- Email: [email protected]
Parity-violating chiral solitons with vector mesons
Volume 214, number 3
PHYSICS LETTERS B
24 November 1988
PARITY-VIOLATING CHIRAL SOLITONS WITH VECTOR MESONS Y. I G A R A S H I ~,n2, A. K O B A Y A S H I u, H. O T S U c a n d S. S A W A D A c '~ Max-Planck Institute for Physics and Astrophysics, D-8000 Munich 40, Fed. Rep. Germany b Faculty of Education, Niigata University, Niigata 950-21, Japan Department of Physics, Nagoya University, Nagoya 464, Japan Received 31 May 1988; revised manuscript received 30 August 1988
The presence of a "phase transition" between the conventional parity-conserving solutions and the spontaneous parity-violating ones is confirmed in the Skyrme model with P, and in a model with P and co. The parity-violating solutions lead to almost degenerate parity-doublets at the semi-classical level. Since this makes Skyrme's picture of low-lying hadrons troublesome, we obtain a non-trivial constraint for couplings in chiral soliton models with vector mesons.
1. Introduction
In the large-No limit [ 1 ] Q C D is supposed to reduce to a non-linear meson theory which contains the N a m b u - G o l d s t o n e pions as a primary field in the extremely low-energy limit, at energies much less than AocD. Since the energy scale which characterizes baryon generation as topological solitons is of order AOCD,the low-lying vector mesons with masses less than those of nucleons seem to play a particularly important role in the unified description o f mesons and baryons originating from Skyrme's pioneering work [ 2 ] ~~. Actually the importance of the vector mesons in hadron physics at the length scale down to 0.5 fm is well recognized by, for example, the success of the vector meson dominance model in the electromagnetic interactions as well as o f the one-boson exchange model for nuclear forces [4]. In the past few years an attractive approach based on hidden local symmetry [ 5 ] ~2, where the p meson emerges as a dynamical gauge boson of a hidden symmetry [ SU (2) ] ~ocal, has been used for the construction o f chiral soliton models coupled with vector mesons [ 7 Humboldt Fellow. On leave of absence from Faculty of Education, Niigata University, Niigata 950-21, Japan. ~ See ref. [3] for recent reviews. ,2 For a recent review, see ref. [6].
0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
9 ]. A variety of vector coupled solitons has been investigated so far ~3 We stress here that the presence o f the p field leads to a non-trivial constraint on parameters in the soliton models in order that they make sense as an effective low-energy theory o f QCD. It is related with a new aspect of the soliton solutions, reported in a previous letter [ 11 ] (hereafter referred to as I), the occurrence of spontaneous parity-violation at the purely classical level or o f almost degenerate parity-doublets at the semi-classical level. The new class of soliton solutions with topological (baryon) number Q = B = 1 was obtained with the most general spherically symmetric ansatz for the p-field which consists both of parity-even and -odd terms. It was found in the pcoupled chiral model with fourth-order derivative terms that the spontaneous parity-violating (SPV) solutions with both parity-even and -odd terms become stable minima when a coupling o f the fourthorder term e is larger than a critical value ec. The conventional parity-conserving ( P C ) solutions with the parity-odd term only, on the other hand, are minima for e < ec. For e > eo the parity-invariance may be restored via tunneling effects at the semi-classical level, resolving the exact degeneracy in the SPV states. Since the mixing transition was found to be small, an almost degenerate parity-doublet may appear in the ~3 See ref. [ 10] for a recent review.
445
Volume 214, number 3
PHYSICS LETTERS B
low-lying baryonic states. In order to avoid the parity doublet, the coupling strength must be e < e~. These results were obtained using a variational calculation in I. In this paper we confirm the presence of the "phase transition" between the PC and SPV solutions in Skyrme's model coupled with the p by exactly solving the Euler-Lagrange equations for static masses. The numerical calculation evaluates the critical Skyrme coupling e~=6.8 with other parameters fixed from low-energy relations. We also examine whether the "phase transition" appears in a model of Meissner et al. [ 12 ] for the n-p-to system where the vector mesons have been introduced as dynamical gauge bosons of a hidden SU (2) × U ( 1 ) symmetry. For the p-to-coupled model, the parameters can be completely fixed in the meson sector. We find that the "phase transition" occurs at the critical value of the gauge coupling g~=4.8. The determined coupling strength g ~ 6 is larger than &, and therefore it is not in the parity-doubling phase, but in the conventional phase. Thus the model makes sense as an effective theory for QCD.
