Initial-Boundary Value Problem and Stability of So

arXiv:hep-ph/9903429v122Mar1999Initial-boundaryvalueproblemandstabilityofsolutionsforstringbaryonmodel“triangle”G.S.Sharov∗,V.P.PetrovTverstateuniversityTver,170002,Sadovyjper.35,Mathem.dep-t.AbstractForthestringbaryonmodel“triangle”theinitial-boundaryvalueproblemisstatedandsolvedingeneral.Thisproblemimpliesdefiningaclassicalmotionofthesystemonthebaseofgiveninitialpositionandinitialvelocitiesofstringpoints.Thepresentedsolutionreducestheinitial-boundaryvalueproblemfortheconsideredmodeltothesystemofordinarydifferentialequationsthatcanbeintegratednumericallyingeneral.Usingthisapproachweascertainthestabilityoftherotationalmotions(flatuniformrotations)forthe“triangle”stringconfiguration.IntroductionFordescribingorbitallyexcitedbaryonsvariousstringmodelsareused.Theydifferfromeachotherinthetopologyofspatialjunctionofthreemassivepoints(quarks)byrelativisticstrings.Fourvariantsofthisjunctionarepossible(Fig.1):(a)thequark-diquarkmodelq-qq[1](fromthepointofviewofclassicaldynamicsitcoincideswiththemesonmodelofrelativisticstringwithmassiveends[2,3]);(b)thelinearconfigurationq-q-q[4,5];(c)the“three-string”modelorY-configuration[6,7]and(d)the“triangle”modelorΔ-configuration[8,9]thatisunderconsiderationinthispaper.tttqqq(a)tttqqq(b)HHHtttqqq(c)JJJJJtttqqq(d)Figure1:Stringbaryonmodels.Theexactsolutionsoftheclassicalequationsofmotiondescribinguniformrotationsofthesystemareknownforallthesemodels.Fortheconfigurationsq-qqandq-q-qtherotatingstringhastheformofarectilinearsegment[1,2,3].Forthe“three-string”modelthereare∗E-mail:[e-mail]1threerectilinearsegmentsjoinedinaplaneattheangles120◦[6,7].Forthemodel“triangle”therotatingstringhastheformoftheclosedcurveconsistingofsegmentsofahypocycloid[8,9].TheconnectionbetweentheenergyofthesystemE=ManditsangularmomentumJforallthesemotionsinthehighenergylimit(foranywayoftakingquarkspinsintoaccountorneglectingquarkspins)hastheform[1,4,9,10]J≃α′E2,J,E→∞.(1)ThisfactallowsustoapplyeachofthementionedmodelstodescribingthebaryonstateswithlargeJontheReggetrajectories[4].Theproblemofchoosingthemostadequatemodelamongthefourmentionedonesisnotsolvedyet.Fortheq-qqandq-q-qconfigurationstheReggeslopeα′inEq.(1)isconnectedwiththestringtensionγbytheNambyrelationα′=(2πγ)−1[11]andthetensionshouldbeequalformesonsandbaryons.TheconfigurationsYandΔdescribetheReggetrajectoriesiftheeffectivetensionγforthemislowerthenthe“mesonic”tensionγ≃0.18GeV2[4].OntheotherhandtheQCD-motivatedbaryonWilsonloopoperatorapproachgivessomeargumentsinfavouroftheY-configuration[12]orthe“triangle”model[13].Whenwechoosetheadequatestringbaryonconfigurationwearealsotakeintoaccountthestabilityofrotationalmotionsforthesesystems.Inparticular,therotationalmotionsoftheq-q-qsystemwiththemiddlequarkatrestareunstablewithrespecttocentrifugalmovingawayofthemiddlequark[5].Anysmalldisturbanceresultsinacomplicatedquasiperiodicmotion,butthesystemdoesn’ttransformintothequark-diquarkone[1].Thelatterq-qqconfiguration(orthemesonstringmodel)seemstobestablebutthisquestionisnotexhaustivelystudied.Smalldisturbancesofthewellknownexactsolution(uniformlyrotatingrectilinearstring)resultinarotationofthestringwithslightlyvaryingshapeandinterquarkdistance[14].SomesmallcorrectionstothemesonstringrotationalmotionsaresearchedinRefs.[15].Thestabilityproblemfortheotherstringbaryonmodels(Fig.1)wasn’tstudied.Inthispaperweinvestigateinthisrespectthehypocycloidalrotationalmotionsinthemodel“triangle”[8,9].ThereistheopinionthatanorbitallyexcitedbaryonsystemwiththegivenmomentumJ“chooses”theonlysteadystatewiththelowestenergy[1].Inaccordingwiththisstatementthesimplestnontrivialrotationalmotioninthemodel“triangle”(threequarksconnectedbysmoothsegmentsofhypocycloids)shouldtransformintothe“quark-diquark”state[9]withtwoquarksmergingintoone.Thesimplestwaytosolvethestabilityproblemistodevelopamethodofsolutionoftheinitial-boundaryvalueproblemwitharbitraryinitialconditionsforthe“triangle”model.Suchamethodissuggestedinthispaper.InSect.1theequationsofmotionwiththeircommonsolutionandtheboundarycon-ditionsforthestringbaryonmodel“triangle”aregiven.InSect.2thesolutionoftheinitial-boundaryvalueproblemissuggested.Itdevelopstheapproachusedearlierforthemesonstringmodelwithmassiveends[16]andfortheq-q-qbaryonconfiguration[5].Inthelastsectionthestabilityoftherotationalmotionsisverified.21Dynamicsofstringbaryonmodel“triangle”Letusconsideraclosedrelativisticstringwiththetensionγcarryingthreepointlikemassesm1,m2,m3.Theactionforthissystemis[8,9]S=−γτ2Zτ1dτσ3(τ)Zσ0(τ)√−gdσ−3Xi=1miτ2Zτ1n[ddτXμ(τ,σi(τ))]2o1/2dτ.(2)HereXμ(τ,σ)arecoordinatesofapointofthestringinD-dimensionalMinkowskispaceR1,D−1withsignature+,−,−,...,g=˙X2X′2−(˙XX′)2(τ,σ)∈Ω=Ω0∪Ω1∪Ω3(Fig.2),(a,b)=aμbμisthe(pseudo)scalarproduct,˙Xμ=∂τXμ,X′μ=∂σXμ,thespeedoflightc=1;σi(τ)areinnercoordinatesofthequarkworldlines1,i=0,1,2,3.Theequationsσ=σ0(τ)andσ=σ3(τ)determinethetrajectoryofthesame(third)quark.Itisconnectedwiththefactthatstringisclosedandmaybewritteninthefollowinggeneralform[8]:Xμ(τ,σ0(τ))=Xμ(τ∗,σ3(τ∗)).(3)Theparametersτandτ∗inthesetwoparametrizationsofju