Solving the inhomogeneous Bethe-Salpeter Equation in Minkowski space: the zero-energy limit
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Solving the inhomogeneous Bethe-Salpeter Equation inMinkowski space: the zero-energy limit
Tobias Frederico a,1 , Giovanni Salm`e b,2 , Michele Viviani c,3 Dep. de F´ısica, Instituto Tecnol´ogico de Aeron´autica, DCTA, 12.228-900 S˜ao Jos´e dos Campos, S˜ao Paulo, Brazil Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P.le A. Moro 2, I-00185 Roma, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo Pontecorvo 3, 56100, Pisa, ItalyReceived: date / Accepted: date
Abstract
For the first time, the inhomogeneous Bethe-Salpeter Equation for an interacting system, composedby two massive scalars exchanging a massive scalar, isnumerically investigated in ladder approximation, di-rectly in Minkowski space, by using an approach basedon the Nakanishi integral representation. In this pa-per, the limiting case of zero-energy states is considered,extending the approach successfully applied to boundstates. The numerical values of scattering lengths, arecalculated for several values of the Yukawa couplingconstant, by using two different integral equations thatstem within the Nakanishi framework. Those low-energyobservables are compared with (i) the analogous quan-tities recently obtained in literature, within a totallydifferent framework and (ii) the non relativistic evalu-ations, for illustrating the relevance of a non perturba-tive, genuine field theoretical treatment in Minkowskispace, even in the low-energy regime. Moreover, dy-namical functions, like the Nakanishi weight functionsand the distorted part of the zero-energy Light-frontwave functions are also presented. Interestingly, a highlynon trivial issue related to the abrupt change in thewidth of the support of the Nakanishi weight function,when the zero-energy limit is approached, is elucidated,ensuring a sound basis to the forthcoming evaluationof phase-shifts.
Keywords
Bethe-Salpeter equation · Minkowskispace · Scattering states · Ladder approximation · Light-front projection · Integral representation a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] Within a field theoretical framework, it is a highly nontrivial challenge to develop non perturbative tools inMinkowski space, but it is quite desirable to devote ef-forts in that direction, in order to gain insights thatcould turn out useful in particle physics. In the last fewyears, solving the homogeneous Bethe-Salpeter equa-tion (BSE) [1], directly in Minkowski space, has made asubstantial step forward [2,3,4,5,6,7,8,9,10,11] due toapproaches based on the so-called Nakanishi perturbation-theory integral representation (PTIR) of the n -leg tran-sition amplitudes [12].The Nakanishi PTIR for the three-leg amplitude isemerging as a very effective tool for studying the boundstate problem [2,3,4,5,6,7,8,10,11], within a rigorousfield-theory framework. Though the Nakanishi PTIR ofthe three-leg amplitude, or vertex function, had beendevised within the perturbative framework of the Feyn-man diagrams (as it happens for any n -leg amplitudePTIR), it has been shown to work extremely well asthe initial Ansatz for obtaining actual solutions of thehomogeneous BSE. It must be recalled that BSE, be-ing an integral equation, belongs to a non perturbativerealm, and therefore the Nakanishi integral representa-tion of the three-leg amplitude can be only an Ansatz,when exploited in this context.The main features of the Nakanishi integral repre-sentation of any n -leg amplitude are basically relatedto the formal infinite sum of the parametric Feynmandiagrams, that contribute to the amplitude under con-sideration. In particular, the n -leg amplitude PTIR hasa well-defined structure, given by the folding of (i) adenominator, containing all the allowed independentinvariants and governing the analytic behavior of theamplitude itself, and (ii) a weight function, that is a a r X i v : . [ h e p - ph ] J u l real function depending upon real variables (one is anon compact variable, while the others are compact).It should be emphasized that, at this stage, the Nakan-ishi weight function has only a formal expression [12]. Ifthere were an equation for explicitly determining such aweight function, then one could quantitatively evaluatethe actual n -leg amplitude, under consideration. Thehomogeneous BSE, that obviously does not belong tothe original framework of PTIR, has inspired a differ-ent usage of the formal expression of a particular n -legamplitude, namely the three-leg one, or vertex func-tion. Indeed, if one assumes that the Nakanishi integralrepresentation of the three-leg amplitude be formallyvalid also for the BS amplitude (still a three-leg am-plitude, but for a bound state), then the weight func-tion could be considered as an unknown function tobe determined. It has to be pointed out that, a pri-ori , there is no guarantee that such an approach forsolving BSE be successful, given the caveat above men-tioned. Fortunately, it works, as shown in Refs. [2,3,4,5,6,7,8,10,11], where the above strategy was applied,but with some differences, for solving the homogeneousBSE directly in Minkowski space. More precisely, byusing the PTIR Ansatz for the BS amplitude