Sym metry comma I n tegra b i l i ty a nd Geometry : .... Methods a nd App l i ca t i ons .... Vo l period 2 open parenthesis 2006 closing \noindent Sym metry , I n tegra b i l i ty a nd Geometry : \ h f i l l Methods a nd App l i ca t i ons \ h f i l l Vo l . 2 ( 2006 ) , Paper 29 , 12pa ges parenthesisSym metry comma , I Pan tegra per b 2 i 9 l commai ty a nd 1 Geometry 2 pa ges : Methods a nd App l i ca t i ons Vo l . 2 ( 2006 ) , Pa per 2 9 , 1 2 pa Largeges hyphen j .. Expansion .. Method \noindentfor .. TwoLarge hyphen Body $ − .. Diracj $ ..\ Equationquad Expansion \quad Method Large −j Expansion Method Askold DUVIRYAK \noindentInstitutefor forf o Condensed r Two\quad Two Matter- Body− PhysicsBody \ ofquad National DiracDirac Academy\quad Equation ofEquation Sciences of Ukraine comma 1 SvientsitskiiAskold DUVIRYAK Str period comma Lviv comma .. 7901 1 .. Ukraine \noindentE hyphenInstitute mailAskold for : .. Condensed d DUVIRYAK u v i r y a Matter k at p h Physicsperiod i cm of .. National p period Academyl v i v period of u Sciences a of Ukraine , Received1 Svientsitskii December 0 Str 1 comma . , Lviv 2005 , comma 7901 in final1 form Ukraine FebruaryE 1 - 5 mail comma : 2006d semicolonu v i r y Published a k @ p online h . i February 28 comma 2006 \noindentOriginalcm articleInstitute p . is l availablev i v .for u at a Condensed http : slash slash Matter w w w Physics period em of i s National period de slash Academy j ournals of slash Sciences SIGM A of slash Ukraine 2006 slash , Paper 29 slash AbstractReceived period December By using 0 1 symmetry , 2005 , in properties final form comma February the 1 two 5 , hyphen2006 ; Published body Dirac online equation February in coordinate 28 , 2006 rep hyphen \noindentresentationOriginal1 is article Svientsitskii reduced is available to the coupled at Strhttp pair . : , /of Lviv / radial w w , second w\quad . em hyphen7901 i s . order 1 de\quad differential / j ournalsUkraine equations / SIGM period A / 2006 Then / Paper E the− 29mail large / hyphen : \quad j expansionduviryak technique is used $@$ to solve ph.icm a bound state\ problemquad p period . l v Linear i v hyphen . u a plus hyphen Coulomb potentialsAbstract of different . spinBy structure using symmetry are examined properties in order , the to two describe - body the Dirac asymptotic equation de in hyphencoordinate rep - \noindentgeneracy andReceivedresentation fine splitting is December reduced of light to 0meson the 1 coupled , spectra 2005 pair period , ofin radial final second form - order February differential 1 5 equations , 2006 . ; Then Published online February 28 , 2006 Key wordsthe : large Breit− equationj expansion comma technique two body is used Dirac to equation solve a bound comma state large problem hyphen . N Linear expansion - plus comma - Coulomb Regge trajectories \noindent2000 MathematicsOriginalpotentials Subject articleof different Classification is spin available structure : 8 1 Q 5are semicolon at examined http 8 : in1 /Q order 1/ 5 www semicolon to describe . em 8 the 1 Qi asymptotic 20s . de / de j - ournals / SIGMA / 2006 / Paper 29 / 1 .. Introduction generacy and fine splitting of light meson spectra . \ centerlineTwo hyphenKey{ Abstract bodywords Dirac : Breit . equations By equation using .... , two symmetry open body parenthesis Dirac properties equation 2 BDE , large closing , the−N parenthesis twoexpansion− body comma , Regge Dirac i trajectories period equation e period in comma coordinate .... the Breit rep − .... } equations .... open square2000 bracket Mathematics 1 closing Subject square bracketClassification .... and : it8 s 1 generalizations Q 5 ; 8 1 Q 1 5 .... ; 8 open 1 Q square20 bracket 2 comma .... 3 comma \ centerline4 comma1 .... Introduction{ resentation 5 comma .... 6 comma is reduced .... 7 comma to the .... coupled8 comma .... pair 9 comma of radial .... 1 0 closing second square− order bracket differential comma are used equations frequently for . Then } the descriptionTwo - body of relativistic Dirac equations bound state problem ( 2 BDE comma ) , i . e . , the Breit equations [ 1 ] and it s \ centerlineespgeneralizations ecially in{ the nuclear l a r open g e square $ − bracketj $ expansion1 1 comma .. technique 1 2 closing [ 2 , square is used bracket to and solve hadronic a bound open state square 3 , bracket problem 1 3 . comma Linear .. 1− 4 plus − Coulomb } comma4 .. , 1 5 5 comma , 6 , .. 7 1 , 6 closing 8 , 9 square , 1 0 bracket ] , are physics used frequently period .. Apart for thefrom description two free hyphen of relativistic particle bound \ centerlineDiracstate terms problem{ commapotentials the, 2 BDE of may different include potentials spin structure which are local are matrix examined hyphen in functions order in to the describe coordinate the asymptotic de − } representationesp ecially period in nuclear .. This form [ 1 1 provides , 1 an 2 ] intuitive and hadronic understanding [ 1 3 of , the1 interaction 4 , 1 and 5 , may 1 suggest 6 ] physics \ centerlinea proper. Apart physical{ generacy from choice two of and the free finepotential - particle splitting in phenomenological Dirac of terms light , the models meson 2 BDE period spectra may include . } potentials which are Butlocal the 2 matrix BDE are - pathological functions in if certain the coordinate interaction representationt erms are not treated . This p erturbationally form provides period an intuitive \ centerlineTheunderstanding .... set{ ....Key of words radially of the : reduced interaction Breit .... equation equations and may ,.... two suggest open body square Dirac bracket equation 1 3 comma , .... large 5 comma $ − .... 7N closing $ expansion square bracket , Regge .... may trajectories } possessa .... proper non hyphen physical physical choice .... ofenergy the hyphen potential dependent in phenomenological models . \ centerlinepoles at finit{2000 e distance MathematicsBut r between the 2 BDE particles Subject are open pathological Classification square bracket if certain 1 7 : comma 8 interaction 1 Q .. 15 8 ; closing 8 t erms 1 Q square 1 are 5 bracket not ; 8 treated 1 period Q 20 p ..} Correspondingly comma an exacterturbationally boundary hyphen . value \noindentproblemThe becomes set1 \quad of incorrectradiallyIntroduction mathematically reduced equations or improper [ 1 3 for , assumed 5 , 7 ] physical may possess treatment non period - physical energy - Heredependent we consider a possibility to avoid pathological peculiarities of 2 BDE using a pseudo hyphen \noindentp erturbativepolesTwo at techniquefinit− body e distance similar Dirac t or equations 1between slash N open particles\ h square f i l l [(2BDE) bracket 1 7 , 1 9 1 comma 8 ,i ] . .e. 2 0 comma Correspondingly , \ 2h 1 f iclosing l l the square , B an r e bracket iexact t \ h f .. i l or l 1equations slash l expansions\ h f i l l [ 1 ] \ h f i l l and it s generalizations \ h f i l l [ 2 , \ h f i l l 3 , .. openboundary square bracket - value 2 2 comma problem .. 2 becomes 3 closing squareincorrect bracket mathematically period .. These or methods improper for assumed physical \noindentaretreatment applicable4 , t\ . oh the f i l case l 5 of , a\ strongh f i l l coupling6 , \ andh f i lare l 7 lit t, le\ h affected f i l l 8 by , boundary\ h f i l l peculiarities9 , \ h f i l l 1 0 ] , are used frequently for the description of relativistic bound state problem , of the boundaryHere we hyphen consider value a possibility problem period to avoid pathological peculiarities of 2 BDE using a pseudo \noindentIn our- p case erturbativeesp natural ecially expansion technique in parameter nuclear similar i [ s t 1 1 o slash 11/N , j[19\ commaquad, 21 where 0, 22 ] 1]j iand s theor hadronic conserved1/` expansions t [ otal 1 3 angular , [\ 2quad2 mo , hyphen1 2 4 3 ,] \quad 1 5 , \quad 1 6 ] physics . \quad Apart from two free − p a r t i c l e Diracmentum. terms Theseperiod , .... methods the After 2 BDE the radial may reduction include i spotentials p erformed comma which the are 2 BDE local takes matrix the form− offunctions the set of eight in the coordinate representationcoupledare applicable first hyphen . t\ o orderquad the case differentialThis of aform strong equations provides coupling open an andsquare intuitive are bracket lit t le 5 affectedunderstanding comma by .... boundary 7 closing of the squarepeculiarities interaction bracket period and Using may a suggest chain of transformationsof the boundary we reduce - it value t o problem . \noindentthe pair ofIna coupled our proper case second natural physical hyphen expansion order choice equations parameter of the and potential apply i s 1 the/j, 1where slash in phenomenological jj expansioni s the conserved t echnique models period t otal .. angular . The method i s appliedmo - t o the potential model of meson based on the 2 BDE period \ hspace2 ..mentum 2 hyphen∗{\ f i l .body l }But After .. Dirac the the 2.. radial equationBDE are reduction .. pathologicaland .. iits s .. p radial erformed if .. reduction certain , the 2 BDE interaction takes the t form erms of are the not set of treated p erturbationally . In theeight centre hyphen of hyphen mass reference frame the 2 BDE has the form : \noindentEquation:coupledThe open first\ parenthesish - f iorder l l s e differential1 t closing\ h f i l parenthesis l of equations radially .. open [ 5 brace,reduced h 7sub ]\ . 1h Usingopen f i l l parenthesisequations a chain of p closing transformations\ h f i l parenthesis l [ 1 3 ,plus we\ h h f subi l l 25 open , \ parenthesish f i l l 7 ] \ h f i l l may p o s s e s s \ h f i l l non − p h y s i c a l \ h f i l l energy − dependent minus preduce closing it parenthesis t o plus U open parenthesis r closing parenthesis minus E closing brace F open parenthesis r closing parenthesis = 0 comma\noindentthe pairpoles of coupled at finit second e distance - order equations $ r $ and between apply particlesthe 1/j expansion [ 1 7 ,t echnique\quad 1 . 8 ] The . \quad Correspondingly , an exact boundary − value problemmethod becomes i s applied incorrect t o the mathematically potential model or of meson improper based for on assumed the 2 BDE physical . treatment . 2 2 - body Dirac equation and its radial reduction Here weIn the consider centre - a of possibility - mass reference to avoid frame the pathological 2 BDE has peculiaritiesthe form : of 2 BDE using a pseudo − perturbativetechniquesimilarto $1 / N [ 1 9 , 2 0 , 2 1 ]$ \quad or $ 1 / \ e l l $ expansions \quad [ 2 2 , \quad 2 3 ] . \quad These methods
{h1(p) + h2(−p) + U(r) − E}F (r) = 0, (1) \noindent are applicable t o the case of a strong coupling and are lit t le affected by boundary peculiarities
\noindent of the boundary − value problem .
\ hspace ∗{\ f i l l } In our case natural expansion parameter i s $ 1 / j , $ where $ j $ i s the conserved t otal angular mo −
\noindent mentum . \ h f i l l After the radial reduction i s p erformed , the 2 BDE takes the form of the set of eight
\noindent coupled first − order differential equations [ 5 , \ h f i l l 7 ] . Using a chain of transformations we reduce it t o
\noindent the pair of coupled second − order equations and apply the $ 1 / j $ expansion t echnique . \quad The method i s applied t o the potential model of meson based on the 2 BDE .
\noindent 2 \quad 2 − body \quad Dirac \quad equation \quad and \quad i t s \quad r a d i a l \quad reduction
\noindent In the centre − o f − mass reference frame the 2 BDE has the form :
\ begin { a l i g n ∗} \{ h { 1 } ( p ) + h { 2 } ( − p ) + U ( r ) − E \} F ( r ) = 0 , \ tag ∗{$ ( 1 ) $} \end{ a l i g n ∗} 2 .... A period Duviryak \noindent 2 \ h f i l l A . Duviryak hline2 A . Duviryak where F open parenthesis r closing parenthesis i s a 16 hyphen component wave function comma \ [ h\ subr u l ae { open3em}{ parenthesis0.4 pt }\ p] closing parenthesis = alpha sub a times p plus m sub a beta sub a equiv minus i alpha sub a times nabla plus m sub a beta sub a comma a = 1 comma 2 comma arewhere Dirac HamiltoniansF (r) i s a 16 of free - component fermions of wave mass m function sub a and , U open parenthesis r closing parenthesis i s an interaction potential period If F\noindent open parenthesiswhere r closing $F parenthesis ( r )$ isa16 − component wave function , i s presented in 4 times 4 hyphenha(p matrix) = αa representation· p + maβa ≡ comma −iαa · the ∇ + operatorsmaβa, a alpha= 1, 2 sub, a and beta sub a act as follows : alpha sub 1 F = \ [ h { a } ( p ) = \alpha { a }\cdot p + m { a }\beta { a }\equiv − i \alpha { a } alpha F comma m U(r) \cdotare\ Diracnabla Hamiltonians+ m { ofa free}\ fermionsbeta { ofa mass} ,a aand = 1i s ,an interaction 2 , \ ] potential . If alphaF (r sub) i 2 s F presented = F alpha into the4 × power4− matrix of T etc representation comma where alpha , the and operators beta are Diracαa and matricesβa act period as follows The: potentialα1F = αF, U open parenthesis r closing parenthesis is the Hermitian matrix hyphen function comma it i s invariant under rotation and space α F = F αT α β \noindent2 are Diracetc , where Hamiltoniansand are of Dirac free matricesfermions . of mass $ m { a }$ and $U ( r ) $ i san interaction potential . If inversionThe transformations potential U open(r) is parenthesis the Hermitian so that matrix the total - Hamiltonian function , itH = i s h invariant sub 1 plus under h sub 2 rotation plus U is tand oo closing parenthesis period ..$ It F s general ( r ) $ i sspace presented in $ 4 \times 4 − $ matrix representation , the operators $ \alpha { a }$ and form i s parametrized by 48 scalar function of r = bar r bar open squareH bracket= h + 1 0h closing+ U square bracket period .. Of physical meaning $ \betainversion{ a } transformations$ act as follows ( so that $ : the\ totalalpha Hamiltonian{ 1 } F = 1 \alpha2 isF t oo , ) $ . It s are potentialsgeneral form i s parametrized by 48 scalar function of r =| r | [10]. Of physical meaning are admitting field hyphen theoretical interpretation of interaction period In particular comma potentials reflecting a spin \noindentpotentials$ \alpha admitting{ 2 field} -F theoretical = F interpretation\alpha ˆ{ T of} interaction$ etc , where . In particular $ \alpha , potentials$ and $ \beta $ are Dirac matrices . structurereflecting of vector a spin and scalar structure relativistic of vector interactions and scalar are used relativistic frequently interactions in potential quark are used frequently in models of mesons period .. We will consider such a model in the Section 5 using few examples of scalar \ hspacepotential∗{\ f i l l quark}The potential models of mesons $U . ( We r will ) $ consider is the such Hermitian a model matrixin the Section− function 5 using , it i s invariant under rotation and space andfew vector examples potentials of known scalar in and a lit vector erature potentials period .. In knownthe present in a section lit erature the structure . In of the potential present i s section notthe essential structure period of potential i s not essential . \noindentIn order t oinversion apply a pseudoperturbative transformations expansion ( so method that to the the total 2 hyphen Hamiltonian body Dirac equation $ H let = h { 1 } + h { 2 } + U $ i sIn t orderoo ) t. o\quad applyI a t pseudoperturbative s g e n e r a l expansion method to the 2 - body Dirac equation us transformlet us transform it to an appropriate it to an appropriate form period form . formFirst i of s all parametrized we p erform a radial by 48 reduction scalar period function .. Following of open $ r square = bracket\mid 5 commar \ ..mid 7 closing[ square 1 bracket 0 ] .. we . put $ the\quad waveOf physical meaning are potentials admittingFirst field of all− wetheoretical p erform a radial interpretation reduction . of interaction Following [ 5 . , In 7 particular ] we put ,the potentials wave reflecting a spin eigenfunction F (r) j P 2 × 2 structureeigenfunction of vector andof the scalar total angular relativistic momentum interactionsand the are parity usedinto frequently the inblock potential - quark F openmatrix parenthesis form : r closing parenthesis of the total angular momentum j and the parity P into the 2 times 2 block hyphen matrix form : modelsEquation: of open mesons parenthesis . \quad 2 closingWe willparenthesis consider .. F open such parenthesis a model r closing in the parenthesis Section = 1 5 divided using by few r Row examples 1 i s sub 1 open of scalar parenthesis rand closing vector parenthesis potentials phi to the known power of in A aopen lit parenthesis erature n . closing\quad parenthesisIn the plus present i s sub section 2 open parenthesis the structure r closing parenthesis of potential phi to i s not essential . A 0 − + the power of 0 open parenthesis n1 closingis parenthesis1(r)φ (n) + t subis2(r 1)φ open(n) parenthesist1(r)φ (n r) closing + t2(r) parenthesisφ (n) phi to the power of minus open parenthesis F (r) = − + A 0 (2) n closing parenthesis plus t sub 2 openr parenthesisu1(r)φ (n) r + closingu2(r)φ parenthesis(n) iv1( phir)φ to(n the) + power iv2(r)φ of( plusn) open parenthesis n closing parenthesis Row 2 uIn sub order 1 open t parenthesis o apply r a closing pseudoperturbative parenthesis phi to the expansion power of minus method open to parenthesis the 2 − n closingbody Dirac parenthesis equation plus u sub let 2 open parenthesis j±1 j rus closing transformfor parenthesis the parity it phi toP to= an the (− powerappropriate) ofstates plus open , form parenthesis and . into n a closing similar parenthesis form for i the v sub parity 1 openP parenthesis= (−) states r closing parenthesis phi to the power ofbut A open with parenthesis superscripts n closing interchanged parenthesis as plus follows i v sub:( 2 openA, 0) parenthesis↔ (−, +). r closingHere parenthesisn = r/r, the phi bispinor to the power of 0 open parenthesis nFirst closingharmo of parenthesis all - we . p erform a radial reduction . \quad Following [ 5 , \quad 7 ] \quad we put the wave eigenfunction A $Ffornics the parity (φ ( rn P) =corresp ) open $ parenthesis ofonds the t o total a minus singlet closingangular state parenthesis with momentum a t to otal the $ spin power j $s of= and j 0 plusminuxand the an parity orbital 1 .. states momentum $P$ comma into .. and the into a $ similar 2 form\times for 0 − + the2 $ parity` block= Pj, =and− openmatrixφ parenthesis(n), φ form(n), minus φ : (n) closingcorrespond parenthesis t o triplet to the power with ofs = j states 1 and but` = j, j + 1, j − 1. Then for withj >superscripts0 the eigenstate interchanged problem as follows ( 1 ) : reduces open parenthesis t o the set A comma of eight 0 closing first - parenthesis order differential leftrightarrow equations open parenthesis minus comma plus\ begin closingwith{ a l parenthesisi the g n ∗} functions periods1 ..