24 November 1988
has been fixed to be unitary as in I. This gives an extension of the Weinberg lagrangian [ 13 ]. The hedgehog ansatz we employ is the standard one
U(x) =exp[i(~.z)F(r) ] ,
(2)
for the chiral field, and the most general spherically symmetric one,
V~(x) =e~jkYcjzkG(r) /2r+ [z~--Yc~(~.z) ]H(r) /2r + £¢~(£¢.r)K(r) /2r ,
(3)
Vo=0 for the vector field, where i = x / r and r = Ixl. Both (2) and (3) are invariant under combined spatial and isospin rotations. Since (3) contains a parity-odd term as well as parity-even terms, it is not parity invariant. The static energy functional is given by
M[F,G,H,K] = - J
~d3x
=MA + M v +Mkin +MsK, oo
MA = 4ref2m~J| dr/½ (r/2F'2+2 sin2F) 0
Mv = 4nf~2 ~ dr/a[ ( ~ + c o s F ) E + H 2 + ½K2 ] ,
2. The model for the n-p system
mp
We consider a chiral SU (2) LX SU (2) R non-linear a-model coupled with the p-field [ 5,6,8 ] in the massless pion limit
Mkin --
d 0
41tf2fdr/a[(1/2r/2)(G2+H2-1)2 mp 0
+ ( ~ ' - H K / r / ) 2 + ( H ' + ~K/r/) 2 ] , (3O
~ 0A
2nmp
1 ~Tr(0~, 2 = ~f UOI'U*) ,
MSK =
0
~v = - ~af 2~Tr(D,,~ * + D~,~*~) 2 ,
(4)
-Wki. = -- ( 1/2g 2)Tr(F2.~) , LPsK= ( 1/ 32e 2) Tr [0a UU*, O. UU* ] 2,
( 1)
where U= (~*)2=exp[2ix(x)/f~], V~=gVa~(x)z ~ /2, D.=O~,-iVa, F a . = i [ D a , D~], andf~= 93 MeV is the pion decay constant. We have taken for simplicity the Skyrme term ~4 as a fourth-order term which is needed to stabilize soliton solutions [ 13 ]. The lagrangian ( 1 ) may be derived using the formulaton of hidden local symmetry in which the p is considered as a dynamical gauge boson. Here the hidden gauge ~4 The model described here was referred as (A) in I.
446
dr/[ ( 1/r/2)sin4F+2(F' sin F) 2 ] ,
e2
where G = G - 1, r/=mpr and the primes denote r/derivatives. We fix a = 2 , g = g p ~ = 5 . 8 5 in such a way that the KSRF relation mp2 =2gp~f~ 2 2 holds for the observed meson mass, rnp = 770 MeV. The only free parameter left is then the Skyrme coupling e. It follows from (4) that there exists a discrete symmetry, M[F,G,H,K] = M [ F , G , - H , - K ] . We may redefine with this a parity transformation as x , - , - x , G-,G, H,--,-H, K,--,-K, so that V ( x ) , - , - V ( - x ) . Obviously all the static solutions fall into two distinct classes: the PC ones in which H(r) =0, K(r) =0, and the SPV ones with non-vanishing H ( r ) and K(r). The
Volume 214, number 3
PHYSICS LETTERS B
latter appear as degenerate states (F,G,H,K) and ( F , G , - H , - K ) converted into each other u n d e r the parity transformation. We consider here the topological ( b a r y o n ) n u m b e r Q = B = I sector with the b o u n d a r y conditions F ( 0 ) = r r , G ( 0 ) = 2 , H ( 0 ) = K ( 0 ) = 0 , and F ( ~ ) = G ( ~ ) = H ( c o ) = K ( o o ) = 0 . We solve the stationary equations for the energy functional by using the relaxation method. The typical SPV solutions are shown in fig. 1. The resulting static masses are given in fig. 2 and table 1. It is found that the PC solutions are stable m i n i m a for e < e o while the SPV ones are so for e > ec. The "phase transition" occurs at the critical value e~= 6.8. This is only slightly smaller than the value obtained in I with a variational calculation, ec = 6.9 ~5 In order to get some insights into the PC a n d SPV solutions, we consider two special cases of the gauge field configurations which lead to (i) My = 0. This is a PC configuration where the Vi is expressed in terms of the V, = ( 1 / 2 i ) ( O , ~ * + 0v~*~) = ei,k2j rk ( 1 -- COSF ) / 2 r .
(5)
In this case the kinetic term of the gauge field Mk~, becomes equivalent to the Skyrme term MsK, i.e., gZMki o = e2MsK. (ii) Mk~,=0. This pure gauge limit is realized in a ~5Although we have not considered here the model (B) of I, the critical value should be almost equal to the one determined in I.
M(GeV)
2.0 PC
1.5
1.0
0.5
0
............... , ............. ; . . . . . . . . . .
20
7
r. . . . . . .
5
~
- e
4
Fig. 2. Static masses versus the couplinge in the model for the ~p system. The solid line represents the PC solutions and the SPV ones for e< e~.=6.8 and for e> ec, respectively. The dashed line represents PC solutions which are local minima or saddle points in the latter region. The partial masses are also shown. The dotted curve and dott-dashed curve represent Mk~,, and Mv, respectively. SPV configuration. For the spherically symmetric gauge function g ( x ) = e x p [ i ( a 2 . r ) f ( r ) ] , Vi= i g ( x ) × 0 i g ( x ) - ' is given by V,(x) = eijk2jrk ( 1 -- cos 2 J ) / 2 r + [ r ~ - - ~ ( ~ ' r ) ] (sin 2f)/2r+Yc,(.~.T)f' .
