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Chiral Soliton Model with Dynamical Restoration of Chiral Symmetry from Quark Flavour Dynamics.

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7. Folge. Band 48. 1991. Heft 4, S.241-304
Chiral Soliton Model with Dynamical Restoration
of Chiral Symmetry from Quark Flavour Dynamics
The Xiels Bohr Institute, University of Copenhagen, Denmark
Joint Inst.itiite for Nuclear Reseurch, Dubna,
A b s t r a c t . The Sambu-Joiia-L:isinio model its a particular model of qunrk fl;tvour dynsniics
is homnized in the low-energy regime using R generalized heat kernel expiinsion with local niiiss
s a l e . The resulting effective meson Lngrangian defines n chirnl soliton model, the solutions of which
show a ptrrtial restoration of chirnl symmetry inside the soliton.
Chir;ilcx Solitoneiiiaotlull iiiit tlynaiiiiseher 1Lcst:iur;ition tlcr c*hir:ileri SjIiinictric
;tiis tler ()iuirk-F1:ivoiir-I)yn:lrrik
1n h a I t s iib e r s i c h t. Dirs ~ilmbu-Lonir-I*lsinio-aIodell als ein spexielles .\Iodell fur die QuarkPlitvour-Dyniimik \\ irtl im n'iederenergiebereich bosonisiert, intlem eine vercillgeiiieiuerte Heat~ernel-Entn.ickIuiigmit loknler M;iseenskala verwendet n i d . Die resultierende effektire 3lrsonLngr;mge-Dichte definiert ein chirilles Solitonen-Modell, dessen Iiiuungen eine pnrtielle Reutaiiratioii
tler chirnlen Symmetrie im Zentrrim des Solitons iinfweisen.
For a large iiiiriiber of colours, QCD re(111cesto a n effective nieson tlieory \\ liere
baryons enierge as solitons of the meson fields [ 11. Unlike i n two clinieiisions, in four
tliniensions the effective nieson Lagranginn of QC!D is not h o t \ 11. One has t Iieret'oi~
pheiionienulogic~llyconstructed effective nieson Lagrangians iniplenienting the kncm n
low-energy features of nieson physics [2]. I t turns out that at low energies the striictui-e
of the effective nieson Lagrangian is to a large extent (leterniinecl by chiral syninietry,
uhile tlic details of the confinement niechanisni of QcD seem to be not relevant. This
suggests that the effective chiral meson Lagrangiaii may be derivable front niucli siurplier chiral invariant quark models than QCD. In particular the Sanibu-.Jona-I,nslni,)
model [3] has been used to motlel t h e quark flavour dynamics E l , 51. The resulting effective meson Lagrangian obtained in the gradient expansion [4,51 arorintl the chirill brolieli
vacuum is in g o ~ dagreement with low-energy meson physics [(i, 71.
The hitherto eniployerl gradient expansion yield local espansiuiin of the (noii-local)
effective action around the homogeneous vacuum configuration of spontaneoiisly brokeii
chiral symmetry (with space-time independent order palnnieter q~ = &)). Such an
expansion should certainly be appropriate for the racuuni sector but could fail in the
soliton (baryon) sector. This is because, due to the presence of valence quarks, c h i d
symmetry is expected to be restored inside the hntlrons (ikt sniall tlistancc). Hence the
order parameter q~ of chiral symmetry breaking is expected to be spare dependent in
the presence of a soliton (baryon) ranging from q~ rn 0 inside to its vnciiuni value T~
--> $2
Ann. Physik Leipzig 48 (199L) 1
oiitsitle the baryon. ?‘herefore the effective chiral nteson action shordcl not be espantlecl
aroitntl the vactiuni configuration y = vo,a s in the familiar graclient expansion, but
aroiiiid a (Iynantical (space-dependent) y ( x ) , such that nonpertiirbittive structures o f
the ~ r t l e rpnranieter y ( z ) niay emerge in the soliton sector. This is acconiplislietl in the
present paper by using a generalizecl Iteiit kernel nietliocl nith local ninss scale [ 8 ] . The
ripshot will he n chiral soliton nioclel with tlynmiical restoration of chiral syninictry
insitle the solitoti.
v(Jr definiteness \\e take as chiral invariant qiiark Lagranginn the following SanibuJona-l,asinio tiioclel
Here q denotes the quark spinor nntl g is a coupling constant. The Lagrangian is invariant iintler global LJU(2),x A ‘ ~ C I ( Z ) transformations.
