xs„ E„ri€EPSGIWWR

he™em˜ er IWWR

f—ryons —s ghir—l ƒolitons

y

in the x—m˜u{ton—Ev—sinio wo del

z

‚F elkoferD rF ‚einh—rdt —nd rF ‡eigel

snstitute for „heoreti™—l €hysi™s

„u˜ingen  niversity

euf der worgenstelle IR

hEUPHU T „u˜ingenD  qerm—ny

y

ƒupp orted in p—rt ˜y the heuts™he pors™hungsgemeins™h—ft @hpqA under ™ontr—™t num˜ er ‚e{VSTGPEPF

z

ƒupp orted ˜y — r—˜ilit—nden{s™hol—rship of the heuts™he pors™hungsgemeins™h—ft @hpqAF I

e˜str—™t

„he des™ription of ˜—ryons —s ™hir—l solitons of the x—m˜u{ton—{v— sinio @xtvA mo del is

reviewedF e motiv—tion for the soliton des™ription of ˜—ryons is provided from l—rge x ghF

g

‚igorous results on the sp ont—neous ˜re—king of ™hir—l symmetry in gh —re dis™ussedF st

is then —rgued th—t the xtv mo del provides — f—ir des™ription of low{energy h—dron physi™sF

„he xtv mo del is therefore employed to mimi™ the low{energy ™hir—l —vor dyn—mi™s of

ghF „he mo del is ˜ osonized ˜y fun™tion—l integr—l te™hniques —nd the physi™—l ™ontent of

the emerging ee™tive meson theory is dis™ussedF sn p—rti™ul—rD its rel—tion to the ƒkyrme

mo del is est—˜lishedF

„he st—ti™ soliton solutions of the ˜ oson ized xtv mo del —re foundD their prop erties disE

™ussedD —nd the inuen™e of v—rious meson elds studiedF „hese ™onsider—tions provide strong

supp ort of ‡itten9s ™onje™ture th—t ˜—ryons ™—n ˜ e understo o d —s soliton solutions of ee™tive

meson theoriesF „he ™hir—l soliton of the xtv mo del is then qu—ntized in — semi™l—ssi™—l f—shE

ion —nd v—rious st—ti™ prop erties of the nu™leon —re studiedF „he domin—ting Iax ™orre™tions

g

to the semi™l—ssi™—lly qu—ntized soliton —re investig—tedF „ime{dep endent meson u™tu—tions

o the ™hir—l soliton —re explored —nd employed to estim—te the qu—ntum ™orre™tions to the

soliton m—ssF pin—llyD hyp erons —re des™ri˜ ed —s ™hir—l solitons of the xtv mo delF „his is done

in ˜ othD the ™olle™tive rot—tion—l —ppro—™h of ‰—˜u —nd endo —s well —s in the ˜ ound st—te

—ppro—™h of g—ll—n —nd ule˜—novF P

gontents

I sntro du™tion S

P ‚igorous results from gh V

PFI v—rge x gh X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X V

g

PFP ghir—l symmetry —nd low{energy theorems X X X X X X X X X X X X X X X X X X X X X IH

PFQ ghir—l —nom—ly —nd the gh v—™uum X X X X X X X X X X X X X X X X X X X X X X X IR

Q „he x—m˜u{ton—{v—sinio mo del IU

QFI hes™ription of the mo del X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X IU

QFP fosoniz—tion X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X IV

QFQ hyn—mi™—l ˜re—king of ™hir—l symmetry X X X X X X X X X X X X X X X X X X X X X X X PH

QFR ghir—l rot—tion —nd hidden lo ™—l symmetry X X X X X X X X X X X X X X X X X X X X X PP

R ie™tive meson theory PS

RFI qr—dient exp—nsion X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X PS

RFP ‚el—tion to the ƒkyrme mo del X X X X X X X X X X X X X X X X X X X X X X X X X X X X PT

RFQ fethe{ƒ—lp eter equ—tions X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X PW

RFR ghir—l —nom—ly X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X QS

S ghir—l ƒolitons QW

SFI „op ologi™—l prop erties X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X QW

SFP imergen™e of the soliton X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RH

SFQ ƒemi™l—ssi™—l qu—ntiz—tion X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RP

T ƒt—ti™ solitons of the x—m˜u{ton—Ev—sinio mo del RR

TFI „he energy fun™tion—l X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RS

TFP ƒelf{™onsistent solutions X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X SI

TFPFI „he pseudos™—l—r hedgehog X X X X X X X X X X X X X X X X X X X X X X X X X SP

TFPFP feyond the ™hir—l ™ir™le X X X X X X X X X X X X X X X X X X X X X X X X X X X X SS

TFPFQ sn™lusion of @—xi—lA ve™tor mesons X X X X X X X X X X X X X X X X X X X X X X SV

TFPFR u—dr—ti™ exp—nsion for time ™omp onents of ve™tor elds X X X X X X X X X TP Q

TFPFS vo ™—l ™hir—l rot—tion X X X X X X X X X X X X X X X X X X X X X X X X X X X X X TT

U f—ryons TW

UFI u—ntiz—tion of the ™hir—l soliton X X X X X X X X X X X X X X X X X X X X X X X X X X TW

UFP ƒt—ti™ nu™leon prop erties X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X UP

UFPFI ile™trom—gneti™ prop erties of the nu™leon X X X X X X X X X X X X X X X X X UQ

UFPFP exi—l ™h—rge of the nu™leon X X X X X X X X X X X X X X X X X X X X X X X X X UT

UFPFQ ‚em—rks on Iax ™orre™tions X X X X X X X X X X X X X X X X X X X X X X X X UU

g

UFQ weson u™tu—tions o the ™hir—l soliton X X X X X X X X X X X X X X X X X X X X X X X UW

UFR u—ntum ™orre™tions to the soliton m—ss X X X X X X X X X X X X X X X X X X X X X X VP

UFS ryp erons X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X VT

UFSFI golle™tive rot—tion—l —ppro—™h X X X X X X X X X X X X X X X X X X X X X X X X VU

UFSFP found st—te —ppro—™h X X X X X X X X X X X X X X X X X X X X X X X X X X X X WP

V ƒumm—ry IHH

epp endix e IHP

epp endix f IHS

epp endix g IHT

epp endix h IHU

epp endix i IIH

‚eferen™es IIP R

I sntro du™tion

st is gener—lly —™™epted th—t u—ntum ghromo hyn—mi™s @ghA is the theory of strong

inter—™tions @for —n intro du™tion see eFgF refsF ‘ID P “AF fesides ™oming in three ™olors the

fermion elds of ghD the qu—rksD —lso ™—rry —vorF „he inter—™tions of gh —re —vor ˜lind

˜ut sensitive to ™olorF gh is —n —symptoti™—lly free theory whi™h me—ns th—t the for™es

˜ etween qu—rks ˜ e™ome we—k for sm—ll qu—rk sep—r—tionsD or equiv—lentlyD l—rge momentum

tr—nsfersF „his —llows one to qu—ntit—tively ™—l™ul—te o˜serv—˜les of strong inter—™tion physi™sD

whi™h —re sensitive to the short dist—n™e ˜ eh—vior of ghD ˜y p ertur˜—tive te™hniquesF es —

m—tter of f—™tD the predi™ted s™—ling viol—tions h—ve ˜ een veried to — high —™™ur—™y —t existing

—™™eler—tors ‘I“F e™™ording to present knowledge gh is the only renorm—liz—˜le theory th—t

™—n —™™ount for these s™—ling viol—tionsF

„he s—me self{inter—™tions of gluons whi™h give rise to —symptoti™ freedom le—d to — strong

qu—rk{qu—rk inter—™tion for medium —nd sm—ll energiesF sing —ddition—lly the empiri™—l f—™t

th—t neither qu—rks nor gluons h—ve ˜ een dete™ted —s ’free4 p—rti™les h—s le—d to the ™onneE

ment hyp othesisX ynly singlets of the g—uge group —pp e—r —s physi™—l p—rti™lesF isp e™i—llyD the

qu—rks ˜ elonging to the fund—ment—l represent—tion —nd the gluons ˜ eing in the —djoint repreE

—

sent—tion ™—n never ˜ e o˜served dire™tlyF „he p ertur˜—tive result th—t the ™oupling ™onst—nt

in™re—ses —s the momentum tr—nsfer ˜ e™omes sm—llD or the dist—n™e l—rgeD is in —™™ord—n™e

with the ™onnement hyp othesisF nfortun—telyD this ˜ eh—vior of the ™oupling ™onst—nt exE

™ludes p ertur˜—tive ™—l™ul—tions for low energiesF „hus it is still unproven th—t gh is re—lly

— ™onning theoryF iven worseD until to d—y prop erties of h—drons h—ve not ˜ een ™—l™ul—ted

from gh without m—king use of severe —ssumptions or simpli™—tionsF

„here is pro˜—˜ly one ex™eption to this st—tementX ‡ith the help of wonte{g—rlo te™hE

niques —ttempts h—ve ˜ een m—de to ™—l™ul—te h—dron prop erties dire™tly from gh using —

dis™rete l—tti™e @for —n intro du™tion to l—tti™e g—uge theories see eFgF refsF ‘Q“ or ‘R“AF hespite

the f—™t th—t these non{p ertur˜—tive ™—l™ul—tions should en—˜le the determin—tion of every

physi™—l qu—ntity the results often devi—te very strongly from the exp eriment—l v—luesF „here

—re sever—l re—sons for thisF iven ˜y using ’high{p erform—n™e4 ™omputers the p ossi˜le l—tti™e

sizes —re still mo destD esp e™i—llyD if one w—nts to in™lude dyn—mi™—l fermionsF purthermoreD —

™—l™ul—tion using m—ssless qu—rks is imp ossi˜leF end there —re still op en ™on™eptu—l questions

™on™erning the ™ontinuum limit of l—tti™e theoryF

qiven this st—te of ——irs it is n—tur—l to resort to ee™tive mo dels of strong inter—™tionsF

„hese —re intended to mimi™ the low{energy ˜ eh—vior of gh —s ™losely —s p ossi˜leF por

this purp ose the —pproxim—te ™hir—l symmetry of gh ‘S“ provides — very useful guidelineF

r—dron phenomenology h—s left no dou˜t th—t this symmetry is ˜roken dyn—mi™—lly ˜y strong

inter—™tionsF ‚equiring the known p—ttern of expli™itD sp ont—neous —nd —nom—lous ™hir—l

symmetry ˜re—king puts signi™—nt ™onstr—ints on p ossi˜le mo dels for the strong inter—™tions

of qu—rksF eddition—l guid—n™e ™—n ˜ e o˜t—ined if one gener—lizes gh to — g—uge theory

with —n —r˜itr—ry num˜ er of ™olors x F „his is ˜ e™—use for l—rge x gh redu™es to —n

g g

ee™tive theory of innitely m—ny we—kly inter—™ting mesons —nd glue˜—lls ‘T“F nfortun—telyD

this ee™tive meson theory ™—nnot ˜ e ™onstru™ted expli™itlyF xeverthelessD ‡itten w—s —˜le to

give —rguments th—t within this ee™tive theory ˜—ryons emerge —s soliton solutions ‘U“F

elthough ‡itten9s ™onje™ture h—s never ˜ een proven rigorously the soliton pi™ture of

˜—ryons h—s turned out quite su™™essful in re™ent ye—rsF „he st—rting p oints h—ve ˜ een pheE

nomenologi™—l ee™tive meson theoriesD whi™h p ossess soliton solutionsF „he most p opul—r

—

ƒee eFgF gh—pter QFRFP in refF‘P“F S

ones p erh—ps —re the ƒkyrme mo del ‘VD W “ —nd the g—uged ' Emo del ‘IHD II “F snvestig—tions

within these mo dels h—ve s—tisf—™torily expl—ined the we—lth of sp e™tros™opi™ ˜—ryon d—t—D see

eFgF refF ‘IP“ for — re™ent ™ompil—tion of referen™es on soliton mo dels for ˜—ryonsF st is worthE

while to mention th—t m—ny of the e—rly di™ulties en™ountered for the ƒkyrme mo delD like eFgF

form f—™tors ˜ eing to o soft ‘IQ D IH “D the missing intermedi—te r—nge —ttr—™tion in the nu™leon{

nu™leon for™e ‘IR“ or the line—rly rising ph—se shifts in pion nu™leon s™—ttering ‘II“D h—ve found

s—tisf—™tory solutionsF €—rtlyD this w—s —™hieved ˜y using ee™tive meson v—gr— ngi—ns ‘ISD IT “

des™ri˜ing the meson physi™s ˜ etter th—n the origin—l simple ƒkyrme v—gr— ngi—nF yn the other

h—ndD this in™re—sed ™omplexity —lso gener—ted more —m˜iguity in the ee™tive —™tionF

et th—t p oint n—tur—lly the question —rises whether — more mi™ros™opi™ pi™ture of the soliton

™—n provide some guide for ™ho osing the ee™tive meson theoryF iven moreD one m—y —im —t

— justi™—tion of the soliton pi™ture of ˜—ryons in gener—lF „herefore not only — mi™ros™opi™

re—liz—tion of the soliton pi™ture in terms of qu—rk degrees of freedom is w—nted ˜ut the

dyn—mi™s of the mo del should ˜ e —˜leD —t le—st in prin™ip—lD to determine its f—vored pi™ture of

the ˜—ryonF sn this sense the x—m˜u{ton—Ev— sinio @xtvA mo del ‘IU “ is uniqueF pirst of —llD it

is simple enough th—t su™h ™ompli™—ted eld ™ongur—tion like solitons ™—n ˜ e determined self{

™onsistentlyD see gh—pter TF ƒe™ondD it ™ont—ins the ™orre™t ™hir—l symmetry ˜re—king p—ttern

—nd repro du™es — lot of meson prop erties like m—ssesD de™—y ™onst—ntsD s™—ttering lengths et™FD

using only — few input p—r—metersD see eFgF ‘IVD IW “ —nd referen™es thereinF „hirdD —nd more

imp ort—ntD it —llows for two ™omplement—ry pi™tures of ˜—ryonsD n—mely either —s ordin—ry

three{v—len™e{qu—rk ˜ ound st—te or —s ™hir—l solitonF „he l—tter will ˜ e the su˜ je™t of this

reviewF sn the other —ppro—™h the ˜—ryon w—ve fun™tion is o˜t—ined —s — solution of — p—ddeev

equ—tion using diqu—rks —s intermedi—te ˜uilding ˜lo ™ks ‘PHD PI “F sing fun™tion—l integr—l

te™hniques one ™—n derive — gener—ting fun™tion—l whi™h —llows one to tre—t ˜ oth pi™tures in

one qu—rk theory without —ny —m˜iguities or dou˜le ™ounting ‘PH “F ƒu™h — theory ™ont—ins the

rel—tivisti™ qu—rk mo del —nd the ƒkyrmion —s —symptoti™ limitsF €relimin—ry results indi™—te

th—t su™h — hy˜rid mo del ™—n displ—y unexp e™ted fe—turesD eFgF the diqu—rk m—ss is dr—sti™—lly

redu™ed in — soliton ˜—™kground ‘PP“F

y˜viouslyD the simpli™ity of the xtv mo del whi™h m—kes it suit—˜le for su™h ™ompli™—ted

investig—tions is —lso its most severe dr—w˜—™kF „he xtv mo del is non{renorm—liz—˜le —nd

only uniquely dened if the ne™ess—ry regul—riz—tion pres™ription is sp e™iedF „his intro du™es

—n ultr—violet ™ut{o whi™h indi™—tes the r—nge of —ppli™—˜ility of the mo delF „he results

for sever—l o˜serv—˜les dier in v—rious regul—riz—tion s™hemes ‘IV “F iven worseD sometimes

the qu—lit—tive ˜ eh—vior ™h—nges when —ltering the regul—riz—tion pres™riptionF portun—telyD

the situ—tion is not —s dr—m—ti™ for most of the o˜serv—˜lesD or ™—n ˜ e understo o d from

the de™ien™ies of the regul—riz—tion pro ™edure like eFgF missing g—uge inv—ri—n™eF „he other

serious dis—dv—nt—ge of the xtv mo del is the —˜sen™e of ™onnementD or more pre™iselyD the

—pp e—r—n™e of two{qu—rk @or qu—rk{—ntiqu—rkA thresholdsF „hese ™—use unphysi™—l im—gin—ry

p—rts in ™orrel—tion fun™tions for l—rge @time{likeA moment—F „herefore the results of the xtv

mo del —re restri™ted to low energies not only ˜y the ultr—violet ™ut{o intro du™ed vi— the

regul—riz—tion ˜ut —lso ˜y the two{qu—rk thresholdsF

„his review is devoted to the soliton des™ription of ˜—ryons within the xtv mo delF st is

not the prim—ry go—l of su™h investig—tions to repro du™e —ll the phenomenologi™—l su™™esses

of ƒkyrme typ e mo delsF „he ˜—si™ motiv—tion h—s r—ther ˜ eenD —nd is stillD to in™re—se our

underst—nding how the soliton emergesD —nd to ˜ etter underst—nd its rel—tion to the underlying

qu—rk dyn—mi™sF isp e™i—llyD the ™on™eption—l e—sy —nd dire™t —™™ess to the qu—rk degrees of

freedom —llows one to study questions whi™h es™—p e our ™onsider—tions when st—rting from

— purely mesoni™ mo delF yn the other h—ndD in order to —rrive —t physi™—l ˜—ryons one h—s T

to use te™hniques whi™h —re well known from the ƒkyrmionX semi™l—ssi™—l qu—ntiz—tion ‘W“

@™r—nking ‘PQ“AD ™—l™ul—tion of qu—ntum ™orre™tions ‘PRD PS “D gener—liz—tion to three —vors ˜y

either ™olle™tive qu—ntiz—tion ‘PTD PU D PV “ or ˜ ound st—te —ppro—™h ‘PW D QH “D —nd so onF „hese

su˜ je™ts will ˜ e dis™ussed in det—il in this reviewD see gh—pter UF „hey should ˜ e ™onsidered

—s — ˜—sis for mo del ™—l™ul—tions where one w—nts to des™ri˜ e ˜—ryons —s ™hir—l solitons —nd

—ddition—lly w—nts to h—ve —™™ess to the qu—rk degrees of freedom in one ™onsistent fr—meF

„he org—niz—tion of this review is —s followsX sn gh—pter P we will present some rigorous

results from gh whi™h —re relev—nt for the soliton des™ription of ˜—ryonsF efter — short

summ—ry of the l—rge x —rguments for ˜—ryons ˜ eing solitons we will outline some low{

g

energy theorems ˜—sed on ™hir—l symmetry —nd the ™hir—l —nom—lyF sn gh—pter Q we will

des™ri˜ e the xtv mo del —nd its ˜ osoniz—tion ‘QI “F sn ƒe™tion QFQ — very imp ort—nt prop erty

of the xtv mo delD the dyn—mi™—l ˜re—king of ™hir—l symmetryD is des™ri˜ edF ‡e will —lso use

— lo ™—l ™hir—l rot—tion to displ—y the hidden g—uge symmetry ‘IT “ of the ˜ osonized mo del ‘QP“F

gh—pter R is devoted to the ee™tive meson theory of the xtv mo delF rere˜yD the gr—dient

exp—nsion do es not only yield —pproxim—te meson m—sses —nd ™oupling ™onst—nts ˜ut —lso

reve—ls the rel—tion to the ƒkyrme mo delF „he determin—tion of meson m—sses with the help

of fethe{ƒ—lp eter equ—tions is expl—ined in ƒe™tion RFQF ƒe™tion RFR ™ont—ins some ™omments

on the ™hir—l —nom—ly in the ee™tive meson theoryF sn gh—pter S the generi™ —sp e™ts of

™hir—l solitons —re dis™ussedX the top ologi™—l prop ertiesD the emergen™e of the soliton —nd its

semi™l—ssi™—l qu—ntiz—tionF gh—pter T is the ™entr—l pie™e of this reviewF efter giving the

energy fun™tion—l of the st—ti™ xtv soliton we dis™uss the self{™onsistent solutions for dierent

meson elds in™luded @or negle™tedAF gh—pter U em˜ o dies the des™ription of ˜—ryons —s xtv

solitonsF „his —lso in™ludes — ™omprehensive expl—n—tion of the ™r—nking metho d for two —nd

three —vors —s well —s the des™ription of time{dep endent meson u™tu—tions o the st—ti™

solitonF sn gh—pter V we give — short summ—ryF ƒome lengthy formul—s whi™h —re nevertheless

ne™ess—ry to m—ke this review re—son—˜ly self{™ont—ined —re given in ve —pp endi™esF U

P ‚igorous results from gh

sn this ™h—pter we will present some rigorous gh results whi™h —re relev—nt to h—dron

physi™s —nd in p—rti™ul—r to the soliton des™ription of ˜—ryonsF es dis™ussed in the intro du™tion

gh ™—nnot ˜ e tre—ted p ertur˜—tively —t low energiesF es — ™onsequen™e there —re very few

su™h resultsF „hey —re either ˜—sed on ™onsider—tions —ssuming the num˜ er of ™olors x to

g

˜ e l—rge or on the ™hir—l symmetry of m—ssless gh —nd its sp ont—neous ˜re—kingF es we will

see ˜ oth provide su˜st—nti—l —rguments for the ™hir—l soliton pi™ture of ˜—ryonsF

PFI v—rge x gh

g

sn the low{energy regime there is no o˜vious exp—nsion p—r—meter to tre—t gh p erturE

˜—tivelyF roweverD in the seventies 9t ro oft ‘T“ —nd ‡itten ‘U“ demonstr—ted th—t gener—lizing

gh from the g—uge group ƒ @QA to ƒ @x A with x ˜ eing l—rgeD Iax might ˜ e ™onsidE

g g g

ered —s —n impli™it exp—nsion p—r—meterF x—tur—llyD one might wonder whether this provides

— sound ˜—sis for — p ertur˜—tive —n—lysis sin™e x a Q in the re—l worldF xeverthelessD we

g

will see th—t this ide— is very fruitfulD esp e™i—lly it is the essenti—l motiv—tion for identifying

˜—ryons with solitons of meson eldsF

essuming ™onnement in the l—rge x world it ™—n ˜ e shown th—t gh with x 3 I

g g

P

—nd g x xed h—s — limit whi™h ™—n ˜ e des™ri˜ ed —s keeping only pl—n—r peynm—n di—gr—ms

g

‘T“F „he re—son is th—t non{pl—n—r di—gr—ms or di—gr—ms with gluon h—ndles —re suppressed

I P

F ƒimple p ower ™ounting —rguments using 9t D —nd the ones with qu—rk lo ops ˜y x ˜y x

g g

ro oft9s di—gr—mm—ti™ —n—lysis reve—l th—t the ™orrel—tion fun™tion of — ™olor singlet op er—tor

is domin—ted ˜y pl—n—r di—gr—ms with ex™lusive gluon insertions —nd — qu—rk lo op on the

exteriorF purthermoreD in le—ding order in the Iax exp—nsion this ™orrel—tion fun™tion ™—nnot

g

f—™torizeD iFeF it is s—tur—ted only ˜y ™olor singlet intermedi—te st—tesF „his then le—ds to 9t

ro oft9s most imp ort—nt result on l—rge x ghX sn this limit gh redu™es to — theory of

g

H

@innitely m—nyA we—kly inter—™ting mesons —nd glue˜—llsF „heir m—sses —re of order x D their

g

muti—l inter—™tion is suppressed ˜y p owers of Iax F

g

„he mesons —re st—˜le o˜ je™ts in the limit x 3 IF „heir de™—y r—tes —re of order

g

P

F „his is very gr—tifying ˜ e™—use it provides — n—tur—l Iax D their ™ross se™tions of order x

g

g

expl—n—tion for the existen™e of n—rrow meson reson—n™es in n—tureF sndeedD it is not o˜vious

—t —ll th—t eFgF the de™—y & 3 P% is n—rrow enough to dete™t the & meson exp eriment—llyF

iven moreD the r—tio of this de™—y width to the & meson m—ss very roughly follows the Iax

g

™ountingF „o ˜ e more sp e™i™ we ™onsider —n n{p oint ™orrel—tion fun™tion of — ™olor singlet

op er—torF „he summ—tion of pl—n—r di—gr—ms le—dsD —mong othersD to —n ee™tive meson

self{™oupling whi™h —t tree level t—kes the form ‘U“

n

CI

P

v a x !  XXX  X @PFIA

n i XXXi i i

n n

g I I

p

‚es™—ling the meson elds  —™™ording to  a x 9 le—ds to

i i g i

v a x ! 9 X X X 9 X @PFPA

n g i XXXi i i

n n

I I

prom eqF @PFPA one ™—n then ™on™lude th—t the gener—ting fun™tion—l of gh ˜ e™omes in

le—ding order of the Iax exp—nsion

g

 

x 3I

g

ie ix e‘9“

g h g

 a h ‘eY q" Y q “e 3 h 9e @PFQA

g h V

where e‘9“ is —n ee™tive meson —™tion th—t involves innitely m—ny mesons —nd is of the

generi™ form @summ—tion over rep e—ted indi™es is —ssumedA

 



I I I

R P

e‘9“ a d x @d 9 A m 9 9 C ! 9 9 9 C XXX X @PFRA

" i ij i j ij k i j k

P P3 Q3

Iax ™orre™tions —re then given in terms of meson lo op termsF

g

sing the l—rge x di—gr—mm—ti™s ‡itten ™onje™tured ‘U“ th—t ˜—ryons ˜ eh—ve —s soliton

g

P

solutions of the ee™tive theory @PFRAF essuming —g—in th—t qu—rks —re ™onned —nd th—t g x

g

is — ™onst—nt the m—ss of — ˜—ryon ™omp osed of x qu—rks in their —ntisymmetri™ ground st—te

g

is given ˜y

I

P P

x @x IAg † % x p @g x Y m A @PFSA w a x @m C „ AC

g g g g q f g q

P

to le—ding order in the Iax exp—nsionF fesides the m—sses m —nd the kineti™ energy „ of

g q

the individu—l qu—rks —lso the one{gluon{ex™h—nge is in™ludedF fy ™om˜in—tori™ —rguments it

P n

™—n ˜ e shown th—t the n{gluon{ex™h—nge ˜ eh—ves like x @x g A for x ) nF „herefore the

g g g

P H

fun™tion p @g x Y mA whose le—ding order is x p—r—metrizes the Iax ™orre™tions in — smo oth

g g

g

f—shionF ƒin™e the strength of the meson ™oupling is prop ortion—l to Iax the ˜—ryon m—ss

g

w is prop ortion—l to the inverse of this meson ™ouplingF es the ˜—ryon r—dius is prop ortion—l

f

to the inverse of the qu—rk kineti™ energy up to suppressed ˜inding ee™ts it is indep endent

of x in le—ding orderF „his is the typi™—l ˜ eh—vior of — soliton eld3 „hese results @—s

g

H

—nd well —s the f—™t th—t the meson{˜—ryon —nd ˜—ryon{˜—ryon inter—™tions —re of order x

g

I

x D resp e™tivelyA —re the ˜—sis for ‡itten9s ™onje™ture th—t ˜—ryons emerge —s solitons of —n

g

ee™tive meson theoryF

y˜viouslyD su™h — ˜ eh—vior ™—n never ˜ e o˜t—ined in p ertur˜—tion theoryF nfortun—telyD

—

the ee™tive meson theory @PFRA ™—nnot ˜ e ™onstru™ted expli™itlyF xeverthelessD this kind of

re—soning provides some insight in the we—lth of exp eriment—l resultsF „he l—rge x —rguments

g

—™™ount for suppression of exoti™sD ™onrms weig9s rule —nd ‚egge phenomenologyF purtherE

moreD it is ™onsistent with phenomenologi™—l mo dels like eFgF the meson ex™h—nge mo delsF

yn the other h—ndD to use the Iax exp—nsion —lso qu—ntit—tively one h—s to resort to

g

ee™tive mo delsF „he most prominent ex—mple for soliton mo dels of ˜—ryons is the ƒkyrme

mo del ‘VD W “D for — re™ent review see refF ‘IP “F sn the following ™h—pters we will present the

des™ription of ˜—ryons —s solitons within the xtv mo delF fesides others it h—s the virtue th—t

‡itten9s ™onje™ture m—y ˜ e tested ˜y the dyn—mi™s of the mo delF xeverthelessD one might

—lso r—ise the question whether it is not more n—tur—l to des™ri˜ e ˜—ryons —s three{qu—rk

˜ ound st—tes if one st—rts from —n ee™tive qu—rk theoryF elso here the Iax exp—nsion

g

might give some hintsF ƒt—rting from —n ee™tive qu—rk inter—™tion des™ri˜ ed ˜y — ™urrent{

™urrent ™oupling of ™olor o ™tet qu—rk ™urrents @see eqF @QFPAA erzing into —ttr—™tive ™h—nnels

—nd su˜sequent fun™tion—l integr—l h—droniz—tion le—ds to —n ee™tive theory whi™h ™ont—ins

˜ esides mesons —lso ˜—ryon elds ™omp osed of — diqu—rk —nd — qu—rk ‘PH “F „he interesting

f—™t now is th—t in this ee™tive theory the qu—rk{qu—rkD whi™h is resp onsi˜le for diqu—rk

form—tionD inter—™tion is suppressed ˜y Iax —s ™omp—red to the —ntiqu—rk{qu—rk inter—™tionD

g

iFeF in the limit x 3 I these expli™it ˜—ryon elds —re removed from the theory ‘QS “ —nd one

g

h—s to nd the ˜—ryons in the left{over ee™tive meson theory whi™hD ˜y the w—yD is form—lly

identi™—l to the ee™tive meson theory o˜t—ined ˜y ˜ osonizing the xtv mo delF „his ee™tive

theory will ˜ e dis™ussed in some det—il in ™h—pter RD —nd its soliton solutions —re ex—™tly the

xtv solitonsF

—

sn ™ontr—st to the situ—tion in four sp—™e{time dimensions gh ™—n ˜ e ˜ osonized ex—™tly ‘QQD QR“F

P W

PFP ghir—l symmetry —nd low{energy theorems

fesides the ex—™t ™olor symmetryD the gh v—gr—ngi—n p ossesses —n —ddition—l symmetry

in the limit of v—nishing qu—rk ™urrent m—ssesD the ™hir—l symmetryF sn order to displ—y this

symmetry it is ™onvenient to split the qu—rk elds in right{ —nd left{h—nded ™omp onentsF

„hese —re dened ˜y

q a € q Y q a € q @PFTA

‚ ‚ v v

where

I

@I   A @PFUA € a

S ‚Yv

P

—re the ™orresp onding pro je™torsF „hen the gh v—gr—n gi—n t—ken —t zero ™urrent m—sses

de™ouples into — sum of two v—gr— ngi—ns ™ont—ining only right{@left{A h—nded eldsD resp e™E

tivelyF „hese two v—gr—ngi—ns —re inv—ri—nt under glo˜—l unit—ry —vor tr—nsform—tions of

the ™orresp onding right{@left{A h—nded eldsD iFeF the gh v—gr— ngi—n is inv—ri—nt under

@x A  @x A where x is the num˜ er of —vors under ™onsider—tionF „he de™omp osition

v f ‚ f f

into semi{simple su˜groups

$

@IA  @IA  ƒ @x A  ƒ @x A @x A  @x A

a

vC‚ v‚ v f ‚ f v f ‚ f

—llows one to dis™uss the —sso ™i—ted ™onserved ™h—rgesF „he inv—ri—n™e under @IA is

vC‚

resp onsi˜le for the ™onserv—tion of ˜—ryon num˜ er where—s @IA is su˜ je™t to —n —nom—ly

v‚

‘QT “F „he impli™—tions of this —nom—ly will ˜ e dis™ussed in the su˜sequent se™tion in more

det—ilF

„he inv—ri—n™e under unit—ry tr—nsform—tions ˜ elonging to the su˜group ƒ @x Aƒ @x A

v f ‚ f

P

I ™urrents implies the ™onserv—tion of the x

f

—

!

—

 q @xA @xA a q" @xA t

" v v

v"

P

—

!

—

 q @xA @PFVA @xA a q" @xA t

" ‚ ‚

‚"

P

—

where the m—tri™es ! —re the gener—tors of ƒ @x AF „he ™orresp onding ™onserved ™h—rges

f



— Q —

 a d xt @xA

v vH



— Q —

 a d xt @xA @PFWA

‚ ‚H

fulll ™ommut—tion rel—tions whi™h —re identi™—l to the one of the gener—tors of the group

I

ƒ @x A  ƒ @x AF roweverD they mix under p—rityD €  € a  F €—rity eigenst—tes —re

v f ‚ f v ‚

given ˜y the ™urrents whi™h —re rel—ted to the di—gon—l su˜group ƒ @x A —nd the ™oset

vC‚ f

@ƒ @x A  ƒ @x AAaƒ @x AX

v f ‚ f vC‚ f

—

!

— — —

 q @xA † @xA a t @xAC t @xA a q"@xA

"

" ‚" v"

P

—

!

— — —

  q @xAX @PFIHA e @xA a t @xA t @xA a q"@xA

" S

" ‚" v"

P IH

nder p—rity these ™urrents tr—nsform like ve™tors or —xi—l{ve™torsD resp e™tivelyF purthermoreD

their time ™omp onents fulll the equ—l{time ™ommut—tion rel—tions

— ˜ —˜™ ™ @QA

‘† @xY tAY † @y Y tA“ a if † @xY tA @x y A

H H H

— ˜ —˜™ ™ @QA

‘† @xY tAY e @y Y tA“ a if e @xY tA @x y A

H H H

— ˜ —˜™ ™ @QA

‘e @xY tAY e @y Y tA“ a if † @xY tA @ x y A @PFIIA

H H H

—˜™

where f —re the stru™ture ™onst—nts of the vie —lge˜r— ƒ @x AF „hese ™ommut—tion rel—tions

f

—re known —s ™urrent —lge˜r— ‘QU“F es the left h—nd side is qu—dr—ti™ in the ™urrents where—s

the right h—nd side is line—r these rel—tions —lso x the norm—liz—tion of the ™urrentsF „his is

the ˜—sis of the phenomenologi™—lly su™™essful ™urrent —lge˜r— sum rulesF

ghir—l symmetry is expli™itly ˜roken ˜y the ™urrent m—ssesF xevertheless it is — go o d

—pproxim—tion for the two light —vors @up —nd downA sin™e their ™urrent m—sses —re of the

order of — few we†F por the str—nge qu—rk whose ™urrent m—ss is of the s—me order of m—gnitude

—s the fund—ment—l gh s™—leD  D ™hir—l symmetry is ™ert—inly not —s go o d — symmetryF

g h

sn spite of thisD —ssuming —pproxim—te ™hir—l symmetry for str—nge qu—rks is useful for —

wide r—nge of —ppli™—tionsF por the he—vy qu—rks the opp osite typ e of exp—nsionD n—mely

in p owers of IamD h—s proven to ˜ e su™™essful ‘QV“F re—vy qu—rks ™—n ˜ e negle™ted in low

energy pro ™esses —s — ™onsequen™e of the epp elquist{g—rr—zzone theorem ‘QW“ whi™h st—tes

th—t p—rti™les de™ouple whose m—ss is mu™h l—rger th—n the typi™—l energy s™—le for the pro ™ess

under ™onsider—tionF ‡e will therefore tre—t only the ™—ses of two or three —vors in the

followingF

ghir—l symmetry is sp ont—neously ˜roken implying the existen™e of pseudos™—l—r @would{

˜ eA qoldstone ˜ oson sF „hese —re identied with the isotriplett of pions or for three —vors

with the o ™tet of pseudos™—l—r mesons % D u —nd  F e further ™onsequen™e is the dyn—mi™—l

gener—tion of — non{p ertur˜—tive qu—rk m—ssF st is exp e™ted th—t this qu—rk m—ssD whi™h is

often l—˜ eled —s ™onstituent qu—rk m—ssD is of the order of  F purthermoreD — nite non{

g h

p ertur˜—tive qu—rk m—ss implies — non{zero qu—rk ™ondens—te whi™h is dened in terms of the

full qu—rk prop—g—tor

trƒ @xY y AX @PFIPA hq" q i a i lim

p

C

y 3x

xote th—t the qu—rk ™ondens—te hq" q i is — g—uge inv—ri—nt qu—ntityF „herefore it ™—n ˜ e ev—luE

—ted in —ny g—ugeF sn — ™ov—ri—nt g—uge the qu—rk prop—g—tor is of the form

i

X @PFIQA ƒ @q A a

p

P P

aqe@q A f @q AC i

P

sn p ertur˜—tion theory in the ™hir—l limit one h—s f @q A a HF „herefore the qu—rk ™ondens—te

would v—nish due to the v—nishing hir—™ tr—™e of  F yn the other h—ndD — dyn—mi™—lly

"

P

gener—ted f @q A Ta H implies — non{v—nishing ™ondens—teF

—

sing hq" q i Ta H —nd the qoldstone theorem one ™—n prove th—t the —xi—l ™urrents e @PFIHA

"

™ouple the qoldstone ˜ osons to the v—™uumF henoting the one p—rti™le st—tes of qoldstone

—

˜ osons with momentum p ˜y j% @pAi one o˜t—ins

— ˜ —˜ ipx

hHje @xAj% @pAi a if p e X @PFIRA

"

"

—˜

„he f —re non{v—nishing ™onst—ntsF sn the isospin symmetri™ ™—se they —re prop ortion—l to

—˜ —˜

the unit m—trixD f a  f F yne usu—lly determines their numeri™—l v—lues from we—k pion

— II

@k—onA de™—yD eFgF % 3 "#"F f @f A is therefore ™—lled pion @k—onA de™—y ™onst—ntF por —

% u

™ompil—tion of re™ent exp eriment—l d—t— see p—ge IRRQ of refF ‘RH “F por the purp ose of this

review it su™es to know th—t f a WQwe† —nd f af a IXPPF

% u %

e™ting with the deriv—tive op er—tor on eqF @PFIRA —nd using the ulein{qordon equ—tion for

P P

the pion st—te @p a m A yields

%

" — ˜ —˜ P ipx

hHjd e @xAj% @pAi a  f m e X @PFISA

%

" %

„he ™onserv—tion of the —xi—l ™urrent implies either f a H or m a HF „hese two p ossi˜ilities

% %

™orresp ond to the ‡igner{‡eyl or x—m˜u{qoldstone re—liz—tion of ™hir—l symmetryD resp e™E

tivelyF es ˜ oth qu—ntities —re nite ™hir—l symmetry h—s to ˜ e expli™itly ˜rokenD iFeF one h—s

to use nite ™urrent qu—rk m—sses in order to des™ri˜ e n—tureF „o further pro ™eed eqF @PFISA

—

is elev—ted to —n op er—tor identityF vet us rst intro du™e the pion eld op er—tor 0 @xA —nd

%

™ho ose its norm—liz—tion with resp e™t to the one pion st—te to ˜ e

— ˜ —˜ ipx

hHj0 @xAj% @pAi a  e X @PFITA

%

„hen the op er—tor identity

— P " —

@xA @PFIUA 0 @xA a f m d e

%

% % "

˜

implies the rel—tion @PFISAF iqF @PFIUA is known —s the €geg @€—rti—lly gonserved exi—lve™tor

gurrentA hyp othesisF essuming —ddition—lly th—t meson —nd ˜—ryon form f—™tors ™—n ˜ e exE

tr—p ol—ted smo othly from the ™orresp onding m—ss shells —llows one to rel—te dierent h—droni™

o˜serv—˜les dep ending on ˜ othD we—k —nd strong inter—™tion p—r—metersF yne f—mous ex—mple

is the qold˜ erger{„reim—n rel—tion

f g a m g @PFIVA

% % x x x e

whi™h ™onne™ts the —xi—l ™oupling of the nu™leon g with the pion nu™leon ™oupling ™onst—nt

e

g @m is the nu™leon m—ssAF „his rel—tion is exp eriment—lly fullled within ten p er™entF

% x x x

„his @inA—™™ur—™y sheds some light on the usefulness of the €geg hyp othesisF

sing the €geg hyp othesis one furthermore dedu™es ‘S“



" ˜ R — P —˜ P

@HA“jHi @PFIWA @xAY d e a i d xhHj @x A‘e f  m

H

" H % %

in the limit of v—nishing meson momentumF „his m—y ˜ e rewritten in terms of the @p—rti—lly

—

™onservedA —xi—l ™h—rges  —nd the r—miltoni—n r@xA —s

S

—˜ P P — ˜

 m f a hH‘  Y ‘ Y r@HA““jHiX @PFPHA

% % S S

es these ™h—rges gener—te innitesim—l —xi—l tr—nsform—tions it is o˜vious th—t only ™hir—l

symmetry ˜re—king terms in the r—miltoni—n ™ontri˜ute to the dou˜le ™ommut—torF sn the

q

H

PaQ Is A ™—se of three —vors the symmetry ˜re—king m—ss term is given ˜y @! a

Q V H

! ! !

H H H

"

q @PFPIA q C ™ q" q C ™ q" m uu" C m dd C m ss" a ™ q"

Q V H

u d s

P P P

˜

xote th—t eqF @PFIUA is not derived from @PFISAF yne h—s —lso to —ssume th—t the m—trix elements

" —

hHjd e @xAjˆ i for —ll st—tes ˆ ex™ept the one pion st—te v—nishF

" IP

where the symmetry ˜re—king p—r—meters ™ —re fun™tions of the ™urrent qu—rk m—sses

i

I

H H H

p

@m ™ a C m C m A

H

u d s

T

I

H H

@m m A ™ a

Q

u d

P

I

H H H

p

@m C m ™ a Pm AX @PFPPA

V

u d s

P Q

„he ™ommut—tors @PFPHA m—y ˜ e ™omputed with the help of the expli™it expressions @PFWA —nd

@PFIHA of the —xi—l ™h—rgesF sing the —lge˜r— of the qell{w—nn m—tri™es —nd esp e™i—lly the

rel—tion

™ ˜ —

! ! !

—˜™

q “a id q"  q @PFPQA   q Y q" ‘q "

S H S

P P P

—˜™

@the d —re the symmetri™ stru™ture ™onst—nts of ƒ @QAA one o˜t—ins the f—mous qellEw—nn{

‚enner{y—kes rel—tions ‘RI“

I

P H P H

"

a AhHjuu" C d djHi m C m f @m

% d % u

P

I

H H P P

AhHjuu" C"ssjHi C m @m a m f

s u u u

P

R I

H H H P P

"

hHjs" sjHiX @PFPRA m AhHjuu" C d djHi C C m @m a m f

s d u  

Q T

por the ™—se of equ—l qu—rk ™ondens—tes

"

hHjuu" jHi a hHjd djHi a hHjss" jHi

the denition of the de™—y ™onst—nts @PFIRA implies th—t —ll de™—y ™onst—nts —re equ—lF por this

P P P

sp e™i—l ™—se eqsF @PFPRA —lso yield the qellEw—nn{yku˜ o m—ss rel—tion ‘RP “ Rm a Qm C m

u  %

whi™h h—s ˜ een o˜served empiri™—llyF sn —ddition one ™—n extr—™t the r—tio of ™urrent qu—rk

m—sses

H H P

C m m m I

d u %

X @PFPSA a %

P

H P

PS Pm Pm m

u

s %

H H

„—king into —™™ount isospin ˜re—king m Ta m —nd ele™trom—gneti™ ™orre™tions the rel—E

u d

tions @PFPRA ™—n ˜ e renedF xow—d—ys the v—lues of —ll ™urrent m—sses ™—n ˜ e estim—ted quite

reli—˜lyF roweverD only the r—tios of ™urrent m—sses —re uniquely dened ˜ e™—use these —re

s™—le inv—ri—ntF giting —n —˜solute v—lue of — ™urrent m—ss hen™e ne™essit—tes referen™e to —

s™—leF et I qe† ™ommonly —™™epted v—lues for the ™urrent m—sses —re ‘RQ“

H

m @Iqe† A % Swe†

u

H

m @Iqe† A % Wwe†

d

H

m @Iqe† A % ITHwe† X @PFPTA

s

yne shouldD howeverD note th—t these d—t— —re to some extend mo del dep endent ˜ e™—use they

—re extr—™ted form low{energy meson phenomenology ‘RR“F IQ

PFQ ghir—l —nom—ly —nd the gh v—™uum

„he prop erties of the gh v—™uum —re still p o orly understo o dF prom h—dron phenomenolE

ogy one dedu™es th—t non{v—nishing ™ondens—tes h—ve to existF ix—mples —re the gluon —nd

— —"#

the qu—rk ™ondens—tesD hq q i —nd hq" q iF „he rel—tion of the l—tter to the sp ont—neous

"#

˜re—king of ™hir—l symmetry h—s ˜ een dis™ussed in the pre™eding se™tionF sn this se™tion we

will lo ok —t it from —nother p oint of viewX „he qu—rk ™ondens—te is rel—ted to the me—n density

of eigenv—lues of the qu—rk hir—™ op er—torF „o see this we ™onsider the qu—rk prop—g—to r for

xed

— xed g—uge eld e

"

y

ˆ

@y A u @xAu

n

n

@PFPUA a a h„ q @xA"q@y Ai @ƒ @xY y AA

xed

xed

p

e

e

"

H

"

m i!

n

n

where u @xA —nd ! —re eigenfun™tions —nd eigenv—lues of the iu™lide—n hir—™ op er—tor

n n

ha u @xA a ! u @xAX @PFPVA

n n n

ix™ept for the zero mo des the eigenfun™tions o ™™ur in p—irs of opp osite ™hir—lity with ™orreE

sp onding eigenv—lues ! F „herefore one o˜t—ins from eqF @PFPUA

n



H

ˆ

I I Pm

R

@PFPWA a d xhq"@xAq @xAi

xed

e

"

H P P

† @m A C ! †

†

n

! bH

n

where the zero{mo de ™ontri˜ution h—s ˜ een negle™tedF „o —rrive —t the qu—rk ™ondens—te one

h—s to —ver—ge over —ll g—uge eld ™ongur—tions —nd then t—ke the limit † 3 IF sn this

limit the sp e™trum ˜ e™omes dense —nd gives — non{v—nishing ™ondens—te if the me—n num˜ er

of eigenv—lues in the interv—l d! is prop ortion—l to the volumeX



I

&@!A

H

hq" q i a Pm @PFQHA d!

H P P

@m A C !

H

H

where &@!A is the me—n sp e™tr—l densityF „—king now the limit m 3 H one —rrives —t ‘RS“

hq" q i a %&@HAX @PFQIA

iFeF the qu—rk ™ondens—te is rel—ted to the level density —t zero virtu—lityD ! a HF

sn deriving this result the innite volume limit is very imp ort—ntF sn — nite volume there

is no sp ont—neous ˜re—king of ™hir—l symmetryD —nd therefore the qu—rk ™ondens—te would

v—nishF „he sp e™trum is dis™rete for nite † —nd the sum in eqF @PFPWA do es not develop —n

infr—red singul—rityF sf the ™hir—l limit h—d ˜ een t—ken —t nite † the ™hir—l symmetry would

H

h—ve ˜ een restoredF y˜viouslyD the two limits m 3 H —nd † 3 I —re not inter™h—nge—˜leF

sn order to —n—lyze the situ—tion —t nite volume one h—s to ™onsider the dimensionless

H

qu—ntity x Xa † m hq" q iF @iqF @PFQIA implies th—t the level sp—™ing for sm—ll ! is
! %

H P P

% a† hq" q iFA ren™e the denomin—tor @m A C ! v—ries only slowly for neigh˜ oring levels if

x ) IF sn this ™—se it is — go o d —pproxim—tion to repl—™e the sum in eqF @PFPWA ˜y —n integr—lD

iFeF eqF @PFQHA st—ys true —s long —s the qu—rk ™urrent m—ss is l—rger th—n Ia† hq" q iF „his suggests

th—t sp ont—neous ™hir—l symmetry ˜re—king is rel—ted to the —pp e—r—n™e of sm—ll eigenv—lues

of ha su™h th—t ! G Ia† F

n

‚e™ently the sp e™trum of this op er—tor h—s ˜ een investig—ted ‘RT “ ˜y studying the role

pl—yed ˜y the winding num˜ er # of the v—™uum g—uge eld ™ongur—tionF e top ologi™—lly IR

non{trivi—l gluon eld ™ongur—tion ne™ess—rily gives rise to qu—rk zero mo desF por sm—ll

@or v—nishingA qu—rk m—sses these zero mo des tend to suppress the fermion determin—nt like

H j# jx

f

@m A @x is the num˜ er of light —vorsAF roweverD —s emph—sized in refF ‘RT “ this supE

f

pression of the winding num˜ er # ™ongur—tion is —lw—ys —™™omp—nied ˜y —n enh—n™ement

j# jx

f

prop ortion—l to † F ren™eD in the physi™—l situ—tion x ) I there is no suppression —t —llF

e further very interesting result o˜t—ined in refF ‘RT “ is — set of sum rules for the sp e™trum

of the hir—™ op er—tor haF por — gluon eld ™ongur—tion with winding num˜ er # there —re

j# j zero mo des of haF „he few lowest non{zero eigenv—lues —re prop ortion—l to Ia† hq" q i —s

—nti™ip—ted from the dis™ussion —˜ oveF „heir distri˜ution is sensitive to the winding num˜ erD

—nd the levels —re pushed up if j# j in™re—sesF „he sum rules of refF ‘RT“ rel—te the inverse

moments of the eigenv—lue distri˜ution to the qu—rk ™ondens—teF iFgF the lowest one is

ˆ

I

I I

H

P

X @PFQPA @† hq" q iA a

P

j# j C x ! R

f

n

n

`

„hese sum rules ree™t the f—™t th—t for — nite volume the eigenv—lues with Ia† hq" q i ! (

n

$

 —re the one rel—ted to sp ont—neous ™hir—l symmetry ˜re—king —nd the o ™™urren™e of the

gh

qu—rk ™ondens—teF

yn the ˜—sis of these results one ™—n ™on™lude th—t the winding num˜ er is irrelev—nt —s

long —s x ) I whi™h is most likely the ™—se in the re—l worldF roweverD the winding num˜ er

density u™tu—tionsD iFeF the top ologi™—l sus™epti˜ility



P

h# i I

R

~ ~

d xhqq@xAq q@HAiY @PFQQA 1 a a

P P

@QP% A †

H

h—s me—sur—˜le ™onsequen™es for h—dron physi™sX it gener—tes the  m—ssF €hr—sed otherwiseD

despite the f—™t th—t the winding num˜ er is irrelev—nt the —xi—l @IA symmetry is still ˜roken

in —n —nom—lous f—shionD —nd this is ree™ted ˜y the gh v—™uumF „he strength of this

˜re—king is determined ˜y the top ologi™—l sus™epti˜ility 1F elso here veutwyler —nd ƒmilg—

‘RT “ found —n —stonishing resultD

H

( m hq" q i

@PFQRA 1 a

H

x ( m hq" q i

f

where ( is the would{˜ e top ologi™—l sus™epti˜ility in the —˜sen™e of qu—rk eldsF sf one

™onsiders l—rge x —nd — xed qu—rk m—ssD ( h—s — nite limit where—s hq" q i G x F „hen one

g g

o˜t—ins 1 a ( D iFeF in the l—rge x world the top ologi™—l sus™epti˜ility would ˜ e given only ˜y

g

H

gluonsF roweverD for x a Q —nd — sm—ll ™urrent qu—rk m—ss @m hq" q i ( ( A the top ologi™—l

g

H

sus™epti˜ility is domin—ted ˜y the qu—rk ™ondens—teX 1 a m hq" q iax @for unequ—l qu—rk

f

H

m—sses m ax is repl—™ed ˜y the redu™ed m—ssAF „he me—n squ—re winding num˜ er is l—rgeD

f

P H

h# i a † m hq" q iax D —nd for —n innite volume —ll winding num˜ ers —re equ—lly likelyF

f

„he physi™—l impli™—tions of these ™onsider—tions ˜ e™ome more tr—nsp—rent when one ™onE

™

siders the —nom—lous ‡—rd identity

x

f

"

— H —"#

~

q d j a Pim j q @PFQSA

" S

S

"#

P

IT%

where

"

"

j a hq"   q i —nd j a hq"  q i @PFQTA

S S S

S

™

por the simpler ™—se of —n —˜ eli—n —nom—ly the deriv—tion of the —nom—lous ‡—rd identity is presente d in

epp endix eF IS

denote the —xi—l singlet —nd pseudos™—l—r ™urrents of the qu—rksD resp e™tivelyF gonsidering the

™orrel—tor of the —xi—l singlet ™urrent this —nom—lous ‡—rd identity m—y ˜ e used to derive —n

H

expression for the  m—ss ‘RU “

 

P P

H P

x (X @PFQUA x ( m hq" q i % m a

H

f f



P P

f f

% %

ƒin™e for x a Q the top ologi™—l sus™epti˜ility 1 is governed ˜y the qu—rk ™ondens—te one

g

H

might h—ve —nti™ip—ted this to ˜ e the ™—se for the  m—ssD to oD there˜y ™ontr—di™ting the result

H

of refF ‘RU “F xoteD howeverD th—t the  m—ss is still domin—ted ˜y the winding num˜ er density

u™tu—tions of the purely gluoni™ theoryF

ƒumm—rizing this ™h—pter we would like to emph—size the following p ointsX pirstD ™onsidE

ering the limit of — l—rge num˜ er of ™olors x indi™—tes th—t ˜—ryons m—y ˜ e des™ri˜ ed —s

g

solitons @‡itten9s ™onje™tureAF nfortun—telyD the expli™it expression of the ee™tive meson

theory @PFRA is unknownF ƒe™ondD from the —pproxim—te ™hir—l symmetry of gh —nd its

sp ont—neous ˜re—king we know th—t low energy h—dron physi™s is domin—ted ˜y the would{˜ e

qoldstone ˜ osons of ™hir—l symmetryD the pionsF „husD —t le—stD these meson elds h—ve to

—pp e—r in —n —ns—tz for the ee™tive meson theoryF „hirdD due to the ™hir—l —nom—ly in the

—vor singlet ™h—nnel we know th—t the ™hir—l symmetry ˜re—king is —n inherent prop erty of

the sm—ll eigenv—lues of the qu—rk hir—™ op er—tor in the gh v—™uumF

„hese ide—s will —t le—st p—rti—lly ˜ e —pplied in the pro ™eeding ™h—ptersF ‡e will present —

mo del whi™h is me—nt —s — simplied version of the low{energy qu—rk dyn—mi™s of ghD the

xtv mo del ‘IU“F st displ—ys dyn—mi™—l ˜re—king of ™hir—l symmetry —nd yields quite — su™™essful

des™ription of meson physi™sF r—ving noti™ed this ™ommon fe—ture with the underlying theory

we will —g—in t—ke —dv—nt—ge of the l—rge x —rguments —nd —ttempt — des™ription of ˜—ryons

g

—s solitons in the xtv mo delF IT

Q „he x—m˜u{ton— {v —sinio mo del

sn this ™h—pter we intro du™e the xtv mo del —s —n ee™tive ™hir—lly inv—ri—nt theory of qu—rk

—vor dyn—mi™sF yrigin—llyD it w—s prop osed to des™ri˜ e the pion —s — m—ssless ˜ ound st—te of

the nu™leon —nd the —nti{nu™leon ‘IU“F x—m˜u —nd ton—Ev— sinio studied — lo ™—l four{fermion

inter—™tion in the s™—l—r{isos™—l—r —nd pseudos™—l—r{isove™tor ™h—nnelF fy ™onstru™tion the

origin—l xtv mo del p ossesses — glo˜—l @IA  ƒ @PA  ƒ @PA symmetryF xow—d—ysD it is

v ‚

™ommon to ™—ll —ll ™hir—lly inv—ri—nt mo dels with lo ™—l four{fermion @or even six{fermionA

inter—™tions —n xtv mo delF

QFI hes™ription of the mo del

„o ˜ e sp e™i™ we will ™onsider the mo del des™ri˜ ed ˜y the v—gr—ngi—n

P

2 3

x I

f

i i

ˆ

! !

