Relevance of complex branch points for partial wave analysis
S. Ceci, M. Döring, C. Hanhart, S. Krewald, U.-G. Meißner, A. Svarc
aa r X i v : . [ nu c l - t h ] A p r Relevance of complex branch points for partial wave analysis
S. Ceci, M. D¨oring, C. Hanhart,
3, 4
S. Krewald,
3, 4
U.-G. Meißner,
2, 3, 4 and A. ˇSvarc Rudjer Boˇskovi´c Institute, Bijeniˇcka 54, HR-10000 Zagreb, Croatia HISKP (Theorie),Universit¨at Bonn, Nußallee 14-16, D-53115 Bonn, Germany Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Institute for Advanced Simulation, Forschungszentrum J¨ulich, D-52425 J¨ulich,Germany
A central issue in hadron spectroscopy is to deduce — and interpret — resonance parameters,namely pole positions and residues, from experimental data, for those are the quantities to becompared to lattice QCD or model calculations. However, not every structure in the observablesderives from a resonance pole: the origin might as well be branch points, either located on the realaxis (when a new channel comprised of stable particles opens) or in the complex plane (when atleast one of the intermediate particles is unstable). In this paper we demonstrate first the existenceof such branch points in the complex plane and then show on the example of the πN P partialwave that it is not possible to distinguish the structures induced by the latter from a true pole signalbased on elastic data alone. PACS numbers: 14.20.Gk, 13.75.Gx, 11.80.Gw, 24.10.Eq,
I. INTRODUCTION
The second and third resonance region of baryonic ex-cited states is currently under intense experimental inves-tigation at various laboratories such as ELSA, MAMI,or JLab [1–4]. Many resonances overlap at these ener-gies, and usually partial wave analyses in different frame-works, such as K -matrix approaches or dynamical cou-pled channel models [5–22] are necessary to disentanglethe resonance content. Furthermore, many resonancesmay couple only weakly to the πN channel, and the in-vestigation of different initial and final states in hadronicreactions is mandatory [18]. Also, at these energies multi-pion intermediate and final states are becoming increas-ingly important and should be included in the analysis ofthe S -matrix. For the corresponding T -matrix, channelswith stable particles like ηN induce a branch point at thethreshold energy ( √ s = m η + M N ), that may be visibleas cusps in the amplitude [23, 24].For effective multi-pion channels with one unstable andone stable particle, such as ρN , the analytic structure ismore complicated. In comparison to the branch pointson the real s axis and the first and second sheet poles,the third type of allowed singularities are branch pointswithin the complex energy plane. They emerge whenamongst groups of particles of an at least three–body de-cay there exists a strong correlations between two parti-cles. For example, a significant fraction of π + π − X inter-mediate and final states typically goes through the ρ me-son. The resulting line shapes are discussed in Ref. [25].Branch points in the complex plane also emerge in the re-cently developed complex-mass scheme for baryonic res-onances [26].Known theoretically for a long time [27, 28], thesebranch points are present in several modern approaches,such as the GWU/SAID analysis [5, 6], the J¨ulich [14–18] and EBAC [20, 21] approaches, or the Bonn-Gatchina [13] analysis. It is the goal of this study to demonstrate the model-independent character of thosecomplex branch points. To do so, we employ generalproperties of the S -matrix only. In a particular exampleit is then shown that the branch points are of relevancein partial wave analyses: if the theoretical partial wavedoes not include them, their absence can easily be sim-ulated by resonance poles. This, of course, distorts theextracted baryon spectrum. Branch points in the com-plex plane are thus important for the reliable extractionof resonance parameters.The paper is organized as follows: in Sec. II the ex-istence of branch points in the complex plane is derivedfrom three-body phase space, in Sec. II A the propertiesof the branch points are determined, and in Sec. III itis