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Few-Body Systems0, 1(cid:21)27 (2008) Few- Body Systems (cid:13) by Springer-Verlag 2008 PrintedinAustria S hwinger fun tions and light-quark bound 7 states 0 0 2 1 2 3 1 1 n M.S. Bhagwat, A. Höll, A. Krassnigg, C.D. Roberts and S.V. Wright a J 1 3 2Physi sDivision, Argonne National Laboratory, Argonne, IL60439-4843, U.S.A. 3Institutfür Physik,Universität Rosto k, D-18051 Rosto k, Germany 1 Institutfür Physik, Karl-Franzens-Universität Graz, A-8010 Graz, Austria v 9 0 0 Abstra t. We examine the appli ability and viability of methods to obtain 1 knowledge about bound-states from information provided solely in Eu lidean 0 7 spa e.Rudimentary methods an beadequate if one only requires information 0 aboutthegroundand(cid:28)rstex ited stateandassumptions madeaboutanalyti / h properties are valid. However, to obtain information from S hwinger fun tions t - abouthighermassstates,somethingmoresophisti atedisne essary.Amethod l c basedonthe orrelatormatrix anbedependablewhenoperatorsare arefully u tuned and errors are small. This method is nevertheless not ompetitive when n : an unambiguous analyti ontinuation of even a single S hwinger fun tion to v omplex momenta is available. i X r a 1 Introdu tion Theprobabilitymeasureplaysa ru ialroleinquantum(cid:28)eldtheory.Thesimplest su h measure is the Gaussian distribution 1 (q q′)2 (q,q′) = exp − , τ K (2πτ)3/2 "− 2τ # (1) whi histhefundamentalsolutionoftheheatequationandtheprobabilitydensity q that indi ates whether a parti le hara terised initially by a oordinate will be q′ τ des ribed by after an interval . This distribution, whi h is related to the free- parti le density matrix in statisti al me hani s, is a probability measure be ause lim (q,q′) = τ→0 τ it is positive de(cid:28)nite and normalisable [a tually, normalised: K δ3(q q′) − ℄. In Eu lidean spa e a theory's generating fun tional an truly be expressed in terms of a probability measure and the properties of su h measures make it likely that the rigorous de(cid:28)nition of intera ting quantum (cid:28)eld theories is possible in that ase. However, in Minkowski spa e the probability density i be omes a probability amplitude, through the appearan e of (cid:16) (cid:17) in the exponent, and that pre ludes the formulation of a measure theory. 2 S hwingerfun tions and light-quarkboundstates n Themoments of aprobability measureare -pointS hwinger fun tions. They orrespond to va uum expe tation values of Eu lidean (cid:28)elds and may loosely be termed Eu lidean spa e Green fun tions. When ertain onditions are met [1℄, analyti ontinuation of the S hwinger fun tions yields the Wightman fun tions and one may prove the re onstru tion theorem; namely, the omplete ontent 1 of a quantum (cid:28)eld theory is re overed from the Wightman fun tions. This is the basis for the ontemporary belief that the evaluation of a theory's S hwinger fun tions is equivalent to solving that theory. While that may be true in prin iple one must nonetheless develop pra ti al means by whi h to extra t physi al information from the S hwinger fun tions. The hallenge may be illustrated by noting that omplete knowledge of a two- point S hwinger fun tion in momentum spa e orresponds only to dire t knowledge of the expe tation value at spa elike momenta. A physi al parti le pole an only appear at timelike momentum. Lo ating su h a pole therefore requires that all the onditions be met for a unique analyti ontinuation. If the S hwinger fun tion is known only at a dis rete set of points; i.e., on a set of measure zero, thenthat isstri tlyimpossible.Inthis ommoninstan eallthatisreally possible is onstrained inferen e and an estimation of the inherent error. For us a ontext for these observations is the problem of determining the spe trumofQCD.Mu hhasbeenlearntaboutthetwo-pointS hwingerfun tions forquarks,gluonsandghosts.A onsensushasbeenrea hed; viz.,thesefun tions annotbedes ribedbyapositivede(cid:28)nitespe traldensity.However,thepointwise behaviour ofthe ontinuation ofthesefun tions totimelikemomenta isun ertain (cid:21) see, e.g., Refs.[2, 3, 4, 5, 6, 7℄ and referen es therein (cid:21) and is the subje t of ontinuing study, e.g., Ref.[8, 9℄. These two-point fun tions appear in the kernels of the Bethe-Salpeter equations whose solutions provide information about the properties of olour-singlet hadrons. The theory and phenomenology of su h appli ations are reviewed in Refs.[10, 11, 12, 13, 14℄. The study of ground state mesons in the pseudos alar and ve tor hannels has met with su ess, as eviden ed by re ent appli ations to heavy-heavy mesons [15℄ and ve tor meson ele tromagneti form fa tors [16℄. A present hallenge is the extension to ex ited states in these and other hannels, e.g., Refs.[17, 18, 19, 20℄, and to hannels in whi h the ground state lies above 1 2 GeV, e.g., Refs.[21, 22, 23, 24℄. A vera ious analyti ontinuation of two-point S hwinger fun tions be omes in reasingly important as the bound-state mass be omes larger [29℄. The possibility that this ontinuation might be absent re- turns us to the question: an reliable information about bound state properties be obtained dire tly from the S hwinger fun tions? Another ontext for this question is the numeri al simulation of latti e- regularised QCD. That approa h is grounded on the Eu lidean spa e fun tional integral.S hwinger fun tionsareallthatit andire tlyprovide.Hen eit anonly be useful if methods are found so that the question an be answered in the a(cid:30)r- mative.Itistherefore unsurprisingthatlatti e-QCD pra titionershaveexpended 1 NB. Minkowski spa e Green fun tions are onstru ted from appropriately time-ordered om- 2binations of Wightmanfun tions. There is also merit in studying the olour antitriplet quark-quark hannel sin e su h diquark orrelations quite likely playanimportantrole in baryonstru ture, e.g, [25, 26,27, 28℄. M.S. Bhagwat, A. Höll, A. Krassnigg, C.D. Roberts and S.V. Wright 3 mu h e(cid:27)ort on this problem (see, e.g., Refs.[30, 31℄ and referen es therein). Our study explores the e(cid:30) a y of the methods devised for latti e-QCD and whether it is worthwhile to adapt them to ontinuum approa hes. In Se t.2 we re apitulate on aspe ts of Dyson-S hwinger equations (DSEs) that are relevant to the study of mesons, fo using in parti ular on the inhomoge- neousBethe-Salpeterequation for olour-singlet three-point S hwinger fun tions. In Se t.3 we explore and illustrate the appli ation of a simple momentum-spa e method to the problem of extra ting meson masses and vertex residues from these S hwinger fun tions. This is followed by an examination of two straightforward on(cid:28)guration-spa e methods for re overing bound-state information from S hwinger fun tions, andastudyoftheimpa tofstatisti alnoiseonthesepro e- dures. The pi ture that emerges is that rudimentary methods are often adequate if one only requires information about the ground and (cid:28)rst ex ited state, and assumptions made about the S hwinger fun tion's analyti stru ture are valid. However, to obtain information from S hwinger fun tions about heavier states in a given hannel, something mu h more re(cid:28)ned is ne essary. In Se t.4 we analyse one su h method; namely, that based on the orrelator matrix [32, 33℄. Se tion 5 is an epilogue. 2 Inhomogeneous Bethe(cid:21)Salpeter Equation A al ulation of the properties of two-body bound states in quantum (cid:28)eld theory may begin with the Poin aré ovariant Bethe-Salpeter equation (BSE). The 3 inhomogeneous BSE for a pseudos alar quark-antiquark vertex is τj Λ Γj(k;P) = Z γ + χj(q;P) Ktu(q,k;P), h 5 itu 4 5 2 Zq h 5 isr rs (2) j Γ (k;P) k P where 5 isthe vertex Bethe-Salpeter amplitude, with the relative and r,...,u the total momentum; represent olour, Dira and (cid:29)avour matrix indi es; j j χ (q;P) = S(q )Γ (q;P)S(q ), 5 + 5 − (3) q = q P/2 Z ± 4 ± ; is the Lagrangian-mass renormalisation onstant, des ribed in Λ onne tion with Eq.(9); and q represents a Poin aré invariant regularisation of Λ K the integral, with the reguRlarisation mass-s ale [34, 35℄. In Eq.(2), is the fully amputated and renormalised dressed-quark-antiquark s attering kernel and S in Eq.