H P P

@"q v a q"@ida m” Aq C Pq q A C@"q i q A

xtv I S

P P

iaH

P

3 2

I x

f

i i

ˆ

! !

P P

X @QFIA  q A C@"q i  q A @"q Pq

" S " P

P P

iaH

H

rere q denotes the qu—rk spinors —nd m” the ™urrent qu—rk m—ss m—trixF „he m—tri™es

q

i H

! aP —re the gener—tors of the —vor group @! a Pax Is AD x ˜ eing the num˜ er of —vors

f f

P

under ™onsider—tionF xote th—t the ™oupling ™onst—nts q —nd q h—ve dimension ‘energy“ F

I P

„hese ™oupling ™onst—nts m—y t—ke dierent v—luesD q Ta q D without sp oiling the glo˜—l

I P

™hir—l @x A  @x A symmetryD iFeF the two sums in eq @QFIA —re indep endently ™hir—lly

v f ‚ f

inv—ri—ntF

„he sp e™i—l form of the v—gr—ng i—n @QFIA ™—n ˜ e motiv—ted from the one{gluon{ex™h—ngeF

sn the lo ™—l limit of the one{gluon{ex™h—nge the qu—rk inter—™tion is given ˜y the ™urrent{

™urrent inter—™tion

"

@QFPA j j

—"

—

"

is the ™olor o ™tet —vor singlet ™urrent of the qu—rksD where j

—

—

!

" P ™ "

 q Y — a IY X X X Y x I a VY @QFQA j a q"

™ —

P

—

the ! aP ˜ eing the gener—tors of the ™olor group ƒ @QA in the fund—ment—l represent—tionF

™

pierzing the inter—™tion @QFPA into the ™olor singlet ™h—nnel le—ds to the inter—™tion of eqF @QFIA

with the —ddition—l rel—tion q a q aP ‘PH“F ƒin™e we —re ™onsidering the xtv mo del —s —n

P I

ee™tive theory we will rel—x this ™onditionF es mentioned in se™tion PFI the inter—™tion @QFPA

—lso ™ont—ins qu—rk{qu—rk inter—™tions le—ding to diqu—rksF „hese —reD howeverD suppressed

for l—rge x ‘QS“F

g

„he inv—ri—n™e of the v—gr—ngi—n @QFIA under ™hir—l rot—tions

i

!

y

i

Y a exp @i A Y  a  q 3 q Y q" 3 q"

† † † †

†

†

P

i

!

i

 @QFRA q 3 q Y q" 3 q" Y a exp @i A Y  a 

S e e e e e

e

P IU

H

in the limit m 3 H is e—sily veriedF ixpressing the qu—rk elds in terms of right{ —nd

left{h—nded qu—rk elds dened ˜y @™fF se™tion PFPA

q a € q Y q a € q @QFSA

‚ ‚ v v

the v—gr—ng i—n @QFIA de™ouples in — sum of two v—gr—n gi—ns ™ont—ining only right{@left{A

h—nded eldsD resp e™tivelyF ynly the m—ss term

H H H

v a q"m” q a @"q m” q C"q m” q A @QFTA

m—ss ‚ v v ‚

™ouples right{ to left{h—nded elds —nd thus ˜re—ks the ™hir—l symmetryF

„he lo ™—l four{fermion inter—™tion of @QFIA with — dimensionful ™oupling ™onst—nt is o˜viE

ously not renorm—liz—˜leF „herefore it is only ™ompletely dened when supplemented with —

regul—riz—tion s™heme in order to ™ut o momentum integr—ls —nd thus —voiding ultr—violet

divergen™iesF sn gh these divergen™ies would h—ve ˜ een —˜sor˜ ed in the renorm—liz—tion

pro ™edureF yn —n op er—tion—l level one m—y interpret the o ™™urren™e of the ™ut{o —s — very

™rude w—y of mimi™ing the —symptoti™ freedom of ghF †—rious s™hemes h—ve ˜ een dis™ussed

—

in the liter—ture X y@QA —nd y@RA inv—ri—nt sh—rp ™ut{osD €—uli{†ill—rs regul—riz—tion —nd so

onF rere we will ex™lusively use the prop er time regul—riz—tion s™heme prop osed ˜y ƒ™hwinger

‘RV “F „his pro ™edure h—s the —dv—nt—ge of ˜ eing g—uge inv—ri—nt if @extern—lA g—uge elds —re

™oupled to the mo delF isp e™i—llyD for the ™—l™ul—tion of ele™trom—gneti™ —nd we—k form f—™tors

this s™heme is sup erior to sh—rp ™ut{os ˜ e™—use it —llows one to —ppropri—tely m—nipul—te

the momentum sp—™e integr—lsF e further ˜ enet of this regul—riz—tion s™heme is the f—™t th—t

it m—y ˜ e dened —t the level of the —™tion r—ther th—n ˜ eing dened vi— its —ppli™—tion to

peynm—n integr—lsF „his —utom—ti™—lly gu—r—ntees th—t dierent qu—ntities —re regul—rized in

— ™onsistent m—nnerF ‡e will —pply this regul—riz—tion to the ˜ osonized —™tionD see ˜ elowF

QFP fosoniz—tion

„he —im is to rewrite the qu—rk @fermionA theory @QFIA into —n ee™tive meson @˜ osonA

theoryF st is ™onvenient to use — ™omp—™t not—tion

i

!

P

 a IY P fIY i Y i Y i  g @QFUA  Y i a HY X X X Y x

 — S " " S —

f

P

—nd

@



Rq  for P fIY i g

I — S



 a X @QFVA



Rq  for P f Y   g

P — " " S

„he —uxili—ry eld  a   D whi™h is intro du™ed vi— the identityD

 

   

   

i i

 I 

exp q" q  q" q a h  exp  @ A  i  q" q Y @QFWA

     

P P

™ont—ins therefore @in the ™—se of three —vorsA nonets of s™—l—rD pseudos™—l—rD ve™tor —nd

—xi—l{ve™tor meson eldsF sing eqF @QFWA the gener—ting fun™tion—l

 

R

 a h q h q" exp@i d xv A

x t v x t v

—

por — re™ent review see refF‘IV“ IV

m—y ˜ e written —s ‘QI“

 

 

i

I

   ‘“Y  a h  exp

p x t v

P

 

 

H

 ‘“ a h q h q" exp i q"@ida m” Aq X @QFIHA

p

por the interpret—tion of  ‘“ we note th—t it is equiv—lent to

p

”

h „

 ‘“ a lim hHje jHi @QFIIA

p

„ 3I

‚

Q y

”

where „ denotes — l—rge iu™lide—n time interv—l —nd h a d xq hq is the se™ond qu—ntized

form of —n one{p—rti™le hir—™ r—miltoni—nF sn winkowski sp—™e this r—miltoni—n is given ˜y

I

H

d Y @QFIPA h a 
p C  @m ” C A Y p a

i

H

—s ™—n ˜ e veried from the rel—tion ida m”  a  @id hAF

t

por su˜sequent ™onsider—tions it is ™onvenient to remove the ™urrent qu—rk m—ss from the

H

hir—™ op er—tor ˜y shifting  3  m” F „his yields the fun™tion—l

 

‚

‚

i

H I R H

R

m”  d x m”

i d xq"@idaA q

A @ A @

P

h q h q" e X @QFIQA  a h  e

x t v

purthermore we de™omp ose the generi™ meson eld  into irredu™i˜le vorentz tensors

 a ƒ C i € i†a iea X @QFIRA

S S

ƒ is — s™—l—rD € — pseudos™—l—rD † — ve™tor —nd e —n —xi—l{ve™tor eldF ell these elds —re

i i

—vor m—tri™esD iFeF ƒ a ƒ @! aPA et™F

sing eqF @QFIRA the ™hir—lly inv—ri—nt inter—™tion term of the xtv mo del is written —s

I I I

H P P " " H I H

tr@@ƒ m” A C € AC tr@† † C e e AX @QFISA @ m” A @ m” A a

" "

P Pq Pq

I P

por su˜sequent ™onsider—tions it is —lso ™onvenient to intro du™e the —ngul—r de™omp osition of

the s™—l—r —nd pseudos™—l—r meson elds ˜y dening — ™omplex eld w

y

w a ƒ C i€ a $  $ Y @QFITA

‚

v

whi™h in turn denes the rermiti—n eld  —nd unit—ry elds $ —nd $ F „he de™omp osition

v ‚

@QFITA is not uniqueY $ —nd $ —re r—ther rel—ted ˜y — ’g—ug e4 ™onditionF

v ‚

sn eqF @QFIQA the qu—rk eld —pp e—rs ˜iline—rly in the exp onent —nd ™—n therefore ˜ e inteE

gr—ted outF sing th—t het a exp „r log one o˜t—ins



ie‘“

 a h e Y

x t v



I

R H I H

d x@ m” A @ m” A C „r log @ida AX @QFIUA e‘“ a

P

xote th—t the —™tion e‘“ is — non{line—rD even non{p o lynomi—l fun™tion of the meson eld F

iven moreD the term „r log @ida A is non{lo ™—lF „he qu—ntum theory dened ˜y eqF @QFIUA IW

isD howeverD equiv—lent to the underlying xtv mo del dened ˜y the v—gr—n gi—n @QFIAF yn the

other h—ndD the gener—ting fun™tion—l @QFIUA h—s the —dv—nt—ge th—t it m—y ˜ e tre—ted in —

semi™l—ssi™—l —pproxim—tion ˜ e™—use the v—™uum exp e™t—tion v—lue of the ˜ osoni™ eld  ™—n

˜ e dierent from zeroD —nd in gener—l will ˜ eD where—s the v—™uum exp e™t—tion v—lue of the

fermioni™ qu—rk eld is ne™ess—rily v—nishing in the —˜sen™e of extern—l fermion sour™esF

@v A @v Ai i

ƒymmetry ™urrents —re ™onstru™ted ˜y —dding extern—l g—uge elds — a — ! aP —nd

" "

@—A @—Ai i

— a — ! aP to the —™tion for the ve™tor @v A —nd —xi—l{ve™tor @—A symmetriesD resp e™tively

" "

 

@v A @—A

q @ida A q 3 q ida i—a i—a  q X @QFIVA

„he symmetry ™urrents —re then identied —s the terms ™oupling line—rly to the extern—l g—uge

elds





 



@v Y—A

X @QFIWA  j a

"

@v Y—A"



 —

@v Y—A"

— aH

„he g—uge elds m—y ˜ e elimin—ted from the fermion p—rt of the —™tion ˜y tr—nsforming the

@—xi—lA ve™tor elds —™™ordingly

@—A @v A

X @QFPHA —nd e 3 e C — † 3 † C —

" " " "

" "

„his —llows one to str—ightforw—rdly ™ompute the deriv—tives in eq @QFIWA yielding the ™urrent

eld identities ‘QU “

I I

@v A @—A

a j a † —nd j e X @QFPIA

" "

" "

Pq Pq

P P

QFQ hyn—mi™—l ˜re—king of ™hir—l symmetry

„he v—™uum exp e™t—tion v—lue @†i†A of the —uxili—ry eld  is found from the st—tion—ry

p oint of the —™tion @QFIUAX



h i a  hq" q iX @QFPPA

 

isp e™i—llyD the qu—rk ™ondens—te hq" q i is rel—ted to the †i† of the s™—l—r eld vi— the ™ouE

pling ™onst—nt q F eltern—tively the ee™tive —™tion e‘“ @QFIUA yields the hyson{ƒ™hwinger

I

equ—tion



 @xA a  tr@q @xY xA AX @QFPQA

  

„he solution of this equ—tion determines the †i† of the meson eld F rere˜y q is the



qu—rk prop—g— tor in the ˜—™kground of the EeldF st is dened ˜y

I

q @xY y A a @ida A @x y AX @QFPRA



fefore ˜ eing —˜le to derive expli™it expressions for @QFPQA in the xtv mo del — regul—riz—tion

s™heme h—s to ˜ e imp osedF sing the de™omp osition @QFIRA the ee™tive —™tion @QFIUA m—y ˜ e

™—st into the form

e a e C e Y

p m

y

e a „r log @ihaA a „r log @ida C i†a C iea @€ w C € w AAY @QFPSA

p S ‚ v

 



I I

y H y H P " " R

tr@w w m @w C w AC@m A AC tr@† † C e e A X e a d x

" " m

Rq Rq

I P PH

es this —™tion is equiv—lent to the non{renorm—liz—˜le xtv mo del it is only ™ompletely deE

ned if — regul—riz—tion s™heme is providedF es —lre—dy st—ted in su˜se™tion QFI we will use

ƒ™hwinger9s prop er time regul—riz—tion‘RV“ whi™h intro du™es —n y @RAEinv—ri—nt ™utEo  —fter

™ontinu—tion to iu™lide—n sp—™eF por this regul—riz—tion pro ™edure it is ne™ess—ry to ™onsider

the re—l —nd im—gin—ry p—rt of e sep—r—tely

p

e a e C e Y

p ‚ s

I

y

ha AY „r log @ha e a

i ‚

i

P

I

y

I

A ha AX @QFPTA „r log @@ha e a

i s

i

P

„he re—l p—rt e diverges for l—rge moment— p where—s the im—gin—ry p—rt e do es not ™ont—in

‚ s

˜

ultr—violet divergen™iesD iFeF it is nite without regul—riz—tionF „herefore one h—s the option

of keeping e unregul—rizedD or to regul—rize it in — w—y ™onsistent with the regul—riz—tion of

s

e F xote th—t this denes two dierent mo delsF

‚

por the re—l p—rt of the —™tion the prop er time regul—riz—tion ™onsists in repl—™ing the

log—rithm ˜y — p—r—meter integr—l

 



I

ds I

y

Y @QFPUA „r exp sha ha e 3

i ‚

i

P

s P

Ia

™

whi™h for  3 I repro du™es the log—rithm up to —n irrelev—nt ™onst—ntF ƒin™e the op er—tor

y

ha ha is rermiti—n —nd p ositive denite this integr—l is well denedF

i

i

por the issues dis™ussed in this se™tion it is su™ient to only insp e™t e F †—rying the

‚

regul—rized ee™tive —™tion with resp e™t to the s™—l—r —nd pseudos™—l—r elds yields the hyson{

ƒ™hwinger or g—p equ—tions

h i a  m Y

ij ij i

H

m a m Pq hq" q i Y

i I i

i

x

™

Q P P

hq" q i a m a A @QFPVA @IY m

i

i i

P

R%

whi™h is the regul—rized version of eqF @QFPQA sp e™i—lized to the ™—se of the s™—l—r meson eldF

„he dyn—mi™—lly gener—ted ™onstituent qu—rk m—sses m D i a uY dD @—nd —lso the qu—rk ™onE

i

"

dens—tes hq" q i A —re equ—lD m a m —nd huu" i a hddiD resp e™tivelyD if —nd only if the qu—rk

i u d

H H

™urrent m—sses —re equ—lD m a m F „hroughout this review we will restri™t ourselves to this

u d

isospin symmetri™ limitD —nd in the following we will use the not—tion m Xa m a m —nd

u d

H H H

m Xa m a m F yf ™ourseD for the str—nge qu—rk we will ™onsider l—rger ™urrent m—sses

u d

—™™ording to @PFPSA le—ding to l—rger ™onstituent m—ssesF por these qu—ntities we will keep the

H

—vor index —nd denote them m —nd m D resp e™tivelyF

s

s

˜

por — more det—iled dis™ussion of — ™orre™t denition of e —nd its prop erties see se™tF TFIF

s

™

„he fun™tions

I



uI (

@uY xA a d( ( e

x

—re known —s in™omplete Efun™tionsF isp e™i—llyD

C

@HY xA a log x C  C y @xA for x 3 H

where  a HXSUUPI is iuler9s ™onst—ntF PI

H

pigure QFIX „he solution of the g—p equ—tion @QFPVA for v—nishing ™urrent m—ss m a H @solid

H

lineA —nd m a IVwe† @d—shed lineAF por the ™hosen v—lue of the ™utEoD  a TQHwe†D

— ™onstituent qu—rk m—ss m a RHHwe† repro du™es the phenomenologi™—l v—lue of the pion

de™—y ™onst—ntD f a WQwe†F

%

es ™—n ˜ e seen from gure QFI in the ™hir—l limit the qu—rk ™ondens—te —nd therefore —lso

the qu—rk ™onstituent m—ss is zero when the ™oupling ™onst—nt q st—ys ˜ elow — ™riti™—l v—lueF

I

e˜ ove this ™riti™—l v—lue the trivi—l solution ™o exists with — non{trivi—l oneF ix—mining the

d

ee™tive p otenti—l for — ™onst—nt s™—l—r eld in the ™hir—l limit

2 3

P

 x I

P P

™

R P P P  a P

 @HY A @  A e  C @QFPWA † @A a

P P

Pq IT% 

I

one ™on™ludes th—t the non{trivi—l solution is energeti™—lly f—voredF

QFR ghir—l rot—tion —nd hidden lo ™—l symmetry

„he freedom in the ™hoi™e of $ in the p—r—metriz—tion @QFITA of the s™—l—r —nd pseuE

vY‚

dos™—l—r eld ree™ts the lo ™—l hidden symmetry ƒ @QA F nder ƒ @QA  ƒ @QA  ƒ @QA

v ‚ h

h

the elds $ —nd  tr—nsform —s

vY‚

y

$ @xA 3 h@xA$ @xAv @xAY

v v

d

„he rst p—rt follows trivi—lly from e where—s the se™ond p—rt ™—n ˜ e o˜t—ined from e ˜y ev—lu—ting

m ‚

the fun™tion—l tr—™eF PP

y

$ @xA 3 h@xA$ @xA‚ @xAY

‚ ‚

y

@xA 3 h@xA@xAh @xA @QFQHA

where

v P ƒ @QA Y ‚ P ƒ @QA —nd h P ƒ @QA X @QFQIA

v ‚ h

„he —ddition—l degrees of freedom ™ont—ined in $ ™—n ˜ e g—uged —w—y —s in the usu—l riggs

vY‚

me™h—nism ‘IT “F v—ter on we will —dopt the unit—ry g—uge

y

a $ @QFQPA $ a $

‚

v

where $ is —n element of the ™osetEsp—™e ƒ @QA  ƒ @QA aƒ @QA F q—uge ™onditions like

v ‚ †

@QFQPA furthermore x h@xA in terms of v@xAD ‚@xA —nd $ @xAF

por su˜sequent ™onsider—tions it is ™onvenient to p erform — ™hir—l rot—tion from the ™urrent

qu—rks q to the ™onstituent qu—rks @™fF eq @PFTAA

vY‚

1 a $ q X @QFQQA

vY‚ vY‚ vY‚

nder — ™hir—l rot—tion of the origin—l qu—rk elds

q 3 vq Y q 3 ‚q @QFQRA

v v ‚ ‚

the ™onstituent qu—rk elds tr—nsform —™™ording to the hidden g—uge symmetry

1 3 h@xA1 X @QFQSA

vY‚ vY‚

efter the ™hir—l rot—tion

~

q" i ha q a 1" i ha 1 @QFQTA

the ™hir—lly rot—ted hir—™Eop er—tor

y

~

iha a „ i ha „ Y with „ a € $ C € $ @QFQUA

‚ ‚ v v

˜ e™omes

~ ~ ~

iha a i@ad C †a C ea  A X @QFQVA

S

st ™ont—ins the ™hir—lly rot—ted ve™tor —nd —xi—l{ve™tor elds

y

~ ~

†  e a $ @†  e C d A$ X @QFQWA

" " vY‚ " " "

vY‚

„hese elds tr—nsform under — leftErightEtr—nsform—tion —™™ording to the hidden symmetry

group

y

~ ~ ~ ~

†  e 3 h@xA@†  e C d Ah @xAX @QFRHA

" " " " "

prom these rel—tions it follows th—t the ve™tor eld tr—nsforms —s — g—uge eld of the hidden

lo ™—l symmetry ƒ @QA dening — ™ov—ri—nt deriv—tive

h

" " "

~

h a d C † X @QFRIA PQ

yn the other h—nd the ™hir—lly rot—ted —xi—l{ve™tor eld tr—nsforms homogeneously under the

hidden lo ™—l symmetry

" " y

~ ~

e @xA 3 h@xAe @xAh @xAX @QFRPA

sn terms of the ™hir—lly rot—ted elds the v—gr—ngi—n ˜ e™omes ‘QID QP “

! 

 

I

y y

P

~ ~

$ $ C $ $  m” tr v a 1" i@ad C †a C ea  A  1

v ‚ H p S

‚ v

Rq

I

i h

I

P P

~ ~

X @QFRQA @† v A C@e — A tr

" " " " p

Rq

P

rere we h—ve dened the ve™tor —nd —xi—l{ve™tor elds

I

y y

A @QFRRA C $ d $ @$ d $ v a

v " ‚ " "

v ‚

P

I

y y

@$ d $ $ d $ A @QFRSA — a

‚ " v " "

‚ v

P

whi™h —re indu™ed ˜y the ™hir—l rot—tionF „he v—gr—ngi—n @QFRQA is the ˜—sis for the investiE

g—tion des™ri˜ ed in se™tion TFPFSF

„he ™hir—l rot—tion of the qu—rk elds do es not le—ve the fermioni™ integr—tion me—sure

inv—ri—nt ˜ut intro du™es — nonElo ™—l t—™o˜i—n ‘RW“

 

h q h q" a t h 1h 1Y" @QFRTA

whi™h ™ont—ins the —nom—lyF sts physi™—l impli™—tions will ˜ e dis™ussed in se™tion RFRF PR

R ie™tive meson theory

sn the previous ™h—pter the xtv mo del h—s ˜ een ™onverted into —n ee™tive meson theoryF

sn the present ™h—pter we will ex—mine its impli™—tions for meson physi™sF

RFI qr—dient exp—nsion

„he qu—nt— of the sm—ll —mplitude u™tu—tions of the ™omp osite meson elds  @see se™tion

QFPA —round their v—™uum exp e™t—tion v—lues represent the mesonsF por the extr—™tion of these

@freeA mesons the ee™tive —™tion h—s to ˜ e exp—nded up to se™ond order in the elds F „his

will ˜ e done in se™tion RFQ —nd le—ds to the fethe{ƒ—lp eter equ—tionF et the moment we will

™on™entr—te on — simplied des™ription of the mesons —t low energiesF sn this se™tion we will

follow refF ‘QI“F

„he ee™tive meson —™tion o˜t—ined in the previous se™tion ˜y ˜ osonizing the xtv mo del

is — highly nonElo ™—l o˜ je™t due to the presen™e of the qu—rk determin—ntD „r log @ida AF ƒin™e

‘idaY “ Ta H this term ™ont—ins innitely high deriv—tives of the meson eldF y˜viouslyD for the

determin—tion of the lowEenergy @eFgF st—ti™A prop erties of the mesons we do not need to keep

—ll the higher order deriv—tives ˜ut — gr—dient exp—nsion of the qu—rk determin—nt seems to ˜ e

—ppropri—teF sn le—ding order gr—dient exp—nsion of the re—l p—rt of the ee™tive meson —™tion

one nds

@

!



I I I

† "# e"# † e R # y

p A p A tr@p tr@p e a d x g @maA C g @maA tr@r w r w A

‚ H P #

"# "#

P P R

A

I

" "

C tr @† † C e e A C XXX @RFIA

" "

Rq

P

where

# # # #

r w a d w C ‘† Y w “  fe Y w g @RFPA

S

denotes the ™ov—ri—nt deriv—tive of the s™—l—r { pseudos™—l—r eld w F purthermoreD

†

a d † d † C ‘† Y † “ C ‘e Y e “ p

" # # " " # " #

"#

e

a d e d e C ‘e Y † “ C ‘† Y e “ @RFQA p

" # # " " # " #

"#

—re the ve™tor —nd —xi—l{ve™tor p—rts of the eld strength tensorF „he qu—ntity g @maA

P

dep ends only on the r—tio ma —nd diverges log—rithmi™—lly —s  3 IF iqF @RFIA denes the

g—uged line—r ' Emo delF „o ™—st this mo del into its st—nd—rd form one h—s to p erform eld

renorm—liz—tions —nd redenitions in order to remove the % e —nd ' † mixings from the

I

v—gr—ng i—nF yne then nds for the ve™tor —nd —xi—l{ve™tor m—sses

P

g

P P P P

†

m a —nd m a m CTm @RFRA

† e †

Rq

P

where

35 2 4

IaP

P

m x

™

@RFSA HY g a

†

P P

 PR% PS

is the univers—l ve™tor meson ™oupling ™onst—ntF purthermore the s™—l—r meson m—ss is given

P P P

˜y m a Rm C m @see —lso eqF @RFRTAAF „he pseudos™—l—r meson m—ss is given ˜y

' %

m m

H

P

X @RFTA m a

%

P

q f

I

%

„he resulting ee™tive meson v—gr—ngi—n s—tises the well{known —nd phenomenologi™—lly

—

very su™™essful ™urrent —lge˜r— results F „his ™omes with no surprise sin™e our st—rting qu—rk

H

theory h—s stri™t ™hir—l symmetry for m” a HF prom eqs @RFSA —nd @RFTA one furthermore nds

the rel—tion

3 2

I

P

m

P P P †

m a —g f @RFUA Y — a I

† %

P

†

m

e

whi™h for — a P represents the uƒ‚p{rel—tion ‘SH “F purthermoreD for this ’m—gi™4 v—lue of —

p

‡ein˜ erg9s rel—tion ‘SI“ m a Pm —lso holds trueF ell these low energy theorems —re quite

e †

re—son—˜ly repro du™ed ˜y the exp eriment—l d—t—F

ƒo f—r we h—ve dis™—rded the im—gin—ry p—rt of the ee™tive meson —™tion @e in eq @QFPTAAF

s

„his p—rt —llows for — simil—r gr—dient exp—nsion where —g—in su˜sequent terms —re suppressed

˜y p owers of the r—tio maF es —lre—dy mentioned in ™h—pter Q the im—gin—ry p—rt of the

ee™tive meson —™tion st—ys nite when the ™utEo is removedF por  3 I only the le—ding

order term of the gr—dient exp—nsion survives —nd this term is then given ˜y the @g—ugedA

‡ess{umino{‡itten —™tion ‘SP“ @see se™tion RFRA

e a e C


@RFVA

s ‡  ‡

„herefore in le—ding order gr—dient exp—nsion we nd for the ee™tive meson —™tion on the

™hir—l ™ir™le @ a hi a m” A

e a e C e @RFWA

nl' ‡  ‡

where e denotes the g—uged nonline—r ' {mo del displ—yed in eqF @RFIAF „his mo del v—E

nl'

˜

gr—ngi—n h—s st—˜le ™hir—l solitons provided th—t the @—xi—l{A ve™tor mesons —re ret—ined F

RFP ‚el—tion to the ƒkyrme mo del

por the soliton des™ription of ˜—ryons the ™hir—l eld is ™ru™i—lF vet us therefore ™on™entr—te

now on the ™hir—l eld while setting —ll the other meson elds to their v—™uum v—lueF eside

from the pseudos™—l—r m—ss term the ee™tive meson —™tion of the ™hir—l eld is then given

y



S

$ is the ™hir—l eldD see eqF @QFITAF sn ˜y the qu—rk lo op „r log @ida m@ A A where a $

‚

v

le—ding order its deriv—tive exp—nsion is given ˜y the nonline—r ' Emo del v—gr—ngi—n

P

f

" y %

trv v Y v a d @RFIHA v a

" " " nl'

R

in —greement with eqF @RFIAF es is well{known this mo del do es not —llow for st—˜le solitons

—s ™—n ˜ e seen ˜y —pplying herri™k9s theorem ‘SQ“F „o o˜t—in st—˜le solitons the nonEline—r

' Emo del h—s to ˜ e supplemented ˜y st—˜ilizing higher order deriv—tive terms like the ƒkyrme

—

por — ™ompil—tion of ™urrent —lge˜r— results see refF‘QU“ —nd referen™ es thereinF

˜

por — review on mesoni™ solitons with ve™tor mesons see refF ‘IH“F PT

term ‘V“D whi™h h—s four deriv—tivesF yne would exp e™t th—t these st—˜ilizing terms emerge in

higher order of the gr—dient exp—nsion of the ee™tive meson —™tionF sn next to le—ding order

of the deriv—tive exp—nsion one nds in the limit  3 I ‘QI“

' &

I I x I

g

" P " P P @RA

@RFIIA @d v A C @v v A @‘v Y v “A tr v a

" " " #

P

Q T QP% IP

„he rst term is in f—™t the desired ƒkyrme term whi™h shows up with — ™o e™ient e a P% F

„his v—lue f—vor—˜ly ™omp—res with the phenomenologi™—l v—lue e a SXRS o˜t—ined ˜y edkins

™

et —lF ‘W“ ˜y tting the ƒkyrmion m—ss @—fter semi™l—ssi™—l qu—ntiz—tionD ™fF se™tion UFIA to

the nu™leon m—ssF nfortun—tely the l—st two terms dest—˜ilize the solitonF „he l—st term

le—ds to — neg—tive denite energy for —ny ™hir—l eld ™ongur—tion —nd for™es the soliton to

™oll—pseF „he se™ond term whi™h provides — t—™hyoni™ p ole in the pion prop—g—to r is even

more d—ngerous ˜ e™—use it le—ds to — v—™uum inst—˜ility ‘SR“F iven if one go es to the next

@sixthA order gr—dient exp—nsion one do es not nd st—˜le solitons ‘SS “F „his ™ert—inly sign—ls

the ˜re—kEdown of the n—ve gr—dient exp—nsion in the soliton se™tor of the ee™tive —™tion of

the ™hir—l eldF „hese r—ther frustr—ting results h—d for some time given rise to the fe—r th—t

the ™hir—l soliton of the ee™tive —™tion might not —t —ll p osses soliton solutionsF

ƒome hints for the existen™e of st—˜le ™hir—l solitons —re provided ˜y the he—t kernel exp—nE

sion of the ee™tive ™hir—l —™tion @QFPSAF „he he—t kernel exp—nsion represents — semiE™l—ssi™—l

typ e of —symptoti™ exp—nsion whi™h ™orresp onds to — resumm—tion of the gr—dient exp—nsion

‘ST “F sn next to le—ding order this exp—nsion givesD ˜ esides the forth order terms given ˜y

equ—tion @RFIIAD —lso — term with six deriv—tives ‘QI“

 

P

I m

I

" y @TA

tr @d P A@d P A @RFIPA A v a @PY

"

P P P

QP% QHm 

whi™h in f—™t over™omes the forth order terms —nd st—˜ilizes the ™hir—l soliton ‘SU“F

vet us —lso mention th—t the next to le—ding order terms of the deriv—tive exp—nsion ™onsidE

er—˜ly improve the predi™tion for the %% Es™—ttering length tow—rds their exp eriment—l v—lues

‘SV “F sn f—™t the ™o e™ients of these terms whi™h —re predi™ted ˜y the gr—dient exp—nsion of

the qu—rk lo op —re in r—ther go o d —greement ‘QI“ with the phenomenologi™—l v—lues determined

in ™hir—l p ertur˜—tion theory ‘SW “F „hus the dest—˜ilizing terms —re o˜viously relev—nt for the

low energy meson physi™sF

xeverthelessD we will show ˜ elow th—t the ƒkyrme mo del do es —rise —s low energy —pE

proxim—tion to the non{lo ™—l ee™tive meson theory provided one in™ludes the ve™tor —nd

—xi—lve™tor mesons in —n —ppropri—te m—nnerF „he imp ort—n™e of the ve™tor mesons for the

st—˜iliz—tion of the ™hir—l soliton —g—inst the ™oll—pse should ™ome with no surprise sin™e the

s™—l—r —nd —xi—lve™tor mesons —re mu™h he—vier th—n the pseudos™—l—r mesons —nd therefore

should ™ontrol the short dist—n™e ˜ eh—vior of the soliton —t low energiesF

por the study of the soliton se™tor of the ee™tive meson theory it is o˜viously —dv—nt—geous

to reg—rd the ™hir—lly rot—ted ve™tor —nd —xi—lve™tor elds intro du™ed in se™tion QFR —s the

physi™—l eldsF „he ™orresp onding tre—tment in the soliton se™tor will ˜ e dis™ussed —t length

in su˜se™tion TFPFSF xote th—t —fter the ™hir—l rot—tion des™ri˜ ed in se™tion QFR —ll higher th—n

le—ding order gr—dient terms of the ™hir—l eld —re the fully —˜sor˜ ed into the rot—ted ve™tor

—nd —xi—lve™tor eldsF „he elimin—tion of these gr—dients is o˜viously ™ru™i—l in „he soliton

se™tor sin™e the gr—dient exp—nsion seems not to ™onverge for the ™hir—l eldD —t le—st not in

nextEtoEle—ding order —nd nextEtoEnextEtoEle—ding orderF yn the other h—nd for the elds of

™

xote th—t etkins et —lF ™onsidered f —s — free p—r—meterF „heir t to ˜—ryon m—sses provided f a TRwe†F

% % PU

the he—vier ve™tor —nd —xi—lve™tor mesons — gr—dient exp—nsion might still ˜ e —ppropri—te —t

low energiesD whi™hD howeverD h—s to ˜ e justied — posterioriF

sn terms of the rot—ted elds the le—ding order of the deriv—tive exp—nsion the v—™uum p—rt

of the —™tion is given ˜y the following v—gr—ng i—n ‘QP“

~

v a v C v @RFIQA

nl' ‡  ‡

where

!

   

P P

— I

P P

~ ~ ~ ~

f tre tr p C p v a

† e nl'

%

P

Pg — I

†

 

!

P

   

P P

I m

P P y †

~ ~

f m tr C P X @RFIRA tr † v C e — C

% %

P

R g

†

xote th—t we h—ve —ssumed the ™hir—l ™ir™le ™ondition  a m” F v is the ‡ess{umino{

‡  ‡

~ ~

‡itten ‘SP D IS “ term expressed now in terms of the rot—ted elds † Y e F „he ve™tor elds

" "

v —nd — h—ve ˜ een dened in eqs @QFRRDQFRSAF purthermore the eld strength tensors

" "

~ ~

@p A Y @p A —re dened in equ—tion @RFQA with the unrot—ted elds repl—™ed ˜y the rot—ted

† "# e "#

~ ~

elds † Y e F

" "

ƒin™e the @—xi—lEA ve™tor meson m—sses m Y m —re l—rge ™omp—red to the pion m—ss it is

† e

~ ~

tempting to integr—te out the @—xi—lEAve™tor elds † Y e in the —˜ ove v—gr— ngi—n in the st—ti™

" "

 

P P

~ ~

—nd higher order terms —s Y @p A limitF „hen one might ™onsider the kineti™ energy @p A

e †

"# "#

well —s the ‡essEumino term —s p ertur˜—tionsF „he resulting equ—tion of motion yield ‘QP“

~

† a v Y @RFISA

" "

— I

~

e a — X @RFITA

" "

—

por these eld ™ongur—tions the eld strength tensors redu™e to

 

— I

#"

P " #

~

A I ‘— Y — “ @RFIUA a @ p

†

—

— I

"#

" " # " # #

~

@‘d C v Y — “ C ‘— Y d C v “A a H @RFIVA a p

e

—

—nd the v—gr—ng i—n @RFIRA pre™isely ˜ e™omes the ƒkyrme v—gr—ng i—n

I I I

P P y P "

m f tr@ C PA @RFIWA tr‘v Y v “ C f trv v C v a

" # "

% % %

P

R R QPe

with

P

—

e a g X @RFPHA

†

jP— Ij

„hus in the st—ti™ limit of the ve™tor —nd —xi—l{ve™tor mesons the tot—l ee™tive meson v—E

gr—ngi—n in f—™t ˜ e™omes the ™ele˜r—ted ƒkyrme mo del ‘V“F

ƒin™e the lowest lying —xi—l{ve™tor mesons @m % IPTH we†A is ™onsider—˜ly he—vier th—n

e

the lowest lying ve™tor mesons @m % m % UUHwe†A one might dis™—rd the —xi—l{ve™tor

& 3

mesons —t low energiesF „hen one o˜t—ins

e a g @RFPIA

† PV

sing the exp eriment—l v—lue of g 9 TXH ‘RH“ for the univers—l ve™tor ™oupling ™onst—ntD the

†

ƒkyrme p—r—meter is not f—r from its phenomenologi™—l v—lue g a SXRS o˜t—ined ˜y edkins

†

et —lF ‘W“ ˜y tting the nu™leon —nd
Em—ssesF

sn the deriv—tion of the ƒkyrme mo del we h—ve used the st—ti™ limit for the ve™tor —nd

—xi—l{ve™tor mesons ™onsidering ˜ oth the kineti™ energy —s well —s the ‡ess{umino{‡itten

term —s p ertur˜—tionsF ‡hen the st—ti™ equ—tions of motion @RFISDRFITA —re used the ‡ess{

umino{‡itten term v—nishesF eltern—tivelyD one ™ouldD howeverD —lso in™lude the ‡ess{

umino{‡itten term into the denition of the st—ti™ equ—tion of motion —nd le—ving only

the kineti™ energy —s — p ertur˜—tionF „his —dds — ™urrentE™urrent inter—™tion to the ƒkyrme

v—gr—ng i—n

I

P "

 v a f f @RFPPA

T "

T

P

from the ™oupling of the 3 Emeson to the top ologi™—l ™urrent f in the ‡ess{umino{‡itten

"

termF sn the ™ontext of the xtv mo del the ™oupling strength is o˜t—ined to ˜ e ‘TH “

P

T% x

g

P

X @RFPQA  a

T

P P P

m @HY m a A

&

„he ™urrent{™urrent inter—™tion @RFPPA mo dies the short dist—n™e ˜ eh—vior of the soliton

resulting in — re—son—˜le des™ription of the short r—nge ˜ eh—vior of the ™entr—l p otenti—l in the

nu™leon{nu™leon inter—™tion ‘TI“F

eltern—tivelyD if the @—xi—lA ve™tor mesons —re kept —s dyn—mi™—l elds there will ˜ e no

re—son to negle™t the ‡ess{umino{‡itten —™tion @™ont—ined in the full v—gr—ngi—nA in ™omE

p—rison to the norm—l p—rity termsF sn f—™tD the ‡ess{umino{‡itten term is then —˜solutely

ne™ess—ry to ensure the existen™e of st—˜le solitonsF yn the other h—ndD if one integr—tes out

the @—xi—lEA ve™tor mesons in the st—ti™ —pproxim—tion in the w—y it w—s done —˜ oveD the

‡ess{umino{‡itten term v—nishes for the ™orresp onding st—ti™ eld ™ongur—tions of two

—vorsF sn this ™—se st—˜ility of the soliton is provided ˜y the indu™ed ƒkyrme termF

RFQ fethe{ƒ—lp eter equ—tions

sn this se™tion we will dis™uss the determin—tion of the meson m—sses —s fun™tions of the

xtv mo del p—r—metersF roweverD this time we will go ˜ eyond the gr—dient exp—nsionD iFeF we

will t—ke into —™™ount the gr—dients of the elds in —n ex—™t m—nnerF „his ™—n ˜ e done ˜y

extr—™ting the fethe{ƒ—lp eter equ—tions for the mesons st—rting with the regul—rized —™tionD

see eqsF @QFPSA —nd @QFPUAF sn order not to disguise the used metho d ˜y te™hni™—l di™ulties

we will displ—y it for the ™—se of pions in the isospin limit onlyF por the other ™—ses we will

d

only ™ite the resultsF por det—ils we refer the re—der to epp endix e of refF‘TP“ F

„he physi™—l meson ex™it—tions —re given ˜y sm—ll —mplitude u™tu—tions —round the tr—nsE

l—tion—l inv—ri—nt v—™uum eld ™ongur—tion o˜t—ined ˜y solving @QFPVAF ixp—nding the ee™E

tive —™tion in iu™lide—n sp—™e

e a e C e

m p

 



I

R y y P

e a d x tr@w w m @w C w AC m A

m H

H

Rq

I

 



I

I ds

y

e a „r exp sha ha

p i

i

P

P s

Ia

y

P y y y

ha ha a d C ‘idaY € w C € w “ C € w w C € w w @RFPRA

i ‚ v ‚ v

i

d

purthermoreD we will restri™t ourselves to prop er time regul—riz—tionF PW

up to se™ond order in the pion u™tu—tions of the pseudos™—l—r eld @xA a P% @xA —llows us

e

to extr—™t the inverse prop—g—tor for the pseudos™—l—r mesonsF ƒin™e we —re not interested in

the u™tu—tions of the s™—l—r eld we su˜stitute it ˜y the v—™uum exp e™t—tion v—lue @†i†A

hi a mIs F „he ™omplex eld w @QFITA is then given ˜y

i@xA iP% @xA

w a m Y a me a me X @RFPSA

y˜viouslyD the †i† of the pseudos™—l—r eld @xA is zeroF „he ™hir—l eld is exp—nded for

i i

sm—ll{—mplitude u™tu—tions of the pseudos™—l—r meson eld % a % @( aPA —s

Pi% P

a e a I C Pi% P% C XXXX @RFPTA

„he f—™tor P is here˜y intro du™ed for l—ter ™onvenien™eF

es the st—tion—ry p oint of the —™tion o ™™urs for % a H there is no line—r termF pirstD we

will ™onsider the m—ss term of the mesoni™ —™tion e

m



 

I

Q R H P H P

C y @% AX @RFPUA d x @m m A CRm m % e a

m

Pq

I

sntro du™ing the pourier tr—nsform of the u™tu—ting eldD % @q AD the ˜iline—r term of @RFPUA is

o˜t—ined to ˜ eX



Q

H R

ˆ

I m m d q

i i

X @RFPVA % @q A% @q A

R

q @P% A P

I

iaI

sn order to exp—nd the fermion determin—nt e we rewrite the op er—tor

p

y

ha ha a e C e C e C XXX @RFPWA

i H I P

i

where e is of k Eth order in the eld % F sing eqsF @RFPRAD @RFPSA —nd @RFPTA we o˜t—in

k

P P

e a d C m

H

e a P m@da% A

I S

e a HX @RFQHA

P

„he l—st rel—tion is v—lid in the isospin limit onlyF y˜viouslyD only deriv—tives of % ™—n o ™™ur

y

—s — ™onsequen™e of the ™hir—l inv—ri—n™e of hetha ha F sing now

i

i

 



I

I

ds

y

„r exp sha ha e a

i p

i

P

s P

Ia

  

I I I

I I

ds

s e s@I Ae se

H H H

d „r e e e ds „re C a

P

P P

P P s

H Ia Ia

  

I I I

I

s e s@I  Ae s e Q

H H H

d „re e e e e C y @% A @RFQIA d dss

I I

P

P

H H Ia

we ™—n system—ti™—lly exp—nd this p—rt of the —™tionF „he —vor @or isospinA tr—™e only gives

—n over—ll f—™tor PD the ™olor tr—™e — f—™tor x —nd the hir—™ tr—™e — f—™tor RF „he fun™tion—l

™

e

et this order the im—gin—ry p—rt the —™tion do es not ™ontri˜uteF QH

tr—™e will ˜ e done using momentum eigenst—tes % @q AF es e a H in the isospin limit we h—ve

P

to ™—l™ul—te the following p—r—meter integr—l

 



I I

P P P P

s@ C A@k Cm A s@I  A@@k Cq A Cm A

d e e d

H H



P P P P

s@ C A@k Cm A s@I  A@@k Cq A Cm A

Ce e



I

P P P P

s@k Cm A s@IA@@k Cq A Cm A

de e X @RFQPA a

H

ƒin™e the momentum k is only —pp e—ring in this exp onenti—l we m—y shift k 3 k @I Aq

in the se™ond term without ™h—nging the v—lue of the integr—l yielding

   

R R

I I

d k d k

P P P P P P

s@k C@IAq Cm A s@@k C@IAq A C@IAq Cm A

d e X @RFQQA d e a

R R

@P% A @P% A

H H

sing



R

x d k

P

™

sk

@RFQRA Rx a e

™

P R

R% @P% A

—s well —s the denition of the in™omplete {fun™tionD the term ˜iline—r in pseudos™—l—r elds

—rising from e re—ds

p

 

Q

R

I

ˆ

I d q x

™

P P P i i P P

d@HY ‘m C @I Aq “a AX @RFQSA % @q A% @q A@q Am

R P

@P% A P R%

H

iaI

„his —llows us now to extr—™t the inverse pion prop—g—to rD

H

m m

P P I

@q AY @RFQTA @q A a h

%

q

I

P

where the p ol—riz—tion op er—tor @q A is given ˜y

P P P P

@q A a q f @q A



 

I

x

™

P P P P P P

X @RFQUA dx HY ‘m C x@I xAq “a f @q A a m

P

R%

H

„he fethe{ƒ—lp eter equ—tion whi™h determines the physi™—l meson m—sses m is equiv—lent

%

to the ™ondition th—t the meson prop—g—tor h—s — p oleX

I P P

h @q a m A a HX @RFQVA

% %

P P P

xote th—t f @q a m A then is the ™orresp onding meson de™—y ™onst—ntF ‡e w—nt to emph—E

%

size here th—t the fethe{ƒ—lp eter equ—tion @RFQTA { @RFQVA is the one in l—dder —pproxim—tion

—nd th—t no other —pproxim—tions —s eFgF gr—dient @su˜se™tion RFIA or he—t kernel exp—nsions

h—ve ˜ een m—deF „his —lso implies th—t the de™—y ™onst—nts —re ev—lu—ted on the ™orresp onding

meson m—ss shellF „he pion de™—y ™onst—nt is then given ˜y



 

I

x

™

P P P P P

dx HY ‘m x@I xAm “a f a m X @RFQWA

% %

P

R%

H QI

„—˜le RFIX w—ss p—r—meters xed in the meson se™tor of the xtv mo delF „he k—on de™—y

™onst—nt f is predi™tedF

u

H H

m @we†A m @we†A m am f @we†A

s u

s

QSH SUU PQFS IHRFR

RHH TIQ PPFV IHHFQ

RSH TSH PPFS WUFR

SHH TVU PPFQ WSFS

yne ™le—rly sees th—t in the ™hir—l limit @m a HA the expression

%

x

™

P P P

f a m @HY @maA A @RFRHA

%

P

R%

™—l™ul—ted ˜y me—ns of — deriv—tive exp—nsion ˜ e™omes ex—™tF sing eqsF @RFQTA —nd @RFQUA it

H

is trivi—l to show th—t in the ™hir—l limit m a H the fethe{ƒ—lp eter equ—tion @RFQVA is solved

P P

a HD iFeF the pions —re qoldstone ˜ osons F „he m—ss shell ™ondition @RFQVA ˜y setting q a m

%

™—n ˜ e expressed —s

H

m m m

H H P P

% Pm huu" i @RFRIA a Pm huu" i a f m

% %

H

q m m

I

where we used the g—p equ—tion @QFPVA to express the ™oupling ™onst—nt q in terms of the

I

qu—rk ™ondens—tesF „his —pproxim—te rel—tion is the qellEw—nn{‚enner{y—kes rel—tion ‘RI“D

see eqF @PFPRAF

„he gener—liz—tion of the —˜ ove ™—l™ul—tion to the ™—se of unequ—l qu—rk m—ssesD —nd

esp e™i—lly the ™—se of three —vorsD ™—n ˜ e found in epp endix e of refF‘TP“F por the inverse

prop—g—to r for the pseudos™—l—r mesons one o˜t—ins

H H

 

@m C m A@m C m A

i j

i j

I P P

h @q A a  @q A   X @RFRPA

ij il k j

ijYk l

Rq

I

P

„he p ol—riz—tion op er—tor  @q A is given ˜y

ij

 

I hq" q i hq" q i

j i

P P P P P P P P P

 @q A a q f @q AC@m m A f @q A @m m A @RFRQA

ij i j

ij ij i j

R m m

i j

wherein



 

I

I x

™

P P P P P P P

@RFRRA dx HY ‘@I xAm C xm C x@I xAq “a @m C m A f @q A a

i j

i j ij

P

R R%

H

while the qu—rk ™ondens—tes —re dened in eqsF @QFPVAF y˜viouslyD these expressions ™oin™ide

with the one given —˜ ove if the —vor symmetri™ limit is ™hosenF por the k—on one simply

uses i a s —nd j a u@dAD eFgF the k—on de™—y ™onst—nt ™—n ˜ e o˜t—ined from eqF @RFRRA ˜y this

P P

™hoi™e of indi™es —nd ev—lu—tion —t q a m whi™h in turn is o˜t—ined —s the ro ot of @RFRPAF

u

xumeri™—l results —re displ—yed in t—˜le RFIF es input serves m a IQSwe†D m a RWSwe†

% u

—nd f a WQwe†F „he up ™onstituent qu—rk m—ss is ™hosen —s indep endent v—ri—˜leF yne

%

o˜serves th—t the xtv mo del in the prop er time regul—riz—tion s™heme underestim—tes the

exp eriment—l v—lue f a IIQwe†F

u QP

por the other mesons @s™—l—rsD ve™tors —nd —xi—lve™torsA we will only give the prop—g—to rs

for the ™—se of two —vors in the isospin symmetri™ limitF por the inverse prop—g—to r of the

s™—l—r mesons ' —nd — one o˜t—ins

H

H

m m

I P P P P

h @q A a @q CRm Af @q AX @RFRSA

'

q

I

xote the simil—rity to the pion prop—g— tor @RFQTAF es — ™onsequen™e the s™—l—r meson m—ss

™—n ˜ e written —s

P P

f @m A

P P P P P %

% Rm C m @RFRTA m a Rm C m

% ' %

P P

f @m A

'

P P

where for the l—st rel—tion we h—ve —ssumed f @q A @RFQUA — slowly v—rying fun™tionF roweverD

there is — pro˜lemF prom @RFRTA one sees th—t the s™—l—r meson m—ss lies —˜ ove the qu—rk{

P P P P I

—ntiqu—rk thresholdD iFeF m b Rm F sn this ™—se f @m AD or equiv—lently h D —™quires —n

' ' '

f

im—gin—ry p—rt

q

x

™

P P P P P P P I

P P

m ICRm aq for Rm q R@m C  AX @RFRUA A a @q CRm A sm@h

'

R%

„herefore the extr—™tion of meson m—sses vi— the fethe{ƒ—lp eter equ—tion —re o˜s™ured ˜y

the unphysi™—l qu—rk{—ntiqu—rk threshold whi™h is present in the @non{™onningA xtv mo delF

por the ™—l™ul—tion of the ve™tor meson prop—g— tor it is ™onvenient to split the expressions

in the exp—nsion of the fermion determin—nt in longitudin—l —nd tr—nsverse onesF e n—ve

extr—™tion of the longitudin—l p—rt of the inverse prop—g—to r yields

 



I

I

P P P P P P P

dx @m C x@I xAq A@IY ‰ A C P@x m @IY m a A A q @HY ‰ A @RFRVA

P

H

where

P P P

‰ a @m C x@I xAq Aa X

P

„his expression identi™—lly v—nishes —s ™—n ˜ e shown ˜y — form—l exp—nsion in p owers of q

P

—nd ev—lu—ting the peynm—n p—r—meter integr—ls for e—™h p ower of q sep—r—telyF xote th—t it

is not su™ient to —rgue th—t the prop er time regul—riz—tion is g—uge inv—ri—nt @—nd therefore

the longitudin—l p—rt h—s to v—nishAF „he sm—ll —mplitude exp—nsion for ve™tor mesons whi™h

keeps only terms up to se™ond order in the ve™tor @g—uge ˜ osonA eld ˜ut —ll deriv—tives do es

expli™itly viol—te @lo ™—lA g—uge inv—ri—n™eF

es the expression @RFRVA v—nishes only the mesoni™ —™tion e ™ontri˜utes to the longituE

m

din—l p—rt of the inverse ve™tor meson prop—g—torF „he l—tter is given ˜y

3 2 3 2

" # " # P

q q I q q I q

I "# "#

@RFRWA @h A @q A a 

P P P P

Rq q Rq q g @q A

P P

where



I

x I

™

P P P

dxx@I xA@HY @m C x@I xAq Aa AX @RFSHA a

P P P

R% g @q A

H

f P P P

por q ! R@m C  A —ddition—l @even more unphysi™—lA im—gin—ry p—rts —pp e—rF QQ

„he ve™tor meson m—ss is determined ˜y the on{shell ™ondition

I

P P P P

m g @q a m a A @RFSIA

† †

Rq

P

P P P P

—nd g a g @q a m A is the univers—l ve™tor meson ™oupling ™onst—nt t—ken —t the on{shell

† †

m—ssF

sn™luding the —xi—lve™tor meson ™h—nges the formul—s for the pion prop—g—torF „his is due

% — mixingD iFeF the o ™™urren™e of — term

I

P P

f @q Aiq % @q Ae @q A

" "

in the sm—ll —mplitude exp—nsionF „his mo dies the inverse pion prop—g—to rX

H

P P

m m

f @q A

I P P

h @q A a q X @RFSPA

%

P P

ICRq f @q A q

P I

isp e™i—llyD the pion de™—y ™onst—nt is then given ˜y

P P

f @m A

P %

f a @RFSQA

%

P P

ICRq f @m A

P

%

where the pion m—ss is determined ˜y the ro ot of eqF @RFSPAF es — further ™onsequen™e of % —

I

mixing the —xi—lve™tor prop—g— tor is dierent from the ve™tor oneF „he —xi—l{ve™tor m—ssD

 

I

P P P P P P P

A a g @q a m m A Y @RFSRA C f @q a m

e e e

Rq

P

is signi™—ntly l—rger th—n the ve™tor m—ss —nd gener—lly lies —˜ ove the qu—rk{—ntiqu—rk threshE

oldF

„his su˜se™tion m—y ˜ e summ—rized —s followsX „he prop—g—tors for the s™—l—r —nd pseuE

dos™—l—r elds m—y generi™—lly ˜ e written —s

P

 @q A

P

@RFSSA h @q A a

P P P

q C m @q A

while those for the ve™tor —nd —xi—lve™tor elds re—d

3 2

" # P

q q  @q A

"# "# P

@RFSTA  h @q A a

P P P P P

m @q A q C m @q A

P P P

with the fun™tions  @q A —nd m @q A given in t—˜le RFPF por the s™—l—r —nd —xi—lve™tor mesons

their m—sses lie —˜ ove the qu—rk{—ntiqu—rk threshold m—king their determin—tion vi— the

fethe{ƒ—lp eter equ—tion dou˜tfulF st should ˜ e rem—rked th—t — pro ™edureD whi™h extr—p ol—tes

the p ol—riz—tion tensors from ˜ elow to —˜ ove the qu—rk{—ntiqu—rk thresholdD h—s ˜ een prop osed

in refF‘TQ“F QR

I P P P

weson  @q A m @q A

P P P H P P

' D— f @q A Rm C m ma@q f @q AA

H I

P P P P H P P

% f @q Aa@I C Rq f @q AA m ma@q f @q AA

P I

P P P P

3 D& Iag @q A g @q AaRq

P

P P P P P P

— Iag @q A g @q A@IaRq C f @q AA

I P

P P P

„—˜le RFPX „he fun™tions  @q A —nd m @q A whi™h determine the meson prop—g—torsD see

P P P P

eqsF @RFSSA —nd @RFSTAF „he qu—ntities f @q A —nd g @q A —re given in eqsF @RFQUA —nd @RFSHAD

resp e™tivelyF

RFR ghir—l —nom—ly

sn this se™tion we will dis™uss how the ™hir—l —nom—ly @see se™tion PFQA is represented in the

ee™tive meson theory @QFPSAF hening the qu—ntum theory vi— p—th integr—ls the —nom—lous

symmetry ˜re—king is re—lized ˜y the non{inv—ri—n™e of the fun™tion—l integr—l me—sure ‘RW“Y

despite the f—™t th—t the ™l—ssi™—l —™tion —pp e—ring —s –weight9 f—™tor under the p—th integr—l

is inv—ri—nt under the ™onsidered tr—nsform—tionF „he integr—l me—sure

h q h q" @RFSUA

is not inv—ri—nt under ™hir—l rot—tions

2 3

—

!