(r Here), ..., vn2( =r) rand slash the r comma energy theE bispinort o be harmo found hyphen . Fnics ( phiIt to r i the s convenient ) power = of\ Af ropento a c present{ parenthesis1 }{ thisr }\ n set closingl e in f t the[ parenthesis\ begin following{ array .... matrix corresp}{ cc } ondsformi t . o s a singlet{ Let1 us} state introduce( with r a t ) theotal spin\phi s =ˆ{ 0 andA } an orbital( n momentum) +8 - i dimensional s { 2 } vector( - r function ) \ :phi ˆX{ 0 (}r) =({s1( nr), s2( )r), & t1(r) t, ...,{ v2(1r)}. (Then r the ) set\ ofphi ˆ{ − } ( n )l = +radial j comma t equa{ and2 } - phi t( to ions the r reads power ) : of 0\phi openˆ parenthesis{ + } ( n closing n parenthesis ) \\ u comma{ 1 } phi( to the r power ) of\ minusphi ˆ open{ − parenthesis } ( n n closing ) + u { 2 } ( r ) \phi ˆ{ + } ( n ) &d i v { 1 } ( r ) \phi ˆ{ A } ( n ) parenthesis comma phi to the power of plus open parenthesis{ H n closing(j) dr + parenthesisV (r, E, correspond j)} X (r t) =o tripletE X with(r), s =(3) 1 and l = j comma j plus 1 +comma iwhere j minus v the{ 1 period2 8} × 8( ..real Then r matrices ) \phiH ˆ({j)0and} V( n(r, E, )j)\ =endG { array(j)/r}\+ rightm +] \ Utag ∗{($r, j () − E 2I ) $} \endfor{possesses jal greater i g n ∗} 0 properties the eigenstate problem open parenthesis 1 closing parenthesis reduces t o the set of eight first hyphen order differential T T equationsH = − H , V = V , the diagonal matrix m = diag (m+I, m−I, −m−I, −m+I)( here I i \noindentwiths the2 × functions2forunit the matrix s sub parity 1 open and $P parenthesism± == m1 r (± closingm2−) parenthesisand) ˆ{j−j and comma\pmP − perioddependent1 }$ period\quad period matricess t comma a t e sH , v\ subquad(j 2), openand parenthesis into a similar r closing form for the parity parenthesis$ PG = and(j () the−are energy constant) ˆE{ t oj be}$ found ( i s t. a period e t e . s , but free of r), and matrix - potential U (r, j) comes from withIt iinteracting s superscripts convenientto term present interchanged of the this equation set in the as ( following 1 follows ) . matrix For $ the form : case period (j A= .. 0Letcomponents , us introduce 0 ) thes2\ leftrightarrow= 8t hyphen2 = u2 = v2 = 0 ( − , + )dimensional .so $ that\quad vectorthe dimensionHere hyphen $n function of the = : problem.. r X open / parenthesis ( 3 r ) reduces ,$ r closing thebispinorharmo from parenthesis 8 t o 4 . = open− brace s sub 1 open parenthesis r closing parenthesis comma s subIt turns 2 open parenthesis out that r closing rank parenthesisH comma= 4 t (2 sub 1for openj parenthesis= 0). rIn closing other parenthesis words comma , only period period period comma v\noindent sub 2four open equations parenthesisn i c s $ \ rphi closing of theˆ{ parenthesisA set} ( 3( ) are closing n differential ) brace $ period\ h fwhile i l ..l Thencorresp remaining the set onds of ones radial t are o equa a algebraic singlet hyphen . state They with a t otal spin $ st ionscan = reads be 0 split : $ and by means an orbital of some momentum orthogonal ( i . e . , of O ( 8 ) group ) transformation . In new braceleftbiggbasis we H have open parenthesis j closing parenthesis d divided by dr plus V open parenthesis r comma E comma j closing parenthesis bracerightbigg\noindent $ X\ opene l l parenthesis= j r closing , $ and parenthesis $ \phi = Eˆ{ X0 open} parenthesis( n ) r closing , \ parenthesisphi ˆ{ − comma } ( open n parenthesis ) , 3\ closingphi ˆ{ + } X parenthesis( n )$ correspond to tripletX(r) = with1 $s, H == 2[J 1$(2) 00 and], $ \ e l l = j , j + 1 , j − X 0 1where . $ the\quad 8 timesThen 8 real matrices H open parenthesis2 j closing parenthesis and V open parenthesis r comma E comma j closing parenthesis f o r $ j > 0 $ the eigenstate problem ( 1 ) reduces t o the set of eight first − order differential equations = G openwhere parenthesisJ(2) i s j the closing symplectic parenthesis4 slash× 4 matrix r plus m . plus Thus U open we parenthesis arrive at r the comma set j closing parenthesis minus E I possesses properties withH to the the power functions of T = minus $ s H{ comma1 } V( to ther power ) of , T = . V comma . . the , diagonal v { matrix2 } m( = diag r open )$ parenthesis andthe m energy sub plus I comma$E$ m t sub o minus be found I comma . minus m sub minus I comma minus m sub plus I closing parenthesis open parenthesis here I i s 2 times 2 unit (2) 0 matrix and m sub plusminux = m sub 1 plusminux2J X1 + mV sub11X 21 closing+ V12X parenthesis2 = 0, .. and j hyphen and P hyphen dependent(4) matrices .. H open parenthesisIt i s convenient j closing parenthesis to present comma this .. G open set parenthesis in the following j closing parenthesis matrix .. form are constant . \quad .. openLet parenthesis us introduce i period the e period 8 − comma dimensionalfree of r closing vector parenthesis− f ucomma n c t i o andn : matrix\quad hyphenX $ potential ( r U )open = parenthesis\{ rs comma{ 1 j} closing( parenthesis r ) comes, s from{ 2 interacting} ( termr of ) the , equation t { open1 } parenthesis( r 1 closing ) , parenthesis . . period . .. , For the v { 2 } ( r ) \} . $ \quad Then the set of radial equa − tcase ions j = reads 0 components : s sub 2 = t sub 2 = u sub 2 = v sub 2 = 0 so that the dimension of the problem open parenthesis 3 closing parenthesis reduces \ hspacefrom 8∗{\ t of 4 i period l l } $ \{ $ H $ ( j ) \ f r a c { d }{ dr } +$V$( r ,E , j ) \} $ X $(It turns r .. )=E$X$( out .. that .. rank H .. = 4 open r parenthesis) , ( 2 .. for3 j = )$ 0 closing parenthesis period .. In .. other words comma .. only four equations .. of the \noindentset open parenthesiswhere the 3 closing $ 8 parenthesis\times are8$ differential realmatricesH while remaining ones $( are algebraic j )$ period andV .. They $( can be r split by, means E of , some j )orthogonal =$G$( open parenthesis j ) i period / e r period +$m$+$U$( comma of O open parenthesis r 8 closing , parenthesis j ) − groupE closing $ I parenthesis possesses transformation properties period In new basis we have \noindentX open parenthesis$ H ˆ{ rT closing} = parenthesis− $ H = Row $ , 1 X V sub ˆ{ 1T Row} 2=$ X sub V 2 ,. comma the diagonal H = 2 bracketleftbigg matrixm J $=$ sub 0 to diag the power $ ( of open m { + } parenthesisI , m 2 closing{ − } parenthesisI, 0 to− them power{ − of } 0 bracketrightbiggI, − commam { + } I ) ($ here $I$ is $2 \times 2 $where unit J to the power of open parenthesis 2 closing parenthesis i s the symplectic 4 times 4 matrix period .. Thus we arrive at the set matrixEquation: and open $ parenthesis m {\pm 4} closing= parenthesis m { 1 }\ .. 2 Jpm to them power{ 2 of} open) parenthesis $ \quad 2and closing $ parenthesis j − $ X suband 1 to $ the P power− $ of prime dependent matrices \quad H plus$ ( V sub j 1 1 ) X sub , 1$ plus\quad V subG 1 2 $ X ( sub 2 j = 0 comma) $ \quad are constant \quad ( i . e . , free of $r ) ,$ andmatrix − potential U $ ( r , j ) $ comes from interacting term of the equation ( 1 ) . \quad For the case $ j = 0$ components $s { 2 } = t { 2 } = u { 2 } = v { 2 } = 0 $ so that the dimension of the problem ( 3 ) reduces from 8 t o 4 .
I t turns \quad out \quad that \quad rank H \quad $ = 4 ( 2 $ \quad f o r $ j = 0 ) . $ \quad In \quad other words , \quad only four equations \quad o f the set ( 3 ) are differential while remaining ones are algebraic . \quad They can be split by means of some orthogonal ( i . e . , of O ( 8 ) group ) transformation . In new basis we have
\ [ X ( r ) = \ l e f t [ \ begin { array }{ c} X { 1 }\\ X { 2 }\end{ array }\ right ] , H = 2 [ J ˆ{ ( 2 ) } { 0 } 0 ˆ{ 0 } ], \ ]
\noindent where $ J ˆ{ ( 2 ) }$ i s the symplectic $ 4 \times 4 $ matrix . \quad Thus we arrive at the set
\ begin { a l i g n ∗} 2 J ˆ{ ( 2 ) } X ˆ{\prime } { 1 } + V { 1 1 } X { 1 } + V { 1 2 } X { 2 } = 0 , \ tag ∗{$ ( 4 ) $} \end{ a l i g n ∗} Large hyphen j Expansion Method for Two hyphen Body Dirac Equation .... 3 \noindent Large $ − j $ Expansion Method for Two − Body Dirac Equation \ h f i l l 3 LineLarge 1 hline− Linej Expansion 2 V sub 2 1 Method X sub 1 forplus Two V sub - 22Body X sub Dirac 2 = 0 Equation period open parenthesis 5 closing parenthesis 3 Eliminating X sub 2 from open parenthesis 4 closing parenthesis by means of open parenthesis 5 closing parenthesis we get a differential set for\ [ \ thebegin 4 hyphen{ a l i g vectorn e d }\ Xr sub u l e 1{3em}{0.4 pt }\\ Vbraceleftbigg{ 2 1 J} toX the{ power1 } of+ open V parenthesis{ 22 } 2 closingX { parenthesis2 } = d 0 divided . by ( dr plus 5 V ) to\end the{ powera l i g n of e d bottom}\ ] open parenthesis r V21X1 + V22X2 = 0. (5) comma E comma j closing parenthesis bracerightbigg X sub 1 open parenthesis r closing parenthesis = 0 comma where V to the power of bottom = openEliminating parenthesis VX sub2 from 1 1 minus ( 4 ) V by sub means 1 2 V subof ( 22 5 to) we the get power a differential of minus 1 V set sub for 2 1 the closing 4 - parenthesis vector X1 slash 2 \noindentwhile X subEliminating 2 then follows from $ X the{ algebraic2 }$ fromrelation ( X 4 sub ) by2 = means minus V of sub ( 22 5 to ) the we power get aof minus differential 1 V sub 2 1 set X sub for 1 period the 4 − vector $ X { 1 }$ (2) d ⊥ ⊥ −1 The elimination of X sub{J 2 causes+ V non(r, E, hyphen j)}X1 physical(r) = 0, energywhere hyphenV = dependent (V11 − V12 singularV22 V21 points)/2 open parenthesis apart of r = 0 and physical singularities ofdr potentials closing parenthesis in matrix elements of V to the power of bottom period \ [ \{ J ˆ{ ( 2 ) }\ f r a c { d }{ dr } + V ˆ{\bot−1 } ( r , E , j ) \} X { 1 } ( Nowwhile we presentX2 then the 4 follows hyphen from vector the X sub algebraic 1 in 2 plus relation 2 blockX form2 = − commaV V21X1. r ) = 0 , where Vˆ{\bot } = ( V { 122 1 } − V { 1 2 } V ˆ{ − 1 } { 22 } V { 2 X sub 1The open elimination parenthesis r of closingX2 causes parenthesis non = - Row physical 1 Capital energy Phi - sub dependent 1 Row 2 Capital singular Phi points sub 2 . ( comma apart V of to the power of bottom = Row1 } 1r V)= sub 0 / 1 1 V 2 sub\ ] 1 2 Row 2 V sub 2 1 V sub 22 . comma eliminateand physical then Capital singularities Phi sub 2 and of potentials arrive at the ) secondin matrix hyphen elements order differential of V⊥. equations for 2 hyphen vector Capital Phi sub 1 : Equation: open parenthesisNow 6 closing we present parenthesis the .. 4 L - openvector parenthesisX1 in 2 E + closing 2 block parenthesis form , Capital Phi sub 1 = braceleftbigg parenleftbigg d\noindent divided by drwhile plus V $ sub X 1{ 2 parenrightbigg2 }$ then open follows square from bracket the V subalgebraic 22 closing relation square bracket $ X to the{ 2 power} = of minus− 1V parenleftbigg ˆ{ − 1 d} { 22 } V { 2 1 } X { 1 } . $ divided by dr minus V sub 2 1 parenrightbigg plus VΦ sub1 1 1 bracerightbigg⊥ V11 CapitalV12 Phi sub 1 = 0 period X1(r) = , V = , The matrix V sub 22 i s diagonal for all potentialsΦ2 considered in SectionV21 5V open22 parenthesis and many other ones closing parenthesis period \ hspaceIn these∗{\ casesf i l wel }The can perform elimination the transformation of $ X : { 2 }$ causes non − physical energy − dependent singular points ( apart of eliminate then Φ2 and arrive at the second - order differential equations for 2 - vector Φ1 : $ rPhi-tilde = sub 0 $ 1 = Capital Phi sub 1 slash radicalbig-line of V sub 22 sub comma tilde-L = radicalbig-line of V sub 22 L radicalbig-line of V sub 22 \noindent and physical singularitiesd of potentialsd ) in matrix elements of $ V ˆ{\bot } . $ providing for the operator L-tildeL(E the)Φ form= {( which+ V is as)[V close]−1( as possible− V ) + t oV 2} hyphenΦ = 0. t erm form : (6) Equation: open parenthesis 7 closing1 parenthesisdr ..12 L-tilde22 opendr parenthesis21 11 E1 closing parenthesis = d to the power of 2 divided by dr to the\ centerline power of 2 minus{Now W we open present parenthesis the r 4 comma− vector E comma $ X j closing{ 1 parenthesis}$ in$2 minus braceleftbigg + 2$ Z blockform, open parenthesis} r comma E comma j closing parenthesisThe matrix commaV22 di divided s diagonal by dr for bracerightbigg all potentials sub plus considered J to the in power Section of open 5 ( parenthesis and many 1 other closing ones parenthesis semicolon \ [Xhere){ W .1 open In} these parenthesis( cases rr we ) comma can = perform E\ l comma e f t [ \ thejbegin closing transformation{ array parenthesis}{ c}\ is :aPhi symmetric{ 1 2}\\\ times 2Phi matrix{ comma2 }\end J to{ array the power}\ right of open] parenthesis , V ˆ{\bot } =1 closing\ l e fparenthesis t [ \ begin { iarray s 2 times}{ cc 2 symplectic} V { 1 matrix 1 } and&V open brace{ 1 times 2 }\\ commaV times{ 2 closing 1 } brace&V plus{ denotes22 }\end{ array }\ right ] , \ ] ˜ p ˜ p p the anticommutator period Φ1 = Φ1/ V22, L = V22L V22 Weproviding are going to for apply the the operator 1 slash jL expansion˜ the form method which t o is the as equation close as open possible parenthesis t o 2 -7 tclosing erm form parenthesis : period .. In many physically interesting cases the function Z vanishes or it is negligible at j large period .. Thus the wave equation has \noindenta 2 times 2eliminate matrix 2 hyphen then t erm $ \ formPhi which{ 2 is} convenient$ and arrive for application at the of the second method− periodorder .. differential In other cases the equations for 2 − vector $ \thirdPhi t{ erm1 of} the: operator $ open parenthesisd2 7 closing parenthesis containsd a first hyphen order derivative via off hyphen diagonal matrix L˜(E) = − W (r, E, j) − {Z(r, E, j), } J (1); (7) elements dr2 dr + \ beginonly{ perioda l i g n ..∗} This form i s tractable t oo comma but with more t edious calculations period .. We do not consider such (1) L(E)equationshere W in( thisr, E, paper j)\isPhi a period symmetric{ Before1 } = proceeding2 × 2\{matrix further( ,J\ f r comma a ci{ s d2 we×}{2 studysymplecticdr a} simpler+ matrix V example{ 1 and of a 2{· single,}·}+)[Vdenotes 2 hyphen t erm{ 22 } ] ˆ{ − 1 }relativisticthe( \ anticommutatorf r equation a c { d }{ perioddr .} − V { 2 1 } ) + V { 1 1 }\}\Phi { 1 } = 0 . \ tag ∗{$ ( 63 .. ) Todorov $}We are .. equation going to .. via apply .. 1 slash the l1 ../j methodexpansion method t o the equation ( 7 ) . In many \endHere{physicallya lwe i g nconsider∗} interesting the Todorov cases hyphen the type function equation describingZ vanishes the or relativistic it is negligible system of at twoj interactinglarge . Thus scalarthe particles wave equation in the centre has hyphen a 2 of× hyphen2 matrix mass 2 reference- t erm frame form open which square is convenient bracket 2 4 comma for application 2 5 comma of 2 .. 6 closing square bracket :The matrixthe method $ V .{ 22 In} other$ i cases s diagonal the third for t ermall ofpotentials the operator considered ( 7 ) contains in Section a first - 5 order ( and many other ones ) . Inbraceleftbig thesederivative cases p to via the we off power can - diagonal perform of 2 plus matrix U the open transformation elements parenthesis only r comma . : This E closing form parenthesis i s tractable minus t oob open , but parenthesis with E closing parenthesis bracerightbigmore tCapital edious Psi calculations open parenthesis . r closing We do parenthesis not consider = 0 period such equations in this paper . Before \ [ Here\ tildeproceeding p ={\ minusPhi} furtheri nabla{ 1 comma} , we= study the\Phi quasipotential a simpler{ 1 } example U/ open\ sqrt parenthesis of{ aV single{ r22 comma 2 -}} t erm{ E closing, relativistic}\ parenthesistilde equation{L} depends= . \ onsqrt energy{ V E{ of the22 system}} L comma\ sqrt3{ andV the{ Todorov binding22 }}\] equation via 1/` method parameterHere we b open consider parenthesis the Todorov E closing - parenthesis type equation is the following describing function the relativistic of E comma system of two inter- 2 acting scalar particles in the centre - of - mass reference frame [ 2 4 , 2 5 , 2 6 ] : \noindentEquation:providing open parenthesis for 8 the closing operator parenthesis $ ..\ btilde open{ parenthesisL} $ the E closingform which parenthesis is as = 1 close divided as by 4 possible E to the power t o 2of− 2 minust erm form : 2 1 divided by 2 parenleftbig 2 m sub 1 plus 2{ mp to+ U the(r, powerE) − b( ofE) 2}Ψ( parenrightbigr) = 0. plus 1 divided by 4 parenleftbig 2 m sub 1 minus m sub 2 \ begin { a l i g n ∗} to theHere power ofp 2= parenrightbig− i ∇, the to quasipotential the power of 2 slashU(r, E) todepends the power on of energy 2 commaE soof that the E systemopen parenthesis , and the b closing parenthesis = sum radicalbig-line\ tilde {L} of a( m to E the power ) = of 2\ plusf r a c b{ subd ˆperiod{ 2 }}{ a = 1dr ˆ{ 2 }} − W ( r , E , j ) − \{ Z ( rbinding , E , j ) , \ f r a c { d }{ dr }\} { + } J ˆ{ ( 1 ) } ; \ tag ∗{$ ( 7 ) $} \end{parametera l i g n ∗} b(E) is the following function of E, 2 \noindent here $W ( r , E , j )$ isasymmetric $2 \times 2$ matrix $, Jˆ{ ( 1 ) }$ i s $ 2 \1times1 2 $ symplectic1 matrix and $ \{\Xcdot , \cdot \} + $ denotes b(E) = E2 − (2 + 22 ) + (2 − m2)2/E2, sothat E(b) = pa2 + b (8) the anticommutator . 4 2 m1 m 4 m1 2 m . a = 1 We are going to apply the $ 1 / j $ expansion method t o the equation ( 7 ) . \quad In many physically interesting cases the function $ Z $ vanishes or it is negligible at $ j $ large . \quad Thus the wave equation has a $ 2 \times 2 $ matrix 2 − t erm form which is convenient for application of the method . \quad In other cases the third t erm of the operator ( 7 ) contains a first − order derivative via off − diagonal matrix elements only . \quad This form i s tractable t oo , but with more t edious calculations . \quad We do not consider such equations in this paper . Before proceeding further , we study a simpler example of a single 2 − t erm relativistic equation .