(3) is the renormalised dressed-quark propagator. It is notable that the SSK produ t is a renormalisation group invariant. The solution of Eq.(2) has the form τj j iΓ (k;P) = γ [iE (k;P) 5 2 5 5 + γ P F (k;P)+γ kk P G (k;P)+σ k P H (k;P)]. 5 5 µν µ ν 5 · · · (4) 3 tWheew(cid:29)aovrkouwristthrutw toudree.gPeInnµeorPuart2eE<quu 0alirdke(cid:29)anavmouertsria n:d{γhµe,nγ νe}P=au2lδiµmνa;tγrµ†i =esγaµre;saun(cid:30)d aie·bnt=toPre4i=pr1easiebnit. Foratimelikeve tor , .Moreaboutthemetri onventions anbefoundinSe t.2.1 of Ref.[11℄. 4 S hwingerfun tions and light-quarkboundstates This is the stru ture ne essary and su(cid:30) ient to ensure Poin aré ovarian e. It follows that quark orbital angular momentum is generally present within pseudos alar and indeed all bound states. This is quanti(cid:28)ed, e.g., in Ref.[15℄. The dressed-quark propagator appearing in the BSE's kernel is determined by the renormalised gap equation S(p)−1 = Z (iγ p+mbm)+Σ(p), 2 · (5) Λ λa Σ(p) = Z g2D (p q) γ S(q)Γa(q,p), 1 µν − 2 µ ν (6) Zq D Γ (q,p) µν ν where is the dressed-gluon propagator, is the dressed-quark-gluon mbm Λ vertex, and is the -dependent urrent-quark bare mass. The quark-gluon- Z (ζ2,Λ2) 1,2 vertex and quark wave fun tion renormalisation onstants, respe - ζ tively, depend on the renormalisation point, , the regularisation mass-s ale and the gauge parameter. The gap equation's solution has the form 1 S(p)−1 = iγ p+M(p2) . Z(p2,ζ2) · (7) (cid:2) (cid:3) It is obtained from Eq.(5) augmented by the renormalisation ondition S(p)−1 = iγ p+m(ζ), p2=ζ2>0 · (8) (cid:12) m(ζ) (cid:12) where is the renormalised (running) mass: Z (ζ2,Λ2)mbm(Λ) = Z (ζ2,Λ2)m(ζ). 2 4 (9) Featuresofthegapequationanditssolutionsbearupontheradiusof onvergen e foraperturbativeexpansioninthe urrent-quark massofphysi alquantities[36℄. The solution of the inhomogeneous equation, Eq.(2), exists for all values of P2 , timelike and spa elike, with ea h bound state exhibited as a pole. This is P2 illustrated in Fig.1, wherein the solution is seen to evolve smoothly with and 4 the pole asso iated with the pseudos alar ground state is abundantly lear. Naturally, the numeri al determination of the pre ise lo ation of the (cid:28)rst E (k2 = 0;P2) 5 pole (ground state) in will generally be di(cid:30) ult. The task be- omes harder if one seeks in addition to obtain the positions of ex ited states. It is for these reasons that the homogeneous equation [39℄ is usually used. The homogeneous pseudos alar BSE is obtained from Eq.(2) by omitting the driving (1/2)Z γ τj 4 5 term; viz., the matrix valued onstant . The equation thus obtained de(cid:28)nes an eigenvalue problem, with the bound state's mass-squared being the eigenvalue andits Bethe-Salpeter amplitude, theeigenve tor. As su h, the homo- P2 geneous equation only has solutions at isolated timelike values of . However, if 4 NB.Thekernelusedforthis al ulation[37℄hashithertoprovidedonlystablebound-states.For aresonan ethepoleisnotlo atedontherealaxis.Nevertheless,inprin iplesu hsystems an alsobehandledviatheBSE.Hereinwewillmainlyoverlooksu h asesbe ausetheasso iated ompli ations are not yet relevant: all ontemporary studies employ kernels that omit the hannelsasso iated with physi alde ays;andunstablestates arestill notmu hstudied using latti e-QCD. M.S. Bhagwat, A. Höll, A. Krassnigg, C.D. Roberts and S.V. Wright 5 200 0.3 160 x e t0.2 r e V Inverse PS BSE Vertex / 120 1 0.1 Inverse Scalar BSE Vertex x e t r e V 0 80 0 0.2 0.4 0.6 0.8 2 2 P (GeV ) PS BSE Vertex Calculation 40 Scalar BSE Vertex Pion Pole 0 0 0.2 0.4 0.6 0.8 1 2 2 P (GeV ) E5(0;P2) ES(0;P2) Figure 1. Dimensionless pseudos alar (PS) and s alar amplitudes, and , respe tively, obtained by solving an inhomogeneous BSE of the type in Eq.(2) using the renormalisation-group-improved rainbow-ladder trun ation introdu ed in Ref.