— i@xA

S

S

q @xA Y  a  q @xA 3 1@xA a @ @xAA q @xA a e

P

p



S

q"@xA 3 1" @xA a q"@xA@ @xAA @RFSVA

˜ut r—ther —™quires — ph—se t @A Ta I

h 1h 1" a t @Ah q h q" X @RFSWA

„his implies th—t for the ee™tive meson theory @QFPSA the —nom—ly is ™ont—ined in the

fermion determin—nt e F „he pro of of —nom—lous ™hir—l symmetry ˜re—kingD ree™ted ˜y the

p

rel—tion t @A Ta ID is given in epp endix e for the simplied ™—se of — hir—™ op er—tor ™ont—ining

˜ esides the usu—l kineti™ —nd m—ss term only — ve™tor eld

H H

iha a i@da C †aA m aX ida m X @RFTHA

sn this ™—se one o˜t—ins

 



x

™

R

~

d x@xAtr @p p A @RFTIA t @A a exp i

p

P

V%

~

where p is the eld strength tensor ™orresp onding to the ve™tor eld † —nd p is the

"# " "#

du—l tensorF xote th—t the t—™o˜i—n t @A is — pure ph—se f—™torD jt @Aj a ID —nd therefore QS

™ontri˜utes to the im—gin—ry p—rt of the ee™tive —™tionF sn the presen™e of — ve™tor eld this

f—™tor will in gener—l dier from oneF

„he expression @RFTIA is the t—™o˜i—n for innitesim—l e˜ eli—n ™hir—l rot—tionsF edditionE

—llyD we w—nt to know the t—™o˜i—n t @A for nite non{—˜ eli—n ™hir—l tr—nsform—tionsF st ™—n

˜ e ™—l™ul—ted ˜y fun™tion—l integr—tion of the dierenti—l —nom—ly



 t @A

t @A a h  a exp@ie AX @RFTPA

‡‡

 

„he term e represents the log—rithm of the integr—ted —nom—ly —nd is the ‡ess{umino{

‡‡

‡itten @‡‡A term ‘SP“F sn gener—l it is — very ™ompli™—ted fun™tion—lF ‡e will therefore

dis™uss in more det—il only the sp e™i—l ™—se of — —vor singlet ve™tor meson @3 {mesonA whi™h

™ouples to the ˜—ryon num˜ er —nd — ƒ @PA ™hir—l eldD

† @xA a 3 @xAIs

" " p

—

(

i@xA —

X @RFTQA @xA a e Y @xA a  @xA

P

„he ‡‡ term is then given ˜y



R "

d x 3 @xAf @xA @RFTRA e a x

" ‡‡ ™

where

I

"# ! y "

 tr@v v v AY v a d @RFTSA f @xA a

# ! # #

P

PR%

is the winding num˜ er ™urrent rel—ted to the ™hir—l eld @xAF hue to its top ologi™—l n—ture

"

the ™urrent f is ™onserved indep endent of the expli™it form of the ™hir—l eld @xAF

sn order to reve—l the physi™—l n—ture of this ™urrent we ™onsider the qu—rk gener—ting

fun™tion—l tre—ting the ve™tor eld † —s —n extern—l sour™e

"

 

 

R H

exp@e A a h q h q" exp d xq"@i@da C †aA m” Aq

p

~

a t @A exp@e‘† “A

 



R "

~

a exp ix d x† @xAf @xAC e‘† “ @RFTTA

™ "

~

where e‘† “ ™ont—ins the non{—no m—lous terms of the ee™tive —™tionF prom the rst equ—tion

we o˜t—in

3 2

 e

p

"

"

@xA @RFTUA a hq"@xA q @xAi aX j

f

i † @xA

"

† aH

iFeF the ˜—ryon ™urrent of the qu—rksF „he l—st equ—tion —llows us to rel—te the ˜—ryon ™urrent

to the top ologi™—l ™urrent

"

"

@xA a f @xAC XXX @RFTVA j

f

~

where the dots indi™—te the ™ontri˜utions from the ’norm—l4 terms e‘† “F „his le—ds to the

following interpret—tionX „he ˜—ryon ™urrent of the origin—l fermion @qu—rkA theory is ™—rried QT

˜y the top ologi™—l ™urrent of the ™hir—l eld in the ˜ osonized ee™tive meson theoryD —t le—st

in le—ding order gr—dient exp—nsionF es mentioned —˜ ove this top ologi™—l ™urrent is ™onserved

indep endent of the sp e™i™ form of the ™hir—l eld where—s the ˜—ryon ™urrent of the underlying

qu—rk theory is the @™onservedA xo ether ™urrent of —vor singlet ph—se tr—nsform—tionsF ƒoD

the dyn—mi™—l prop erty of the qu—rk theory h—s turned into — purely top ologi™—l prop erty in

the ee™tive theoryF

st should ˜e rem—rked th—t ™ontinuing to iu™lide—n sp—™e is equiv—len t to using peynm—n

˜ound—ry ™onditions in winkowski sp—™eF „his implies th—t the gener—ting fun™tion—l ™orreE

sponds to the ’v—™uum to v—™uum4 tr—nsition —mplitudeF por innitely l—rge iu™lide—n times

"

thus represents the v—™uum p—rt of the qu—rk ˜—ryon ™urrent onlyF j

f

„his immedi—tely le—ds to the question how the pseudos™—l—r meson eld @xA whi™h is

˜ osoni™ —nd spinless ™—n des™ri˜ e prop erties of ˜—ryons whi™h —re fermions with spin s a IaPF

‡e will see in se™tion SFP th—t — su™iently strong top ologi™—l non{trivi—l ™hir—l eld with

winding num˜ er n p ol—rizes the v—™uum or hir—™ se— of the qu—rks so strongly th—t n of

the v—len™e qu—rk levels —re ˜ ound tightly enough th—t they join the neg—tive hir—™ se—X

their energy is neg—tiveF es the physi™—l v—™uum is dened —s the st—te with lowest energy

—ll neg—tive energy levels —re o ™™upied in the v—™uumF „he physi™—l v—™uum ™—rries — non{

v—nishing ˜—ryon num˜ er if qu—rk st—tes —re ˜ ound in the hir—™ se—F ƒo it is not re—lly the

™hir—l eld itself whi™h ™—rries the ˜—ryon num˜ er ˜ut r—ther the p ol—rized v—™uumF es we

™—nnot o˜serve the v—™uum ˜ut only the p ol—rizing meson eld we rel—te the ˜—ryon num˜ er

to the ™hir—l eldF sn th—t sense meson elds ™—rry ˜—ryoni™ ™h—rgeF „his f—™t is the underlying

fe—ture for the des™ription of the ˜—ryons —s ™hir—l solitons —s will ˜ e dis™ussed in more det—il

in the following ™h—pterF

vet us ™lose this se™tion with — ™ommentF sing the inv—ri—n™e of the ™l—ssi™—l —™tion —nd

the t—™o˜i—n @RFTIA the —nom—lous ‡—rd identity



x

™

"

R H

~

d xtr h@p p Ai @RFTWA d j a Pim j i

p " S

S

P

V%

H

™—n str—ightforw—rdly ˜ e derived @see epp endix eAF xote th—t even for m a H the —xi—l

"

singlet ™urrent j is not ™onserved if ve™tor elds —re present despite the f—™t th—t the ™l—ssi™—l

S

H

v—gr—ng i—n is inv—ri—nt under ™hir—l rot—tions for m a HF „he —xi—l —nom—ly —s formul—ted

˜y the —nom—lous ‡—rd identity @RFTWA is known —s edler{fell{t—™kiw —nom—ly ‘QT “F su—lly

it is required th—t in fund—ment—l @g—ugeA theories —nom—lies should ˜ e not present or should

˜ e ™—n™eled ˜y other ee™tsF es we —re ™onsidering ee™tive low{energy mo dels —nom—lies ™—n

H

˜ e presentF „hey even h—ve me—sur—˜le ™onsequen™es —s eFgF the de™—y % 3 P F

ristori™—llyD the —nom—ly w—s dis™overed in p ertur˜—tion theoryF „he one{lo op di—gr—m

H

whi™h is resp onsi˜le for the de™—y % 3 P is shown in gure RFIF „he p—th integr—l deriv—tion

of the —nom—ly is — non{p ertur˜—tive one —nd demonstr—tes th—t higher order terms do not

™ontri˜ute to the —nom—lyF sf one ™—l™ul—tes the edler{fell{t—™kiw —nom—ly in — renorm—lizE

—˜le theory @—s ghA in whi™h the ™ut{o  is t—ken to innity —t the end of the ™—l™ul—tion

only the tri—ngle di—gr—m gure RFI ™ontri˜utesF roweverD if one works within —n ee™tive

non{renorm—liz—˜le theory where the r—tio ma h—s to ˜ e kept niteD not only the tri—ngle

˜ut —ll higher order di—gr—ms ™ontri˜uteF ‚emem˜ er th—t the limit  3 ID iFeF ( 3 HD @see

epp endix eA w—s ne™ess—ry in order to m—ke the ™ontri˜utions from h D n ! Q v—nishF sn

n

the liter—ture ‘TR“ there —re ™l—ims th—t the xtv mo del is not —ppropri—te for the tre—tment

H

of the —nom—ly —nd the % de™—y ˜ e™—use the tri—ngle di—gr—m gives only —˜ out two thirds

of the exp eriment—lly determined de™—y —mplitude using — nite ™utoF roweverD together

with the nite ™ut{o —lso higher order terms should ˜ e in™luded —nd presum—˜ly yield the QU

     










H

%

T d

d

d‚

d

d

     

 d








pigure RFIX „he tri—ngle di—gr—m whi™h is resp onsi˜le for the —nom—lyF

missing third of the de™—y —mplitude ‘TS“F xote —lso th—t not regul—rizing the im—gin—ry p—rt

H

of the —™tion le—ds to — suppression of higher order terms for the % de™—yD iFeF in this ™—se the

tri—ngle di—gr—m gives —lre—dy the ™orre™t de™—y —mplitudeF QV

S ghir—l ƒolitons

„his ™h—pter is devoted to — ˜rief dis™ussion on the interpret—tion of solitons —s ˜—ryonsF

elso —n outlo ok is provided on ™—l™ul—tions whi™h —re p erformed —t length in ™h—pters T —nd

UF por these intro du™tory rem—rks on soliton physi™s we —ssume th—t some ee™tive ™hir—lly

inv—ri—nt meson theory is given whi™h —llows for st—˜le soliton solutionsF ‡hen studying low

energy prop erties it is suggestive to t—ke into —™™ount only those mesons with the l—rgest

gompton w—ve{lengthD iFeF with the sm—llest m—ssF es expl—ined in the pre™eding ™h—pters

these —re the pseudos™—l—r would{˜ e qoldstone ˜ osons % @xA of sp ont—neous ™hir—l symmetry

˜re—kingF eg—in the non{line—r re—liz—tion will ˜ e —dopted

@xA a exp @i@xAA @SFIA

whi™h denes the ™hir—l eld @xAF

SFI „op ologi™—l prop erties

por two —vors — st—ti™ ™hir—l eld m—ps the ™o ordin—te sp—™e into the group of isospin

Q

X s‚ 3 ƒ @PAX @SFPA

ƒolitons —re nite energy eld solutions to the iuler{v—gr—ng e equ—tionsF piniteness of the

energy requires the energy density to v—nish for r a jr j 3 IF „his implies th—t the soliton

™ongur—tion @xA —symptoti™—lly —ppro—™hes — ™onst—nt v—lue @indep endent of the orient—tion

”

r A whi™hD —s — ™onsequen™e of ™hir—l symmetryD ™—n ˜ e ™hosen to ˜ e unityX

@xA 3 Is for r 3 IX @SFQA

Q

ren™e —ll p oints —t sp—ti—l innity —re identied there˜y ™omp—™tifying s‚ to — three dimenE

Q Q

sion—l sphere ƒ F ƒin™e the group m—nifold of ƒ @PA is —lso ƒ — st—ti™ ™hir—l eld ™ongur—tion

Q Q

with the ˜ ound—ry ™ondition @SFQA represents — m—pping from ƒ to ƒ

Q Q

X ƒ 3 ƒ X @SFRA

„hese m—ppings —re distinguished ˜y the winding num˜ er # P s whi™h is — top ologi™—lly

inv—ri—nt fun™tion—l of the ™hir—l eld



Q H

# ‘ “ a d xf @xAX @SFSA

"

rere f @xA is the top ologi™—l ™urrent @RFTSAF

e nonEtrivi—l m—pping @r A is given ˜y the soE™—lled hedgehog —ns—tz @™fF su˜se™tion

H

TFPFIA

”

@r A a exp @i(
r @r AA X @SFTA

H

por this eld ™ongur—tion the isospin ve™tor p oints into the r—di—l dire™tionF niqueness of

the ™hir—l eld —t r 3 H requires the ™hir—l —ngle to o˜ ey the ˜ ound—ry ™ondition

@r a HA a n% Y n P s X @SFUA QW

por the hedgehog ™ongur—tion @SFTA the sp—ti—l ™omp onents of the top ologi™—l ™urrent v—nish

while

P

sin @r A I

H H

X @SFVA  @r A f @r A a

P P

r P%

‡ithout loss of gener—lity one m—y ™ho ose @r 3 IA a H to s—tisfy @SFQA yielding

# ‘ “ a nX @SFWA

vet us emph—size —t this p oint th—t the winding num˜ er # a n is — purely top ologi™—l prop erty

of the ™hir—l eldD whi™h — priori is not rel—ted to —ny physi™—lly relev—nt qu—ntityF

SFP imergen™e of the soliton

sn gener—l the ee™tive meson theory results from integr—ting out the qu—rk —nd gluon

degrees of freedom ™ont—ined in ghF „hen the ˜—ryonsD whi™h —re origin—lly ˜uilt from x

g

qu—rksD emerge —s solitons of meson degrees of freedomF sn order to expl—in how the ˜—ryoni™

™h—r—™ter is en™o ded in the top ologi™—l prop erties of the soliton we ex—mine the sp e™trum of

™onstituent qu—rks in the ˜—™kground of the ™hir—l eldF „he ™orresp onding hir—™ r—miltoni—n

re—ds



S

Y @SFIHA h a 
p C  m@ A

where m denotes the m—ss of the ™onstituent qu—rksF „he hedgehog eld ™ongur—tion @SFTA

—

viol—tes the spin —nd isospin symmetries D iFeF

‘hY j “ a ‘hY t“ Ta HY @SFIIA

˜ut preserves the gr—nd spin

q a j C t @SFIPA

symmetryD iFeF ‘hY q“ a HF ren™e the eigenfun™tions of h ™—rry go o d gr—nd spin —nd p—rityF

sn order to ex—mine the sp e™trum of @SFIHA we p—r—metrize the ™hir—l —ngle ˜y ‘TT“

@

Pr

I for r —

Q—

@r A a n% X @SFIQA

P

—

for r ! —

P

Qr

„his sp e™i—l form is motiv—ted ˜y the f—™t th—t for sm—ll r the ™hir—l —ngle is line—r where—s

P

in the ™hir—l limit it is prop ortion—l to Iar for l—rge r F „he ™hir—l —ngle —nd its deriv—tive —re

™ontinuous —t the m—t™hing p oint r a —D whi™h p—r—metrizes the sp—ti—l extension of the ™hir—l

—ngleF pigure SFI shows the eigenv—lues  of the hir—™Er—miltoni—n @SFIHA for — ™hir—l eld

#

with winding num˜ er n a I —s — fun™tion of the strength of the ™hir—l eld me—sured ˜y —
mF

% C

es the strength of the ™hir—l eld in™re—sesD the lowest v—len™e qu—rk st—te in the q a H

™h—nnel ˜ e™omes strongly ˜ ound —nd eventu—lly joins the hir—™ se—F pigure SFP shows the

sp e™trum of ™onstituent qu—rks in the ˜—™kground eld of — hedgehog with winding num˜ er

%

n a PF sn this ™—se — se™ond v—len™e qu—rk st—te q a H ˜ e™omes ˜ ound in the hir—™ se— for

su™iently l—rge —
mF sn gener—lD strong ™hir—l elds with winding num˜ er n ˜ind ex—™tly

—

„he qu—ntities j —nd t denote the single qu—rk tot—l —ngul—r momentum —nd isospin op er—torsD resp e™ tivelyF RH

pigure SFIX „he eigenv—lues of the hir—™ r—miltoni—n @SFIHA in the ˜—™kground of — ™hir—l

C

eld @SFIQAF hispl—yed —re the lowest eigenv—lues in the H @full linesA —nd H @d—shed linesA

™h—nnelsF

pigure SFPX ƒ—me —s gure SFI for n a PF RI

n v—len™e qu—rk or˜its in the neg—tive hir—™ se—F y˜viously the ˜—ryon ™h—rge of the v—len™e

qu—rks gets en™o ded in the top ologi™—l stru™ture of the soliton when des™ri˜ing ˜—ryons in the

fr—mework of ee™tive meson theoriesF

fy denitionD —ll p ositive energy st—tes —re empty in the v—™uum ™ongur—tion while —ll

neg—tive energy st—tes —re o ™™upiedF ren™e for su™iently strong ™hir—l elds the v—™uum

p ossesses — ˜—ryoni™ ™h—rge whi™h is identi™—l to the winding num˜ erF es the p ol—rizing

meson eld is o˜serv—˜le r—ther th—n the v—™uumD the ˜—ryoni™ prop erties of the ™h—rged

v—™uum —re —ttri˜uted to this meson ™loudF „his justies the st—tement th—t ˜—ryons emerge

—s solitons in the ee™tive meson theoryF ƒtri™tly sp e—kingD howeverD the ˜—ryoni™ ™h—rge is

™—rried ˜y the p ol—rized hir—™ v—™uumF „o m—ke this s™en—rio more expli™it it is helpful to

™onsider the ˜—ryon num˜ er ™—rried ˜y the v—™uumD whi™h is dened —s the —symmetry of the

hir—™ sp e™trum @see eq @TFIPAA

ˆ

I

v—™

f a sgn@ AX @SFIRA

#

P

#

v—™

por — v—nishing ™hir—l eld the hir—™ sp e™trum is symmetri™ —nd therefore f a HF roweverD

—s the strength of ™hir—l eld in™re—ses the energy eigenv—lue of the v—len™e qu—rk or˜its

v—™

eventu—lly reverse their signs le—ding to f a nF

„hese studies provide the following re—soning for the solitoni™ pi™ture of ˜—ryonsX „he

v—len™e qu—rks of the ˜—ryons with ˜—ryon num˜ er f gener—te — strongly lo ™—lized ™hir—l eld

of winding num˜ er n a f F sn the ˜—™kground of this eld the v—len™e qu—rks —re ˜ ound in

v—™

su™h — w—y th—t the ˜—ryoni™ ™h—rge is ™—rried ˜y the p ol—rized v—™uumD iFeF f a nF yn™e

the v—len™e qu—rks h—ve joined the hir—™ se—D the p ol—rizing ™hir—l meson ™loud is interpreted

—s the ˜—ryonF

SFQ ƒemi™l—ssi™—l qu—ntiz—tion

es —lre—dy mentioned —fter eq @SFIHA neither spin nor isospin represent go o d qu—ntum

num˜ ers of the solitonF rere we will ˜riey des™ri˜ e — tre—tment whi™h gener—tes st—tes

™—rrying qu—ntum num˜ ers of physi™—l ˜—ryonsF por det—ils we refer to se™tion UFIF

„ime{indep endent rot—tions in ™o ordin—te{ —ndGor isosp—™e do not —lter the energy of

the solitonF sn order to p erform the semi™l—ssi™—l qu—ntiz—tion one requires time dep endent

solutions to the equ—tions of motionF „hese —re —pproxim—ted ˜y —llowing these rot—tions to

—di—˜—ti™—lly v—ry in timeF hue to the gr—nd spin symmetry the rot—tions in ™o ordin—te{ —nd

iso{sp—™e —re equiv—lentF st therefore su™es to write

y

@r Y tA a ‚@tA @r A‚ @tAY @SFISA

H

where ‚@tA denotes —n ƒ @PA isospin m—trixF hening the —ngul—r velo ™ity  ˜y

d i

y

‚ @tA ‚@tA a 
( Y @SFITA

d t P

the energy fun™tion—l i ‘ “ of the rot—ting soliton ™ongur—tion ™—n ˜ e exp—nded in p owers of

˜



I

ij y

  ‘ “ C


X @SFIUA i ‘ a ‚ ‚ “ a i ‘ “ C

i H j H H

P

˜

ƒin™e isospin ˜re—king is negle™ted the rot—tion m—trix ‚ do es not —pp e—r expli™itlyF RP

„he term line—r in  is missing due to time ree™tion symmetry whi™h holds for the isospin

symmetri™ two —vor mo delF „he moment of inerti— tensor



P



d i ‘ “



ij ij P

 ‘ “ a a   @SFIVA



H



d  d 

i j

 aH

is di—gon—l ˜ e™—use no dire™tion in isosp—™e is distinguishedF „he ™—noni™—l qu—ntiz—tion of

the ™o ordin—tes ‚ ™orresp onds to the repl—™ement

P

  3 t Y @SFIWA

with t ˜ eing the ™olle™tive spin op er—torF „he qu—ntized energies of the ˜—ryons —re n—lly

o˜t—ined to ˜ e

t @t C IA

@SFPHA i ‘ “ a i ‘ “ C

H

P

P

P

where t @t C IA is the eigenv—lue of t F

hue to the gr—nd spin symmetry of the hedgehog the spin of the qu—ntized soliton ™oin™ides

with the ™olle™tive isospin s up to — rot—tionF „he qu—ntized hedgehog soliton therefore yields

— tower @rot—tion—l ˜—ndA of st—tes with s a t F qener—lizing this tre—tment to three —vor

mo dels shows th—t for —n o dd @evenA num˜ er of ™olors the spin is ™onstr—ined to h—lf{integer

I Q

@integerA v—lues ‘SP D PT D PU “F por x a Q the nu™leon @s a t a A —nd the
@s a t a A

g

P P

reson—n™e thus represent ground —nd rst ex™ited st—tesD resp e™tivelyF

fothD the ™l—ssi™—l soliton m—ss —nd the moment of inerti— —re of order x F „herefore the

g

P

™l—ssi™—l soliton m—ss i ‘ “ —nd the rot—tion—l energy s @s C IAaP —re of order x —nd Iax D

H g g

resp e™tivelyF roweverD — system—ti™ exp—nsion of the ˜—ryon energy in p owers of Iax

g

@IA @HA @IA

i ‘ “ a i C i C i C


@SFPIA

H @HA

™ont—ins in —ddition to @SFPHA — term of order @Iax A D i F „his termD whi™h presum—˜ly

g

domin—tes over the rot—tion—l energyD is gener—ted from sm—ll —mplitude mesoni™ u™tu—tions

o the soliton —nd is —sso ™i—ted with the qu—ntum ™orre™tions to i ‘ “F x—vely one would

H

exp e™t

3 2

ˆ ˆ

I

@HA

@HA

Y @SXPPA 3 3 i a

i

i

P

i i

@HA

where 3 —nd 3 refer to the eigen{frequen™ies of the meson u™tu—tions in presen™e —nd

i

i

—˜sen™e of the solitonD resp e™tivelyF roweverD the expression @SFPPA is † divergent —nd

thus su˜ je™t to regul—riz—tionF xeverthelessD @SFPPA is illumin—ting sin™e it indi™—tes th—t the

@HA

domin—nt ™ontri˜ution to i stems from the zero{mo des @3 a HA ˜ e™—use there —re no

i

™ounterp—rts in the —˜sen™e of the solitonF woreoverD the ™ontri˜utions of the zero{mo des is

neg—tive —nd hen™e ™—uses — su˜st—nti—l redu™tion of the ™l—ssi™—l m—ssF „his is — desired fe—ture

sin™e the ™l—ssi™—l m—ss ™ommonly overestim—tes the ˜—ryon m—sses on™e the p—r—meters —re

tted to mesoni™ d—t— @see ™h—pter RAF ‚e™entlyD it h—s ˜ een shown for the ƒkyrme mo del

th—t — renorm—lized version of @SFPPA indeed predi™ts ˜—ryon m—sses whi™h well —gree with

exp eriment—l d—t— ‘PRD PS “F „he —n—logous ™—l™ul—tion ‘TU“ for the xtv mo del is presented in

se™tion UFRF RQ

T ƒt—ti™ solitons of the x—m˜u{ton— Ev—si nio mo del

sn the present se™tion we will dis™uss the emergen™e —s well —s prop erties of st—ti™ solitoni™

meson ™ongur—tions within the xtv mo delF „he most imp ort—nt ingredient of the ee™tive

meson —™tion of the ˜ osonized xtv mo del is the fermion determin—nt @QFPSAF sn pr—™ti™e we

h—ve to ev—lu—te this determin—nt in the presen™e of non{p ertur˜—tive meson eld ™ongur—E

tionsF „his determin—nt is given in terms of the eigenv—lues of the hir—™ op er—tor haF roweverD

sin™e det@i A a I one m—y equ—lly well ™onsider

i ha a id h @TFIA

t

whi™h intro du™es the one{p—rti™le r—milton op er—tor hF por the time ˜ eing we ™onsider h

to ˜ e st—ti™ —nd rermiti—nF „he dis™ussion of more gener—l ™—ses will ˜ e p ostp oned to l—ter

su˜se™tionsF por st—ti™ elds the eigenv—lues of @TFIA sep—r—te into the eigenv—lues of id

t

—nd hF „he fermion elds —ssume —nti{p erio di™ ˜ ound—ry ™onditions on the time interv—l „ F

„herefore the eigenv—lues of id —re given ˜y the w—tsu˜—r— frequen™ies  a @Pn C IA% a„ with

t n

n a HY IY PYXXXF henoting furthermore the eigenv—lues of h ˜y  the fermion determin—nt

#

—

is o˜t—ined —s the pro du™t ‘TV D UH“

H I

4 5

P

 

‰ ‰ ‰ ‰

Pn CI  „

#

d e

het @haA a I a g %  Y @TFPA

#

„ @Pn C IA%

# n #

n!H

sin™e for st—ti™ ™ongur—tions id —nd h m—y ˜ e di—gon—lized simult—neouslyF „he ™onst—nt

t

2 3

!

P

‰ ‰

Pn CI

% @TFQA g a

„

#

n!H

do es not dep end on the dyn—mi™—l prop erties of the system —nd m—y hen™e ˜ e —˜sor˜ ed into

the integr—tion me—sureF „he pro du™t over n is re—dily ™—rried out

 

‰

 „

#

het @haA a g ™os

P

#

4 5

ˆ ‰

i

~

a g exp j j„ @I C exp ‘i„ j j“A X @TFRA

# #

P

# #

‡ith the intro du™tion of o ™™up—tion num˜ ers  a HY I the pro du™t over # m—y n—lly ˜ e

#

expressed —s

4 5

‰ ˆ ˆ

@I C exp ‘i„ j j“A a exp i„  j j Y @TFSA

# # #

# #

f g

#

where the sum go es over —ll p ossi˜le ™om˜in—tions of  a HY IF „hen the fermion determin—nt

#

—™quires the form

i h

ˆ

f g

#

~

@TFTA het @haA a g exp ‘ie “ exp ie

H

†

f g

#

—

ƒee —lso ™h—pter W of refF ‘TW“F RR

whi™h provides — n—tur—l de™omp osition into v—™uum

ˆ

I

„ e a j j @TFUA

H #

P

#

—nd v—len™e @—nti{A qu—rk

ˆ

f g f g

# #

e a „  j j a „ i @TFVA

# #

† †

#

™ontri˜utions to the fermion determin—ntF

sn order to equip the o ™™up—tion num˜ ers  with — physi™—l me—ning let us ™onsider the

#

˜—ryon num˜ er ™urrent whi™h is dened —s the —ver—ge



i



" "

log het @ha i†aA  X @TFWA j @xA a hq"@xA q @xAi a

"

† @xAaH

 † @xA

"

„re—ting † —s — p ertur˜—tion in the eigenv—lue pro˜lem @h C  †aA2 a  2 reve—ls th—t

" # # #



 

#



" y

@xA  2 @xA @TFIHA  a 2

#

#

"

† @xAaH

 † @xA

"

with 2 @xA —nd  ˜ eing the eigenst—tes —nd {v—lues of hF st is then e—sy to see th—t —™™ording

# #

to @TFTA the ˜—ryon num˜ er ™urrent is —dditive in v—™uum —nd v—len™e p—rts ‘UH“

" "

"

@xA j @xA a j @xAC j

H

†

ˆ

I

"

"

"

sgn @ A2 @xA 2 @xA j @xA a

# # #

H

P

#

ˆ

"

"

"

@xA a j  sgn@ A2 @xA 2 @xAX @TFIIA

# # # #

†

#

„—king into —™™ount th—t the eigenfun™tions 2 —re prop erly norm—lized one o˜t—ins the ˜—ryon

#

num˜ er

 

ˆ

I

sgn @ AX @TFIPA  f a

# #

P

#

„husD ™onsidering — sp e™i—l set f g of o ™™up—tion num˜ ers ™onnes the system to — se™tor

#

with denite ˜—ryon num˜ erF

p to now we h—ve ignored the f—™t th—t the v—™uum p—rt of the fermion determin—nt

@TFUA is divergent —nd hen™e needs regul—riz—tionF ‡e will —ddress this pro˜lem in the next

su˜se™tion when extr—™ting the energy fun™tion—l from e F purthermoreD — more rigorous

H

tre—tment of symmetry ™urrents for the regul—rized theory will ˜ e presented in ™h—pter UF

TFI „he energy fun™tion—l

‡e h—ve —lre—dy o˜served th—t the ™ontri˜ution to the energy fun™tion—l due to the expli™it

o ™™up—tion of the v—len™e qu—rk or˜its is given ˜y @™fF eq @TFVAA

ˆ

i a  j j @TFIQA

† # #

# RS

with  ˜ eing the eigenv—lues of — one{p—rti™le rermiti—n hir—™ r—miltoni—n hF „his qu—ntity

#

m—y ˜ e extr—™ted from the hir—™ op er—tor vi— eq @TFIAF

„he v—™uum @or ground st—teA ™ontri˜ution i to the energy is extr—™ted from the fun™E

H

tion—l integr—l ˜y ™ontinuing to iu™lide—n times ( a ix —nd o˜serving th—t for l—rge iu™lide—n

H

time interv—lsD „ 3 ID the ground st—te provides the domin—nt ™ontri˜utionD iFeF

lim het@i ha A G exp @i „ A X @TFIRA

i H

„ 3I

rere ha denotes the iu™lide—n hir—™ op er—tor whi™h is o˜t—ined from ha @TFIA ˜y —n—lyti™

i

™ontinu—tionF st is imp ort—nt to note th—t in iu™lide—n sp—™e ( h—s to ˜ e ™onsidered — re—l

qu—ntityF

en rermiti—n one{p—rti™le hir—™ r—miltoni—n h—s ˜ een the st—rting{p o int of the pre™eding

™onsider—tionsF ‡e will w—ive this —ssumption from now on —nd —llow h to ™ont—in —nti{

rermiti—n p—rts —s wellF xeverthelessD the iu™lide—n hir—™ op er—tor for st—ti™ meson elds is

de™omp osed into — temp or—l p—rt —nd — st—ti™ iu™lide—n r—miltoni—n

i ha a d hX @TFISA

i (

gommonly the non{rermiti™ity of h stems from ™ontinuing the time ™omp onents of @—xi—lA ve™E

tor elds to iu™lide—n sp—™eX † 3 i† —nd e 3 ie F „he iu™lide—n hir—™ r—miltoni—n

H R H R

then re—ds

y

h a 
p C † C  e C i
† C i 
e C  @€ w C € w AX @TFITA

R S R S ‚ v

purthermore our not—tion for the @—xi—lA ve™tor meson implies to t—ke † —nd e —nti{

" "

rermiti—nF isp e™i—lly the —ntiErermiti™ity of † —nd e ™—uses the eigenv—lues of @TFITA

R R

s ‚

F ren™e the eigenv—lues ! of the op er—tor d C h —re given ˜y C i to ˜ e ™omplex  a 

nY# ( #

# #

s ‚

X @TFIUA C i ! a i C  a i C 

nY# n # n

# #

vet us @for the momentA ignore regul—riz—tion —nd p erform m—nipul—tions —n—logous to those

des™ri˜ ed —˜ ove for the sp e™i—l ™—se of h ˜ eing rermiti—nF „he fermion determin—nt in iuE

™lide—n sp—™e ™—n —g—in ˜ e shown to sep—r—te into v—™uum —nd v—len™e ™ontri˜utions

3 2

ˆ ˆ

~

@TFIVA het @ha A a g exp @„ i A exp „  "

i H # #

#

f g

#

€

" F „he qu—ntities " —re dened in terms of the with the v—™uum energy i a @x aPA

# # H g

#

re—l —nd im—gin—ry p—rts of the eigenv—lues of h

‚ ‚ s

" a j j C i sgn @ A X @TFIWA

#

# # #

„he ™ontri˜ution of the v—len™e or˜its to the @iu™lide—nA energy fun™tion—l ™—n ˜ e re—d o

@TFIVA —s

ˆ ˆ

‚ s ‚ ‚ s

i a i C ii a x  j j C ix  sgn @ A @TFPHA

† g # g #

† † # # #

# #

in the ™—se th—t h ™ont—ins —nti{rermiti—n p—rtsF ‡e h—ve —lso m—de expli™it the dep enden™e

on x F

g RT

et this p oint — few rem—rks on the —n—lyti™ prop erties of the eigenv—lues  —re in orderF

#

gonsidering the gener—liz—tion of the iu™lide—n hir—™ r—miltoni—n @TFITA dened ˜y su˜stiE

tuting † 3 z † —nd e 3 z e the eigenv—lues  —™tu—lly ˜ e™ome fun™tions of the ™omplex

H R H R #

v—ri—˜le z ‘UID UP “F st h—s ˜ een o˜served th—t for physi™—lly motiv—ted eld ™ongur—tions these

eigenv—lues —re —n—lyti™ in z F „his h—s ˜ een —™hieved ˜y numeri™—lly verifying th—t the v—urent

series for  @z A indeed redu™e to „—ylor series‘UI“F „he relev—nt g—u™hy integr—ls —re ™omputed

#

i9

˜y p—r—metrizing the p—th  @9A a  e z F rere z refers to the ™enter of the v—urent exp—nE

H H

sion while — v—ri—tion of  provides — me—ns to extr—™t the r—dius of ™onvergen™eF „hen the

eigenv—lues of the gener—lized hir—™ r—miltoni—n —re ™omputed —long the p—th @H ` 9 ` P% A

—nd su˜stituted into the g—u™hy integr—lsF „he eigenv—lues —re then found to exhi˜it —n —n—E

lyti™ stru™ture for — r—dius of ™onvergen™e of the order unity or even l—rgerF „his —n—lyti™ity

˜

is not — priori ™le—r ˜ e™—use the eigenv—lues represent ro ots of the ™h—r—™teristi™ p olynomi—lD

whi™h ™—rries — l—rge degree @innitely l—rge in the ™ontinuum ™—seAF por the relev—nt eld ™onE

 

gur—tion it h—s furthermore ˜ een shown th—t  @z A a  @z AF y˜viously —ll qu—ntities whi™h

# #

‚   s  

dep end on  @z Y z A a @ @z AC  @z AA aP —nd  @z Y z A a i @ @z A  @z AA aP sep—r—tely —re

# # # #

# #

no longer —n—lyti™ fun™tions of z F ƒu™h qu—ntities —re eFgF " F xext we ™onsider the sp e™i—l

#

i9

p—r—metriz—tion z a e D whi™h for H 9 % aP des™ri˜ es the p—th ™onne™ting iu™lide—n

 ‚

@z Y z A is reversed —long this p—th the —n—lyti™ —nd winkowski sp—™esF sn ™—se the sign of 

#

stru™ture of i is o˜viously destroyedF sn refF‘UI“ it h—sD howeverD ˜ een demonstr—ted th—t the

H

tot—l energy for — su˜system with unit ˜—ryon num˜ erD f a ID ™—n ˜ e written —s

H I

ˆ

x

g

d e

 " X @TFPIA

v—l #

P

# Tav—l

‚

jF „hus — ™h—nge „he v—len™e qu—rk level @v—lA refers to the st—te with the sm—llest mo dule j

#

 

‚

of sgn  do es —™tu—lly not destroy the —n—lyti™ prop ertiesF es the other st—tes @# Ta v—lA

v—l

i9

v—ry only mildly —long the p—th z a e D —n—lyti™ity is form—lly m—int—ined for the unregul—rE

ized energy fun™tion—l when ™onstr—ining oneself to ™ongur—tions with unit ˜—ryon num˜ erF

nfortun—telyD — dierent typ e of non{—n— lyti™ity existsF vevel ™rossings —long the p—th ™onE

ne™ting iu™lide—n —nd winkowski sp—™es will —pp e—r if the time ™omp onents † —ndGor e

R R

—re strong enoughF „his m—kes — denition of — winkowski energy fun™tion—l ˜y —n—lyti™

™ontinu—tion imp ossi˜leF „husD in order to —tt—™h physi™—l signi™—n™e to — given eld ™ongE

ur—tion one h—s to ™he™k th—t no su™h level ™rossing o ™™ursF ‡hen these level ™rossings —re

—voided the unregul—rized energy fun™tion—l is —n—lyti™ —nd the ™ontinu—tion forth —nd ˜—™k

from iu™lide—n to winkowski sp—™es ™—n ˜ e p erformedF sn this ™ontext it is then evident th—t

mo dels like the ™hir—l qu—rk mo del with the 3 {meson in™luded ‘UR“ indeed exhi˜it —n—lyti™—l

stru™turesF

xext we h—ve to f—™e the pro˜lem of regul—riz—tion whi™h h—s ˜ een ignored for the st—ti™ enE

ergy fun™tion—l up to hereF iq @TFIVA demonstr—tes th—t for „ 3 I only the v—™uum @groundA

st—te ™ontri˜utes to the fun™tion—l integr—l sin™e only the term with —ll  a H survives in the

#

sum @TFIVAF roweverD the expression for i is divergent —nd thus needs regul—riz—tionF es in

H

the study of the meson se™tor @™fF ™h—pter RA we will employ the prop er{time regul—riz—tion

s™heme ‘RV“ whi™h su˜stitutes the log—rithm ˜y — p—r—meter integr—l @QFPUAF nfortun—telyD

this pro ™edure is only —ppli™—˜le when the —rgument of the log—rithm is p ositiveF ‡e therefore

de™omp ose the ™ontri˜ution of the fermion determin—nt to the mesoni™ —™tion into re—l @e A

‚

˜

ƒee eFgF —pp endix g of refF‘UQ“F RU

—nd im—gin—ry @e A p—rts

s

   

I I

y y

I

X @TFPPA A ha „r log @ha C ha „r log ha e a e C e a

i i p ‚ s

i i

P P

sn terms of the eigenv—lues ! @TFIUA this de™omp osition is expressed —s

nY#

2 3

 

ˆ ˆ

I ! I

nY#



e a log ! ! log X @TFPQA —nd e a

‚ nY# s

nY#



P P !

#Yn #Yn

nY#

xote th—t —t this p oint the rules for m—nipul—ting the log—rithm h—ve ˜ een used in the sense

y

A h—ve ˜ een ™omputed indep endentlyF „his o˜viously represents th—t „r log @ha A —nd „r log @ha

i

i

y

—n —pproxim—tion ˜ e™—use the tr—™es of h —nd h —re —sso ™i—ted with dierent ril˜ ert sp—™esF

sn refF‘US“ it h—s ˜ een demonstr—ted th—t su™h — tre—tment m—y indu™e —n error whi™h is of

qu—dr—ti™ or higher order in the time ™omp onent of the @—xi—lA ve™tor eldsF

„he prop er{time pres™ription ™—n o˜viously ˜ e —pplied to this form of e yielding

‚

 

ˆ

I

P P ‚ s

e a @TFPRA A A C@ log @ 

‚ n

# #

P

#Yn



o n 

I

ˆ

I ds

P P ‚ s

X @TFPSA A A C@ exp s @  3

n

# #

P

P s

Ia

#Yn

e™™ording to the —˜ ove dis™ussion the expression @TFPRA ™—n only ˜ e ™onsidered —s —n —pproxE

im—tion to the —™tion in the presen™e of † —ndGor e r—ther th—n ˜ eing ex—™tF

R R

por l—rge iu™lide—n time interv—ls @„ 3 IA the sum over n in @TFPSA m—y now ˜ e repl—™ed

˜y —n integr—l

 

o n 

I I

ˆ

„ dz ds

P P ‚

e a @TFPTA A exp s z C@

‚

#

P

P P% s

Ia I

#

™ s

3 z F es mentioned —˜ oveD where we h—ve furthermore shifted the integr—tion v—ri—˜le z 

#

the limit „ 3 I —llows one to extr—™t the v—™uum ™ontri˜ution to the re—l p—rt of the energy

‚

whi™h ™oin™ides with the ™orresp onding expression derived from fun™tion—l from e 3 „ i

‚

H

— rermiti—n r—miltoni—n ‘UH“X

I H

3 2

P

‚

ˆ

 I x

g

#

‚ ‚

e d

p

Y j j i a

# H

R % P 

#

W V

I H

3 2

P

‚

a `

ˆ

x  

g

#

‚

e d

p

j jx @TFPUA a exp

#

#

Y X

P % 

#

where

I H

 

2 3 3 2

P

‚ ‚

 

  I I

 

# #

e d

p

a erf™   @TFPVA Y x a

#

 

% P  

—re the ’v—™uum o ™™up—tion num˜ ers4 in the prop er time regul—riz—tion s™hemeD whi™h for

 3 I redu™e to x a IF

#

™

ƒin™e the integr—l @TFPTA ™onverges —˜solutelyD the sum over # —nd the integr—l over z m—y ˜ e ex™h—ngedF RV

por the im—gin—ry p—rt @TFPQA we o˜t—in @—g—in to ˜ e ™onsidered with some ™—ution —s eq

@TFPRAA

2 3

I I I

ˆ ˆ ˆ ˆ ˆ ˆ

i  I I

n #



@TFPWA log log @! A a log @! A e a

#Yn s

#Yn



P i  P

n

# # #

#

naI naI naI

where we h—ve reversed the sign in the rst sum over the integer v—ri—˜le nF xext we express

e in terms of — p—r—meter integr—l

s



I

s

I

ˆ ˆ

I i

#

e a X @TFQHA d!

s

‚ s

P i  i!