\noindent 3 \quad Todorov \quad equation \quad via \quad $ 1 / \ e l l $ \quad method
\noindent Here we consider the Todorov − type equation describing the relativistic system of two interacting scalar particles in the centre − o f − mass reference frame [ 2 4 , 2 5 , 2 \quad 6 ] :
\ [ \{ p ˆ{ 2 } + U ( r , E ) − b ( E ) \}\Psi ( r ) = 0 . \ ]
\noindent Here $ p = − $ i $ \nabla , $ the quasipotential $U ( r , E ) $ depends on energy $ E $ of the system , and the binding
\noindent parameter $b ( E ) $ is the following function of $E , $
\ centerline {2 }
\ begin { a l i g n ∗} b ( E ) = \ f r a c { 1 }{ 4 } E ˆ{ 2 } − \ f r a c { 1 }{ 2 } ( 2 { m { 1 }} + 2 { m }ˆ{ 2 } ) + \ f r a c { 1 }{ 4 } ( 2 { m { 1 }} − m ˆ{ 2 } { 2 } ) ˆ{ 2 } / E ˆ{ 2 } , so that E ( b ) = \sum \ sqrt { a { m }ˆ{ 2 } + b } { . }\ tag ∗{$ ( 8 ) $}\\ a = 1 \end{ a l i g n ∗} 4 .... A period Duviryak \noindent 4 \ h f i l l A . Duviryak hline4 A . Duviryak The corresp onding radial equation takes the form \ [ Equation:\ r u l e {3em open}{0.4 parenthesis pt }\ ] 9 closing parenthesis .. braceleftbigg d to the power of 2 divided by dr to the power of 2 minus W open parenthesis r comma E comma l closing parenthesis bracerightbigg Capital Psi open parenthesis r closing parenthesis = 0 comma whereThe l is corresp the angular onding momentum radial quantumequation number takes commathe form and \noindentEquation:The open corresp parenthesis onding 10 closing radial parenthesis equation .. W open takes parenthesis the form r comma E comma l closing parenthesis = U open parenthesis r comma E closing parenthesis plus l open parenthesisd2 l plus 1 closing parenthesis slash r to the power of 2 minus b open parenthesis E closing \ begin { a l i g n ∗} { − W (r, E, `)}Ψ(r) = 0, (9) parenthesis period dr2 \{\Let usf r consider a c { d ˆmotion{ 2 }}{ of thedr system ˆ{ 2 in}} the −neighbourhoodW ( of r classical , stable E ,circular\ e lorbit l period) \}\Psi ( r ) = 0 , \Giventagwhere∗{ l$ greater (` is 09 the comma angular ) $} .. the momentum radius r sub c quantum = r sub c numberopen parenthesis , and l closing parenthesis .. of the stable circular orbit and the corresp onding\end{ a energy l i g n ∗}
Line 1 E sub c = E sub c open parenthesis l closing parenthesis satisfy2 conditions : Line 2 W open parenthesis r sub c comma E sub c comma\noindent l closingwhere parenthesis $ \ e = l l 0$ comma isW partialdiffthe(r, E, angular `) = WU open(r, momentum E) parenthesis + `(` + 1) quantum/r r sub− b c(E comma number). E sub , and c comma l closing(10) parenthesis slash partialdiff r sub c = 0Let open us parenthesis consider 1 motion 1 closing of parenthesis the system in the neighbourhood of classical stable circular orbit \ beginand partialdiff{ a l i g n ∗} to the power of 2 W open parenthesis r sub c comma E sub c comma l closing parenthesis slash partialdiff c r to the power W. ( Given r ,` > E0, ,the radius\ e l l rc)= =rc(`) Uof ( the stable r , circular E ) orbit + and\ thee l l corresp( \ ondinge l l + 1 ) / r ˆ{ 2 } of 2 greaterenergy 0 semicolon here partialdiff W open parenthesis r sub c comma E sub c comma l closing parenthesis slash partialdiff r sub c equiv −partialdiffb W ( open E parenthesis ) . \ rtag comma∗{$ E ( comma 10 l closing ) $} parenthesis slash partialdiff r vline sub E r = sub = E sub c to the power of r sub c ..\end etc{ perioda l i g n ∗} Ec = Ec(`)satisfyconditions : One puts r = r sub c plus Capital Delta r and E = E sub c plus Capital Delta E where Capital Delta r and Capital Delta E are small in W (r ,E , `) = 0, ∂W (r ,E , `)/∂r = 0 (11) someLet usmeaning consider comma motion of the systemc c in the neighbourhoodc c c of classical stable circular orbit . Given $ \2 e l l > 02 , $ \quad the radius $ r { c } = r rc { c } ( \ e l l ) $ \quad of the stable circular orbit and the corresp onding energy andand expand∂ W the(r functionc,Ec, `)/∂ ..c Wr > open0; here parenthesis∂W (rc,E r subc, `) c/∂r plusc ≡ Capital∂W (r, Delta E, `)/∂rvline r commaEr E= sub=E cc plusetc Capital . Delta E comma l closing parenthesis .. in powerOne series puts withr respect= r + .. ∆ Capitalr and E Delta= E r+ and ∆E Capitalwhere Delta∆r and E period∆E are small in some meaning , and \ [ \ begin { a l i g n e d } E c{ c } = E c{ c } ( \ e l l ) satisfy conditions : \\ Thenexpand due t o the the function conditions ..W open(rc + parenthesis ∆r, Ec + ∆ 1E, 1 ` closing) in parenthesispower series .. the with leading respect t erms∆ ofr thisand expansion∆E. Then represent the harmonic Woscillatordue ( t potential or the{ conditionsc and} other,E ones ({ 1arec 1 anharmonic )} , the leading\ e t l erms l t period) erms = .. of If this 0 the conditions expansion , \ partial open represent parenthesisW the ( 1 harmonic .. 1 r closing{ c parenthesis} ,E hold{ forc } any , large\oscillatore l value l of)/ l potential it i s possible\ partial and by other renormalizationr ones{ arec } anharmonic of= Capital 0 Delta ( t erms r 1 and .1 Capital If ) the\ Deltaend conditions{ Ea l to i g singlen e d (}\ 1 out] 1in ) the hold equation open parenthesis 9 closingfor parenthesis any large value of ` it i s possible by renormalization of ∆r and ∆E to single out in the theequation l hyphen independent ( 9 ) the ` harmonic− independent oscillator harmonic problem and oscillator anharmonic problem perturbations and anharmonic which disappear perturbations if \noindent and $ \ partial ˆ{ 2 } W ( r { c } ,E { c } , \ e l l )/ \ partial c { r }ˆ{ 2 } l rightwhich arrow disappear infinity period if ` ..→ This ∞. i sThis the idea i s of the 1 slash idea l expansionof 1/` expansion method period method .. Application . Application of pseudoperturbative of t echniques > of0 thispseudoperturbative typ ; $ e open here square $ \ bracketpartial t echniques 1 .. 9W comma of this ( 2 typ.. r 0 comma e{ [c 1} 2 9 ..,E , 1 2 comma 0{ , .. 2c 2} 2 1 comma, 2\ 2e 2 l .. ,l 32 closing)/ 3 ] t square o our\ partial bracket t o ourr case{ c meets} two\equiv p eculiaritiescase\ meetspartial : .. two the p equationW eculiarities ( open r parenthesis : , the E equation 9 closing , \ parenthesise ( l 9 l ) represents)/ represents\ apartial nonlinear spr ectral v l i n eproblem{ E } r = { = E }ˆ{ r { c }} { c }$ \quada nonlinear,e t c and . sp an ectral exact problem solution comma of.. and the an equations exact .. solution ( 1 of1 )the equations may appear .. open t o parenthesis b e unknown 1 1 closing or t parenthesis .. may appear t o b e oo cumbersome for practical use . Thus we modify the t echnique . Oneputs $r = r { c } + \Delta √ r $ and $ E = E { c } + \Delta E $ where $ \Delta unknownLet or us t oo introduce cumbersome the for parameter practical useλ period= 1/ ..` Thuswhich we i modify s small the at t echnique` large period . Since the exact r $ and $ \Delta E $ are small in some meaning , Letform us introduce of the the functions parameterr lambda(`) and =E 1 slash(`) is square unknown root of in l general which i s , small we first at l large determine period .. asymptotics Since the exact form of and expand the function \cquad $ Wc ( r { c } + \Delta r , E { c } + \Delta E, ther functions∼ r (λ r) sub, b c open= parenthesisb(E ) ∼ l closingb (λ) parenthesisat λ and→ E sub0 cwhich open parenthesis may b e l found closing much parenthesis easier is unknown . in general comma \ e l l c ) $ ∞ \quadc in powerc series∞ with respect \quad $ \Delta r $ and $ \Delta E . $ we firstThen determine the functionsasymptoticsr rc( sub`) and c thicksimEc(`) rcan sub be infinity presented open parenthesis in the form lambda : closing parenthesis comma Thenb sub due c = b t open o the parenthesis conditions E sub c\ closingquad ( parenthesis 1 1 ) \quad thicksimthe b sub leading infinity open t erms parenthesis of this lambda expansion closing parenthesis represent .. at the .. lambda harmonic rightoscillator arrow 0 which potential may b e found and much other easier ones period are .. anharmonic Then the functions t erms r sub . c\ openquad parenthesisIf the conditions l closing parenthesis ( 1 \quad 1 ) hold for any large value of $ \ e l l $ it i s possible by renormalization of $ \Delta r $ and $ \Delta E $ to single out in the equation ( 9 ) and E sub c open parenthesis l closing parenthesisr canc(λ) be = presentedr∞(λ)ρ(λ) in, the bc(λ form) = b :∞(λ)µ(λ), the $ \ e l l − $ independent harmonic oscillator problem and anharmonic perturbations which disappear if r sub c open parenthesis lambdaρ(λ) = closing 1 + λρ(1) parenthesis+ λ2ρ(2) + =··· r sub, infinity µ(λ) = open 1 + λµ parenthesis(1) + λ2µ(2) lambda+ ···, closing parenthesis(12) rho open parenthesis lambda closing$ \ e parenthesis l l \rightarrow comma b sub c\ openinfty parenthesis. $ lambda\quad closingThis iparenthesis s the idea = b sub of infinity $1 open / parenthesis\ e l l $ lambda expansion closing method parenthesis . \quad Application of pseudoperturbative t echniques of this typ e [ 1 \quad 9 , 2 \quad 0 , 2 \quad 1 , \quad 2 2 , 2 \quad 3 ] t o our case meets two p eculiarities : \quad the equation ( 9 ) represents mu openwhere parenthesis expansion lambda coefficients closing parenthesisρ(n), µ(n comma), n = 1, Equation:2, ... ( and open thus parenthesis the analytical 1 2 closing functions parenthesisρ(λ ..) and rho open parenthesis lambda a nonlinear sp ectral problem , \quad and an exact \quad solution of the equations \quad ( 1 1 ) \quad may appear t o b e closingµ parenthesis(λ)) can b = e 1 found plus lambda , st ep rho by to step the power , from of the open conditions parenthesis : 1 closing parenthesis plus lambda to the power of 2 rho to the power ofunknown open parenthesis or t oo 2 closing cumbersome parenthesis for plus practical times times use times . comma\quad muThus open we parenthesis modify lambda the t closing echnique parenthesis . = 1 plus lambda mu to the power of open parenthesis 1 closing parenthesis plus lambda to the power of 2 mu to the power of open parenthesis 2 closing parenthesis plusLet times us introduce times times comma the parameterW¯ $ (ρ,\lambda µ, λ) = 0,= ∂W¯ (ρ, 1 µ, λ /)/∂ρ\=sqrt 0{\ e l l }$ which i s small (13) at $ \ e l l $ l a r g e . \quad Since the exact form of thewhere functions expansion coefficients $ r { c rho} to the( power\ e l l of open) $ parenthesis and $ n E closing{ c parenthesis} ( \ commae l l mu) to $ the is power unknown of open in parenthesis general n closing , we first determine asymptotics 2 ¯ 2 ¯ parenthesis$ r {andc comma}\∂ W (simρ, n µ, = λ 1)/∂ρ commar {\> 2infty comma0; here} period( the period\ dimensionlesslambda period open) parenthesisfunction , $ andW ( thusρ, µ, the λ) analyticali s constructed functions by rho open parenthesis lambda ¯ closing$ bthe parenthesis{ directc } = and useb of ( 1 ( 2 ) E in{ ( 1c 0} ) and) normalizing\sim b {\ in orderinfty that} (W (ρ,\ µ,lambda λ) to b e) regular $ \quad at at \quad $ \lambda \rightarrowmuλ open→ 0 parenthesis, 0 $ lambda which closing may b parenthesis e found closing much parenthesis easier . can\quad b e foundThen comma the functions st ep by step comma $ r { fromc } the conditions( \ e l l : ) $ andEquation: $ E open{ c parenthesis} ( \ e 13 l l closing) $ parenthesis can be .. presented W-macron open in the parenthesis form rho: comma mu comma lambda closing parenthesis = 0 ¯ 4 2 2 comma partialdiff macron-W open parenthesisW (ρ, µ, rho λ) comma = λ ∞r muW [r comma∞ρ, E(b lambda∞µ), 1/λ closing]. parenthesis slash partialdiff rho = 0 \ begin { a l i g n ∗} and partialdiffNow to we the go power t o the of 2dimensionless W-macron open variable parenthesisr → rhoξ and comma sp ectral mu comma parameter lambdab closing(E) → parenthesis, slash partialdiff rho to ther power{ c of} 2( greater\lambda 0 semicolon) .. here = the r dimensionless{\ infty } function( \ ..lambda macron-W) open\ parenthesisrho ( rho\lambda comma mu) comma , lambda b { closingc } parenthesis( \lambda .. i s constructed) = by b the{\ infty } ( \lambda ) \mu ( \lambda ), \\\rho ( \lambda ) = 1 + \lambda \rho ˆ{ ( 1 ) } + \lambda ˆ2{ 2 }\rho ˆ{ ( 2 ) } + \cdot \cdot direct use of open parenthesis 1 2 closingr = r∞( parenthesisλ)[ρ(λ) + λξ in], open b = parenthesisb∞(λ)[µ(λ) 1 + 0λ closing], parenthesis and normalizing(14) in order that W-macron open\cdot parenthesis, \ rhomu comma( mu\lambda comma lambda) =closing 1 parenthesis + \lambda to b e regular\mu atˆ lambda{ ( right 1 arrow ) } 0+ comma\lambda ˆ{ 2 }\mu ˆ{ ( 2W-macron ) in} t erms+ open\ ofcdot parenthesis which\ thecdot rho equation comma\cdot (mu 9 )comma takes, \ tag lambda the∗{$ form ( closing 1 parenthesis 2 ) $ =} lambda to the power of 4 infinity r to the power of 2 W\end bracketleftbig{ a l i g n ∗} r sub infinity rho comma E open parenthesis b sub infinity mu closing parenthesis comma 1 slash lambda to the power of 2 bracketrightbig period d2 1 \noindentNow we gowhere t o the expansion dimensionless coefficients variable r{ right− arrow $ \wrho(ξ, xi , and λˆ){}ψ sp((ξ ectral) = n 0 parameter ) } , b open\mu parenthesisˆ{ ( n E closing (15)) } parenthesis, n = right 1 arrow , dξ2 λ2 epsilon2 , comma . . . ($ andthus the analytical functions $ \rho ( \lambda ) $ and $Equation:\mu ( open\ parenthesislambda 14) closing ) $ parenthesis can b e.. r found = r sub , infinity st ep open by parenthesis step , from lambda the closing conditions parenthesis : open square bracket rho open parenthesis lambda closing parenthesis plus lambda xi closing square bracket comma b = b sub infinity open parenthesis lambda closing parenthesis\ begin { a l bracketleftbig i g n ∗} mu open parenthesis lambda closing parenthesis plus lambda to the power of 2 epsilon bracketrightbig comma \barin t{W erms} of( which\rho the equation, open\mu parenthesis, \lambda 9 closing parenthesis) = 0 takes , the form\ partial \bar{W} ( \rho , \mu , Equation:\lambda open)/ parenthesis\ partial 1 5 closing parenthesis\rho = .. braceleftbigg 0 \ tag ∗{$ d to ( the 13 power ) of $ 2} divided by d xi to the power of 2 minus 1 divided by\end lambda{ a l i g to n ∗} the power of 2 w open parenthesis xi comma epsilon comma lambda closing parenthesis bracerightbigg psi open parenthesis xi closing parenthesis = 0 \noindent and $ \ partial ˆ{ 2 }\bar{W} ( \rho , \mu , \lambda )/ \ partial \rho ˆ{ 2 } > 0 ; $ \quad here the dimensionless function \quad $ \bar{W} ( \rho , \mu , \lambda ) $ \quad i s constructed by the direct use of ( 1 2 ) in ( 1 0 ) and normalizing in order that $ \bar{W} ( \rho , \mu , \lambda ) $ to be regular at $ \lambda \rightarrow 0 , $
\ [ \bar{W} ( \rho , \mu , \lambda ) = \lambda ˆ{ 4 }\ infty { r }ˆ{ 2 } W [ r {\ infty } \rho , E ( b {\ infty }\mu ) , 1 / \lambda ˆ{ 2 } ]. \ ]
\ centerline {Now we go t o the dimensionless variable $ r \rightarrow \ xi $ and sp ectral parameter $ b ( E ) \rightarrow \ epsilon , $ }
\ begin { a l i g n ∗} r = r {\ infty } ( \lambda )[ \rho ( \lambda ) + \lambda \ xi ] , b = b {\ infty } ( \lambda )[ \mu ( \lambda ) + \lambda ˆ{ 2 }\ epsilon ], \ tag ∗{$ ( 14 ) $} \end{ a l i g n ∗}
\noindent in t erms of which the equation ( 9 ) takes the form