one canderive, in a formally exact way, an equation for theNakanishi weight function, starting from the homoge-neous BSE, and look for solutions. If the new equa-tion for the weight function has solution, then one canclaim that BS amplitudes, actual solutions of the ho-mogeneous BSE in Minkowski space, can be (i) for-mally written like the PTIR three-leg amplitude, and(ii) numerically determined. In order to achieve a for-mally exact integral equation for the weight functionfrom BSE, it is very useful and effective adopting aLight-front (LF) framework. This has been done bothin the covariant version of the LF framework [4] andin the non-explicitly covariant one [9]. In particular,the bound states of a massive two-scalar system inter-acting through the exchange of a massive scalar havebeen studied by adopting both ladder [4,6,7,10,11] andcross-ladder approximations of the BS kernel [5]. No-tably, the extension to a bound fermionic system havebeen also undertaken [8]. It has to be recalled thatnumerical investigations of the homogeneous BSE hasbeen performed also by considering the standard 4-dimensional variables [2,3].The successful achievements for the homogeneousBSE encourage the extension of the Nakanishi inte-gral representation to the study of the inhomogeneousBSE, i.e. the integral equation that determines the scat-tering states. Our aim is to present a new applica-tion of our general approach [9], based on the so-calledLF projection of the BS amplitude, i.e. the exact in- tegration on the minus component of the relative four-momentum that appears in the BS amplitude. After ap-plying this formally exact step to BSE, we have numer-ically investigated the zero-energy limit of the inhomo-geneous BSE, for a massive two-scalar system interact-ing through the exchange of a massive scalar, in ladderapproximation. The calculated scattering lengths havebeen compared in great detail with the analogous ob-servables recently obtained [13,14] within a completelydifferent framework. We have also compared our re-sults with the non relativistic scattering lengths, withthe intent to yield a possible guidance for lowering themodel dependence in the treatment of interacting fi-nal states, pertaining to relevant hadronic decay modes.Indeed, improving and widening our study could con-tribute to achieve an actual evaluation of the covari-ant off-shell T-matrix, that represents a key ingredientfor describing, e.g., the heavy meson decay amplitudes,like in D → Kππ processes [15,16], and could also havean impact in the development of final-state-interactionmodels, needed in the analysis of the CP violation incharmless three-body B decays [17]. Moreover, we haveproperly analyzed the distorted part of the zero-energywave function, putting in evidence the relation betweena non smooth behavior of the Nakanishi weight functionand the expected singularities of the LF 3D wave func-tion, like the one that brings the information relativeto the global propagation of the interacting two-scalarsystem. Finally, the integral equation for the Nakan-ishi weight function obtained by applying the so-calleduniqueness theorem [12], is carefully analyzed for thegeneral case of positive energy. Such an in-depth anal-ysis allows us to illustrate a surprising change in thewidth (from ( −∞ , ∞ ) to [0 , ∞ )) of the support of theNakanishi weight function with respect to its non com-pact variable, when the zero-energy limit is considered.Clarifying this feature allows us to put the forthcomingcalculation of the phase-shifts on a sound basis, sincetheir calculation requests a careful analysis from boththe theoretical and numerical points of view, as it willbe illustrated elsewhere [18]. By concluding this Intro-duction, it could be useful to remind that developinggenuine non perturbative descriptions of the scatteringprocesses within the Minkowski space, possibly apply-ing formally exact frameworks, is an appealing goal, inview of attempts of extracting tiny, but fundamentalsignals once very accurate experimental data will be-come available.The paper is organized as follows. In Sec. 2, weshortly introduce both the definitions and the generalformalism, and we thoroughly discuss the problem ofthe support of the Nakanishi weight function, given itsrelevance for the zero-energy limit. Sec. 3 illustrates how to evaluate the scattering length from the Nakan-ishi weight function, in ladder approximation. In Sec.4, the numerical studies of the scattering length arepresented and compared with the existing calculationsfound in literature; moreover the scattering 3D LF wavefunction (indeed the distorted part) is analyzed. Finally,in Sec. 5, the conclusions are drawn. In this Section, (i) we quickly recall the general formal-ism of Ref. [9], for obtaining two integral equations thatallows one to determine the Nakanishi weight functionneeded for scattering processes, and (ii) we demonstratea relevant feature of the weight-function support, thatit turns out to be very important also for numericallysolving the inhomogeneous BSE.In our investigation, we considered an interactingsystem composed by two massive scalars that exchangea massive scalar. This is a generalization of the hon-orable Wick-Cutkosky model[20,21] in two respects: (i)the interaction takes place through a massive-scalar ex-change and (ii) the scattering states is our focus.2.1 General formalismFor scattering states, the incoming particles are on their-own mass-shell and we indicate their total and relativefour-momenta with p and k i , respectively. By assum-ing that the inhomogeneous BS amplitude Φ + ( k, p, k i )be expressed in terms of the Nakanishi weight-function g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ; κ , z i ), then one can write (cf Ref. [9]) Φ (+) ( k, p, k i ) == (2 π ) δ (4) ( k − k i ) − i (cid:90) − dz (cid:48) (cid:90) − dz (cid:48)(cid:48) (cid:90) ∞−∞ dγ (cid:48) × g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ; κ , z i ) (cid:2) γ (cid:48) + m − M − k − p · k z (cid:48)(cid:48) − k · k i z (cid:48) − i(cid:15) (cid:3) == (2 π ) δ (4) ( k − k i ) − i (cid:90) − dz (cid:48) (cid:90) − dz (cid:48)(cid:48) (cid:90) ∞−∞ dγ (cid:48) × g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ; κ , z i )[ D − i(cid:15) ] , (1)where the total four-momentum is p ≡ { M, } and D = γ (cid:48) + γ + κ − k − ( k + + M z (cid:48)(cid:48) − M z i z (cid:48) ) − k + M z (cid:48)(cid:48) + z i z (cid:48) ) + 2 z (cid:48) cosϕ √ γγ i . The power of the denominator is the same one adoptedfor describing a bound state (cf Refs. [4,5,9,10,11]). Ex-ploiting a standard formalism introduced in Ref. [4], one defines z i = − k + i /M and gets z i = 2 k − i /M , since theincoming particles are on their-own mass shell: ( p/ ± k i ) = m . Moreover, one has 1 ≥ | z i | , since the in-coming particles have positive longitudinal momenta,i.e. p + / ± k + i ≥
0. In Eq. (1), the following notationshave been used: (i) cosϕ = (cid:98) k ⊥ · (cid:98) k i ⊥ , (ii) γ = | k ⊥ | and γ i = | k i ⊥ | , and (iii) κ = m − M /
4. For the initialstate one has( p/ ± k i ) = m = M k + i k − i − γ i = (1 − z i ) M − γ i , (2)with necessarily ( z i ) <
1. Hence one gets M = 4 ( m + γ i )(1 − z i ) κ = − γ i − z i M k i ≤ , (3)To complete the generalities, we also give the expres-sion for the inhomogeneous BSE, without self-energyinsertions and vertex corrections, in the present stageof our approach,. Then, one can write Φ (+) ( k, p, k i ) = (2 π ) δ (4) ( k − k i )+ G (12)0 ( k, p ) (cid:90) d k (cid:48) (2 π ) i K ( k, k (cid:48) , p ) Φ (+) ( k (cid:48) , p, k i ) , (4)where i K is the interaction kernel (where the vertexcorrections should appear), and G (12)0 is the free two-particle Green’s function given by G (12)0 ( k, p ) = G (1)0 G (2)0 == i ( p + k ) − m + i(cid:15) i ( p − k ) − m + i(cid:15) . (5)It is worth noting that the bosonic symmetry of theBS amplitude, Eq. (1), when 1 →
2, (i.e. p → p , k → ( − k ) and k i → ( − k i )) has to be fulfilled, as in the caseof bound states [10]. Therefore, the Nakanishi weightfunction must have the following property g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ; κ , z i ) = g (+) ( γ (cid:48) , z (cid:48) , − z (cid:48)(cid:48) ; κ , − z i ) . (6)Moreover, as shown in details in in Appendix A, onehas g (+) ( γ (cid:48) , z (cid:48) = ± , z (cid:48)(cid:48) ) = g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) = ±
1) = 0 . (7)As well-known (see e.g. Refs. [22,23,9]), by projectingthe BS amplitude onto the null-plane, i.e. integrating on k − , one exactly gets the 3D LF scattering wave func-tion ψ (+) , that is proportional to the valence component ψ (+) n =2 /p appearing in the Fock expansion of a two-scalarstate, namely ψ (+) = √ ψ (+) n =2 /p (given the normaliza-tions assumed in Refs. [9,10]). The 3D LF scattering wave function reads ψ (+) (cid:0) z, γ, cosϕ ; κ , z i (cid:1) == p + (1 − z )4 (cid:90) dk − π Φ (+) ( k, p, k i ) = p + (1 − z )4 × (2 π ) δ (3) (˜ k − ˜ k i ) + ψ dist (cid:0) z, γ, cosϕ ; κ , z i (cid:1) (8)where ˜ k ≡ { k + , k ⊥ } and ψ dist (cid:0) z, γ, cosϕ ; κ , z i (cid:1) is thedistorted part of the 3D LF scattering wave function,that in the CM frame, where p + = p − = M/ p ⊥ = 0, reads ψ dist (cid:0) z, γ, cosϕ ; κ , z i (cid:1) = (1 − z )4 (cid:90) − dz (cid:48) × (cid:90) ∞−∞ dγ (cid:48) g (+) ( γ (cid:48) , z (cid:48) , z ; κ , z i )[ D − i(cid:15) ] . (9)with D = γ (cid:48) + γ + z m + (1 − z ) κ + z (cid:48) ( M z z i + 2 cosϕ √ γγ i ) . In what follows, without loss of generality, we choosea head-on scattering process, namely a z -axis along theincoming three-momenta. In this case the variable γ i is zero and therefore the dependence upon cosϕ disap-pears. As a matter of fact, the distorted wave functionbecomes ψ dist (cid:0) z, γ ; κ , z i (cid:1) = (1 − z )4 (cid:90) − dz (cid:48) (cid:90) ∞−∞ dγ (cid:48) × g (+) ( γ (cid:48) , z (cid:48) , z ; κ , z i )[ D − i(cid:15) ] (10)with D = γ (cid:48) + γ + z m + (1 − z ) κ + z (cid:48) M z z i (11)and z i = ± M (cid:112) − κ . Remarkably, ψ dist (cid:0) z, γ ; κ , z i (cid:1) displays a cut, originatedby the free propagation of the two constituents, just asin the non relativistic case. In particular, the distortedpart of the scattering wave function can be rearrangedin order to make explicit the free propagation, obtain-ing (see details in Appendix B) ψ dist (cid:0) z, γ ; κ , z i (cid:1) = i (1 − z )4 × κ (1 − z ) + m z + γ − i(cid:15) ] (cid:90) − dζ (cid:48)(cid:48) (cid:90) − dζ (cid:48) × (cid:90) ∞−∞ dγ (cid:48)(cid:48) (cid:101) G + ( γ (cid:48)(cid:48) , ζ (cid:48)(cid:48) , ζ (cid:48) ; κ , z i ) θ (1 − | ζ (cid:48)(cid:48) | − | ζ (cid:48) | ) × (cid:20) (1 + z )(1 + ζ (cid:48) − ζ (cid:48)(cid:48) z i ) θ ( ζ (cid:48) − z − ζ (cid:48)(cid:48) z i ) D ( z, ζ (cid:48) , ζ (cid:48)(cid:48) ) − i(cid:15) ++ (1 − z )(1 − ζ (cid:48) + ζ (cid:48)(cid:48) z i ) θ ( z + ζ (cid:48)(cid:48) z i − ζ (cid:48) ) D ( − z, − ζ (cid:48) , − ζ (cid:48)(cid:48) ) − i(cid:15) (cid:21) , (12) where D ( z, ζ (cid:48) , ζ (cid:48)(cid:48) ) = κ (1 − z ) + m z + γ + (1 + z )(1 + ζ (cid:48) − ζ (cid:48)(cid:48) z i ) (cid:18) M zζ (cid:48)(cid:48) z i + γ (cid:48)(cid:48) (cid:19) (13)and (cid:101) G + ( γ (cid:48) , ζ (cid:48)(cid:48) , ζ (cid:48) ; κ , z i ) is the Nakanishi weight func-tion for the half-off-shell T-matrix (see Ref. [9]). In par-ticular, the relation between the two Nakanishi weightfunctions is given by g + ( γ (cid:48) , z (cid:48) , z ; κ , z i ) == i (cid:90) dαα (cid:90) − dζ (cid:48) (cid:101) G + ( γ (cid:48) α , z (cid:48) α , ζ (cid:48) a ; κ , z i ) × θ ( α − | z (cid:48) | − | ζ (cid:48) | ) θ (1 − α − | ζ (cid:48) − z − z (cid:48) z i | ) . (14)with all the constrains on the variables explicitly writ-ten. Indeed, notice that the dependence upon z in theweight function g (+)( Ld ) ( γ (cid:48) , z (cid:48) , z ; κ , z i ) should be read as z + z (cid:48) z i (cf Eq. (66) in [9] and Appendix B of the presentpaper). From Eq. (12), the analogy with the non rela-tivistic case appears evident, once the familiar form ofthe global propagation is recognized. As a matter offact, one has (1 > z )1 γ + z m + (1 − z ) κ − i(cid:15) == (1 − z )4 1 M − M − i(cid:15) , (15)where M is the free mass of the two-body system givenby M = 4 ( m + γ )(1 − z ) . (16)It should be pointed out that the cut in ψ dist is mir-rored in the integral equation determining the Nakan-ishi weight function, in particular in the part governedby the dynamics (see Eq. (18), below). It is useful toanticipate that the cut is canceled by the proper factorin the evaluation of the scattering amplitude.An issue of fundamental relevance related to ψ dist inEq. (10) (or to ψ dist in Eq. (12)) is to determine the sup-port of the Nakanishi weight function g (+) ( γ (cid:48) , z (cid:48) , z ; κ , z i )(or equivalently (cid:101) G + ( γ (cid:48) , z (cid:48) , ζ (cid:48) ; κ , z i )) with respect to thenon compact variable γ (cid:48) . While the variable γ = k ⊥ in ψ dist (cid:0) z, γ ; κ , z i (cid:1) is such that γ ∈ [0 , ∞ ) and thesame holds for γ (cid:48) in the Nakanishi weight function whenthe bound state is discussed (see [10]), in the case ofa scattering state one has a different interval, namely γ (cid:48) ∈ ( −∞ , ∞ ). Then, a question rises about the widthof the support when κ → − , i.e. the zero-energy limitwhich we are interested in. One should expect that therelevant support of γ (cid:48) had to shrink in order to matchthe one pertaining to a bound state. This can be accomplished if lim κ → − g (+) ( γ (cid:48) , z (cid:48) , z ; κ , z i ) = 0for γ (cid:48) <
0. Notably, this is what happens, as shown indetail in the following subsection. It should be pointedout that such a result is relevant for what follows, sincewe are going to consider the limit of a scattering statefor κ → − , and one could be puzzled by the abrupttransition of the lower extremum for γ (cid:48) from an un-bound value, for κ < − (cid:15) , to a bound one, for the zero-energy limit.2.2 The support of the Nakanishi weight function forthe inhomogeneous BSEIn order to address the support issue above introduced,let us consider the first meaningful approximation toEq. (4), namely the approximation where the kernel i K is substituted by its ladder contribution, given by i K ( Ld ) ( k, k i , p ) = i ( − ig ) ( k − k (cid:48) ) − µ + i(cid:15) . (17)First, one inserts the Nakanishi Ansatz for the BS am-plitude, Eq. (1), in the ladder BSE. Then, one canperform the integration over k − without any approxi-mation, and obtain the ladder inhomogeneous BSE pro-jected onto the null-plane, i.e. an integral equation thatrelates ψ dist given by Eq. (10), to the dynamics dictatedby the ladder kernel (see details in Ref. [9]). Namely, onegets (cid:90) ∞−∞ dγ (cid:48) (cid:90) − dz (cid:48) g (+)( Ld ) ( γ (cid:48) , z (cid:48) , z ; κ , z i )[ D − i(cid:15) ] == g [ γ + z m + (1 − z ) κ − i(cid:15) ] (cid:104) W ( Ld ) ( γ, z ; κ , z i )+ 12(4 π ) (cid:90) ∞−∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) g (+)( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ , z i ) (cid:90) ∞ dy F ( y, γ, z ; γ (cid:48) , ζ, ζ (cid:48) ) (cid:21) , (18)where W ( Ld ) is W ( Ld ) ( γ, z ; κ , z i ) = 1( z − z i ) × θ ( z − z i ) M (1 + z ) − M (1 + z i ) + µ + γ ( z − z i ) − i(cid:15) + − θ ( z i − z ) M (1 − z ) − M (1 − z i ) + µ + γ ( z i − z ) − i(cid:15) , (19) and F is F ( y, γ, z ; γ (cid:48) , ζ, ζ (cid:48) ) = (1 + z ) (1 + ζ (cid:48) − z i ζ ) × θ ( ζ (cid:48) − z − z i ζ )[ D ( y, γ, z ; γ (cid:48) , ζ, ζ (cid:48) ; z i ) − i(cid:15) ] + (1 − z ) (1 − ζ (cid:48) + z i ζ ) × θ ( z + z i ζ − ζ (cid:48) )[ D ( y, γ, − z ; γ (cid:48) , ζ, − ζ (cid:48) ; − z i ) − i(cid:15) ] , (20)where D ( y, γ, z ; γ (cid:48) , ζ, ζ (cid:48) ; z i ) = γ + z m + κ (1 − z )+ Γ ( y, z, z i , ζ, ζ (cid:48) , γ (cid:48) ) + Z ( z, ζ, ζ (cid:48) ; z i ) M zz i (21)with Γ ( y, z, z i , ζ, ζ (cid:48) , γ (cid:48) ) = (1 + z )(1 + ζ (cid:48) − z i ζ ) × (cid:26) y A ( ζ, ζ (cid:48) , γ (cid:48) , κ ) + µ y + µ + γ (cid:48) (cid:27) , (22) Z ( z, ζ, ζ (cid:48) ; z i ) = (1 + z )(1 + ζ (cid:48) − z i ζ ) ζ , (23)and A ( ζ, ζ (cid:48) , γ (cid:48) , κ ) = ζ (cid:48) M κ (1 + ζ ) + γ (cid:48) . (24)Because of the presence of the theta functions in Eq.