I

n

#

# #

naI

sn —n—logy to @TFPUA we m—y ™—rry out the temp or—l tr—™e in the limit „ 3 IX

 

s

I I

ˆ

dz  i

#

d! „ € X @TFQIA e a

s

s ‚

P P% i@z ! A 

I I

#

# #

g—re h—s to ˜ e t—ken when p erforming the z {integr—tion ˜ e™—use only its prin™iple v—lue @€ A

s

is prop erly denedF xext the shift in the integr—tion v—ri—˜le z ! 3 z is p erformedF „hisD

#

of ™ourseD —lso ee™ts the ˜ ound—ries

 

s

s

w! I

ˆ

#

 i

dz

#

X @TFQPA e a d! „ lim

s

‚

s

w3I

P% iz  P

w! I

#

#

#

„—king into —™™ount th—t ! ™overs — symmetri™ r—nge of integr—tionD whi™h —llows one to reverse

the sign in the ! integr—l without ™h—nging the —sso ™i—ted ˜ ound—riesD the shift —pp e—ring in

the z {integr—l ˜ ound—ries ™—n ˜ e shown not to ™ontri˜ute —s w 3 IF „his yields

 

s

w I

ˆ

i dz 

#

e a X @TFQQA d! „ lim

s

‚

w3I

P P% iz 

w I

#

#

xow the integr—l over the p—r—meter ! m—y ˜ e doneF hue to the prin™iple v—lue pres™ription for

the z integr—tion the terms of o dd p owers in z ™—n™elF „his results in — ™onvergent expression



‚

I

ˆ

dz i P

s #

„  X @TFQRA e a

s

#

P ‚ P

P P% z C@ A

I

#

#

st should ˜ e mentioned th—t the divergen™eD whi™h dis—pp e—rs in the prin™iple v—lue pres™ripE

tionD h—s the dr—sti™ ™onsequen™e th—t the energy fun™tion—l is not inv—ri—nt when shifting the

3 eld ˜y — ™onst—nt 3 F sn th—t ™—se the im—gin—ry p—rt ot the iu™lide—n —™tion is —ugmented

™

˜y „ x 3 f where f denotes the ˜—ryon num˜ er of the origin—l ™ongur—tionF „his result is

g ™

very gr—tifyingD —s will ˜ e seen ˜ elowF

elthough e is nite in the prin™iple v—lue formul—tionD the prop er time regul—riz—tion

s

m—y ˜ e imp osed ˜y expressing the integr—nd —s — p—r—meter integr—l ‘UT “



n  o

I

I

P ‚ P

ds exp s z C@ @TFQSA A 3

#

P P ‚

P

A z C@

Ia

#

whi™h do es o˜viously not diverge —s  3 IF gompleting the ev—lu—tion of e in —n—logy

s

to eqsF @TFPSETFPUA we nd for the ™ontri˜ution of the hir—™ se— to the im—gin—ry p—rt of the

„ 3I

s s

iu™lide—n energy i from e 3 i„ i

s

H H

&

ˆ

x

IY e not regul—rized

g

s

s s ‚

i a  sgn @ A X @TFQTA

H # #

x Y e regul—rized

P

# s

# RW

„he upp er ™—se in eq @TFQTA ™orresp onds to the limit  3 IF y˜viously only the re—l p—rt of

the oneEp—rti™le energy eigenv—lue is relev—nt for the regul—riz—tion of e F iq @TFQTA represents

s

— regul—riz—tion for e th—t only involves qu—ntities whi™h —re stri™tly p ositive deniteF

s

„he iu™lide—n ’energy4 fun™tion—l

 

‚ s ‚ ‚ s s

@TFQUA i C ii a i C i C i i C i

H † H †

exhi˜its the interesting fe—ture th—tD —s — qu—rk st—te ˜ e™omes p—rt of the v—™uumD ˜ oth re—l

‚ s

@i A —nd im—gin—ry @i A —re ™ontinuous under the ™ondition th—t the ˜—ryon num˜ er @TFIPA

‚

rem—ins un™h—ngedF por — st—te j# i to ˜ e™ome p—rt of the v—™uum me—ns th—t  ™h—nges

#

signF essume th—t for the ™ongur—tion under ™onsider—tion one sp e™i—l or˜itD s—y jv—liD h—s

‚

 a I —s long —s  b HF sn order to st—y in the s—me ˜—ryon num˜ er se™tor  h—s

v—l v—l

v—l

 

‚ ‚

to v—nish —s  reverses its sign @™fF eq @TFIPAAF „hus we dem—nd  a I C sgn@ A aPF

v—l

v—l v—l

xoting th—t erf™@HA a I one e—sily veries th—t for  % H the terms whi™h dep end on the

v—l

‚

sign of  ™—n™el in the sum @TFQUAF prom this dis™ussion it is —lso o˜vious th—t the v—len™e

v—l

qu—rk or˜it o ™™up—tion num˜ ers  —re dyn—mi™—l qu—ntities whi™h fun™tion—lly dep end on the

#

soliton ™ongur—tionF

„he determin—tion of the iu™lide—n energy fun™tion—l is now ™ompleted —nd we h—ve to

o˜t—in — winkowski energy fun™tion—l from thisF sn the dis™ussion pro ™eeding eq @TFPHA we h—ve

—lre—dy rem—rked th—t the one{p—rti™le energy eigenv—lues  @z A ™—n ˜ e ™onsidered —n—lyti™ in

#

the ™omplex v—ri—˜le z F nfortun—telyD su™h — st—tement w—s shown ‘UI“ not to hold for the

energy fun™tion—l

 ‚  s 

i @z Y z A a i @z Y z AC ii @z Y z AX @TFQVA



rere the dep enden™e on z —nd its ™omplex ™onjug—te z is due to the impli™it dep enden™ies

  ‚Ys

@z Y z AF „he —pp e—r—n™e of z is ™—used ˜y the f—™t th—t the regul—riz—tion tre—ts re—l —nd 

#

im—gin—ry p—rts of the —™tion in the ™omplex pl—ne sep—r—telyF ‚emem˜ er th—t @form—llyA the

unregul—rized —™tion h—s ˜ een o˜served to ˜ e —n—lyti™ in z F „he fun™tion—l @TFQVA h—s ˜ een

investig—ted with resp e™t to its —n—lyti™—l prop erties ‘UI “F st h—s ˜ een found th—t the v—urent

series ™entered —t the iu™lide—n p oint @z a iA h—s v—nishing r—dius of ™onvergen™e —nd the

™o e™ients of the singul—r terms —re non{v—nishingF ƒt—ted more dr—sti™—llyX „he —n—lyti™

™ontinu—tion of @TFQUA do es not existF ren™e the winkowski energy fun™tion—l for st—ti™D non{

p ertur˜—tive eld ™ongur—tions involving time ™omp onents of @—xi—lA ve™tor elds ™—nnot ˜ e

o˜t—ined ˜y me—ns of —n—lyti™—l ™ontinu—tionF

st h—s then ˜ een —rgued ‘UI“ th—t the gener—lized energy fun™tion—l @TFQVA ™—n well ˜ e

‚ s

—pproxim—ted ˜y i @iY iAC z i @iY iA —t the iu™lide—n p oint @z a iAF „his —pproxim—tion

g—ins further supp ort ˜y the f—™t th—t the regul—riz—tion —s dis™ussed —˜ ove is not without

—m˜iguities —t the qu—dr—ti™ order in the time ™omp onents of the @—xi—lA ve™tor elds @see

se™tion TFPFRAF edopting this —pproxim—tion the —n—lyti™ ™ontinu—tion to the winkowski p oint

@z a IA is trivi—lD yielding

‚ s ‚ s

i a i C i C i C i C i X @TFQWA

m

H H † †

„he mesoni™ ™ontri˜ution i is str—ightforw—rdly o˜t—ined ˜y su˜stituting the soliton ™ongE

m

ur—tion into the purely mesoni™ p—rt of the —™tion e F yne might wonder whether there is —

m

s

sign —m˜iguity for i in the denition @TFQWAF „his is not the ™—se for the energy fun™tion—l

HY†

of — self{™onsistent soliton ™ongur—tion ˜ e™—use the eigenv—lues of h —re —n—lyti™—l —nd hen™e

su™h — sign —m˜iguity m—y ˜ e —˜sor˜ ed in the denition of the time ™omp onents † —nd e F

R R SH

vet us next mention two ™onsisten™y ™onditions whi™h h—ve to ˜ e s—tised ˜y the winkowski

energy fun™tion—lF es —lre—dy noted —fter eq @TFQRA shifting the 3 eld ˜y — ™onst—nt —mount

@3 A le—ds to — ™h—nge of the @unregul—rized —nd thus —n—lyti™—lA energy fun™tion—l of ix f 3 F

™ g ™

‡e dem—nd this to hold for the regul—rized energy fun™tion—l in winkowski sp—™e —s wellF es

— m—tter of f—™t this ™ondition is — ™onsequen™e of glo˜—l g—uge inv—ri—n™e @iFeF — ™onst—nt

† ˜ eh—ves like — ™hemi™—l p otenti—lAF purthermoreD the ™urrent eld identity @QFPIA for the

H

˜—ryon ™urrent imp oses — norm—liz—tion on the prole fun™tion of the isos™—l—r 3 mesonF „his

norm—liz—tion is o˜t—ined ˜y integr—ting the st—tion—ry ™ondition for this eld over the whole

™o ordin—te sp—™eF es this st—tion—ry ™ondition is o˜t—ined from extremizing the winkowski

sp—™e energy fun™tion—l we h—ve —v—il—˜le — se™ond ™onsisten™y ™onditionF st is imp ort—nt to

note th—t the energy fun™tion—l @TFQWA s—tises these two ™onsisten™y ™onditionsY —t le—st when

the im—gin—ry p—rt rem—ins unregul—rizedF „hus @TFQWA provides — well suited o˜ je™t for the

investig—tion of soliton solutions ‘UTD UQ D US “F

et this p oint it should ˜ e mentioned th—t two other —ppro—™hes to in™lude time ™omp onents

of ve™tor mesons —re dis™ussed in the liter—tureF sn refF ‘UU“ these ™omp onents —re not ™ontinued

—™™ording to the dis™ussion pre™eding @TFITAF sn th—t ™—se the energy fun™tion—l in the whole

z {pl—ne dep ends on the eigenv—lue  @z A only —nd is thus —n—lyti™—lF roweverD due to the

#

regul—riz—tion the glo˜—l g—uge inv—ri—n™e —s well —s the ™urrent eld identity for the ˜—ryon

num˜ er ™urrent —re lost in winkowski sp—™eF „his —ppro—™h h—s therefore to ˜ e —˜—ndoned

˜ e™—use of physi™—l re—sonsF sn the se™ond tre—tment ‘UVD UW“ the time ™omp onents h—ve indeed

˜ een ™ontinuedF „hese —uthors h—ve expressed the iu™lide—n energy fun™tion—l @TFQUA in terms

 

of  @z A —nd its ™omplex ™onjug—te  @z A a  @z AF „hen the energy fun™tion—l dep ends on z —nd

# #

#

 

z X i a i @z Y z AF st should ˜ e evident from the pre™eding dis™ussions th—t the dep enden™e on

 

 @z A —nd thus z purely origin—tes from regul—riz—tion —nd tre—ting re—l —nd im—gin—ry p—rts

#



of the —™tion dierentlyF es in iu™lide—n sp—™e z a i a z the —uthors of refF‘UVD UW “ h—ve

~ ~

then set i a i @z Y z A —nd ™l—imed th—t the ™ontinu—tion of i to z a I yielded the winkowski

energy fun™tion—lF st should ˜ e noted th—t this —ppro—™h preserves the glo˜—l g—uge inv—ri—n™eF



roweverD —s —lre—dy indi™—ted —˜ oveD the ™ontinu—tion of i @z Y z A do es not exist —nd hen™e

the tre—tment of refF‘UVD UW “ h—s to ˜ e ™onsidered —s — physi™—lly motiv—ted denition ˜ut not

—s — m—them—ti™—lly ™orre™t deriv—tion of the winkowski energy fun™tion—lF

†ery re™ently it h—s ˜ een demonstr—ted th—t ™ounting p owers of the time ™omp onents of

ve™tor elds in the eigenv—lues of the hir—™ r—miltoni—n do es not ™orre™tly repro du™e the

™orresp onding exp—nsion for e ‘UI“F „he re—son is th—t the m—nipul—tions le—ding to the sum

‚

y

@TFPRA —re in f—™t ill{dened —s —lre—dy indi™—ted ˜ e™—use the tr—™es of h —nd h —re ™omputed

in dierent ril˜ ert sp—™esF „his topi™ will further ˜ e illumin—ted in su˜se™tion TFPFRF

TFP ƒelf{™onsistent solutions

‡e denote meson eld ™ongur—tions whi™h extremize the winkowski energy fun™tion—l

@TFQWA self{™onsistentF vet 9 represent the whole set of meson elds involved in the soliton

™ongur—tionF „hen the st—tion—ry ™ondition m—y ˜ e expressed —s

 i

a HX @TFRHA

 9

„he mesoni™ p—rt of the energy fun™tion—lD i D is —t le—st qu—dr—ti™ in 9 while the hir—™

m

op er—tor provides — ‰uk—w— typ e ™oupling of 9 to the qu—rk eldsF „hus @TFRHA rel—tes the SI

meson elds to the qu—rk eldsF „he l—tter di—gon—lize the st—ti™ r—miltoni—n h —nd the —sE

so ™i—ted eigenv—lues —g—in enter the energy fun™tion—lF st is this interpl—y ˜ etween eigenst—tes

—nd {v—lues of h whi™h le—ds to the notion of self{™onsisten™yF gommonly the self{™onsistent

soliton solution is numeri™—lly o˜t—ined with the use of —n iter—tive pro ™edure ‘VQ D VH “F e test

prole is employed to di—gon—lize hF „he resulting eigenv—lues —nd {ve™tors —re then su˜stiE

tuted into the st—tion—ry ™ondition @TFRHA yielding —n up d—ted prole fun™tionF „his up d—ted

prole serves —g—in —s input for hF „his pro ™edure is rep e—ted until ™onvergen™e is g—inedF

TFPFI „he pseudos™—l—r hedgehog

„he simplest meson eld ™ongur—tion whi™h —llows for soliton solutions involves the ™hir—l

eld onlyF „his ™ongur—tion h—s —lre—dy ˜ een employed in ™h—pter S to dis™uss gener—l

prop erties of the solitonF „hen —ll ve™tor meson elds —re set to zero —nd for the pseudos™—l—r

eld the ™ele˜r—ted hedgehog —ns—tz is —ssumed

”

w a m exp @i(
r @r AA X @TFRIA

y˜viously the s™—l—r elds —re ™onstr—ined to the ™hir—l ™ir™le  a hi a mF „he r—di—l

fun™tion @r A is ™ommonly referred to —s the ™hir—l —ngleF ƒu˜stituting this —ns—tz into the

hir—™ r—miltoni—n @TFITA yields

”

h a 
p C  m @™os @r AC i (
r sin@r AA X @TFRPA

S

es h is rermiti—n the eigenv—lues  o˜t—ined from

#

h a  @TFRQA

# # #

—re re—lF „e™hni™—llyD the dis™retiz—tion of the eigenv—lues  is —™hieved ˜y restri™ting the

#

Q

™o ordin—te sp—™e s‚ to — spheri™—l ™—vity of r—dius h —nd dem—nding ™ert—in ˜ ound—ry ™ondiE

tions —t r a h F iventu—lly the ™ontinuum limit h 3 I h—s to ˜ e ™onsideredF ‡e releg—te the

dis™ussion of the sp e™i—l form of the ˜ ound—ry ™onditions to —pp endix f where the ™o ordin—te

represent—tions of the st—tes j# i —re givenF st is nevertheless worthwhile to dis™uss the stru™E

ture of these st—tesF hue to the sp e™i—l form of the hedgehog @TFRIA the hir—™ r—miltoni—n

™ommutes with the gr—nd spin op er—tor

' ( (

a l C C @TFRRA q a j C

P P P

where j l—˜ els the tot—l spin —nd l the or˜it—l —ngul—r momentumF ( aP —nd ' aP denote the

isospin —nd spin op er—torsD resp e™tivelyF „he eigenst—tes of h —re then —s well eigenst—tes of qF

„he l—tter —re ™onstru™ted ˜y rst ™oupling spin —nd or˜it—l —ngul—r momentum to the tot—l

spin whi™h is su˜sequently ™oupled with the isospin to the gr—nd spin ‘VI“F „he resulting st—tes

—re denoted ˜y jl j qw i with w ˜ eing the pro je™tion of qF „hese st—tes o˜ ey the sele™tion

rules

V &

q CI

b

b

q CIaPY l a b

`

q

j a X @TFRSA

&

b

q

b

b

X

q IaPY l a

q I SP

es the one{p—rti™le eigenenergies  —re re—lD the iu™lide—n energy do es not p ossess —n

#

im—gin—ry p—rtF „he winkowski energy fun™tion—l —rising from eq @TFPUA is then given ˜y

3 2

 

P

ˆ ˆ

x  I

g #

p

i ‘“ a x C i Y  j j C j j

g m # # #

R % P 

# #

I H

3 2

P

H

ˆ

x  I

g

#

H

e d

p

@TFRTA Y j j

#

R % P 

#

with the mesoni™ p—rt



P P P

i a R% m f dr r @I ™os @r AA X @TFRUA

m

% %

elso the energy fun™tion—l —sso ™i—ted with the trivi—l meson ™ongur—tion   H is su˜tr—™ted

in @TFRTA —nd @TFRUAF „he st—tion—ry ™ondition  i ‘“a @r A is o˜t—ined with the help of the

™h—in ruleF ‡hen™e we require the fun™tion—l deriv—tive of 

#



 

#

y

”

@r A @sin@r AC i (
r ™os @r AA @r AX @TFRVA a m d

S #

#

 @r A

„hen the st—tion—ry ™ondition ˜ e™omes the equ—tion of motion for  ‘VQ “

@ A

 

P P

R%

f m

% %

”

™os@r A tr d & @r Y r Ai (
r a sin@r A tr d & @r Y r A @TFRWA

ƒ S ƒ

x m

g

where the tr—™es —re over —vor —nd hir—™ indi™es onlyF e™™ording to the sum @TFRTA the s™—l—r

qu—rk density m—trix & @xY y A a hq @xA"q @y Ai is de™omp osed into v—len™e qu—rk —nd hir—™ se—

ƒ

p—rtsX

† H

& @xY y A a & @xY y AC & @xY y A

ƒ

ƒ ƒ

ˆ

†

"

@y Asgn@ A @xY y A a @xA &

# # # #

ƒ

#

 

 

ˆ

 

 I

#

H

"

 

@y Asgn @ AX @TFSHA @xAerf™ @xY y A a &

# # #

ƒ

 

P 

#

es indi™—ted —˜ ove the st—tion—ry ™ondition rel—tes the soliton prole fun™tion @r A to the

eigenfun™tions of the hir—™ r—miltoni—n hF

#

‡e —re now —t the p oint to dis™uss the numeri™—l solutions in the unit ˜—ryon num˜ er se™tor



‚



”

@f a IAF por the eigenve™tors @fFREfFTA one ™—n e—sily verify th—t tr d & @r Y r Ai (
r  a

ƒ S

r aH

HF „his tr—nsfers to the ˜ ound—ry ™ondition for the ™hir—l —ngleX @HA a l % F yn the other

h—ndD for r 3 I @TFRWA h—s — solution with  a HF sn the @f a IA se™tor it turns out th—t

the soliton proles with the ˜ ound—ry ™onditions @HA a % —nd @IA a H @mo dulu P% A

extremize the st—ti™ energy fun™tion—l @TFRTAF st should ˜ e mentioned th—t for ™ongur—tions

whi™h do not o˜ ey these ˜ ound—ry ™onditions the energy rem—ins nite —lthough it is not —

minimumF „his is in ™ontr—st to top ologi™—l soliton mo dels ‘VP“ where the energy diverges for

innitesim—l devi—tions from these ˜ ound—ry ™onditionsF „he xtv soliton is not — top ologi™—l

one3 SQ

pigure TFIX „he r—di—l dep enden™e of the self{™onsistent prole fun™tion @r A for v—rious

™onstituent qu—rk m—sses mF „he pion m—ss is m a IQS we†F

%

„—˜le TFIX „he soliton energy i —s well —s its v—rious ™ontri˜utions —™™ording to the sum

tot

@TFRTA —s fun™tions of the ™onstituent qu—rk m—ss mF „he pion m—ss is t—ken to ˜ e m a

%

IQSwe†F ell num˜ ers —re in we†F

m QSH RHH SHH THH UHH VHH

i IPQT IPQW IPPI IIWQ IITI IIQH

tot

i URS TQQ RTH PWQ IPI ESS

†

i RSW SUI UPV VTW IHIP IIHQ

H

i QI QR QQ QI PV PT

m

ƒelf{™onsistent solutions were numeri™—lly o˜t—ined for m ! QPSwe† ‘VQ“F sn pigure TFI

the r—di—l dep enden™e of the self{™onsistent prole is displ—yed for v—rious v—lues of the ™onE

stituent qu—rk m—ss mF „he prole fun™tion o˜viously exhi˜its only — very mild dep enden™e

on mF „he soliton energy i D iFeF the energy fun™tion—l ™orresp onding to the self{™onsistent

tot

™hir—l —ngleD shows the s—me ™h—r—™teristi™s —s ™—n ˜ e seen from t—˜le TFIF y˜viously the

`

soliton m—ss is l—rger th—n the three qu—rk threshold —s long —s m RPHwe†F sn t—˜le TFI —lso

$

the v—rious ™ontri˜utions to the soliton energy —re displ—yedF por sm—ll ™onstituent qu—rk

m—sses the expli™it o ™™up—tion of the v—len™e qu—rk or˜it @ a IA provides the domin—nt

v—l

™ontri˜ution to the energyF es m in™re—ses the situ—tion is reversedF por m b USHwe† i

†

even ˜ e™omes neg—tiveF e™™ording to our previous dis™ussions this implies  a HF ƒt—ted

v—l

otherwiseX „he v—len™e qu—rk ˜ e™omes p—rt of the p ol—rized v—™uum whi™h then ™—rries the

˜—ryon num˜ erF „his situ—tion —™tu—lly ™orresp onds to the ƒkyrmion pi™ture of the ˜—ryonF

purthermoreD the ™ontri˜ution i of the purely mesoni™ p—rt of the —™tion to the energy is

m

shown in t—˜le @TFIAF „his qu—ntity ™—n —™tu—lly ˜ e interpreted —s the pion{nu™leon  term

P

—t zero momentum tr—nsfer  @q a HAF „he results for i seem to ˜ e somewh—t lower

% x m SR

th—n the d—t— extr—™ted from — ™riti™—l ex—min—tion of the existing d—t— on % x s™—ttering

P P

‘VR “X  @q a HA % RSwe†F st shouldD howeverD ˜ e noted th—t  @q A is ™l—imed ‘VR“ to

% x % x

P

h—ve — strong momentum dep enden™e whi™h m—kes the extr—™tion of the  @q a HA from

% x

% x s™—ttering d—t— r—ther involvedF

TFPFP feyond the ™hir—l ™ir™le

ƒo f—r the s™—l—r mesons h—ve ˜ een ™onstr—ined to their v—™uum exp e™t—tion v—lueD the

™onstituent qu—rk m—ss mF „he re—son for ™ho osing this —ns—tz is not only th—t of simpli™—tion

˜ut r—ther the —pp e—r—n™e of — sp e™i—l inst—˜ility of the soliton ™ongur—tion when the s™—l—r

eld is —llowed to ˜ e sp—™e dep endentY —t le—st in the prop er time regul—riz—tion s™heme

‘UU D VS “F sn this ™—se one ™—n prove the existen™e of — meson eld ™ongur—tionD whi™h h—s

v—nishing energyF ‡hen v—rying the sp—ti—l extension of this ™ongur—tion ˜ elow — ™riti™—l size

the system p—sses from — unit ˜—ryon num˜ er to — zero ˜—ryon num˜ er ™ongur—tionF „his

v—ri—tion ™—n ˜ e p—r—metrized —s @w a ' C %
( A

r r r r

”

A @TFSIA A ™os @ A“Y % @r A a r 0 ‡ f @ A sin @ ' @r A a 0 ‘I C ‡ f @

H H

‚ ‚ ‚ ‚



with ‡
@‚0 A a ™onst F sn refF‘VS“ — ‡o o dEƒ— xon sh—p e w—s ™hosen for f @r a‚A —nd — line—r

H

prole for @r a‚AF „he ™onst—nt 0 $ m ™h—r—™terizes the v—™uum exp e™t—tion v—lue of the

H

s™—l—r eldF por very n—rrow ™ongur—tionsD iFeF ‚ 3 HD the mesoni™ p—rt of the xtv soliton

p

P P

energyD whi™h is m—xim—lly qu—dr—ti™ in the s™—l—r eld 0 a ' C % @to whi™h we will

refer to —s the ™hir—l r—diusAD ™—n ˜ e shown to v—nish for  ` QaPF „hus only the fermion

determin—nt ™ontri˜utes to the energy in this ™—seF por su™h — v—lue of  the sp e™trum of

the hir—™ r—miltoni—n h—s the interesting fe—ture th—t the v—len™e qu—rk level gets tr—nsferred

from the lower ˜ ound—ry of the p ositive hir—™ sp e™trum to the upp er ˜ ound—ry of neg—tive

% C

hir—™ sp e™trumF ell other levels in the q a H ™h—nnel follow —s — ™onsequen™e of ’—voided

™rossings4 ‘VS“F y˜viously the ˜—ryon num˜ er is ™—rried ˜y the hir—™ se— —nd hen™e the soliton

energy is solely given ˜y the v—™uum p—rt i F „his is plotted —s — fun™tion of ‚
0 in gure TFP

H H

for  a RaQF ividently the tot—l energy is identi™—l to zero in the limit ‚ 3 HF roweverD sin™e

these lo ™—lized meson eld ™ongur—tions ™—rry no ˜—ryon num˜ er for sm—ll ‚D this inst—˜ility

is unphysi™—l indi™—ting merely th—t for ‚ 3 H the system ™h—nges from the f a I to the

f a H se™torF „his is —lso indi™—ted ˜y the f—™t th—t these ™ongur—tions —ssume zero energyD

whi™h is the energy of the ground st—te in the f a H se™torF

yne w—y to ™ir™umwent these pro˜lems is eFgF indi™—ted the p ossi˜ility to mo ™k up the

tr—™e —nom—ly of gh in ee™tive meson theoriesF sn su™h —n —ppro—™h the s™—l—r dil—ton

"#

eld 1 $ hq q i is in™orp or—ted —s —n order p—r—meter to —˜sor˜ the m—ss dimension of

"#

the p—r—meters ‘VU“F isp e™i—lly one m—y intro du™e — ™oupling to the ™hir—l r—dius 0 su™h th—t

— fourth order term —pp e—rs in the meson p—rt of the energy ‘VVD VT “

 



I

P P R P P

i a R% dr r — 0 @r A  1 0 @r A ™onst X @TFSPA

m

H

P

rere 1 denotes the v—™uum exp e™t—tion v—lue of the dil—ton eldF „he two new p—r—meters

H

— —nd  —re rel—ted vi— the @extendedA g—p equ—tionF ren™e one m—y equ—lly well ™onsider

P

 1 —s the only free p—r—meter @˜ esides the ™onstituent qu—rk m—ss mAF „he meson energy

H

with the dil—tion in™luded @TFSPA only v—nishes for  ` QaR when the p—r—metriz—tion @TFSIA is

—dopted ‘VT“F ql—n™ing —g—in —t gure TFP one o˜serves th—t in this ™—se the energy do es not SS

pigure TFPX „he energy of the fermion determin—ntD i C i D for the v—ri—tion—l meson eld

† H

™ongur—tion @TFSIAF „he p—r—meter  is dened —fter eq @TFSIAF @pigure t—ken from refF‘VT“FA

pigure TFQX „he self{™onsistent soliton ™ongur—tion with the s™—l—r eld in™ludedF ’xon{

line—r4 refers to the ™—se with the s™—l—r elds xed —t their v—™uum exp e™t—tion v—luesF

@pigure t—ken from refF‘VT“FA ST

v—nish for ‚ 3 H ˜ut r—ther ˜ e™omes very l—rgeF sndeedD the solution to the self{™onsistent

pro˜lem exists —nd is displ—yed in gure TFQF prom this gure one —lso re™ognizes th—t the

™oll—pse —pp e—rs when the fourth order inter—™tion is swit™hed oF es — ™onsequen™e of the

P P

g—p equ—tion the limit — 3 H ™orresp onds to  1 a0 3 IF pigure TFQ —lso demonstr—tes

H H

th—t for in™re—sing ™oupling to the dil—ton eldD the ™hir—l —ngleD  gets more ™on™entr—ted —t

r a HF „his ree™ts the ˜—g form—tion ™—used ˜y the dil—ton eld ‘VW“F

„he —˜ ove ™onsider—tions m—de use of —n extended denition of the energy fun™tion—l in

order to —void the ™oll—pse of the meson prolesF es the net result of the limit ‚ 3 H in the

% C

™ongur—tion @TFSIA for  a QaR is to shift —ll eigenv—lues  in the q a H ™h—nnel ˜y one

#

levelD the v—nishing energy is just — m—tter of regul—riz—tion whi™h ignores the —symmetries of

the sp e™trum of the hir—™ r—miltoni—n —t l—rge energiesF roweverD the ˜—ryon num˜ er @TFIPA

is sensitive to these l—rge energies ˜ e™—use no regul—riz—tion is involvedF „—king the p oint of

view th—t the im—gin—ry p—rt of the iu™lide—n —™tion should —lso ˜ e regul—rizedD —utom—ti™—lly

le—ds to — regul—rized ˜—ryon num˜ er f



 

  

ˆ

 

I 

#

 

X @TFSQA erf™ f ‘0Y “ a sgn@ A 

 # #

 

P 

#

sn the s—me w—y —s the energy v—nishesD f go es to zero for the —˜ ove ™onsidered limit ‚ 3 HF



sn refF‘WH“ therefore the ide— h—s ˜ een pursued to ™onstr—in f to unity in order to —void the



™oll—pseF „hen the energy fun™tion—l @TFRTA is supplemented ˜y — v—gr—n ge multiplier for f





P

P

! X @TFSRA i ‘0Y “ 3 i ‘0Y “ C ! @f ‘0Y “ IA C



P

sn this ™—se no qu—rti™ term in 0 is needed in the mesoni™ p—rt of the —™tionF „he l—st term

in @TFSRA is intro du™ed for ™onvenien™eY eventu—lly the limit  3 H should ˜ e —ssumedF et

rst pl—™e it should ˜ e noted th—t for the ™hir—l soliton with the s™—l—r elds ™onstr—ined to

the v—™uum exp e™t—tion v—lues @see su˜se™tion TFPFIA f is not ex—™tly unityY ˜ut r—ther only



HXWUF iven ˜y in™luding the ™onstr—int @TFSRA this num˜ er ™—nnot ˜ e in™re—sed signi™—ntly —s

long —s 0 is not —llowed to ˜ e sp—™e dep endentF „—king 0 a 0@r A to ˜ e — r—di—l fun™tion indeed

d

in™re—ses f F xumeri™—llyD howeverD — solution with f a I h—s not ˜ een found for nite h F

 

por nite h —lw—ys — non{v—nishing lower limit for  exists ˜ elow whi™h no solution h—s ˜ een

foundF „his limit ™—n ˜ e tr—nsl—ted into — v—lue for f whi™h slightly devi—tes from unity



˜ut —ppro—™hes it —s h in™re—sesF „he —sso ™i—ted self{™onsistent soliton proles —re displ—yed

in gure TFRF „he most imp ort—nt result isD of ™ourseD th—t st—˜le solutions do exist when

the regul—rized ˜—ryon num˜ er is xedF purthermoreD one o˜serves th—t the l—rge dist—n™e

@r ! IfmA ˜ eh—vior of the proles is —lmost unee™ted ˜y the ™onstr—int @TFSRAF roweverD in

the vi™inity of the origin dr—sti™ ™h—nges o ™™urF por f 3 I the ™hir—l r—diusD 0 v—nishes —t



the originD iFeF the ™hir—l symmetry is restoredF ƒimult—neouslyD the slop e of the ™hir—l —ngleD

 in™re—ses —nd eventu—lly go es to innityF sn refF‘WH“ it h—s —lso ˜ een shown th—t ™onstr—ining

the regul—rized ˜—ryon num˜ er le—ds to — v—len™e qu—rk domin—ted pi™ture of the solitonF

isp e™i—llyD the ™orresp onding energy eigenv—lue —ppro—™hes the one o˜t—ined in the ˜—ryon

num˜ er zero se™torF xeverthelessD the v—len™e qu—rk w—ve{fun™tion w—s found to ˜ e strongly

lo ™—lizedF

e further —ppro—™h to prevent the meson proles from ™oll—psing is to in™lude ve™tor mesons

‘UW D US “Y esp e™i—lly ˜ e™—use the 3 {meson provides — siz—˜le repulsion whi™h is supp osed to

st—˜ilize the systemF „his will —lso ˜ e dis™ussed in the pro ™eeding su˜se™tionF

d

h denotes the r—dius of the spheri™—l ˜ ox for the numeri™—l ™—l™ul—tionsD ™fF the dis™ussion —fter eq @TFRQAF SU

pigure TFRX „he self{™onsistent soliton ™ongur—tion with the s™—l—r eld in™luded —nd the

regul—rized ˜—ryon num˜ er f @TFSQA ˜ eing ™onstr—inedF hierent v—lues for f ™orresp ond

 

to v—rious r—dii of the spheri™—l ˜ ox used for the numeri™—l ™—l™ul—tionsF ’xon{line—r4 denotes

the ™—se without ™onstr—int —nd the s™—l—r eld —t its v—™uum exp e™t—tion v—luesF @pigure

t—ken from refF‘WH“FA

TFPFQ sn™lusion of @—xi—lA ve™tor mesons

sn re™ent ye—rs the xtv soliton h—s exp erien™ed sever—l typ es of extensionsF sn the present

se™tion we will des™ri˜ e the ee™ts of v—rious ve™tor —nd —xi—l{ve™tor meson elds on the soliton

solution —nd put some emph—sis on the histori™—l developmentF €ro ™eeding in this m—nner

we —re —lso en—˜led to illumin—te the ee™ts of dierent @—xi—lA ve™tor meson elds sep—r—telyF

e ™onvenient p—r—metriz—tion of the —nti{rermiti—n elds † —nd e is given ˜y sep—r—ting

" "

isos™—l—r —nd isove™tor p—rts

( (

X @TFSSA Y e a if C i—
† a i3 C i&

" I" I" " "

"

P P

„he iu™lide—n hir—™ r—miltoni—n for the most gener—l gr—nd spin zero eld ™ongur—tion

™onsistent with the p—rity prop erties of the meson elds is given in eq @TFTQAF eddition—lly the

™omplete set equ—tions of motion resulting from extremizing the winkowski energy fun™tion—l

@TFQWA is displ—yed in —pp endix gF „he r—miltoni—n ™orresp onding to the su˜systems whi™h

will su˜sequently ˜ e dis™ussed ˜ elow ™—n e—sily ˜ e o˜t—ined from @TFTQA ˜y setting those eldsD

whi™h —re not involvedD to their v—™uum exp e™t—tion v—luesF „he det—iled numeri™—l results

will ˜ e presented —t the end of this se™tionF „his —llows us to dire™tly ™omp—re the inuen™e

of the v—rious eldsF

sn refF‘WI“ the in™lusion of the &{meson vi— the ‡u{‰—ng —ns—tz

& a  r” q@r A @TFSTA

i— ik — k

h—s ˜ een studiedF ix™ept for the pseudos™—l—r eldD for whi™h the hedgehog sh—p e @TFRIA

w—s —doptedD —ll other elds were set to their v—™uum exp e™t—tion v—luesF elthough this SV

™ongur—tion do es not exhi˜it glo˜—l ™hir—l inv—ri—n™e it —llows one to investig—te the —ttr—™tive

™h—r—™ter of the &{mesonF sndeedD the size of the solitonD —s me—sured ˜y the ˜—ryoni™ ro ot

me—n squ—red r—diusD w—s found to de™re—seF purthermore the v—lue for mD —˜ ove whi™h soliton

solutions do existD de™re—sed to PUHwe†F st should ˜ e stressed th—t the in™orp or—tion of the

&{meson neither —lters —ny of the p—r—meters involved nor intro du™es —ddition—l onesF „hisD

howeverD is not the ™—se when —lso the — {meson is in™ludedF es the % — mixing ™h—nges

I I

the rel—tion ˜ etween the pion de™—y ™onst—ntD f D —nd the ™ut{oD D the v—lue of  for —

%

given ™onstituent qu—rk m—ssD mD is signi™—ntly in™re—sed ‘QI“D see —lso eq @RFSQAF por det—ils

the re—der is referred to ™h—pter RF xeedless to rem—rk th—t the in™lusion of the — meson is

I

m—nd—tory to preserve ™hir—l symmetryF „he —ns—tz for the — meson whi™h is ™onsistent with

I

the gr—nd spin zero —ssumption for the st—ti™ elds —nd exhi˜its the pseudove™tor ™h—r—™ter

involves two r—di—l fun™tions

— a  r @r A Cr ” r” p @r AX @TFSUA

Ii— i— i —

ƒelf{™onsistent solutions were found for m ! QHHwe† ‘WP D WQ“F prom the equ—tion of motion

for r @r A @™fF eq @gFWAA one e—sily veries th—t r @HA is non{v—nishingF „hus the —xi—l{ve™tor

meson h—s — dire™t inuen™e on the v—len™e qu—rk w—ve{fun™tion whi™h is lo ™—lized —t the

originF es — resultD the energy of the v—len™e qu—rk st—te is signi™—ntly lowered —nd —™tu—lly

˜ e™omes neg—tiveF prom the dis™ussion in se™tion TFI it is then o˜vious th—t the tot—l ˜—ryon

™h—rge is ™—rried ˜y the p ol—rized hir—™ se— —nd no or˜it is expli™itly o ™™upiedD iFeF —ll   HD

#

"

in the unit ˜—ryon num˜ er se™torF „he ˜—ryon ™h—rge is solely due to j in eq @TFIIAF „his

H

result strongly supp orts the ƒkyrmion pi™ture of ˜—ryons —s ™—n ˜ e seen from the following

"

™onsider—tionsF „he gr—dient exp—nsion of j yields in le—ding order the top ologi™—l ™urrent

H

e

@RFTSA ‘WR “ whi™hD in the ƒkyrme mo delD is identied with the ˜—ryon ™urrent F e priori the

p oint of view th—t — distorted hir—™ se— is resp onsi˜le for — non{v—nishing ˜—ryon ™h—rge is

integr—ted in the ƒkyrme mo delF „his p oint of opinion is ™ommonly referred to —s ‡itten9s

™onje™tureF y˜viouslyD it is supp orted ˜y the xtv mo del ‘WP “F purthermoreD the ƒkyrmion

phenomenology h—s est—˜lished th—t short r—nge ee™ts —re des™ri˜ ed ˜y either expli™it qu—rk

degrees of freedom or @—xi—lA ve™tor mesons ‘WSD WT “Y the ’or4 ˜ eing ex™lusiveF sn the xtv

mo delD howeverD this m—tter of opinion represents — dire™t result3

p to now the 3 {meson h—s ˜ een left outD mostly for te™hni™—l re—sonsF es dis™ussed

in se™tion TFI it intro du™es — non{rermiti—n hir—™ r—miltoni—n @™fF eq @TFTQAA —ndD more

trou˜lesomeD in — non{p ertur˜—tive —ppro—™h — unique extr—™tion of — winkowski energy fun™E

tion—l is not —v—il—˜leF roweverD the in™orp or—tion of this meson is extremely desir—˜le from

the p oint of ve™tor meson domin—n™e @the 3 {meson h—s signi™—nt inuen™e on the isos™—l—r

r—diusA —nd the ™urrent eld identities @QFPIA @the isos™—l—r p—rt of the ele™trom—gneti™ ™urrent

is prop ortion—l to the 3 eldAF eddition—lly the repulsive ™h—r—™ter of the 3 {meson is exp e™ted

to provide st—˜ility even when the s™—l—r eld is —llowed to v—ry in sp—™eF ‡e will therefore

rep ort on investig—tions whi™h —re ˜—sed on the physi™—lly motiv—ted denition @TFQWA for the

winkowski energy fun™tion—lF

pirst —ttempts in this dire™tion h—ve ™on™erned the % 3 system with —ll other elds put

to their v—™uum exp e™t—tion v—lues ‘UV D UT “F ynly — r—di—l fun™tion for the time{™omp onent

of the isos™—l—r ve™tor meson is —llowed ˜y the gr—nd spin symmetry

3 a 3 @r A X @TFSVA

" "R

e

„he top ologi™—l ™urrent @RFTSA ™—n —lso ˜ e o˜t—ined —s the xo ether ™urrent —sso ™i—ted with the singlet

ve™tor symmetry on™e the ƒkyrme mo del is extended to —vor ƒ @QA —nd the ‡ess{umino term is in™ludedF SW

es the r—miltoni—n is non{rermiti—n we h—ve to distinguish ˜ etween left —nd right eigenst—tes

y 

~ ~ ~ ~

hj i a  j i h jh a  h j i Xe X h j i a  j i @TFSWA

# # # # # # # #

#

~

with the norm—liz—tion ™ondition h j i a  F sn gener—l it is p ossi˜le to sep—r—te the

" # "#

non{rermiti—n p—rt ˜y writing

h a h C i3 @TFTHA



with h ˜ eing rermiti—nF essuming the ˜—sis @fFRDfFSA it is e—sy to see th—t the m—trix





~

elements of 3 —re re—l —nd symmetri™F „his further implies j i a j iF st is then useful

#

#

‚ s

to de™omp ose the eigenst—tes into re—l —nd im—gin—ry p—rts j i a j i C ij iF rere the

#

# #

sup ers™ript refers to re—l @‚A —nd im—gin—ry @sA p—rts of the exp—nsion ™o e™ients † in eq

# k

‚ s

~

@fFTAF „—king —™™ount of the f—™t th—t j i a j i ij i —llows one to extr—™t re—l —nd

#

# #

im—gin—ry p—rts of the one{p—rti™le energy eigenv—lues

‚ ‚ ‚ s s s ‚ ‚ s

 a h jh j i h jh j i h j3 j i h j3 j iY

 

# # # # # # # # #

s ‚ ‚ s s s ‚ ‚ s

 a h j3 j i h j3 j i C h jh j i C h jh j iX @TFTIA

 

# # # # # # # # #

@‚Ys A

„hese rel—tions —re suit—˜le to ev—lu—te the fun™tion—l deriv—tives of  with resp e™t to the

#

meson eldsF iFgF one nds



s

 

d  

P # ‚ ‚ s s

a r hr j ih jr i hr j ih jr i X @TFTPA

# # # #

 3 @r A R%

ixpressions like this enter the equ—tions of motion for the soliton prole fun™tions @™fF —pE

p endix gAF es ™—n ˜ e inferred from the equ—tion of motion for the 3 {meson @gFTA this prole

fun™tion is dire™tly rel—ted to the ˜—ryon densityF es — m—tter of f—™t this ™—uses — non{

‚

P Q P

is xed when e is not aRm v—nishing 3 eldF sn p—rti™ul—rD the integr—l d r 3 a g

s

† †

regul—rized —nd s—tises the norm—liz—tion ™ondition on the 3 eld imp osed ˜y the ™urrent

eld identities @QFPIAF

xumeri™—llyD solutions h—ve ˜ een found for m ! QSHwe†D —lthough the determin—tion of —

lower ˜ ound h—s not ˜ een the ™entr—l issue in refF‘UT“F wore imp ort—ntly it h—s to ˜ e rem—rked

th—t these solutions were o˜t—ined for the physi™—l 3 meson m—ssD m a UUHwe†F „his is in

3

f

™ontr—st to the tre—tment dis™ussed in refF‘UV“ where st—˜le solutions —pp e—red only when m

3

w—s ™hosen —˜ out four times —s l—rgeF sn su˜se™tion TFPFR we give — p ossi˜le expl—n—tion for

the non{existen™e of solutions in th—t tre—tmentF

xext it should ˜ e noted th—t the v—lue of the ™onstituent qu—rk m—ss m —t whi™h the

v—len™e qu—rk energy ™h—nges its sign is —t SRSwe† —nd thus ™onsider—˜ly lower th—n in the

purely pioni™ systemF „his result is e—sy to underst—nd sin™e the repulsive ™h—r—™ter of the

3 meson yields — l—rger extension of the soliton prole whi™h ™—uses the v—len™e qu—rk to

˜ e more strongly ˜ oundF „his repulsive ee™t is —lso o˜served when ev—lu—ting the ˜—ryoni™

r—dius —lthough its extr—™tion is somewh—t h—mp ered ˜y nite size ee™tsF „hese ee™ts do

not show up in the determin—tion of the energyD howeverD higher moments of the 3 prole

fun™tion m—y ˜ e o˜s™uredF st h—s —lso ˜ een —s™ert—ined th—t regul—riz—tion of the im—gin—ry

p—rt of the —™tion do es not le—d to qu—lit—tive ™h—nges of the —˜ ove presented resultsF

sn the next step —ll ve™tor mesons were in™luded ‘UQ “F „his em˜r—™es the —ns—tze @TFSTDTFSUA

—nd @TFSVAD only the isos™—l—r{s™—l—r eld ˜ eing kept —t its v—™uum exp e™t—tion v—lueF „he

f

ƒee —lso se™tion TFI for the dis™ussion of this —ppro—™hF TH

—˜ ove dis™ussed stru™ture of the iu™lide—n hir—™ r—miltoni—n is not —lteredD howeverD it is

more ™omplex ˜ e™—use — l—rger num˜ er of elds is involvedF sn this ™—se the interesting

question —rises whether or not ‡itten9s ™onje™tures rem—ins v—lidF „he previous explor—tions

seem to indi™—te th—t ee™ts on the v—len™e qu—rk energy eigenv—lue of the isove™tor —nd

isos™—l—r @—xi—lA ve™tor mesons —dd up ™oherentlyF xumeri™—lly self{™onsistent solutions to

this extended pro˜lem h—ve ˜ een found in the interv—l QHHwe† m RHHwe† —lthough the

—uthors of refF‘UQ“ do not ex™lude the existen™e of solutions in — l—rger r—ngeF sndeed — further

‚

de™re—se of the v—len™e qu—rk energy eigenv—lueD  h—s ˜ een o˜servedD eFgF for m a QSHwe†

v—l

‚

 ™h—nges from EIQQwe† to EISRwe† when supplementing the % & — system ˜y the 3

I

v—l

mesonF „hisD howeverD is only the ™—se when the im—gin—ry p—rt of the —™tion is su˜ je™t to

‚

regul—riz—tionY when this regul—riz—tion is dis™—red  in™re—ses even slightly —s —n ee™t of

v—l

in™luding 3 F yn the wholeD howeverD the inuen™e of regul—rizing the im—gin—ry p—rt h—s ˜ een

found to ˜ e sm—llF

†ery re™ently the energy fun™tion—l @TFQWA h—s ˜ een studied for the ™—se th—t —lso the

isos™—l—r{s™—l—r eld is sp—™e dep endent ‘US“F „his intro du™es the —ddition—l r—di—l fun™tion

0@r AD ™fF su˜se™tion TFPFPF es this represents the most gener—l ™—se we use this opp ortunity to

displ—y the —sso ™i—ted iu™lide—n hir—™ r—miltoni—n

”

h a 
p C i3 @r AC m0@r A @™os @r AC i (
r sin@r AA

S

I I I

” ” ”

@'
r A@(
r Ap @r AC @'
( Ar @r AX @TFTQA @  r A
( q@r AC C

P P P

es long —s the s™—l—r eld 0 h—s not ˜ een in™luded —s — dyn—mi™—l degree of the freedom

the question of whether or not regul—rizing the im—gin—ry p—rt only pl—yed — su˜le—ding roleF

„—kingD howeverD 0 to ˜ e sp—™e dep endent le—ds to — ™ompletely dierent situ—tionF es —lE

re—dy seen in su˜se™tion TFPFP — eld ™ongur—tion whi™h is sh—rply p e—ked —t r a H yields

— v—nishing soliton energyF „his rem—ins true even when ve™tor mesons —re in™luded —s long

—s no me™h—nism prevents these ve™tor mesons from ˜ eing zeroF „here is no su™h me™h—nism

for the st—ti™ & —nd — elds ˜ut the st—ti™ 3 eld is dire™tly ™orrel—ted to the ˜—ryon num˜ er

I

densityF nfortun—telyD for the sh—rply p e—ked eld ™ongur—tion @TFSIA —lso the regul—rized

˜—ryon num˜ er v—nishes —nd thus —lso 3  H is —llowed in this ™—seF

b

‡hen the im—gin—ry p—rt is not regul—rized st—˜le solutions h—ve ˜ een o˜t—ined for m QSH

$

we† ‘US “F „hen the 3 meson provides su™ient repulsion to prevent the s™—l—r eld from

™oll—psingF „he lower ˜ ound for m is somewh—t l—rger @RIHwe†A when & —nd — elds —re

I

ignoredF e™tu—lly the devi—tion of the s™—l—r eld from its v—™uum exp e™t—tion v—lues is —lmost

negligi˜leF „he —ltern—tive metho d of st—˜iliz—tion @TFSRA h—s not ˜ een studied for the ™—se

th—t @—xi—lA ve™tor mesons —re in™ludedF

sn t—˜le TFP we summ—rize the numeri™—l results for the v—rious ™—ses dis™ussed —˜ oveF

„here the sp e™i™ ™ontri˜utions to the energy —re ™omp—red for the ™onstituent qu—rk m—ss

m a QSHwe†F

pigure TFS is well suited to dis™uss the ee™ts of v—rious mesons elds on the ™hir—l —ngleF

yne o˜serves th—t the & meson is repulsive —t sm—ll dist—n™es —nd —ttr—™tive for r ! HXQfmF e

mu™h stronger —ttr—™tion is provided ˜y the —xi—l{ve™tor mesonF elso the repulsive ™h—r—™ter

`

of the 3 {meson is repro du™ed ˜y the xtv mo delF sn the inner region of the soliton IXSfm

$

the s™—l—r eld provides — sm—ll repulsion while —t l—rger dist—n™es @not shown in gure TFSA

the s™—l—r eld ™—uses the ™hir—l —ngle to swell slightlyF elthough this —ttr—™tion —pp e—rs to ˜ e

quite we—k it indi™—tes th—t the xtv soliton with s™—l—r mesons ™ont—ins —t le—st some interE

medi—te r—nge —ttr—™tion ™—lled for ˜y the ™entr—l p otenti—l of the nu™leon{nu™leon inter—™tionF

nfortun—telyD no qu—ntit—tive st—tement on this su˜ je™t ™—n ˜ e m—de presently in the xtv TI

„—˜le TFPX „he soliton energy for v—rious tre—tments of the xtv solitonF „he meson elds listed

in the rst line represent those meson proles whi™h —re —llowed to devi—te from their v—™uum

exp e™t—tion v—luesF ell num˜ ers —re ev—lu—ted for — ™onstituent qu—rk m—ss m a QSHwe† —nd

m a IQSwe†F

%

% %Y& % Y 3 % Y &Y — % Y 3 Y &Y — 0Y % Y 3 Y &Y —

I I I

i @we†A IPQT WTT IRHR IHII IIRR IIPS

‚

i @we†A RSW SVR SRT TIT SSV PPUI

H

s

i @we†A H H EIR H IVT IUU

H

i @we†A QI IUQ H QWS RHH EIQPQ

m

‚

 am HFUI HFPH HFST EHFQU EHFPU EHFPV

v—l

s

 am H H HFPU H HFIT HFIS

v—l

mo delF

TFPFR u—dr—ti™ exp—nsion for time ™omponents of ve™tor elds

„he pre™eding dis™ussions ™on™erning the —n—lyti™ ™ontinu—tion of the —™tion in the presE

en™e of the 3 meson were ˜—sed on the —n—lysis of the —n—lyti™ prop erties of the eigenv—lues

of the iu™lide—n hir—™ r—miltoni—n h @TFTQAF st h—s ˜ een —ssumed th—t the exp—nsion of

these eigenv—lues in p owers of the 3 {eld ™—n str—ightforw—rdly ˜ e tr—nsferred to the —™tion

fun™tion—lF roweverD it h—s re™ently ˜ een demonstr—ted th—t this is —™tu—lly not the ™—se

y

‘UI “F sn order to prove the identity one h—d to —ssume th—t h —nd h ™—n ˜ e di—gon—lized

simult—neouslyF ƒt—rting o —t

y

P P P

ha ha a d C h CPi3 d C i ‘h Y 3 “ C 3 @TFTRA

i R (  R

i

(  R

—nd imp osing the prop er{time pres™ription —t the op er—tor level



o n

I

ds I

P P P

@TFTSA CPi3 d C i ‘h Y 3 “ C 3 C h exp d „r e 3

R (  R ‚

R  (

P

s P

Ia

the energy fun™tion—l @TFPSA ™—n only ˜ e o˜t—ined with the —˜ ove mentioned —ssumption of

 

y

d F yne simult—neous di—gon—liz—˜ility ˜ e™—use the term line—r in d origin—tes from h h

( (

s

m—y o˜t—in eq @TFPSA from @TFTSA ˜y su˜stituting the ™o e™ient of d with Pi F „he ˜ efore{

(

#

mentioned —m˜iguitiesD whi™h —rise from the in—dequ—te —ppli™—tion of the rules for m—nipuE

l—ting the log—rithm to derive @TFPRAD —re —voided when st—rting from @TFTSA without further

—ssumptionsF st is —lso o˜vious th—t ˜ e™—use of these —ssumptions dierent exp—nsion s™hemes

will le—d to dierent resultsF st is the purp ose of the present se™tion to p oint out the dieren™es

˜ etween the —ppro—™h whi™h relies on exp—nding the eigenv—lues  of h on the one h—nd —nd

#

the exp—nsion of the op er—tor @TFTSA on the otherY ˜ oth exp—nsions —re understo o d in terms

of 3 F

R

imploying te™hniques whi™h h—ve ˜ een worked out in the ™ontext of the semi{™l—ssi™—l

qu—ntiz—tion of the soliton @se™tion UFIA ‘UH“ —nd ˜ een extended to the tre—tment of sm—ll

—mplitude u™tu—tions o the soliton @see se™tion UFQA ‘WU“ one o˜t—ins — winkowski energy TP

pigure TFSX „he prole fun™tion @r A for the self{™onsistent ™hir—l —ngle in v—rious —ppro—™hes

to the xtv mo delF „hose elds whi™h —re —llowed to ˜ e sp—™e dep endent —re indi™—tedF „he

p—r—meters used —re m a QSHwe† —nd m a IQSwe†F

%

fun™tion—l up to se™ond order in the 3 eld ‘UI“

 

3 2

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rere i ‘“ refers to the energy fun™tion—l —sso ™i—ted with the rermiti—n p—rt of the r—milE

H

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toni—n h F purthermore  —nd j# i denote the eigenv—lues —nd {ve™tors of h F elso the m—ss

 

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terms —sso ™i—ted with the & —nd — mesons —re in™luded in i ‘“F „he im—gin—ry p—rt of the

I H

—™tion h—s ˜ een —ssumed in regul—rized formF sf one wishes to —˜—ndon this regul—riz—tion the

™omplement—ry error fun™tion in @TFTTA h—s to ˜ e repl—™ed ˜y unityF pin—lly the regul—riz—tion

fun™tion

   

   

    

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h—s — smo oth limit ‘UH“

f @ Y  Y A a HX @TFTVA lim

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 3

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st is —™tu—lly ˜ e™—use of this limit th—t the ™onsisten™y ™onditions for energy fun™tion—lD whi™h

—re dis™ussed in se™tion TFID —re s—tised ˜y the fun™tion—l @TFTTAF es w—s to ˜ e exp e™ted this

regul—tor fun™tion is identi™—l to the one for the moment of inerti— ‘UH “F sn ˜ oth ™—ses the re—l TQ

pigure TFTX veftX „he tot—l energy —s — fun™tion of the s™—ling p—r—meter ! dened in eq

@TFTWA for v—rious v—lues of the ™onstituent qu—rk m—ss mF ‚ightX „he prole fun™tions whi™h

minimize the energy fun™tion—l @TFTTA under v—ri—tion of the s™—ling p—r—meter !F „he qu—rk

y

˜—ryoni™ density ˜ $ q q is —rti™i—lly s™—led su™h th—t the sp—ti—l integr—ls over 3 am —nd