\ begin { a l i g n ∗} \{\ f r a c { d ˆ{ 2 }}{ d \ xi ˆ{ 2 }} − \ f r a c { 1 }{\lambda ˆ{ 2 }} w ( \ xi , \ epsilon , \lambda ) \}\ psi ( \ xi ) = 0 \ tag ∗{$ ( 1 5 ) $} \end{ a l i g n ∗} Large hyphen j Expansion Method for Two hyphen Body Dirac Equation .... 5 \noindent Large $ − j $ Expansion Method for Two − Body Dirac Equation \ h f i l l 5 hlineLarge −j Expansion Method for Two - Body Dirac Equation 5 with \ [ psi\ r uopen l e {3em parenthesis}{0.4 pt xi}\ closing] parenthesis = Capital Psi open square bracket r sub infinity open parenthesis rho plus lambda xi closing parenthesiswith closing square bracket comma and \noindentEquation:with open parenthesis 16 closing parenthesis .. w open parenthesis xi comma epsilon comma lambda closing parenthesis = W-macron ψ(ξ) = Ψ[r (ρ + λξ)], open parenthesis rho plus lambda xi comma mu plus lambda∞ to the power of 2 epsilon comma lambda closing parenthesis period \ [ If\ psi theand functions( \ rhoxi open) parenthesis = \Psi lambda[ closing r {\ parenthesisinfty } .. and( mu\rho open parenthesis+ \lambda lambda\ closingxi parenthesis)], ..\ satisfy] the conditions open parenthesis 1 3 closing parenthesis comma the equation open parenthesis 1 5 closing parenthesis i s nonsingular at lambda right arrow 0 period .. This is true even if we use the first hyphen order approximate solution to open parenthesis 1 3 closing w(ξ, , λ) = W¯ (ρ + λξ, µ + λ2, λ). (16) parenthesis\noindent inand open parenthesis 14 closing parenthesis comma Equation:If the open functions parenthesisρ(λ) 1 7and closingµ( parenthesisλ) satisfy .. rhothe open conditions parenthesis ( 1 lambda 3 ) , the closing equation parenthesis ( 1 = 5 1 ) plus i s lambda rho to the power \ begin { a l i g n ∗} of opennonsingular parenthesis 1 at closingλ → parenthesis0. This is comma true mueven open if we parenthesis use the lambda first - order closing approximate parenthesis = solution1 plus lambda to ( mu to the power of open parenthesisw1 ( 3 ) 1\ in closingxi ( 14, parenthesis ) , \ epsilon period , \lambda ) = \bar{W} ( \rho + \lambda \ xi , \mu + \lambdaIndeed commaˆ{ 2 }\ usingepsilon the notation, partialdiff\lambda W-macron). to the\ tag power∗{$ of ( open 16 parenthesis ) $} 0 closing parenthesis sub slash partialdiff mu = limint\end{ lambdaa l i g n ∗} right arrow 0 partialdiff macron-W slash partialdiff mu = partialdiff W-macron slash partialdiff mu open parenthesis rho = 1 (1) (1) comma mu = 1 comma lambda = 0 closingρ parenthesis(λ) = 1 + λρ .. etc, period µ(λ) = .. 1 we + λµ . (17) \noindenthave If the functions $ \rho ( \lambda ) $ \quad and $ \mu ( \lambda ) $ \quad satisfy the conditions ( 1 3 ) , the equation ( 1 5 ) i s nonsingular at Indeed , using the notation ∂W¯ (0) = lim ∂W¯ /∂µ = ∂W¯ /∂µ(ρ = 1, µ = 1, λ = 0) etc . we $1 divided\lambda by lambda\rightarrow to the power0 of 2 w ./∂µ open$ \quad parenthesisλ→0 This xi is comma true epsilon even comma if we lambda use the closing first parenthesis− order = 1 approximate divided by lambda solution to ( 1 3 ) in ( 14 ) , to the powerhave of 2 to the power of W-macron bracketleftbig rho open parenthesis lambda closing parenthesis plus lambda xi comma mu open parenthesis\ begin { a l lambda i g n ∗} closing parenthesis plus lambda to the power of 2 epsilon comma lambda bracketrightbig = 1 divided by lambda to the \rho ( \lambda ) = 1 + \lambda \rho ˆ{ ( 1 ) } , \mu ( \lambda ) = 1 power of 2 to the power of W-macron to the power1 of open parenthesis1 W¯ 0 closing parenthesis plus 1 divided by lambda open brace partialdiff + \lambda \mu ˆ{ ( 1 ) } . \ tag ∗{$ ( 1 7 ) $} 2 macron-W to the power of open parenthesis 0 closing2 w parenthesis(ξ, , λ) = divided2 [ρ(λ by) + partialdiffλξ, µ(λ) + rhoλ , parenleftbig λ] rho to the power of open parenthesis 1 \end{ a l i g n ∗} λ λ closing parenthesis plus xi parenrightbig(0) plus partialdiff W-macron to the power of open parenthesis 0 closing parenthesis divided by partialdiff 1 W¯ 1 ∂W¯ (0) ∂W¯ (0) ∂W¯ (0) mu parenleftbig mu to the power of open parenthesis 1 closing(1) parenthesis plus(1) lambda epsilon parenrightbig plus partialdiff macron-W to the = 2 + { (ρ + ξ) + (µ + λ) + } power\noindent of openIndeed parenthesis , using 0 closing theλ parenthesis notationλ divided∂ρ $ \ bypartial partialdiff∂µ lambda\bar{W closing} ˆ{ ( brace∂λ 0 plus ) 1} divided{ / by\ 2partial partialdiff to\mu the} power= of\ 2lim {\lambda W-macron\rightarrow to the power0 }\ of openpartial parenthesis\bar1 0∂{ closing2WW¯}(0) parenthesis/ \ partial divided1 ∂2W¯ (0) by\mu partialdiff=1 ∂ rho2\W¯partial(0) to the power\bar of 2{ parenleftbigW} / rho\ partial to the power\mu of ( \rho = 1 , \mu =+ 1 , (ρ(1)\lambda+ ξ)2 + = 0[µ(1)]2 ) + $ \quad e t c . \quad we open parenthesis 1 closing parenthesis plus xi2 parenrightbig∂ρ2 to the power2 ∂µ2 of 2 plus 1 divided2 ∂λ2 by 2 partialdiff to the power of 2 macron-W to the have power of open parenthesis 0 closing∂2W¯ (0) parenthesis divided∂ by2W¯ partialdiff(0) mu to∂2 theW¯ (0) power of 2 bracketleftbig mu to the power of open parenthesis 1 closing parenthesis bracketrightbig+ 2( plusρ(1) + 1ξ divided)µ(1) + by 2 partialdiff(ρ(1) + ξ to) + the powerµ(1) of 2+ W-macronO(λ). to the power(18) of open parenthesis 0 closing ∂ρ∂µ ∂ρ∂λ ∂µ∂λ parenthesis\ begin { a l divided i g n ∗} by partialdiff lambda to the power of 2 Equation: open parenthesis 18 closing parenthesis .. plus partialdiff to the power of\ f2 r a W-macron c Singular{ 1 }{\ to tlambda the erms power areˆ{ of absent2 open}} parenthesis ifw the ( following 0\ closingxi set parenthesis, of equations\ epsilon divided holds by, partialdiff : \lambda rho partialdiff) = mu\ f parenleftbig r a c { 1 }{\ rholambda to the powerˆ{ 2 of}}ˆ{\bar{W}} open[ \ parenthesisrho ( 1 closing\lambda parenthesis) plus + xi\lambda parenrightbig\ xi mu to, the power\mu of( open\ parenthesislambda 1) closing + parenthesis\lambda plusˆ{ partialdiff2 }\ epsilon to the power, \ oflambda 2 macron-W] \\ to the= power\ f r a c of{ open1 }{\ parenthesislambda 0ˆ closing{ 2 }} parenthesisˆ{\bar{W divided} ˆ{ by( partialdiff 0 ) }} rho partialdiff+ \ f r a clambda{ 1 }{\ parenleftbiglambda rho}\{ to W¯ (0) = 0, ∂W¯ (0) = 0, (19) the\ f r apower c {\ ofpartial open parenthesis\bar{W 1} closingˆ{ ( parenthesis 0 ) plus}}{\ xipartial parenrightbig/∂ρ \ plusrho partialdiff} ( \ torho theˆ power{ ( of 1 2 W-macron ) } + to the\ xi power) of open + parenthesis\ f r a c {\ partial 0 closing parenthesis\bar{W} dividedˆ{ ( by partialdiff 0∂W )¯ (0)}}{\ mu partialdiffpartial∂W¯ (0) lambda\mu mu} to( the power\mu ofˆ{ open( parenthesis 1 ) } 1+ closing\lambda parenthesis\ plusepsilon O ) + \ f r a c {\ partial \bar{W} ˆ{ ( 0µ(1) + ) }}{\ =partial 0. \lambda }\}\\ + \(20)f r a c { 1 }{ 2 }\ f r a c {\ partial ˆ{ 2 } open parenthesis lambda closing parenthesis period∂µ ∂λ \barSingular{W} ˆ{ t erms( are 0 absent ) }}{\ if thepartial following set\ ofrho equationsˆ{ 2 holds}} :( \rho ˆ{ ( 1 ) } + \ xi ) ˆ{ 2 } + \ f r a c { 1 }{ 2 }\ f r a c {\ partial ˆ{ 2 } \barEquation:{WBesides} ˆ{ open( , zero parenthesis 0 - order ) }}{\ 19 t closingermspartial which parenthesis are\mu linear ..ˆ W-macron{ 2 in}}ξ disappear to[ the\ powermu ˆ if{ of open( 1parenthesis ) } 0] closing 2 parenthesis + \ f r a c={ 01 comma}{ 2 partialdiff}\ f r a c {\ partial ˆ{ 2 } macron-W\bar{W} ˆ to{ the( power 0 of ) open}}{\ parenthesispartial 0 closing\lambda parenthesisˆ{ 2 sub}}\\ slash+ partialdiff\ f r a c {\ rhopartial = 0 commaˆ{ Equation:2 }\bar open{W} parenthesisˆ{ ( 0 20 closing ) }}{\ partial \rho \ partial \mu } ( \rho ˆ{ ( 1 ) } + \ xi ) \mu ˆ{ ( 1 ) } + \ f r a c {\ partial ˆ{ 2 } parenthesis .. partialdiff W-macron to the∂2 powerW¯ (0) of open∂2W parenthesis¯ (0) ∂ 02 closingW¯ (0) parenthesis divided by partialdiff mu mu to the power of open \bar{W} ˆ{ ( 0 ) }}{\ partial ρ\(1)rho+ \ partialµ(1) + \lambda= 0. } ( \rho ˆ{ ( 1(21) ) } + \ xi ) parenthesis 1 closing parenthesis plus partialdiff∂ρ2 macron-W∂ρ∂µ to the power∂ρ∂λ of open parenthesis 0 closing parenthesis divided by partialdiff lambda += 0\ periodf r a c {\ partial ˆ{ 2 }\bar{W} ˆ{ ( 0 ) }}{\ partial \mu \ partial \lambda }\mu ˆ{ ( 1 ) }Besides+Notice comma O that ( zero the\ hyphenlambda equations order). t erms ( 1 9 which\ )tag ∗{ andare$ linear ( ( 20 18in ) – xi ( disappear 2 ) 1 $ )} represent if the conditions ( 1 3 ) \endEquation:{ina l i the g n ∗} zerothopen parenthesis and first 2 orders 1 closing of parenthesisλ, respectively .. partialdiff . Thus to the the power equations of 2 W-macron ( 1 9 ) hold to the identically power of open parenthesis 0 closing parenthesisand divided ( 2 0 ) by – (partialdiff 2 1 ) are rho linear to the power of 2 rho to the power of open parenthesis 1 closing parenthesis plus partialdiff to the power of (1) (1) 2\noindent macron-Wequations toSingular the with powerρ of t open ermsand parenthesis areµ t absent o be 0 closing found if parenthesisthe . following divided set by partialdiff of equations rho partialdiff holds mu : mu to the power of open parenthesis 1 closingIn zero parenthesis - order plus approximation partialdiff to the the power equation of 2 W-macron ( 1 5 ) reduces to the power t o the of open harmonic parenthesis oscillator 0 closing problem parenthesis divided by partialdiff rho\ begin partialdiff{ a l i g lambdan ∗} = 0 period \bar{W} ˆ{ ( 0 ) } = 0 , \ partial \bar{W} ˆ{ ( 0 ) } { / \ partial \rho } = 0 Notice that the equations .. open parenthesisd 12 9 closing parenthesis .. and open parenthesis 20 closing parenthesis endash open parenthesis , \ tag ∗{$ ( 19 ) $}\\\ f r a c {\ partial{ + κ − ν\bar− ω2{Wξ2}}ψˆ(ξ{) =( 0 0 ) }}{\ partial \mu (22)}\mu ˆ{ ( 1 ) } 2 1 closing parenthesis .. represent the conditionsdξ2 .. open parenthesis 1 3 closing parenthesis in the zeroth and + first\ f r aorders c {\ ofpartial lambda comma\bar respectively{W} ˆ{ ( period 0 .. ) Thus}}{\ thepartial equations open\lambda parenthesis} = 1 9 closing 0 . parenthesis\ tag ∗{$ hold ( identically 20 ) $and} open parenthesis\end{witha l i g 2n 0∗} closing parenthesis endash open parenthesis 2 1 closing parenthesis are linear equations with rho to the power of open parenthesis 1 closing parenthesis and mu to the power of open parenthesis 1 closing parenthesis t o \noindent Besides , zero − order t erms which are linear in $ \ xi $ disappear if be found period ∂W¯ (0) 1 ∂2W¯ (0) κ = − ω2 = (23) In zero hyphen order approximation the equation open parenthesis 1 5 closing parenthesis2 reduces t o the harmonic oscillator problem ∂µ , 2 ∂ρ , \ beginEquation:{ a l i g open n ∗} parenthesis 22 closing parenthesis .. braceleftbigg d to the power of 2 divided by d xi to the power of 2 plus kappa epsilon \ f r a c {\ partial ˆ{ 2 }\bar1 ∂2{W¯}(0)ˆ{ (1 0∂2W¯ )(0)}}{\ partial1 ∂2W¯ (0) \rho∂2W¯ (0)ˆ{ 2 }}\rho ˆ{ ( 1 ) } + \ f r a c {\ partial ˆ{ 2 } minus nu minus omega to theν power= − of 2 xi to[ρ(1) the]2 power+ of 2 bracerightbigg[µ(1)]2 + psi open+ parenthesisµ(1), xi closing parenthesis(24) = 0 \barwith{W} ˆ{ ( 0 ) }}{\2partial∂ρ2 \rho2 ∂µ\2partial 2 \mu∂λ2 }\mu∂µ∂λˆ{ ( 1 ) } + \ f r a c {\ partial ˆ{ 2 } \bar{W} ˆ{ ( 0 ) }}{\ partial(0) \rho \ partial \lambda } = 0 . \ tag ∗{$ ( 2 1 ) $} Equation: open parenthesis 23 closing2 ¯ parenthesis .. kappa = minus2 partialdiff(0) W-macron2 (0) to the power of open parenthesis 0 closing ∂ W/∂λ 1 ∂ W¯ ∂ W¯ parenthesis\end{ a l i g dividedn ∗} by partialdiffµ(1) = − mu sub commaρ omega(1) = − to the power{ of 2 = 1µ(1) divided+ by 2} partialdiff. to the(25) power of 2 macron-W to the ∂2W¯ (0) ∂2W¯ (0) 2 ∂ρ∂µ ∂ρ∂λ power of open parenthesis 0 closing parenthesis/∂µ , divided by partialdiff/∂ρ rho to the power of 2 sub comma Equation: open parenthesis 24 closing parenthesis\noindent ..Notice nu = minus that 1 divided the equations by 2 partialdiff\quad to the( power 1 9 of ) 2\ W-macronquad and to ( the 20 power ) −− of( open 2 1 parenthesis ) \quad 0represent closing parenthesis the conditions divided \quad ( 1 3 ) in the zeroth and byfirst partialdiff orders rho to of the $power\lambda of 2 bracketleftbig, $ respectively rho to the power . of\quad openThus parenthesis the 1 equations closing parenthesis ( 1 9 bracketrightbig ) hold identically 2 plus 1 divided and ( 2 0 ) −− ( 2 1 ) are linear by 2 partialdiff to the power of 2 macron-W to the power of open parenthesis 0 closing parenthesis divided by partialdiff mu to the power of 2\noindent bracketleftbigequations mu to the power with of $ open\rho parenthesisˆ{ ( 1 1 closing ) } parenthesis$ and $ bracketrightbig\mu ˆ{ ( 2 1 plus 1) divided}$ t by o 2 be partialdiff found to. the power of 2 W-macron to the power of open parenthesis 0 closing parenthesis divided by partialdiff lambda to the power of 2 plus partialdiff to the power of\ centerline 2 macron-W{ toIn the zero power− oforder open parenthesis approximation 0 closing the parenthesis equation divided ( 1 by 5 partialdiff ) reduces mu t partialdiff o the harmonic lambda mu oscillatorto the power of problem open } parenthesis 1 closing parenthesis comma Equation: open parenthesis 25 closing parenthesis .. mu to the power of open parenthesis 1 closing parenthesis\ begin { a l = i g minus n ∗} partialdiff to the power of 2 W-macron to the power of open parenthesis 0 closing parenthesis sub slash partialdiff lambda divided\{\ byf r a partialdiff c { d ˆ{ to2 the}}{ powerd of\ 2xi macron-Wˆ{ 2 }} open+ parenthesis\kappa 0 closing\ epsilon parenthesis− sub \ slashnu partialdiff− \omega mu subˆ{ comma2 }\ rhoxi to theˆ{ power2 } of\}\ open parenthesispsi ( 1\ closingxi parenthesis) = 0 =\ minustag ∗{ 1$ divided ( 22 by partialdiff ) $} to the power of 2 W-macron open parenthesis 0 closing parenthesis sub\end slash{ a l i partialdiff g n ∗} rho 2 open brace partialdiff to the power of 2 macron-W to the power of open parenthesis 0 closing parenthesis divided by partialdiff rho partialdiff mu mu to the power of open parenthesis 1 closing parenthesis plus partialdiff to the power of 2 W-macron to the power\noindent of openwith parenthesis 0 closing parenthesis divided by partialdiff rho partialdiff lambda closing brace period \ begin { a l i g n ∗} \kappa = − \ f r a c {\ partial \bar{W} ˆ{ ( 0 ) }}{\ partial \mu } { , }\omega ˆ{ 2 } = \ f r a c { 1 }{ 2 }\ f r a c {\ partial ˆ{ 2 } \bar{W} ˆ{ ( 0 ) }}{\ partial \rho ˆ{ 2 }} { , }\ tag ∗{$ ( 23 ) $}\\\nu = − \ f r a c { 1 }{ 2 }\ f r a c {\ partial ˆ{ 2 } \bar{W} ˆ{ ( 0 ) }}{\ partial \rho ˆ{ 2 }} [ \rho ˆ{ ( 1 ) } ] 2 + \ f r a c { 1 }{ 2 }\ f r a c {\ partial ˆ{ 2 } \bar{W} ˆ{ ( 0 ) }}{\ partial \mu ˆ{ 2 }} [ \mu ˆ{ ( 1 ) } ] 2 + \ f r a c { 1 }{ 2 }\ f r a c {\ partial ˆ{ 2 } \bar{W} ˆ{ ( 0 ) }}{\ partial \lambda ˆ{ 2 }} + \ f r a c {\ partial ˆ{ 2 }\bar{W} ˆ{ ( 0 ) }}{\ partial \mu \ partial \lambda }\mu ˆ{ ( 1 ) } , \ tag ∗{$ ( 24 ) $}\\\mu ˆ{ ( 1 ) } = − \ f r a c {\ partial ˆ{ 2 } \bar{W} ˆ{ ( 0 ) } { / \ partial \lambda }}{\ partial ˆ{ 2 }\bar{W} ( 0 ) { / \ partial \mu }} { , }\rho ˆ{ ( 1 ) } = − \ f r a c { 1 }{\ partial ˆ{ 2 }\bar{W} ( 0 ) { / \ partial \rho } 2 }\{\ f r a c {\ partial ˆ{ 2 }\bar{W} ˆ{ ( 0 ) }}{\ partial \rho \ partial \mu }\mu ˆ{ ( 1 ) } + \ f r a c {\ partial ˆ{ 2 }\bar{W} ˆ{ ( 0 ) }}{\ partial \rho \ partial \lambda }\} . \ tag ∗{$ ( 25 ) $} \end{ a l i g n ∗} 6 .... A period Duviryak \noindent 6 \ h f i l l A . Duviryak hline6 A . Duviryak The higher hyphen order t erms in the expansion open parenthesis 1 8 closing parenthesis can b e considered as p erturbations of the oscillator\ [ \ r u l e {3em}{0.4 pt }\ ] problem .. open parenthesis 22 closing parenthesis period .. They depend comma .. in general comma .. on the spectral parameter epsilon and canThe be taken higher into - order t erms in the expansion ( 1 8 ) can b e considered as p erturbations of the \noindentaccountoscillator byThe means problem higher of the− p erturbativeorder ( 22 ) t. erms procedure They in depend the open expansion square , in bracket general ( 21 .. 8 , 5 ) closing oncan the b square spectrale considered bracket parameter is appropriate as p erturbations t o this case period of the .. oscillator Otherwiseproblemand the can\quad be taken( 22into ) . \ accountquad They by means depend of the , \quad p erturbativein g e n e procedure r a l , \quad [ 2on 5 the] is appropriate spectral parameter $ \ epsilon $ andtreatment cant o thisbe i takens case similar . into t o Otherwise open square the bracket treatment 1 9 comma i s .. similar 2 0 comma t o [ 2 1 .. 9 1 , comma 2 0 2 , 2 comma 1 , 2 2 2 .. , 3 2 closing 3 ] square . bracket period account by means of the p erturbative procedure = [ [ω 2(2n\quad+ 1) +5ν] ]/κ, is appropriaten = 0, 1, ... t o this case . \quad Otherwise the TheThe eigenvalues eigenvalues in zero in hyphen zero - order order approximation approximation epsilonnr sub n subr r = open squarewhere bracketr omega openi s parenthesis a 2 n sub r plus 1 closingtreatmentradial parenthesis quantum i s plus similar nu number closing t o square, [ are 1 bracket 9 to , b\ equad slash corrected kappa2 0 , bycomma 2 means\quad where1 ofn , sub higher 2 r 2 = , 0 orders 2comma\quad 1 of comma3 p ] erturbative . period period period i s ..procedure a radial quantum . Then number , using comma ofthe .. are 2 to nd b equation e corrected of by ( means14 ) in .. ( of 8 higher ) gives orders us the .. of energy p erturbative spectrum \noindentprocedure. The period eigenvalues .. Then comma in using zero of− theorder 2 nd approximation equation of open parenthesis $ \ epsilon 14 closing{ n parenthesis{ r }} = in open [ parenthesis\omega 8 closing( 2 parenthesisn { 4r } gives+ Breit us 1 the - energy ) type + spectrum\nu equation period]/ \kappa via ,1/ $j wheremethod $ n { r } = 0 , 1 , . . . $ ˜ i4 s ..At\ Breitquad this hyphena point radial type we .. return quantum equation t o .. number the via ..radial 1 slash, \ 2quad j BDE .. methodare in the to b form e correctedL˜(E)Φ1 = 0 by, where means the\quad2 × 2 ofmatrix higher orders \quad of p erturbative procedureAt thisoperator point . we\L˜quad(E return) isThen given t o the , by using radial equation 2 of BDE the ( in7 ) the2 with nd form equation the L-tilde last opent erm of parenthesis ( neglected 14 ) in E . (closing 8 Let ) parenthesis gives us put us to the the energy power of spectrum tilde-Phi 1 =. 0 comma where the 2 times 2 matrix \noindent 4 \quad B r e i t − type \quad equation \Ψquad1 via \quad $ 1 / j $ \quad method operator L-tilde open parenthesis E closing parenthesisΦ1 = is given, by equation open parenthesis 7 closing parenthesis with the last t erm neglected period .. Let us put Ψ2 \noindentCapital PhiAt sub this 1 = Row point 1 Capital we return Psi sub t 1 oRow the 2 Capital radial Psi 2 sub BDE 2 . in comma the form $ \ tilde {L} ( E ) ˆ{\ tilde {\Phi}} 1 =where 0Ψ ,$1 and wheretheΨ2 are components $2 \ oftimesΦ1. Then2 $ the matrix equation ( 6 ) can b e presented as a pair whereof Capital coupled Psi ordinary sub 1 and second Capital - Psi order sub 2 differential are components equations of Capital : Phi sub 1 period .. Then the equation open parenthesis 6 closing parenthesisoperator can $ b\ etilde presented{L} as a( pair ofE ) $ is given by equation ( 7 ) with the last t erm neglected . \quad Let us put coupled ordinary second hyphen order differential equations : \ [ Equation:\Phi { open1 } parenthesis= \ l e f t26[ \ closingbegind2 parenthesis{ array }{ c ..