Volume 208, number 2 PHYSICS LETTERS B 14 July 1988

M U L T I P L E S K Y R M I O N S O L U T I O N S IN T H E U(2) H I D D E N GAUGE T H E O R Y

Yves BRIHAYE Physique Thborique et Mathbmatique, Universitb de I'Etat ~ Mons, B- 7000 Mons, Belgium

and

Jutta K U N Z NIKHEF-K, Postbus 41882, NL-I O09 DB Amsterdam, The Netherlands

Received 4 February 1988

We construct soliton solutions with generalized spherical symmetry for a U (2) hidden gauge theory. Two types of solutions are found. The usual skyrmion solutions are based on the Wu-Yang ansatz for the isovector p-fields. Parity doubling solutions based on the more general Witten ansatz exist only for unrealistic parameter choices.

1. Introduction

In the Skyrme model [1] baryons arise as the quantum states of topological soliton solutions, whose topological winding number corresponds to baryon number (B) [2,3]. The effective lagrangian o f Skyrme [ 1 ] is based on the non-linear a-model for the pseudoscalar pions, supplemented by a stabiliz- ing term. This simple model describes the properties o f nucleon and delta within 30% accuracy [ 4 ].

In the large Arc limit QCD is equivalent to a theory with infinitely many mesons. The corresponding true effective action of QCD is unkown. However, a more realistic model should contain at least all mesons with M~< 1 GeV. Phenomenologically satisfactory me- sonic effective lagrangians contain also the low-lying vector mesons, i.e. p-mesons and co-mesons [5,6], as well as the axial vector Al-mesons [7,11 ]. In these effective lagrangians the vector mesons are either in- troduced via the hidden gauge principle [ 5-7 ] or by directly gauging the global chiral symmetry [ 8-11 ]. Both types of effective lagrangians have been used to construct the non-strange baryons and lead to re- markable improvement o f baryon properties such as radii, magnetic moments and form factors [ 12-18 ].

Work supported by DFG.

The quantization of the classical soliton solutions requires them to be minima in energy except for the existence of zero modes associated with spin and iso- spin rotations and with translations. These zero modes are then quantized via the collective coordi- nate method [4 ], leading to states with good spin and isospin.

The interpretation of the lowest quantum states as nucleon and delta presumes that the corresponding classical solution is the absolute min imum in the B = 1 sector. For the pure Skyrme model this seems to be the case [ 19 ]. For effective lagrangians with vector mesons, however, the configuration space can con- tain a multitude o f classical solutions in a given sector.

In the SU (2) hidden gauge theory [ 5 ] a whole se- ries o f classical solutions exists [ 20 ], where the chiral angle representing the 9-field develops an increasing number of nodes, while the n-field configuration hardly changes. A stability analysis however shows, that all these solutions are saddle points [ 21,22 ]. The inclusion o f a Skyrme stabilizing term makes the low- est solution stable while the solutions with nodal ex- citations disappear. In the U (2) hidden gauge theory the presence of the c0-meson has analogous effects [ 17 ]. Based on such an co-stabilized skyrmion solu- tion rather good baryon properties are obtained [ 15- 17].

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Whereas the inclusion of the stabilizing Skyrme term leads to the disappearance of the states with no- dal excitations, it allows for the occurrence of new solutions in the SU(2) hidden gauge theory, as a re- cent variational calculation suggests [ 23 ]. These new solutions are parity doubling solutions, which could potentially spoil the good baryon phenomenology of such effective lagrangians.

In this paper we construct parity doubling solu- tions for a U (2) hidden gauge theory. Since the ex- istence of these solutions depends crucially on the presence of a stabilizing mechanism, we study these solutions as a function of the m-n coupling strength, which is conveniently parametrized via the number of colours (Arc).

of the m-meson to the baryon current [ 24 ]. After gauge fixing

~[ =~R = { (4)

the chiral matrix U is related to the variable ~ via

U(X) = ~ ( X ) ~ R ( X ) ---~ [ ~ ( X ) ] 2 . ( 5 )

The mass of the vector mesons is due to the Higgs mechanism of the hidden local symmetry and satis- fies the KSFR relation, m 2 = 2gv=j=.2 2

Since we do not consider hyperons here, we restrict our calculations to the U (2)v subgroup of U (3)v.