(6)
Table 1 Static masses (in MeV) versus the coupling e for the PC and SPV solution in the model for the n-p system
F
e 1.0
0
10
Roughly speaking, apart from the transition region e ~ ec, the PC and SPV solutions are approximately
3.0
2.0
24 November 1988
1.0
2.0 r/
~0
.
Fig. 1. The SPV solutions for the functions of (2), (3) in the model for the n - 9 system with e = 10.
4 6 8 10 15 20
Solution PC
SPV
2030 1574 1373 1266 1147 1100
1257 1019 689 523
447
Volume 214, number 3
PHYSICS LETTERS B
described by the field configurations (i) and (ii), respectively, as shown in fig. 2. For e > eo the SPV solutions become stable. At the semi-classical level, two degenerate states mix with one another via tunneling effects, making the mass difference between parityeven and parity-odd states, AM. Since A M / M is less than 1% as estimated in I with a variational method, the two states form an almost degenerate parity-doublet which is not relevant to the observed hadron spectrum.
24 November 1988
U(x) = e x p [ i ( . ~ ' z ) F ( r ) ] , V~(x) =%kYc/rkG(r)/2r+ [zi-:ci(~'z) ]H(r)/2r + Yci(.fc'OK( r) /2r ,
Vo(x) = 0 , ~o (x) = to [F,G,H,K] , o~(x) = 0 ,
(8)
leads to the energy functional
M[F,G,H,K] = - J £ad3x=Mo +M,o,
3. The model for the 7[-p-co system
Mo=-
Let us turn to the chiral soliton models with the co meson in addition to the p meson. They have been constructed by enlarging the hidden local symmetry to S U ( 2 ) v × U ( 1 ) or to U ( 3 ) v . For our purpose of discussing the occurrence of new soliton solutions, we consider here the model of Meissner et al. [ 12 ] in which the Wess-Zumino term is included in a simple manner:
f [£PA+~V(~-p-part)
+ ~ki. (p-part) ] d3x,
M,~=- J [A~(c0-mass) + ~ i n ( ~ - p a r t ) + ,,~¢wz] d 3 x , oo
(9)
Mo = 4~rf---~2j dr/{ ½(qZF'2 + 2 sin~F) mp
o
+a[ (~+cosF)Z+H2@½K
£,e= £#A+ ~V (n-p-part) + £aki.(p-part)
2]
+ (a/22) [ ( 1 / 2 q 2 ) ( ~ 2 + H 2_ 1)2
+ L#(co-mass) + £#ki.(w-part) + £awz,
+ (G'-HK/q)2+ (H' +(~K/r/)2] ) ,
£PA= ~/~Tr(0u UOuU*),
oo
£#v(x-p-part) = - ~af2=Tr(Du~ * + Ou~?~) 2 ,
m---o
r/ ~--,~2r/2
(2)
£#ki.(p-part) = - ( 1/2g 2)Tr(Fzu.) , L#(c0-mass) = £aui. (re_part) =
1"1°2 ¢"2 ~"'2
+~ ( i F ' sin2F 87[ \
i4 (.0/~v 2
_ _ l d [sin 2F((~2 + H 2 _ 1 ) ] ) c o ] , 2 dr/
Aawz= ½Ncgwu( 1/24x z) e u~,~#
x ( U*O. UU*O~UU*O#U) + ½Ncg( 1 / 167[2 ) ¢ uVa#OOuv ×Tr[iV¢,(O#UU*-8#U*U)+V~U*V#U] ,
(7)
where o~/,. = 0/,o~- 0.oJ/, is the field strength for the ¢o field, and the covariant derivative reads as before D/,=O/,-iVu(x ). We take a = 2 , N~=3. The gauge coupling constant g may be fixed by use of the co~ 3x decay width, or of the KSRF relation as before. We consider the gauge coupling g only in some range around the physical value g ~ 6. The spherically symmetric ansatz for U(x), Vu(x) and o~/,(x) 448
(10)
where 2=g/go, and go = 5.85 which is determined by the KSRF relation. The radial Green function if(t/, t/')
1 f#(q, r/' ) - 222qr/, × [ e x p ( - 2 1 r/-r/' I ) - e x p ( - A I r / + r/' I ) ] ,
(11)
solves the co field equation (-
d
2 d +22r/2)o)(q) =J(?])
(12)
Volume 214, number 3
PHYSICS LETTERS B
24 November 1988
Table 2 Static masses (in MeV ) as a function o f g in the model for the n p-t0 system
3.o I -500
F
g
g/go
2.0,
-300 >~ IE
2.93 4.10 4.68 5.85 8.78
1.0 -IO0
0.5 0.7 0.8 1.0 1.5
Solution PC
SPV
2182 1692 1567 1418 1283
1235 1469 1518 -
,
0
1.0
T/---~
3.0
ZO
Fig. 3. SPV solutions F, G, H (full curves ) and oJ (dashed curve ) in the model for the x-p-t0 system with g = 3.
table 2, which indicates that the "phase transition" appears at gc=4.8, below which the SPV solutions become stable. The physical value g ~ 6 is not in the parity-doublet phase but in the PC phase.
as
4. Concluding remarks
co(u) = i du' ~(n, ~' )J(u'),
(13)
o
where the source current is given by
3gmp
J(r/) = ~
{2F' sin2F
- ½(d/d~/) [sin
2F(O2+H 2- 1 ) ] } .