[37℄. (We set ω = 0.33 mu,d(ζ19) = 3.7 the model's mass-s ale GeV and used a urrent-quark mass MeV, ζ19 = 19 ω ∈ [0.3,0.5] GeV. NB. For GeV, ground state observables are onstant along a tra- ωD = (0.72GeV)3 =: m3 g je tory [38℄.) The verti al dotted-line indi ates the position of the π ground state mass pole; whereas the verti al solid line marks only the oordinate zero. The 1/E5(0;P2) 1/ES(0;P2) inset shows and . one is employing a framework that an only provide reliable information about the form of the Bethe-Salpeter vertex at spa elike momenta; i.e., a framework that only provides S hwinger fun tions, then a s heme must be devised that will yield the pole positions from this information alone. 3 Masses from S hwinger Fun tions: Simple Methods 3.1 Momentum spa e: Padé approximant One straightforward approa h is to fo us on 1 P (P2)= E E(k2 = 0;P2) (10) and lo ate its zeros. It is plain from Fig.1 that this method an at least be su essful for the ground state in the pseudos alar hannel. It is important to determine whether the approa h is also pra ti al for the determination of some properties of ex ited states. For the purpose of exploration and illustration, onsider a model for an inhomogeneous vertex (three-point S hwinger fun tion) whose analyti stru ture is 6 S hwingerfun tions and light-quarkboundstates Table1.Parameters hara terisingthevertexAnsatz,Eq.(11).Theywere hosenwithoutprej- udi e,subje tto the onstraint in quantum(cid:28)eldtheorythatresiduesofpoles inanobservable proje tion of a three-point S hwinger fun tion should alternate in sign [17℄, and ordered su h mi < mi+1 i = 0 i = 1 that , with denoting the ground state and the (cid:28)rst ex ited state, et . b=0.78 Z4(ζ19,Λ=200 ) is the al ulated valueof GeV usedto obtain the urves in Fig.1. i m a 2 i i Mass, (GeV) Residue, (GeV ) 0 0.14 4.23 1 1.06 -5.6 2 1.72 3.82 3 2.05 -3.45 4 2.2 2.8 known pre isely; namely, M−1 a V(P2) = b+ i , P2+m2 (11) i=0 i X i m a i i where: for ea h , is the bound state's mass and is the residue of the bound state pole in the vertex, whi h is related to the state's de ay onstant b (see Appendix A:); and is a onstant that represents the perturbative ba k- ground that is ne essarily dominant at ultraviolet total momenta. The parti ular parameter values we employ are listed in Table1. This Ansatz provides an information sample that expresses essential qualitative features of true DSE solutions for olour-singlet three-point S hwinger fun tions. N To pro eed we employ a diagonal Padé approximant of order to analyse the information sample generated by Eq.(11); viz., c +c x+...+c xN f (x) = 0 1 N , x= P2, N 1+c x+...+c xN (12) N+1 2N 1/V(P2) P2 is used to (cid:28)t a sample of values of de(cid:28)ned on a dis rete grid, su h as would be employed in a numeri al solution of the BSE. The known ultraviolet 5 behaviour of typi al verti es requires a diagonal approximant. There are some similarities between this approa h and that of the inverse amplitudes method in Refs.[40, 41℄. In a on(cid:28)ning theory it is likely that a olour-singlet three-point fun tion exhibits a ountable in(cid:28)nity of bound state poles. Therefore no (cid:28)nite order approximant an be expe ted to re over all the information ontained in that fun - M tion. The vertex model presented in Eq.(11) possesses bound state poles. It is N < M reasonable to expe t that an approximant of order an at most provide N 1 reliable information about the (cid:28)rst − bound states, with the position and 5 NB. A real-world data sample will exhibit logarithmi evolution beyond the renormalisation point.NosimplePadéapproximant anre overthat.However,thisisnotaprobleminpra ti al appli ations be ause the approximantis never applied onthat domain. M.S. Bhagwat, A. Höll, A. Krassnigg, C.D. Roberts and S.V. Wright 7 3 N = 3 ; Ground State 2.5 st N = 3 ; 1 excited state nd N = 3 ; 2 excited state 2 ) V e G (1.5 s s a M 1 0.5 0 0 0.2 0.4 0.6 0.8 1 2 2 P (GeV ) max N =3 Figure 2. Pole positions (mass values) obtained through a (cid:28)t of Eq.(12) with to data 1/V(P2) for generated from Eq.