(20), one has1 ≥ | ζ | ≥ | Z ( ± z, ζ, ± ζ (cid:48) , z i ) | . (25)It should be pointed out that Eq. (18) is relevant for thecalculation of the phase shifts and, in the zero-energylimit, of the scattering lengths (cf Sec 3).After combining the global propagation with the de-nominator in W ( Ld ) ( γ, z ; κ , z i ) and repeating the samestep for F (see Ref. [9]) one can apply the Nakanishitheorem on the uniqueness of the weight-function foran n -leg transition amplitude [12]. It should be recalledthat the uniqueness theorem has been proven within aperturbative framework, while in the present context,a non perturbative one, the uniqueness is conjecturedand numerically checked. Eventually, one gets a new in-tegral equation for the Nakanishi weight function, that allows us to discuss the support issue, viz [9] g (+)( Ld ) ( γ, z (cid:48) , z ; κ , z i ) = g θ ( − z (cid:48) ) δ ( γ − γ a ( z (cid:48) )) × (cid:110) θ ( z − z i ) θ [1 − z + z (cid:48) (1 − z i )]+ θ ( z i − z ) θ [1 + z + z (cid:48) (1 + z i )] (cid:111) + − g π ) (cid:90) ∞−∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) g (+)( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ , z i ) × (cid:20) (1 + z ) θ ( ζ (cid:48) − z − z i ζ )(1 + ζ (cid:48) − z i ζ ) h (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ )+ (1 − z ) θ ( z − ζ (cid:48) + z i ζ )(1 − ζ (cid:48) + z i ζ ) h (cid:48) ( γ, z (cid:48) , − z, − z i ; γ (cid:48) , ζ, − ζ (cid:48) , µ ) (cid:21) (26)where γ a ( z (cid:48) ) = z (cid:48) (2 κ − µ ) ≥ , (27)and h (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) is given by h (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) = (1 + z )(1 + ζ (cid:48) − z i ζ ) × (cid:110) ∂∂λ (cid:90) ∞ dy (cid:90) dξ δ [ z (cid:48) − ξZ ( z, ζ, ζ (cid:48) ; z i )] × δ (cid:104) F ( λ, y, ξ ; γ (cid:48)(cid:48) , z, ζ, ζ (cid:48) , γ (cid:48) ; z i , κ , µ ) (cid:105)(cid:111) λ =0 . (28)with F ( λ, y, ξ ; γ (cid:48)(cid:48) , z, ζ, ζ (cid:48) , γ (cid:48) ; z i , κ , µ ) == γ (cid:48)(cid:48) − ξ (1 + z )(1 + ζ (cid:48) − z i ζ ) × (cid:18) y A ( ζ, ζ (cid:48) , γ (cid:48) , κ ) + y ( µ + γ (cid:48) ) + µ y (cid:19) − ξλ (29)Notice that the inhomogeneous term vanishes both at z = ± z (cid:48) = −
1, as expected (cf Eq. (7)). Indeed,for z = 1, one has θ ( − z (cid:48) ) (cid:110) θ (1 − z i ) θ [ z (cid:48) (1 − z i )]+ θ ( z i − θ [2 + z (cid:48) (1 + z i )] (cid:111) , (30)that vanishes. For z = −
1, one gets θ ( − z (cid:48) ) (cid:110) θ ( − − z i ) θ [ z (cid:48) (1 − z i )]+ θ ( z i + 1) θ [ z (cid:48) (1 + z i )] (cid:111) , (31)and again the theta functions produce a vanishing out-come. Finally, if z (cid:48) = −
1, the inhomogeneous term isvanishing, since θ ( z − z i ) θ [1 − z − (1 − z i )]+ θ ( z i − z ) θ [1 + z − (1 + z i )] = 0 . (32)The integral equation based on the uniqueness the-orem (that has been numerically verified for the boundstates case in Ref. [10], and for the zero-energy limit in the present work, cf Sec. 4) leads to understand indetail the sharp transition of the support in γ .For κ < −∞ , ∞ ), and one cansplit the integral equation (26) in two coupled integralequations: one is inhomogeneous, while the other is ho-mogeneous. To show this, let us introduce the followingdecomposition of the weight function g (+)( Ld ) ( γ, z (cid:48) , z ; κ , z i ) g (+)( Ld ) ( γ, z (cid:48) , z ; κ , z i ) = θ ( γ ) g (+) p ;( Ld ) ( γ, z (cid:48) , z ; κ , z i )+ θ ( − γ ) g (+) n ;( Ld ) ( γ, z (cid:48) , z ; κ , z i ) . (33)Inserting such a decomposition in Eq. (26) one gets g (+) p ;( Ld ) ( γ, z (cid:48) , z ; κ , z i ) = g θ ( − z (cid:48) ) δ ( γ − γ a ( z (cid:48) )) × (cid:110) θ ( z − z i ) θ [1 − z + z (cid:48) (1 − z i )]+ θ ( z i − z ) θ [1 + z + z (cid:48) (1 + z i )] (cid:111) + − g π ) θ ( γ ) (cid:104)(cid:90) ∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) × H (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) g (+) p ;( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ , z i )+ (cid:90) −∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) × H (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) g (+) n ;( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ , z i ) (cid:105) , (34)and g (+) n ;( Ld ) ( γ, z (cid:48) , z ; κ , z i ) = − g π ) θ ( − γ ) × (cid:104)(cid:90) ∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) × H (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) g (+) p ;( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ , z i )+ (cid:90) −∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) × H (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) g (+) n ;( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ , z i ) (cid:105) , (35)with H (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) == (cid:104) (1 + z ) θ ( ζ (cid:48) − z − z i ζ )(1 + ζ (cid:48) − z i ζ ) h (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ )+ (1 − z ) θ ( z − ζ (cid:48) + z i ζ )(1 − ζ (cid:48) + z i ζ ) × h (cid:48) ( γ, z (cid:48) , − z, − z i ; γ (cid:48) , ζ, − ζ (cid:48) , µ ) (cid:105) . (36)If κ → − , the off-shell kernel in the homogeneous in-tegral equation, namely the one with γ < γ (cid:48) > pled equations. As a matter of fact, one has for κ → − θ ( − γ ) θ ( γ (cid:48) ) H (cid:48) ( γ, z (cid:48) , z, z i = 0; γ (cid:48) , ζ, ζ (cid:48) , µ ) == θ ( − γ ) θ ( γ (cid:48) ) (1 + z )(1 + ζ (cid:48) ) (cid:110) ∂∂λ (cid:90) ∞ dy × (cid:90) dξ δ (cid:20) z (cid:48) − ξζ (1 + z )(1 + ζ (cid:48) ) (cid:21) × δ (cid:2) F ( λ, y, ξ ; γ (cid:48)(cid:48) , z, ζ, ζ (cid:48) , γ (cid:48) ; z i , κ = 0 , µ ) (cid:3)(cid:111) λ =0 == 0 , (37)since the delta function is always vanishing, given γ < ξ (1 + z )(1 + ζ (cid:48) ) (cid:18) y A ( ζ, ζ (cid:48) , γ (cid:48) , κ = 0) + y ( µ + γ (cid:48) ) + µ y (cid:19) + ξ λ == ξ (1 + z )(1 + ζ (cid:48) ) y (cid:16) ζ (cid:48) m + γ (cid:48) (cid:17) + y ( µ + γ (cid:48) ) + µ y + ξ λ > . Then, for κ → − , Eq. (35) becomes g (+) n ;( Ld ) ( γ, z (cid:48) , z ; κ = z i = 0) = − g π ) θ ( − γ ) (cid:90) −∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) H (cid:48) ( γ, z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) × g (+) n ;( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ = z i = 0) . (38)The above homogeneous integral equation, valid in thezero-energy limit, is expected to have as a solution only g (+) n ;( Ld ) ( γ, z (cid:48) , z ; κ = z i = 0) = 0, given the freedomin choosing g for scattering states. Let us recall thatfor the bound state case, where κ ≥ γ >
0, onegets a homogeneous integral equation and deals with aneigenvalue problem. In particular, one finds a discretespectrum for g , once a value is assigned to κ and µ (see, e.g., Ref. [10], where the bound state case isdiscussed, within the present approach).It is also instructive to trace the behavior of theprevious coupling term when κ approaches 0 − . If κ is different from zero, than the delta function in Eq.(37) can give a finite contribution, since its argumentcan vanish. To achieve such a possibility, one must have(remind that γ (cid:48) > A ( ζ, ζ (cid:48) , γ (cid:48) , κ ) = M ζ (cid:48) + κ (1 + ζ ) + γ (cid:48) < , (39)since the other terms, µ + γ (cid:48) and λ , always yield apositive contribution ( λ approaches zero from positivevalues). The above constraint leads to a volume of theintegration in the space { γ (cid:48) , ζ (cid:48) , ζ } (it is a hyperboloid),that shrinks to zero for κ → − , viz M ζ (cid:48) + κ ζ + γ (cid:48) < − κ In conclusion, for scattering states in the limit κ → − , the corresponding Nakanishi weight function re-duces to the component g (+) p ;( Ld ) ( γ, z (cid:48) , z ; κ = z i = 0)and fulfills the following inhomogeneous integral equa-tion g (+) p ;( Ld ) ( γ, z (cid:48) , z ; κ = z i = 0) = g θ ( − z (cid:48) ) δ ( γ − γ a ( z (cid:48) )) × (cid:110) θ ( z ) θ [1 − z + z (cid:48) ] + θ ( − z ) θ [1 + z + z (cid:48) ] (cid:111) − g π ) θ ( γ ) (cid:90) ∞ dγ (cid:48) (cid:90) − dζ × (cid:90) − dζ (cid:48) H (cid:48) ( γ, z (cid:48) , z, z i = 0; γ (cid:48) , ζ, ζ (cid:48) , µ ) × g (+) p ;( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ = z i = 0) (40)This is sharply different from the general case κ < In the CM frame, the differential cross section for theelastic scattering of two scalars can be written as follows[19] dσdΩ = 164 π s | T elii | = | f el ( s, θ ) | , (41)with T elii the invariant matrix element of the T-matrix,that is dimensionless (recall that in a φ theory thecoupling constant g has the dimension of a mass), and f el ( s, θ ) the elastic scattering amplitude. It turns outthat f el ( s, θ ) = − π √ s T elii , (42)where s = M and cosθ = ˆ k f · ˆ k i . To introduce the rela-tion with the phase shifts δ (cid:96) , let us expand the scatter-ing amplitude on the basis of the Legendre polynomials, P (cid:96) ( cosθ ), as follows f el ( s, θ ) = 1 k r (cid:88) (cid:96) (2 (cid:96) + 1) f el(cid:96) P (cid:96) ( cosθ ) , (43)where the relative three-momentum is k r = (cid:112) s/ − m ,or k r = − κ , and the projected amplitudes are givenby f el(cid:96) = e iδ (cid:96) sinδ (cid:96) . (44)Finally, in the zero-energy limit, only the amplitudewith (cid:96) = 0 survives and one obtains f el (cid:39) δ (cid:39) − a k r , (45)where a is the s-wave scattering length . Therefore, inthe zero-energy limit one getslim s → m f el ( s, θ ) = − a (46) On the other hand, the scattering amplitude can becalculated through the BS amplitude as follows (seeRef. [9] for details) f el ( s, θ ) = − i √ s π lim k (cid:48) → k f (cid:104) k (cid:48) , p | G − ( p ) | Φ (+) ; p, k i (cid:105) == 1 √ s π lim ( γ,z ) → ( γ f ,z f ) (cid:2) γ + z m + (1 − z ) κ − i(cid:15) (cid:3) × − z ) ψ dist ( z, γ ; κ , z i ) , (47)where k (cid:48) = ( p (cid:48) − p (cid:48) ) / p (cid:48) + p (cid:48) = p (recall that p = M = s ).In ladder approximation and choosing γ i = 0, fromEqs. (10) and (18) one gets f el ( Ld ) ( s, θ ) = 2 α m √ s (cid:104) W ( Ld ) ( γ f , z f ; κ , z i )+ 12(4 π ) (cid:90) ∞−∞ dγ (cid:48)(cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) × (cid:90) ∞ dy F ( y, γ f , z f ; γ (cid:48)(cid:48) , ζ, ζ (cid:48) ) g (+)( Ld ) ( γ (cid:48)(cid:48) , ζ, ζ (cid:48) ; κ , z i ) (cid:105) , (48)where α = g m π . If z → ± M → ∞ , and one can seethat W ( Ld ) ( γ, z = ± κ , z i ) →