˜ ™oin™ideF rere the ™onstituent qu—rk m—ss is —ssumed to ˜ e SHHwe†F @pigure t—ken from

refF‘UI“FA

p—rt of the —™tion is exp—nded in terms of st—ti™ —nd —nti{rermiti—n op er—tors @in the ™—se of

the moment of inerti— these —re the isospin gener—torsAF

sn refF‘UI“ sever—l p—r—metri™—l des™riptions h—ve ˜ een —dopted for the proles other th—n

3 @r A —nd su˜sequently the l—tter h—s ˜ een determined from the st—tion—ry ™ondition ™orreE

sp onding to @TFTTAF es —n ex—mple we quote the s™—ling —ns—tz

 @r A a  @!r A @TFTWA

! sX™X

for the mo del ™onsisting only of the pion —nd the 3 mesonF sn eq @TFTWA  @r A refers to

sX™X

the self{™onsistent soliton solution to the pro˜lem without the 3 mesonF ƒu˜sequently the

st—tion—ry ™ondition —sso ™i—ted with the fun™tion—l @TFTTA w—s solved in order to o˜t—in the

prole fun™tion 3 @r AF „his pro ™edure of rst p—r—metrizing the ™hir—l —ngle —nd su˜sequently

!

solving for the 3 prole ex—™tlyD is —lso justied ˜y the f—™t th—t ™ommonly the st—tion—ry

™ondition for 3 @r A represents — ™onstr—int r—ther th—n only —n iuler{v—gr—nge equ—tion of

motionF „hen the prole fun™tions —s well —s the energy fun™tion—l dep end on the p—r—meter

!F „his dep enden™e is displ—yed in gure TFTF „his gure indi™—tes th—t the fun™tion—l @TFTTA

b

indeed p ossesses — unique lo ™—l minimum for ™onstituent qu—rk m—sses m RHHwe†F „his

$

gure —lso ™ont—ins the prole fun™tion whi™h minimize @TFTTA when m a SHHwe† is —doptedF

y˜viously the 3 prole gets supp ort —t l—rger dist—n™es th—n do es the qu—rk ˜—ryon ™urrent

y

˜ $ q q F „his ™onrms the repulsive ™h—r—™ter of the 3 mesonF „he minim—l energy v—lues —re

given —s fun™tions of m in t—˜le TFQF elso shown —re the ™ontri˜utions due to the line—r —nd

qu—dr—ti™ terms in 3 F xote the viri—l f—™torD EPD ˜ etween these two pie™esF „he presen™e of the TR

wink

„—˜le TFQX ƒoliton energy i @TFTTA when  —nd 3 —re the only sp—™e dep endent elds

P

for v—rious ™onstituent qu—rk m—ssesF elso shown —re the ™ontri˜utions from the terms line—r

@linFA —nd qu—dr—ti™ @qu—dFA in 3 F ell num˜ ers —re in we†F @‚esults —re t—ken from refF‘UI“FA

wink

m i linF qu—dF

P

RHH ISVS TSH EQPS

RSH ITVS UQR EQTU

SHH IUTW VST ERPV

3 meson then in™re—ses the soliton energy ˜y —˜ out QHH RHHwe†F „his ™ontri˜ution grows

with the ™onstituent qu—rk m—ssF st should ˜ e noted th—t the qu—dr—ti™ term splits into two

pie™esD one from the purely mesoni™ p—rt of the —™tion @PWTwe† for m a SHHwe†A —nd one

stemming from the exp—nsion of e @IPWwe† for m a SHHwe†AF „he sm—llness of the l—st

‚

qu—ntity ™omp—red to the tot—l energy gives eviden™e for the f—™t th—t this exp—nsion of e

‚

in terms of 3 ™onverges qui™klyF purthermoreD it h—s ˜ een shown th—t in™luding the 3 meson

R

vi— the exp—nsion @TFTTA indeed provides — siz—˜le repulsionD — fe—ture ™ommonly —ttri˜uted

to this meson eldF „his is expressed ˜y the f—™t th—t the v—lueD whi™h ! —ssumes —t the

minimum of the energyD is signi™—ntly sm—ller th—n oneF

„he —ppro—™h h—ving just ˜ een des™ri˜ ed h—s to ˜ e ™omp—red with —n exp—nsion of the

eigenv—lues of the whole hir—™ r—miltoni—n h C i3 @TFTHA in iu™lide—n sp—™eF epplying

 R

st—nd—rd p ertur˜—tion te™hniques results in

ˆ

h# j3 j"ih"j3 j# i

R R

s H ‚

a ih# j3 j# i XXX X @TFUHA XXX —nd  a  

R

# # #

H H

 

# "

"Ta#

ƒu˜stitution of this exp—nsion into the expression for re—l —nd im—gin—ry p—rts of the iu™lide—n

energy @TFPU —nd TFQTA —nd su˜sequent ™ontinu—tion to winkowski sp—™e yields —t se™ond order

—n energy fun™tion—l simil—r to @TFTTAD howeverD with f @ Y  Y A repl—™ed ˜y

" #

V



"



#

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sgn@ Aerf™ sgn@ Aerf™

# @j jA " @j jA

  I

Y  Ta 

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P   f @ Y  Y A a X @TFUIA

# "

# "

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~

‡e will refer to the ™orresp onding energy fun™tion—l ˜y i F sn refF‘UI“ it h—s ˜ een demonstr—ted

P

~

th—t for — well motiv—ted eld ™ongur—tion i devi—tes only ˜y —˜ out Q7 from @TFQUA whi™h

P

results from di—gon—lizing @TFTHAF

~

roweverD the regul—tor fun™tions f —nd f dier signi™—ntlyF pirst of —llD they only —gree

in the limit  3 IF es the ™ut{o  is quite sm—ll this dieren™e is siz—˜le numeri™—llyF xext

~

it h—s to ˜ e noted th—t f is dis™ontinuous —s  3  F „his h—s the unple—s—nt ™onsequen™e

" #

th—t the 3 ™ontri˜ution to the energy is not p ositive denite for ™ongur—tions whi™h s—tisfy

~

the st—tion—ry ™onditionF purthermoreD the se™ond order ™ontri˜ution —sso ™i—ted with f is

SSQwe†Y to ˜ e ™omp—red with the —˜ ove mentioned IPWwe† for f F wost imp ort—ntlyD

~

howeverD i do es not p ossess — lo ™—l minimumF „his expl—ins why for empiri™—l p—r—meters

P

the —uthors of refF‘UV“ did not nd — self{™onsistent solution in the mo del ™ont—ining only the

pion —nd the 3 degrees of freedomF

„hese investig—tions demonstr—te th—t ™ounting p owers of 3 in the iu™lide—n energy fun™E

R

tion—l @TFQUA m—y not ˜ e the ™orre™t —ppro—™h to derive — winkowski energy fun™tion—l in TS

the presen™e of isos™—l—r eld 3 F ‚—ther the fun™tion—l @TFTTA —pp e—rs to ˜ e the —ppropri—te

st—rting p ointF „his st—tement is —lso supp orted ˜y previous ™—l™ul—tions in the ™ontext of

the semi{™l—ssi™—l qu—ntiz—tion of the ™hir—l soliton —s well —s other nu™leon o˜serv—˜les @see

™h—pter UAF „hese ™omput—tions —ll imp ose the prop er{time regul—riz—tion —t the —™tion level

in iu™lide—n sp—™e @TFTSAF

TFPFS vo™—l ™hir—l rot—tion

~

sn se™tions QFR —nd RFP the ee™tiveness of the ™hir—l tr—nsform—tion a „ with „ a

€ $ C € $ h—s ˜ e™ome evident when exploring the meson se™tor of the xtv mo delF sn

v v ‚ ‚

p—rti™ul—rD it provides the link ˜ etween the m—ssive ‰—ng{wills —nd hidden g—uge —ppro—™hes

to ve™tor mesonsF st is therefore interesting to illumin—te the role of this tr—nsform—tion in

y

a $ will ˜ e —doptedF „his the soliton se™tor —s wellF por this study the unit—ry g—uge $ a $

‚

v

”

implies „ @A a ™os@aPA C i (
r sin@aPAF „he hir—™ r—miltoni—n for the ™hir—lly rot—ted

S

~

qu—rk elds then ˜ e™omes ‘QPD WV “

 

I I

H y

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r A@(
r A  @r A sin @r A h a „ @Ah„ @A a 
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@r  ( A sin @'
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eg—in one o˜serves th—t the ™hir—l eld h—s ˜ een elimin—ted from the m—ss term —t the exp ense

of indu™ed ve™tor mesonsF fefore dis™ussing the physi™—l impli™—tions of this tr—nsform—tion

in the soliton se™tor — few rem—rks on its te™hni™—l fe—si˜ility —re in orderF pirst of —llD one

o˜serves th—t the ™o ordin—te singul—rities do not dis—pp e—r —t r a H —s long —s @r a HA a l % F

„hese singul—rities —pp e—r sin™e „ @A is top ologi™—lly distin™t from the unit tr—nsform—tionF

por „ @PA these ™o ordin—te singul—rities —re —˜sentD howeverD the sp e™trum o˜t—ined from

y % C

di—gon—lizing „ @PAh„ @PA in — nite ˜—sis is not identi™—l to th—t of hF sn the q a H

™h—nnel the lowest st—te is missing while —n —ddition—l one shows up —t the upp er ˜ ound—ry

in momentum sp—™eF „his shift of eigenst—tes is rep e—ted for e—™h n in „ @PnAD with n ! PF

ren™eD one o˜serves —nother ree™tion of the top ologi™—l ™h—r—™ter of the tr—nsform—tion „ F

ƒe™ondlyD one should remem˜ er th—t for the di—gon—liz—tion @TFRQA ™ert—in ˜ ound—ry ™onditions

on the eigenst—tes h—ve ˜ een imp osed @™fF the dis™ussion —fter eq @fFTAAF st should ˜ e o˜vious

th—t these ˜ ound—ry ™onditions —re —e™ted ˜y the ™hir—l rot—tion „ whi™h is not uniquely

dened —t r a HF ‡ith the ™orresp onding redenition of the ˜—sis spinors the ™o ordin—te

singul—rities —re ™—n™eledF „he numeri™—l di—gon—liz—tion

~ ~

h a ~ @TFUQA

‚ # # #

y

~

then indeed yields ~ a  —nd a „ @A with  —nd ˜ eing the solutions to the

# # # # # #

eigenv—lue pro˜lem @TFRQAF

„his form—lism h—s in turn ˜ een employed to ex—mine the situ—tion when the ™hir—l rot—tion

is p erformed on the @—xi—lA ve™tor elds ‘QI“

y

~ ~

† C e a $ @d C † C e A$ Y

" " ‚ " " "

‚

y

~ ~

† e a $ @d C † e A$ X @TFURA

" " v " " "

v TT

„—˜le TFRX gontri˜utions to the energy for self{™onsistent solution in v—rious tre—tments of

the —xi—l{ve™tor meson in the xtv mo delF „hose meson elds whi™h —re —llowed to ˜ e sp—™e

dep endent —re indi™—tedF „he ™onstituent qu—rk m—ss m aRHHwe† is ™ommonF ell num˜ ers

—re in we†F @‚esults —re t—ken from refF‘WV“FA

~

% & % & — % †

I

 QIQ EPPP EQSI

v—l

det

i UII SRQ PRH

i IRW QWQ IUS

m

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tot

~

—nd the tr—nsformed —xi—l{ve™tor eld is negle™tedD iFeF e a HF „his —ns—tz do es not viol—te

"

the lo ™—l ™hir—l symmetry in ™ontr—st to the —ppro—™h e a HD ™fF eq @QFRPAF „his ™ongur—tion

"

is of sp e™i—l interest sin™e this —pproxim—tion h—s frequently ˜ een —pplied to extended ƒkyrme

g

mo dels F „he imp ort—nt question —rises whi™h @or whether —t —ll —nyA fe—tures of the unrot—ted

—xi—l{ve™tor eld @e A —re m—int—inedF por these studies the ee™ts of the 3 meson h—ve ˜ een

"

ignoredF efter ™—rrying over the ™hir—l rot—tion onto the qu—rk spinors only ve™tor meson

degrees of freedom —re ™ont—ined in the hir—™ r—miltoni—nF por these the ‡uE‰—ng —ns—tz h—s

˜ een —ssumed yielding ‘WV“

q@r A

”

h a 
p C  m C 
@r  ( AX @TFUSA

‚

Pr

„he ™oupling of the ve™tor —nd pseudos™—l—r elds is then ™ompletely ™ont—ined in the mesoni™

p—rt of the energy fun™tion—l ‘WV “

' &



I %

P

P

P H P

r @ @r AA C sin @r A X @TFUTA dr @q@r ACI ™os@r AA C i a

m

P q

P

y˜viously @r a HA a % implies th—t q@r a HA a PF yne thus h—s to de—l with — hir—™

r—miltoni—n whi™h ™ont—ins — top ologi™—lly non{trivi—l ve™tor meson eldF „hus @TFUSA is

singul—r —nd ™—nnot ˜ e tre—ted using the st—nd—rd ˜—sis ‘VI “ ˜ut r—ther ˜y employing te™hniques

whi™h —re —n—logous to those develop ed to di—gon—lize the rot—ted r—miltoni—n @TFUPAF

st h—s ˜ een demonstr—ted th—t self{™onsistent solutions whi™h minimize the tot—l energy

det det

exist ‘WV“F rere the tot—l energy is dened —s the sum i a i C i with i ˜ eing the

tot m

™ontri˜ution of the fermion determin—nt @TFPUA in terms of the eigenv—lues of @TFUSAF elso the

v—len™e qu—rk ™ontri˜ution h—s to ˜ e —dded if ne™ess—ry to —™™ommo d—te unit ˜—ryon num˜ erF

sn t—˜le TFR the results of these ™—l™ul—tions —re ™omp—red with two dierent tre—tments of the

—xi—l{ve™tor degrees of freedom in the unrot—ted formul—tionF ‚eg—rding the strong ˜inding

of the v—len™e qu—rk st—te it is ™le—r th—t this imp ort—nt ee™tD whi™h is ™ommonly —sserted to

~

e Ta HD is ret—ined in the mo del with e a HF purthermoreD the v—len™e qu—rk w—ve{fun™tions

" "

exhi˜its fe—tures whi™h —re —sso ™i—ted with — lo ™—lized —ntiqu—rk —s eFgF — domin—ting lower

™omp onentF nfortun—tely the tot—l energy ™omes out quite low m—king —n —ppli™—tion of

this mo del for the des™ription of ˜—ryons dou˜tfulF sn refF‘WV“ is h—s —lso ˜ een demonstr—ted

g

por — ™ompil—tion of relev—nt —rti™les see refF‘IP“F TU

th—t the sm—llness of the tot—l energy is linked to the missing repulsion in this mo delF „his is

—lso ree™ted ˜y the meson prole fun™tions whi™h p ossess only — very sm—ll sp—ti—l extensionF

eg—in this is —n ee™t origin—ting from the presen™e of —n —xi—l{ve™tor eld in the unrot—ted

fr—meD e Ta HF purthermore the ee™ts —sserted to the 3 {meson were mo deled ˜y the in™lusion

"

th

of — T order term @RFPPA usu—lly used in the ƒkyrme mo del to simul—te the 3 {mesonF „hen

— siz—˜le repulsion w—s o˜t—ined —nd the energy of the v—len™e qu—rk or˜it de™re—sed even

more to —pproxim—tely the neg—tive ™onstituent m—ss ‘WV“F „hus this tre—tment provides —

strong supp ort of ƒkyrme typ e mo dels whi™h rely on the —ssumption th—t the v—len™e qu—rk

h—s joined the neg—tive hir—™ se—F TV

U f—ryons

sn the pre™eding se™tions we h—ve ex—mined the soliton solutions in the xtv mo delF ix™ept

for the ˜—ryon num˜ er these solutions do not ™—rry the qu—ntum num˜ ers of physi™—l ˜—ryonsF

sn p—rti™ul—rD these solutions —re neither eigenst—tes of the —ngul—r momentum t nor of isospin

s F es these solitons h—ve v—nishing gr—nd spin @q a HA they represent line—r ™om˜in—tions of

st—tes with jt j a js jF sn order to des™ri˜ e physi™—l ˜—ryons the soliton ™ongur—tion h—s to

˜ e pro je™ted onto st—tes with go o d —ngul—r momentum —nd isospin qu—ntum num˜ ersF es —lE

re—dy mentioned in se™tion SFQ this is ™ommonly —™hieved ˜y intro du™ing ™olle™tive ™o ordin—tes

whi™h des™ri˜ e the orient—tion of the soliton in ™o ordin—te{ —nd isosp—™eF ƒu˜sequently these

™o ordin—tes —re ™—noni™—lly qu—ntized yielding the —ngul—r momentum —nd isospin op er—torsF

es the gr—nd spin symmetry of the hedgehog @TFRIA ™—uses the equiv—len™e of rot—tions in

™o ordin—te{ —nd isosp—™e only one set of ™olle™tive ™o ordin—tes is neededF por ™onvenien™eD one

™ho oses the isospin orient—tionF „his pro ™edure h—s rst ˜ een —pplied ˜y edkinsD x—ppi —nd

‡itten to the ƒ @PA ƒkyrmion ‘W“F sn the pro ™eeding se™tion we will expl—in the —n—logous

tre—tment for the xtv ™hir—l soliton of pseudos™—l—r elds ‘UH “F „he gener—liz—tion to three

—vors is more involved not only due to the f—™t th—t the moment of inerti— tensor @SFIVA is no

longer prop ortion—l to the unit m—trix ˜ut —lso ˜ e™—use of symmetry ˜re—king ˜ eing presentF

„wo ™omplement—ry —ppro—™hes @‘TP “ —nd ‘WW “A will ˜ e presented in su˜se™tion UFSF

UFI u—ntiz—tion of the ™hir—l soliton

„he time{dep endent ™olle™tive ™o ordin—tes des™ri˜ing the isospin orient—tion of the soliton

—re p—r—metrized with the help of —n P  P unit—ry m—trix ‚@tA

y

w @r Y tA a ‚@tAw @r A‚ @tA @UFIA

H

where w @r A denotes the st—ti™ hedgehog ™ongur—tion @TFRIAF „he s™—l—r{pseudos™—l—r p—rt

H

of e do es not ™ont—in —ny time deriv—tivesF „hus it is indep endent of ™olle™tive ™o ordin—tes

m

—s long —s symmetry ˜re—king is ignoredF „hen the dep enden™e of the —™tion on ‚@tA —nd its

time deriv—tives ™ompletely origin—tes from the fermion determin—nt

 !

y

„r log i da  € w @r Y tAC € w @r Y tA X @UFPA

v ‚

„he fun™tion—l tr—™e is most ™onveniently ev—lu—ted ˜y tr—nsforming to the —vor rot—ting

H

system q 3 q a ‚q ‘UH“

 !

y

„r log i da  € w @r Y tAC € w @r Y tA

v ‚



!

I

y

a „r log i da (
  € w @r AC € w @r A

v ‚

H

P

 

I

(
 h @UFQA a „r log id

t

P

whi™h intro du™es the —ngul—r velo ™ities

i d

y

(
 X @UFRA ‚@tA a ‚ @tA

P d t TW

sn eq @UFQA h denotes the st—ti™ hir—™ r—miltoni—n dened in eq @TFRPAF sn the —di—˜—ti™ —pE

proxim—tionD iFeF when the time deriv—tives of  —re negle™tedD one might dene —n intrinsi™

I

H

r—miltoni—n h a h C (
 whi™h ™ould form—lly ˜ e tre—ted —n—logous ly to the pre™eding ™—lE

P

™ul—tions —s long —s regul—riz—tion is ignoredF st is then o˜vious th—t the fermion determin—nt

H

™—n ˜ e de™omp osed into v—™uum —nd v—len™e qu—rk p—rtsF nfortun—telyD the eigenv—lues of h

—re not known —nd thus — p ertur˜—tion exp—nsion in  h—s to ˜ e ™—rried outF por the v—len™e

qu—rk p—rt this is — str—ightforw—rd —ppli™—tion of st—nd—rd p ertur˜—tion theory resulting in

Q

 

ˆ

„

v—l R

X @UFSA    C y  e a „ i C

i j † †

ij

P

iYj aI

v—l

sn — symmetri™ two —vor mo del only even p owers of  —re —llowedF  denotes the v—len™e

—˜

qu—rk ™ontri˜ution to the moment of inerti— ‘UH“

ˆ ˆ

x hv—lj( j"ih"j( jv—li x h"j( j# ih# j( j"i

g i j g i j

v—l

X @UFTA  a  a  @I  A

v—l " #

ij

  P   P

" v—l # "

"#

"Tav—l

„he l—tter equ—lity —rises ˜ e™—use only the v—len™e qu—rk or˜it is o ™™upiedF „he fermion

v—™

‘UH“F sn determin—nt gives —lso rise to — v—™uum ™ontri˜ution to the moment of inerti— 

ij

order to extr—™t this p—rt from the fermion determin—nt one —g—in h—s to ™ontinue to iu™lide—n

sp—™e —nd ™onsider the limit „ 3 IF sn this ™ontext it is imp ort—nt to note th—t  represents

the time ™omp onent of —n indu™ed ve™tor eldF „hus  h—s to ˜ e ™ontinued —s

 3 i @UFUA

i

with  ˜ eing rermiti—nF hue to isospin symmetry only even p owers of  —pp e—r in —n

i i

exp—nsionF „hus we only need to ™onsider the re—l p—rt of the fermion determin—ntF st is

useful to dene

 





H H y

 a exp @se@z AA @UFVA u @sY z A a exp s@d h A@d h A

( (

d 3iz

(

where we h—ve indi™—ted th—t the temp or—l p—rt of the tr—™e is su˜stituted ˜y — sp e™tr—l integr—l

over z for „ 3 IF „his —llows us to express the re—l p—rt of the fermion determin—nt

 

I I

I

dz ds

tr u @sY z A @UFWA e a

‚

P

s P% P

I Ia

in the prop er{time regul—riz—tionF sn @UFWA the tr—™e in™ludes sp—ti—l —nd intern—l degrees of

freedomF purthermoreD

 

P



I i



P H H y

(
 C ‘(
 Y h“ C h X @UFIHA e@z A a @d h A@d h A  a z C

i i ( (

d 3iz

(

P P

sn order to extr—™t the moment of inerti— we rst exp—nd u @sY z A up to se™ond order in 

i

—nd dene the ™o e™ient of the qu—dr—ti™ term —s



P



d u @sY z A



P

X @UFIIA  u @sY z A a

—˜

— ˜



d  d 

i i

 aH

i UH

„his deriv—tive m—y ˜ e expressed with the help of — peynm—n p—r—meter integr—l

!  ! ! 



 

I

( (

˜ —

H H P P

Y h u @s@I xAY z A iz ( C Y h u @sxY z A dx tr iz ( C tr u @sY z A a s

˜ —

—˜

P P

H

'  &

( (

˜ —

H

u @sY z A @UFIPA Y Ctr

P P

H P P

where u @sY z A a exp @s@z C h AA denotes the zeroth{order he—t kernelF „he sp e™tr—l integr—l

in @UFWA is of q—ussi—n typ e —nd m—y ˜ e ™—rried out str—ightforw—rdlyF „he rem—ining tr—™e is

p erformed using the eigenst—tes of h @TFRQAF „his then —llows one to —lso ™—rry out the peynm—n

p—r—meter integr—lF „he v—™uum p—rt of the moment of inerti— m—y n—lly ˜ e extr—™ted from

 

R

„

v—™ — ˜

e a „ i in the limit „ 3 I ‘UH “    C y 

‚ H

—˜ i i i

P

ˆ

x

g

v—™

f @ Y  Y Ah"j( j# ih# j( j"i @UFIQA  a

 " # — ˜

—˜

R

"#

with the ™utEo fun™tion @whi™h —™tu—lly is the s—me —s in @TFTUAA given ˜y

   

   

    

"  P P

#

@ aA @ aA

# "

sgn @ Aerf™ sgn @ Aerf™

   

" #

 e e

 

p

f @ Y  Y A a X @UFIRA

 " #

P P

%   P@  A

" #

# "

hue to isospin inv—ri—n™e no dire™tion in isosp—™e is distin™t —nd the tot—l moment of inerti—

is isotropi™

 

P v—l v—™

 ‘“ a  C  X @UFISA

—˜

—˜ —˜

P

 is — fun™tion—l of the ™hir—l —ngle  sin™e the eigenst—tes j# i —nd {v—lues  fun™tion—lly

#

P —

dep end on F xumeri™—l results for  m—y eFgF ˜ e found in refsF‘IHH D IHI “ F p to qu—dr—ti™

order in  the ™olle™tive v—gr—n gi—n v@ A in winkowski sp—™e is n—lly given ˜y

I

P P

 ‘“ X @UFITA v@ A a i ‘“ C

P

yne e—sily veries th—t the innitesim—l ™h—nge of the meson elds w @r Y tA under sp—ti—l

rot—tions m—y ˜ e written —s

•

d w @r Y tA

X @UFIUA ‘w @r Y tAY r  d “ a

d 

„hus the tot—l spin is given ˜y the xo ether ™h—rges

A @



•

d v d w d v@w Y d w A

"

P Q

a   X @UFIVA C hX™X a t a d r tr

•

d  d 

d w

„he hedgehog stru™ture of the soliton rel—tes isospin s —nd spin t vi— the —djoint represent—tion

 

I

y

h a of ‚ tr ( ‚( ‚

ij i j

P

Q

ˆ

h t X @UFIWA s a

ij j i

j aI

—

yur numeri™—l results —gree with refF‘IHH“ ˜ut dis—gree with refF‘IHI“F UI

sn this resp e™t the spin of the soliton m—y ˜ e reg—rded —s the isospin in the isorot—ting fr—me

—nd vi™e vers—F es in the ƒkyrme mo del the resulting ™olle™tive r—miltoni—n is qu—ntized —s

— rigid top

I I d v

P P

t a i C s @UFPHA v a i C r a t

P P

P P d 

whi™h yields — tower of ˜—ryons with identi™—l spin —nd isospin like eFgF the nu™leon or the

reson—n™eF ‡e will p ostp o ne the dis™ussion of numeri™—l results for the ˜—ryon m—sses

t @t CIA

w a i C to se™tion @UFSFIA where the —ppli™—tion of this —ppro—™h to three —vors

f

P

P

is dis™ussedF roweverD the qu—ntiz—tion @UFIVA is equ—lly imp ort—nt for the study of st—ti™

nu™leon prop erties whi™h will ˜ e dis™ussed in the next se™tionF „hen frequent use will ˜ e

m—de of one more rel—tion ˜ etween ™olle™tive ™o ordin—tes —nd op er—tors ‘W“

Q

h @UFPIA s t a

ij i j

R

P

whi™hD howeverD is v—lid only when s—ndwi™hed ˜ etween nu™leon st—tes ˜ e™—use then t a

P

s a QaRF

UFP ƒt—ti™ nu™leon prop erties

sn this se™tion we will dis™uss the results for nu™leon o˜serv—˜les su™h —s m—gneti™ moments

—nd ™h—rge r—dii @su˜se™tion UFPFIA —nd the —xi—l ™h—rge of the nu™leonD g @su˜se™tion UFPFPAF

e

es these qu—ntities ™orresp ond to ™ert—in moments of symmetry ™urrents the ™—l™ul—tions —re

p erformed in two stepsF pirstlyD these ™urrents —re ™onstru™ted from the xtv mo del —™tion

@QFPSAF ƒe™ondlyD m—trix elements of these ™urrents with resp e™t to nu™leon st—tes —re ev—lu—ted

using the —pp—r—tus develop ed in se™tion @UFIAF

„he results of su™h ™—l™ul—tions h—ve —lre—dy ˜ een rep orted in —nother review —rti™le ‘IHP“F

„herefore we only ˜riey rep ort —nd ™omment on these resultsF „he re—der m—y ™onsult th—t

referen™e for more det—ilsF

#

@xA —nd @xA —re extr—™ted ˜y intro du™ing extern—l g—uge elds — „he symmetry ™urrents j

#

identifying their line—r ™oupling to the meson eldsD see —lso eqs @QFIVA —nd @QFIWAF denotes

the symmetries under ™onsider—tionF „hese —reX @IA the ve™tor symmetry in ele™trom—gneti™

I I

dire™tion  a ( C —nd @PA the —xi—l symmetryF porm—lly we write

Q

P T





 e‘9Y — “

 &

#

@UFPPA j @xA a





@xA  —

#

aH —

"

where 9 denotes the set of meson elds involvedF sn the present ™—se this only refers to the

™hir—l eld @UFIAF „he g—uge elds — @xA only —pp e—r in the fermion determin—nt

#





 





# &

j @xA a „r log ha iw @r Y tAC —  

&



 — @xA

#

aH —

"



 



I 



y &

@UFPQA (
 h i‚  — „r log id  ‚ a



t

&



 — P @xA

#

aH —

" UP

where the tr—nsform—tion to the —vor rot—ting fr—me h—s ˜ een p erformed @™fF se™tion @UFIAAF

es demonstr—ted for the ˜—ryon num˜ er ™urrent @TFIIA the ™urrents —re —dditive in v—len™e

#

—nd v—™uum p—rtsF „he v—len™e qu—rk p—rt j @xA is o˜t—ined ˜y — p ertur˜—tive exp—nsion

v—lY

I

y

of the single p—rti™le eigenv—lues of h C (
 C ‚  —  ‚ up to line—r order in ˜ oth  —nd

&

&

P

— F „his ends up to

&

4



y & #

hv—l j‚  —  ‚jv—l i j @xA a  x

v—l g

& v—lY

 — @xA

#



5

y &



o n

ˆ

hv—lj( j"ih"j‚  —  ‚jv—li

I

—

 &

@UFPRA   Y C

—



P  

" v—l

"Tav—l

aH —

"

where ’v—l4 l—˜ els the v—len™e qu—rk or˜itF „he ev—lu—tion of the v—™uum p—rt of the ™urrents

#

j @xA pro ™eeds in — f—shion simil—r to the ™omput—tion of the moment of inerti— ‘UH “ @see

v—™Y

se™tion UFIAD iFeF we —g—in h—ve to ™ontinue to iu™lide—n sp—™e —nd ™onsider the limit „ 3 IF

roweverD in the present ™—se —lso the im—gin—ry p—rt of the iu™lide—n —™tion ™ontri˜utesF

fesides the —ngul—r velo ™ities  —lso the time ™omp onent of the extern—l g—uge eld h—s to

R

rermiti—nF purthermoreD in order to —pply the prop er time D with — 3 i— ˜ e ™ontinued —s —

R H

regul—riz—tion s™hemeD one h—s to distinguish ˜ etween re—l —nd im—gin—ry p—rts of the fermion

determin—ntF por the re—l p—rt the regul—riz—tion is dened in eq @TFPSA while for the im—gin—ry

p—rt — pro ™edure —n—logous to @TFQSA is employedF „he p—rt of the fermion determin—nt whi™h

k

do es not ™ont—in the —ngul—r velo ™ity re™eives ™ontri˜utions prop ortion—l to — @k a IY XXY QA

R

from the re—l p—rt —nd prop ortion—l to — from the im—gin—ry p—rtF „—king into —™™ount th—t

(
 @UFUA —lso ˜ eh—ves like — time ™omp onent of — ve™tor eld one nds th—t the p—rts

i

R —

stemming from the re—l p—rt — whi™h —re line—r in the iu™lide—n —ngul—r velo ™ity h—ve 

i

k —

from the im—gin—ry p—rtF pin—llyD the tot—l expression for the v—™uum p—rt of the — —nd 

i

™urrent is ™ontinued ˜—™k to winkowski sp—™e

4

 

 

ˆ

 

x  

g "

# & y

 

@xA a j  ‚j"i h"j‚  — sgn @ Aerf™

"

v—™Y &

 

P  — @xA 

"

#



5

& '



ˆ

I



& y

X @UFPSA f @ Y  Y A  ‚j"i  Y h"j( j# ih# j‚  — 

 " # — —

&



P

"#

aH —

"

„he regul—tor fun™tion f is dened in eq @UFIRAF sts —pp e—r—n™e in @UFPSA is not —™™ident—l ˜ut



r—ther gu—r—ntees the prop er norm—liz—tion of the nu™leon ™h—rge when the qu—ntiz—tion rule

@UFIVA is employedF „he expressions @UFPRDUFPSA were o˜t—ined in refF ‘IHI “ for the ev—lu—tion

of m—gneti™ moments —nd the —xi—l ™h—rge of the nu™leonF

UFPFI ile™trom—gneti™ properties of the nu™leon

eXmX

sn order to extr—™t the ele™trom—gneti™ ™urrent we put — @xA a — a — @xA where 

"

# #

is dened —˜ ove eq @UFPPAF y˜viously  ™ont—ins isove™tor $ ( aP —nd isos™—l—r $ IaT p—rtsF

Q

"

"

@xA —nd @xA ™urrent m—y ˜ e de™omp osed into isove™tor j e™™ordinglyD the ele™trom—gneti™ j

† eXmX

"

@xA ™ontri˜utionsF isos™—l—r j

ƒ

vet us rst dis™uss the m—gneti™ moment op er—torF es in —ll st—ti™ soliton mo dels these

‚

" Q

—re o˜t—ined —s the sp—ti—l integr—l of the sp—™e ™omp onents of j @xA vi— " a @IaPA d r r 

eXmX

j @xAF st is then o˜vious th—t " involves the m—trix elements of r    a r   ˜ etween

eXmX UQ

„—˜le UFIX „he m—gneti™ moments in nu™leon m—gnetons —s fun™tions of the ™onstituent qu—rk

m—ss m ™omp—red to the exp eriment—l d—t—F @„—ken from refF‘IHI“AF

m@we†A QSH RHH SHH THH UHH VHH exptF

" HFTV HFTI HFSR HFSI HFRV HFRT HFVV

ƒ

" PFVU PFTR PFPW IFWW IFUS IFST RFUH

†

" IFUV IFTQ IFRP IFPS IFIP IFHI PFUW

p

" EIFIH EIFHP EIFQV EHFUR EHFTV EHFSS EIFWI

n

the eigenst—tes of h @TFRQAF st is most ™onvenient to ™onsider " F por the isove™tor p—rt of

" z

y

the m—gneti™ moment one h—s ‚ ( ‚ a h ( F por the isos™—l—r p—rt the expli™it dep enden™e

Q Qi i

on ‚ drops outF „he qu—ntiz—tion rules @UFPIA —nd @UFIVA @for isove™tor —nd {s™—l—r p—rtsD

resp e™tivelyA —re used to repl—™e the expli™it dep enden™e on the ™olle™tive ™o ordin—tes ˜y

™olle™tive op er—tors whi™h —™t on nu™leon st—tes with spin{pro je™tion t a IaPF st is then

Q

I

o˜vious th—t the m—gneti™ moments op er—tor ™—n ˜ e written —s " a " C s " F foth "

ƒ Q † ƒ

P

—nd " m—y ˜ e de™omp osed into v—len™e —nd v—™uum p—rts —™™ording to @UFPRDUFPSA " a

† ƒY†

v—™ v—l

F ixpli™it ev—lu—tion shows th—t only the rst terms on the ‚r ƒ of eqs @UFPRDUFPSA C " "

ƒY† ƒY†

™ontri˜ute to "

†

w

x

v—l

hv—lj( @r  A jv—li a x  "

Q g v—l

†

Q

Q

 

 

ˆ

 



w

"

x

v—™

 

" @UFPTA a x h"j( @r  A j"isgn @ Aerf™

g Q "

 † 

Q

T 

"

˜

where w a WQWwe† denotes the exp eriment—l nu™leon m—ss F yn the other h—nd the se™ond

x

terms on the ‚r ƒ of eqs @UFPRDUFPSA ™ontri˜ute to "

ƒ

ˆ

w hv—lj( j"ih"j @ r  A jv—li

x Q

v—l

Q

a x  "

g v—l

ƒ

P

  P

" v—l

"Tav—l

ˆ

w

x

v—™

a x " f @ Y  Y Ah"j( j# ih# j @ r  A j"iX @UFPUA

g  " # Q

ƒ

Q

P

R

"#

sn —ll ™—ses use h—s ˜ een m—de of the f—™t th—t the ™hoi™e of the z {™omp onent for the m—gneti™

moment op er—tor pro je™ts out the z {™omp onent in m—trix elements like h"j( j# i —fter summing

i

over the gr—nd spin pro je™tionF

sn t—˜le UFI the numeri™—l results —s o˜t—ined in refF ‘IHI “ for the isove™tor —nd {s™—l—r

moments —re displ—yed for v—rious ™onstituent qu—rk m—sses mD the only free p—r—meter in

the ˜—ryon se™tor of the mo delF foth " —nd " —re seen to de™re—se with in™re—sing mF

ƒ †

purthermore the ™orresp onding v—lues for the proton —nd nu™leon m—gneti™ moments " a

pYn

@IaPA@"  " A —re ™omp—red to the exp eriment—l d—t—F elthough the isos™—l—r p—rt of the

ƒ †

m—gneti™ moment is re—son—˜ly well repro du™edD the isove™tor p—rt ™omes out to o sm—llF st

should ˜ e —dded th—t in the ™—l™ul—tions of refF ‘IHQ “ — somewh—t l—rger " is o˜t—ined sin™e

†

these —uthors do not regul—rize terms whi™h stem from the im—gin—ry p—rt of the —™tionF ‡e

will ™omment on — p ossi˜le solution to the pro˜lem of to o sm—ll — " in se™tion UFPFQF

†

sn order to o˜t—in the ™h—rge r—dii one needs the se™ond moment of the ele™trom—gneti™

‚

P Q P H

™h—rge density hr i a d r r j @xAF eg—in — de™omp osition into isos™—l—r —nd {ve™tor p—rts

eXmX

˜

„he m—gneti™ moments —re me—sured in nu™leon m—gnetons UR

P

„—˜le UFPX „he me—n squ—red ™h—rge r—dii @in fm A —s fun™tions of the ™onstituent qu—rk m—ss

m ™omp—red to the exp eriment—l d—t—F @„—ken from refF‘IHQ“AF sn p—renthesis the results for

— regul—rized im—gin—ry p—rt —re givenF

m@we†A QUH RPH RSH exptF

P

hr i HFTQ @HFTRA HFSP @HFSHA HFRV @HFRSA HFTP

ƒ

P

hr i IFHU HFVW HFVR HFVT

†

P

hr i HFVS @HFVTA HFUI @HFUHA HFTT @HFTSA HFUR

p

P

hr i EHFPP@EHFPPA EHFIV@EHFIWA EHFIV@EHFIWA EHFIP

n

™—n ˜ e ™—rried out

I

P P P

hr i C s hr i @UFPVA hr i a

ƒ Q †

P

with

@ A

 

 

ˆ

 

 I x

" g

P P P

 

h"jr j"isgn @ Aerf™  hv—l jr jv—li @UFPWA hr i a

" v—l ƒ

 

P Q 

"

@

P

ˆ

x

hv—l jr ( j"i
h"j( jv—li

g

P

 hr i a

v—l †

P

T  

" v—l

"Tav—l

A

ˆ

I

P

C f @ Y  Y Ah"jr ( j# i
h# j( j"i X @UFQHA

 " #

P

"#

„hese expressions ™orresp ond to the ™—se th—t the im—gin—ry p—rt of the iu™lide—n —™tion is

P

regul—rizedF xoting th—t hr i origin—tes from those terms in @UFPRDUFPSA whi™h —re line—r in 

†

P

it should ˜ e o˜vious th—t only hr i re™eives ™ontri˜utions from the im—gin—ry p—rtF gho osing

ƒ

not to regul—rize this p—rt then ™orresp onds to repl—™ing the ™omplement—ry error fun™tion

in @UFPWA ˜y unityF sn th—t ™—se numeri™—l results —re —lso —v—il—˜le ‘IHQ “ whi™h —re displ—yed

in t—˜le UFPF y˜viously the question of regul—rizing the im—gin—ry p—rt only pl—ys — minor

roleF por the r—nge of m displ—yed in t—˜le UFP the v—len™e ™ontri˜utions strongly domin—teF

P

isp e™i—lly the v—™uum ™ontri˜ution to hr i is only —˜ out IEP7D however it in™re—ses with m

ƒ

P P

for hr i —nd hr i ‘IHQ“F

ƒ †

xu™leon gompton s™—ttering provides —™™ess to ele™trom—gneti™ p ol—riz—˜ilities whi™h thus

m—y ˜ e t—ken —s — me—sure for the squ—re of dip ole tr—nsitions ˜ etween ˜—ryon st—tesF „e™hE

ni™—llyD the p ol—riz—˜ilities m—y ˜ e extr—™ted from the resp onse of the soliton to extern—l

ele™trom—gneti™ eldsF por this purp ose the —™tion for the rot—ting soliton @UFIA is exp—nded

up to qu—dr—ti™ order in the sour™e — a e D iFeF one higher order th—n for the ™urrent

"

"

@UFPQAF st is su™ient to only ™onsider homogeneous ele™trom—gneti™ elds in this ™ontextX

I

 x f A F ‡hen exp—nding the —™tion the ele™tri™ @m—gneti™A p ol—riz—˜ilities e a @i z Y

ij k j k " "

P

P P

 @ A —re then re—d o from the ™o e™ients of i @f AF sn the xtv mo del so f—r only the

I

isos™—l—r ele™tri™ p ol—riz—˜ility of the nu™leonD  a @ C  A h—s ˜ een investig—ted ‘IHR“F

s aH p n

P

„his qu—ntity is sensitive to the l—rge dist—n™e ˜ eh—vior of the ™hir—l —ngle @r AF purtherE

P

moreD the le—ding term in — ™hir—l exp—nsion is known ‘IHS “ to ˜ e prop ortion—l to g @g

e

e

denotes the —xi—l ™h—rge of the nu™leonFAF „he —uthors of refF‘IHR“ —lso to ok into —™™ount the

su˜le—ding @in Iax A ™ontri˜utions to  D whi™h unfortun—tely —re su˜ je™t to ordering —m˜iE

g s aH

guities in the qu—ntiz—tion pro ™edure —nd might ˜ e —rti™i—lF sn —ny eventD these ™ontri˜utions US

„—˜le UFQX „he —xi—l ™h—rge of the nu™leon g —s — fun™tion of the ™onstituent qu—rk m—ss mF

e

m@we†A QSH RHH SHH THH UHH VHH exptF

g HFVH HFUT HFUH HFTT HFTP HFSW IFPT

e

h—ve the desired ee™t of rendering the exp eriment—l v—lue for g @see se™tion UFPFQAF por

e

the ™onstituent qu—rk m—ss m a RPHwe† the isos™—l—r ele™tri™ p ol—riz—˜ility w—s found to ˜ e

R Q R Q

 a IW  IH fm to whi™h the su˜le—ding terms ™ontri˜ute —˜ out Q  IH fm F elthough

s aH

Q

this result overestim—tes the exp eriment—l v—lue @WXT  IXV  PXPAfm ˜y —˜ out — f—™tor of P

it is still signi™—ntly sm—ller th—n in other soliton mo dels whi™h predi™t g ™orre™tly —s eFgF

e

the ' {mo del ‘IHS “F st should ˜ e —dded th—t the ™—l™ul—tion of refF‘IHR“ ™ont—ins the further

simpli™—tion th—t the Iax su˜le—ding terms h—ve ˜ een —pproxim—ted ˜y the le—ding order

g

R Q

expression of — gr—dient exp—nsionF „he numeri™—l result  a IW  IH fm m—y further

s aH

˜ e lowered when one t—kes —™™ount of the qu—ntum ™h—r—™ter of the pion eldsF sn th—t ™—se

the se—gull ™ontri˜ution to  h—s ˜ een ™l—imed to ˜ e —˜sent —s — ™onsequen™e of ™urrent

s aH

™onserv—tion ‘IHT “F sn the ™—l™ul—tion of refF‘IHR“ the se—gull terms ™ontri˜utes —˜ out TH7D

thoughF

UFPFP exi—l ™h—rge of the nu™leon

(

—

—

sn order to extr—™t the —xi—l prop erties of the nu™leon we put — @xA a — @xA  F iqns

# S

#

P

"Y—

@UFPRDUFPSA then provide the —xi—l ™urrent j F hue to isospin inv—ri—n™e the —xi—l ™h—rge g

e

S

QYQ

of the nu™leon m—y ˜ e o˜t—ined —s the m—trix element of Pj ˜ etween proton st—tes with spin

S

I

F gonsidering the —ngul—r velo ™ities  —s ™ommuting ™{num˜ ers one o˜serves pro je™tions C

P

th—t there —re no ™ontri˜utions to g from the terms line—r in  @seeD howeverD su˜se™tion

e

UFPFQAF „hen one only needs to ev—lu—te the m—trix element hp 4 jh jp 4i a IaQ —s —

QQ

™onsequen™e of @UFPIAF purther use of    a ' yields ‘IHV “

Q S Q

A @

 

 

ˆ

 

x I 

g "

 

g a X @UFQIA  hv—l j' ( jv—li h"j' ( j"isgn @ Aerf™

e v—l Q Q Q Q "

 

Q P 

"

es g is o˜t—ined to ˜ e the m—trix element of sp—ti—l ™omp onents of — ™urrent —nd do es

e

not ™ont—in —ny  dep enden™e it is o˜vious th—t g ™ompletely stems from the re—l p—rt of

e

the fermion determin—nt —nd thus ne™ess—rily undergo es regul—riz—tionF st should ˜ e noted

th—t in the ™hir—l limit @m a HA the ‚r ƒ of eq @UFQIA —™quires —n —ddition—l f—™tor @QGPA

%

™orresp onding to the symmetri™ zero momentum tr—nsfer limit ‘W“F

yne m—y —s well employ €geg @PFIUA in order to rel—te g to the prole fun™tion @r A

e

‘WD IHW “





V% d  R%



P Q P P Q

 f lim ‚ f m dr r sinX @UFQPA g a

e

% % %

r a‚

‚3I

Q d r W

es — m—tter of f—™t these two formul—s for ev—lu—ting g provide —n ex™ellent ™he™k on the

e

—™™ur—™y of the numeri™—l results whi™h —re displ—yed in t—˜le UFQF „he results extr—™ted from

eqsF @UFQIA —nd @UFQPA dier ˜y I{P7 onlyF y˜viously the exp eriment—l v—lue is underestim—ted

˜y —˜ out RH7 for —ll v—lues of the ™onstituent qu—rk m—ssF

„he ™—l™ul—tion of nu™leon —xi—l ™h—rges h—s ˜ een extended to those ™orresp onding in™luding

str—nge qu—rk ™urrentsF „hen there —re two more —xi—l ™h—rges of the nu™leon whi™h re™ently UT

V

re™eived sp e™i—l —ttention in the ™ontext of the ’proton spin puzzle4 ‘IHU “F „he rst oneD g

e

V

is o˜t—ined ˜y —ssuming — @xA a — @xA ! aP in the denition of the ™urrents @UFPPA with !