3. Classical equations

2. U(2) hidden gauge theory

We consider the phenomenologically successful ef- fective action of Fujiwara et al. [6]. This U ( 3 ) L × U ( 3 ) R/U ( 3 ) v hidden gauge action is given in terms of the matrix value fields ~ and V

f ~f~ [Tr (Du~L~ - - D , ~ R ~ ) 2 F = { - ' 2

+ 2 Tr (D,~L~_ + D , ~ R ~ ) 2 ]

_ (½g2) [Tr(F , . )2]} d4x+Fwz, (1)

with covariant derivatives

D~dL(R) (X) = [G -- iG(x)]~L(R) (X) (2a)

and field strength tensor

F , , = O u V ~ - a , V , - i [ V ~ , V,] . (2b)

The Wess-Zumino term (see ref. [ 6 ] for notations) is defined by

o f Fwz = F w z - ci~/,, (3) M 4

where F ° z is the usual irreducible Wess-Zumino ac- tion and the additional terms in eq. (3) are solutions of the homogeneous anomaly equation. The coeffi- cients cs are determined by requiring good mesonic decays and (partial) vector meson dominance [ 6,15- 17]. Here we consider only the "minimal model" choice of parameters c~ [ 15,16 ], where the homoge- neous part of Fwz reduces to the standard coupling

To construct classical skyrmion-type solutions, which possess a generalized spherical symmetry, we use the hedgehog ansatz for the n-field

U(r) = cos F + ilr. r sin F . (6a)

For the U (2) v vector field

Vu =gPT, z~/2 + go, J 2 , ( 6b )

we choose the Witten ansatz for the isovector part

p~ = ( G/gr)~aib/c ~

+ (H/gr ) (t~ai __.~a~i) ~_ (K/gr )2a2 , , (6C)

p~=0 , ~Oi=0, 09o=~O/g, (6d,e,f)

where G, H, K and ~o are functions of r. It includes the Wu-Yang ansatz ( H = K = 0) as a special case.

These ans~tze then lead to the radially symmetric energy functional

E = j'{½fZ(rZF' 2+2 sinZF)

+ 2f~ [ ( G - 1 +cos F) 2 + H 2 + ½K 2 - ½r2fo 2 ]

+ (2g2) - l{ r -2 [ ((7-- 1 ) 2 + H 2 - 1 ]2

+ 2 ( G ' - H K / r ) 2 + 2 [H ' + ( G - 1 )K/r]2-r2~o '2

+ (Nc/4/r 2) rnF' siniF} dr . ( 7 )

The corresponding equations of motion for the func- tions F, G, H and ~o read

r2F" = - 2 r F ' +sin 2 F - 4 ( G - 1 +cos F)sin F

2 2 - ( N c / 4 f , ~ )09 sin2F, (8a)

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Volume 208, number 2 PHYSICS LETTERS B 14 July 1988

rZG '' = m ~ r 2 ( G - 1 + c o s F) + r ( 2 H ' K + HK' ) - H K

+ ( G - 1 ) [ ( G - 1 ) Z + H Z + K 2 - 1 ] , (8b)

r2H '' = m 2 r 2 H - r [ 2 G ' K + ( G - 1 ) K ' ] + ( G - 1 ) K

+ H [ ( G - 1 ) 2 + H Z + K 2 - 1 ] , (8c)

r2(o " = - - 2roy + mZ r2o~- (Ncg2/4n 2)F' s in2F,

(8d)

while the absence of K' in the energy functional leads to the const ra int equat ions

K = r [ G ' H - ( G - 1 ) H ' ]

1 2 , ( B e ) X [ ~ m v r 2 + ( G - 1 ) 2 + H 2 ] - 1

rK' = - K+ 2H cos F , (8f)

where the last equat ion is a consequence of eqs. (8b) , (8c) , (8e) .