shown that these branch points are relevant in the ex-traction of the resonance content of partial waves. II. ANALYTIC STRUCTURE OF THE S -MATRIX AND COMPLEX BRANCH POINTS Every channel opening introduces a new branch pointand with it a new sheet to the S -matrix, located at s =( P m i ) , with m i the masses of the stable particles inthat channel. The first sheet is always the physical one,i.e. where the physical amplitude is situated. The onlysingularities allowed on the first sheet are poles on thereal s axis below the lowest threshold (=bound states)or branch points on the real axis. On other sheets, polesand branch points can be located anywhere. Poles onthe second sheet are called resonances if their real part islocated above the lowest threshold, and they are calledvirtual states, if they are located below the threshold, buton the real axis. It is also possible to have poles on thesecond sheet inside the complex plane with a real partlower than the threshold [29], or on other hidden sheetswhich are often referred to as shadow poles.In this study we are interested in branch points on thesecond sheet in the complex plane, i.e. on the same sheeton which the resonance poles are situated. To prove theemergence of these branch points, let us start from theoptical theorem T ( j → i ) − T † ( j → i )= i (2 π ) X f Z d Φ f T † ( i → f ) T ( j → f ) (1)where T ( j → i ) denotes the T -matrix connecting chan-nels i and j and d Φ f denotes the phase space of channel f .To simplify the argument we assume that the T -matrix isin a particular partial wave; below we focus on the singu-larities that stem from the unitarity cuts only. Singulari-ties like the left-hand cuts, the short nucleon cut [17, 30]or the circular cut, induced by the partial wave projec-tion, are ignored in the following for they are irrelevantfor the argument given.To be specific we use the normalization of phase spaceas proposed by the particle data group [31]. Then wehave for the n –particle phase space d Φ n ( P ; p , .., p n ) = δ (4) P − X i p i ! Y i d p i (2 π ) E i (2)where P is the overall center-of-mass (c.m.) four-momentum.To avoid complications, that are irrelevant for the va-lidity of the present argumentation, we now focus onthe diagonal channel i = j . To be concrete we assume i = πN . To further simplify the argument, in additionwe focus on f = ρN as the only relevant intermediate ππN channel. The latter assumption allows us to write T ( πN → ππN ) = iW ( m ππ ) D ( m ππ ) T ( πN → ρN ) , (3)where D ( m ππ ) denotes the physical ρ propagator as afunction of the ππ invariant mass m ππ and W ( m ππ ) isthe partial wave projected decay vertex, that containsalso the information on the orbital angular momentum ℓ of the decay into ππ . In the following we abbreviate m ≡ m ππ .One can decompose the three-body phase space intotwo subspaces [31], d Φ n ( P ; p , .., p n ) = d Φ j ( q ; p , .., p j ) × d Φ n − j +1 ( P ; q, p j +1 , .., p n )(2 π ) dm . (4) For the example of the ρ [ ππ ] N system considered here,the first factor d Φ refers to the ππ phase space in the ρ subsystem at four-momentum q (note that m = q ), thesecond is the ρN phase space at four-momentum P , and n = 3, j = 2. With this decomposition, Z d Φ j ( q ; p , .., p j ) | D ( m ) W ( m ) | = − π Im( D ( m )) = ρ ( m ) , (5)where ρ ( m ) denotes the spectral density for the reso-nance normalized via Z ∞ m π dm ρ ( m ) = 1 . (6)We get for the discontinuity of the πN amplitude fromthe ππN channel1 i (cid:0) T ( πN → πN ) − T † ( πN → πN ) (cid:1) = (2 π ) Z dm ρ ( m ) Z d Φ ( P ; q, p ) × | T ( πN → ρN )( s, m ) | + ... , (7)where the ellipses denote contributions from the otherchannels omitted here. The two-body phase space canbe calculated explicitly. One finds d Φ ( P ; q, p ) = 1256 π p ( √ s, m, m ) √ s d Ω , (8)with p ( √ s, m, m )= 12 √ s p ( s − ( m + m ) )( s − ( m − m ) ) (9)for the c.m. momentum of the nucleon (particle 3) andthe pion pair with invariant mass m .Using Eq. (7) we may thus express the T -matrixthrough a dispersion integral and obtain T ( πN → πN ) = 14 ∞ Z ( M N +2 m π ) ds ′ √ s ′ ( √ s ′ − M N ) Z