}\ d toPsi the power{ 1 }\\\ of 2 dividedPsi by{ dr2 to}\ theend power{ array of 2}\ Capitalright Psi], sub\ 1] open parenthesis Ψ (r) − W (r, E, j)Ψ (r) = Y (r, E, j)Ψ (r), (26) r closing parenthesis minus W sub 1 opendr2 1 parenthesis1 r comma1 E comma j closing2 parenthesis Capital Psi sub 1 open parenthesis r closing parenthesis = Y open parenthesis r commad2 E comma j closing parenthesis Capital Psi sub 2 open parenthesis r closing parenthesis comma \noindent where $ \Psi { 1 }$Ψ ( andr) − W $(r,\ E,Psi j)Ψ{(r)2 =}Y$(r,are E, j)Ψ components(r). of $ \Phi { (27)1 } . $ \quad Then the equation ( 6 ) can b e presented as a pair of Equation: open parenthesis 27 closingdr parenthesis2 2 ..2 d to the2 power of 2 divided1 by dr to the power of 2 Capital Psi sub 2 open parenthesis coupled ordinary second − order differential equations : r closing parenthesis minus W sub 2 open parenthesis r comma E comma j closing parenthesis Capital√ Psi sub 2 open parenthesis r closing parenthesisWe will = Y treat open parenthesis this system r comma perturbationally E comma j closing using parenthesis the pseudosmall Capital Psi parameter sub 1 openλ parenthesis= 1/ j. r closing parenthesis period \ beginWe will{ aLet l treat i g n us∗} this suppose system for perturbationally a moment that using the the right pseudosmall - hand parameter side of the lambda system = 1 ( slash 2 6 )square – ( 2 root 7 ) canof j sub b period \ f rLet a c e{ us ignoredd suppose ˆ{ 2 , for}}{ so a momentthatdr ˆ these{ that2 }}\ equations the rightPsi hyphen decouple{ 1 hand} ( . side rofThen the ) system we− can openW apply parenthesis{ t1 o} each( 2 6 of closing r the equations , parenthesis E , endash j open ) parenthesis\Psi { 1 } 2(r)=Y(r,E,j) 7 closingthe parenthesis scheme of can the b e ignored comma \Psi { 2 } ( r ) , \ tag ∗{$ ( 26 ) $}\\\ f r a c { d ˆ{ 2 }}{ dr ˆ{ 2 }} \Psiso thatSection{ 2 these} 3 equations( . r We decouple define ) − radii periodW and ..{ Then2 energies} we( can of r apply circular , t o each E orbits of , the by equationsj means ) of the\ thePsi scheme conditions{ 2 of} the ( : r ) = Y ( rSection , E 3 period , .. j We define ) \ radiiPsi and{ energies1 } ( of circular r ) orbits . by\ tag means∗{$ of ( the conditions 27 ) $ :} ∂W (r ,E , j) ∂2W (r ,E , j) \endW{ suba l i ig openn ∗} parenthesis r sub i comma E subi i commai i j closing parenthesisi i i = 0 comma partialdiff W sub i open parenthesis r sub i comma Wi(ri,Ei, j) = 0, = 0, 2 > 0, i = 1, 2. E sub i comma j closing parenthesis divided by partialdiff∂r r = 0 comma partialdiff∂r to the power of 2 W sub i open parenthesis r sub i comma E sub\noindent i commaThen jWe we closing single will parenthesis treat out asymptotics this divided system by of partialdiff these perturbationally functions r to the power of λ ofby using 2 greater means the 0 of comma pseudosmall the relations i = 1 comma :parameter 2 period $ \lambda = 1 / Then\ sqrt we{ singlej } out{ . asymptotics}$ of these functions of lambda by means of the relations : Line 1 r sub i open parenthesis lambda closingri(λ parenthesis) = ri∞(λ)ρi =(λ r), sub bi i(λ infinity) = bi∞ open(λ)µi( parenthesisλ), lambda closing parenthesis rho i open Let us suppose for a moment that(1) the2 right(2) − hand side of(1) the2 system(2) ( 2 6 ) −− ( 2 7 ) can b e ignored , parenthesis lambda closing parenthesisρi(λ) = 1 +commaλiρ + bλ subiρ i open+ ···, parenthesis µ(λ) = 1 lambda + λiµ + closingλ iµ parenthesis+ ···, = b sub i infinity open parenthesis lambda closingso that parenthesis these mu equations i open parenthesis decouple lambda . \ closingquad parenthesisThen we can comma apply Line 2 t rho o eachi open parenthesisof the equations lambda closing the parenthesis scheme of = 1 the plus lambdaand i rho ,using to the thepower relations of open parenthesis 1 closing parenthesis plus lambda to the power of 2 i rho to the power of open parenthesis 2 closing\noindent parenthesisSec tion plus times 3 . \ timesquad timesWe comma define mu radii open parenthesis and energies lambda of closing circular parenthesis orbits = 1 plus by meanslambda i of mu the to the conditions power of open : parenthesis 1 closing parenthesis plus lambda to the power of 2 i mu to the power of open parenthesis 2 closing parenthesis plus times times r = r (λ)[ρi(λ) + λξi], b = b (λ)[µi(λ) + λ2 ], i = 1, 2 (28) times\ [W comma{ i } ( r { i } i∞,E { i } , ji∞ ) = 0i , \ f r a c {\ partial W { i } ( r { i } ,Eand comma{ i } using, the j relations ) }{\ partial r } = 0 , \ f r a c {\ partial ˆ{ 2 } W { i } ( r { i } , we reformulate the equation ( 26 ) in terms of the dimensionless variable ξ1 and the E Equation:{ i } , open j parenthesis ) }{\ 28partial closing parenthesisr ˆ{ 2 ..}} r => r sub0 i infinity , open i parenthesis = 1 lambda , 2 closing . \ ] parenthesis open square bracket sp ectral parameters while the equation ( 2 7 ) – in terms of ξ2 and . Finally , we p rho i open parenthesis lambda closing1 parenthesis plus lambda xi i closing square bracket comma2 b = b sub i infinity open parenthesis lambda erform expansion of the equations into powers of λ and solve them separately . closing parenthesisNow we open are going square t bracket o take mu actual i open coupling parenthesis of thelambda equations closing parenthesis ( 26 ) and plus ( 2 lambda 7 ) into to account the power . of 2 epsilon sub i closing square\noindent bracketThen comma we i = single 1 comma out 2 asymptotics of these functions of $ \lambda $ by means of the relations : First of all , we note that the variables ξ1 and ξ2 are not of one another , and the spectral we reformulate the equation .. open parenthesis 26 closing parenthesis .. in terms .. of the dimensionless variable xi 1 .. and the sp ectral parameters \ [ \parametersbegin { a l iepsilon g n e d }1 subr 1 while{ i } the equation( \lambda open parenthesis) = 2 7 closing r { parenthesisi \ infty endash} ( in terms\lambda of xi 2 and) epsilon\rho sub 2 periodi ( Finally\lambda ) ,and b 2{ arei } also( not\lambda independent) . = b Thus{ i we should\ infty choose} ( common\lambda variables) \mu in bothi ( \lambda ) commaequations we p erform . expansion of , \\the equations into powers of lambda and solve them separately period Let us first choose ξ = ξ1, = . Then the set ( 2 6 ) – ( 2 7 ) reduces t o the form : \Nowrho we arei going ( t o\ takelambda actual coupling) = of1 the 1 equations + \lambda open parenthesisi {\ 26 closingrho } parenthesisˆ{ ( 1 and ) open} parenthesis+ \lambda 2 7 closingˆ{ 2 parenthesis} i {\rho }ˆ{ ( into2 account ) } + period\cdot .. First \cdot \cdot , \mu ( \lambda ) = 1 + \lambda i {\mu }ˆ{ ( 1of ) all} comma+ we\lambda note thatˆ{ the2 variables} i {\ xi 1mu and1 } xiˆ{ 2 are( not 2 of one ) } another+ comma\cdot and\ thecdot spectral\cdot parameters, \ epsilonend{ a sub l i g n 1 e d }\ ] ψ00(ξ) − w (ξ, , λ)ψ1(ξ) = y(ξ, , λ)ψ2(ξ), (29) and epsilon sub 2 are also not independent1 periodλ2 1 .. Thus we should choose common variables in both equations period Let us first choose xi = xi 1 comma epsilon =1 epsilon sub 1 period .. Then the set open parenthesis 2 6 closing parenthesis endash open \noindent and , using the relationsψ00(ξ) − w (ξ, , λ)ψ2(ξ) = y(ξ, , λ)ψ1(ξ), (30) parenthesis 2 7 closing parenthesis reduces2 t o theλ2 form2 : Equation: open parenthesis 29 closing parenthesis .. psi sub 1 to the power of prime prime open parenthesis xi closing parenthesis minus 1\ begin divided{ a by l i glambda n ∗} to the power of 2 w sub 1 open parenthesis xi comma epsilon comma lambda closing parenthesis psi 1 open parenthesis xir closing = parenthesis r { i = y\ infty open parenthesis} ( \ xilambda comma epsilon)[ comma\rho lambdai closing ( parenthesis\lambda psi 2) open + parenthesis\lambda xi closing\ xi parenthesisi ] comma, b Equation: = b open{ i parenthesis\ infty 30} closing( parenthesis\lambda .. psi)[ sub 2 to the\mu poweri of prime ( prime\lambda open parenthesis) + xi\ closinglambda parenthesisˆ{ 2 }\ minusepsilon { i } 1] divided , by i lambda = to 1 the , power 2 of\ tag 2 w∗{ sub$ 2 ( open 28 parenthesis ) $} xi comma epsilon comma lambda closing parenthesis psi 2 open parenthesis xi\end closing{ a l i parenthesis g n ∗} = y open parenthesis xi comma epsilon comma lambda closing parenthesis psi 1 open parenthesis xi closing parenthesis comma \noindent we reformulate the equation \quad ( 26 ) \quad in terms \quad of the dimensionless variable $ \ xi 1 $ \quad and the sp ectral parameters $ \ epsilon { 1 }$ while the equation ( 2 7 ) −− in terms o f $ \ xi 2 $ and $ \ epsilon { 2 } . $ Finally , we p erform expansion of the equations into powers of $ \lambda $ and solve them separately .
Now we are going t o take actual coupling of the equations ( 26 ) and ( 2 7 ) into account . \quad F i r s t of all , we note that the variables $ \ xi 1 $ and $ \ xi 2 $ are not of one another , and the spectral parameters $ \ epsilon { 1 }$
\noindent and $ \ epsilon { 2 }$ are also not independent . \quad Thus we should choose common variables in both equations .
\ centerline { Let us first choose $ \ xi = \ xi 1 , \ epsilon = \ epsilon { 1 } . $ \quad Then the set ( 2 6 ) −− ( 2 7 ) reduces t o the form : }
\ begin { a l i g n ∗} \ psi ˆ{\prime \prime } { 1 } ( \ xi ) − \ f r a c { 1 }{\lambda ˆ{ 2 }} w { 1 } ( \ xi , \ epsilon , \lambda ) \ psi 1 ( \ xi ) = y ( \ xi , \ epsilon , \lambda ) \ psi 2 ( \ xi ), \ tag ∗{$ ( 29 ) $}\\\ psi ˆ{\prime \prime } { 2 } ( \ xi ) − \ f r a c { 1 }{\lambda ˆ{ 2 }} w { 2 } ( \ xi , \ epsilon , \lambda ) \ psi 2 ( \ xi ) = y ( \ xi , \ epsilon , \lambda ) \ psi 1 ( \ xi ), \ tag ∗{$ ( 30 ) $} \end{ a l i g n ∗} Large hyphen j Expansion Method for Two hyphen Body Dirac Equation .... 7 \noindent Large $ − j $ Expansion Method for Two − Body Dirac Equation \ h f i l l 7 hlineLarge −j Expansion Method for Two - Body Dirac Equation 7 where \ [ Equation:\ r u l e {3em open}{0.4 parenthesis pt }\ ] 31 closing parenthesis .. psi i open parenthesis xi closing parenthesis = Capital Psi sub i open square bracket r sub 1 infinitywhere open parenthesis rho 1 plus lambda xi closing parenthesis closing square bracket comma i = 1 comma 2 comma Equation: open parenthesis 32 closing parenthesis .. w sub i open parenthesis xi comma epsilon comma lambda closing parenthesis = lambda to the power of 4\noindent 1 r sub infinitywhere to the power of 2 W sub i bracketleftbig r sub 1 infinity open parenthesis rho 1 plus lambda xi closing parenthesis comma E parenleftbig b sub 1 infinity open parenthesis mu 1 plusψi(ξ lambda) = Ψi[r to1∞ the(ρ1 power + λξ)], of 2 i = epsilon 1, 2, closing parenthesis(31) parenrightbig comma 1 slash lambda\ begin to{ a the l i g n power∗} of 2 bracketrightbig comma Equation: open parenthesis 33 closing parenthesis .. y open parenthesis xi comma epsilon w (ξ, , λ) = λ412 W [r (ρ1 + λξ),E(b (µ1 + λ2)), 1/λ2], (32) comma\ psi lambdai closing ( \ parenthesisxi )i = = lambda\Psi tor the∞{ powerii 1}∞ of[ 2 1 r r sub{ infinity1∞ \ toinfty the power} of( 2 Y\ bracketleftbigrho 1 +r sub 1\lambda infinity open\ parenthesisxi ) ] , i = 1 , 2 , \ tag2∗{2 $ ( 31 ) $}\\ w { i2 } ( 2 \ xi , \ epsilon , \lambda ) rho 1 plus lambda xi closing parenthesisy(ξ, , λ) comma = λ 1r∞ EY parenleftbig[r1∞(ρ1 + λξ b) sub,E(b 11∞ infinity(µ1 + λ open)), parenthesis1/λ ]. mu 1 plus lambda(33) to the power of 2 epsilon =closing\lambda parenthesisˆ{ parenrightbig4 } 1 { commar }ˆ{ 12 slash} {\ lambdainfty to the} powerW { ofi 2} bracketrightbig[ r { 1 period\ infty } ( \rho 1 + \lambda \ xi )The , functions E ( ( b 3 2{ )1 – ( 3\ infty 3 ) are} regular( \ atmuλ →10. +Moreover\lambda , theˆ{ general2 }\ structureepsilon of ) ) , 1 / Thethe functions function open parenthesis 3 2 closing parenthesis endash open parenthesis 3 3 closing parenthesis are regular at lambda right arrow 0 period\lambda .. Moreoverˆ{ 2 } comma], the\ generaltag ∗{$ structure ( 32 of the ) function $}\\ y ( \ xi , \ epsilon , \lambda ) = \lambda ˆ{ 2 } w i s the same as that of w in the Section 3 ( see equations ( 1 6 ) , ( 1 8 ) ) . In particular 1 w{ subr 1 } 1ˆ{ i s2 the} same{\ infty as that} of wY in the [ Section r { 31 open\ parenthesisinfty } see( equations\rho open1 parenthesis + \lambda 1 6 closing\ xi parenthesis),E( comma open , w = O(λ2). Thus the equation ( 29 ) i s similar to ( 1 5 ) ( but with non - zero right - hand parenthesisb { 1 1 1\ infty 8 closing} parenthesis( \mu closing1 parenthesis + \lambda periodˆ In{ particular2 }\ epsilon comma w sub) 1 =) O open , 1parenthesis / \lambda toˆ{ the2 power} ] of 2 side ) . It admits similar expansion in λ. closing. \ tag parenthesis∗{$ ( 33period ) $} \end{ a l i g n ∗} On the contrary , the function w2 may have a different behaviour at λ → 0. Here we Thusconsider the equation open parenthesis 29 closing parenthesis i s similar to open parenthesis 1 5 closing parenthesis open parenthesis but with non hyphen zero right hyphen hand side closing parenthesis period .. It admits similar \ hspacethree∗{\ casesf i l l } .The functions ( 3 2 ) −− ( 3 3 ) are regular at $ \lambda \rightarrow 0 . $ \quad Moreover , the general structure of the function expansion in lambda period −n 1 . Let r2∞ 6= r1∞ and b2∞ 6= b1∞. Then w2 = O(λ ), n ≥ 0( except perhaps very special Onexamples the contrary which comma we the do function not consider w sub 2 ) may . have In thisa different case onebehaviour can solve at lambda formally right thearrow equation 0 period Here( 3 we consider \noindentthree cases period$ w { 1 }$ i s the same as that of $w$ in the Section 3 ( see equations ( 1 6 ) , ( 1 8 ) ) . In particular 0 ) in favour of ψ2(ξ) as follows : $ ,1 period w { .. Let1 } r sub= 2 infinity O ( equal-negationslash\lambda ˆ{ 2 r} sub) 1 infinity . $ and b sub 2 infinity equal-negationslash b sub 1 infinity period .. Then wThus sub 2 the= O openequation parenthesis ( 29 lambda ) i s to similar the power to of minus ( 1 5 n closing) ( but parenthesis with non comma− zero n greater r i g h equal t − 0hand open sparenthesis i d e ) . except\quad perhapsIt admits similar expansion in $ \lambda . $ n=0 very special λ2 ∂2 λ2 X λ2 ∂2 λ2 ψ2 = −(1 − 2 ) − 1 yψ1 = − ( 2 )n yψ1. (34) examples which we do not consider closingw2 parenthesis∂ξ w2 period .. In thisw2 case∂ξ onew2 can solve formally the equation open parenthesis 3 .. 0 closing\ hspace parenthesis∗{\ f i l l } inOn the contrary , the function $ w ∞{ 2 }$ may have a different behaviour at $ \lambda \rightarrowfavourThis of representationpsi 2 open0 parenthesis . $ leads Here xi to weclosing the consider loss parenthesis of solutions as follows for :ψ2 which are not analytical in λ and thus Equation:have nothing open parenthesis to do with 34 closing the perturbation parenthesis .. procedure psi 2 = minus . The parenleftbigg use of ( 3 1 4 minus ) in the lambda r . h to . thes . power of of 2 divided by w sub 2 partialdiff\noindent( 2 to 9 )thethree p power ermits cases of 2us divided to . eliminate by partialdiffψ2 from xi to (the 2 power9 ) and of2 thus parenrightbigg to obtain minus a close 1 lambda wave equation to the power for of 2 divided by w sub 2 y psi 1 =ψ minus1. The sum structure from n = 0 and to infinity treatment parenleftbigg of this lambda equation to the are power the same of 2 divided as those by wof sub the 2 equation partialdiff ( to 1 the power of 2 divided by partialdiff1 . \quad5 ) xi . toLet the Moreover power $ r of{ , 22 it parenrightbigg i s\ obviousinfty }\ n from lambdane ( 3 tor 4 the ){ that power1 at of\ infty least 2 divided the}$ by zero and w sub - and $ 2 y b psithe{ 1 first2 period -\ orderinfty t}\ne b { 1 \ infty } . $Thiserms\quad representation ofThenψ2 vanish $leads w .to{ the2 Thus} loss= of the solutions O r . h ( for . s psi\ .lambda of2 which ( 2 9ˆ are{ ) − notdoes analyticaln not} contribute) in lambda , n in and lower\geq thus orders0 of ( $ except perhaps very special exampleshaveperturbation nothing which to do we procedure with do the not perturbation . consider In zero procedure ) - .order\quad period approximationIn The this use ofcase open we one have parenthesis can the solve oscillator 3 4 closing formally problem parenthesis the . inequation the r period ( 3 h period\quad s0 ) in favour o f $ \ psi 2 ( \ xi ) $ as follows : period .. of open parenthesis2 . Let 2 9 closingr2∞ = parenthesisr1∞ and b p2∞ ermits= b1∞ but ρ2 − ρ1 = O(λ) and µ2 − µ1 = O(λ). Then us tow2 eliminate= O(λ). psi 2 from open parenthesis 2 9 closing parenthesis and thus to obtain a close wave equation for psi 1 period .. The structure \ begin { a l i g n2∗} and Since λ /w2 = O(λ) the perturbative treatment ( 3 4 ) of the equation ( 3 0 ) i s still valid \ psitreatment.2 The of = this only equation− difference( are 1 the from− same \ thef as r a those c case{\ oflambda1 thei s equation thatˆ{ 2 the open}}{ r . parenthesisw h .{ s2 .}}\ 1 5of closingf requation a c {\ parenthesispartial ( 2 9 ) periodˆ may{ 2 ..}}{\ Moreoverpartial comma it\ xi i s ˆ{ 2 }} obvious) −contribute1 \ f r a in c {\ thelambda first orderˆ{ 2 of}}{λ. w { 2 }} y \ psi 1 = − \sum ˆ{ n = 0 } {\ infty } ( \ f rfrom a c {\ openInlambda both parenthesis theˆ{ above2 3}}{ 4 closing casesw { parenthesis we2 used}}\ f the thatr a c dimensionless{\ at leastpartial the zeroˆ variable{ hyphen2 }}{\ andξ1partial theand first obtained hyphen\ xi a orderˆ closed{ 2 t erms}} eigen- of) psi 2n vanish\ f r a period c {\lambda .. Thus ˆ{ 2 }}{ w { 2 }} they r period\statepsi h equationperiod1 . s\ periodtag ∗{ ( of which$ open ( parenthesis we 34 will ) reference $ 2} 9 closing to parenthesis as the problem 1 ) for the wave function \end{ a l i g n ∗} doesψ1( notξ1) contributeand the in lower sp ectral orders parameterof perturbation1. procedureWe can period proceed .. In zero with hyphen the variable order approximationξ2 and obtain we havethe the problem oscillator 2problem for the period function ψ2(ξ2) and the parameter 2. One might be inclined t o \noindent2 periodthink ..This that Let r both sub representation 2 problems infinity = r sub 1 leads 1 and infinity 2 to are and the equivalent b sub loss 2 infinity of and solutions = lead b sub t o 1 the infinity for same but$ \ spectrum rhopsi 2 minus2 ( $ inrho whichterms 1 = O open are parenthesis not analytical lambda in closing$ \lambdaof parenthesis energy$ andE and). thus muActually 2 minus , mu different 1 = O open problems parenthesis complement lambda closing one another parenthesis . period This .. i Then s evident w sub 2 = O open parenthesis lambdahavefrom closing nothing equation parenthesis to do ( 2 periodwith 8 ) leading the perturbation to the relation procedure : . The use of ( 3 4 ) in the r . h . s . \quad of ( 2 9 ) p ermits usSince to lambda eliminate to the power $ \ psi of 2 slash2 $ w sub from 2 = ( O 2open 9 )parenthesis and thus lambda to obtain closing parenthesis a close the wave perturbative equation treatment for $ open\ psi parenthesis1 . $ \quad The structure and 1 b b 3 .. 