown in table 1. To address, the question of classical stability of these solutions, we introduced the more general two-parameter Ansatz for the vector field (A,:=A~): V~'~'P)= ( G / 2 r ) { ( 1 - c o s
c~ cos # ) ( i × A ) ~ + i sin ot cosfl [(~×A)o, Ar]_ }
+ [ H sin fl/2r] {2 cos ot ~cA2 - cos o~ [Ac, At] + - sin oL [ ( P × A ) c, Ar] + } , V~'~'m = 0 .
(12)
For o~= ½n = - f l V~",p) reduces to the Ansatz (6b). We calculated the second derivatives of the energy with respect to ot and fl, 02E/O# 2 and OZE/OoL2. The results are shown in table 1. As expected the limiting H/S conI
I
I
I
I
I
I
I
I
I
" < < ~ ..
0
-1 I
I
I
I
0.2
0.4
0.6
0.8
r [fm] Fig. 1. Shown are the vector functions G(r) (solid) and H(r) (dashed) and the chiral functions O(r) (dotted) as well as O(r) (dash-dotted) for the e = ~ case with no nodes. The fields are in dimensionless units, r is in fm.
176
]..
.......;-.> ....
\.
I
-I
,
\ I
~
/ ~r
0.2
I
I
I
0.4
0.6
0.8
r [fm] Fig. 2. Same as fig. 1 for the solution with one node in both chiral functions (e = oo).
Volume 191, number 1,2
PHYSICS LETTERSB
4 June 1987
2
1
22
0
-I
0
0.2
0.4
0.6
0.8
[fm]
Fig. 3. Same as fig. 1 for the hadroid/sphaleron solution. Note that the chiral functions take the value of n at r - 0 and 0 elsewhere.
figuration is unstable. The B = 2 solutions with no and one node are also unstable with respect to this variation. one can reduce the energy c o n t r i b u t i o n from the vector field to zero, by changing the parameters from a = - f l - ½ ~ z to a = f l = 0 . The chiral fields will then collapse to a spike, Adding the fourth-order Skyrme term leads to a lower topological b o u n d for the energy:
E>~6~2(f~/e)B . For small enough e the solutions become probably globally stable. The influence of the Skyrme term on the classical solutions is also demonstrated in table 1. Comparing the energies of these solutions with the SU (2) solutions [ 14 ], we observe the approximate relation EB=2,~ 1.98EB= l, Classical solutions." B = 4. The classical solutions in the B = 4 sector have O ( r ) - H ( r ) = 0 ; the equations for O(r) and G(r) thus reduce to the equations for the S U ( 2 ) soliton, if the replacement O(r)--20su(z)(r), G(r) = 2G sU(2) (r) is made. Thus all earlier results [ 14], correspondingly modified, hold. We also obtain the energy relation Es= 4 = 4E8= ~, consequently these solutions can decay into B = 2 solutions [ 8 ].
4. Quantization The standard way to quantize the classical solutions of a chiral lagrangian is the collective coordinate method applied by Adkins, Nappi a n d Witten [ 19] to the S U ( 2 ) baryons. For S O ( 3 ) c S U ( 3 ) the classical solutions to the quantized energy become [ 8] Equantized= Ed,ssical + J( J + 1 )/21+ 2 [p2 + q2 + 3 (p + q) +pq]/3I',
(13)
Table 1 Shown here are the energiesE and the second derivatives E~, Ea~ of the energieswith respect to the parameters in eq. (12), all are given in MeV. Also the quantal corrections due to vibrations are listed as AEq.Additionally given are the RMS radii in fm. The hadroid/sphaleron is labelled H/S and the other solutions by S~ where e refers to the Skyrme parameter and n to the number of nodes in the chiral functions.
H/S S~ S~ S~0
Ed
AEq
E~
Eaa
RMS
3361 2068 2931 2496
72 97 76
-831 -272 -514 119
-627 -154 -370 82
0.51 0.15 0.58
177
Volume 191, number 1,2
PHYSICS LETTERSB
4 June 1987
where (p, q) labels the appropriate irreducible representation of SU (3). For the flavour singlet (p, q) = (0, 0) we get a J = 0 ground state, such that Equantize d = Eclassica I •
This corresponds to the H-dibaryon. By considering vibrational excitations of the type [20] ~(r)--,~(r e~) a n d V~,(r)~e ~ V~,(r ez) we obtain small q u a n t u m corrections to the energy shown in table 1.
5. Conclusions In the SO(3) subgroup of S U ( 3 ) we have constructed finite-energy solutions for a chiral lagrangian with vector mesons. The non-trivial b o u n d a r y conditions of the vector field functions in the h i d d e n gauge model at the origin lead to solutions even without stabilization term. But these solutions do not correspond to m i n i m a . They converge towards saddle point solutions of a massive S U ( 3 ) theory. Global m i n i m a can be obtained by adding a Skyrme term. To obtain reliable predictions for the properties of the constructed H-dibaryon, an SU (3) calculation of the baryon spectrum with symmetry breaking mass terms in the presence of vector mesons is necessary first.
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