m π dm ρ ( m ) p ( √ s ′ , m, m ) Z d Ω | T ( πN → ρN )( s ′ , m ) | s ′ − s + iǫ + ... , (10)where now the ellipses stand for the unitarity cut con-tributions from other channels as well as left-hand cutcontributions. First of all, there is the three–body cut,which drives the inelasticity of the T -matrix. To be con-crete, we may write ρ ( m ) = − Nπ Im 1 m − m ρ + im ρ ˜Γ , ˜Γ = Γ ˜ p ( m, m π , m π ) ℓ +1 p ℓ +10 (11)where p is the three-momentum at the nominal res-onance mass and N is a normalization factor so thatEq. (6) is fulfilled. The factor (˜ p/p ) ℓ +1 accounts forthe centrifugal barrier and ˜ p is the pion momentum inthe ρ rest frame. Note, ˜ p = p ( m, m π , m π ) at thresh-old, √ s = 2 m π + M N , i. e. the ρ is at rest and the ρ rest frame and overall rest frame coincide. Note also theexplicit form of Eq. (11) is only for illustration. The m -dependence of the denominator is more complicated ingeneral (see, e.g., the Appendix), but the only propertyneeded in the following is the presence of poles in thespectral function.Indeed, the spectral function ρ ( m ) of Eq. (11) con-tains a pair of poles located at m = m , where m denotes the pole position of the ρ meson, located in thecomplex plane. We may write m = m ρ ± i Γ /
2, where Γdenotes the width of the ρ –meson.For the existence of branch points in the complexplane, it is sufficient to consider the imaginary part ofEq. (10) in the following, or, more correctly, we considerthe analytic function δT which is δT = Im T for √ s ∈ R ,but of course δT = Im T for √ s / ∈ R (e.g., δT develops animaginary part for complex √ s , whereas Im T does not).The function δT can be straightforwardly evaluated, δ T = − π √ s ( √ s − M N ) Z m π dm ρ ( m ) × p ( √ s, m, M N ) p ( √ s, m, M N ) L g ( √ s, m ) (12)with p from Eq. (9). In Eq. (12), we have explicitly de-noted a factor of p L that comes from the L = 0 , , · · · transition T ( πN → ρN ). The function g ( √ s, m ) con-tains the integral R d Ω over the part of | T | without thesecentrifugal barrier factors. In general, g ( m + M N , m ) = 0.The overall process we consider here as an example isshown in Fig. 1.A function f ( √ s ) has a branch point z b at √ s = z b , whenever in its integral representation f ( √ s ) = R ba dq ˜ f ( √ s, q ) the function ˜ f has a simple pole at q = q and a √ s = z b exists such that q = a or q = b . For ex-ample, the integrand of the two-body phase space integral R ∞ dq q / ( √ s − E − E + iǫ ), where E i = p m i + q ,has a simple pole at q = p ( √ s, m , m ) with the on-shell momentum p from Eq. (9). Then, the branch pointis given for the √ s for which q = 0 (lower integration FIG. 1: The quasi-particle ( ρ ) coupling to the stable particle N with orbital angular momentum L ; the decay of the quasi-particle into stable particles (2 π ) is in ℓ -wave with respect tothe quasi-particle c.m. frame. limit). This is the case for √ s ≡ z b = m + m , i.e. thebranch point is at the two-body threshold.With this knowledge, it is straightforward to determinethe branch points of δ T : as discussed before, the simplepoles of the integrand (spectral function) are located atthe complex m = m which equal the upper integrationlimit of Eq. (12) for √ s = M N + m .Thus, without loss of generality, we have shown thatpoles in the spectral function at m = m lead to branchpoints of the amplitude at the complex scattering energy √ s = M N + m or √ s ≡ z b , = M N + m ρ ± i Γ / . (13)More general, the model-independent result is that z b is given by the sum of the mass of the stable particleplus m , where m is the pole position in the scatteringamplitude of the subsystem, in this case given by ππ which resonates through a ρ meson. Eq. (13) has also beobtained in Ref. [17], starting from an explicit expressionfor the ππN system, derived from field theory, and inwhich the ππ subsystem is boosted. In Appendix A wewill come back to the connection of that formalism to thepresent one.The branch points z b in Eq. (13) have been obtainedby considering the upper integration limit in Eq. (12).However, also the lower integration limit can coincidewith a