(14)
The typical SPV solutions are shown in fig. 3. The static M as a function of g are shown in fig. 4 and M(GeV)
2.0
xx
1.0
,g 4.0
6D
B.0
10.0
Fig. 4. Static masses versus the coupling g in the model for the n p-to system. The solid line represents the PC solutions and the SPV ones for g > g c = 4 . 8 and for g
We have confirmed in this note the existence of a "phase transition" between the PC solutions and the SPV ones in the p-coupled Skyrme model, and also in a model with p and to. In the former the critical Skyrme coupling turns to be ec= 6.8. When 1/e, proportional to the size of the Skyrme term, is smaller than 1/ec, the soliton states with topological (baryon) number = Q = B = 1 are in the parity-doubling phase. The critical gauge coupling in the latter model of Meissner et al. is gc = 4.8, which is smaller than the physical value g ~ 6 fixed from experimental data in the meson sector. Therefore, the model with g ~ 6 is not in the parity-doubling phase but in the conventional phase relevant to hadron physics. The vector field configuration we have obtained here can be interpreted as follows: in the SPV solutions it is approximately described by the pure gauge form so that the kinetic term of the gauge field vanishes, while the vector field in the PC ones has a similar configuration to that of the pionic vector current. Which configuration is realized depends on the strength of the soliton stabilizing term. When the contribution from the stabilizing term to the total static mass is too small, the SPV solutions become absolute minima. Our investigation indicates that the SPV solutions appear universally in chiral soliton models coupled with the p field, not depending on the stabilization mechanism. 449
Volume 214, number 3
PHYSICS LETTERS B
We h a v e c o n s i d e r e d o n l y the Q=B= 1 sector in w h i c h the c o u p l i n g to the p field s e e m s to be essential to the p r e s e n c e o f the SPV. H o w e v e r , the s i t u a t i o n m i g h t change for chiral solitons for m o r e t h a n three flavors w i t h h i g h e r t o p o l o g i c a l n u m b e r s . F o r e x a m ple, in the S U ( 3 ) - S k y r m e m o d e l the spherically symm e t r i c ansatz for the chiral variable carrying Q = B = 2 d o e s not respect n a i v e parity i n v a r i a n c e , a n d S P V solutions m a y a p p e a r e v e n w i t h o u t t h e c o u p l i n g to vectors.
Acknowledgement O n e o f us ( Y . I . ) w o u l d like to t h a n k the M a x P l a n c k - I n s t i t u t e for hospitality.
References [ 1] G. 't Hooft, Nuel. Phys. B 72 (1974) 461; E. Witten, Nucl. Phys. B 160 (1979) 57. [ 2 ] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 1961 ) 127; Nucl. Phys. 31 (1962) 556. [3] I. Zahed and G.E. Brown, Phys. Rep. 142 (1986) 1; A.P. Balachandran, Proc. Yale Theor. A.S.I., eds. M.J. Bowick and F. Gtirsey (World Scientific, Singapore, 1985 ) p.l.
450
24 November 1988
[ 4] J.J. Sakurai, Currents and mesons (Chicago U.P., Chicago, IL, 1969). [5 ] M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54 (1985) 1215. [6] M. Bando, K. Yamawaki and T. Kugo, Phys. Rep. 164 (1988) 217. [ 7 ] G.S. Adkins and C.R. Nappi, Phys. Lett. B 137 ( 1984 ) 251; G.S. Adkins, Phys. Rev. D 33 (1986) 193. [8] Y. Igarashi, M. Johmura, A Kobayashi, H. Otsu, T. Sato and S. Sawada, Nucl. Phys. B 259 (1985) 721; T. Fujiwara, Y. Igarashi, A. Kobayashi, H. Otsu, T. Sato and S. Sawada, Prog. Theor. Phys. 74 (1985) 128. [9] U.G. Meissner and I. Zahed, Phys. Rev. Lett. 56 (1986) 1035; U.G. Meissner, N. Kaiser, A. Wirzba and W. Weise, Phys. Rev. Lett. 57 (1986) 1676. [10] U.G. Meissner, Phys. Rep. 161 (1988) 213. [ 11 ] Y. Igarashi, A. Kobayashi, H. Otsu and S. Sawada, Phys. Lett. B 195 (1987)479. [ 12] U.G. Meissner, N. Kaiser and W. Weise, Nucl. Phys. A 466 (1987) 685. [ 13] F.R. Klinkhamer, Z. Phys. C 31 (1986) 623; Z.F. Ezawa and T. Yanagida, Phys. Rev. D 33 (1986) 247; J. Kunz and D. Masak, Phys. Lett. B 179 (1986) 146; Y. Igarashi, A. Kobayashi, H. Otsu and S. Sawada, Prog. Theor. Phys. 78 (1987) 358. [ 14] S. Weinberg, Phys. Rev. 166 (1968) 1568.