4 closing parenthesis of the equation open− parenthesis= { 1∞ µ 31 ..− 0µ closing2} + { parenthesis1∞ − 1} . i s still valid period .. The only treatment of this equation are2 the1 same2 as those of the equation1 ( 1 5 ) . \quad Moreover , it i s obvious difference from the case 1 i s that the r periodλ h periodb2∞ s period ..b of2∞ equation open parenthesis 2 9 closing parenthesis may contribute in from ( 3 4 ) that at least the zero − and the first − order t erms of $ \ psi 2 $ vanish . \quad Thusthe r . h . s . of (29) the firstIndeed order , in both 1 and 2 cases | 2 − 1 |→ ∞ if λ → 0. It does mean that an arbitrary energy does not contribute in lower orders of perturbation(0) procedure . \quad In zero − order approximation we of lambdalevel E periodcalculated by means of eigenvalue 1 of zero - order oscillator problem 1 ( with the haveIn bothuse the ofthe oscillator above cases we problem used the. dimensionless variable xi 1 and obtained a closed eigenstate (0) equationequations .. open ( parenthesis 2 8 ) and which ( 8 ) )we cannot will reference be obtained to as the by problem means 1 ofclosing any parenthesis finite eigenvalue .. for the2 wave of function the psi 1 open parenthesis xi\ hspace 1 closingproblem∗{\ parenthesisf i l 2 l }2 . ..\ andquad the Let $ r { 2 \ infty } = r { 1 \ infty }$ and $ b { 2 \ infty } = b { 1 \ infty }$ but $ \rho 2 − \rho 1 = O ( \lambda ) $ and $ \mu 2 − \mu sp ectral parameter epsilon sub 1 period .. We can proceed with(0) the variable(0) xi 2 and obtain the problem 2 for the 1 =and O vice ( versa\lambda . Higher) - order . $ corrections\quad Then t o $1 w ( {or2 2} )=are O small ( and\lambda do not change) . $ functionqualitatively psi 2 open parenthesisthis picture xi 2 . closing Thus parenthesis different .. and problems the parameter generate epsilon different sub 2 period branches .. One of might the be inclined t o think that both problems .. 1 \noindentenergySince sp ectrum $ \lambda of the originalˆ{ 2 } set/ of equation w { 2 } .= In thisO resp ( ect\lambda the following) $ special the perturbative case treatment ( 3 \quad 4 ) of the equation ( 3 \quad 0 ) i s still valid . \quad The only anddiffers 2 are equivalent essentially and from lead thet o the same spectrum open parenthesis in terms of energy E closing parenthesis period .. Actually comma differentdifference from the case 1 i s that the r . h . s . \quad of equation ( 2 9 ) may contribute in the first order o fprevious $ \lambda ones . $ problems complement one another period .. This i s evident from equationn open parenthesis 2n 8 closing parenthesis leading to the relation : 3 . Let r2∞ = r1∞ and b2∞ = b1∞ but ρ2 − ρ1 = O(λ ) and µ2 − µ1 = O(λ ), n ≥ 2. Then epsilon sub 22 minus epsilon sub 1 = 1 divided by lambda to the power of 2 braceleftbigg b sub 1 infinity divided by b sub 2 infinity mu 1 w2 = O(λ ). Both equations ( 2 9 ) and ( 3 0 ) have similar structure and should b e treated minusIn both mu 2 the bracerightbigg above cases plus braceleftbigg we used the b sub dimensionless 1 infinity divided variable by b sub 2 infinity $ \ xi minus1 1 $ bracerightbigg and obtained epsilon a sub closed 1 period eigenstate equationon the\quad same( footing which we . will Use reference of common to variables as the problemξ, defined 1 ) by\quad ( 1 4for ) and the ( wave 1 7 ) function is $ \ psi 1 Indeedappropriate comma in both to this 1 and case 2 cases . bar epsilon sub 2 minus epsilon sub 1 bar right arrow infinity if lambda right arrow 0 period .... It does mean( \ thatxi an1 arbitrary ) $ energy\quad and the splevel ectral E calculated parameter by means $ of\ epsilon eigenvalue 1{ epsilon1 } to. the $ power\quad ofWe open can parenthesis proceed 0 closingwith the parenthesis variable of zero $ hyphen\ xi order2 $ oscillator and obtain the problem 2 for the problemf u n c t i 1 o nopen $ parenthesis\ psi 2 with ( the use\ xi of 2 ) $ \quad and the parameter $ \ epsilon { 2 } . $ \quad One might be inclined t o think that both problems \quad 1 andequations 2 are open equivalent parenthesis and 2 8 closinglead t parenthesis o the same and open spectrum parenthesis ( in 8 terms closing parenthesis of energy closing $ E parenthesis ) .$ cannot\quad be obtainedActually by , different meansproblems of any complementfinite eigenvalue one 2 epsilon another to the . power\quad of openThis parenthesis i s evident 0 closing from parenthesis equation of the ( problem2 8 ) leading 2 to the relation : and vice versa period Higher hyphen order corrections t o 1 epsilon to the power of open parenthesis 0 closing parenthesis open parenthesis or\ [ 2\ epsilonepsilon to the{ power2 } of − open \ parenthesisepsilon 0{ closing1 } parenthesis= \ f r a c closing{ 1 }{\ parenthesislambda areˆ{ small2 }}\{\ and do notf change r a c { b qualitatively{ 1 \ infty }}{ b { 2 \ inftythis picture}}\ periodmu ..1 Thus− different \mu problems2 \} generate+ different\{\ f branches r a c { b of{ the1 energy\ infty sp ectrum}}{ ofb the{ 2 \ infty }} − 1 \} \ epsilonoriginal set{ of1 equation} . \ period] .. In this resp ect the following special case differs essentially from the previous ones period 3 period .. Let r sub 2 infinity = r sub 1 infinity and b sub 2 infinity = b sub 1 infinity but rho 2 minus rho 1 = O open parenthesis lambda to\noindent the power ofIndeed n closing , parenthesis in both and 1 and mu 2 2 minus cases mu 1 $ =\ Omid open parenthesis\ epsilon lambda{ 2 to} the − power \ epsilon of n closing{ parenthesis1 }\mid comma\rightarrow n greater equal\ infty 2 period$ i .. f Then $ \lambda \rightarrow 0 . $ \ h f i l l It does mean that an arbitrary energy w sub 2 = O open parenthesis lambda to the power of 2 closing parenthesis period .. Both equations open parenthesis 2 9 closing parenthesis and\noindent open parenthesislevel 3 $ .. E0 closing $ calculated parenthesis have by means similar structure of eigenvalue and should $ b e1 treated{\ epsilon on the }ˆ{ ( 0 ) }$ o f zero − order oscillator problem 1 ( with the use of same footing period .. Use of common variables xi comma epsilon defined by open parenthesis 1 4 closing parenthesis and open parenthesis 1\noindent 7 closing parenthesisequations is appropriate ( 2 8 ) to and this ( case 8 ) period ) cannot be obtained by means of any finite eigenvalue $ 2 {\ epsilon }ˆ{ ( 0 ) }$ of the problem 2
\noindent and vice versa . Higher − order corrections t o $ 1 {\ epsilon }ˆ{ ( 0 ) } ( $ or $ 2 {\ epsilon }ˆ{ ( 0 ) } ) $ are small and do not change qualitatively this picture . \quad Thus different problems generate different branches of the energy sp ectrum of the original set of equation . \quad In this resp ect the following special case differs essentially from the
\noindent previous ones .
\ hspace ∗{\ f i l l }3 . \quad Let $ r { 2 \ infty } = r { 1 \ infty }$ and $ b { 2 \ infty } = b { 1 \ infty }$ but $ \rho 2 − \rho 1 = O ( \lambda ˆ{ n } ) $ and $ \mu 2 − \mu 1 = O ( \lambda ˆ{ n } ) , n \geq 2 . $ \quad Then
\noindent $ w { 2 } = O ( \lambda ˆ{ 2 } ) . $ \quad Both equations ( 2 9 ) and ( 3 \quad 0 ) have similar structure and should b e treated on the same footing . \quad Use of common variables $ \ xi , \ epsilon $ defined by ( 1 4 ) and ( 1 7 ) is appropriate to this case . 8 .... A period Duviryak \noindent 8 \ h f i l l A . Duviryak hline8 A . Duviryak In the zero hyphen order approximation we obtain the coupled pair of wave equations open parenthesis on the contrary \ [ t\ or theu l e cases{3em 1}{ and0.4 2 pt where}\ ] we had a single wave equation closing parenthesis period .. In physically meaningful cases open parenthesis of Section 5 comma for example closing parenthesis they have the form : Equation:In the open zero parenthesis - order approximation35 closing parenthesis we .. obtain braceleftbig the coupled d to the power pair of of 2 wave slash d equations xi to the power ( on of the2 plus kappa epsilon minus nu sub\noindent 1 minuscontrary omegaIn the t to o the the zero power cases− oforder1 2 xiand to approximation the2 powerwhere of 2 we bracerightbig had we obtain a single psi 1the wave open coupled parenthesis equation pair xi). closing of Inwave parenthesis physically equations = chi psi ( 2 on open the parenthesis contrary xit closing omeaningful the parenthesis cases 1cases comma and ( 2 of Equation: where Section we open 5 had , parenthesis for a example single 36 closing) wave theyparenthesis equation have the form .. ) braceleftbig . :\quad In d to physically the power of 2 meaningful slash d xi to the cases power ( of of 2 plusSection kappa epsilon 5 , for minus example nu sub 2 minus) they omega have to the the power form of :2 xi to the power of 2 bracerightbig psi 2 open parenthesis xi closing parenthesis = chi psi 1 open parenthesis xi closing parenthesis comma {d2/dξ2 + κ − ν − ω2ξ2}ψ1(ξ) = χψ2(ξ), (35) \ beginwhere{ a chi l i g = n limint∗} lambda right arrow 0 y = const comma1 and parameters nu sub i comma kappa and omega are related t o the functions \{ d ˆ{ 2 } / d \ xi ˆ{ 2 } 2 + \kappa2 2 \ epsilon − \nu { 1 } − \omega ˆ{ 2 }\ xi ˆ{ 2 } w sub i open parenthesis i = 1 comma 2{ closingd /dξ parenthesis+ κ − ν2 − ω ξ }ψ2(ξ) = χψ1(ξ), (36) \}\by thepsi equation1 of ( the\ typxi e of) open = parenthesis\ chi 2\ 3psi closing2 parenthesis ( \ xi comma), .... open\ tag parenthesis∗{$ ( 2 35 4 closing ) $}\\\{ parenthesisd and ˆ{ open2 } parenthesis/ dwhere\ 2xi 5χ closingˆ={ lim2 λ} parenthesis→0 y+= const\kappa period , and .... parameters\ Theepsilon equationsν−i, open κ and \ parenthesisnuω are{ 2 related} 3 5 − closing t o \ theomega parenthesis functionsˆ{ 2 comma}\wi(i xi= .... 1,ˆ open2){ 2 parenthesis}\}\ 3psi 6 closing2 parenthesis( \byxi the can) equation b e = evidently\ chi of the typ\ psi e of (1 2 3 ( ) ,\ (xi 2 4 )), and ( 2\ 5tag )∗{ .$ The ( equations 36 ) $ (} 3 5 ) , ( 3 6 ) \endreduced{cana l i g bto n ∗} e the evidently pair of similar equations but with parameters nu-tilde sub i = open brace nu sub 1 plus nu sub 2 plusminux radicalbig-line p 2 2 of openreduced parenthesis to nuthe sub pair 1 minus of similar nu sub equations 2 closing parenthesis but with parametersto the power ofν˜i 2= plus{ν1 4+ chiν2 ± to the(ν1 power− ν2) of+ 2 4χ closing}/2 brace slash 2 \noindentopen(i parenthesis= 1,where2) and i = $ 1\χ˜chi comma= 0. = 2Thus closing\lim they parenthesis{\ becomelambda .. and split chi-tilde\rightarrow equations = 0 period of the0 ..} form Thusy ( they =$2 2 become ) const . split The , equations and parameters of the form $ open\nu { i } parenthesis, \kappaeigenvalues 2 2$ closing and corresponding parenthesis $ \omega period$ to .. are the The first related eigenvalues and second t epsilon o the equations functions are separated $ w { i by} finit( e constant i = 1 , 2 ) $ correspondingν˜1 − ν˜2. Thus to the first the and corresponding second equations states are mix separated in higher by finit orders e constant of p nu-tildeerturbation sub 1 procedure minus tilde-nu . sub 2 period .. Thus \noindentthe5 correspondingby Application the states equation mix in : higherof the Regge orders typ of e p of erturbation trajectories ( 2 3 ) procedure , \ h f i period l l of(24)and(25) mesons . \ h f i l l The equations ( 3 5 ) , \ h f i l l ( 3 6 ) can b e evidently 5 ..Here Application we apply : .. Regge the pseudoperturbative .. trajectories .. of mesons treatment of 2 BDE in meson sp ectroscopy . \noindentHere weIt apply ireduced s known the pseudoperturbative to [ 2 the 7 ] that pair spectra of treatment similar of heavy of 2 equations BDE mesons in meson are but sp described withectroscopy parameters well period by the $ nonrelativistic\ tilde {\nu} { i } = \{\nu { 1 } + \nu { 2 }\pm \ sqrt { ( \nu { 1 } − \nu { 2 } ) ˆ{ 2 } + 4 \ chi ˆ{ 2 }}\} / 2 $ It ipotential s known open model square with bracket QCD 2 7 - closing motivated square funnel bracket potential that spectrau( ofr) heavy = ul(r mesons) + uC ( arer), where described well by the nonrelativistic potential model with QCD hyphen motivated funnel potential u open parenthesis r closing parenthesis = u sub l open parenthesis r closing parenthesis plus\noindent u sub C open$ ( parenthesis i = r closing 1 parenthesis, 2 ) comma $ \quad whereand $ \ tilde {\ chi } = 0 . $ \quad Thus they become split equations of the form ( 2 2 ) . \quad The eigenvalues $ \Equation:epsilon open$ parenthesis 37 closing parenthesis ..uC u( subr) = C− openα/r, parenthesis α = 0.27, r closing parenthesis = minus alpha(37) slash r comma alpha = 0 corresponding to the first and second equations are separated2 by finit e constant $ \ tilde {\nu} { 1 } period 2 7 comma Equation: open parenthesisul(r) 38 = closingar, a parenthesis= 0.25 ÷ .. u0. sub3GeV l open. parenthesis r closing parenthesis(38) = ar comma a = 0 period −25 divided \ tilde by{\ 0 periodnu} { 3 GeV2 to . the}$ power\quad of 2Thus period theTheThe corresponding .... Coulomb Coulomb part part states .... open ( 3 7mix parenthesis ) of in this higher 3 potential7 closing orders parenthesis describ of p es erturbation.... a of nonrelativistic this .... potential procedure limit describ of. es the .... vector a nonrelativistic limit .... of the vector one hyphen - \noindentgluongluon exchange5 exchange\quad interactionApplication interaction while the while linear : \quad the part linear openRegge parenthesis part\quad ( 3trajectories 83 8 ) closing i s suggested parenthesis\quad by i the so suggested f areamesons law by the in the area law in the lattice approximationlattice approximation of QCD and has of presumably QCD and scalar has presumably or scalar hyphen scalar vector or nature scalar period - vector nature . \noindentDescriptionDescriptionHere of light we meson of apply light sp themeson ectroscopy pseudoperturbative sp ectroscopyneeds application needs of applicationtreatment appropriate relativistic of appropriate 2 BDE models in meson relativistic period sp ectroscopy models . Most. of Most them of are them related are t o related the string t otheory the stringperiod From theory the . theoretical From the viewpoint theoretical the most viewpoint interesting the most Itare iinteresting sQCD known hyphen [ 2 aremotivated 7 QCD ] that relativistic - motivated spectra models ofrelativistic embracingheavy mesons models properties are embracing of described both heavy properties and well light by of mesons both the heavynonrelativisticperiod and potential modelSuchlight models with mesons QCD should− . reflectmotivated Such the models scalar funnel hyphen should potential vector reflect structure the $u scalar of interaction ( - vector r and ) structure should = lead u of t{ o interaction funnell } hyphen( and r ) + u { C } (typ rshould e potential ) lead , in $ t the o where funnel nonrelativistic - limit period A naturaltyp e potentialcandidate for in the the relativistic nonrelativistic potential limit model . i s the 2 BDE with a short hyphen range vector \ beginpotential{ aA l i andnatural g n ∗} a long candidate hyphen range for scalar the relativistic one period .. potential At least three model general i s the structures 2 BDE of with vector a potential short - range are u used{ vectorC in} the lit( potential erature r comma ) and = a long− - range \alpha scalar/ one r . , At least\alpha three= general 0 structures . 2 7 of vector , \ tag ∗{$ ( 37 ) $}\\ u { l } (Equation: rpotential )=ar open are parenthesis used ,a=0 in 39 the closing lit erature parenthesis , . .. 25 U sub\ vdiv open parenthesis0 . 3r closing GeV parenthesis ˆ{ 2 } =. u\ tag sub∗{ v open$ ( parenthesis 38 ) r $ closing} parenthesis\end{ a l i g comman ∗} Equation: open parenthesis 40 closing parenthesis .. U sub v open parenthesis r closing parenthesis = open brace 1 minus alpha sub 1 times alpha sub 2 closing brace u sub v open parenthesis r closing parenthesis comma Equation: open parenthesis 41 closing parenthesis\noindent ..The U sub\ vh openf i l l parenthesisCoulomb rpart closing\ h parenthesis f i l l ( 3 = 7 braceleftbig ) \ h f i l lU 1vo( minusr f) = t hu i1v s( dividedr)\,h f i l by l 2potential alpha sub 1 timesdescrib(39) alpha es sub\ 2h bracerightbig f i l l a nonrelativistic limit \ h f i l l of the vector one − u sub v open parenthesis r closing parenthesis plus 1 divided byUv 2(r open) = {1 parenthesis− α1 · α2}u nv( timesr), alpha sub 1 closing parenthesis(40) open parenthesis n \noindent gluon exchange interaction while the linear part ( 3 8 ) i s suggested by the area law in the lattice times alpha sub 2 closing parenthesis ru sub v to1 the power of prime1 open parenthesis r closing parenthesis comma approximation of QCD andU has(r) = presumably{1 − α · α } scalaru (r) + or(n · scalarα )(n · α− )ruvector0 (r), nature . (41) with u sub v open parenthesis r closingv parenthesis2 1 =2 u subv C open2 parenthesis1 2 r closingv parenthesis or another short hyphen range potential semicolon here u to the power of prime open parenthesis r closing parenthesis = du open parenthesis r closing parenthesis slash dr period .. The Description of light meson sp ectroscopy needs application0 of appropriate relativistic models . potentialwith openu parenthesisv(r) = uC (r 3) or 9 closing another parenthesis short - range potential ; here u (r) = du(r)/dr. The potential ( Mosti s only3 of 9) a them static i s only are part a of related static vector part interaction t oof the vector open string interaction parenthesis theory see( see . .. From open [ 5 square the ] ) . theoretical bracket The 5 relativistic closing viewpoint square vector bracket the field closing most parenthesis interesting period .. Theare relativistic QCDkinematics− motivated vector i s field taken kinematics relativistic into account i s models in the potential embracing ( 4 properties 0 ) ( see of [both 1 6 , heavy 8 ] and ) which light mesons . Suchtaken, models into for account the should Coulomb in the reflect potential case , the open was scalar parenthesis first− proposedvector 4 0 closing by structure parenthesis Eddington of .. and open interaction Gaunt parenthesis [ 2 8 seeand , 2 .. 