singularity for a certain √ s : this is the case for √ s ≡ z b = 2 m π + m N (14)for which the lower integration limit coincides with thebranch point singularity coming from the factors of p inthe integrand. The overall analytic structure is shownin Fig. 2. The first, physical sheet has the branch point z b with an associated cut. If the cut is chosen along thereal √ s axis like in the figure, the discontinuity of theamplitude is given by 2 δT from Eq. (12). The branchpoints z b and z b are in δT , i.e. on the sheet that isobtained by analytically continuing the discontinuity ofthe first sheet. They are, thus, on the second sheet, wherealso resonance poles are normally situated. The branchpoints z b and z b induce the new sheets 3 and 4; they areanalytically connected to the second sheet along the cutsinduced by z b and z b . In Fig. 2 these cuts are chosenparallel to the real √ s axis; in Ref. [17] they are chosen FIG. 2: Analytic structure of the amplitude. There are threebranch points z b , z b = z ∗ b , and z b . z b and z b are struc-tures in δT and thus on the second sheet. parallel to the imaginary √ s axis, which is a convenientchoice to search for poles. For the numbering of sheets,see also Ref. [17]. A. Threshold behavior
Apart from determining the existence and position ofbranch points, one can also deduce their threshold behav-ior, i.e. the functional form of δT close to the three z b . InFig. 1, the three-body decay is schematically shown. Letthe quasi-particle ( ρ ) couple to the stable particle ( N )in L -wave in the overall c.m. system, while the quasi-particle decays into stable particles (2 pions) in ℓ -wavewith respect to the quasi-particle c.m. frame.In the following we will use the explicit form of Eq. (11)to determine the threshold behavior. It is clear, however,that the final results do not depend on this particularform for the spectral function, but only on the fact thatthe spectral function has poles [right side of Eq. (15)] andthe presence of factors of p in Eq. (12) that follow fromthe previously given phase space derivation.To study the behavior of the amplitude in the complexenergy plane close to the branch points z b , , complexvalues of m will be needed, and thus the (non-analytic)function Im in Eq. (11) needs to be evaluated to obtaina meromorphic expression, ρ ( m ) = Nπ m ρ ˜Γ (cid:0) m − m ρ (cid:1) + m ρ ˜Γ m → m −−−−−→ Γ h ( m ) m − m . (15)The right-hand side shows the behavior of ρ ( m ) close tothe pole at m = m ; the function h does not contain anypoles or zeros close to m and thus does not influence thethreshold behavior. In particular, ˜ p ℓ +1 that appears inthe numerator [cf. Eq. (11)], has no zero close to m andcan be absorbed in h . Thus the threshold dependenceof the branch points z b and z b does not depend on ℓ ,which may appear a surprising result.To obtain the threshold behavior of the branch points z b , in the complex plane, one inserts the right-hand side FIG. 3: Branch point z b in Re δT in the upper √ s half plane,for a realistic ρN intermediate state. The cut is chosen herein the positive Re √ s direction. of Eq. (15) into Eq. (12), δT ∼ ( √ s − M N ) Z m π dm Γ ( m − m ) L +12 h ( m ) m − m (16)where we have expanded the argument of the square rootof the p factors of Eq. (12) in m , at the point p ( √ s = z b = M N + m , m, M N ) to obtain the power of the leadingzero from these factors. The function h is again analytic,free of zeros close to m = m , and does not influence thethreshold behavior. The integral may now be evaluatedsetting this numerator and h constant. The result forthe threshold behavior of the branch points z b , is [seealso Eq. (13)] δT ( z b , ) ∼ (cid:0) √ s − z b , (cid:1) L +12 ∼ p ( √ s, m , M N ) L +1 . (17)In Fig. 3 we show the branch point z b in the upper √ s half plane [see Fig. 2] for a realistic ρN intermediate stateand L = 0. The branch point is clearly visible, togetherwith the cut that in this picture is chosen in the positiveRe √ s direction.To obtain the threshold behavior for the third branchpoint at z b = 2 m π + M N [see Fig. 2], we inspect againEq. (15). As discussed