PHYSICS LETTERS B
24 November 1988
PARITY-VIOLATING CHIRAL SOLITONS WITH VECTOR MESONS Y. I G A R A S H I ~,n2, A. K O B A Y A S H I u, H. O T S U c a n d S. S A W A D A c '~ Max-Planck Institute for Physics and Astrophysics, D-8000 Munich 40, Fed. Rep. Germany b Faculty of Education, Niigata University, Niigata 950-21, Japan Department of Physics, Nagoya University, Nagoya 464, Japan Received 31 May 1988; revised manuscript received 30 August 1988
The presence of a "phase transition" between the conventional parity-conserving solutions and the spontaneous parity-violating ones is confirmed in the Skyrme model with P, and in a model with P and co. The parity-violating solutions lead to almost degenerate parity-doublets at the semi-classical level. Since this makes Skyrme's picture of low-lying hadrons troublesome, we obtain a non-trivial constraint for couplings in chiral soliton models with vector mesons.
1. Introduction
In the large-No limit [ 1 ] Q C D is supposed to reduce to a non-linear meson theory which contains the N a m b u - G o l d s t o n e pions as a primary field in the extremely low-energy limit, at energies much less than AocD. Since the energy scale which characterizes baryon generation as topological solitons is of order AOCD,the low-lying vector mesons with masses less than those of nucleons seem to play a particularly important role in the unified description o f mesons and baryons originating from Skyrme's pioneering work [ 2 ] ~~. Actually the importance of the vector mesons in hadron physics at the length scale down to 0.5 fm is well recognized by, for example, the success of the vector meson dominance model in the electromagnetic interactions as well as o f the one-boson exchange model for nuclear forces [4]. In the past few years an attractive approach based on hidden local symmetry [ 5 ] ~2, where the p meson emerges as a dynamical gauge boson of a hidden symmetry [ SU (2) ] ~ocal, has been used for the construction o f chiral soliton models coupled with vector mesons [ 7 Humboldt Fellow. On leave of absence from Faculty of Education, Niigata University, Niigata 950-21, Japan. ~ See ref. [3] for recent reviews. ,2 For a recent review, see ref. [6].
0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
9 ]. A variety of vector coupled solitons has been investigated so far ~3 We stress here that the presence o f the p field leads to a non-trivial constraint on parameters in the soliton models in order that they make sense as an effective low-energy theory o f QCD. It is related with a new aspect of the soliton solutions, reported in a previous letter [ 11 ] (hereafter referred to as I), the occurrence of spontaneous parity-violation at the purely classical level or o f almost degenerate parity-doublets at the semi-classical level. The new class of soliton solutions with topological (baryon) number Q = B = 1 was obtained with the most general spherically symmetric ansatz for the p-field which consists both of parity-even and -odd terms. It was found in the pcoupled chiral model with fourth-order derivative terms that the spontaneous parity-violating (SPV) solutions with both parity-even and -odd terms become stable minima when a coupling o f the fourthorder term e is larger than a critical value ec. The conventional parity-conserving ( P C ) solutions with the parity-odd term only, on the other hand, are minima for e < ec. For e > eo the parity-invariance may be restored via tunneling effects at the semi-classical level, resolving the exact degeneracy in the SPV states. Since the mixing transition was found to be small, an almost degenerate parity-doublet may appear in the ~3 See ref. [ 10] for a recent review.
445
Volume 214, number 3
PHYSICS LETTERS B
low-lying baryonic states. In order to avoid the parity doublet, the coupling strength must be e < e~. These results were obtained using a variational calculation in I. In this paper we confirm the presence of the "phase transition" between the PC and SPV solutions in Skyrme's model coupled with the p by exactly solving the Euler-Lagrange equations for static masses. The numerical calculation evaluates the critical Skyrme coupling e~=6.8 with other parameters fixed from low-energy relations. We also examine whether the "phase transition" appears in a model of Meissner et al. [ 12 ] for the n-p-to system where the vector mesons have been introduced as dynamical gauge bosons of a hidden SU (2) × U ( 1 ) symmetry. For the p-to-coupled model, the parameters can be completely fixed in the meson sector. We find that the "phase transition" occurs at the critical value of the gauge coupling g~=4.8. The determined coupling strength g ~ 6 is larger than &, and therefore it is not in the parity-doubling phase, but in the conventional phase. Thus the model makes sense as an effective theory for QCD.
24 November 1988
has been fixed to be unitary as in I. This gives an extension of the Weinberg lagrangian [ 13 ]. The hedgehog ansatz we employ is the standard one
U(x) =exp[i(~.z)F(r) ] ,
(2)
for the chiral field, and the most general spherically symmetric one,
V~(x) =e~jkYcjzkG(r) /2r+ [z~--Yc~(~.z) ]H(r) /2r + £¢~(£¢.r)K(r) /2r ,
(3)
Vo=0 for the vector field, where i = x / r and r = Ixl. Both (2) and (3) are invariant under combined spatial and isospin rotations. Since (3) contains a parity-odd term as well as parity-even terms, it is not parity invariant. The static energy functional is given by
M[F,G,H,K] = - J
~d3x
=MA + M v +Mkin +MsK, oo
MA = 4ref2m~J| dr/½ (r/2F'2+2 sin2F) 0
Mv = 4nf~2 ~ dr/a[ ( ~ + c o s F ) E + H 2 + ½K2 ] ,
2. The model for the n-p system
mp
We consider a chiral SU (2) LX SU (2) R non-linear a-model coupled with the p-field [ 5,6,8 ] in the massless pion limit
Mkin --
d 0
41tf2fdr/a[(1/2r/2)(G2+H2-1)2 mp 0
+ ( ~ ' - H K / r / ) 2 + ( H ' + ~K/r/) 2 ] , (3O
~ 0A
2nmp
1 ~Tr(0~, 2 = ~f UOI'U*) ,
MSK =
0
~v = - ~af 2~Tr(D,,~ * + D~,~*~) 2 ,
(4)
-Wki. = -- ( 1/2g 2)Tr(F2.~) , LPsK= ( 1/ 32e 2) Tr [0a UU*, O. UU* ] 2,
( 1)
where U= (~*)2=exp[2ix(x)/f~], V~=gVa~(x)z ~ /2, D.=O~,-iVa, F a . = i [ D a , D~], andf~= 93 MeV is the pion decay constant. We have taken for simplicity the Skyrme term ~4 as a fourth-order term which is needed to stabilize soliton solutions [ 13 ]. The lagrangian ( 1 ) may be derived using the formulaton of hidden local symmetry in which the p is considered as a dynamical gauge boson. Here the hidden gauge ~4 The model described here was referred as (A) in I.
446
dr/[ ( 1/r/2)sin4F+2(F' sin F) 2 ] ,
e2
where G = G - 1, r/=mpr and the primes denote r/derivatives. We fix a = 2 , g = g p ~ = 5 . 8 5 in such a way that the KSRF relation mp2 =2gp~f~ 2 2 holds for the observed meson mass, rnp = 770 MeV. The only free parameter left is then the Skyrme coupling e. It follows from (4) that there exists a discrete symmetry, M[F,G,H,K] = M [ F , G , - H , - K ] . We may redefine with this a parity transformation as x , - , - x , G-,G, H,--,-H, K,--,-K, so that V ( x ) , - , - V ( - x ) . Obviously all the static solutions fall into two distinct classes: the PC ones in which H(r) =0, K(r) =0, and the SPV ones with non-vanishing H ( r ) and K(r). The
Volume 214, number 3
PHYSICS LETTERS B
latter appear as degenerate states (F,G,H,K) and ( F , G , - H , - K ) converted into each other u n d e r the parity transformation. We consider here the topological ( b a r y o n ) n u m b e r Q = B = I sector with the b o u n d a r y conditions F ( 0 ) = r r , G ( 0 ) = 2 , H ( 0 ) = K ( 0 ) = 0 , and F ( ~ ) = G ( ~ ) = H ( c o ) = K ( o o ) = 0 . We solve the stationary equations for the energy functional by using the relaxation method. The typical SPV solutions are shown in fig. 1. The resulting static masses are given in fig. 2 and table 1. It is found that the PC solutions are stable m i n i m a for e < e o while the SPV ones are so for e > ec. The "phase transition" occurs at the critical value e~= 6.8. This is only slightly smaller than the value obtained in I with a variational calculation, ec = 6.9 ~5 In order to get some insights into the PC a n d SPV solutions, we consider two special cases of the gauge field configurations which lead to (i) My = 0. This is a PC configuration where the Vi is expressed in terms of the V, = ( 1 / 2 i ) ( O , ~ * + 0v~*~) = ei,k2j rk ( 1 -- COSF ) / 2 r .
(5)
In this case the kinetic term of the gauge field Mk~, becomes equivalent to the Skyrme term MsK, i.e., gZMki o = e2MsK. (ii) Mk~,=0. This pure gauge limit is realized in a ~5Although we have not considered here the model (B) of I, the critical value should be almost equal to the one determined in I.
M(GeV)
2.0 PC
1.5
1.0
0.5
0
............... , ............. ; . . . . . . . . . .
20
7
r. . . . . . .
5
~
- e
4
Fig. 2. Static masses versus the couplinge in the model for the ~p system. The solid line represents the PC solutions and the SPV ones for e< e~.=6.8 and for e> ec, respectively. The dashed line represents PC solutions which are local minima or saddle points in the latter region. The partial masses are also shown. The dotted curve and dott-dashed curve represent Mk~,, and Mv, respectively. SPV configuration. For the spherically symmetric gauge function g ( x ) = e x p [ i ( a 2 . r ) f ( r ) ] , Vi= i g ( x ) × 0 i g ( x ) - ' is given by V,(x) = eijk2jrk ( 1 -- cos 2 J ) / 2 r + [ r ~ - - ~ ( ~ ' r ) ] (sin 2f)/2r+Yc,(.~.T)f' .