The boundary condi t ions for the winding number n = 1 solutions are

F ( o v ) = F ( 0 ) - n = 0 , (9a)

G ( ~ ) = G ( 0 ) - 2 = 0 , (9b)

H ( o o ) = H ( 0 ) = 0 , (9c)

¢ o ( ~ ) =~o' (0) = 0 , (9d )

K ( ~ ) = K ( 0 ) = 0 . (9e)

4. Classical solutions

We solve the eqs. ( 8 a ) - ( 8 d ) subject to the con- straint equat ions (8e) , ( 8 f ) for K and the boundary condi t ion eqs. (9) . We choose the physical values of the parameters f ~ = 9 3 MeV and g = 5 . 8 5 (corre- sponding to rnp = m,o = 770 MeV) , while we treat Arc as a parameter . We then invest igate the solut ions as functions of this " p a r a m e t e r " No.

The first type of solut ion ( I ) uti l izes the W u - Y a n g ansatz for the p-field, i.e. the funct ions H and K are ident ical ly zero. This solut ion is based on the exis- tence of the S U ( 2 ) h a d r o i d / s p h a l e r o n saddle poin t solut ion [20,21,25 ], a pure p-field exci tat ion with a singular n-field configurat ion ( F ( r ) = 0 everywhere except at the origin, where F ( 0 ) = n) . This p-field ex- c i ta t ion also allows for the cont inuous n-field config- urations, where the chiral field F has k nodes (k>~ 0 ).

However , all these solutions are saddle points. The solut ion with lowest energy (k = 0 ) can be stabil ized. The presence of the co-field (or o f a Skyrme- te rm) changes the nature of this solut ion from being a sad- dle poin t to becoming a m i n i m u m above a certain coupling strength [17,21 ]. We exhibit this solut ion for Arc = 2 in fig. 1.

The second type of solut ion ( I I ) uti l izes the full Wit ten ansatz for the p-field. The existence of this type of solut ion crucially depends on the presence of a sta- bi l izing mechanism as p rov ided by the co-field (or by a Skyrme- te rm) . This solution may be viewed (at least for relat ively small Nc) as being based on a n-co soliton (or a pure skyrmion) with smooth p-field functions, satisfying the boundary condi t ion eqs. (9) .

There are always two type II solut ions degenerate in energy, corresponding to _+ H and _+ K, which are in terpre ted as the components of a par i ty doublet [ 23 ]. In addi t ion , in a certain pa ramete r range there exist two different configurat ions of type II solutions. We exhibit these two configurat ions for the parame- ter choice Arc = 2 in figs. 2a, 2b.

In fig. 3 we il lustrate the dependence of the energy eq. (7) on the pa ramete r Arc. The energy of the type I solut ion increases smoothly start ing f rom the finite value of E~= 1.047 GeV at Nc=0 . At the physical value ofN~ = 3 we find E1 = 1.425 GeV [ 15 ]. In con- trast, at Arc = 0 the type II solut ion collapses to zero size and E , = 0, since the stabil izing n-to interact ion is then switched off. The energy o f the type II solu-

o ~2 o m

~.. , , , , , , ~ , , , . , , , , , , - -

0 , q , , , ~ , , ~ i ~ ' 7 7 - ~ - - T - - ~ - - ~ - + - ~ - ,

0 . 5

r [~ml

Fig. 1. The functions F(r) (dashed) of the n-field, G(r) (solid) of the p-field and -oJ (r)/g (dash-dotted) are shown for the type I configuration based on the Wu-Yang ansatz for N~=2 (El = 1.272 GeV ). The fields F and G are in dimensionless units, ca is in fm- ~ and r is in fm (f~= 93 MeV, g= 5.85 ).

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o 7~

0

0 0 . 5 1

r [ fro]

~ ~ b

o 2

u c :3 t~

0 . 5 I

r [ fra]

Fig. 2. The functions F(r) (dashed) of the ~-field, G(r) and H(r) (both solid) of the p-field and - ~o (r)/g (dash-dotted) are shown for the two type II configurations based on the Witten ansatz for N~=2. The energetically lower solution (Eu= 1.262 GeV) is shown in (a), the higher solution (EH= 1.275 GeV) in (b). The fields F, G and H are in dimensionless units, 09 is in fm ~ and r is in fm (f~=93 MeV, g=5.85).