following Eq. (11), close to √ s = z b = 2 m π + M N [see Eq. (14)] the ρ c.m. frame coincideswith the overall c.m. frame, i.e. ˜ p = p ( m, m π , m π ), andthus the ℓ -wave decay in the ρ subsystem is also an ℓ -wave decay in the overall c.m. system. For √ s in thevicinity of z b , the denominator of Eq. (15) is free of zeros;however, in contrast to the case of z b , , the numerator˜Γ ∼ ˜ p ℓ +1 = p ℓ +1 = ( m − m π ) (2 ℓ +1) / does have azero that contributes to the threshold behavior. InsertingEq. (15) (including this factor) in Eq. (12) and expandingthe arguments of the square roots of the p factors aroundthe zero [cf. Eq. (9)] one obtains δT ∼ ( √ s − M N ) Z m π dm Γ (cid:0) m − m π (cid:1) ℓ +12 × ( m − m π ) L +12 h ( m ) , (18)with a function h free of zeros and poles in the vicinityof z b . Integration leads now to the threshold behaviorof z b , δT ( z b ) ∼ (cid:0) √ s − (2 m π + M N ) (cid:1) ℓ + L +2 ∼ p ( √ s, M N , m π ) ℓ +2 L +4 . (19)This corresponds to the opening of the three-body thresh-old. Note that even if ℓ = L = 0, the threshold behavioris still ∼ p , i.e. the standard three-body phase space;thus, this threshold opening is always smooth. B. The limit of vanishing width
It is instructive to study the limit of a vanishing widthof the ρ –meson in Eq. (7). Then ρ ( m ) −→ δ ( m − m ρ ) for Γ → . This allows us to perform the m integration to get1 i (cid:0) T ( πN → πN ) − T † ( πN → πN ) (cid:1) = (2 π ) Θ(( √ s − M N ) − m ρ ) Z d Φ ( P ; q, p ) × | T ( πN → ρN )( s, m ρ ) | + ... , (20)such that Eq. (10) reduces to the dispersion integral overthe standard two-body cut T ( πN → πN ) → Z ∞ ( M N + m ρ ) ds ′ √ s ′ p ( √ s ′ , m ρ , M N ) × Z d Ω | T ( πN → ρN )( s ′ , m ρ ) | s ′ − s + iǫ + ... . (21)The imaginary part which is given by δ T Γ → = − π/ (4 √ s ) p ( √ s, m ρ , M N ) L +1 g ( √ s, m ρ ) (22)has a branch point at √ s = m ρ + M N , which is simplythe ordinary two-body threshold on the real √ s axis. AsΓ →
0, the two branch points z b , in the complex planemove towards the real √ s axis until they coincide andform this single branch point at √ s = m ρ + M N . Notethat there is a factor of Γ in the numerator of Eq. (16), but in the limit Γ →
0, another factor ∼ Γ appears in thedenominators from the two poles moving to the real axis,that cancels the Γ of the numerator. Thus, indeed thebranch point persists in the limit Γ → z b = 2 m π + M N , Eq. (18)shows that there are no poles that can prevent the termfrom disappearing in the limit Γ →
0; thus, as Γ → → ρ decouples from ππ and thus, in ourexample, the ππN channel decouples from πN . III. THE RELEVANCE OF BRANCH POINTSIN THE COMPLEX PLANE
As shown in the previous section, whenever there isa multi-particle intermediate state with pairwise strongcorrelations, unavoidably branch points show up in thecomplex plane. As we will demonstrate on a particularexample in this section, their influence on the data mightwell be visible. However, as will be also shown, it isin general not possible to deduce the origin of such astructure from elastic data only.The first model we use is the so-called J¨ulichmodel [14–18]. It is a coupled channel meson exchangemodel including the channels πN, ηN, K Λ , K Σ as wellas 3 effective ππN channels, namely π ∆, σN , and ρN .All these two-pion channels show the mentioned kind ofbranch points [17]. In Appendix A we show the connec-tion of the formalism of the J¨ulich model to the one ofthe previous section. The J¨ulich model allows for a gooddescription of the available πN data in all partial waveswith j ≤ / √ s ∼ . N (1710) P , havebeen found in several analyses [31]. It is, however, re-markable that in recent analyses of the GWU/SAIDgroup [6], there is no sign for this resonance any more.Like the GWU/SAID analysis, the J¨ulich model containsexplicitly the branch points z b , in the complex plane at √ s = M N + m ρ ± i Γ / ∼ ± i MeV. However,there are no poles around these energies (the only gen-uine pole term in the P11 partial wave is the nucleon,while the poles of the Roper resonance are dynamicallygenerated [14]). For the purpose of this study we