(6)
Table 1 Static masses (in MeV) versus the coupling e for the PC and SPV solution in the model for the n-p system
F
e 1.0
0
10
Roughly speaking, apart from the transition region e ~ ec, the PC and SPV solutions are approximately
3.0
2.0
24 November 1988
1.0
2.0 r/
~0
.
Fig. 1. The SPV solutions for the functions of (2), (3) in the model for the n - 9 system with e = 10.
4 6 8 10 15 20
Solution PC
SPV
2030 1574 1373 1266 1147 1100
1257 1019 689 523
447
Volume 214, number 3
PHYSICS LETTERS B
described by the field configurations (i) and (ii), respectively, as shown in fig. 2. For e > eo the SPV solutions become stable. At the semi-classical level, two degenerate states mix with one another via tunneling effects, making the mass difference between parityeven and parity-odd states, AM. Since A M / M is less than 1% as estimated in I with a variational method, the two states form an almost degenerate parity-doublet which is not relevant to the observed hadron spectrum.
24 November 1988
U(x) = e x p [ i ( . ~ ' z ) F ( r ) ] , V~(x) =%kYc/rkG(r)/2r+ [zi-:ci(~'z) ]H(r)/2r + Yci(.fc'OK( r) /2r ,
Vo(x) = 0 , ~o (x) = to [F,G,H,K] , o~(x) = 0 ,
(8)
leads to the energy functional
M[F,G,H,K] = - J £ad3x=Mo +M,o,
3. The model for the 7[-p-co system
Mo=-
Let us turn to the chiral soliton models with the co meson in addition to the p meson. They have been constructed by enlarging the hidden local symmetry to S U ( 2 ) v × U ( 1 ) or to U ( 3 ) v . For our purpose of discussing the occurrence of new soliton solutions, we consider here the model of Meissner et al. [ 12 ] in which the Wess-Zumino term is included in a simple manner:
f [£PA+~V(~-p-part)
+ ~ki. (p-part) ] d3x,
M,~=- J [A~(c0-mass) + ~ i n ( ~ - p a r t ) + ,,~¢wz] d 3 x , oo
(9)
Mo = 4~rf---~2j dr/{ ½(qZF'2 + 2 sin~F) mp
o
+a[ (~+cosF)Z+H2@½K
£,e= £#A+ ~V (n-p-part) + £aki.(p-part)
2]
+ (a/22) [ ( 1 / 2 q 2 ) ( ~ 2 + H 2_ 1)2
+ L#(co-mass) + £#ki.(w-part) + £awz,
+ (G'-HK/q)2+ (H' +(~K/r/)2] ) ,
£PA= ~/~Tr(0u UOuU*),
oo
£#v(x-p-part) = - ~af2=Tr(Du~ * + Ou~?~) 2 ,
m---o
r/ ~--,~2r/2
(2)
£#ki.(p-part) = - ( 1/2g 2)Tr(Fzu.) , L#(c0-mass) = £aui. (re_part) =
1"1°2 ¢"2 ~"'2
+~ ( i F ' sin2F 87[ \
i4 (.0/~v 2
_ _ l d [sin 2F((~2 + H 2 _ 1 ) ] ) c o ] , 2 dr/
Aawz= ½Ncgwu( 1/24x z) e u~,~#
x ( U*O. UU*O~UU*O#U) + ½Ncg( 1 / 167[2 ) ¢ uVa#OOuv ×Tr[iV¢,(O#UU*-8#U*U)+V~U*V#U] ,
(7)
where o~/,. = 0/,o~- 0.oJ/, is the field strength for the ¢o field, and the covariant derivative reads as before D/,=O/,-iVu(x ). We take a = 2 , N~=3. The gauge coupling constant g may be fixed by use of the co~ 3x decay width, or of the KSRF relation as before. We consider the gauge coupling g only in some range around the physical value g ~ 6. The spherically symmetric ansatz for U(x), Vu(x) and o~/,(x) 448
(10)
where 2=g/go, and go = 5.85 which is determined by the KSRF relation. The radial Green function if(t/, t/')
1 f#(q, r/' ) - 222qr/, × [ e x p ( - 2 1 r/-r/' I ) - e x p ( - A I r / + r/' I ) ] ,
(11)
solves the co field equation (-
d
2 d +22r/2)o)(q) =J(?])
(12)
Volume 214, number 3
PHYSICS LETTERS B
24 November 1988
Table 2 Static masses (in MeV ) as a function o f g in the model for the n p-t0 system
3.o I -500
F
g
g/go
2.0,
-300 >~ IE
2.93 4.10 4.68 5.85 8.78
1.0 -IO0
0.5 0.7 0.8 1.0 1.5
Solution PC
SPV
2182 1692 1567 1418 1283
1235 1469 1518 -
,
0
1.0
T/---~
3.0
ZO
Fig. 3. SPV solutions F, G, H (full curves ) and oJ (dashed curve ) in the model for the x-p-t0 system with g = 3.
table 2, which indicates that the "phase transition" appears at gc=4.8, below which the SPV solutions become stable. The physical value g ~ 6 is not in the parity-doublet phase but in the PC phase.
as
4. Concluding remarks
co(u) = i du' ~(n, ~' )J(u'),
(13)
o
where the source current is given by
3gmp
J(r/) = ~
{2F' sin2F
- ½(d/d~/) [sin
2F(O2+H 2- 1 ) ] } .