S V —

# #

H

˜ eing the qell{w—nn m—tri™esF „he se™ondD g is rel—ted to the —xi—l singlet ™urrent dened

e

H

vi— — @xA a — @xA aQF

S

# #

V H

sn order to ™ompute g —nd g the mo del h—s to ˜ e extended to —vor ƒ @QAF „his is

e e

—™hieved most str—ightforw—rdly ˜y ™ho osing the ™olle™tive ™o ordin—tes ‚@tA in eq @UFUPA to

˜ e Q  Q unit—ry m—tri™esF „his is often referred to —s the ™olle™tive —ppro—™h to in™lude

str—ngeness in ™hir—l soliton mo delsF st will ˜ e dis™ussed extensively in se™tion UFSFIF rere we

will not go into det—il ˜ut r—ther rem—rk two essenti—l dieren™es to the two —vor ™—seF pirstD

one h—s to t—ke into —™™ount th—t the ƒ @QA nu™leon w—ve{fun™tion diers from the ƒ @PA ™—se

esp e™i—lly in the —vor symmetri™ ™—seF yne ™onsequen™e is th—t hp 4 jh jp 4i a UaQH ‘IIH “

QQ

iFeF — signi™—nt redu™tion from @UFPIAF „hus the —xi—l ™h—rge for nu™leon  de™—y is —ltered

when ™onsidering —n ƒ @QA ™olle™tive qu—ntiz—tion even without ™h—nging the ™hir—l solitonF

Q

‡e will refer to this —xi—l ™h—rge ˜y g F ƒe™ondly one h—s to in™lude the —vor symmetry

e

˜re—king —rising from the dierent qu—rk m—ssesF „his gives rise to further ™h—nges of the

™

nu™leon w—ve{fun™tion —s will ˜ e des™ri˜ ed ˜ elow F purthermore the extension to three —vors

le—ds to —ddition—l terms for the ™urrents not ™ont—ined in @— three —vor gener—liz—tion of A

@UFPRDUFPSA ‘IIP “F „hese ee™ts h—ve ˜ een t—ken into —™™ount in — p ertur˜—tive s™heme with

the str—nge ™urrent qu—rk m—ss —s the p ertur˜—tion p—r—meter while non{str—nge —nd str—nge

™onstituent qu—rk m—sses were identied ‘IIQ “F „his s™heme diers somewh—t from the more

el—˜ or—te one dis™ussed ˜ elowD howeverD —s the ˜—ryon sp e™trum ™omes out simil—r @when

physi™—l v—lues for the p—r—meters —re —ssumedA there is enough re—son to t—ke the results on

Q V H

the —xi—l ™h—rges seriouslyF prom g D g —nd g one m—y n—lly extr—™t the ™ontri˜utions of

e e e

d

the individu—l —vors to the —xi—l ™h—rges RuD Rd —nd RsF essuming — ™onstituent qu—rk

m—ss m a RPHwe† the —uthors of refF‘IIQ“ o˜t—in

Ru a HXTR Rd a HXPR Rs a HXHPX @UFQQA

„hese should ˜ e ™omp—red with the ’exp eriment—l4 d—t— given ˜y illis —nd u—rliner ‘IIR“

Ru a HXVI Rd a HXRR Rs a HXIPX @UFQRA

„he —˜solute v—lues for Ru —nd Rd turn out to ˜ e somewh—t to o sm—ll —s — result of the

Q

to o sm—ll g F ‡hen ™omp—ring @UFQQA with @UFQRA one should ˜ e somewh—t ™—reful sin™e the

e

extr—™tion of the l—tter from semi{leptoni™ hyp eron de™—ys involves the —ssumption of —vor

symmetri™ ˜—ryon w—ve{fun™tions ‘IIS “F ‡ithin su™h — fr—mework usu—lly the —˜solute v—lue

of Rs ™omes out to o l—rgeF „his —ssumption h—s ˜ een dropp ed in the ™olle™tive —ppro—™hF

prom studies in ƒkyrme typ e mo dels it is well known th—t nevertheless the ˜r—n™hing r—tios

for the semileptoni™ hyp eron de™—ys —re well des™ri˜ ed in the ™olle™tive —ppro—™h together with

— sm—ll jRsj ‘IIT “F es most ™hir—l soliton mo dels ‘WTD RR “ the xtv mo del re—son—˜ly expl—ins

the sm—llness of the qu—rk ™ontri˜ution to the proton spin Ru C Rd C Rs % HXQVF

UFPFQ ‚em—rks on Iax ™orre™tions

g

sn —ll pre™eding ™onsider—tions we h—ve ™onsidered the —ngul—r velo ™ities  —s ™ommuting

™Enum˜ ersF por the ™omput—tion of nu™leon o˜serv—˜les  is repl—™ed ˜y the spin op er—tor

™ Q

por innitely l—rge str—nge qu—rk m—sses one h—s g a g D iFeF the two —vor limit ‘III“F

e

e

d Q

iFgF g a Ru RdF

e UU

t —™™ording to @UFIVAF roweverD in non{le—ding ™ontri˜utions ordering —m˜iguities exist —nd

dierent results m—y ˜ e o˜t—ined ˜ e™—use of non{v—nishing ™ommut—tors

Q

ˆ

I i

 h X @UFQSA ‘ Y h “ 3 ‘t Y h “ a

i˜m —m i —˜ i —˜

P P

 

maI

„his su˜stitution le—ds to non{v—nishing m—trix elements ˜ etween nu™leon st—tes

i

 X @UFQTA  hx jh  h jx i 3

ij lmn il m j n

P



„he pref—™tor on the right h—nd side is —m˜iguousD in p—rti™ul—rD it is v—nishing in the ordering

—ssumed in se™tion UFIF iqn @UFQTA then yields —ddition—l ™ontri˜utions to " —nd g whi™h we

† e

@IA @IA

denote ˜y " —nd g D resp e™tivelyF „hese ™orre™tions —re su˜le—ding in —n Iax exp—nsion

g

† e

P

˜ e™—use the moment of inerti—  is of order x F ƒin™e the st—rting p oint for the ™omput—tion

g

of the ˜—ryon o˜serv—˜les is ™ompletely ™l—ssi™—l the ordering ˜ etween the —ngul—r velo ™ities

—nd the rot—tions m—tri™es in the pres™ription @UFQTA is —m˜iguousF edopting — ’n—tur—l4

ordering ‘IIU “D whi™h is suggested ˜y the p ertur˜—tion exp—nsion of the qu—rk w—ve fun™tion

in the —vor rot—ting fr—me

P Q P Q

P

ˆ ˆ

h"j(
 j# i h"j(
t a j# i

R S R S

‚@tA C

3 ‚@tA C

@UFQUA

" # " #

   

# " # "

# Ta" # Ta"

@IA @IA

the numeri™—l results for " —nd g —re —˜ out QH7 of the le—ding order ‘IIV“F „hus it

† e

is suggestive ˜ut not o˜vious th—t the series h—s —lre—dy ™onverged —t this orderF ƒimil—r

™—l™ul—tions h—ve ˜ een p erformed ‘IIQ“ for the three —vor —xi—l ™ouplings dis™ussed in the

pre™eding su˜se™tionF por these qu—ntities the qu—ntiz—tion pres™ription @UFQSA le—ds to Iax

g

™orre™tions of the s—me orderF

et the time when these ™orre™tions were rst o˜served they were highly wel™ome sin™e they

help ed to solve @—mong othersA the pro˜lem of the to o sm—ll g @see t—˜le UFQAF xevertheless

e

the —ppli™—˜ility of @UFQSA from the very ˜ eginning rem—ins dou˜tfulD esp e™i—lly sin™e — sp e™i—l

ordering of the ™olle™tive ™o ordin—tes —nd op er—tors h—s to ˜ e —doptedF sn order to o˜t—in

some restri™tions on p ossi˜le orderings the ™omp—ti˜ility of the —sso ™i—ted Iax ™orre™tions

g

to g with symmetries of the underlying mo del h—ve ˜ een ex—minedF

e

es the Iax ™orre™tions —pp e—r to ˜ e — sp e™i—l fe—ture of the semi™l—ssi™—l qu—ntiz—tion

g

pro ™edure the self{™onsistent soliton prole rem—ins unee™tedF „hus g ev—lu—ted with eqn

e

@IA

@UFQPA —ssumes the le—ding order v—lueF „he ™orre™tion g m—y then ˜ e interpreted —s to

e

viol—te €geg ˜y QH7 whi™hD of ™ourseD represents —n undesired ˜re—king of one of the fund—E

ment—l symmetries of the mo delF st h—s ˜ een shown th—t this pro˜lem ™—n ˜ e solved ˜y simply

—dopting €geg —s the equ—tion of motion inste—d of the st—tion—ry ™ondition @TFRHA ‘IHW“F

„his then le—ds to —n extended soliton prole fun™tion ™—using the moment of inerti— —nd the

r—dii to in™re—seF nfortun—telyD until now this equ—tion of motion h—s not ˜ een derived from

—n —™tion prin™ipleF

purthermoreD requiring the prop er ˜ eh—vior of the symmetry ™urrents under p—rti™le ™onE

jug—tion m—y provide —ddition—l ™onstr—ints on orderings of the ™olle™tive ™o ordin—tesF „his

h—s re™ently ˜ een done in the fr—mework of the ™hir—l qu—rk mo del ‘IIW “D whi™h —t le—st for

the v—len™e qu—rk p—rt of the —™tion is simil—r to the xtv soliton mo delF st h—s ˜ een shown

‘IPH “ th—t the —xi—l ™urrent tr—nsforms under this symmetry prop erly only when —n ordering UV

@IA

is ™hosen su™h th—t the ™orre™tion g —™tu—lly is zeroF por ex—mple the rermiti—n ordering

e

pres™ription

I

@h t C t h A X @UFQVA h  3

—˜ j j —˜ —˜ j

P

P

le—ds to symmetry ™urrents whi™h tr—nsform prop erly under p—rti™le ™onjug—tion while there

—re no ™orre™tions to g —nd " —t the su˜le—ding order in the Iax exp—nsionF sn refF‘IIP“ —n

e † g

@IA

expression for g h—s ˜ een presentedD whi™h involves the —˜solute v—lue Ia@j  jA r—ther

" #

e

@IA

th—n Ia@  A —s in the ™hir—l qu—rk mo delF ren™e this result for g ™omplies with p—rti™le

" #

e

™onjug—tionF st shouldD howeverD ˜ e rem—rked th—t the —ltern—tive qu—ntiz—tion pres™ription

@UFQVA le—ds to v—nishing Iax ™orre™tions in the xtv mo del —s wellF

g

UFQ weson u™tu—tions o the ™hir—l soliton

e

sn order to explore ˜—ryon prop erties like eFgF % x s™—ttering D ele™tro{m—gneti™ p oE

l—riz—˜ilities @˜ eyond the isos™—l—r ele™tri™ ™h—nnelA ‘IHS“ or for the investig—tion of qu—ntum

u™tu—tions it is m—nd—tory to go ˜ eyond the zero{mo de qu—ntiz—tion des™ri˜ ed in se™tion

@UFIAF por —ppli™—tions like these time{dep endent u™tu—tions o the soliton h—ve to ˜ e inE

™luded —nd qu—ntized ™—noni™—llyF „his form—lism is —lso of relev—n™e for the dis™ussion of

qu—ntum ™orre™tions to the soliton m—ss ‘PS“ @se™tion UFRA —s well —s the des™ription of hyp erE

ons in the ˜ ound st—te —ppro—™h ‘PW“ @™omp—re su˜se™tion UFSFPAF

sn order to form—lly intro du™e the u™tu—tions we ™onsider the —ns—tz ‘WU“

w a $ $ hi$ $ X @UFQWA

H f f H

 

i

”

(
r @r A denotes the hedgehog soliton ™ongur—tion while the sp—™e{time rere $ a exp

H

P

dep endent u™tu—tions  @xA —re given ˜y

—

3 2

V

ˆ

 @xA! aP X @UFRHA $ @xA a exp i

— — f

—aI

„he m—in t—sk now is to exp—nd the —™tion up to qu—dr—ti™ order in the u™tu—tions  F sn

—

order to —void pro˜lems with st—˜ility the meson elds h—ve ˜ een ™onstr—ined to the ™hir—l

™ir™leF elthough the p—r—metriz—tion @UFQWA devi—tes from the ™ommonly —dopted unit—ry

y y

g—uge w a $ hi$ Y $ a $ it h—s proven to ˜ e ™onvenient when m—trix elements ˜ etween

‚ ‚

v v

eigenst—tes of the st—ti™ r—miltoni—n @TFRPA —re ™omputedF „his ™—n ˜ e o˜served e—sily ˜y

™onsidering the iu™lide—n hir—™ op er—tor ha

i

 

y y

y

„ @UFRIA i ha a d 
p „  $ hi$ € C $ hi$ €

i ( f f ‚ v

f f

wherein ( a ix is the iu™lide—n timeF „he unit—ry m—trix

H

y

€ @UFRPA „ a $ € C $

‚ H v

H

e

por — review on % x s™—ttering in soliton mo dels see refF‘II“F UW

™ont—ins —ll the inform—tion on the ™hir—l solitonF „husD whenever — m—trix element involving

the u™tu—tions h—s to ˜ e ev—lu—ted it ™—n ˜ e simplied ˜y ™hir—lly tr—nsforming the st—tes

˜ etween whi™h the op er—tor is s—ndwi™hedF

es the rst step tow—rds exp—nding the —™tion we write the iu™lide—n hir—™ op er—tor —s

 

i ha a d h a d h C h C h C


@UFRQA

i ( (

@HA @IA @PA

wherein the su˜s™ript l—˜ els the p ower of the meson u™tu—tionsF „he dots indi™—te p owers of

the u™tu—tions l—rger th—n twoF p on noting th—t h —s given ˜y @TFRPA is time indep endent

@HA

one o˜t—ins for the —rgument of the re—l p—rt of the fermion determin—nt e @TFPPA

‚

y

P P P

ha a d ha C h ‘d Y h “ C fh Y h g ‘d Y h “ C fh Y h g C h C


X @UFRRA

i ( (

@IA @IA @HA @PA @PA @HA

i

( @HA @IA

‡hen the v—len™e qu—rk ™ontri˜ution —s well —s the terms origin—ting from the mesoni™ p—rt of

the —™tion —re in™luded it is o˜vious th—t the zeroth{order @in the u™tu—tionsA just renders the

st—ti™ energy fun™tion—l while the expression line—r in  @xA v—nishes su˜ je™t to the equ—tion

—

of motion for the st—ti™ solitonF „herefore only the se™ond order expression is of interest

for the ™urrent dis™ussionF „he ™orresp onding ™ontri˜ution from the re—l p—rt of the fermion

determin—nt re—ds

  

 

s I I

I I

@PA

H H P

” ”

u @s s A ds ds „r dsu @sA fh Y h g C h „r e a

H H

@PA @HA

@IA ‚

P P

R P

H Ia Ia

 

H H

” ”

 ‘d Y h “u @s A‘d Y h “ C fh Y h gu @s Afh Y h g X @UFRSA

( H ( H

@IA @IA @IA @HA @IA @HA

 

P P

”

h A h—s turned out to „he intro du™tion of the zerothEorder he—t kernel u @sA a exp s@d

H

(

@HA

˜ e useful for this ™omput—tionF por the im—gin—ry p—rt of the fermion determin—ntD e D the

s

™utEo  is intro du™ed ˜y the su˜stitution

 

I

@d h h A@d C h C h A

( (

@HA @IA @HA @IA



 

I

ds exp s@d h h A@d C h C h A @UFRTA 3

( (

@HA @IA @HA @IA

P

Ia

whi™h is legitim—te sin™e the —rgument is neg—tive deniteD iFeF it ™onverges for l—rge moment—F

„he resulting im—gin—ry p—rt st—rts o —t se™ond order in the u™tu—tions

 

s I

H H H

” ”

ds u @s s Ad h u @s Ah h C


X @UFRUA ds e a „r

H ( H s

@IA @HA @IA

P

H Ia

„o p erform the temp or—l p—rt of the fun™tion—l tr—™e in iu™lide—n sp—™e the meson u™tuE

—tions —re pourier tr—nsformed



CI

d3

i3 (

 @r Y i( A a ~ @r Y i3 Ae @UFRVA

— —

P%

I

„his tr—nsform—tion dire™tly tr—nsfers to the r—miltoni—nsX



CI

d3

i3 (

~

h @r Y i3 Ae —nd h @r Y i( A a

@IA @IA

P%

I

 

H

CI CI

d3 d3

H

H i@3 C3 A(

~

h @r Y i3 Y i3 Ae @UFRWA h @r Y i( A a

@PA @PA

P% P%

I I VH

sin™e the ™hir—l tr—nsform—tion @UFRPA is time indep endentF sn order to pro je™t out the v—™uum

p—rt of the fermion determin—ntD e D one ™onsiders the limit of innitely l—rge iu™lide—n

v—™

timesF xoting th—t ‘UH “

3 2

H P

@( ( A I

P H

”

p

@UFSHA exp@sh A exp h( ju @sAj( i a

H

@HA

Rs

R% s

the temp or—l p—rt of the tr—™e then —mounts to ™—rrying out q—ussi—n integr—ls involving the

pourier frequen™y 3 F „he sp—ti—l p—rt of the tr—™e —s well —s the tr—™es over hir—™ —nd —vor

indi™es —re ev—lu—ted using the eigenst—tes of the st—ti™ oneEp—rti™le r—miltoni—n h @TFRQAF

@HA

pin—llyD the frequen™y 3 h—s to ˜ e ™ontinued ˜—™k to winkowski sp—™e in order to o˜t—in

physi™—lly relev—nt expressionsF „he se™ond order @in meson u™tu—tionsA ™ontri˜ution to the

v—™uum p—rt of the fermion determin—nt n—lly results in ‘WU“

 

I CI

ˆ

x ds d3

P

g

@PA @PA

@PA s

~ "

p

e a e C e a h"jh @r Y 3 Y 3 Aj"i P e

"

@PA

v—™ ‚ s

P

P P%

Ia I

R% s

"

r

 

I CI

ˆ

s d3 x

g

~ ~

ds h"jh @r Y 3 Aj# ih# jh @r Y 3 Aj"i @UFSIA C

@IA @IA

P

R R% P%

Ia I

"#

@ A

P

P

s

s

"

#

C e e

P P

 C ‘3 @ C  A “‚ @sY 3 Y  Y  A R3  ‚ @sY 3 Y  Y  A X

" # H " # # I " #

s

„he inform—tion on the orderings of the op er—tors in eqnsF @UFRSDUFRUA is ™ont—ined in the

peynm—n p—r—meter integr—ls



 

I

P P i P

x@I xA3 “ @UFSPA C x x dx exp s‘@I xA ‚ @sY 3 Y  Y  A a

i " #

# "

H

whi™h represent moments of the qu—rk lo op in the presen™e of the solitonF

fesides the p ol—rized v—™uum ™ongur—tion —lso the expli™it o ™™up—tion of the v—len™e qu—rk

level ™ontri˜utes to the —™tion —s long —s the —sso ™i—ted energy eigenv—lue  is p ositiveF ƒin™e

v—l

no regul—riz—tion is involved the ™omput—tion is ™ompletely p erformed in winkowski sp—™eF

„re—ting the meson u™tu—tions —s timeEdep endent p ertur˜—tions the —sso ™i—ted rst order

™h—nge  of the v—len™e qu—rk w—veEfun™tion is o˜t—ined to ˜ e

v—l v—l

 

I

 @r Y tA a id h @r A h @r Y tA @r Y tAX @UFSQA

v—l t v—l

@HA @IA

„he ™orresp onding ™ontri˜ution to the se™ond order p—rt of the —™tion re—ds ‘WU“





CI

d3

@PA

~

hv—ljh @r Y 3 Y 3 Ajv—li e a  x

v—l g

@PA

v—l

P%

I



~ ~

ˆ

hv—ljh @r Y 3 Aj"ih"jh @r Y 3 Ajv—li

@IA @IA

C X @UFSRA

 3 

v—l "

"Tav—l

rere  a HY I —g—in denotes the o ™™up—tion num˜ er of the v—len™e qu—rk —nd —ntiEqu—rk

v—l

st—tesF

„he ™omplete se™ond order ™ontri˜ution to the —™tion is then given ˜y

@PA

@PA @PA @PA

C e X @UFSSA e a e C e

v—l m v—™ VI

@PA

„he mesoni™ ™ontri˜utionD e D is o˜t—ined ˜y su˜stituting the —ns—tz @UFQWA into the expresE

m

sion for the mesoni™ p—rt of the —™tion @QFPSA

 

n

CI

d3 I

Q P P @PA

~ ~

™os   @3 A
 @3 A d r m f e a

% % m

P% P

I

U

H

o   

ˆ

I m m

s

s

~ @3 A ~ @3 A @UFSTA C IC ™os C

 

H

R m m

aR

H P P @PA

wherein use h—s ˜ een m—de of the rel—tion q a m mam f @RFRIAF y˜viously e ™ont—ins

I

% %

terms of o dd p owers in 3 F „hese ™orresp ond to the im—gin—ry p—rt in iu™lide—n sp—™e —nd

h—ve the imp ort—nt prop erty of removing the degener—™y ˜ etween solutions with 3 ‘PW“F sn

the ƒkyrme mo del these terms origin—te from the ‡ess{umino —™tion ‘SP“ whi™h is identi™—l

to the le—ding order term of the gr—dient exp—nsion of the im—gin—ry p—rt ‘QI“D ™fF se™tion RFIF

sn gener—l the se™ond order ™ontri˜ution to the —™tion ™—n ˜ e written —s — fun™tion—l of the

u™tu—tions 

—

  

n

CI

I

d3

@PA

Q Q H H H @PA

d r d r  @3 Y r Y r A ~ @r Y 3 A ~ @r Y 3 A e ‘ “ a

— ˜ —

—˜

P% P

I



o

@IA

Q

C d r  @r A ~ @r Y 3 A ~ @r Y 3 A X @UFSUA

— ˜

—˜

@IAY@PA

„he lo ™—l —nd ˜ilo ™—l kernels  h—ve to ˜ e ™omputed —s mo de sums involving the eigenE

—˜

st—tes —nd Efun™tions of h @TFRQAF

@HA

sing the ‡igner{i™k—rt theorem it ™—n ˜ e shown th—t due to the symmetry of the soliton

under gr—nd spin —nd p—rity tr—nsform—tionsD the u™tu—tions de™ouple with resp e™t to their

@IAY@PA

gr—nd spin —nd p—rity qu—ntum num˜ ersF sFeF the kernels  —re di—gon—l in these qu—ntum

—˜

num˜ ersF „his prop erty turns out to ˜ e helpful when investig—ting qu—ntum ™orre™tions to

the soliton m—ss ‘TU “ —nd hyp erons in the ˜ ound st—te —ppro—™h ‘WW“ to the xtv solitonF

UFR u—ntum ™orre™tions to the soliton m—ss

„he —™tion fun™tion—l @UFSUA of the meson u™tu—tions in the ˜—™kground eld of the xtv

™hir—l soliton h—s origin—lly ˜ een derived to m—ke p ossi˜le — des™ription of hyp erons in the

xtv mo del within the so{™—lled ˜ ound st—te —ppro—™h @™fF su˜se™tion UFSFPAF †ery re™ently it

h—s ˜ een shown in the two —vor redu™tion th—t the —™tion @UFSUA —lso —llows one to estim—te

@PA

the qu—ntum ™orre™tion to the soliton m—ss ‘TU“F „hen the ˜ilo ™—l kernel  only dep ends on

P

3 D iFeF the ™ontri˜utions to the —™tion ™orresp onding to o dd p owers in the frequen™y v—nishF

f

„his h—s the ™onsequen™e th—t the solutions to the fethe{ƒ—lp eter equ—tion



@PA @IA

Q H H H

d r  @r Y r Y 3 A ~ @r Y 3 A C  @r A ~ @r Y 3 A a H @UFSVA

˜ ˜

—˜ —˜

—pp e—r in p—irs 3 F henoting the ™orresp onding w—ve{fun™tions whi™h solve @UFSVA ˜y

i

@iA

~ @r Y 3 A the u™tu—ting eld m—y ˜ e de™omp osed —s

i

—

' &

ˆ

I

y

@iA i3 t @iA i3 t

i i

p

~ ~

 @r Y tA a X @UFSWA  @r Y 3 Ae —  @r Y 3 Ae C —

i i i

i

P3

i

i

f

e metho d for numeri™—lly solving this equ—tion is provided in refF‘IPI“F VP

!

y

a  A the efter ™—noni™—l qu—ntiz—tion @iFeF —ssuming the ™ommut—tion rel—tions — Y —

ij i

j

r—miltoni—n —sso ™i—ted with the u™tu—tions is th—t of —n h—rmoni™ os™ill—tor

 

ˆ

I

y

X @UFTHA — C — 3 r a

i i

i

P

i

„his result @UFTHA is quite non{trivi—l sin™e the —™tion @UFSUA involves —ll orders of time deriv—E

tives r—ther th—n termin—ting —t qu—dr—ti™ order —s eFgF in the ƒkyrme mo delF „his new

fe—ture le—ds to —n involved energy fun™tion—l @in ™o ordin—te sp—™eA —s well —s — ™ompli™—ted

orthonorm—liz—tion ™ondition for the meson u™tu—tionsF iFgF the norm—liz—tion ™ondition for

~

— solution  @r Y 3 A to the fethe{ƒ—lp eter equ—tion re—ds

i



 

—˜ H



d  @r Y r Y 3 A

@iA 

Q Q H @iA P

d r d r ~ @r Y 3 A ~ @r Y 3 A a IX @UFTIA 

i i

— ˜

P



d 3

3 a3

i

nfortun—telyD the orthogon—lity ™ondition for solutions with dierent frequen™ies ™—nnot ˜ e

presented in su™h — ™losed formF

€

3 aPA is st h—s —lso form—lly ˜ een shown th—t the v—™uum ™ontri˜ution to the energy @

i

i

o˜t—ined to ˜ e of the form @UFTHA whenever the ˜—™kground eld is st—ti™ —nd the eigenv—lues

of the fethe{ƒ—lp eter equ—tion —pp e—r in p—irs 3 F

i

„he form @UFTHA implies the existen™e of —n op er—tor —™ting in the sp—™e sp—nned ˜y the

eigenst—tes of @UFSVA

P P

r a r C † @UFTPA

H

with eigenv—lues 3 F „his op er—tor ™—n ˜ e expressed —s the sum of the ™orresp onding one

i

P

in the —˜sen™e of the solitonD r D —nd — ’p ertur˜—tion4D † D whi™h dep ends on the solitonF sn

H

refF‘PS“ it h—s ˜ een shown for the ƒkyrme mo del th—t — nite @renorm—lizedA energy ™orre™tionD

Ri D is o˜t—ined from these op er—tors vi—

 

I I I

P I Q

@UFTQA † C † „r r r r r Ri a

H

H H

P P V

whi™h —™tu—lly represents the gener—liz—tion of the qu—ntum ™orre™tions to the kink m—ss ‘IPP “

to QCI dimensionsF elthough the xtv mo del is quite dierent from the ƒkyrme mo del the

tr—™e m—y simil—rly ˜ e ™omputed ˜y expressing it in terms of the ™orresp onding eigenv—lues

@

 

ˆ ˆ

P

I I

 @HA  @HA

@HA

~ ~

h @r Y 3 Aj @r Y 3 Ai @UFTRA 3 3 Ri a

i i

j j

P V

i j

P Q

H I H I

P R

A

3 3

i i

T U

d e d e

 QCT X

R S

@HA @HA

3 3

j j

@HA @HA

@HA

~

rere 3 —nd  @r Y 3 A denote the eigenfrequen™ies —nd {w—ve{fun™tions to the fethe{

j j

ƒ—lp eter equ—tion @UFSVA in the —˜sen™e of the solitonF „he overl—p in eq @UFTRA h—s o ™™urred

˜ e™—use the op er—tors r —nd r —™t in distin™t ril˜ ert sp—™esF e re—son—˜le denition of

H

these overl—ps is g—ined ˜y in™luding the metri™





Q

—˜ H



   

p p

ˆ

—™ ˜™

d  @r Y r Y 3 A



—˜ H Q P H

a w @r Y r Y 3 A a d x  w @r Y xY 3 A w @r Y xY 3 A @UFTSA

i i i

P



d 3

™aI

3 a3

i VQ

into the w—ve{fun™tion



Q

 

p

ˆ

™—

— Q

w @xY r Y 3 A ~ @xY 3 AX @UFTTA 0 @r Y 3 A a d x

i ™ i i

™aI

„he mo di™—tied w—ve{fun™tion 0 turns out to ˜ e indep endent of the p—r—metriz—tion @UFQWA

‘IPQ “F pin—lly the relev—nt overl—p m—trix element is given ˜y



@HA @HA

@HA Q @HA

~ ~

h @r Y 3 Aj @r Y 3 Ai Xa d r 0 @r Y 3 A
0 @r Y 3 A @UFTUA

i i

j j

@HA

where 0 denotes the —n—logue of 0 in the —˜sen™e of the solitonF

prom eq @UFTRA it is intuitively ™le—r th—t the zero mo des @3 a HA provide the m— jor

i

™ontri˜ution to Ri ˜ e™—use no ™ounterp—rt exists in the —˜sen™e of the solitonF „he zero mo des

g

—rise ˜ e™—use the soliton ˜re—ks the rot—tion—l —nd tr—nsl—tion—l inv—ri—n™eF sn th—t sense the

~

zero mo des m—y ˜ e reg—rded —s qoldstone ˜ osonsF „he —sso ™i—ted w—ve{fun™tionsD  @r AD

zXmX

™orresp ond to — w—ve{fun™tion in the € {w—ve ™h—nnel for the rot—tion—l zero mo de while the

tr—nsl—tion—l zero mo de ™ont—ins ƒ { —nd h {w—ve p—rtsF e™™ording to these stru™tures —ns—tze

™—n ˜ e m—de for the w—ve fun™tions in the zero mo de ™h—nnelsF „hen the fethe{ƒ—lp eter

equ—tion redu™es to @™oupledA homogeneous integr—l equ—tions for purely r—di—l fun™tionsF es

this pro ™edure is quite te™hni™—l we refer the re—der to refF‘TU“ for det—ils on this ™—l™ul—tion

—s well —s on the expli™it ™onstru™tion of the mo died w—ve fun™tion 0 F es —lre—dy noted the

™ontri˜ution of the zero mo des to the energy ™orre™tion

 

ˆ

P

Q

 @HA  @HA

@HA

~ ~

h @r Aj @r Y 3 Ai @UFTVA 3

zXmX j j

IT

j

is neg—tive deniteF ren™e it —utom—ti™—lly le—ds to the desired result of redu™ing the tot—l

energyF

fefore presenting the numeri™—l results one rem—rk ™on™erning the non{™onning ™h—r—™ter

of the xtv mo del h—s to ˜ e m—deF y˜viously the tr—nsform—tion from iu™lide—n to winkowski

sp—™e is only well dened —s long —s the exp onent in the peynm—n p—r—meter integr—ls @UFIPA

v—nishes for l—rge s —long the p—th ™onne™ting these two sp—™esF sn the —˜sen™e of the soliton

this is only the ™—se for 3 ` PmF feyond this threshold the regul—riz—tion fun™tions develop

im—gin—ry p—rts whi™h me—sure the de™—y of the meson u™tu—tions into qu—rk{—ntiqu—rk

p—irsF gonsequently the qu—ntum ™orre™tion to the soliton m—ss whi™h stems from the zero

mo des is estim—ted ˜y the trun™—ted sum

 

ˆ

P

Q

@HA @HA

 

@HA

~ ~

Ri a 3 X @UFTWA h @r Aj @r Y 3 Ai

 

zXmX

j zXmX j

IT

@HA

3 `Pm

j

sn order to judge the qu—lity of this trun™—tion it is useful to dene — sum of overl—ps

 

ˆ

P

 @HA 

@HA

~ ~

h @r Aj @r Y 3 Ai @UFUHA ƒ a

j

zXmX

@HA

3 `Pm

j

whi™h should —ppro—™h unity if the mo del were insensi˜le to the trun™—tionF

g

hue to the gr—nd spin symmetry of the hedgehog —ns—tz ™o ordin—te sp—™e rot—tions —nd isospin tr—nsforE

m—tions —re equiv—lentF VR

„—˜le UFRX „he qu—ntum ™orre™tions to the soliton m—ss due to the rot—tion—l zero mo de ‘TU“F

m a H m a IQSwe†

% %

m@we†A RHH SHH THH RHH SHH THH

ƒ HFVV HFWR HFWS HFVQ HFVW HFWI

Ri @we†A EPHI EPUR EPWH EPRR EPWU EQPQ

zXmX

„—˜le UFSX „he qu—ntum ™orre™tions to the soliton m—ss due to the tr—nsl—tion—l zero mo deF

„he ™ontri˜utions stemming from the ƒ @l a HA{ —nd h @l a PA{w—ves —re disent—ngled ‘TU “F

m a H m a IQSwe†

% %

m@we†A RHH SHH THH RHH SHH THH

ƒ HFQP HFRI HFSP HFPV HFQU HFRT

Ri @we†A EIV EPP EQP EIP EPP EQP

laH

Ri @we†A EIPU EIRH EPHU EVP EIPV EIVU

laP

Ri @we†A EIRS EITP EPQW EWR EISH EPIV

zXmX

„he numeri™—l results o˜t—ined in refF‘TU“ for the ™ontri˜utions of the rot—tion—l —nd tr—nsE

l—tion—l zero mo des to the energy ™orre™tion —re displ—yed in t—˜les UFR —nd UFSD resp e™tivelyF

elso shown —re the ™orresp onding sums of overl—psD ƒ F „he ™hir—l limit @m a HA —s well —s

%

the physi™—l ™—se @m a IQSwe†A —re ™onsideredF xote th—t the num˜ er of st—tes whi™h lie

%

˜ elow 3 a Pm de™re—ses —s the pion m—ss in™re—sesF sn ™—se of the rot—tion—l zero mo de ƒ

—ppro—™hes unity ™lose enough to ™onsider Ri % PSH $ QHHwe† —s — reli—˜le estim—te of

the qu—ntum ™orre™tion to the energyF „he situ—tionD howeverD is not —s go o d for the tr—nsE

l—tion—l zero mo deF „he ™orresp onding w—ve{fun™tion is more strongly p e—ked —t r a H th—n

in ™—se of the rot—tion—l zero mo deF „herefore the pourier tr—nsform —™quires ™ontri˜utions

from mo des ™—rrying higher frequen™ies whi™h lie —˜ ove PmF ren™e ƒ h—rdly re—™hes HFSF „he

™orresp onding Ri ™—n then only ˜ e ™onsidered —s — lower ˜ oundF es in the ƒkyrme mo del it

turns out th—t the h {w—ve ™ontri˜utions strongly domin—te the ƒ {w—ve p—rtF

sn refF‘TU“ it h—s furthermore veried th—t the s™—ttering mo des provide — negligi˜le ™onE

tri˜ution of — few we†F

„—king the num˜ ers displ—yed in t—˜les UFR —nd UFS seriously @improving the mo del su™h

—s to over™ome the pro˜lem of non{™onnement would in —ny event —lso —lter the predi™tions

on the ™l—ssi™—l m—ssD moment of inerti—D et™FA one o˜t—ins — m—ss formul— for ˜—ryons

t @t C IA

@UFUIA w a i C Ri C

™l

P

P

with Ri estim—ted ˜y the sum of the rot—tion—l —nd tr—nsl—tion—l zero mo de ™ontri˜utionsF

„he l—st term in @UFUIA —rises from the semi{™l—ssi™—l ™r—nking pro ™edure dis™ussed in se™tion

UFIF sn gener—l there —re —lso qu—ntum ™orre™tion to this rot—tion—l termF roweverD these —re

P

of the order y @x A —nd hen™e they —re omittedF „he numeri™—l results for the ˜—ryon m—sses

g

—re given in t—˜le UFTF ‚e—son—˜le —greement with the exp eriment—l d—t— for the m—sses of the

nu™leon @WQWwe†A —nd the
{reson—n™e @IPQPwe†A —re only o˜t—ined for ™onstituent qu—rk

m—sses m % RHHwe†F nfortun—tely ƒ then is —s low —s HFR for the tr—nsl—tion—l zero mo deF

es — ™on™lusive st—tement of this se™tion we would like to mention th—t the m—ss of the

nu™leon is signi™—ntly lower th—n the soliton m—ss due to qu—ntum ™orre™tionsF sn gener—l VS

„—˜le UFTX „he predi™tions for the m—sses of the nu™leon @x A —nd
{reson—n™eF „he empiri™—l

d—t— —re WQWwe† —nd IPQPwe†D resp e™tivelyF @‚esults t—ken from ‘TU“FA

m a H m a IQSwe†

% %

m@we†A RHH SHH THH RHH SHH THH

i @we†A IPIP IIWQ IITT IPSH IPPI IIWQ

™l

Ri @we†A EQHP ERQT ESPS EQQV ERRV ESQV

P

 @IGqe†A TFPT RFUQ QFVU SFVH RFIU QFRQ

w @we†A WUH VQT UQV WUT VTQ UTR

x

w @we†A IPIH IISQ IIPT IPQT IPPQ IPHI

these ™orre™tions redu™e the m—sses of ˜—ryons ˜y — few hundred we† —s ™omp—red to the

™r—nking result @UFPHAF „he ee™t of the qu—ntum ™orre™tions on o˜serv—˜les other th—n the

m—sses h—s not yet ˜ een studied in the xtv mo delF

UFS ryp erons

sn this se™tion we will des™ri˜ e the tre—tment of str—nge degrees of freedom within the xtv

mo del of pseudos™—l—r eldsF „he m—in go—l is to nd — des™ription of the hyp eron sp e™trum in

this mo delF es h—s —lre—dy ˜ een p ointed outD the pro je™tion of the soliton onto st—tes with go o d

spin —nd —vor qu—ntum num˜ ers ™—n only ˜ e p erformed —pproxim—telyF „his is —lre—dy the

™—se for the @symmetri™A two —vor mo delF por str—nge degrees of freedom the situ—tion is even

worse due to the presen™e of symmetry ˜re—kingF g—l™ul—tions in ƒkyrme typ e mo dels provide

two dierent —ppro—™hes whi™h —re frequently viewed —s opp osite limits of symmetry ˜re—kingF

foth tre—tments represent —pproxim—tions to the ex—™t time dep endent solution whi™h is yet

un{knownF „he rst one represents — gener—liz—tion of the zero mo de qu—ntiz—tion of ƒ @PA

—nd requires the intro du™tion of ™olle™tive ™o ordin—tes des™ri˜ing rot—tions in the whole ƒ @QA

—vor sp—™e ‘PT D PU “F ‡e will therefore refer to this tre—tment —s the ™olle™tive —ppro—™hF es

these ™olle™tive ™o ordin—tes des™ri˜ e l—rge —mplitude u™tu—tions the restoring for™e is —ssumed

to ˜ e sm—llF ƒt—ted otherwiseD the —vor symmetry ˜re—king is ™onsidered to ˜ e sm—ll —nd

—n exp—nsion in the p—r—meters me—suring symmetry ˜re—king is p erformedF „he ™olle™tive

tre—tment h—s undergone ™onsider—˜le improvement when ‰—˜u —nd endo ‘PV“ o˜served th—t

the resulting ™olle™tive r—miltoni—n in™luding symmetry ˜re—king terms ™—n ˜ e di—gon—lized

ex—™tly ˜y numeri™—l metho dsF v—ter on this tre—tment w—s seen to represent —dmixtures of

st—tes from higher dimension—l ƒ @QA represent—tions to the ˜—si™ o ™tet —nd de™uplet st—tes

‘III “F sn the ™omplement—ry tre—tmentD whi™h w—s initi—ted ˜y g—ll—n —nd ule˜—nov ‘PW“

—nd to whi™h we will refer —s the ˜ ound st—te —ppro—™hD the st—rting p oint is to ™onsider

symmetry ˜re—king l—rgeD —lthough the tre—tment yields the ™orre™t results in the symmetri™

limit —s wellF rere only sm—ll —mplitude u™tu—tions —re —llowed @™fF se™tion UFQAF „hese

u™tu—tions ™—n ˜ e qu—ntized ™—noni™—llyF „he ˜ ound st—te —ppro—™h he—vily relies on the

sp e™i—l fe—ture th—t — solution to the fethe{ƒ—lp eter equ—tion @UFWTA emerges in the zero{

mo de ™h—nnel when the symmetry ˜re—king is swit™hed onF „he eigenfrequen™y of this mo de

is dierent from zero ˜ut —lso signi™—ntly lower th—n the k—on m—ssD iFeF it represents — ˜ ound

st—teF eddition—lly the ˜ ound st—te w—ve{fun™tion is well lo ™—lizedF sFeF the ˜ ound st—te is the

’would{˜ e4 qoldstone ˜ oson of the str—nge —vor tr—nsform—tions in the soliton ˜—™kgroundF

fy ™onstru™tion the o ™™up—tion num˜ er of this ˜ ound st—te is identi™—l to the str—ngeness of VT

the ˜—ryon under ™onsider—tionF pin—lly the re—l zero mo des ™orresp onding to spin —nd isospin

—re tre—ted within the ™olle™tive —ppro—™h yielding the hyp erne splitting —nd thus removing

the degener—™y of ˜—ryons with identi™—l str—ngeness —s eFgF the  —nd the  ‘PW“F

rere we will dis™uss ˜ oth —ppro—™hes in the fr—mework of the xtv mo del —nd ™riti™—lly

™omp—re the resultsF

UFSFI gol le™tive rot—tion—l —ppro—™h

sn the ™olle™tive —ppro—™h symmetry ˜re—king is ™onsidered to ˜ e sm—ll —nd ™onsequently

str—nge degrees of freedom —re intro du™ed —s if they were zero mo desF yn top of thisD ee™ts

due to symmetry ˜re—king —re tre—ted ˜y exp—nding the fermion determin—nt in terms of the

dieren™e of the ™onstituent qu—rk m—ssesF st should ˜ e stressed th—t for the mesoni™ p—rt of

the —™tion e @QFPSA no exp—nsion is p erformedF

m

e™™ording to the —˜ ove dis™ussionD ™olle™tive ™o ordin—tes for rot—tions —re dened in the

whole —vor sp—™e in order to —pproxim—te the time dep endent solutionF es the ee™ts due to

—vor symmetry ˜re—king h—ve to ˜ e t—ken into —™™ount we h—ve to go ˜ eyond the —ns—tz for

the two —vor ™—se @UFIAF ‡e ™onsider ‘TP “

y y

w @r Y tA a ‚@tA$ @r A‚ @tAhi‚@tA$ @r A‚ @tA ‚@tA P ƒ @QAX @UFUPA

H H

”

rere $ a exp @i(
r @r AaPA denotes the st—ti™ soliton ™ongur—tionF y˜viouslyD only the

H

pseudos™—l—r elds rot—te in —vor sp—™e while the s™—l—r elds —re kept —t their v—™uum

exp e™t—tion v—luesF por the ongoing explor—tion it is helpful to dene eight —ngul—r velo ™ities

h

 Y @— a IY XXY VA whi™h me—sure the time dep enden™e of the —vor rot—tion

—

V

ˆ

i

y

•

!  a ‚ @tA‚@tAX @UFUQA

— —

P

—aI

„hese —re the extensions of the previously intro du™ed —ngul—r velo ™ities @UFRA to the ƒ @QA

—vor groupF eg—in it is ™onvenient to tr—nsform to the —vor rot—ting system in order to

H

ev—lu—te the fermion determin—ntX q a ‚q F „his —llows to elimin—te the ’outer4 rot—tions in

@UFUPA —t the exp ense of —n indu™ed rot—tion—l p—rt

V

ˆ

I

!  X @UFURA h a

— — r ot

P

—aI

sn the rot—ting fr—me the hir—™ op er—tor —™quires the form

H

i ha a id h h h X @UFUSA

t r ot ƒ f

@HA

rere h represents the st—ti™ one{p—rti™le r—miltoni—n dened in eq @TFRPAF „he intro du™tion

@HA

of the ™hir—l tr—nsform—tion „ @UFRPA —llows one to e—sily displ—y the symmetry ˜re—king p—rt

in the hir—™ op er—tor

 

y y

h a „  ‚ hi‚ hi „

ƒ f

3 2

U Q

ˆ ˆ

m m

s

y

p

„ X @UFUTA h ! C@h IA! „  a h ! C

V  VV V Vi i

Q

aR

iaI

h

„he dot indi™—tes the deriv—tive with resp e™t to the time ™o ordin—teF VU

‡e h—ve —lso indi™—ted the ƒ @PA inv—ri—nt pie™esF purthermore use h—s ˜ een m—de of the

 

I

y

F ! ‚! ‚ —djoint represent—tion for the —vor rot—tions h a

— ˜ —˜

P

„he m—in t—sk ™onsists of exp—nding the fermion determin—nt e in terms of h —nd h F

p r ot ƒ f

sn order to regul—rize e — ™ontinu—tion to iu™lide—n sp—™e is requiredF sn this ™ontext it is

p

imp ort—nt to t—ke into —™™ount th—t h ™orresp onds to the time ™omp onent of —n indu™ed

r ot

ve™tor eldF ren™e h h—s to ˜ e ™onsidered —s —n —nti{rermiti—n qu—ntityF „he re—l p—rt of

r ot

e therefore ™ontri˜utes terms of even p ower in  to the ee™tive —™tionF ƒin™e the exp—nsion

p

is ™onstr—ined to the se™ond order in h C h we ™onsider

r ot ƒ f

Hy

H P P P P

ha ha a d C h C fh Y h g C h C ‘h Y h “ h X @UFUUA

ƒ f r ot

@HA @HA

i

i ( @HA ƒ f r ot

sn gener—l there ™ould —lso ˜ e ™ontri˜utions from the ™ommut—tor ‘d Y h “D howeverD these

( ƒ f

will ™ontri˜ute to the moments of inerti— —t se™ond order in symmetry ˜re—king —nd h—ve to ˜ e

dis™—rded for ™onsisten™yF ƒin™e the im—gin—ry p—rt of the fermion determin—ntD e D ™ont—ins

s

only terms of o dd p owers in the —ngul—r velo ™ityD it will re™eive ™ontri˜utions from the mixed

terms of the form h h F xoting th—t the —™tion is only exp—nded up to se™ond order in the

r ot ƒ f

—ngul—r velo ™ityD e m—y ˜ e o˜t—ined vi—

s

I

Hy

I H

„rlog @ha A ha e a

s

i i

P

& '

i   h

I

I

„r d h h fh Y h C h g C XXXX@UFUVA d h h a

( ƒ f r ot ƒ f ( ƒ f

@HA @HA @HA

P

„he time{dep enden™e of h —™tu—lly yields — non{v—nishing ™ommut—tor ‘d Y h “ whi™hD

ƒ f ( ƒ f

Q

A —nd m—y therefore ˜ e dis™—rded for the howeverD will not ™ontri˜ute to e ˜ elow y @

s

—

further ™—l™ul—tionF elthough the im—gin—ry p—rt is nite the ongoing ev—lu—tion of e ™—n ˜ e

s

m—de ™onsistent with the prop er{time regul—riz—tion of the re—l p—rtF „his is —™hieved ˜y the

repl—™ement @UFRTAF

xow the fun™tion—l tr—™e ™—n ˜ e ev—lu—tedF „he temp or—l p—rt is p erformed ˜y intro du™ing

i

eigenst—tes exp@i3 ( A of d whi™h s—tisfy —nti{p erio di™ ˜ ound—ry ™onditions in the iu™lide—n

n (

time interv—l „ F „he sp—ti—l p—rt —s well —s the tr—™es over hir—™ —nd —vor indi™es —re

ev—lu—ted using eigenst—tes of the st—ti™ one{p—rti™le r—miltoni—n h @TFRQAF

@HA

„he le—ding term in the exp—nsion of the fermion determin—nt is the v—™uum ™ontri˜ution

to the soliton m—ss —nd stems from the re—l p—rt @™fF ™h—pter TAF „he im—gin—ry p—rt st—rts

o with —n expression rel—ted to the —nti{™ommut—tor fh Y h g

r ot

@HA



h    i

I

I

ds exp s d h d C h fh Y h g „r

( ( r ot

@HA @HA @HA

P

P

Ia

 

 

ˆ

 

x 

g "

 

h"jh j"iX @UFUWA a „ sign@ Aerf™

r ot "

 

P 

"

„he sum over the eigenst—tes j"i of h o˜viously pro je™ts out the gr—nd spin zero p—rt of h

r ot

@HA

whi™h is rel—ted to the eighth ™omp onent of the —ngul—r velo ™ity

p

 

 

ˆ

 

x Q 

g "

v—™

 

p

a „ f  @UFVHA sign @ Aerf™ „ 

V " V

 

 P

R Q

"

i

3 a @Pn C IAa„ —re the w—tsu˜—r— frequen™iesD ™fF ™h—pter TF

n VV

v—™

wherein f denotes the ™ontri˜ution to the ˜—ryon num˜ er origin—ting from the p ol—riz—tion

of the hir—™ se—F „he rem—inder of the im—gin—ry p—rt is line—r in ˜ othD the —ngul—r velo ™ity

—nd the ™onstituent qu—rk m—ss dieren™e m m —nd we denote it ˜y @see —pp endix f for

s

the denition of
A

—˜

I

v—™

„
 h X @UFVIA

— V˜

—˜

P

„he fe—tureD whi™h w—s —lre—dy o˜served for the im—gin—ry p—rtD th—t the rst order in the

exp—nsion of the re—l p—rt in terms of h C h only —llows for gr—nd spin symmetri™ expresE

ƒ f

@HA

sions —lso holds for the re—l p—rtF „—king into —™™ount th—t e is even in the —ngul—r velo ™ities

‚

there rem—ins only one gr—nd spin symmetri™ term involving h

VV

I

v—™

„  @I h A @UFVPA

VV

P

sin™e the devi—tion of the h {m—trix from unity is ™onsidered to ˜ e the dyn—mi™—lly relev—nt

qu—ntity @UFUTAF purthermore the re—l p—rt of the fermion determin—nt ™ont—ins terms whi™h

—re qu—dr—ti™ in either the —ngul—r velo ™ity or the symmetry ˜re—king

I I

v—™ v—™

  Y h h X @UFVQA „  „

— ˜ V— V˜

—˜ —˜

P P

v—™ v—™

eg—in we refer to refF‘TP“ for the —™tu—l ™—l™ul—tion of the qu—ntities  —nd @see —lso

—˜ —˜

se™tion UFI for denitions —nd —pp endix h for the expli™it expressionsAF xeedless to mention

th—t the resulting expression for the —™tion h—s to ˜ e ™ontinued ˜—™k to winkowski sp—™e in

order to extr—™t the ™olle™tive v—gr—ng i—nF

p to this p oint we h—ve ™onsidered the limit of l—rge iu™lide—n times whi™h h—s pro je™ted

out the v—™uum ™ontri˜ution to the —™tionF eddition—lly the ™o e™ients of the ™olle™tive

qu—ntities  —nd h re™eive ™ontri˜utions from the expli™it o ™™up—tion of the v—len™e qu—rk

— V—

levelF „hese —re o˜t—ined ˜y ™onsider—tion of the extented hir—™ equ—tion

 

a  @UFVRA h C h C h

v—l v—l v—l r ot ƒ f

@HA

in st—tion—ry p ertur˜—tion theoryF ƒin™e the v—len™e qu—rk p—rt of the —™tion is not regul—rized

there is no need to ™ontinue forth —nd ˜—™k to iu™lide—n sp—™eF „hus the relev—nt ™—l™ul—tions

—re ™ompletely p erformed in winkowski sp—™eF „he resulting expressions —re displ—yed in

—pp endix hF pin—lly there rem—ins the ™ontri˜ution from the meson p—rt of —™tion e F „his

m

™—n ˜ e ev—lu—ted str—ightforw—rdly ˜y su˜stituting the —ns—tz @UFUPA into eq @QFPSA yielding

U Q

ˆ ˆ

I I I I I

m m m m

h h @I h h A @UFVSA h h  @I h A e a

V V VV VV Vi Vi VV m

ƒ V „

P P P P „

aR iaI

H

with the ™o e™ients ™ont—ining the dire™t inform—tion on the ™urrent qu—rk m—sses m —nd

H

m F eg—in we refer to —pp endix h for their expli™it formsF

s

‡e h—ve now ™olle™ted —lmost —ll ingredients for the ™olle™tive v—gr— ngi—nF ‡e still need

to dis™uss meson eld ™omp onents whi™h v—nish ™l—ssi™—lly ˜ut —re indu™ed ˜y the ™olle™tive

rot—tion into str—nge dire™tionF €—r—metrizing the ™orresp onding k—on u™tu—tion ˜y @™fF the

previous se™tion UFQA

2 3

U

ˆ

H u @r A

 @r A! a @UFVTA

 

y

u @r A H

aR VW

the im—gin—ry p—rt of the iu™lide—n —™tion provides ™ouplings whi™h —re line—r in ˜ othD the

k—on eld u —s well —s the —ngul—r velo ™ity  @ a RY XXY UAF ixp—nding the re—l p—rt of



the —™tion in terms of u only ™ont—ins even p owers of the k—on eldF ren™e — non{trivi—l

solution for u @r A m—y existF e suit—˜le —ns—tz whi™h ™—rries the —ppropri—te —vor —nd p—rity

qu—ntum num˜ ers re—ds ‘IPR“

 

 i

R S

”

X @UFVUA u @r A a r
( ‡ @r A

 i

T U

sn gener—l ‡ @r A is — ™omplex r—di—l fun™tionF „he fr—mework of the ™olle™tive —ppro—™h

o˜viously requires to exp—nd the —™tion up to qu—dr—ti™ order in  F „his pro ™edure —dds

—

U

ˆ

„

P

 @UFVVA  ‘‡ “

s



P

aR

to the —™tionF „he equ—tion of motion o˜t—ined from the v—ri—tion   a ‡ @r A a H for the

s

r—di—l fun™tion ‡ ™ont—ins — sour™e term stemming from the line—r ™oupling to the ™olle™tive

rot—tion dis™ussed —˜ ovedF „his sour™e term is ™ompletely given ˜y the st—ti™ @hedgehogA

eldF ren™e the solution requires ‡ Ta HF st turns out th—t only the re—l p—rt of ‡ is

ex™itedF sn the xtv mo del up to now the ex™ited elds h—ve not ˜ een tre—ted ex—™tly ˜ut

r—ther in the gr—dient exp—nsionF „hen the im—gin—ry p—rt of the —™tion is —pproxim—ted

˜y the ‡ess{umino termF „he re—l p—rt of the —™tion is —pproxim—ted ˜y the non{line—r

' {mo del —llowingD howeverD the k—on de™—y ™onst—nt f to ˜ e dierent from f F por f the

u % u

predi™tion of the xtv mo del is used @™fF t—˜le RFIAF ‡henever the ™hir—l —ngle shows up

the self{™onsistent solution is su˜stitutedF „he re—der m—y ™onsult refF‘TP“ for det—ils on this

™—l™ul—tionF „he imp ort—nt result is th—t the indu™ed p—rt of the str—nge moment of inerti—

 is not negligi˜leF st should —lso ˜ e stressed th—t there —re no —n—logous ex™ited elds for

s

the moment of inerti— —sso ™i—ted with rot—tions in iso{sp—™e sin™e the im—gin—ry p—rt of the

j

—™tion v—nishes identi™—lly in — two —vor mo del —s long —s only pion elds —re in™luded F

„he deriv—tion of the ™olle™tive three —vor v—gr—ngi—n is now termin—ted —nd its n—l

form m—y ˜ e dis™ussedF ssospin —nd rot—tion—l inv—ri—n™e provide rel—tions ˜ etween ™ert—in

™omp onents of the moment of inerti—

P P

 Xa  a  a  Y  Xa  a  a  a  —nd  a H @UFVWA

II PP QQ RR SS TT UU VV

while —ll other ™omp onents v—nishF „he l—st equ—tion in @UFVWA stems from the v—nishing

™ommut—tor ‘h Y! “F en—logous rel—tions hold for
—nd F

V —˜ —˜

@HA

sn the next step the ™olle™tive r—miltoni—n —sso ™i—ted with the v—gr—ngi—n @hFIQA h—s to

˜ e ™onstru™tedF „his is —™hieved ˜y gener—lizing the tre—tment of se™tion @UFIA to three —vorsF

„his deriv—tion is —lso des™ri˜ ed in —pp endix hY it results in the m—ss formul— for ˜—ryon f

™—rrying spin t

Q I I I I

At @t C IA C  X @UFWHA @ i a i C

ƒ f f tot

P P P P

 V P P 

P P

„he moments of inerti—s  —nd  m—y ˜ e found in —pp endix h —s wellF „he qu—ntity 

ƒ f

denotes the eigenv—lue of the ƒ @QA op er—tor @hFIVA whi™h ™ont—ins the inform—tion on the

j

en iso{singlet  {eldD howeverD m—y yield — non{v—nishing ˜ut sm—ll ™ontri˜ution to the non{str—nge

moment of inerti—F WH

C C

Q I

—nd ˜—ryons with resp e™t to the nu™leonF „—˜le UFUX „he m—ss dieren™es of the lowElying

P P

‡e ™omp—re the predi™tions of the ™olle™tive —ppro—™h to the xtv mo del with the exp eriment—l

d—t—F „he upEqu—rk ™onstituent m—ss m is ™hosen su™h th—t the
{nu™leon m—ss dieren™e is

repro du™ed ™orre™tlyD see textF „he l—st ™olumn refers to the ™—se th—t the symmetry ˜re—ker

predX exptX

P

A ell d—t— @from refF‘WW“A —re in we†F af  is s™—led ˜y @f

u u

™orrX

f xed f xed ixptF f

% u

u

 IHS IHW IUU IUS

 IRV ISI PSR PRV

 PQT PRQ QUW QWT

PWQ PWQ PWQ PWI



 QVU QWI RRT RRW



 RVP RVW SWI THV

 SUT SVT UQQ UTS

spin —nd str—ngeness of ˜—ryon f F es dis™ussed in —pp endix h the eigenst—tes of the ™olle™tive

r—miltoni—n le—ding to @UFWHA —re ™onstr—ined to ™—rry h—lf{integer spinD iFeF they —re fermionsF

‡e —re now en—˜led to dis™uss the numeri™—l results on the hyp eron m—ss sp e™trumF es