haveslightly changed the parameters of the model comparedto the results of Ref. [15] to obtain a good descriptionof the GWU/SAID solution. This is shown in Fig. 4by the dashed lines. The important point here is thatthe theoretical amplitude in the complex plane around √ s ∼ . ρN branchpoint.To illustrate the difficulties in determining the origin [MeV]00.20.40.6 R e , I m P FIG. 4: Fit of the CMB Zagreb model (solid lines) to theP11 amplitude provided by the J¨ulich model (dashed lines).The “data” points represent the Single Energy Solution of theGWU/SAID group [6]. of structures in the amplitude we fit this J¨ulich modelamplitude with another model, which does not containthe ρN branch point in the complex plane. For this,we use a Carnegie-Mellon-Berkeley (CMB) type of modelthat has been developed by the Zagreb group [7, 8, 32,33]. In this unitary coupled channel model which respectsanalyticity, background plus resonances are provided, butall branch points are on the real axis. The result of the fit,using two resonance terms, is shown in Fig. 4 by the solidlines. As the figure shows, the fit is very precise and, inparticular, shows no visible discrepancy to the amplitudeof the J¨ulich model in the energy range shown.However, the behavior in the complex plane is quitedifferent: as mentioned before, there is no complexbranch point in the CMB fit by construction; instead,a pole is found at 1698 − i MeV which in this casemight simulate the branch point missing in that model.Thus, at a realistic scale of precision, the ρN branchpoint does not manifest itself in a unique structure on thephysical axis; it can be simulated by resonance terms thatproduce poles in the complex plane. Still, the ρN branchpoint is a required structure of the S -matrix, as shown inthis study, and we have demonstrated that in an analysisof partial waves, this and other branch points have to beincluded to avoid false resonance signals, which of coursecan totally distort the spectrum of excited baryonic res-onances.In such circumstances, one clearly has to consider otherfinal states in which the resonance candidate shows aclearer signal. As already proposed in Ref. [34], perform-ing global analyses of many different reaction channelswithin one theoretical ansatz is a much cleaner way todetermine the resonance spectrum than increasing theprecision of a partial wave for one reaction.First steps within the coupled-channel J¨ulich modelhave been undertaken in this direction through the in- clusion of some ρN data [14], ηN data [15], and, mostrecently, K + Σ + data [18]. For the isospin I = 1 / K Λ data to further clarify therole of the N (1710) P [see also Ref. [35]].Thus, the aim of the present short exercise is not todiscard the existence of the much-debated N (1710) P as such; rather, we have shown that branch points in thecomplex plane are relevant; in their absence, resonancesmay be needed to simulate them, and, thus, the extractedbaryon spectrum can be easily distorted. IV. CONCLUSIONS
Using only general properties of the S -matrix we haveshown the existence and determined the position of threebranch points induced by intermediate quasi-two bodystates. Those are three-body states in which two particlesare so strongly correlated that the scattering amplitudeof this subsystem has a pole. A pole in the subsystemnecessarily leads to the appearance of branch points inthe complex √ s plane of the overall πN amplitude. Thisresult is model-independent because it does not dependon any particular parameterization, but only on analytic-ity and general properties of the three-body phase space.We have also determined the threshold behavior of allbranch points, which depends on the orbital angular mo-menta of the two decay processes involved. Finally, onthe example of the P
11 partial wave, it has been shownthat branch points in the complex plane are relevant inpartial wave analysis: if a theoretical amplitude does notcontain the branch points, false resonance signals maybe obtained. To allow for a reliable extraction of thebaryon spectrum, it is thus mandatory to include alsothese branch points in the analysis.