t ion rises rapidly as a funct ion of N¢. At the critical point ~ ~ Nc ~ 2.06 both solutions are equal in energy. Thus for N ~ < N ~ the type II solution is the stable so- lution, type I represents only a local m i n i m u m in configuration space. A further increase of Nc leads to a further increase of the energy of the type II solution, rendering the type I solution stable for N~>N~c. However, the type II solution can be cont inued as a function of N¢ only till a second critical point is reached, N 2 ~ 2.13, which is the end point of a spike. Here the solution can be cont inued backwards to- wards smaller N~ and lower energies. Finally at a third critical point 3 ~ Nc ~ 1.81 the type II solution merges with the type I solution. During this descent from N 2 to N 3 the type II configuration drastically changes

1 , 5

~ 0 . 5 ~

1 8 2 2 . 2

0 1 2 3

Number o f G o ] o u r s

Fig. 3. The energy is shown as a function of the "parameter" N~ for the two types of solutions. The "parity conserving" type I configuration based on the Wu-Yang ansatz is indicated by the dashed curve. The "'parity violating" type II configuration based on the Winen ansatz is indicated by the solid curve. An enlarged graph of the spike is shown in the lower right corner of the figure. The energy is given in GeV (f~ = 93 MeV, g= 5.85 ).

it character due to a rapid decrease in the magnitude of the functions H and K.

Thus in the U (2) hidden gauge theory studied, the type II solutions do not exist for the physical values of the parameters f~, m , and N¢. To study the rele- vance of these solutions further we varied also f~,

while keeping the mass of the vector particles m o fixed. The relevant critical point is N ~, beyond which due to E~ > E~ the type I solutions are stable. By de-

creasing f~ by 10% we find also a decreased critical point N~ ~ 1.67, by increasingf~ by 10% we find also an increased critical point N~ ~ 2.49. In order to in- crease the critical point up to the value ofN~ = 3, the parameterf~ must be increased by as much as 20.6%, corresponding to a classical energy of the solution of

E I = E . = 1.866 GeV. We finally note that we also constructed the corre-

sponding set of solutions in the Skyrme-term stabi- lized model. The variational solutions of Igarashi et al. [23 ] are in very good agreement with the true so- lutions. For example, the energy of the type II solu- t ion for a Skyrme parameter of e = 2 0 is for the variat ional solution E[~r= 0. 525 GeV, while the cor- responding exact numerical solution has an energy of E~I um =0.518 GeV.

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5. Conclusions

We have cons t ruc t ed classical so lu t ions wi th gen-

e ra l i zed spher ica l s y m m e t r y for U (2 ) v h i d d e n gauge

theory. The re are two types o f classical solut ions: a

" p a r i t y c o n s e r v i n g " type I so lu t ion based on the W u -

Yang ansatz, wh ich was s tud ied p rev ious ly [ 15-17 ],

and a " s p o n t a n e o u s l y par i ty v i o l a t i n g " type II solu-

t ion based on the m o r e genera l W i t t e n ansatz.

F o r smal l va lues o f Arc the type II so lu t ion is lower

in energy and the re fore co r re sponds to the global

m i n i m u m . B e y o n d Nc ~ ~ 2.06, however , the type I so-

lu t ion b e c o m e s the lowes t energy conf igu ra t ion and

thus the global m i n i m u m . T h e type II con f igu ra t ion

app roaches jus t above this cr i t ical va lue o f No a spike,

and it c anno t be c o n t i n u e d fu r ther towards h igher Arc.

T h u s it exists no m o r e for the phys ica l va lue Arc = 3.

Also a smal l change o f the p ion decay cons t an t f~

does no t ex t end the occur rence o f the " s p o n t a -

neous ly par i ty v i o l a t i n g " type II so lu t ions to high

enough Arc, to be physica l ly re levant .

Consequen t ly , the good b a r y o n p h e n o m e n o l o g y ,

based on q u a n t i z i n g the type I so lu t ion , is no t af-

fected. T h e r e are no par i ty doub l ing states in the vi-

c in i ty for a physica l ly reasonab le cho ice o f

pa ramete r s . Such states may, however , be o f rele-

vance for the weak in t e rac t ions [ 26 ].

Acknowledgement

Y.B. is grateful to C. S e m a y for useful discussions,

J .K. gra teful ly acknowledges d i scuss ions wi th A.

L a n d e and P. Mulders .

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