(11) with the parameters listed in Table 1. The oordinate P2 max isdes ribedinthetext.Horizontaldottedlinesindi atethethreelightestmassesinTable 1. The ground state mass (solid line) obtained from the Padé approximantlies exa tly on top of thedotted line representing thetrue value. Nth residue asso iated with the pole providing impure information that repre- M (N 1) sents a mixture of the remaining − − signals and the ontinuum. We anti ipate that this is the pattern of behaviour that will be observed with any N N rank- approximation to a S hwinger fun tion. The -dependen e of the Padé (cid:28)t an provide information onthis aspe t of the pro edure. The domain of spa e- like momenta for whi h information is available may also a(cid:27)e t the reliability of bound-state parameters extra ted via the (cid:28)tting pro edure. Information on this possibility is provided by (cid:28)tting Eq.(12) to the Ansatz data on a domain [0,P2 ] P2 max max , and studying the -dependen e of the (cid:28)t parameters. 1/V(P2) We(cid:28)ndthataPadéapproximant(cid:28)ttedto ana uratelyre overthe pole residues and lo ations asso iated with the ground and (cid:28)rst ex ited states. P2 max This is illustrated in Fig.2, whi h exhibits the -dependen e of the mass- N = 3 parametersdeterminedviaa Padéapproximant.Plateauxappearforthree isolatedzeros,whi histhemaximumnumber possible,andthemassesde(cid:28)nedby these zeros agree very well with the three lightest values in Table 1. This appears to suggest that the pro edure has performed better than anti ipated. However, P2 max that inferen e is seen to be false in Fig.3, whi h depi ts the -dependen e a 0,1 of the pole residues. While the results for are orre t, the result inferred a a 2 2 from the plateau for is in orre t. Plainly, if the value of were not known a priori, then one would likely have been misled by the appearan e of a plateau N = 3 and produ ed an erroneous predi tion from the (cid:28)t to numeri al data. A approximant an truly at most only provide reliable information for the (cid:28)rst N 1= 2 − bound states. 8 S hwingerfun tions and light-quarkboundstates 4 N = 3 ; Ground state 2 2V ) N = 3 ; 1st excited state e nd G N = 3 ; 2 excited state ( 0 e u d i es-2 R -4 -6 0 0.2 0.4 0.6 0.8 1 2 2 P (GeV ) max N = 3 1/V(P2) Figure 3. Pole residues obtained through a (cid:28)t of Eq.(12) with to data for generated from Eq.(11) with the parameters in Table 1. Horizontal dotted lines indi ate the residues asso iated with the three lightest masses in Table 1. The residue asso iated with the ground state (solid line) lies exa tly atop the dotted line representing the true value. The residue for the se ond pole exhibits a plateau at the orre t (negative) value. However, the plateau exhibitedbytheresult for theresidue of thethird pole is wrong. 3.2 A realisti test Following this exploration of the viability in prin iple of using spa elike data and the Padé method to extra t bound state information, we onsidered the DSE- al ulated pseudos alar vertex that is in depi ted Fig.1. The masses and residues (see the appendix) for the ground state pseudos alar and its (cid:28)rst radial ex itationwereobtainedfromthehomogeneousBSEinRef.[17℄.The omparison between these masses and those inferred from the Padé approximant ispresented in Fig.4. It appears that with perfe t (e(cid:27)e tively noiseless) spa elike data at hand, reliable information on the masses an be obtained. However, while the ground state residue is orre tly determined, that of the (cid:28)rst ex ited state is not. We will subsequently explain this result. In the rainbow-ladder trun ation of QCD's DSEs the next heaviest ground- state meson, after the pseudos alar, is the s alar [11℄. The inhomogeneous s alar BSEvertex anbe al ulated[42℄andtheanalysisdes ribedaboverepeated. The relative magnitude of the s alar vertex ompared with the pseudos alar is shown P2 = 0 inFig.1.Inthevi inity of thesignalforabound stateismu h suppressed be ause in this model the mass-squared of the quark- ore in the ground-state P2 = (0.675 )2 = 0.456GeV2 s alar meson is − GeV − . The ontribution to the signal from a more massive state will be further suppressed. Figure 5 shows the N = 3 1/E (k2 = 0;P2) S results from an (cid:28)t of Eq.