Acknowledgements
This work is supported by theDAAD (Deutscher Akademischer Austauschdienst) grantNo. D/08/00215. It is also supported by the DFG(Deutsche Forschungsgemeinschaft, Gz.: DO 1302/1-2and SFB/TR-16) and by the EU-Research Infrastruc-ture Integrating Activity “Study of Strongly InteractingMatter” (HadronPhysics2, grant n. 227431) under theSeventh Framework Program of the EU.
Appendix A: Spectral representation of the J¨ulichmodel
In this Appendix the connection of the field theoreticalformalism, used in the J¨ulich model of hadron exchange,to the formalism used in this study is outlined, up tooverall normalization factors. For further details of theformalism used in the J¨ulich model, we refer to Ref. [17].For the example of the ρN propagator, that is consideredhere, the propagator on the real axis is given by g ρN ( √ s, k ) = 1 √ s − E N ( k ) − E ρ ( k ) − Σ( z ρ ( √ s, k ) , k )(A1)where E N is the nucleon energy, E ρ is the ρ energy usingthe bare ρ mass and Σ is the ρ self energy, where z ρ ( √ s, k )is the boosted energy for the ρ subsystem. The explicitform of z ρ ( √ s, k ) is quoted in Ref. [17] but for the presentdiscussion the only needed property is that z ρ ( √ s, k =0) = √ s − M N . The propagator g ρN is iterated in themultichannel scattering equation, but to investigate theanalytic structure it is sufficient to consider the one-loopamplitude G ρN ( √ s ) = ∞ Z dk k g ρN ( √ s, k ) (A2)where for simplicity we have omitted the form factorsthat regularize this divergent expression. One can rewritethe Dyson-Schwinger representation of Eq. (A1) with thespectral function S ( ω, k ) = − π Im 1 ω − E N ( k ) − E ρ ( k ) − Σ( z ρ ( ω, k ) , k )(A3)resulting in the Lehmann representation g ρN ( √ s, k ) = ∞ Z m π + M N dω S ( ω, k ) √ s − ω + iǫ . (A4)For the imaginary part of the ρN loop G ρN ( √ s ), oneobtains:Im G ρN ( √ s ) = Im ∞ Z dk k g ρN ( √ s, k )= − π ∞ Z dk k S ( √ s, k ) = − π k Z dk k S ( √ s, k ) . (A5)The last equality shows that the integration can be cutat k = k as for k > k the spectral function is zero be-cause then z ρ ( √ s, k ) < m π . In particular, k is givenby z ρ ( √ s, k ) = 2 m π . Note that the explicit evaluationof the integration limits as done here is necessary if onewants to use the spectral representation in the complex √ s plane. This has been shown recently in the context ofFeynman parameterized loops [36]: the integration limitshave to be analytically continued for complex √ s to ob-tain the analytic continuation of the loop itself, and forthis they need to be known explicitly. Eq. (A5) can be rewritten asIm G ρN ( √ s ) = √ s − m π Z m dm S ( √ s, k on ( m )) mE on m × ( − π ) k on ( m ) E on π E on m √ s (A6)with k on ( m ) = p ( √ s, m, m π ) , E on π = p m π + ( k on ) ,E on m = p m + ( k on ) . (A7)and p from Eq. (9). The lower integration limit m isgiven as the solution of z ρ ( √ s, k on ( m )) = 2 m π . Thesecond fraction in Eq. (A6) can be compared to the imag-inary part of the well-known [17] propagator of two stableparticles M and N ,Im G stable = − π k on ( m = √ s ) E on M E on N √ s . (A8)Thus, the imaginary part of a loop with one stable andone unstable particle can be expressed as an integral overa distribution of imaginary parts of the form of Eq. (A8).Comparing Eq. (A6) to Eq. (12), one sees the formalsimilarity: there is an integral of a spectral function,that has poles [cf. Eq. (A3)], together with the factor k on ( m ) = p ( √ s, m, m π ), and both ingredients producethe three branch points z b , , as has been shown in themain text (we have omitted here the additional 2 L pow-ers of p for simplicity). There is a difference in the cho-sen parameterization in terms of the spectral function[compare p ( √ s, m, m π ) in Eq. (A7) vs. p ( √ s, m, M N ) inEq. (12)], but this does not change the position of thebranch points.Indeed, k on = 0 for the upper integration limit m = √ s − m π and thus z ρ ( √ s,
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