Following the stantlard procedure
o f pnth integral hosonisatioti (see e.g. ref. [51) we rewrite the quark theory ( 1 ) i n terms
of rotnposite niesoti fields. The effective tilesonic action which is eqiiivalent t o ( 1 ) reatls
S = - -$d4xv2 - i Tr log i D , *)
i D = i I3 - y(U)Ys.
Here v is i i neutral real scaliir fieltl (the SIJ-Citllcd ratlitis fieltl) aiitl U denotes the cltiral
nugle fielcl living in the coset spnce A ~ U ( 2 ) , x s a ( Z ) , / s c r ( ~ ) ~Since
the group R U ( 2 )
is l;no\vti to be atioitialy-free it suffices to calculate the reill piwt o f T r log (in k:riclitleaii
spwe) \\hicli cilti be ritten iis - Tr log B b B. The operiitor uncler interest follows
ft.oni eq. (3) to be given by
For the evaliintion o f Tr log D . B u.e use an itiiprovetl heat kernel ~iietliotl[S] (cf. also
I\ liicli USCS tlie Scillat. field as local, space dependent iiiass scale. Yollouing ref. [S],
~c represent the cleterniiiiitnt a s ii proper time integral
Tr log B+B = -
J” dt t esp{-tDr
\\.-liereA is a sltort tlisttince cutoff. The lieat kernel esp(-t B t B) is then espaiitletl
an unperturbed kernel
K,(t) = esp(--t
a,, P}- esp{-t@}
in Fowers of tlie proper-time
exp{--t B+B} = Ko(t)2 t”h,:
Inserting this espansion into eq. (4)the proper-time integral can be carried out and the
tleterniinant is expressed in terms of the heat coefficients h,. The use of tlie local niass
scale in the unpert urbetl kernel (.i) consitlerably improves the convergence of the heat
kernel expansion ( 6 ) .For instance, the effective potential of the scalar field q~ (see below)
is obtained in the usual heat kernel espansion (with constant mass scale) as a power
expansion which does not terminate in finite order of t h e expansion (i.e. arbitrary high
order heat coefficients h, contribute). I n the generalized heat kernel expansion (6)
*) The symbols d,
B repreaent t t e usual inclinedly crossed 8,D.
B. K:LmFm, Chirid Soliton Model
the whole potential is obtainetl already in zero’s order. We hare perfornietl the improved
lieat kernel expansion (6) up to second order derivative ternis in the scalar field cp anti
iip to fourth order derivative ternis in the c h i d field U l ) . Then one obtains for the
Lagraiigian (in Minkowski spice) corresponcling to the action (2) the following exprehsion
2’= -C(cp) 8,“p P r p - V(rp) - -F , ( Y ) ~TrL2
Tr([L,,L,,][L”,L”] 2(L2)2 - l(aL)2},
\vith e = 2n ; ~ n t L,
l = U ’ a,CJ being the left current. Here
F ( z )= esp{-z) - z[exp{-z) - zI‘(0, z ) ] ,
is tlic potential for spontaneous breaking of cliiral ayniiiietry. It lias the typicnl “niesimi
lint” shape. Furtherriiore
field clcpentlent. kinetic coefficient of t.lie scalar field and
is a tlynaniicnl (cp-clepentlent) pion decay constant. 111 the above etliiatioris T ( ~z I) ,
clenotes the inconiplete gnniniib fiinrtioii antl N is the nuniber of c o h i r s . The last t tvo
terriis in eel. ( 7 ) spoil the stability o f the chical soliton. In particular, the last term lends
t o an iinphysical tachyonic pcile in the pion propagztor. One expects that these terms
c i ~ nbe overcortipeusatetl by contributions frorti ot her meson fieltls, like the vector mesons. We shall therefore onlit these two ternis in the nunierical analysis presented below.