(12) to al ulated using the inhomogeneous s alar BSE. The two plateaux give masses in agreement with m = 0.675 results obtained dire tly from the homogeneous BSE; i.e., 13L0 GeV M.S. Bhagwat, A. Höll, A. Krassnigg, C.D. Roberts and S.V. Wright 9 0 1.4 First Excited State -4 First Excited State 1.2 V) 1 2eV) -1-28 e G Mass (G 00.2.85 Ground State sidue ( -168 Ground State 0.2 Re 0.15 4 0.1 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 2 2 2 2 P (GeV) P (GeV) max max N =3 Figure 4. Mass and residue obtained from a (cid:28)t of Eq.(12) with to the DSE result for 1/E5(k2 = 0;P2) in Fig.1. Horizontal dotted lines indi ate the masses and residues obtained for theground and (cid:28)rst ex ited state via a dire tsolution of the homogeneous BSE. 2 4 1.8 2 Ground State ) ) 1.6 First Excited State 2V V e 0 Ge 1.4 G ( s ( 1.2 ue -2 Mas sid 1 e -4 R 0.8 -6 0.6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 2 2 2 2 P (GeV ) P (GeV ) max max N =3 Figure 5. Mass and residue obtained from a (cid:28)t of Eq.(12) with to the DSE result for 1/ES(k2 = 0;P2) ; i.e., for the s alar hannel, also depi ted in Fig.1. As in Fig.4, horizontal dottedlinesindi atethemassesandresiduesobtainedforthegroundand(cid:28)rstex itedstatevia a dire tsolution of thehomogeneous BSE. m = 1.16 and 23L0 GeV. However, aswith thepseudos alar, theresidue asso iated with the (cid:28)rst ex ited state is wrong. The pattern exposed here is not in a ord with the expe tations raised in Se t.3.1. However, this is only evident be ause we an simultaneously use timelike information to al ulate the residues. Su ess with the method explained in Se t.3.1 is predi ated upon the assumption that the olour-singlet inhomogeneous vertex an be des ribed by a positive de(cid:28)nite spe tral density; e.g., this validates the form of Eq.(11). The method may fail, therefore, if the intera tion produ es a vertex that is in onsistent with this physi al requirement. To explore that possibility we employed matrix inversion methods to solve Eq.(2)dire tlyfortimeliketotalmomenta.Theresult,presentedinFig.6,reveals asignaldefe tintheintera tion modelproposedinRef.[37℄.Themodelprodu es P2 = 1.15 2 E (0,P2) 5 a vertex that possesses a zero at − GeV , and a zero in is 10 S hwingerfun tions and light-quarkboundstates 1000 500 ) 2 P , 0 0 ( 5 E -500 -1000 -1.5 -1 -0.5 0 2 2 P [GeV ] Figure 6. Solution of Eq.(2) obtained dire tly at timelike momenta. Three singularities that would normally indi ate bound state poles are readily apparent. However, the ir le at P2=−1.15 2 GeV marks anunanti ipated zero. only possible if the spe tral density in the olour-singlet pseudos alar hannel is negative on a measurable domain. Similar behaviour is found in the s alar hannel. 1/E (0,P2) 5 An explanation of Figs.4 and 5 is now evident. The zeros in provide a strong signal and may readily be re overed via a Padé approximant. 1/E (0,P2) 5 However, it is plain that the singularity in , produ ed by the zero ir led in Fig.6 and lying just a little further into the timelike region than the position of the (cid:28)rst ex ited state, an distort the value inferred for that state's residue. We have thus arrived via a ir uitous route at important news. In attempting toobtainknowledgeofboundstatesfrominformationprovidedsolelyatspa elike momenta,wehaveun overed aweaknessintheintera tion modelofRef.[37℄that an plausibly be said to limit its domain of reliability to bound systems with < 1.1 mass GeV. However, in order to establish this we had to forgo the notion ∼ of using spa elike information alone. Without knowing the orre t values of the residuesa priori,onemayreasonably havea eptedtheresultsinFigs.4and5as true measures of the (cid:28)rst ex ited states' properties. The analysis in this se tion emphasises that in relying solely on spa elike information one rests heavily on assumptionsmadeabouttheanalyti propertiesoftheS hwinger fun tionsunder examination. 3.3 Con(cid:28)guration spa e analysis It is natural to ask whether more information about bound states an be extra ted from S hwinger fun tions spe i(cid:28)ed in on(cid:28)guration spa e. These are,