—lre—dy expl—ined in se™tion RFQ the xtv mo del underestim—tes the r—tio f af ˜y —˜ out IS7

u %

for non{str—nge ™onstituent qu—rk m—sses of the order m % RHHwe†F „his short™oming of the

mo del is —lso exp e™ted to show up in the ˜—ryon se™torF sn order to demonstr—te th—t the r—tio

f af r—ther th—n the —˜solute v—lue for f is the ingredient to whi™h the results —re sensitive

u % u

we ™omp—re ™—l™ul—tions for two dierent sets of p—r—metersF pirstD f a WQwe† is kept —t its

%

empiri™—l v—lue —nd m a RHUwe† is ™hosen su™h —s to repro du™e the exp eriment—l
{nu™leon

m—ss dieren™e @PWQwe†AF por this set f a WWXVwe† is predi™tedD iFeF f af a IXHUF „he

u u %

se™ond set of p—r—meters is xed su™h —s to ™orre™tly give f a IIRwe†F eg—in m a RQQwe† is

u

determined ˜y dem—nding the exp eriment—l
{nu™leon m—ss dieren™eF „hen f a IHRXWwe†

%

is in™re—sed ™onsider—˜lyD howeverD the r—tio f af a IXHW rem—ins —lmost un—lteredF „he

u %

™orresp onding numeri™—l results for the ˜—ryon m—ss dieren™e —re displ—yed in t—˜le UFUF

y˜viously the ™h—nge in the m—ss dieren™es for the two sets of p—r—meters is not l—rger

th—n the ™h—nge for the r—tio f af D iFeF P7F es exp e™ted the ƒ @QA symmetry ˜re—king in

u %

the ˜—ryon m—ss dieren™es is ™onsider—˜le underestim—tedF st h—s ˜ een demonstr—ted th—t

this ee™t is strongly ™orrel—ted to pro˜lem of in™orre™tly predi™ting f F „he le—ding order

u

terms in — gr—dient exp—nsion to the domin—ting symmetry ˜re—king p—r—meter  —re given

˜y @UFVPDUFVQA

2 3 A @



P

P P

Psin  f f R

v—™ plus m

HP u % Q P P P P

™os   C C XXX d r @m f m f A@I ™osA C a 

u u % %

gr—dX expX

P

Q P r



R

Q P P

d r @I ™osAY @UFWIA m f %

u u

Q

where the l—tter rel—tion is veried numeri™—llyF prom ƒkyrme mo del ™—l™ul—tions it is well

known ‘IPR “ th—t this s™—ling of  with f represents the most imp ort—nt dep enden™e on the

u

k—on de™—y ™onst—ntF sn order to estim—te the ee™ts of this s™—ling in the fr—mework of the xtv

soliton the ™orresp onding qu—ntities @ Y A in the ™olle™tive v—gr—ng i—n @hFIQA —re s™—led

„ YƒYV

exptX pr edX P

P

˜y @f af A % @IIRaIHH A —nd the hyp eron sp e™trum is re{ev—lu—ted ‘TP“F eg—in m is

u u

—djusted to repro du™e w w F „he results for the m—ss dieren™esD whi™h —re —lso shown

x
WI

in the l—st ™olumn of t—˜le UFUD —gree ex™ellently well with the exp eriment—l d—t—F yf ™ourseD

it should ˜ e stressed th—t this estim—te h—s the purp ose of rel—ting the to o low predi™tion

for the ƒ @QA symmetry ˜re—king in the ˜—ryon se™tor to the meson se™tor r—ther th—n ˜ eing

™onsidered — restri™tive predi™tion of the xtv mo delF ‡e m—y ™on™lude this se™tion ˜y st—ting

th—t the xtv mo del in the ™olle™tive —ppro—™h provides the ™orre™t hyp eron sp e™trum —s f—r —s

the qu—ntum num˜ ers —nd the ordering of the hyp eron st—tes within — sp e™ied spin multiplet

—re ™on™ernedD howeverD the qu—ntit—tive ƒ @QA ˜re—king is underestim—ted due to the to o

sm—ll predi™tion for f F

u

st should ˜ e stressed th—t the ™olle™tive —ppro—™h together with the —˜ ove des™ri˜ ed —pE

proxim—tion is not sensitive to the —˜solute v—lue of the str—nge ™urrent qu—rk m—ss ˜ut r—ther

H H

to the r—tio m am —s ™—n ˜ e seen from eqs @hFIHD hFII —nd hFIPAF „his pro ™edure —voids

s

un™ert—inties rel—ted to the ™hoi™e of — sp e™i—l regul—riz—tion pres™ription ˜ e™—use the r—tio is

known to ˜ e insensitive on the regul—riz—tion pres™ription while the —˜solute v—lues for the

k

™urrent qu—rk m—sses —re not F

„here h—ve ˜ een simil—r ™—l™ul—tions ˜y —nother group using the ™olle™tive —ppro—™h to the

H

xtv mo del ‘IPS “F „hese —uthors ™ho ose not to p erform the shift ƒ 3 ƒ m” for the s™—l—r

elds @™fF se™tion QFPAF yf ™ourseD when exp—nding the fermion determin—nt to —ll orders th—t

tre—tment should ˜ e identi™—l to the one presented hereF roweverD this is not the ™—se when

—pproxim—tions —re involvedF sn refF‘IPS“ furthermore —vor indep endent ™onstituent qu—rk

H

F st should ˜ e m—sses @m a mA —re —ssumed —nd the determin—nt is exp—nded in terms of m

s

s

H

in the sense of ™hir—l p ertur˜—tion theory mentioned th—t —n exp—nsion of the —™tion in m

s

H

is ™l—imed not to ™onverge for the ’physi™—l4 v—lue of m ‘IV“F sn —ny eventD it is o˜vious

s

H

th—t this tre—tment is sensitive to the —˜solute v—lue of m —nd thus —lso to the regul—riz—tion

s

H

pres™riptionF sn order to —™™ommo d—te the empiri™—l v—lue m % ISHwe† the —uthors of

s

refF‘IPS“ employ — dou˜le step prop erEtime pres™riptionF „he —pproxim—tion @m a mA —lso

s

implies the identity f a f @RFRRAF st is therefore o˜vious th—t in th—t tre—tment the hyp eron

u %

m—ss splittings ™—nnot ˜ e des™ri˜ ed prop erlyF sn order to get re—son—˜le —greement with the

H H

—nd h—d to ˜ e in™re—sed ˜y —˜ out PH7F „he rel—tion ˜ etween m exp eriment—l sp e™trum m

s s

the k—on m—ss m is —lmost line—rF „he gr—dient exp—nsion @UFWIA reve—ls th—t this in™re—se

u

H

just ™orresp onds to su˜stituting the physi™—l k—on de™—y ™onst—ntF purthermore the of m

s

dou˜le step regul—riz—tion s™heme of refF‘IPS“ involves two more undetermined p—r—metersF

xot surprisingly this m—kes p ossi˜le — further in™re—se of the symmetry ˜re—king p—r—meter

 F st should ˜ e rem—rked th—t within the prop er time regul—riz—tion — ™onsistent ™omput—tion

H

of the p—r—meters of the ™olle™tive r—miltoni—n yields — l—rger  when the shift ƒ 3 ƒ m”

l

is —™tu—lly p erformed F „his indi™—tes th—t the results on the hyp eron m—ss splittings —re

somewh—t sensitive to the regul—riz—tion pres™ription r—ther th—n the p—r—metriz—tion of the

rot—ting eldsF

UFSFP found st—te —ppro—™h

„he ˜ ound st—te —ppro—™h represents —n —ltern—tive tre—tment for the des™ription of hyE

p erons in soliton mo delsF st h—s origin—lly ˜ een —pplied to the ƒkyrme mo del ‘PW “ —nd l—ter

on exp erien™ed siz—˜le extensions ‘IPTD IPU D IPV “F sn the ™—se of the xtv mo del the ˜ ound

st—te —ppro—™h providesD ˜ esides the des™ription of hyp eronsD —n —pp—rent —ppli™—tion of the

k

por — ™ompil—tion of relev—nt —rti™les see refF‘IV“F

l I

yur numeri™—l ™omput—tions in the prop er time s™heme show th—t the y @m A ™ontri˜ution to  is —˜ out

s

IH7 l—rger when the shift is p erformed —nd dierent ™onstituent qu—rk m—sses —re —ssumedF WP

form—lism for tre—ting mesoni™ u™tu—tions o the ™hir—l soliton ‘WU“F „he gener—l —sp e™ts of

this form—lism h—ve ˜ een expl—ined in se™tion UFQF

es shown in refF‘WW“ —nd mentioned in se™tion UFQ the mesoni™ u™tu—tions de™ouple with

resp e™t to their gr—nd spin —nd p—rity qu—ntum num˜ ersF ƒin™e k—ons ™—rry isospin IaP the

gr—nd spin of these u™tu—tions is h—lf{integerF st is therefore su™ient to only ™onsider k—on

mo des in eqs @UFRHD UFRVA —nd disreg—rd other pseudos™—l—r u™tu—tions iFeF

2 3

U

~

ˆ

H u @r Y 3 A

~

~ @r Y 3 A! a @UFWPA  @r Y 3 A a H —nd

 

y

~

u @r Y 3 A H

aR

~

wherein u @r Y 3 A is — two{™omp onent isospinorF „he ˜ ound st—te is exp e™ted to —pp e—r in

H H

ex—™tly th—t ™h—nnel whi™h ™ont—ins — zero mo de in the symmetri™ @m a m A ™—seF en

s

innitesim—l ve™tor tr—nsform—tion in str—nge dire™tion m—y ˜ e —sso ™i—ted with

2 3

@r A

sin

P

”

u @r A a r
( X @UFWQA

H H

H

rere denotes —n —r˜itr—ry P  P sp—™e{time indep endent unit—ry m—trix xing the isospin

H

m

orient—tionF „his form of the ’would{˜ e4 zero mo de suggests the following —ns—tz for the

k—on ˜ ound st—te w—ve{fun™tion

 

—@rY 3 A

~

”

@UFWRA u @r Y 3 A a r
( @3 Y r A with @rY 3 A a

˜@rY 3 A

wherein @rY 3 A is — two{™omp onent isospinor @not to ˜ e ™onfused with the —ngul—r velo ™ity  A

whi™h only dep ends on the r—di—l ™o ordin—te r —nd the frequen™y 3 F „he —ngul—r dep enden™e is

”

™ompletely given ˜y r
( —nd thus @UFWRA represents — €{w—ve k—onF hue to isospin inv—ri—n™e

the kernels @UFSUA ™orresp onding to  —re unit m—tri™es in iso{sp—™e

  

n

CI

d3

@PA P H HP @PA H y H

e ‘“ a dr r dr r  @3 Y rY r A @rY 3 A@r Y 3 A

P%

I



o

P @IA y

C dr r  @r A @rY 3 A@rY 3 A X @UFWSA

@IYPA

ixpli™it expressions for the kernels  —re given in —pp endix iF

†—rying @UFWSA with resp e™t to  yields — homogeneous line—r integr—l equ—tion

& '



P H HP @PA H H @IA

r dr r  @3 Y rY r A@r Y 3 A C  @r A@rY 3 A a H @UFWTA

whi™h in f—™t is the fethe{ƒ—lp eter equ—tion for the k—on ˜ ound st—te in the soliton ˜—™kE

groundF st is the —n—log of the ˜ ound st—te equ—tion in the g—ll—n{ule˜—nov —ppro—™h ‘PW“

@IAY@PA

to the ƒkyrme mo delF „he f—™t th—t the kernels  —re unit m—tri™es in isosp—™e le—ds to

identi™—l fethe{ƒ—lp eter equ—tions for the isospinor ™omp onents —@rY 3 A —nd ˜@rY 3 A implying

th—t ˜ oth h—ve the s—me r—di—l dep enden™e  @r AF „hus pourier —mplitudes — @3 A m—y ˜ e

3 i

dened —s

—@rY 3 A a  @r A— @3 A —nd ˜@rY 3 A a  @r A— @3 AX @UFWUA

3 I 3 P

m

e™tu—llyD the ˜ ound st—te is nothing ˜ut the ’would{˜ e4 qoldstone ˜ oson for —vor rot—tions into str—nge

dire™tionF WQ

sn the pro ™ess of qu—ntiz—tion the pourier —mplitudes — @3 A —™quire the st—tus of ™re—tionE —nd

i

—nnihil—tion op er—torsF

sn order to determine the norm—liz—tion of  @r A @whi™h ™—nnot ˜ e dedu™ed from the fethe{

3

ƒ—lp eter equ—tion @UFWTAA it is helpful to ™ompute the str—ngeness ™h—rge ƒ —sso ™i—ted with

the k—on ˜ ound st—teF et the mi™ros™opi™ level ƒ is dened —s the exp e™t—tion v—lue

  

Q y R

”

ƒ a h q"h q d r q @ƒ Aq exp@i d xvA @UFWVA

”

wherein v is the xtv { v—gr—ng i—n of eq @QFIA —nd ƒ a di—g @HY HY IA is the pro je™tor onto

str—ngenessF xote th—t in the st—nd—rd ™onvention — str—nge qu—rk p ossesses str—ngeness IF

„he expression @UFWVA is exp—nded up to se™ond order in the u™tu—tionsF „his ™—l™ul—tion h—s

˜ een p erformed in refF‘WW“ —nd m—y ˜ e summ—rized —s

 

  

d3

y y

H H  H

@3 A— @3 A X @UFWWA @3 A— @3 AC — dr dr  @3 Y rY r A @r A @r A — ƒ a

P I ƒ 3

P I

3

P%

H

„he symmetri™ ˜ilo ™—l kernel  @3 Y rY r A is given in —pp endix iF ƒin™e the str—ngeness ™h—rge

ƒ

h—s to ˜ e qu—ntized to ˜ e integer{v—luedD eqnF @UFWWA provides — suit—˜le norm—liz—tion ™onE

dition for the r—di—l fun™tion  @r A

3

 

 

 

H H H 

@r A @r A a IX @UFIHHA  dr dr  @3 Y rY r A

3 ƒ

3

y˜viouslyD the sign of the exp e™t—tion v—lue ƒ is determined ˜y the dyn—mi™sF sn n—ture

only hyp erons with neg—tive str—ngeness —re o˜servedF „hus — m— jor ™riterion whi™h the

xtv mo del h—s to s—tisfy is the existen™e of — ˜ ound st—te with neg—tive str—ngeness while

the fethe{ƒ—lp eter equ—tion @UFWTA should not p ossess solutions with j3 j m —nd p ositive

u

str—ngenessF

sn gure UFI the r—di—l dep enden™e of the ˜ ound st—te w—ve{fun™tion norm—lized —™™ording

to @UFIHHA is displ—yed for sever—l v—lues of the ™onstituent qu—rk m—ss m while the ™orreE

sp onding ˜ ound st—te energy 3 is listed in t—˜le UFVF

H

hierent o ™™up—tions of the ˜ ound st—te remove the degener—™y of st—tes with dierent

str—ngenessF roweverD the system ™onsisting of the st—ti™ soliton —nd the k—on ˜ ound st—te

h—s still to ˜ e pro je™ted onto st—tes with go o d spin —nd isospin qu—ntum num˜ ers in order to

—lso remove the degener—™y of ˜—ryons with identi™—l str—ngenessD s—y the  —nd the F „his

pro je™tion is —™hieved ˜y the usu—l ™r—nking pro ™edure for the ex—™t zero mo desD whi™h ™orreE

sp ond to isospin rot—tionsF „he —sso ™i—ted ™olle™tive ™o ordin—tes —re intro du™ed —n—logous ly

to the pro ™edure expl—ined in se™tion UFID howeverD —lso the u™tu—tions h—ve to ˜ e t—ken into

—™™ountF „his is —™™omplished ˜y writing

y

w a ‚@tA$ $ hi$ $ ‚ @tA with ‚@tA P ƒ @PAX @UFIHIA

H f f H

y˜viouslyD — glo˜—l isospin rot—tion ™orresp onds to the su˜stitution ‚@tA 3 ‚@tAF „his

g g

indi™—tes th—t the tot—l isospin is ™—rried ˜y ‚@tAF ƒin™e hi ™ommutes with the isospin

gener—tors the —ns—tz @UFIHIA is equiv—lent to

y

~ ~

$ 3 ‚@tA$ ‚ @tA —nd u 3 ‚@tAu X @UFIHPA

H H

~

„his demonstr—tes th—t u h—s lost its isospinF st should ˜ e rem—rked th—t —lthough the —ns—tz

@UFIHIA is quite dierent from the one —ssumed in the origin—l ƒkyrme mo del ™—l™ul—tion ‘PW“ the WR

pigure UFIX „he r—di—l dep enden™e of the ˜ ound st—te w—ve{fun™tion  @r A for v—rious ™onE

3

stituent qu—rk m—sses mF  @r A is norm—lized —™™ording to @UFIHHAF @„—ken from refF ‘WW“FA

3

su˜stitution @UFIHPA is identi™—lF es for the two —vor version of the mo del —ngul—r velo ™ities

 —re intro du™edD see eq @UFRA

i

y

•

(
 a ‚ @tA‚@tAX @UFIHQA

P

elthough the ™olle™tive rot—tions —re not the only time{dep endent eld ™omp onents when

u™tu—tions —re presentD the identity

!

•

d w i

( Y w a h @UFIHRA

i ij

P d 

j

nevertheless holdsF „he rot—tion m—trix h for two —vors is dened in se™tion UFI ˜ efore eq

ij

@UFIWAF edopting the previous —rgument—tion on xo ether ™h—rges @™fF eqs @UFIVD UFIWAA the

tot—l isospin is given ˜y

d v@‚Y  A

s a h @UFIHSA

i ij

d 

j

wherein v is the v—gr—ng e fun™tion whi™h is — fun™tion—l of the ™hir—l —ngle  —s well —s the

~

k—on u™tu—tions u F st is furthermore instru™tive to t—ke into —™™ount the hedgehog stru™ture

of the soliton —nd dene the momentum ™onjug—ted to the —ngul—r velo ™ity 

d v

@UFIHTA t a



d 

P P

—s the spin ™—rried ˜y the solitonF „his le—ds to the identity s a t F ƒtill t needs to ˜ e





rel—ted to the tot—l spinF yn the mi™ros™opi™ level the l—tter is dened —s the exp e™t—tion WS

v—lue

 

Q y

ht i a h q" h q d r q t q exp @ie A @UFIHUA

xtv

‚

R

wherein t is the spin op er—tor for — hir—™ spinor —nd e a d xv denotes the —™tion

xtv xtv

—sso ™i—ted with the xtv v—gr—ng i—n @QFIAF ƒin™e the spin op er—tor ™ommutes with the isoE

H

rot—tions ‚@tA the tr—nsform—tion into the rot—ting fr—me q a ‚q is str—ightforw—rd

 

H H Q Hy H H

ht i a h q" h q d r q t q exp @ie A X @UFIHVA

xtv

H

rere e represents the xtv —™tion in the rot—ting fr—me whi™h —lso ™ont—ins the goriolis

xtv

term

 



I

Hy H H R

X @UFIHWA q (
 q e a d x v

xtv

xtv

P

ƒu˜stituting the denition of the gr—nd spin q yields

 

 

(

H H H H Q Hy

ht i a h q" h q q A X @UFIIHA q exp @ie d r q

xtv

P

sn this expression the soliton ™ontri˜ution to the spin t m—y ˜ e identied ˜y dierenti—ting



H

with resp e™t to the —ngul—r velo ™ity  e

xtv



H

I d e

H H H xtv

ht i a hq i C h q" h q A a hqi C t X @UFIIIA exp @ie



xtv

„ d 

ƒt—ted otherwiseX the spin ™—rried ˜y the k—ons

t a ht i t a hq i @UFIIPA

u 

is identi™—l to the gr—nd spin exp e™t—tion v—lueF „his is gener—lly exp e™ted ˜—sed on the Iax

g

exp—nsion for ˜—ryonsD see eFgF refF‘IPW“F ƒin™e the eigenst—tes of the st—ti™ r—miltoni—n —re

—lso eigenst—tes of the gr—nd spin op er—torD iFeF q j"i a w j"i the result for the k—oni™

Q "

@PA

spin is e—sily o˜t—in—˜leF ‚ep e—ting the ™—l™ul—tion le—ding to e D howeverD in™luding the

p

gr—nd spin pro je™tion qu—ntum num˜ er w when t—king m—trix elements provides —fter —

"

str—ightforw—rd ™—l™ul—tion ‘WW“ the third ™omp onent t D whi™h in turn is prop ortion—l to

u Q

€

y y y

— @( A — F ixploiting rot—tion—l inv—ri—n™e one nds — @3 A— @3 A — @3 A— @3 A a

Q ij j I P

ij

i I P

I H



P

ˆ

I d3

y

e d

— @3 A( — @3 A d@3 A X @UFIIQA t a

ij j u

i

P P%

iYj aI

„he sp e™tr—l fun™tion d@3 A is given in —pp endix iF rere it is interesting to dis™uss — ™on™eptu—l

dieren™e in ™omp—rison with ƒkyrme typ e mo delsF sn these mo dels the ˜ ound st—te —ppro—™h

involves ™l—ssi™—l st—ti™ elds —s well —s k—on u™tu—tionsF „he former h—ve v—nishing gr—nd

spin while the l—tter ™—rry gr—nd spin IaPF „his yields the identity d@3 A a I ‘QH“F sn the

xtv mo del the situ—tion is dierent ˜ e™—use the fun™tion—l tr—™e involves qu—rk spinors with

—r˜itr—ry gr—nd spinF „hese spinors get p ol—rized ˜y the soliton eld —s well —s the k—on ˜ ound

st—te ™—using d@3 A to devi—te from unityF WT

sn the ™ontext of the ™olle™tive qu—ntiz—tion the m—in t—sk now ™onsists of determining the

™oupling of the k—on u™tu—tions to the —ngul—r velo ™ities  F por this purp ose the fermion

determin—nt is rst exp—nded in 

Q



ˆ

d e

p

 @HA @IA

P P

 e a e @ a HA C  C y @ A a e C e C y @ AX @UFIIRA

— p p

p p

 aH

d 

—

—aI

P

P

where the ™ontri˜utions of order  yield the non{str—ng e moment of inerti—  @UFISAF rere

the interest is on the line—r term whi™hD in ™ontr—st to the two —vor mo delD do es not v—nish

˜ut r—ther provides the ™oupling ˜ etween the u™tu—tions —nd the ™olle™tive rot—tionsF sn the

@IA

se™ond step of the ™—l™ul—tionD e is therefore exp—nded up to qu—dr—ti™ order in the k—on

p

eldsF „o this end the  {dep endent terms of the v—gr—nge fun™tion m—y ˜ e extr—™ted

I H



P

ˆ

I d3 I

y

P P

e d

  @3 A( — @3 A C


X @UFIISA — v a ™@3 A

ij j

i



P P P%

iYj aI

eg—in we refer to refF‘WW“ for det—ils on this ™—l™ul—tion while the expli™it expression for the

sp e™tr—l fun™tion ™@3 A is presented in —pp endix iF

es —lre—dy mentioned hyp erons with dierent str—ngeness —re ™onstru™ted ˜y v—rious o ™E

™up—tions of the k—on ˜ ound st—te iFeF the solution to the fethe{ƒ—lp eter equ—tion @UFWTA with

j3 j ` m F „hus

H u

™ Xa ™@3 A —nd d Xa d@3 A @UFIITA

H H

—re pro je™ted out from the sp e™tr—l integr—ls @UFIISDUFIIQAF ‡ith this restri™tion to the ˜ ound

st—te the ™olle™tive r—miltoni—n o˜t—ined from v re—ds



I H

P

P

ˆ

™ I I

y

P

e d

— ( — @t C 1t A @UFIIUA t C a r a

ij j  u 

i



P P

P P P

iYj aI

where the p—r—meter 1 a ™ad h—s ˜ een intro du™edF „he —rguments of the —nnihil—tion

@™re—tionA op er—tors h—ve ˜ een omittedF e™™ording to the —˜ ove dis™ussion t —nd t m—y

 u

˜ e rel—ted to spin t —nd isospin s qu—ntum num˜ ers of the ™onsidered ˜—ryonF „his yields

2 3

d1 I

P

1t @t C IA C @I 1As @s C IA C @ A ƒ @ƒ PA @UFIIVA r a



P

P P

 

P

€

y

P

˜ e™—use it ™—n ˜ e shown th—t — ( — a ƒ @ƒ PA ‘QH“F „his term is —lre—dy of fourth

ij j

i

iYj aI

order in the ˜ ound st—te w—ve{fun™tion —nd h—s to ˜ e dropp ed for ™onsisten™yF i—™h su˜stituE

tion of — nonEstr—ng e v—len™e qu—rk ˜y — str—nge one ™h—nges the v—len™e qu—rk ™ontri˜ution

to the energy from  to  3 F „he m—ss formul— for physi™—l ˜—ryons thus ˜ e™omes

v—l v—l H

I

@1t @t C IA C @I 1As @s C IAA @UFIIWA w a i C ƒ 3 C

f ™l H

P

P

wherein i is the ™l—ssi™—l energy @TFRTAF „his expression is simil—r to the ƒkyrme mo del

™l

result ‘QH“D howeverD here the r—tio 1 a ™ad h—d to intro du™ed with d Ta IF „he term line—r

in 3 should —lso ™ome out ˜y p erforming — ™—l™ul—tion simil—r to th—t le—ding to eq @UFTHA

H

when the terms of o dd p owers in 3 —re prop erly —™™ounted forF WU

„—˜le UFVX €—r—meters for des™ri˜ing the hyp eron sp e™trum —s fun™tions of the ™onstituent

m—ss mF elso listed —re the empiri™—l v—lues whi™h —re o˜t—ined ˜y the ™onsider—tion of

™ert—in m—ss dieren™esF @h—t— t—ken from refF‘WW“FA

m@we†A QSH RHH RSH SHH empirF

3 @we†A EPHUFI EIVPFT EITQFT EIRVFV EIVWFS

H

™ EHFPH EHFQT EHFRT EHFSQ |

d HFWH HFVW HFVW HFVW |

1 a ™ad HFPP HFRH HFSP HFTH HFTP

P

 @Ia@qe† AA VFQH SFVH RFUV RFIU SFIP

n

fefore going into the det—ils of the numeri™—l results it is imp ort—nt to note th—t ˜ ound

st—tes h—ve only ˜ een o˜t—ined in the region m ` 3 ` H —nd th—t these ™—rry neg—tive

u H

str—ngeness —™™ording to eq @UFWWAF xo ˜ ound st—te h—s ˜ een o˜served in the interv—l H ` 3 `

m F

u

es in the ™olle™tive tre—tment only the m—ss dieren™es with resp e™t to the nu™leon m—ss

P

—re ™onsideredF sn this ™—se eq @UFIIWA ™ont—ins three qu—ntities @ Y 3 Y 1A whi™h determine

H

P

seven m—ss dieren™esF „hus the ˜—ryon m—ss formul— @UFIIWA m—y ˜ e inverted rel—ting  Y 3

H

—nd 1 to hyp eron m—ss dieren™esF „hese rel—tions m—y ˜ e employed to o˜t—in empiri™—l d—t—

P

for  Y 3 —nd 1F ‡e list these together with the ™—l™ul—ted v—lues for v—rious ™onstituent

H

qu—rk m—sses m in t—˜le UFVF „he empiri™—l d—t— —re re—son—˜ly well —™™ommo d—ted in the

region RHHwe† m RSHwe† —lthough 1 is somewh—t to o sm—llF

eg—inD the only free p—r—meter m is ™hosen to get — ˜ est t to the exp eriment—l m—ss

P

dieren™esF „he
{nu™leon m—ss dieren™e is the s—me —s in the two —vor mo delD QaP F

sn order to repro du™e this m—ss splitting k—on u™tu—tions need not ˜ e ™onsideredF „his

P

simpli™—tion h—s ˜ een used in refF‘WW“ to x m a RQHwe† su™h th—t  a SXIPaqe† F „hen the

six other m—ss dieren™es —re predi™tedF „he results —re listed in t—˜le UFWF „here —lso the ™—se

is presented when f is xed to its exp eriment—l v—lue r—ther th—n f F „his yields — dierent

u %

P

v—lue for the ™utEo  —nd thus m needs to ˜ e re—djusted in order to get  a SXIPaqe† F

„his do es not signi™—ntly —lter the predi™tions on the m—ss dieren™es sin™e the r—tio f af

u %

is —lmost identi™—l in ˜ oth ™—sesF gomp—ring with t—˜le UFU it is found th—t the ˜ ound st—te

—ppro—™h improves on the m—ss dieren™esD esp e™i—lly for the spin IaP hyp eronsF

„he ˜ ound st—te —ppro—™h m—y ˜ e extended to the investig—tion of k—oni™ ™h—nnels others

th—n the €{w—veF e prominent ™—se is represented ˜y the ƒ{w—ve sin™e it in prin™iple —llows

one to explore the o dd p—rity  hyp eron whi™h is o˜served —t IRHS we† exp eriment—llyF sn the

ƒkyrme mo del this st—te h—s ˜ een studied intensivelyD see eFgF refsF‘PWD IPU “F „he ™orresp onding

xtv mo del ™—l™ul—tions —re rep orted in refF‘IQH“F „he k—on ƒ{w—ve is des™ri˜ ed ˜y the —ns—tz

 

ƒ

— @rY 3 A

~

u @r Y 3 A a X @UFIPHA

ƒ

ƒ

˜ @rY 3 A

~

elthough u p ossesses the s—me gr—nd spin @q a IaPA —s the €{w—ve @UFWRA these two ™h—nnels

ƒ

nevertheless de™ouple due to the opp osite p—rityF es for the €{w—veD — fethe{ƒ—lp eter equ—tion

ƒ

for the r—di—l fun™tion  @r A in the pourier de™omp osition

3

ƒ ƒ ƒ ƒ ƒ ƒ

— @rY 3 A a — @3 A @r A —nd ˜ @rY 3 A a — @3 A @r A @UFIPIA

I 3 P 3

n

„he numeri™—l metho d is expl—ined in refF‘IPI“F WV

C C

Q I

—nd ˜—ryons with resp e™t to the nu™leon „—˜le UFWX „he m—ss dieren™es of the lowElying

P P

in the ˜ ound st—te —ppro—™h to the xtv mo delF „he upEqu—rk ™onstituent m—ss m is ™hosen

su™h th—t the
{nu™leon m—ss dieren™e is repro du™edF ell d—t— @from refF‘WW“A —re in we†F

f xed f xed ixptF

% u

 IQP IQU IUU

 PQR PRU PSR

 QRI QSU QUW

PWQ PWQ PWQ



 QUR QUS RRT



 RVI RVS SWI

 TIQ TPP UQQ

™—n ˜ e derivedF „he solution of this equ—tion determines the ˜ ound st—te energyD 3 F „he

ƒ

qu—ntiz—tion of spin —nd isospin pro ™eeds —s for the €{w—ve yielding ™ —nd d D whi™h denote

ƒ ƒ

the ™oupling of the ƒ{w—ve ˜ ound st—te to the ™olle™tive rot—tions —nd the ™ontri˜ution of the

ƒ{w—ve ˜ ound st—te to the spinD resp e™tivelyF pollowing the qu—ntiz—tion of the €{w—ve ˜ ound

st—te it is then str—ightforw—rd to derive the m—ss formul— for the o dd p—rity 

Q1

ƒ

w @o dd p—rity A a i 3 C @UFIPPA

™l ƒ

P

V

ƒ

wherein the r—tio 1 a ™ ad do es not dep endent on the norm—liz—tion of  @r AF

ƒ ƒ ƒ

3

ƒin™e the @IRHSA is only —˜ out RHwe† ˜ elow the k—on{nu™leon threshold the ˜ ound st—te

eigenv—lue for the ƒ{w—ve is exp e™ted to lie —t 3 % RSHwe†F plu™tu—tions p ossessing enerE

ƒ

gies —s l—rge —s this v—lue m—y r—ise pro˜lems ˜ e™—use the xtv mo del is not — ™onning theoryF

sn ™—se the v—len™e qu—rk is only slightly ˜ oundD iFeF  m is sm—llD k—on energies of sever—l

v—l

hundred we† m—y s™—tter the o ™™upied v—len™e qu—rk st—te into the str—nge ™ontinuumF „his

situ—tion ™orresp onds to h—ving —n undesired nu™leon{str—nge qu—rk thresholdD i D ˜ elow the

x s

k—on{nu™leon thresholdF sn th—t ™—se the se™ond term on the ‚r ƒ of eq @UFSRA develops — p oleD

~

the p osition of whi™h determines i F ƒin™e for k—on u™tu—tions the p ertur˜—tion h ™—rries

x s

@IA

unit str—ngenessD  in @UFSRA is the eigenv—lue of — free hir—™{r—miltoni—n in the str—nge

"

se™torF „herefore  % m is the sm—llest eigenv—lue whi™h determines ji j % m  F st

" s x s s v—l

turns out th—t ji j is — monotonous ly rising fun™tion of the non{str—ng e ™onstituent qu—rk

x s

m—ss m —nd th—t for m ! RSHwe† ji j lies —˜ ove the k—on{nu™leon threshold ‘IQH“F „he

x s

predi™tions o˜t—ined in this region on the o dd p—rity  hyp eron —re displ—yed in t—˜le UFIHF

„he resulting m—ss dieren™e ˜ etween the o dd p—rity  hyp eron —nd the nu™leon is insensiE

tive to the ™onstituent m—ss mF es shown —˜ oveD the
{nu™leon m—ss dieren™e is repro du™ed

for m % RQHwe†F „henD unfortun—telyD the threshold i lies slightly ˜ elow the physi™—l

x s

k—on{nu™leon thresholdF xevertheless — re—son—˜le des™ription for ˜ oth the rel—tive p osition

of this hyp eron —s well —s the
™—n ˜ e o˜t—ined for m % RSHwe†F WW

„—˜le UFIHX €—r—meters for the ˜—ryon m—ss formul— @UFIPPA —nd the predi™tion for the m—ss

of the o dd p—rity  hyp eron rel—tive to the nu™leon w w —s fun™tions of the ™onstituent

 x

ƒ

m—ss mF elso the resulting v—lues for the
nu™leon m—ss splitting —re presentedF @h—t— t—ken

from refF‘IQH“FA

P

m @we†A  @IGqe†A 3 @we†A 1  x @we†A
x @we†A

ƒ ƒ

RSH RFUV ERTI HFRT RIV QIR

SHH RFIW ERTI HFSR RIW QSV

SSH QFUR ERSW HFSU RIT RHI

THH QFRP ERST HFTH RII RQW

V ƒumm—ry

‡e h—ve presented the des™ription of ˜—ryons —s ™hir—l solitons within the xtv mo delD

— mi™ros™opi™ mo del for the qu—rk —vor dyn—mi™sF ‡ithin gh we h—ve stressed the role

of ™hir—l symmetryD its sp ont—neous ˜re—king —nd its ™onsequen™es in h—dron physi™s —s —

˜—sis for mo deling the strong inter—™tionF purthermoreD we h—ve reviewed some imp ort—nt

fe—tures of gh in the limit of innitely m—ny ™olor degrees of freedomF „his provides the

˜—si™ motiv—tion for the pi™ture th—t ˜—ryons emerge —s soliton solutions of —n ee™tive meson

theoryF sn the fr—mework of gh — re—liz—tion of the soliton pi™ture of ˜—ryons is not fe—si˜leF

„herefore we h—ve —pproxim—ted the qu—rk —vor dyn—mi™s ˜y the xtv mo del whi™hD like

ghD is inv—ri—nt under glo˜—l ™hir—l tr—nsform—tions —nd ˜re—ks this symmetry dyn—mi™—llyD

whi™h is ree™ted ˜y — non{v—nishing qu—rk ™ondens—te in the ground st—teF

sing fun™tion—l integr—l te™hniques the xtv mo del ™—n ˜ e ™onverted —s —n ee™tive meson

theoryF „his ee™tive meson theory is highly non{lo ™—lF et low energies one ™—nD fortun—telyD

resort to —n gr—dient exp—nsionD whi™h des™ri˜ es the low energy @light —vorA meson dyn—mi™s

re—son—˜ly wellF sn le—ding order the gr—dient exp—nsion of the regul—r p—rity p—rt of the

ee™tive meson theory yields the g—uged line—r ' {mo delF „he —sso ™i—ted m—ssive g—uge ˜ osons

—re interpreted —s ve™tor —nd —xi—l{ve™tor mesonsF sn the limit of innitely l—rge @—xi—lA ve™tor

meson m—sses this —pproxim—tion to the ee™tive theory redu™es to the ƒkyrme mo delF „he

gr—dient exp—nsion of the irregul—r p—rity p—rt of the ee™tive meson theoryD whi™h involves

only o dd num˜ ers of sp—ti—l ™omp onents of vorentz{ve™torsD provides the ‡ess{umino —™tion

in le—ding orderF „he l—tter result furthermore —llows one to identify the ˜—ryon ™urrent with

the top ologi™—l ™urrent of the ƒkyrme mo delF st should ˜ e emph—sized th—t this gr—dient

exp—nsion refers to the v—™uum @or se—A p—rt of the —™tion onlyF ren™e the ƒkyrmion pi™ture

implies th—t the ˜—ryon num˜ er is ™—rried ˜y the p ol—rized hir—™ se— of the qu—rksF ƒt—ted

otherwiseD — soliton of ˜—ryon num˜ er f requires x
f v—len™e qu—rks to ˜ e ˜ ound in the

g

hir—™ se— ˜y the solitoni™ meson eldsF „his is —n inherent —ssumption of —ll purely mesoni™

soliton mo dels of ˜—ryons @‡itten9s ™onje™tureAF ‚estr—ining from the gr—dient exp—nsion the

xtv mo del —llows one to test this ™onje™turesF por this investig—tion we h—ve ™onsidered the

self{™onsistent soliton solutions of v—rious eld ™ongur—tions in the xtv mo delF elthough in

order to des™ri˜ e — unit ˜—ryon num˜ er ™ongur—tion expli™it v—len™e qu—rks h—ve to ˜ e —dded

to the ™hir—l soliton of the pseudos™—l—r elds onlyD the ˜—ryon num˜ er is indeed ™—rried ˜y the

hir—™ se— on™e —lso the —xi—l{ve™tor meson elds —re in™ludedF „his result strongly supp orts

the ƒkyrmion pi™ture of the ˜—ryonF st furthermore provides — mi™ros™opi™ expl—n—tion for

the f—™t th—t expli™it v—len™e qu—rks —nd @—xi—lA ve™tor meson degrees of freedom represent

—ltern—tive —ppro—™hes to des™ri˜ e ˜—ryons —s solitonsF IHH

sn order to simplify the investig—tion of st—ti™ ˜—ryon prop erties we h—ve resorted to the

mo del ™ont—ining only the pseudos™—l—r eld while expli™it v—len™e qu—rks —re presentF sn

order to gener—te st—tes with go o d spin —nd isospin qu—ntum num˜ ers we h—ve —dopted the

semi{™l—ssi™—l qu—ntiz—tion s™hemeF sn this s™heme the time dep endent ™o ordin—tesD whi™h

p—r—metrize the @isoA rot—tion—l zero mo desD —re qu—ntized ™—noni™—llyF ƒin™e this tre—tment

is —n—logous to the qu—ntiz—tion of — rigid top it yields @isoA rot—tion—l ˜—nds of ˜—ryonsF

„he energy splitting of the low{lying ˜—nd mem˜ ers is of order Iax while the soliton m—ss

g

is of order x F vike in ƒkyrme typ e mo dels the —˜solute v—lues for the ˜—ryon m—sses —re

g

H

gener—lly predi™ted to o l—rge if y @x A qu—ntum ™orre™tions ™orre™tions —re negle™tedF ‡e h—ve

g

estim—ted these ™orre™tions —nd found th—t they m—y ™onsider—˜ly redu™e the ˜—ryon m—sses

tow—rds their exp eriment—l v—luesF

‡hen extending this qu—ntiz—tion pres™ription to the ™—se of three —vors —nd in™luding

ƒ @QA symmetry ˜re—king expli™itly — re—son—˜le des™ription of the hyp erons is o˜t—inedF

purthermoreD —n ex™ellent des™ription of hyp erons within this —ppro—™h is somewh—t restri™ted

˜ e™—use the xtv mo del predi™ts to sm—ll — r—tio ˜ etween the k—on —nd pion de™—y ™onst—ntsY —

pro˜lem inherited from the meson se™torF e —ltern—tive des™ription of hyp erons within soliton

mo dels relies on the existen™e of — strongly ˜ ound k—on meson in the ˜—™kground of the ™hir—l

solitonF es — m—tter of f—™t this meson ˜ e™omes — zero mo de if the ƒ @QA symmetry ˜re—king

is ignoredF por the xtv mo del this tre—tment —pp—rently provides — ˜ etter des™ription of

C

I

%

the sp e™trum of the low{lying hyp eronsD esp e™i—lly for the t a st—tesF ‡hen —pplying

P

this —ppro—™h to ex™ited st—tes one is ™onfronted with the existen™e of unphysi™—l qu—rk{

@—ntiAqu—rk thresholdsF „hese ree™t the non{™onning ™h—r—™ter of the xtv mo delF yf

™ourseD —ny —ttempt to extend or rene the xtv mo del in su™h — w—y th—t this pro˜lem is

—voided would ˜ e interestingF

xevertheless the xtv soliton mo del —pp e—rs to ˜ e — well{est—˜lished —ppro—™h to study

the low{energy prop erties of the nu™leonF isp e™i—lly it provides —n —pp—rent opp ortunity to

™omp—re the ™ontri˜utions of v—len™e —nd se— qu—rks to v—rious ˜—ryon o˜serv—˜lesF

e™knowledgements

yn of us @r‚A would like to th—nk hF i˜ ert for the ™o op er—tion on the mesoni™ prop ertiesF

‡e —re gr—teful to F u™  kert —nd tF ƒ™hlienz for the ™o op er—tion on p—rts of the m—teri—l

presented in this rep ortF IHI

epp endix e

sn this —pp endix we demonstr—te the ™—l™ul—tion of the t—™o˜i determin—nt @RFSWA using

pujik—w—9s fun™tion—l integr—l metho d ‘RW “F sn order to m—ke the ™omput—tion of the ™h—nge

of the integr—l me—sureD ree™ted ˜y the rel—tion t @A Ta ID more tr—nsp—rent we will use —

few simpli™—tionsF pirstD we will only ™onsider innitesim—l ™hir—l tr—nsform—tionsD iFeF sm—ll

F ƒin™e the —xi—l —nom—ly o ™™urs only for the —˜ eli—n su˜group @IA we —ssume th—t 

e

is prop ortion— l to the unit m—trix in —vor sp—™eD  G Is F „his —ssumption will m—ke the

p

™omput—tion more fe—si˜le —s  ™ommutes with other —vor m—tri™esF „he whole ™—l™ul—tion

m—y ˜ e done ret—ining these ™ommut—tors with the result th—t the tr—™eless p—rts of  do not

™ontri˜uteF ƒe™ondD we will work with — hir—™ op er—tor ™ont—ining ˜ esides the usu—l kineti™

—nd m—ss term only — ve™tor eld

H H

iha a i@da C †aA m aX ida m X @eFIA

sn iu™lide—n sp—™e we —re working with hermiti—n hir—™ m—tri™es —nd —ntihermiti—n ve™tor

elds the op er—tor ida is hermiti—n with resp e™t to the ordin—ry s™—l—r pro du™tF „herefore the

eigenfun™tions of the op er—tor ida ™—n ˜ e ™hosen to form — ™omplete —nd orthonorm—l set

ida9 @xA a ! 9 @xA

n n n



R y

@xA9 @xA a  d x9

m nm

n

ˆ

y

@y A a  @x y AX @eFPA 9 @xA9

n

n

n

‡e exp—nd the dyn—mi™—l qu—rk eldD iFeF the fun™tion—l integr—tion v—ri—˜leD in terms of the

fun™tions 9

n

ˆ

q @xA a — 9 @xA

n n

n

ˆ



q"@xA a — 9" @xA @eFQA

n

n

n



—re indep endent —nti™ommuting qr—ssm—nn v—ri—˜lesF „he exp—nsion ™o e™ients — —nd —

n

n

„he integr—tion me—sure ™—n now ˜ e written —s

‰



@eFRA h q"h q a d— d—

m

m

m

„he ™hir—lly tr—nsformed qu—rk eld @RFSVA is given ˜y

ˆ ˆ

i@xA

S

1@xA a —~ 9 @xA a — e 9 @xA

m m m m

m m

ˆ ˆ

  i@xA

S

1"@xA a 9" @xA— ~ a 9" @xA— e X @eFSA

m m

m m

m m

where the exp—nsion ™o e™ients for the tr—nsformed qu—rk eld —re rel—ted to the origin—l ones

˜y

ˆ

—~ a g —

m mn n

n

ˆ

 

—~ a — g

nm

m n

n



i@xA R y i@xA

S S

g a hmje jni a d x9 @xAe 9 @xA

mn n

m



R y P

a  C i d x@xA9 @xA 9 @xAC y @ AX @eFTA

mn S n

m IHP

 

rere we h—ve used the orthonorm—lity of the eigenfun™tions 9 F es the sets — Y — —nd ~— Y ~—

n n m

n m

—re qr—ssm—nn v—ri—˜les we o˜t—in

P

t @A a @hetg A X @eFUA

por innitesim—lly sm—ll  the m—trix g is given ˜y

P

g a  C hmji @xAjni C y @ AX @eFVA

mn mn S

hening the m—trix

 a hmji @xAjni @eFWA

mn S

—nd shorth—ndly writing g a I C  the fun™tion—l determin—nt of this m—trix is ev—lu—ted to

˜ e in rst order in 

 

P

hetg a het@I C A a exp „r@AC y @ A

3 2



ˆ

R y

 9 @xA C XXX X @eFIHA d x@xA 9 a exp i

S n

n

n

es it st—nds the op er—tor —pp e—ring in the exp onent is not well{denedF sn order to —tt—™h —

suit—˜le denition we h—ve to regul—rize @eFIHA —nd to remove the regul—riz—tion —t the end of

the ™—l™ul—tion

ˆ

P

@! aA y

n

9 @xA  e 9 f @xA Xa lim

n S

n

3I

n

y

P

da daa

jxiX @eFIIA a lim tr hxje

S

3I

P

„o —pply the he—t kernel exp—nsion we intro du™e the ’prop er time4 ( a Ia —s —n exp—nsion

" P

p—r—meter —nd use e a d d C m —s unp ertur˜ ed op er—torF ixp—nding now the m—trix

H "

element

I

ˆ

y

(a d da ( e k

H

jy i hxje jy i a hxje h @xY y A( @eFIPA

k

k aH

we o˜t—in

I

ˆ

I

k P

f @xA a lim ( tr@ h @xY xAAX @eFIQA

S k

P

( 3H

@R% A

k aH

xote th—t only the he—t ™o e™ients h Y h —nd h in the sum @eFIQA ™ontri˜ute to f @xA ˜ e™—use

H I P

k P

the ™o e™ients of the higher order terms —re suppressed ˜y — f—™tor ( —nd therefore v—nish

in the limit ( 3 H @ 3 IAF es

tr h @xY xA a tr a H

S H S

I

tr@   Ap a H @eFIRA tr h @xY xA a

S " # "# S I

P

only one term ™ontri˜utes

"#

~

tr h @xY xA a x tr p p X @eFISA

S P ™ p "# IHQ

~

rere p is the eld strength tensor ™orresp onding to the ve™tor eld † —nd p is the du—l

"# " "#

tensorF xote th—t — —vor tr—™e h—s to ˜ e t—ken in @eFISAF iqsF @eFIQA —nd @eFISA ™—n now

˜ e ™om˜ined to the result

x

™

"#

~

tr p p @eFITA f @xA a

p "#

P

IT%

n—lly yielding the expression @RFTIA for the t—™o˜i—n t @AX

 



x

™

R P

~

d x@xAtr @p p A X @eFIUA t @A a @hetg A a exp i

p

P

V%

sing the inv—ri—n™e of the ™l—ssi™—l —™tion under @RFSVA one ™—n e—sily derive the —nom—lous

‡—rd identityF nder innitesim—l ™hir—l rot—tions the qu—rk ˜iline—r in the —™tion tr—nsforms

—s

H i@xA H i@xA

S S

q"@ida m Aq a 1e" @ida m Ae 1

H " H P

a 1" @ida m A1 C d @xA"1  1 CPim @xA"1 q~ C y @ A @eFIVA

" S S

whi™h le—ds to the following identities for the ee™tive qu—rk —™tion

 

 

H R H

e a „r log @ida m A a log h q h q" exp i d xq"@ida m Aq @eFIWA

p

a log t @A

 

 

R H " H

C log h 1h 1" exp i d x1" @ida m A1 C d @xA"1  1 CPim @xA"1 1 X

" S S

es the rst equ—tion is indep endent of  the l—st one h—s to ˜ eD to oF „hereforeD



 e x

p ™



" H

~

htr @p p Ai d h1"  1i CPim h1" 1iX @eFPHA H a  a i

p " S S

P

aH

V%  

hening the —xi—l singlet ™urrent —nd density

"

"

j a h1"  1i

S

S

j a h1" 1i @eFPIA

S S

we n—lly —rrive —t the —nom—lous ‡—rd identity @RFTWA

x

™

"

H

~

a Pim j i d j tr h@p p AiX @eFPPA

S " p

S

P

V% IHR

epp endix f

sn this ˜rief —pp endix we present the expli™it represent—tion of the eigenst—tes j# i of the

hir—™ r—miltoni—n h @TFRPDTFTQA in ™o ordin—te sp—™eF es indi™—ted in ™h—pter T we rst ™onE

stru™t eigenst—tes to the gr—nd spin op er—tor

( '

Y @fFIA C q a l C

P P

whi™h is the ve™tor sum of or˜it—l —ngul—r momentum lD spin ' aP —nd isospin ( aPF q —™ts on

the qu—rk spinorsF „hese —re o˜t—ined ˜y rst ™oupling st—tes with or˜it—l —ngul—r momentum

l with s a IaP st—tes to form st—tes ™—rrying tot—l —ngul—r momentum j —nd pro je™tion j

Q

ˆ

I

j j

Q

' iX @fFPA jl mij jl j j i a g

I

Q Q

lm '

Q

P

P

mY'

Q

ƒu˜sequently these st—tes —re ™oupled with isospin t a IaP st—tes yielding the gr—nd spin

eigenst—tes

ˆ

I

qw

( iX @fFQA jl j j ij jl j qw i a g

I

Q Q

( j j

Q Q

P

P

j Y(

Q Q

„his ™oupling s™heme o˜viously le—ds to the sele™tion rules @TFRSAF ƒin™e the upp er —nd lower

™omp onents of hir—™ spinors tr—nsform opp ositely under the p—rity tr—nsform—tion the gener—l

form of hir—™ spinorsD whi™h —re eigenst—tes of q —nd p—rity  a ID —re given ˜y

   

I I

@qYCYIA @qYCYPA

@r Ajqq C ig qw i qw i @r Ajqq ig

@qYCA

# #

P P

a

C @fFRA

I I

# @qYCYIA @qYCYPA

@r Ajq CIq C @r Ajq Iq f qw i f qw i

# #

P P

   

I I

@qYYIA @qYYPA

qw i qw i @r Ajq CIq C ig @r Ajq Iq ig

@qYA

# #

P P

a C X @fFSA

I I

# @qYYIA @qYYPA

@r Ajqq C @r Ajqq f qw i f qw i

# #

P P

„he se™ond sup ers™ript l—˜ els the intrinsi™ p—rity  —nd enters the p—rity eigenv—lue vi—

intr

q

 a @IA   F  represents — go o d qu—ntum num˜ er sin™e ˜ oth gr—nd spin —nd p—rE

intr intr

ity op er—tors ™ommute with hF purthermoreD the gr—nd spin inv—ri—n™e is very helpful when

™ho osing —ns—tze for meson elds other th—n the ™hir—l eldD see se™tion TFPF „he di—gon—liz—E

@qYCYIA

@r AD et™F —s line—r ™om˜in—tions of spheri™—l tion @TFRQDTFSWA yields the r—di—l fun™tions g

#

fessel fun™tions @the solutions to the free pro˜lemAF iFgF

q

ˆ

@qYCYIA

g @r A a IC mai j @k r AY † ‘“x

k q q k q # k k

#

k

q

ˆ

@qYCYIA

f @r A a † ‘“x sgn @i A I mai j @k r A @fFTA

# k k k q k q qCI k q

#

k

—

where the eigenve™tors † ‘“ —re o˜t—ined ˜y di—gon—lizing hF sn ™—se th—t the r—miltoni—n is

# k

not rermiti—n @TFTQA these eigenve™tors —re ™omplexF x —re norm—liz—tion ™onst—ntsF i a

k k q

q

P

P

 m C k denote the energy eigenv—lues in the —˜sen™e of the solitonF „he momentum

k q

eigenv—lues k —re su˜ je™t to the ˜ ound—ry ™ondition j @k h A a H ‘VI “F „here —re —lso

k q q k q

other ˜ ound—ry ™onditions dis™ussed in the liter—ture ‘TPD WV “ —nd the p ertinent ˜ ound—ry