The reriiaining Lagrangian defines a chiral soliton nioclel \vhicli is similar to the one
introclitcetl plienonienologically in ref. [LO] on the basis of QCI) scaling argumentsz).
Fiii~therrnore,for g~ = yo it reduces to the standard Skyrnie niorlel [ 11, 121 with fixed
coupling strength e = ‘Jnof the Skyrnie term. Our present niodel contains two parameters: the cutoff A antl the coupling constant g. They can be fixed choosing the mininiuni 9 = cpo of the potential corresponding to the phase with broken chiral syninietry
occ‘ur at the constituent quark mass m = cpo and by giving the vacuuni value of the
pion decay constant F,(~I,)its physical value. (For F,(cp,) M 93 MeV and m M 310 MeV
750 MeV, a reasonable value for a low-energy cutoff).
this yields A
We have studied nunierically the above presented chiral soliton niotlel for a strbtjc
cliiral field with winding number one exploiting the hedgehog ansatz
l ) For the rather “massive” scalar field, fluctuations are suppressed by the mass term (i.e. by
the potential, see below) while the c h i d field, being a Goldstone field, has more freedom to fluctuate.
*) It differs from the one in ref. [lo] in several respects: (i) by the presence of the kinetic factor
C ( q ) ,(ii) by the presence of the dynamicel factor r 0, - in F,, (iii) in the potcntial V , and (iv)
the cutoff has 8 different physical meaning.
Ann. Physik Leipzig 48 (1991) 4
The Euler Lagrange eqnations for the chiral angle @ ( r ) follow as
+ 2 sin2 @/x) + 0 ' 2 sin Z@/x2
+ 2e2X 0'15- sin 2@(e2X + sin2 @ / x 2 ) / x 2
+ e*X c p ' ~ '
log X) = 0,
and we have introtlucecl the dimensionless variable x = rF;, F: = Fx(qo).For cp =
qo = const ( X = 1) this equation reduces to the standard Skyrme model [ll, 1.21. The
I.:riler-Lagrange equation for a static cp can be cast into the form
represents a n effective space tlepentlent potential for q. Fig. 1 shows this potential for
several riiclii x for the c h i d angle profile function @(x) = 2 arccos ax. For large x the
estra term in eq. (15) tends to zero clue to the asyniptotic fall off of the chiral angle
O(x),antl the effective potential 7 is essentially given by the hare one V ( z ) ,which favours the state with broken chiral symmetry, 'p = qo. For small x the estra tern1 i n
eq. ( 15) tlomin;ttes ant1 tlie effective potential V(q)has the chiral symmetric niiniilium
at cp = 0. Hence one espects at first glance for q ( x ) bag shaped solitons interpolating
betneen cp a 0 inside antl q = 'po outside the chiral soliton. The csact bag (step function) sliapt: expected for q ( x ) from the above giveii discussion of the potential is, however, sitioothetl out by the kinetic gradient terms. Furthermore the presence of the kinetic
coefficient C ( q ) in eq. (7) which is logarithmically singular for tp+ 0, will prevent q
from becoming exactly zero. The numerical solution of the coupled equations ( 12) nntl
(14) is sliown in Fig. 2 for two values of the constituent quark mass n~ = 320 MeV
: ~ n dtn = I50 MeV, respectively. In both cases '1 and g were fisecl to reproduce the esperiniental value of the pion decay constant Pz(cp,-,)= 93 MeV. The obtained solutions
sho\v in fact tlic espected behaviour: For large x, cp approaches its vaciiuni value cp
while for sniall z it is reduced considerably t o about 10% of its vacnrrm value qo indicating a pwtinl restoration of c h i d symmetry inside the soliton. On the other hand,
as the explicit numerical solution shows, t h e chiral angle O(z) is not milch changed
by the presence of the soltion sollition rp(x) compared to the usual Skyrniion cp = yo =
const solution. Finally, Fig. 3 show the mass of the soliton, which is given by
JI = 4zF: jdx x2
[5 F , ( V ) ~
+ 2 sin2 @/x2)
+ sin2 @(Ot2+ sin2 @ /x 2 )2
H. l~ELSITAIIDT,B. I<iJIPF.eR, Chird Soliton Model
4) IMeVl
L 00
Fig. 1. The rffcctivc potentiid AS fiiiictioii of the scaliir field cp for srvernl values of the radius 1.