9 should open ] . square In lead bracket t o 1 funnel 6 comma− .. 8 closingthe square generalization bracket closing ( parenthesis 4 1 ) of the .. which Breit comma potential .. for [ 1the ] Coulomb retardation case commaterms have .. was b first een added [ 8 ] \noindentproposed. Two bytyp Eddington different e potential and scalar Gaunt in potentials open the square nonrelativistic , bracket 2 8 comma limit 2 9 closing . square bracket period .. In the generalization open parenthesis 4 1 closing parenthesis of the Breit potential open square bracket 1 closing square bracket A natural candidate for the relativistic potential model i s the 2 BDE with a short − range vector retardation terms have b een added open square bracket 8 closingu square bracket period .. Two different scalar potentials comma potentialEquation: open and parenthesis a long − 42range closing scalar parenthesis one ..U U .s( subr\)quad = sβ open1βAt2 parenthesiss( leastr), three r closing general parenthesis structures = beta 1 beta(42) of 2 vector to the power potential of u s open are parenthesisused in r the closing lit parenthesis erature comma , Equation: open parenthesis1 43 closing parenthesis .. U sub s open parenthesis r closing parenthesis = Us(r) = (β1 + β2)us(r), (43) 1 divided by 2 open parenthesis beta 1 plus beta 2 closing2 parenthesis u sub s open parenthesis r closing parenthesis comma \ begincome{ froma l i g differentn ∗} couplings of scalar mediating field with fermionic fields period .. The first one open parenthesis 4 2 closing parenthesis U { comev } from( different r ) couplings = u { ofv scalar} ( mediating r ) field , \ tag with∗{ fermionic$ ( 39 fields ) $.}\\ TheU { firstv } one( r ) = \{ arises( 4 from 2 ) the arises Yukawa from interaction the Yukawa open parenthesis interaction see ( open see [ square 6 ] )bracket while the 6 closing second square one bracket ( 4 3 ) closing corresp parenthesis while the second one1 open− parenthesis \alpha 4{ 31 closing}\ parenthesiscdot \alpha corresp onds{ 2 t}\} o so calledu { v } ( r ) , \ tag ∗{$ ( 40 ) $}\\ U { v } ( ronds ) t o = so called\{ 1 − \ f r a c { 1 }{ 2 }\alpha { 1 }\cdot \alpha { 2 }\} u { v } ( r ) + \ f r a c { 1 }{ 2 } ( n \cdot \alpha { 1 } ) ( n \cdot \alpha { 2 } ) ru ˆ{\prime } { v } ( r ) , \ tag ∗{$ ( 41 ) $} \end{ a l i g n ∗}
\noindent with $ u { v } ( r ) = u { C } ( r ) $ or another short − range potential ; here $ u ˆ{\prime } ( r )=du ( r ) / dr .$ \quad The potential ( 3 9 ) i s only a static part of vector interaction ( see \quad [ 5 ] ) . \quad The relativistic vector field kinematics i s taken into account in the potential ( 4 0 ) \quad ( see \quad [ 1 6 , \quad 8 ] ) \quad which , \quad for the Coulomb case , \quad was f i r s t proposed by Eddington and Gaunt [ 2 8 , 2 9 ] . \quad In the generalization ( 4 1 ) of the Breit potential [ 1 ] retardation terms have b een added [ 8 ] . \quad Two different scalar potentials ,
\ begin { a l i g n ∗} U { s } ( r ) = \beta 1 \beta 2 ˆ{ u } s ( r ) , \ tag ∗{$ ( 42 ) $}\\ U { s } ( r ) = \ f r a c { 1 }{ 2 } ( \beta 1 + \beta 2 ) u { s } ( r ) , \ tag ∗{$ ( 43 ) $} \end{ a l i g n ∗}
\noindent come from different couplings of scalar mediating field with fermionic fields . \quad The first one ( 4 2 ) arises from the Yukawa interaction ( see [ 6 ] ) while the second one ( 4 3 ) corresp onds t o so called Large hyphen j Expansion Method for Two hyphen Body Dirac Equation .... 9 \noindent Large $ − j $ Expansion Method for Two − Body Dirac Equation \ h f i l l 9 hlineLarge −j Expansion Method for Two - Body Dirac Equation 9 quotedblleft minimal quotedblright .. coupling open square bracket 1 5 closing square bracket period .. The latter and also two following potentials\ [ \ r u l e can{3em be}{ treated0.4 pt as}\ static] approximation of various QFT hyphen motivated scalar quasipotentials open square bracket 3 .. 0 comma .. 3 1 comma .. 3 2 comma .. 1 7 closing“ minimal square bracket ” coupling: [ 1 5 ] . The latter and also two following potentials can be treated \noindentEquation:as static open‘‘ minimal approximation parenthesis ’’ 44\quad closing of variouscoupling parenthesis QFT ..[ -U 1 motivated sub 5 ] s open . \quad parenthesis scalarThe quasipotentials latterr closing parenthesis and also [ 3 = two 0 1 ,divided following 3 1 by , 2 open potentials parenthesis 1 can plus be treated as static betaapproximation 13 beta 2 , 2 closing 1 7 ] of parenthesis : various u QFTsub s− openmotivated parenthesis scalar r closing quasipotentials parenthesis comma Equation: [ 3 \quad open0 parenthesis , \quad 453 closing 1 , \quad parenthesis3 2 .. , U\quad 1 7 ] : sub s open parenthesis r closing parenthesis = 1 divided by 4 open parenthesis 1 plus beta 1 closing parenthesis open parenthesis 1 plus beta 2 closing\ begin parenthesis{ a l i g n ∗} u sub s open parenthesis r closing parenthesis1 period U (r) = (1 + β1β2)u (r), (44) U The{ ..s perturbative} ( r .. treatment ) = \ ..f r of a cBreit{ 1 hyphen}{ 2s } type(2 .. equations 1 + ..s has\beta .. been1 used\ forbeta .. calculating2 ) .. a fine u { s } ( r ) , \splittingtag ∗{$ in ( spectra 44 of ) heavy $}\\ mesonsU { opens } square(1 bracket r ) 1 4 = comma\ f r a c..{ 11 6}{ closing4 } square( bracket 1 + period\beta .. Light1 meson ) sp ( ectra 1 are U (r) = (1 + β1)(1 + β2)u (r). (45) +essentially\beta relativistic2 )and u { s } ( rs )4 . \ tag ∗{$ (s 45 ) $} \end{ a l i g n ∗} need aThe nonperturbative perturbative statement of treatment the problem which of Breit i s inconsistent - type because equations of non hyphen has physical been used for singularities .. of radial equations period .. To .. avoid these .. difficulties in numerical calculations .. one i s The \quadcalculatingperturbative a fine\ splittingquad treatment in spectra\quad of heavyo f Bmesons r e i t − [type 1 4 ,\quad 1 6 ]equations . Light\ mesonquad has sp \quad been used for \quad calculating \quad a f i n e forcedectra t o are invent essentially sophisticated relativistic potentials and and needimpose a rather nonperturbative artificial boundary statement conditions of the open problem square bracketwhich i 1 s .. 5 closing square bracket periodsplitting in spectra of heavy mesons [ 1 4 , \quad 1 6 ] . \quad Light meson sp ectra are essentially relativistic and needinconsistent a nonperturbative because of statement non - physical of the singularities problem which of radial i s equations inconsistent . To because avoid of these non − p h y s i c a l Usingdifficulties the pseudoperturbative in numerical treatment calculations of 2 BDE one with i different s forced combinations t o invent of sophisticated potentials potentials and singularitiesopen parenthesis\ 3quad 7 closingof parenthesis radial equations endash open . parenthesis\quad To 4 5\ closingquad avoid parenthesis these we obtain\quad analyticaldifficulties expressions in for numerical meson mass calculationssp \quad one i s forcedimpose t o rather invent artificial sophisticated boundary potentials conditions [ and 1 impose 5 ] . rather artificial boundary conditions [ 1 \quad 5 ] . ectra and estimateUsing the a role pseudoperturbative of general treatment of 2 BDE with different combinations of potentials structure and input parameters of potentials in the model period We consider mass spectra of lightest Using( the 3 7 ) pseudoperturbative – ( 4 5 ) we obtain analytical treatment expressions of 2 BDE for with meson different mass sp ectra combinations and estimate of a potentials role of mesonsgeneral open structureparenthesis and containing input u parameters and d quarks of only potentials closing parenthesis in the model and try . We t o consider reproduce mass their followingspectra general features : (i 3 closing 7 ) −− parenthesis( 4 5 .. ) Mass we obtain spectra of analytical light mesons fall expressions into the family for of straight meson lines mass in sp the ectra open parenthesis and estimate E to the a power role of of 2 comma general structureof lightest and mesons input parameters ( containing ofu and potentialsd quarks in only the ) modeland try . t We o reproduce consider their mass following spectra of lightest j closinggeneral parenthesis features hyphen : plane known mesonsas Regge ( traj containing ectories period $ u $ and $ d $ quarks only ) and try t o reproduce their following general features : i ) Mass spectra of light mesons fall into the family of straight lines in the (E2, j)− plane ii closingknown parenthesis as Regge .. Regge traj ectories traj ectories . are parallel semicolon slop e parameter sigma i s an universal quantity comma sigma = 1 period 1i 5 ) GeV\quad to theMass power spectra of 2 = of light mesons fall into the family of straight lines in the $ ( E ˆ{ 2 } , ii ) Regge traj ectories are parallel ; slop e parameter σ i s an universal quantity jopen ) parenthesis− $ plane 4 divided known by 4 period 5 closing parenthesis a period , σ = 1.15GeV2 = asiii Regge closing parenthesis traj ectories .. Nonrelativistic . classification of light mesons as parenleftbig n to the power of 2 s plus 1 l sub j parenrightbig endash states of quark hyphen antiquark system (4 ÷ 4.5)a. \ hspaceis adequate∗{\ f semicolon i l l } i i ) i period\quad eRegge period comma traj radialectories quantum are number parallel n sub; r = slop n minus e parameter l minus 1 enumerates $ \sigma leading$ open i s parenthesis an universal n quantity $ , \sigma = 1 . 1 5 GeV ˆ{ 2 } = $ 2s+1 sub r = 0 closing parenthesisiii ) Nonrelativistic and classification of light mesons as (n `j) – states of quark - daughterantiquark open parenthesissystem n sub r = 1 comma 2 comma period period period closing parenthesis Regge traj ectories comma spin s = 0 \ [ ( 4 \div 4 . 5 ) a . \ ] comma 1 correspondsis adequate t o ; mass i . e singlets . , radial and quantum number nr = n − ` − 1 enumerates leading (nr = 0) triplesand etc daughter period (nr = 1, 2, ...) Regge traj ectories , spin s = 0, 1 corresponds t o mass singlets and iv closingtriples parenthesis etc . .. Spectrum is l s hyphen degenerated comma i period e period comma masses are distinguished by l open parenthesis \ hspace ∗{\ f i l l } i i i ) \quad Nonrelativistic classification of light mesons as $ ( n ˆ{ 2 s + 1 } not byiv j )closing Spectrum parenthesis is and`s− ndegenerated sub r period , i . e . , masses are distinguished by `( not by j) and nr. \ e lv l closing{ vj parenthesis)} ) States $ −− .. of Statesstates different of different of` quarkpossess l possess− anantiquark an accidental accidental system degeneracy degeneracy which which fact causes fact causes a t ower a structure t ower of thestructure spectrum of period the spectrum . isvi adequate closing parenthesis ;vi i ) . eHyperfine .. Hyperfine. , radialss ss− hyphen quantumsplitting splitting i number s relatively i s relatively $ n small small{ r , comma} about= about5 n ÷ 5− divided6 \%eof l by l σ. 6 percent− 1 of $ sigma enumerates period leading $ (For this n For{ purposer this} purposewe= use 0 the we nonrelativistic ) use$ and the nonrelativistic potential function potential .. open function parenthesis 3 ( 7 3 closing 7 ) parenthesis and ( 3 .. and 8 open parenthesis 3 .. 8 closingdaughter) parenthesis in $ vector ( .. in n and vector{ scalarr and} = potentials 1 , of different2 , spin . structure . . )$ ( 3 9 Reggetrajectories,spin ) – ( 45 ) and calculate $s = 0 ,scalar 1pseudoperturbative $ potentials corresponds of different t sp o spin ectrum mass structure singlets in zero open - and parenthesis order approximation 3 9 closing parenthesis . Classification endash open of parenthesisstates then 45 closing parenthesis and calculatetriplesi s pseudoperturbative done etc using . singlet spectrum - triplet properties of large - large component of wave function ( 2 ) in in zerothe hyphen nonrelativistic order approximation limit . period .. Classification of states then i s done using singlet hyphen triplet properties \ centerlineof largeIf hyphen the{ i vvector large ) \quad component shortSpectrum - range of wave interaction is function $ \ e open l i l s ignoredparenthesiss − and$ 2 closing scalardegenerated parenthesis potentials , in i ( the .42 e nonrelativistic ) . – ( , 4 masses 5 ) are limit are period distinguished by $ \ e l l ($ notby $j )$ and $n { r } . $ } If theused vector with shortus hyphen(r) = ar rangethe interactionpseudoperturbative i s ignored and mass scalar ( i potentials . e . , energy open parenthesis ) of meson 42 closing in zero parenthesis - order endash open parenthesis 4 5 closingapproximation parenthesis are has used the with following form : v )u sub\quad s openStates parenthesis of different r closing parenthesis $ \ e l l =$ ar the possess pseudoperturbative an accidental mass open degeneracy parenthesis which i period fact e period causes comma a energy t ower closing structure of the spectrum . parenthesis of meson in zero hyphen order1 approximation1 )] has√ √ the following form : E2 = ka[` + + η(n + + ζm 2a` + δ +2 −δ m m + O(1/ `), (46) A 2 r 2 + 1 m 2 1 2 \ centerlineEquation: open{ v i parenthesis ) \quad 46Hyperfine closing parenthesis $ ss .. E− sub$ A splitting to the power i of s 2 = relatively ka bracketleftbig small l plus , 1 about divided by $ 2 5 plus\ etadiv parenleftbig6 \% $ E2 = E2 E2 = E2 ± κa; nof sub$ r plus\ sigma 1 divided . by$ 2} to the power of parenrightbig bracketrightbig0 plusA, zeta± m subA plus square root of 2 a l plus delta sub 1 plus m to the power of 2 minus delta sub 2 m sub 1 m sub 2 plus O parenleftbig 1 slash square root of l parenrightbig comma E sub 0 to the power of 2 here m+ = m1 + m2, and k, η, ζ, δ1, δ2, κ are dimensionless constants depending on the potential =For E sub this A comma purpose to the we power use of the 2 E nonrelativistic sub plusminux to the potential power of 2 = function E sub A to\ thequad power( 3 of 72 plusminux ) \quad kappaand ( a semicolon 3 \quad 8 ) \quad in vector and scalarchosen potentials . of different spin structure ( 3 9 ) −− ( 45 ) and calculate pseudoperturbative sp ectrum here m sub plus = m sub 1 plus m sub 2 comma and k comma eta comma zeta comma delta2 sub 1 comma delta sub 2 comma kappa are in zeroFour− order families approximation of energy levels . \Equadi(i = A,Classification0, −, +) form traj of ectories states in then the (E i, s `)− doneplane using which singlet − triplet properties dimensionlessare nearly constants straight depending . on Indeed the potential, ζ = 0 ÷ 2 for all the potentials considered and rest o fchosen l a r g period e − large component of wave function ( 2 ) in the nonrelativistic limit . masses ma √ Four(a families= 1, 2) of energylightest levels mesons E sub are i open small parenthesis compared i = to A comma the energy 0 commascale minusσ commaThus plus the closing parameter parenthesis form traj ectories in theIf open the parenthesis vector√ short E to the− range power of interaction 2 comma l closing i s parenthesis ignored hyphen and scalar plane which potentials. ( 42 ) −− ( 4 5 ) are used with ζm+ a determining a curvature of traj ectories i s small . Parameters δ1 and δ2 determine $are u nearly{ s straight} ( period r .. ) Indeed = comma ar $ zeta the = 0 pseudoperturbative divided by 2 for all the potentials mass ( iconsidered . e . ,and energy rest masses ) of m meson sub a in zero − order approximation has thea following com - mon form shift : of all the traj ectories and are not important for the present discussion . openBelow parenthesis we discuss a = 1 comma the calculated 2 closing values parenthesis of parameters of lightest mesonsk, κ and are smallη determining compared tothe the slop energy e of scale traj square root of sigma sub periodectories .... Thus theand parameter their degeneracy properties . \ beginzeta m{ a sub l i g plusn ∗} square root of a determining a curvature of traj ectories i s small period .. Parameters delta sub 1 and delta sub 2 determine aE com ˆ{ hyphen2 } { A } = ka [ \ e l l + \ f r a c { 1 }{ 2 } + \eta ( n { r } + \ f r a c { 1 }{ 2 }ˆ{ ) ] }mon+ shift\ ofzeta all them traj{ ectories+ }\ andsqrt are{ not2 important a \ e for l l the} present+ \ delta discussion{ period1 } ..+ Below{ m we}ˆ{ 2 } − \ delta { 2 } m { 1 } m discuss{ 2 } the+ calculated O ( values 1 of parameters / \ sqrt {\ k commae l l } kappa), and eta\ tag determining∗{$ ( the 46 slop ) e $of}\\ trajE ectories ˆ{ 2 and} { 0 } = E ˆ{ 2 } { A , }theirE degeneracy ˆ{ 2 } {\ propertiespm } = period E ˆ{ 2 } { A }\pm \kappa a ; \end{ a l i g n ∗}
\noindent here $ m { + } = m { 1 } + m { 2 } , $ and $ k , \eta , \zeta , \ delta { 1 } , \ delta { 2 } , \kappa $ are dimensionless constants depending on the potential chosen .