™ondition dep ends on the pro˜lem under ™onsider—tionF

—

qr—nd spin —nd p—rity indi™es —re omittedF IHS

epp endix g

sn this —pp endix the equ—tions of motion for the meson elds —s t—ken from refF‘UQD US “

—re listedF „hese equ—tions result from the st—tion—ry ™ondition  i a 9 for the winkowski

energy fun™tion—l @TFQWAF „he fun™tion—l deriv—tives of the one{p—rti™le energy eigenv—lues 

#

—re extr—™ted from the iu™lide—n hir—™ r—miltoni—n @TFTQAF

—

st is —ppropri—te to dene densities

n o n o

v—l v—™ v—l v—™

˜ @xY y AC ˜ @xY y A & @xY y AC & @xY y A —nd ˜@xY y A a x &@xY y A a x

™ ™

n o

v—l

" "

`2 @xA2 @y AC a2 @xA2 @y A & @xY y A a 

v—l v—l v—l v—l v—l

n o

ˆ

v—™

" "

& @xY y A a f @ aA`2 @xA2 @y AC f @ aAa2 @xA2 @y A

‚ # # # s # # #

#

' &

y y

v—l

@y A @y A a2 @xA2 `2 @xA2 ˜ @xY y A a 

v—l v—l v—l

v—l v—l

& '

ˆ

v—™ y y

˜ @xY y A a f @ aA`2 @xA2 @y A f @ aAa2 @xA2 @y A X @gFIA

s # # ‚ # #

# #

#

‚e—l @`A —nd im—gin—ry @aA p—rts refer to the exp—nsion ™o e™ients † ‘9“ of the free ˜—sis

# k

@™fF eq @fFTAAF „he regul—tor fun™tions

V

 

 

 

 I

‚

#

b

b

 e not regul—rizedY sgn @ Aerf™ 

s

`

#

P 

f @ aA a @gFPA

‚ #

 

 

b

 

b

I  I

P ‚ ‚ s

#

X

p

  C aA AY e regul—rized Aerf™ aAexp@ @ sgn @ @

s

# # #

P  %

V

IY e not regul—rized

b

s

`

I

‚

@gFQA A sgn @ f @ aA a

 

 

s #

#

b

 



P

# X

erf™  e regul—rized 

s



ree™t the deriv—tive of the energy fun™tion—l with resp e™t to the single qu—rk energy eigenE

v—luesF „he sp e™i™ form of the equ—tions of motion  i a 9 n—lly ˜ e™omes



H H

m m d

”

@r A a ™os@r A @™os@r AC i (
r sin@r AA &@xY xA @gFRA tr

S

P P

m m f R%

% %



d m

”

@sin@r A i (
r ™os @r AA &@xY xA @gFSA tr sin @r A a

S

P P

R% m f

% %



P

g d

†

3 @r A a ˜@xY xA @gFTA tr

P

Rm R%

†



P

g d

†

”

tr q@r A a @@  r A
( A &@xY xA @gFUA

P

Rm R%

†



P

d g

†

” ”

tr  @Q@'
r A@(
r A @'
( AA &@xY xA @gFVA p @r A a

P

Rm R%

†



P

d g

†

” ”

tr r @r A a  @@'
r A@(
r A @'
( AA &@xY xAX @gFWA

P

Rm R%

†

‚

d ˜@rY r A rere the tr—™e is t—ken with resp e™t to hir—™ —nd —vor indi™es onlyF xote th—t tr

represents the ˜—ryon ™h—rge densityF sn eq @gFTA isospin inv—ri—n™e h—s ˜ een —ssumedD iFeF

m a m a m F

3 & †

—

2 @xA repres ents the ™o ordin—te sp—™e repres ent—tion of j# iF

# IHT

epp endix h

sn this —pp endix we present the expli™it expressions for the pref—™tors of the ™olle™tive

op er—tors whi™h —re relev—nt for the des™ription of the hyp erons —s des™ri˜ ed in se™tion UFSFIF

„he expli™it expression for
m—y re—dily ˜ e found in refF‘TP“ where the ™orresp onding

—˜

™—l™ul—tion is des™ri˜ ed

ˆ

x

g

v—™ y

p

a
w f @ Y  Y Ah"j! j# ih# j„  ! „ j"i @hFIA

" # — ˜

—˜

P Q

"#

with the ™utEo fun™tion

   

   

    

 "

#

sgn @ Aerf™ sgn@ Aerf™

   

# "

 

f @ Y  Y A a X @hFPA

" #

 

" #

„he ™ontri˜ution of the hir—™ se— to the ™o e™ient  re—ds

 

 

ˆ

 

 Px
w

" g

y v—™

 

p

sgn @ Aerf™ h"j„  ! „ j"iX @hFQA  a

" V

 



Q

"

„he se™ond order terms in the exp—nsion of the v—™uum ™ontri˜ution to the fermion deE

termin—nt m—y ˜ e expressed —s dou˜le mo de sums over the eigenst—tes of the st—ti™ hir—™

r—miltoni—n @TFRPA

ˆ

x

g

v—™

 a f @ Y  Y Ah"j! j# ih# j! j"i @hFRA

 " # — ˜

—˜

R

"#

where the ™utEo fun™tion f @ Y  Y A is dened in @UFIRAF „he symmetry ˜re—king terms —re

 " #

found to ˜ e

ˆ

x

g

P y y v—™

@m m A f @ Y  Y Ah"j„  ! „ j# ih# j„  ! „ j"i @hFSA a

s " # — ˜

—˜

Q

"#

with the ™utEo fun™tion f @ Y  Y A ˜ eing dened in @hFPAF

" #

„he —n—logous expressions —sso ™i—ted with the expli™it o ™™up—tion of the v—len™e qu—rk

level —re found to ˜ e

P

y v—l

x @m m A hv—lj„  „ jv—liY @hFTA  a

g s v—l

Q

ˆ

hv—l j! j"ih"j! jv—li x

— ˜ g

v—l

  a Y @hFUA

v—l

—˜

P  

" v—l

"Tav—l

y

ˆ

hv—lj! j"ih"j„  ! „ jv—li x

— ˜ g

v—l

p

@m m A Y @hFVA
a

s v—l

—˜

 

Q

" v—l

"Tav—l

y y

ˆ

hv—lj„  ! „ j"ih"j„  ! „ jv—li P

— ˜

P v—l

x @m m A  Y @hFWA a

g s v—l

—˜

Q  

" v—l

"Tav—l

wherein jv—li denotes the v—len™e qu—rk st—te F pin—lly the ™o e™ients —pp e—ring in the

v—l

mesoni™ p—rt of the ™olle™tive —™tion @UFVSA —re given ˜y

5 3 2 3 42

   



H H

m V% m m m

s s

P s P m P s

dr r @I ™osAY@hFIHA I m  a C PC f I PC

% %

H H

m W m m m IHU

2 3

 



H

V% m m

s

P P s m m P

m f I I a a dr r @I ™os A Y @hFIIA

% % „ V

H

W m m

2 3

    



H

 V% m m

s

P P m P s

X @hFIPA dr r I ™os m I f a I

% ƒ %

H

Q m P m

„hese v—rious ™ontri˜utions n—lly —dd up to the ™olle™tive v—gr—ng i—n whi™h m—y ˜ e ™—st

into the form

p

Q U

ˆ ˆ

I Q I I

P P P P

v a i C     C f   @I h A

tot V VV

i 

P P P P

iaI aR

Q U

ˆ ˆ

I I I

h h h h @I h h A

Vi Vi V V „ ƒ V VV VV

P P P

aR

iaI

U Q

ˆ ˆ

 h X @hFIQA  h C
C

 V i Vi ƒ „

aR

iaI

„he ™o e™ients —re sums of the qu—ntities listed —˜ ove

v—™ v—™ P v—l v—™ P v—l

C  Y f a  C f C  Y  a  C   a 

s v—l

RR RR QQ QQ

v—l v—™ v—l v—™ m v—™ v—™ m

 a  C  C P C P C  Y a C C Y

„

VV VV QQ QQ „

m v—™ v—™ m v—™ v—™

Y C Y a C C a

V ƒ

V VV VV ƒ RR RR

v—™ v—™ v—™ v—™

X @hFIRA C
Y
a
C

a

ƒ „

ƒ ƒ „ „

‡henever ne™ess—ryD the ™ontri˜ution of the trivi—l @$ a IA eld ™ongur—tion h—s to ˜ e

H

su˜tr—™ted sin™e the fermion determin—nt is norm—lized —™™ordinglyF

sn order to o˜t—in the ™olle™tive r—miltoni—n from @hFIQA it is imp ort—nt to note th—t the

•

timeEderiv—tive of the ™olle™tive rot—tions h a f  h only —e™ts the right indi™es of the

l˜ ˜™d ™ ld

—djoint represent—tion @see —lso eq @UFIUAAF ren™e the moment— —sso ™i—ted with d vad  —re

—

the right gener—tors ‚ of ƒ @QAF „hese o˜ ey the —lge˜r— ‘‚ Y ‚ “ a f ‚ F xoting th—t

— — ˜ —˜™ ™

‘‚ Y ‚“ a ‚! whi™h —lso implies th—t ‘‚ Y h “ a if h one e—sily veries th—t

— — — l˜ —˜™ l™

•

d $ @xY tA i

X @hFISA ‚ Y $ @xY tA“ a ‘

—

d  P

—

„hus the qu—ntiz—tion pres™ription re—ds

V

P

@  C
h A a t Y —aIDPDQ

b

— „ V— —

`

d v

P

@  C
h AY —aRDFFDU

@hFITA ‚ a a

— ƒ V—

—

p

b

d 

X

—

Q

f Y —aV

P

wherein t @i a IY PY QA denote the spin op er—torsF „his rel—tion is the —vor ƒ @QA extension

i

of @UFIVAF

„he r—miltoni—n op er—torX

V

ˆ

‚  v @hFIUA r a

— —

—aI

m—y ˜ e di—gon—lized ex—™tly ˜y gener—lizing ‘IQI“ the ‰—˜uEendo ‘PV“ —ppro—™h to more ™omE

pli™—ted symmetry ˜re—king termsF sn the origin—l ‰—˜uEendo —ppro—™h only the  @I h A

VV IHV

typ e symmetry ˜re—king term w—s ™onsideredF st should ˜ e noted th—t the ™onstr—int ‚

V

p

Qf aP a H ™ommutes with the r—miltoni—n —nd therefore is rst ™l—ssF ren™e the ˜—ryon

p

w—ve{fun™tions live on the hyp ersphere ‚ a Q aP for f a IF yn this hyp ersphere only

V

st—tes with h—lf{integer spin existF „hus the r—miltoni—n h—s the —ppre™i—ted fe—ture th—t

its eigenst—tes —re fermions ‘SPD PT D PU“F „he ™orresp onding energy eigenv—lue for ˜—ryon f

is found to ˜ e given ˜y the formul— @UFWHAF „he qu—ntity  —pp e—ring in th—t equ—tion

ƒ f

represents the eigenv—lue of the ƒ @QA op er—tor

€ €

Q U

P P P

g C   @I h AC  @
a A h @P‚ C
h A C
h @P‚ C
h A

P VV „ Vi i „ Vi ƒ V  ƒ V

iaI aR

€ €

U Q

P P P

h h Y @hFIVA h h C  C @I h h AC 

V V Vi Vi ƒ V VV VV „

aR iaI

€

V

P

‚ denotes the qu—dr—ti™ g—simir op er—tor of ƒ @QAF veft ƒ @QA gener—tors where g a

P

—aI —

p

€

V

Q ˜ eing the h ‚ with s a v D @— a IY PY QA —nd ‰ a Pv a —re ™onstru™ted vi— v a

—˜ ˜ — — V —

˜aI

isospin —nd hyp er™h—rge op er—torsD resp e™tivelyF e p ertinent p—r—metriz—tion of the ™olle™tive

rot—tions in terms of eight ’iuler —ngles4 is given ˜y ‘PV“

p

i# ! H H H i@&a QA!

R V

‚ a ‚ @Y  Y  Ae ‚ @ Y  Y  Ae X @hFIWA

s t

rere ‚ represents pure isospin tr—nsform—tions —ndD due to the hedgehog stru™ture of the

s

solitonD ‚ ™orresp onds to sp—ti—l rot—tionsF por the p—r—metriz—tion @hFIWA the gener—tors

t

‚ —re expressed —s dierenti—l op er—tors in the ’iuler —ngles4 Y  Y XXXY &F imploying the

—

expli™itly forms of the ‚ D —s given in —pp endix e of refF‘IQI“D the op er—tor @hFIVA ™—n ˜ e

—

formul—ted —s — se™ond order dierenti—l op er—tor for the ’iuler —ngle4 # F „his suggests the

—ns—tz for the eigenfun™tions ‘PV“

ˆ

@s A @t A

t t i‰ & H H H

Q

‚

@‚A a @IA h @Y  Y  Af @# Ae h @ Y  Y  AY @hFPHA

w w ‰ s s Yt t

Q Q s w w t

v ‚

Q Q

v ‚

w w

v ‚

H

where the h s denote the ƒ @PA ‡igner fun™tionsF „he sum over the intrinsi™ pro je™tion

p

num˜ ers w is su˜ je™t to the ™ondition w w a ‰ aP Q for f a IF sn order to

vY‚ v ‚

n—lly o˜t—in the eigenv—lue  @—nd thus the ˜—ryon sp e™trumA — system of ™oupled se™ond{

ƒ f

order dierenti—l equ—tions for the isos™—l—r fun™tions f @# A h—s to ˜ e integr—tedF „he

w w

v ‚

f @# A de™ouple for the v—rious isospin ™h—nnels —nd the integr—tion is —™™omplished ˜y

w w

v ‚

st—nd—rd @numeri™—lA te™hniquesF IHW

epp endix i

@IYPA

rere we ˜riey displ—y the kernels  entering the fethe{ƒ—lp eter equ—tion @UFWTA whi™h

represents the st—rting p oint for the ˜ ound st—te des™ription of the hyp eronsF elso presented

—re the kernel  for the str—ngeness ™h—rge @UFWWA —s well —s qu—ntities relev—nt for the semi{

ƒ

™l—ssi™—l qu—ntiz—tion of the k—on ˜ ound st—teF „hese expressions —re ™omputed in refF‘WU“ —nd

the results —re re—dily t—ken overF

@IA

„he lo ™—l kernel  @r A —™quires ™ontri˜utions from the meson p—rt of the —™tion —s well

—s those terms involving h

@PA

3 2

 

H

m % m

s

s

P P @IA

IC m f  @r A a ™os C

% %

H

P m m



d x

y g

@r Au @r A u @r A2 @r A @iFIA  @m C m A 2

H H v—l v—l s

v—l

R R%

 

n

I

ˆ

d x

P ds

g

y s

"

p

2 @r Au @r A u @r A2 @r A  e @m C m A

H H " " s

"

P

R% R

Ia

R% s

"ans

o

ˆ

P

y s

&

C @r A 2 @r A X 2  e

& &

&

&as

‚

„he integr—l @daR% A indi™—tes th—t the —ver—ge with reg—rd to the intern—l degrees of freedom

h—s ˜ een t—kenF por ™onvenien™e the unit—ryD selfE—djoint tr—nsform—tion m—trix u D whi™h

H

represents — mo died form of the ™hir—l rot—tion @UFRPA h—s ˜ een intro du™ed

 

 

” ”

X @iFPA i r
( ™os u @r A a i  r
( „ a  sin

S H S

P P

@PA H

y˜viously u @r A —™ts —s — unit m—trix on the str—nge spinorsF „he ˜ilo ™—l kernel  @3 Y rY r A

H

H

origin—tes from the terms qu—dr—ti™ in h —nd is symmetri™ in r —nd r

@IA

@

y y

 

H H H H

ˆ

@r Au @r A2 @r A 2 @r Au @r A2 @r A2 d d x

&

H v—l H & g

v—l

P @PA H

 @m C m A  @3 Y rY r A a

v—l s

R R% R%  3 

v—l &

&as

A

ˆ

H H H y y

@r Au @r A2 @r A‚ @3 A @iFQA @r Au @r A2 @r A2 2

H " "Y& H &

& "

"ans

&as

„he regul—tor fun™tion —pp e—ring eq @iFQA h—s ˜ een o˜t—ined —s

P P

r



s s

n

& " I

C e s e

P P

C ‘3 @ C  A “‚ @sY 3 Y  Y  A ds ‚ @3 A a

" & H " & "Y&

P

s %

Ia

o

P3  ‚ @sY 3 Y  Y  ACP3  ‚ @sY 3 Y  Y  A X @iFRA

& I " & " I & "

„he peynm—n p—r—meter integr—ls —re dened in eq @UFSPA



 

I

i P P P

x dx exp s‘@I xA C x x@I xA3 “ X @iFSA ‚ @sY 3 Y  Y  A a

i " #

" #

H

H

ƒimil—rly the ˜ilo ™—l kernel  @3 Y rY r A for the str—ngeness ™h—rge @UFWWA is given ˜y

ƒ

y

 

H H H H

ˆ

d d 2 @r Au @r A2 @r A2 @r Au @r A2 @r A

H & & H v—l

v—l

H

 @3 Y rY r A a  @iFTA

ƒ v—l

P

R% R% @ 3  A

v—l &

&as IIH

  

H

n o

I

ˆ

ds d d

P

s P y y H

&

p

e @I Ps A2 @r A  2 @r A s 2 @r A 2 @r A  @r r A C

& & & & &

&

P

R% R%

Ia

R% s

&as

r

  

H

I

ˆ

s d d

y H y H H

ds 2 @r A u @r A2 @r A2 @r A u @r A2 @r A@ C  3 A

" H & & H " " &

P

R% R% R%

Ia

"ans

&as

n  o

 ‚ @sY 3 Y  Y  A s@3 C  C  A @ 3 A‚ @sY 3 Y  Y  AC 3 ‚ @sY 3 Y  Y  A X

H " & " & & I " & P " &

„o ™on™lude this —pp endix we —lso list the sp e™tr—l fun™tions ™@3 A —nd d@3 A whi™h enter the

rel—tion ˜ etween the spin —nd k—on spin exp e™t—tion v—lues @™fF eqs @UFIIQA —nd @UFIISAAF st

is —ppropri—te to list the ™ontri˜utions stemming from the expli™it o ™™up—tion of the v—len™e

~

qu—rk level —nd the p ol—rized hir—™ se— sep—r—telyF sn wh—t follows h —re understo o d —s

@iA

the p ertur˜—tive p—rts of the hir—™ r—miltoni—n @UFRQA with the prop erly norm—lized solution

 @r A to the fethe{ƒ—lp eter equ—tion @UFWTDUFIHHA su˜stituted

3

™@3 A a ™ @3 AC ™ @3 A

v—l v—™

h i

~

ˆ

hv—ljh @3 Y 3 Aj"ih"j( jv—li

Q

@PA

™ @3 A a  C hX ™X

v—l v—l

 

v—l "

"ans

i h

~ ~

ˆ

hv—l jh @3 Aj&ih&jh @3 Aj"ih"j( jv—li

Q

@IA @IA

C hX ™X C

v—l

@  A@ 3  A

"ans

v—l " v—l &

&as

@

P

P



s

s

I

"

#

ˆ

 e ds  e

# "

~

p

@iFUA h"j( j# ih# jh @3 Y 3 Aj"i ™ @3 A a

Q v—™

@PA

P

 

Ia

R% s

" #

"Y# anXsX

A

P

P

s

s

i h

"

#

ˆ

e e

~ ~

C s‚ @3 A X C h"j( j# ih# jh @3 Aj&ih&jh @3 Aj"i 

"Y#Y& Q #

@IA @IA

P P

 

"Y# anXsX

" #

&asX

„he relev—nt regul—tor fun™tion is given ˜y — peynm—n p—r—meter integr—l

@



 

I

P P P

dx @3 C  C  A exp s‘@I xA ‚ @3 A a C x x@I xA3 “

# & "Y#Y&

# &

H



Q P P

C P@I xA3 C ‘ C  “3 ‘P@I xA@ C  A@ C  A @  A “3 @iFVA

" # # & " & " #

A

P

P

sx

sx

  

"

#

e e

P P

x@I xA3 “ X exp s‘@I xA C@ C  A@ C  A@ C  A

" # " & # &

&

P P

 

" #

pin—lly d@3 A is found to ˜ e

~ ~

ˆ

hv—ljh @rY 3 Aj&ih&jh @rY 3 Ajv—li

@IA @IA

d @3 A a P w

v—l v—l &

P

@ 3  A

v—l &

&as



n

I

ˆ

ds P

P s

" ~

p

@I Ps w e d @3 A a P A h"jh @rY 3 Y 3 Aj"i

" v—™

@PA

"

P

Ia

R% s

"ans

o

~ ~

s h"jh @rY 3 Ah @rY 3 Aj"i

"

@IA @IA

r



I

ˆ

s

~ ~

ds w h"jh @rY 3 Aj# ih# jh @rY 3 Aj"i@ C  3 A @iFWA P

# " #

@IA @IA

P

R%

Ia

"#

n  o

 ‚ @sY 3 Y  Y  A s@3 C  C  A @ 3 A‚ @sY 3 Y  Y  AC 3 ‚ @sY 3 Y  Y  A X

H " # " # # I " # P " # III

‚eferen™es

‘I“ pF tF ‰ndur—inD „he „heory of u—rk —nd qluon snter—™tionsD ƒpringer{†erl—gD ferlin

E reidel˜ erg E xew ‰orkD IWWQF

‘P“ „F wut—D pound—tions of u—ntum ghromodyn—mi™sD ‡orld ƒ™ienti™D ƒing—p oreD IWVUF

‘Q“ rF tF ‚otheD v—tti™e q—uge „heories { en sntrodu™tionD ‡orld ƒ™ienti™D ƒing—p oreD

IWWPF

‘R“ sF wontv—y —nd qF wunsterD  u—ntum elds on — l—tti™eD g—m˜ridge nivF €rF @g—mE

˜ridge monogr—phs on m—them—ti™—l physi™sAD IWWRF

‘S“ „F €F gheng —nd vF pF viD q—uge „heory of ilement—ry €—rti™lesD ™h—pter SD gl—rdenon

€ressD IWVVF

‘T“ qF 9t ro oftD xu™lF €hysF fUP @IWURA RTIY fUS @IWUSA RTIF

‘U“ iF ‡ittenD xu™lF €hysF fITH @IWUWA SUF

‘V“ „F rF ‚F ƒkyrmeD €ro ™F ‚F ƒo ™ IPU @IWTIA PTHF

‘W“ qF ƒF edkinsD gF ‚F x—ppiD —nd iF ‡ittenD xu™lF €hysF fPPV @IWVQA SSPF

‘IH“ FEqF wei%nerD €hysF ‚epF ITI @IWVVA PIQF

‘II“ fF ƒ™hwesingerD rF ‡eigelD qF rolzw—rthD —nd eF r—y—shiD €hysF ‚epF IUQ @IWVWA IUQF

‘IP“ qF rolzw—rthD editorD f—ryons —s ƒkyrme ƒolitonsD ‡orld ƒ™ienti™D IWWRF

‘IQ“ FEqF wei%nerD xF u—iserD —nd ‡F ‡eiseD xu™lF €hysF eRTT @IWVUA TVSF

‘IR“ „F ‡—lhout —nd tF ‡—m˜—™hD €hysF ‚evF vettF UT @IWWIA QIRF



‘IS“ yF u—ym—k™—l—nD ƒF ‚— jeevD —nd tF ƒ™he™hterD €hysF ‚evF hQH @IWVRA SWRY



yF u—ym—k™—l—n —nd tF ƒ™he™hterD €hysF ‚evF hQI @IWVSA IIHWF

‘IT“ wF f—ndoD „F uugoD —nd uF ‰—m—w—kiD €hysF ‚epF ITR @IWVVA PIUF

‘IU“ ‰F x—m˜u —nd qF ton—Ev—s inioD €hysF ‚evF IPP @IWTIA QRSY IPR @IWTIA PRTF

‘IV“ „F r—tsud— —nd „F uunihiroD €hysF ‚epF PRU @IWWRA PPIF

‘IW“ iF i˜ ertD rF ‚einh—rdtD —nd wF †olkovD ie™tive r—dron „heoryD in €rogress in €—rti™le

—nd xu™le—r €hysi™sD volF QQD edF eF p—%lerD €erg—mon €ressD yxfordD IWWRF

‘PH“ rF ‚einh—rdtD €hysF vettF fPRR @IWWHA QITF

‘PI“ eF fu™kD ‚F elkoferD —nd rF ‚einh—rdtD €hysF vettF fPVT @IWWPA PWY

xF sshiiD ‡F fentzD —nd uF ‰—z—hiD €hysF vettF fQHI @IWWQA ITSY fQIV @IWWQA PTY

‚F „F g—hillD gF hF ‚o˜ ertsD —nd tF €r—s™hifk—D eustr—li—n tourn—l of €hysi™sD RP @IWVWA

IPWF

‘PP“ F u™  kertD ‚F elkoferD rF ‚einh—rdtD —nd rF ‡eigelD work in progressF IIP

‘PQ“ hF ‚F snglisD €hysF ‚evF WT @IWSRA IHSWF

‘PR“ fF wouss—ll—m —nd hF u—l—f—tisD €hysF vettF fPUP @IWWIA IWTF

‘PS“ qF rolzw—rthD xu™lF €hysF eSUP @IWWRA TWF

‘PT“ iF qu—d—gniniD xu™lF €hysF fPQT @IWVRA ISF

‘PU“ eF €F f—l—™h—ndr—nD eF f—rdu™™iD pF vizziD †F ‚o dgersD —nd eF ƒternD xu™lF €hysF fPST

@IWVSA SPSF

‘PV“ rF ‰—˜u —nd uF endoD xu™lF €hysF fQHI @IWVVA THIF

‘PW“ gF g—ll—n —nd sF ule˜—novD xu™lF €hysF fPTP @IWVSA QTSY

sF ule˜—nov in r —dr ons —nd r —dr oni™ w —tter D p—ge PPQD pro ™eedings of the xe„y

edv—n™ed ƒtudy snstituteD g—rgeseD IWVWD edited ˜y hF †—utherinD tF xegele —nd pF

venz @€lenum €ress IWVWAF

‘QH“ gF g—ll—nD uF rorn˜ ostelD —nd sF ule˜—novD €hysF vettF fPHP @IWVVA PWTF

‘QI“ hF i˜ ert —nd rF ‚einh—rdtD xu™lF €hysF fPUI @IWVTA IVVF

‘QP“ rF ‚einh—rdt —nd fF †F h—ngD xu™lF €hysF eSHH @IWVWA STQF

‘QQ“ iF ‡ittenD gommF w—thF €hysF WP @IWVRA RSSF

‘QR“ hF qepnerD xu™lF €hysF fPSP @IWVSA RVIF

‘QS“ rF ‚einh—rdtD ie™tive r—dron „heory of ghD €ro ™eedings of the igypti—n{qerm—n

ƒpring ƒ™ho ol —nd gonferen™e on €—rti™le —nd xu™le—r €hysi™sD p—ge ISID g—iroD epril

IWWPF

‘QT“ ƒF vF edlerD €hysF ‚evF IUU @IWTWA PRPTY

tF ƒF fell —nd ‚F t—™kiwD xuovF gimF THe @IWTWA RUF

‘QU“ tF tF ƒ—kur—iD gurrents —nd wesonsD niversity of ghi™—go €ressD IWTWF

‘QV“ iF ii™hten —nd pF pein˜ ergD €hysF ‚evF hPQ @IWVI A PUPRY

wF fF †oloshin —nd wF eF ƒhifm—nD ‰—dF pizF RS @IWVUA RTQ @ƒovF tF xu™lF €hysF RS

@IWVUA PWPAY

xF ssgur —nd wF fF ‡iseD €hysF vettF fPQP @IWVWA IIQY fPQU @IWWHA SPUF

‘QW“ „F epp elquist —nd tF g—r—zzoneD €hysF ‚evF hII @IWUSA PVSTF

‘RH“ €—rti™le h—t— qroupD €hysF ‚evF hSH @IWWRA IIUIF

‘RI“ wF qell{w—nnD ‚F tF y—kesD —nd fF ‚ennerD €hysF ‚evF IUS @IWTVA PIWSF

‘RP“ wF qell{w—nnD €hysF ‚evF IPS @IWTPA IHTUY

ƒF yku˜ oD €rogF „heorF €hysF PU @IWTPA WRWF

‘RQ“ tF q—sser —nd rF veutwylerD €hysF ‚ep VU @IWVPA UUF

‘RR“ tF ƒ™he™hterD eF ƒu˜˜—r—m—nD —nd rF ‡eigelD €hysF ‚evF hRV @IWWQA QQWF IIQ

‘RS“ „F f—nks —nd eF g—sherD xu™lF €hysF fITW @IWVHA IHQF

‘RT“ rF veutwyler —nd eF ƒmilg—D €hysF ‚evF hRT @IWWPA STHUF

‘RU“ iF ‡ittenD xu™lF €hysF fIST @IWUWA PTWY

€F hi †e™™hi— —nd qF †enezi—noD xu™lF €hysF fIUI @IWVHA PSQY

gF ‚osenzweigD tF ƒ™he™hterD —nd qF „r—hernD €hysF ‚evF hPI @IWVHA QQVVF

‘RV“ tF ƒ™hwingerD €hysF ‚evF VP @IWSIA TTRF

‘RW“ uF pujik—w—D €hysF ‚evF hPI @IWVHA PVRVF

‘SH“ uF u—w—r—˜—y—shi —nd wF ƒuzukiD €hysF ‚evF vettF IT @IWTTA PSSY

‚i—zuddin —nd p—yy—zuddinD €hysF ‚evF IRU @IWTTA PSSF

‘SI“ ƒF ‡ein˜ ergD €hysF ‚evF vettF IV @IWTUA SHUF

‘SP“ iF ‡ittenD xu™lF €hysF fPPQ @IWVQA RPPD RQQF

‘SQ“ qF rF herri™kD tF w—thF €hysF S @IWTRA IPSPF

‘SR“ rF ‚einh—rdt —nd eF ‡irz˜—D gontri˜ution to the €exsg gonferen™eD „okyoD IWVUF

‘SS“ tF ukD F €hysF gPW @IWVSA QHQF

‘ST“ rF ‚einh—rdtD €hysF vettF fIVV @IWVUA RVPF

‘SU“ fF u—mpfer —nd rF ‚einh—rdtD ennF €hysF I @IWWPA PWTF

‘SV“ gF ‚iggen˜—™hD tF q—sserD tF pF honoghueD —nd fF ‚F rolsteinD €hysF ‚evF hRQ @IWWIA

IPUY

gF hF ‚o˜ ertsD ‚F „F g—hillD wF iF ƒeviorD —nd xF s—nnell—D €hysF ‚evF hRW @IWWRA IPSF

‘SW“ tF q—sser —nd rF veutwylerD ennF €hysF @x‰A ISV @IWVRA IRPF

‘TH“ rF ‚einh—rdt —nd hF i˜ ertD €hysF vettF fIUQ @IWVTA RSWF

‘TI“ eF t—™ksonD eF hF t—™ksonD eF ƒF qoldh—˜ erD qF iF frownD —nd vF gF g—stillejoD €hysF

vettF ISRf @IWVSA IHIF

‘TP“ rF ‡eigelD ‚F elkoferD —nd rF ‚einh—rdtD xu™lF €hysF fQWU @IWWPA TQVF

‘TQ“ wF t—minonD ‚F wendezEq—l—inD qF ‚ipk—D —nd €F ƒt—ss—rtD xu™lF €hysF eSQU @IWWPA

RIVF

‘TR“ eF flinD fF rillerD —nd wF ƒ™h—denD F €hysF eQQI @IWVV A USF

‘TS“ ‚F elkofer —nd rF ‚einh—rdtD „he ghir—l enom—ly —nd ile™trom—gneti™ €ion he™—y in

xtv{like wodelsD „u˜ingen  niversity preprint IWWPD unpu˜lishedF

‘TT“ xF ƒF w—nton —nd €F tF ‚u˜—™kD €hysF vettF fIVI @IWVTA IQUF

‘TU“ rF ‡eigelD ‚F elkoferD —nd rF ‚einh—rdtD istim—te of u—ntum ™orre™tions to the w—ss

of the ghir—l ƒoliton in the x—m˜u{ton—{v—sinio wodelD „u˜ingen  niversity preprintD

hepEphGWRHV QRQ D eugust IWWRD xu™lF €hysF eD to ˜ e pu˜lishedF IIR

‘TV“ ‚F pF h—shenD fF r—ssl—™herD —nd eF xeveuD €hysF ‚evF hIP @IWUSA PRRQF

‘TW“ ‚F ‚— j—r—m—nD ƒolitons —nd snst—ntonsD xorth roll—ndD IWVPF

‘UH“ rF ‚einh—rdtD xu™lF €hysF eSHQ @IWVWA VPSF

‘UI“ rF ‡eigelD F u™  kertD ‚F elkoferD —nd rF ‚einh—rdtD yn the —n—lyti™ properties of ™hir—l

solitons in the presen™e of the 3 mesonD „u˜ingen  niversity preprintD hepEphGWRHUQ HRD

tuly IWWRD xu™lF €hysF eD to ˜ e pu˜lishedF



‘UP“ gF ƒ™hurenD  gontri˜ution —t the i„g workshop on „he u—rk ƒtru™ture of f—ryons in

„rentoD y™to˜ er IWWQD unpu˜lishedF

‘UQ“ F u™  kertD ‚F elkoferD rF ‚einh—rdtD —nd rF ‡eigelD xu™lF €hysF eSUH @IWWRA RRSF

‘UR“ „F hF gohen —nd wF f—nerjeeD €hysF vettF fPQH @IWVWA IPWF

‘US“ F u™  kertD ‚F elkoferD rF ‚einh—rdtD —nd rF ‡eigelD „he ƒkyrmion limit of the x—m˜u{

ton—Ev—sinio solitonD „u˜ingen  niversity preprintD hepEphGWRHUQ HSD tuly IWWR F

‘UT“ ‚F elkoferD rF ‚einh—rdtD rF ‡eigelD —nd F u™  kertD €hysF vettF fPWV @IWWQA IQPF

‘UU“ „F ‡—t—˜ e —nd rF „okiD €rogF „heorF €hysF VU @IWWPA TSIF

‘UV“ gF ƒ™hurenD  iF ‚F erriol—D —nd uF qo ekeD €hysF vettF fPVU @IWWPA PVQF

‘UW“ gF ƒ™hurenD  pF horingD iF ‚uizEerriol—D —nd uF qo ekeD xu™lF €hysF eSTS @IWWQA TVUF

‘VH“ ‚F elkofer —nd rF ‡eigelD gompF €hysF gommF VP @IWWRA QHF

‘VI“ ƒF u—h—n— —nd qF ‚ipk—D xu™lF €hysF eRPW @IWVRA RTPF

‘VP“ fF uF fh—duriD wodels of the xu™leonD ™h—pter VD eddisonE‡esleyD IWVVF

‘VQ“ rF ‚einh—rdt —nd ‚F ‡uns™  hD €hysF vettF PIS @IWVVA SUUY f PQH @IWVWA WQY

„F wei%nerD pF qrummer  —nd uF qo ekeD €hysF vettF f PPU @IWVWA PWTY

‚F elkoferD €hysF vettF f PQT @IWWHA QIHF

‘VR“ tF q—sserD rF veutwylerD —nd wF iF ƒ—nioD €hysF vettF PSQ @IWWIA PSPD PTHF

‘VS“ €F ƒie˜ erD „F wei%nerD pF qrummerD  —nd uF qo ekeD xu™lF €hysF eSRU @IWWPA RSWF

‘VT“ gF ‡eissD ‚F elkoferD —nd rF ‡eigelD wo dF €hysF vettF eV @IWWQA UWF

‘VU“ tF ƒ™he™hterD €hysF ‚evF hPI @IWVHA QQWQ F

‘VV“ qF ‚ipk— —nd wF t—minonD ennF €hysF @x‰A PIV @IWWPA SIF

‘VW“ rF qommD €F t—inD ‚F tohnsonD —nd tF ƒ™he™hterD €hysF ‚evF hQQ @IWVTA VHIF

‘WH“ tF ƒ™hlienzD rF ‡eigelD rF ‚einh—rdtD —nd ‚F elkoferD €hysF vettF QIS @IWWQA TF

‘WI“ ‚F elkofer —nd rF ‚einh—rdtD €hysF vettF fPRR @IWWIA RTIF

‘WP“ ‚F elkoferD rF ‚einh—rdtD rF ‡eigelD —nd F u™  kertD €hysF ‚evF vettF TW @IWWPA IVURF IIS

‘WQ“ pF horingD iF ‚uizEerriol—D —nd uF qo ekeD F €hysF eQRR @IWWPA ISWF

‘WR“ tF qoldstone —nd pF ‡il™zekD €hysF ‚evF vettF RU @IWVIA WVTF

‘WS“ €F t—inD ‚F tohnsonD xF ‡F €—rkD tF ƒ™he™hterD —nd rF ‡eigelD €hysF ‚evF hRH @IWVWA

VSSF

‘WT“ ‚F tohnsonD xF ‡F €—rkD tF ƒ™he™hterD †F ƒoniD —nd rF ‡eigelD €hysF ‚evF hRP @IWWHA

PWWVF

‘WU“ rF ‡eigelD rF ‚einh—rdtD —nd ‚F elkoferD €hysF vettF fQIQ @IWWQA QUUF

‘WV“ ‚F elkoferD rF ‚einh—rdtD tF ƒ™hlienzD —nd rF ‡eigelD „opologi™—l ly non{trivi—l ™hir—l

tr—nsform—tionsX „he ™hir—l inv—ri—nt elimin—tion of the —xi—l ve™tor mesonD „u˜ingen 

niversity preprintD hepEphGWRH TR PHD tune IWWRF

‘WW“ rF ‡eigelD ‚F elkoferD —nd rF ‚einh—rdtD xu™lF €hysF eSUT @IWWRA RUUF

‘IHH“ uF qo ekeD eF F qorskiD pF qrummerD  „F wei%nerD rF ‚einh—rdtD —nd ‚F ‡uns™  hD €hysF

vettF fPST @IWWIA QPIF

‘IHI“ wF ‡—k—m—tsu —nd rF ‰oshikiD xu™lF €hysF eSPR @IWWIA STIF

‘IHP“ „F wei%nerD iF ‚uizEerriol—D eF flotzD —nd uF qo ekeD f—ryons in ie™tive ghir—l u—rk

wodels with €ol—rized hir—™ ƒe—D fo ™hum niversity preprint ‚ fE„€s sERPGWQD t—nu—ry

IWWRF

‘IHQ“ eF F qorskiD gF †F ghristovD —nd uF qo ekeD ile™trom—gneti™ xu™leon €roperties —nd

u—rk ƒe— €ol—riz—tion in the x—m˜uEton—Ev—siono wodelD fo ™hum niversity preprint

‚ fE„€s sESTGWQD he™em˜ er IWWQF

‘IHR“ iF xF xikolovD ‡F froniowskiD —nd uF qo ekeD xu™lF €hysF eSUW @IWWRA QWVF

‘IHS“ ‡F froniowski —nd „F hF gohenD €hysF ‚evF hRU @IWWQA PWWF

‘IHT“ ƒF ƒ—ito —nd wF eh—r—D €hysF vettF fQPS @IWWRA PHF

‘IHU“ tF eshm—n et —lFD €hysF vettF fPHT @IWVVA QTRD xu™lF €hysF fQPV @IWVWA IY

hF vF enothony et —lFD €hysF ‚evF vettF UI @IWWQA WSWY

fF edev— et —lFD €hysF vettF fQHP @IWWQA SQQD €hysF vettF fQPW @IWWQA QWWF

‘IHV“ wF ‡—k—m—tsuD €hysF vettF fPQR @IWVWA PPQF

‘IHW“ ‚F elkofer —nd rF ‡eigelD €hysF vettF fQIW @IWWQA IF

‘IIH“ qF ƒ edkins —nd gF ‚F x—ppiD xu™lF €hysF fPRW @IWVSA SHUF

‘III“ xF ‡F €—rkD tF ƒ™he™hterD —nd rF ‡eigelD €hysF vettF fPPR @IWVWA IUIF

‘IIP“ eF flotzD wF €r—s—2lowi™zD —nd uF qo ekeD €hysF vettF fQIU @IWWQA IWSF

‘IIQ“ eF flotzD wF €r—s—2lowi™zD —nd uF qo ekeD exi—l €roperties of the xu™leon with Iax

g

gorre™tions in the ƒolitoni™ ƒ @QA xtv wodelD fo ™hum niversity preprint ‚ fE„€s sE

RIGWQD w—r™h IWWRF IIT

‘IIR“ tF illis —nd wF u—rlinerD €hysF vettF fQIQ @IWWQA IQIF

‘IIS“ wF fourquin et —lFD F €hysF gPI @IWVQA PUF

‘IIT“ xF ‡F €—rkD tF ƒ™he™hterD —nd rF ‡eigelD €hysF ‚evF hRI @IWWHA PVQTF

‘IIU“ wF ‡—k—m—tsu —nd „F‡—t—˜ eD €hysF vettF fQIP @IWWQA IVRF

‘IIV“ gF †F ghristovD uF qo ekeD €F €o˜ilits—D †F €etrovD wF ‡—k—m—tsuD —nd „F‡—t—˜ eD €hysF

vettF fQPS @IWWRA RTUF

‘IIW“ ƒF u—h—n—D qF ‚ipk—D —nd †F ƒoniD xu™lF €hysF eRIS @IWVRA QSIY

wF gF firse —nd wF uF f—nerjeeD €hysF ‚evF hQI @IWVSA IIVF

‘IPH“ tF ƒ™he™hter —nd rF ‡eigelD €—tri™le gonjug—tion —nd the Iax gorre™tions to g D

g e

ƒyr—™use{„u˜ingen  preprint hepEphGWRIH QP HD y™to˜ er IWWRF

‘IPI“ rF ‡eigel —nd ‚F elkoferD gompF €hysF gommF VP @IWWRA SUF

‘IPP“ uF g—hillD eF gomtetD —nd ‚F tF ql—u˜ erD €hysF vettF TRf @IWUTA PVQF

‘IPQ“ qF rolzw—rthD qF €—riD —nd fF uF tenningsD xu™lF €hysF eSIS @IWWHA TTSF

‘IPR“ rF ‡eigelD tF ƒ™he™hterD xF ‡F €—rkD —nd FEqF wei%nerD €hysF ‚evF hRP @IWWHA QIUUF

‘IPS“ eF flotzD hF hi—konovD uF qo ekeD xF ‡F €—rkD †F €etrovD —nd €F †F €o˜ylits—D xu™lF

€hysF eSSS @IWWQA UTSF

‘IPT“ xF xF ƒ™o ™™ol—D rF x—de—uD wF eF xow—kD —nd wF ‚hoD €hysF vettF fPHI @IWVVA RPSF

‘IPU“ F flomD uF h—nn˜ omD —nd hF yF ‚isk—D xu™lF €hysF eRWQ @IWVWA QVRF

‘IPV“ fF ‰F €—rkD hF €F winD —nd wF ‚hoD xu™lF €hysF eSSI @IWWQA TSUF

‘IPW“ gF ‚F h—shenD iF tenkinsD —nd eF †F w—noh—rD €hysF ‚evF hRW @IWWRA RUIQF

‘IQH“ rF ‡eigelD ‚F elkoferD —nd rF ‚einh—rdtD €hysF ‚evF hRW @IWWRA SWSVF

‘IQI“ xF ‡F €—rkD tF ƒ™he™hterD —nd rF ‡eigelD €hysF ‚evF hRQ @IWWIA VTWF IIU

vist of pigures

H

QFI „he solution of the g—p equ—tion @QFPVA for v—nishing ™urrent m—ss m a H

H

@solid lineA —nd m a IVwe† @d—shed lineAF por the ™hosen v—lue of the ™utE

oD  a TQHwe†D — ™onstituent qu—rk m—ss m a RHHwe† repro du™es the

phenomenologi™—l v—lue of the pion de™—y ™onst—ntD f a WQwe†F X X X X X X X X PP

%

RFI „he tri—ngle di—gr—m whi™h is resp onsi˜le for the —nom—lyF X X X X X X X X X X X QV

SFI „he eigenv—lues of the hir—™ r—miltoni—n @SFIHA in the ˜—™kground of — ™hir—l

C

eld @SFIQAF hispl—yed —re the lowest eigenv—lues in the H @full linesA —nd H

@d—shed linesA ™h—nnelsF X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RI

SFP ƒ—me —s gure SFI for n a PF X X X X X X X X X X X X X X X X X X X X X X X X X X X X RI

TFI „he r—di—l dep enden™e of the self{™onsistent prole fun™tion @r A for v—rious

™onstituent qu—rk m—sses mF „he pion m—ss is m a IQS we†F X X X X X X X X X SR

%

TFP „he energy of the fermion determin—ntD i C i D for the v—ri—tion—l meson eld

† H

™ongur—tion @TFSIAF „he p—r—meter  is dened —fter eq @TFSIAF @pigure t—ken

from refF‘VT“FA X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ST

TFQ „he self{™onsistent soliton ™ongur—tion with the s™—l—r eld in™ludedF ’xon{

line—r4 refers to the ™—se with the s™—l—r elds xed —t their v—™uum exp e™t—tion

v—luesF @pigure t—ken from refF‘VT“FA X X X X X X X X X X X X X X X X X X X X X X X X ST

TFR „he self{™onsistent soliton ™ongur—tion with the s™—l—r eld in™luded —nd the

regul—rized ˜—ryon num˜ er f @TFSQA ˜ eing ™onstr—inedF hierent v—lues for



f ™orresp ond to v—rious r—dii of the spheri™—l ˜ ox used for the numeri™—l



™—l™ul—tionsF ’xon{line—r4 denotes the ™—se without ™onstr—int —nd the s™—l—r

eld —t its v—™uum exp e™t—tion v—luesF @pigure t—ken from refF‘WH“FA X X X X X X SV

TFS „he prole fun™tion @r A for the self{™onsistent ™hir—l —ngle in v—rious —pE

pro—™hes to the xtv mo delF „hose elds whi™h —re —llowed to ˜ e sp—™e dep enE

dent —re indi™—tedF „he p—r—meters used —re m a QSHwe† —nd m a IQSwe†F TQ

%

TFT veftX „he tot—l energy —s — fun™tion of the s™—ling p—r—meter ! dened in eq

@TFTWA for v—rious v—lues of the ™onstituent qu—rk m—ss mF ‚ightX „he prole

fun™tions whi™h minimize the energy fun™tion—l @TFTTA under v—ri—tion of the

y

s™—ling p—r—meter !F „he qu—rk ˜—ryoni™ density ˜ $ q q is —rti™i—lly s™—led

su™h th—t the sp—ti—l integr—ls over 3 am —nd ˜ ™oin™ideF rere the ™onstituent

qu—rk m—ss is —ssumed to ˜ e SHHwe†F @pigure t—ken from refF‘UI“FA X X X X X X TR

UFI „he r—di—l dep enden™e of the ˜ ound st—te w—ve{fun™tion  @r A for v—rious ™onE

3

stituent qu—rk m—sses mF  @r A is norm—lized —™™ording to @UFIHHAF @„—ken from

3

refF ‘WW “FA X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X WS IIV

vist of „—˜les

RFI w—ss p—r—meters xed in the meson se™tor of the xtv mo delF „he k—on de™—y

™onst—nt f is predi™tedF X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X QP

u

P P P

RFP „he fun™tions  @q A —nd m @q A whi™h determine the meson prop—g—tors D see

P P P P

eqsF @RFSSA —nd @RFSTAF „he qu—ntities f @q A —nd g @q A —re given in eqsF @RFQUA

—nd @RFSHAD resp e™tivelyF X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X QS

TFI „he soliton energy i —s well —s its v—rious ™ontri˜utions —™™ording to the sum

tot

@TFRTA —s fun™tions of the ™onstituent qu—rk m—ss mF „he pion m—ss is t—ken to

˜ e m a IQSwe†F ell num˜ ers —re in we†F X X X X X X X X X X X X X X X X X X X SR

%

TFP „he soliton energy for v—rious tre—tments of the xtv solitonF „he meson elds

listed in the rst line represent those meson proles whi™h —re —llowed to deE

vi—te from their v—™uum exp e™t—tion v—luesF ell num˜ ers —re ev—lu—ted for —

™onstituent qu—rk m—ss m a QSHwe† —nd m a IQSwe†F X X X X X X X X X X X X TP

%

wink

TFQ ƒoliton energy i @TFTTA when  —nd 3 —re the only sp—™e dep endent elds

P

for v—rious ™onstituent qu—rk m—ssesF elso shown —re the ™ontri˜utions from

the terms line—r @linFA —nd qu—dr—ti™ @qu—dFA in 3 F ell num˜ ers —re in we†F

@‚esults —re t—ken from refF‘UI“FA X X X X X X X X X X X X X X X X X X X X X X X X X X TS

TFR gontri˜utions to the energy for self{™onsistent solution in v—rious tre—tments of

the —xi—l{ve™tor meson in the xtv mo delF „hose meson elds whi™h —re —llowed

to ˜ e sp—™e dep endent —re indi™—tedF „he ™onstituent qu—rk m—ss m aRHHwe†

is ™ommonF ell num˜ ers —re in we†F @‚esults —re t—ken from refF‘WV“FA X X X X TU

UFI „he m—gneti™ moments in nu™leon m—gnetons —s fun™tions of the ™onstituent

qu—rk m—ss m ™omp—red to the exp eriment—l d—t—F @„—ken from refF‘IHI“AF X X UR

P

UFP „he me—n squ—red ™h—rge r—dii @in fm A —s fun™tions of the ™onstituent qu—rk

m—ss m ™omp—red to the exp eriment—l d—t—F @„—ken from refF‘IHQ“AF sn p—renE

thesis the results for — regul—rized im—gin—ry p—rt —re givenF X X X X X X X X X X US

UFQ „he —xi—l ™h—rge of the nu™leon g —s — fun™tion of the ™onstituent qu—rk m—ss

e

mF X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X UT

UFR „he qu—ntum ™orre™tions to the soliton m—ss due to the rot—tion—l zero mo de

‘TU“F X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X VS

UFS „he qu—ntum ™orre™tions to the soliton m—ss due to the tr—nsl—tion—l zero mo deF

„he ™ontri˜utions stemming from the ƒ @l a HA{ —nd h @l a PA{w—ves —re disenE

t—ngled ‘TU“F X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X VS

UFT „he predi™tions for the m—sses of the nu™leon @x A —nd
{reson—n™eF „he

empiri™—l d—t— —re WQWwe† —nd IPQPwe†D resp e™tivelyF @‚esults t—ken from

‘TU“FA X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X VT IIW

C C

Q I

—nd ˜—ryons with resp e™t to the UFU „he m—ss dieren™es of the lowElying

P P

nu™leonF ‡e ™omp—re the predi™tions of the ™olle™tive —ppro—™h to the xtv

mo del with the exp eriment—l d—t—F „he upEqu—rk ™onstituent m—ss m is ™hoE

sen su™h th—t the
{nu™leon m—ss dieren™e is repro du™ed ™orre™tlyD see textF

„he l—st ™olumn refers to the ™—se th—t the symmetry ˜re—ker  is s™—led ˜y

predX exptX

P

A ell d—t— @from refF‘WW“A —re in we†F X X X X X X X X X X X X X X X X WI af @f

u u

UFV €—r—meters for des™ri˜ing the hyp eron sp e™trum —s fun™tions of the ™onstituent

m—ss mF elso listed —re the empiri™—l v—lues whi™h —re o˜t—ined ˜y the ™onsidE

er—tion of ™ert—in m—ss dieren™esF @h—t— t—ken from refF‘WW“FA X X X X X X X X X WV

C C

I Q

UFW „he m—ss dieren™es of the lowElying —nd ˜—ryons with resp e™t to the nuE

P P

™leon in the ˜ ound st—te —ppro—™h to the xtv mo delF „he upEqu—rk ™onstituent

m—ss m is ™hosen su™h th—t the
{nu™leon m—ss dieren™e is repro du™edF ell

d—t— @from refF‘WW“A —re in we†F X X X X X X X X X X X X X X X X X X X X X X X X X X WW

UFIH €—r—meters for the ˜—ryon m—ss formul— @UFIPPA —nd the predi™tion for the m—ss

of the o dd p—rity  hyp eron rel—tive to the nu™leon w w —s fun™tions

 x

ƒ

of the ™onstituent m—ss mF elso the resulting v—lues for the
nu™leon m—ss

splitting —re presentedF @h—t— t—ken from refF‘IQH“FA X X X X X X X X X X X X X X X IHH