The shape function is asuiimecl to he O(.r)= 4 iirccos nr uith rr = Y . W (the value which mininiises
the ciirrgy of the stiiridiird Skyrillion). p
' , = 320 JIcY. .1 = M;i MeV, g = 0.9i1. Tlic w c i i i i m encirgy
V(rpo) is not subtracted.
Fig. 2 . The shape function O(z) and the scalar field 'p(z) as functions of the radius z (full lines:
= 340 MeV, A = GD7 MeV, g = 0.9il; dashed lines: 'po = 150 MeV, A = 2 510 MeV, g = 0.535).
The shape function of the standard Skyrmion model coincides nearly on the given scnle with the full
line Q(r).
Ann. Physik Leipzig 48 (1991) 4
(Po [MeV]
Fig. 3. The soliton mass as function of the vacuum value of the scalar field v,, being the constituent
quiwk mass. The m-ow denotes the value obtained in the standard Skyrme model with F, = 93 MeV
and e = 2 3 . The light line indicates the nucleon tnitss.
function of qo with fisetl vacuum value
the pion tlecay constant F z ( y o )= 93 MeV.
i\. Sltyrniion solution with rp = qo, which
for fisetl F t is intlepentlent of cpo.As it is reittl off froni this figure, for po-+ 00 the niass
of the chiral soliton defined as solution of eqs. (12) ant1 ( 14) appronches asytnptoticnlly
froni helou the niiiss of the corresponding Sliyrniion, 111 e 1080 MeV. One observes
that for the physically relevant vidues of yo % 300 Me\' the so!iton niass is considerably
siunller than the one of the Skymiion and is close to thee.\periniental valueof the iiricleon
The inrestigntitms i n the present paper show tliat the freening of the chiral rntliris
field q~ to its VilCUllm value yo is not justified in the soliton sector. The above presented
soliton solutions show a (partial) restoration of cliird syinnietry iiisitle the soliton v here
the chiral rathis fieltl rp is reduced t o about one teittli of its v a c u u m value. This belinviour of rp iniplies according to eq. (LO) a drastic reduction of tlie effective pion tlecay
constant F,(q) iiisitle tlie soliton and may esplain why in almost all chiral soliton niodels,
in particular, in the Skyrnie model [12] the pion decay constant F , has to he reduced
strongly (by about 3Oo/,) in order to bring the soliton energy tlown to the niass of the
Thc arrow intlicates the niass corresponding to
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H. REIXIIARDT,13. I < i l I P F E R , Chir;il Soliton JIotlcl
[S] Z U K ,J. A : J. Phys. A I S (1'385) 1T951.
1101 Gonni, H.; JMX, P.; JOIINSOX,
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C.; WITTEX,E.: Sucl. Phys. H 2% (198.3) 5 . 2 .
Rei der Redaktion eingegangen am 21. April I!)YS.
Anschr. d. Verf.: Dr. H. REINIIARDT,
1 list itii t f fir Theorrtischr P h ysik
Cniversitiit Tiibingrn
Xrif der Morgt~nstdlc14
IV-7400 TBbingcm
Dr. B. Kk.nrm.:R
Centrd lnstitute for Niiclcar Rcscnrcli,
0-8051 Dresden, Germany
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chiral, soliton, symmetry, mode, flavour, dynamical, dynamics, restoration, quark
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