Four families of energy levels $ E { i } ( i = A , 0 , − , + ) $ form traj ectories in the $ ( E ˆ{ 2 } , \ e l l ) − $ plane which are nearly straight . \quad Indeed $ , \zeta = 0 \div 2 $ for all the potentials considered and rest masses $ m { a }$
\noindent $ ( a = 1 , 2 ) $ of lightest mesons are small compared to the energy $ scale \ sqrt {\sigma } { . }$ \ h f i l l Thus the parameter
\noindent $ \zeta m { + }\ sqrt { a }$ determining a curvature of traj ectories i s small . \quad Parameters $ \ delta { 1 }$ and $ \ delta { 2 }$ determine a com − mon shift of all the traj ectories and are not important for the present discussion . \quad Below we discuss the calculated values of parameters $ k , \kappa $ and $ \eta $ determining the slop e of traj ectories and their degeneracy properties . 1 0 .... A period Duviryak \noindent 1 0 \ h f i l l A . Duviryak hline1 0 A . Duviryak In the open parenthesis 42 closing parenthesis case k = 4 so that the slope sigma = ka matches quite well t o that of property ii closing parenthesis\ [ \ r u l e { semicolon3em}{0.4 pt }\ ] eta = 2 .. causes accidental degeneracy typical for the harmonic oscillator semicolon .. but .. kappa = 4 leads t o j hyphenIn dependence the ( 42 ) caseof energyk = open 4 so that parenthesis the slope not lσ hyphen= ka matches dependence quite closing well parenthesis t o that of so property that the ls ii hyphen ) ; degeneracy i s absent periodIn theη (= 42 2 ) casecauses $k accidental = degeneracy 4$ so that typical the for slope the harmonic $ \sigma oscillator= ; ka but$ matchesκ = quite 4 well t o that of property ii ) ; $In\ theleadseta open t= o parenthesisj− 2dependence $ \ 4quad 3 closing ofcauses energyparenthesis accidental ( not case`− k =dependence 4 degeneracy and eta = ) 2 so comma typical that theso that forls− thedegeneracy the slope harmonic and the i s accidental absent oscillator degeneracy ; \quad are thebut \quad $ \samekappa. as in the= open 4 $ parenthesis l e a d s t4 2 o closing parenthesis case semicolon kappa = 4 minus 3 square root of 2 thickapprox minus 0 period 243 $ j − $ dependence of energy ( not $ \ e l l − $ dependence ) so that the $ ls − $ degeneracy i s absent . provides anIn approximate the ( 4 3 ) l case s hyphenk = degeneracy 4 and η = comma 2, so√ that with the slope and the accidental degeneracy are the accuracysame 6 as percent in the semicolon ( 4 2 ) the case splitting; κ = i 4 s− of3 order2 ≈ of−0 the.243 actualprovides ss hyphen an approximate splitting open parenthesisls− degeneracy see property , vi closing parenthesis closing\ hspacewith parenthesis∗{\ accuracyf i l l } periodInthe(43)case6 %; the splitting i s$k of order = of 4$ the actual and $ss\−etasplitting= ( 2 see property , $ so vi that ) ) the slope and the accidental degeneracy are the In the. open parenthesis 44 closing parenthesis√ case k = kappa√ = 3 square root of 3 thickapprox 5 period 1 96 comma eta = square root of 3 thickapprox\noindentIn 1same the period ( 44as 732 ) in semicolon case thek (= none 4κ 2= of 3) these case3 ≈ values5.196 $, η; match= \kappa3 well≈ 1 t.732; o =none 4 of these− 3 values\ sqrt match{ 2 }\ wellthickapprox t o − 0 .properties 243properties $ ii provides closing ii ) parenthesis– vi an ) . approximate endash vi closing $ l parenthesis s − period$ degeneracy , with accuracy $ 6 \%; $ the splittingp √ i s of√ order2 of the actual $ ss − $ splitting ( see property vi ) ) . In theIn .. open the parenthesis ( 4 5 ) 4 case 5 closingk parenthesis= 23 − ..17 case(7 + k =17 radicalbig-line) /128 ≈ 4.2 ofprovides 23 minus the square b est root fit of 1σ 7t sub o open parenthesis 7 plus square root of 1 7 sub closing parenthesis to the power of 2 sub slash 1 28 thickapprox 4 period 2 provides the b est fit of sigma t o that that √ √ Inof the property (44) ii closing case parenthesis $k = semicolon\kappa etap == open parenthesis3 \ sqrt { square3 }\ rootthickapprox of 1 7 minus 3 closing 5 parenthesis . 1 96 radicalbig-line , \eta of 1 2= plus of property ii ); η = ( 17 − 3) 12 + 26 17/8 ≈ 2.3 leads to nearly precise oscillator - 26\ sqrt square{ 3 root}\ of 1thickapprox 7 sub slash 8 thickapprox 1 . 2 period 732 3 ;leads $ to none nearly of precise these oscillator values hyphen match like well t o propertieslike degeneracy ii ) −− ,v with i ) accuracy . 1.5 %; κ = 0 provides exact ls− degeneracy . degeneracyTaking comma into with account accuracy 1 period the 5 vector percent semicolon short - range kappa = 0interaction provides exact ( ls one hyphen of degeneracy potentials period Taking into .. account .. the .. vector .. short hyphen range .. interaction .. open parenthesis one .. of potentials .. open parenthesis 3 9 ( 3 9 ) – ( 4 1 ) with uv(r) = uC (r)) results in a parallel shift of Regge traj ectories closing\ hspace parenthesis∗{\ f i l l } endashIn the open\quad parenthesis( 4 5 4 1 ) closing\quad parenthesiscase $ .. k with = \ sqrt { 23 − \ sqrt { 1 7 }} { ( } 7 + \ sqrt { 1 7 } { .) } Theˆ{ 2 value} { / of} the1 shift 28 i s of\ thickapprox the order αa, it 4 depends . 2 on $ the provides vector potential the b est chosen fit and of $ \sigma $ t o that u subis differentv open parenthesis ( in general r closing ) for parenthesis different = u sub C open parenthesis r closing parenthesis closing parenthesis .. results in a parallel shift .. of Regge traj ectories period .. The value of the shift i s of \noindentthe order alphaof property a comma it dependsii $ ) on the ; vector\eta potential= chosen ( \ andsqrt is{ different1 7 open} parenthesis − 3 in) general\ sqrt closing{ 1 parenthesis 2 + for 26 different\ sqrt { 1 7 }} { / } 8 \ thickapprox 2 . 3 $ leads to nearly precise oscillator − l i k e traj ectories E sub i open parenthesis i = Atrajectories comma 0 commaEi(i = A, minus0, −, comma+). plus closing parenthesis period degeneracyIt has b een proved , with in theaccuracy framework $ of 1 single . hyphen 5 particle\%; Dirac\ equationkappa a = possibility 0 $ ofprovides confi hyphen exact $ l s − $ degeneracy . nementIt by has means b een ofvector proved and in equally the framework mixed vector of single hyphen - scalarparticle long Dirac hyphen equation range interactions a possibility open of square confi bracket 3 3 comma .. 3 4 commaTaking- .. nement i 3 n 5 t o closing\quad by square meansaccount bracket of vector\ periodquad andthe equally\quad mixedvector vector\quad - scalarshort long− range - range\quad interactionsinteraction [ 3 3 \quad ( one \quad of potentials \quad ( 3 9 ) −− ( 4 1 ) \quad with $We u, examined{ 3v 4} , these( 3 5 cases r ] . in ) 2 BDE = approach u { C using} ( different r )vector ) potentials $ \quad ....results open parenthesis in a parallel 3 9 closing shift parenthesis\quad endashof Regge open traj ectories . \quad The value of the shift i s of parenthesistheWe order examined 4 1 closing $ \alpha parenthesis these casesa .... in , with $ 2 BDE it depends approach on using the different vector vectorpotential potentials chosen ( and 3 9 ) is – ( different 4 1 ( in general ) for different u sub) v open parenthesis r closing parenthesis = ar period .. Corresponding zero hyphen order pseudo hyphenwith perturbative spectra have a \ begin { a l i g n ∗} form similaruv(r) =t oar. openCorresponding parenthesis 4 6 closing zero- parenthesis order pseudo period - perturbative spectra have a form similar t o traj ectories E { i } ( i = A , 0 , − , + ) . The( difference 4 6 ) . Thei s that difference k = k sub i i s = that 8 dividedk = k byi = 1 8 2 is÷ two12 t imesis two or more t imes larger or more than desired larger comma than desired and i s different , \endfor{and ia =l i g A ni commas∗} different 0 comma for minusi = A, comma0, −, +( plusi . e open . , traj parenthesis ectories i period are not e period parallel comma ) . traj ectories are not parallel closing parenthesis period 6 Summary It6 has ..The Summary b eenBreit proved equation in and the it framework s generalizations of single ( 2− BDEparticle ) possess Dirac non equation- physical a singularities possibility . of confi − nementTheIn Breit some by equation means cases andof these vectorit s points generalizations and lay equally far open from parenthesis mixed the physically vector 2 BDE− important closings c a l a parenthesis r long domain− possessrange but non interactions they hyphen make physical a [ singularities 3 3 , \quad period3 ..4 , \quad 3 5 ] . In someboundary problem incorrect or physically improper [ 1 7 , 1 6 ] . In order t o avoid \noindentcasesthis these difficultyWe points examined lay and far tofrom these use the the physically cases 2 BDE in important 2 in BDE the approach .. relativistic domain but using bound they make different state a boundary problem vector , especially potentials for \ h f i l l ( 3 9 ) −− ( 4 1 ) \ h f i l l with problemthe case incorrect of strong or physically coupling improper , we opendevelop square the bracket1/j expansion 1 7 comma method .. 1 .. 6 . closing square bracket period .. In order t o avoid this \noindent $ u { v } ( r ) = ar . $ \quad Corresponding zero − order pseudo − perturbative spectra have a form similar t o ( 4 6 ) . difficulty andThe to method use is based on the large −N or large −` t echniques applicable to the radial Schr The difference i s that $k = k { i } = 8 \div 1 2 $ is two t imes or more larger than desired , and i s different theo¨ 2dinger BDE in equation the relativistic . bound In our state case problem the 2 BDE comma i s especially reduced tfor o the case coupled of strong pair coupling of quasipotential comma we - f odevelop rtype $ i the equations 1 = slash A j expansion which , 0structure method , period− causes, principal + ( $ modification i . e . , of traj known ectories t echniques are . not Other parallel ) . Thechanges method is are based related on the t large o the hyphen fact N that or large the hyphen equations l t echniques represent applicable a nonlinear to the radial sp ectral Schr problemdieresis-o dinger \noindentequationwith period cumbersome6 \quad .. InSummary our quasipo case the 2 - BDE t entials i s reduced . t o the coupled pair of quasipotential hyphen type equations which structureWe apply causes this principal pseudoperturbative modification of known method t echniques t o the period 2 BDE .. Other with changes the linear are related+ Coulomb \noindentt opotential the factThe that of Breit the different equations equation scalar represent and - vector a it nonlinear s structure generalizations sp ectral . In problem all cases ( with 2 in BDE cumbersome the zero) possess - order quasipo non approximation hyphen− physical singularities . \quad In some casest entialsit wasthese period obtained points the lay Regge far from traj the ectories physically which are important linear asymptotically\quad domain . Linear but they potentials make a boundary problemWeof apply two incorrect this scalar pseudoperturbative structures or physically ( 43 method ) and improper t o ( the 4 25 BDE ) [ which 1 with 7 was, the\quad linear discarded1 plus\quad Coulomb in [6 1 7] potential ] . as\quad nonphysical of In order ( b t o avoid this difficulty and to use thedifferentecause 2 BDE scalar ofin singularities hyphen the relativistic vector in structure 2 BDE bound period ) reproduce In state all cases problem well in inthe our zero , caseespecially hyphen general order propertiesapproximation for the case of itlight was of meson obtainedstrong the coupling , we develop the $ 1 / j $ expansion method . Reggesp traj ectra ectories . which In particular are linear asymptotically , the slop e periodσ = ka Linearof light potentials meson of two traj scalar ectories structures fit well open t parenthesis o the 43 closing parenthesis andexperimental open parenthesis value 4 5 closingif the parameter parenthesis whicha is taken was discarded from the in open nonrelativistic square bracket potential 1 7 closing model square [ 2 bracket 7 as nonphysical open parenthesisThe method] . b The ecause is third based of singularities linear on the potential in large 2 BDE of closing$ [ 1− 7 ] parenthesisN$ ( with or no reproducel a r singularities g e $ − \ ine l l2 BDE$ t ) echniques does not match applicable to the radial Schr $ \wellddott in o{ ouro experimental} case$ dingergeneral data properties . of light meson sp ectra period .. In particular comma the slop e sigma = ka of equationlightAcknowledgements meson . traj\quad ectoriesIn fit our well case t o the the experimental 2 BDE i value s reduced if the parameter t o the a is coupled taken from pair the of quasipotential − type equations whichnonrelativisticThe structure author potential would causes model like principalt open o thank square Professors modification bracket 2 V 7 closing . Tretyak of square known and bracket I t . Simenogechniques period .. , The . Dr\ thirdquad . Yu linearOther . Yaremko potential changes of open are square related bracket 1 7t closing oand the square fact Referees bracket that for .. the open helpful equations parenthesis suggestions with represent no and singularities critical a nonlinear comments in sp . ectral problem with cumbersome quasipo − t2 e BDE n t i a closing l s . parenthesis does not match t o experimental data period Acknowledgements WeThe apply author this would pseudoperturbative like t o thank Professors method V period t Tretyak o the and 2 BDE I period with Simenog the linear comma .. $ Dr + period $ Coulomb Yu period potential Yaremko and of differentReferees for scalar helpful suggestions− vector and structure critical comments . In all period cases in the zero − order approximation it was obtained the Regge traj ectories which are linear asymptotically . Linear potentials of two scalar structures ( 43 ) and ( 4 5 ) which was discarded in [ 1 7 ] as nonphysical ( b ecause of singularities in 2 BDE ) reproduce well in our case general properties of light meson sp ectra . \quad In particular , the slop e $ \sigma = ka $ o f light meson traj ectories fit well t o the experimental value if the parameter $ a $ is taken from the nonrelativistic potential model [ 2 7 ] . \quad The third linear potential of [ 1 7 ] \quad ( with no singularities in 2 BDE ) does not match t o experimental data .
\noindent Acknowledgements
\noindent The author would like t o thank Professors V . Tretyak and I . Simenog , \quad Dr . Yu . Yaremko and Referees for helpful suggestions and critical comments . Large −j Expansion Method for Two - Body Dirac Equation 1 1
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GThe, effect of retardation on the interaction of two electrons comma Phys\ [ \ periodr u l e {3em Rev}{ period0.4 pt comma}\ ] 1 92 9 comma V period 34 comma 553 endash 5 73 period open2 2 square , V . 28 bracket , 2795 – 2 2809 closing ; nucl square - th / bracket 206048 . .. [ Barut 27 ] A Lucha period W O . ,period Schoberl comma F . F . Komy, Gromes S periodD . , Bound comma states Derivation of of nonperturbative reativisticquarks two , hyphenPhys . Rep body . , equation 1 99 1 , V from . 200 the , Issue action 4 , 1 principle 27 – 240 . [ 28 ] Eddington A . S . , The charge of an electron \ centerline { [ 1 ] \quad Breit G . , The effect of retardation on the interaction of two electrons , Phys . Rev . , 1 92 9 , V . 34 , 553 −− 5 73 . } in quantum electrodynamics, R . commaSoc . Lond Fo. r-t Proc schr . Serperiod . A Phys, 1 92 period 9 , V . comma 1 22 , N 1 789 985 , comma 358 – 369 V . period 33 comma 309 endash 31 8 period open square bracket 3 closing square bracket .. Barut A period O period comma U-dieresis nal N period comma A new approach to bound hyphen\ hspace state∗{\ quantumf i l l } [electrodynamics 2 ] \quad Barut comma A Phys . O period . , Komy A comma S . 1 , 987 Derivation comma V period of nonperturbative 142 comma reativistic two − body equation from the action principle 467 endash 487 period \ centerlineopen square{ bracketin quantum 4 closing electrodynamics square bracket .. Grandy , Fo W $period r−t T $ period schr Jr .period Phys comma . , 1Relativistic 985 ,V. quantum 33 , mechanics 309 −− of31 leptons 8 . } and fields comma Dordrecht endash Boston endash London comma \ hspaceKluwer∗{\ Academicf i l l } [ Publishers 3 ] \quad commaBarutA.O 1 99 1 period $. , \ddot{U} $ nal N . , A new approach to bound − state quantum electrodynamics , Phys . A , 1 987 , V . 142 , open square bracket 5 closing square bracket .. Darewych J period W period period Di Leo L period comma Two hyphen fermion Dirac hyphen\ centerline like eigenstates{467 −− of487 the Coulomb . } QED Hamiltonian comma J period Phys period A : Math period Gen period comma 1 996 comma V period 29 comma 681 7 endash 6841 period \ hspaceopen square∗{\ f i bracket l l } [ 4 6 closing ] \quad squareGrandyW bracket .. . Darewych T . Jr J . period , Relativistic W period comma quantum .. Few hyphen mechanics particle of eigenstates leptons in the and Yukawa fields model , Dordrecht −− Boston −− London , comma .. Condensed Matter Physics comma .. 1 998 comma V period 1 comma \ centerlineN 3 open parenthesis{Kluwer 1 Academic 5 closing parenthesis Publishers comma , 5931 99 endash 1 . 604} period open square bracket 7 closing square bracket .. Darewych J period W period comma Duviryak A period comma Exact few hyphen particle eigenstates\ hspace ∗{\ in partiallyf i l l } [ 5reduced ] \quad QEDDarewych comma Phys J period .W. 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Haysak I period comma Lengyel V period comma Shpenik A period comma Challupka S period\ centerline comma Salak{ f o r M quark period− commaantiquark Quark masses systems in the , relativistic Phys . Rev analytic . D model , 1 982 comma , V . 26 , 2420 −− 2429 . } Ukra dieresis-dotlessi n period Fiz period Zh period comma 1 996 comma V period 41 comma 370 endash 372 open parenthesis in Ukrainian closing\ hspace parenthesis∗{\ f i l l period} [ 35 ] \quad Haysak I . , Lengyel V . , Shpenik A . , Challupka S . , Salak M . , Quark masses in the relativistic analytic model , \ centerline {Ukra $ \ddot{\imath} $ n. Fiz .Zh. ,1996 ,V. 41 ,370 −− 372 ( in Ukrainian ) . }