(14)
The typical SPV solutions are shown in fig. 3. The static M as a function of g are shown in fig. 4 and M(GeV)
2.0
xx
1.0
,g 4.0
6D
B.0
10.0
Fig. 4. Static masses versus the coupling g in the model for the n p-to system. The solid line represents the PC solutions and the SPV ones for g > g c = 4 . 8 and for g
We have confirmed in this note the existence of a "phase transition" between the PC solutions and the SPV ones in the p-coupled Skyrme model, and also in a model with p and to. In the former the critical Skyrme coupling turns to be ec= 6.8. When 1/e, proportional to the size of the Skyrme term, is smaller than 1/ec, the soliton states with topological (baryon) number = Q = B = 1 are in the parity-doubling phase. The critical gauge coupling in the latter model of Meissner et al. is gc = 4.8, which is smaller than the physical value g ~ 6 fixed from experimental data in the meson sector. Therefore, the model with g ~ 6 is not in the parity-doubling phase but in the conventional phase relevant to hadron physics. The vector field configuration we have obtained here can be interpreted as follows: in the SPV solutions it is approximately described by the pure gauge form so that the kinetic term of the gauge field vanishes, while the vector field in the PC ones has a similar configuration to that of the pionic vector current. Which configuration is realized depends on the strength of the soliton stabilizing term. When the contribution from the stabilizing term to the total static mass is too small, the SPV solutions become absolute minima. Our investigation indicates that the SPV solutions appear universally in chiral soliton models coupled with the p field, not depending on the stabilization mechanism. 449
Volume 214, number 3
PHYSICS LETTERS B
We h a v e c o n s i d e r e d o n l y the Q=B= 1 sector in w h i c h the c o u p l i n g to the p field s e e m s to be essential to the p r e s e n c e o f the SPV. H o w e v e r , the s i t u a t i o n m i g h t change for chiral solitons for m o r e t h a n three flavors w i t h h i g h e r t o p o l o g i c a l n u m b e r s . F o r e x a m ple, in the S U ( 3 ) - S k y r m e m o d e l the spherically symm e t r i c ansatz for the chiral variable carrying Q = B = 2 d o e s not respect n a i v e parity i n v a r i a n c e , a n d S P V solutions m a y a p p e a r e v e n w i t h o u t t h e c o u p l i n g to vectors.
Acknowledgement O n e o f us ( Y . I . ) w o u l d like to t h a n k the M a x P l a n c k - I n s t i t u t e for hospitality.
References [ 1] G. 't Hooft, Nuel. Phys. B 72 (1974) 461; E. Witten, Nucl. Phys. B 160 (1979) 57. [ 2 ] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 1961 ) 127; Nucl. Phys. 31 (1962) 556. [3] I. Zahed and G.E. Brown, Phys. Rep. 142 (1986) 1; A.P. Balachandran, Proc. Yale Theor. A.S.I., eds. M.J. Bowick and F. Gtirsey (World Scientific, Singapore, 1985 ) p.l.
450
24 November 1988
[ 4] J.J. Sakurai, Currents and mesons (Chicago U.P., Chicago, IL, 1969). [5 ] M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54 (1985) 1215. [6] M. Bando, K. Yamawaki and T. Kugo, Phys. Rep. 164 (1988) 217. [ 7 ] G.S. Adkins and C.R. Nappi, Phys. Lett. B 137 ( 1984 ) 251; G.S. Adkins, Phys. Rev. D 33 (1986) 193. [8] Y. Igarashi, M. Johmura, A Kobayashi, H. Otsu, T. Sato and S. Sawada, Nucl. Phys. B 259 (1985) 721; T. Fujiwara, Y. Igarashi, A. Kobayashi, H. Otsu, T. Sato and S. Sawada, Prog. Theor. Phys. 74 (1985) 128. [9] U.G. Meissner and I. Zahed, Phys. Rev. Lett. 56 (1986) 1035; U.G. Meissner, N. Kaiser, A. Wirzba and W. Weise, Phys. Rev. Lett. 57 (1986) 1676. [10] U.G. Meissner, Phys. Rep. 161 (1988) 213. [ 11 ] Y. Igarashi, A. Kobayashi, H. Otsu and S. Sawada, Phys. Lett. B 195 (1987)479. [ 12] U.G. Meissner, N. Kaiser and W. Weise, Nucl. Phys. A 466 (1987) 685. [ 13] F.R. Klinkhamer, Z. Phys. C 31 (1986) 623; Z.F. Ezawa and T. Yanagida, Phys. Rev. D 33 (1986) 247; J. Kunz and D. Masak, Phys. Lett. B 179 (1986) 146; Y. Igarashi, A. Kobayashi, H. Otsu and S. Sawada, Prog. Theor. Phys. 78 (1987) 358. [ 14] S. Weinberg, Phys. Rev. 166 (1968) 1568.