0, by taking into ac-count also the constraints generated by the theta func-tions. In general, the denominators in Eq. (19) do notvanish, since (i) only minus components of on-mass-shell particles are present there, and (ii) the momentumconservation law does not hold for those components(one can also explicitly check that the denominatorsdo not have real roots). Moreover, since γ i = 0 then M = m /z i = ( m + γ f ) /z f . It is useful to introducesome kinematical relations relevant for describing thescattering process. In particular, the initial and finalCartesian three-momenta, k i and k f , has to be com-pleted giving the third components, viz k iz = 12 (cid:0) k + i − k − i (cid:1) = − z i M ,k fz = 12 (cid:16) k + f − k − f (cid:17) = − z f M , (49)Then, one can write down the relation between the scat-tering angle θ and the LF variables z f and z i , given by k i · k f = z i z f M − κ cosθ , (50)where p · k i = p · k f = 0 has been used (those constraintsare imposed by the on-mass-shellness of the particles inthe elastic channel). It also follows that | k i | = | k f | = − κ . Finally, by exploiting the relation κ = − z i M /
4, thatholds for γ i = 0, one gets z f = z i cosθ , (51)For κ = ( m − s/ → − , both z i and z f vanish(as well as γ f = | k f ⊥ | ), and one loses the depen-dence upon the scattering angle θ in the scatteringamplitude, namely one has a s-wave scattering, as itmust be. The two functions, W ( Ld ) ( γ f , z f ; κ , z i ) and F ( y, γ f , z f ; γ (cid:48)(cid:48) , ζ, ζ (cid:48) ), becomelim κ → − W ( Ld ) ( γ f , z f = z i cosθ ; κ , z i ) == lim κ → − z f − z i ) (cid:110) θ ( z f − z i ) M ( z f − z i ) + µ + γ f ( z f − z i ) − i(cid:15) + θ ( z i − z f ) − M ( z i − z f ) − µ + γ f ( z i − z f ) + i(cid:15) (cid:111) = 1 µ , (52)andlim κ → − F ( y, γ f , z f = z i cosθ ; γ (cid:48)(cid:48) , ζ, ζ (cid:48) ) == y (cid:2) y ( m ζ (cid:48) + γ (cid:48)(cid:48) ) + y ( µ + γ (cid:48)(cid:48) ) + µ − i(cid:15) (cid:3) , (53)Then, in the zero-energy limit Eq. (48) reduces to (seealso Appendix C)lim s → m f el Ld ( s, θ ) = − a == m α (cid:26) µ + 12(4 π ) (cid:90) ∞ dγ (cid:48)(cid:48) (cid:90) − dζ (cid:48) g (+)0 Ld ( γ (cid:48)(cid:48) , ζ (cid:48) ) × (cid:90) ∞ dy y [ y A ( ζ (cid:48) , γ (cid:48)(cid:48) ) + y ( µ + γ (cid:48)(cid:48) ) + µ − i(cid:15) ] (cid:41) , (54)where the first term in the curly brackets leads to thescattering length in Born approximation, viz a BA = − m αµ (55)Moreover, g (+)0 Ld ( γ (cid:48)(cid:48) , ζ (cid:48) ) is the Nakanishi weight functionin the zero-energy limit. It can be obtained by solvingtwo different integral equations as discussed in detailin Appendix C, where the whole matter is presentedin a substantially simpler way than the one in Ref. [9](notice that a mistyping in Eq. (103) of [9] has beenfixed). In particular, the integral equation that links ψ dist to the dynamics governed by the BS kernel in ladder approximation is given by (cid:90) ∞ dγ (cid:48)(cid:48) g (+)0 Ld ( γ (cid:48)(cid:48) , z )[ γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] == g µ (cid:90) ∞−∞ dγ (cid:48)(cid:48) θ ( γ (cid:48)(cid:48) )[ γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] × (cid:110) θ ( z ) θ (cid:2) − z − γ (cid:48)(cid:48) /µ (cid:3) + θ ( − z ) θ (cid:2) z − γ (cid:48)(cid:48) /µ (cid:3)(cid:111) − g π ) (cid:90) ∞ dγ (cid:48) (cid:90) − dζ (cid:48) g (+)0 Ld ( γ (cid:48) , ζ (cid:48) ) (cid:90) ∞−∞ dγ (cid:48)(cid:48) × γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] θ ( γ (cid:48)(cid:48) ) × (cid:104) (1 + z )(1 + ζ (cid:48) ) θ ( ζ (cid:48) − z ) h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ )+ (1 − z )(1 − ζ (cid:48) ) θ ( z − ζ (cid:48) ) h (cid:48) ( γ (cid:48)(cid:48) , − z ; γ (cid:48) , − ζ (cid:48) , µ ) (cid:105) (56)Notably, h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ) is the proper kernel for abound state with vanishing energy, as one can check inRef. [10].The expression of h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ) is given by (seedetails in Appendix C) h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ) == θ (cid:20) − B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) − µ (cid:113) ζ (cid:48) m + γ (cid:48) (cid:21) × (cid:104) − B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) A ( ζ (cid:48) , γ (cid:48) ) ∆ ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) 1 γ (cid:48)(cid:48) + (1 + ζ (cid:48) )(1 + z ) (cid:90) y + y − dy y [ y A ( ζ (cid:48) , γ (cid:48) ) + y ( µ + γ (cid:48) ) + µ ] (cid:105) − (1 + ζ (cid:48) )(1 + z ) (cid:90) ∞ dy y [ y A ( ζ (cid:48) , γ (cid:48) ) + y ( µ + γ (cid:48) ) + µ ] , (57)with A ( ζ (cid:48) , γ (cid:48) ) = ζ (cid:48) m + γ (cid:48) = ζ (cid:48) m + γ (cid:48) > , B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) = µ + γ (cid:48) − γ (cid:48)(cid:48) (1 + ζ (cid:48) )(1 + z ) ≤ ,∆ ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) = B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) − µ A ( ζ (cid:48) , γ (cid:48) ) ≥ ,y ± = 12 A ( ζ (cid:48) , γ (cid:48) ) × (cid:2) −B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) ± ∆ ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) (cid:3) . (58)The zero-energy limit of Eq. (26), i.e. the integralequation based on the uniqueness of the Nakanishi weight function, reads (cf Ref. [9,10]) is g (+)0 Ld ( γ, z ) = g µ θ ( γ ) θ (cid:2) µ (1 − | z | ) − γ (cid:3) − g π ) θ ( γ ) (cid:90) − dζ (cid:48) (cid:90) ∞ dγ (cid:48) g (+)0 Ld ( γ (cid:48) , ζ (cid:48) ) × (cid:104) (1 + z )(1 + ζ (cid:48) ) θ ( ζ (cid:48) − z ) h (cid:48) ( γ, z ; γ (cid:48) , ζ (cid:48) , µ )+ (1 − z )(1 − ζ (cid:48) ) θ ( z − ζ (cid:48) ) h (cid:48) ( γ, − z ; γ (cid:48) , − ζ (cid:48) , µ ) (cid:105) . (59)It should be pointed out that the presence of a nonsmooth behavior, like the discontinuity around γ ∼ µ (1 − | z | ), is expected if one has to reproduce the sin-gular behavior of the distorted part of the scatteringwave function (cf Eqs. (10) and (12)).As illustrated in the next Sec. 4, we have taken profitof the general structure of the weight function suggestedby Eq. (59) for obtaining numerical solutions of bothEq. (56) and Eq. (59), and eventually calculating thescattering lengths.It is worth noting that the scattering length givenby Eq. (54) represents a normalization for g (+)0 Ld ( γ (cid:48)(cid:48) , ζ (cid:48) ) ,when µ ≤ m . As a matter of fact, from Eq. (59), onerealizes that the inhomogeneous term is different fromzero only for 0 ≤ γ ≤ µ (1 − | z | ) . Moreover, within the previous interval and µ ≤ m , thecontribution to the kernel h (cid:48) that contains θ (cid:20) γ (1 ± ζ (cid:48) )(1 ± z ) − γ (cid:48) − µ − µ (cid:113) ζ (cid:48) m + γ (cid:48) (cid:21) disappears, since γ (1 ± ζ (cid:48) )(1 ± z ) − γ (cid:48) − µ − µ (cid:113) ζ (cid:48) m + γ (cid:48) ≤≤ µ (1 − | z | )(1 ± z ) (1 ± ζ (cid:48) ) − µ − µm | ζ (cid:48) | << µ (1 ± ζ (cid:48) ) − µ − µm | ζ (cid:48) | < µ | ζ (cid:48) | ( ± µ − m ) , (60)The final step in the above expression is always negativewhen µ < m .Therefore, for 0 ≤ γ ≤ µ (1 − | z | ) and µ ≤ m , onehas g (+)0 Ld ( γ, z ) = g µ + g π ) (cid:90) − dζ (cid:48) (cid:90) ∞ dγ (cid:48) (cid:90) ∞ dy y g (+)0 Ld ( γ (cid:48) , ζ (cid:48) )[ y A ( ζ (cid:48) , γ (cid:48) ) + y ( µ + γ (cid:48) ) + µ ] == − πm a , (61)where Eq. (54) has been exploited in the last step. In this Section, the numerical studies of both the scat-tering length and the distorted part of the 3D wavefunction are presented. First of all, let us illustrate ournumerical method for solving the two integral equa-tions in (56) and (59). The main ingredient is the fol-lowing decomposition of the Nakanishi weight functionthat takes into account the singular behavior shown inEq. (59), but also the result in Eq. (61), that holds for µ ≤ m (this is always fulfilled for realistic cases) g (+)0 Ld ( γ, z ) = β θ ( − t ) + θ ( t ) N z (cid:88) (cid:96) =0 N g (cid:88) j =0 A (cid:96)j G (cid:96) ( z ) L j ( t ) , (62)where (i) t = γ − µ (1 − | z | ), (ii) the functions G (cid:96) ( z )are given in terms of even Gegenbauer polynomials, C (5 / (cid:96) ( z ) (recall that g (+)0 Ld ( γ, z ) must be even in z ) by G (cid:96) ( z ) = 4 (1 − z ) Γ (5 / × (cid:115) (2 (cid:96) + 5 /
2) (2 (cid:96) )! πΓ (2 (cid:96) + 5) C (5 / (cid:96) ( z ) , (63)and (iii) the functions L j ( t ) are expressed in terms ofthe Laguerre polynomials, L j ( bt ), by L j ( t ) = √ b L j ( bt ) e − bt/ . (64)The following orthonormality conditions are fulfilled (cid:90) − dz G (cid:96) ( z ) G n ( z ) = δ (cid:96)n , (cid:90) ∞ dt L j ( t ) L (cid:96) ( t ) == b (cid:90) ∞ dt e − bt L j ( bt ) L (cid:96) ( bt ) = δ j(cid:96) . (65)In Appendix D, some details are given for illustrat-ing how Eq. (59) can be numerically solved by usingthe previous decomposition. In order to speed up theconvergence, in the actual calculations the parameter b = 15 . /m has been adopted. The finite-range inte-grations (as those with respect to the variable z and thevariable γ when integrated up to µ (1 − | z | )) have beenperformed using a Gauss-Legendre quadrature rule. Theinfinite-range integrations, on the other hand, have beenperformed adopting a Gauss-Laguerre quadrature method.Finally, the convergence of the expansion given in Eq.(62)is very rapid, and adopting the values N z = 10 and N g = 24 well converged values have been obtained. Allthe results presented in this Section have been obtainedfor such a choice. Notice that at the end of the calcula-tion β resulted to be in agreement with the normaliza-tion − πa shown in Eq. (61). In Tables 1, 2 and 3, the scattering lengths, Eq.(54), calculated by using the Nakanishi weight functionobtained by solving both the integral equation (56), a F V S , and the integral equation (59), a UNI , are shownfor µ/m = 0 . , . , . α = g / (16 πm ), that range from a weak-interaction regime to a strong one. Moreover, for thesake of comparison, the results of Ref. [13], a CK , eval-uated within a totally different framework, based on adirect calculations of the half-off-shell scattering ampli-tude taking explicitly into account contributions fromthe singularities affecting the amplitude itself, are pre-sented in the second column. For reference, also theBorn values of the scattering lengths are given in thefifth column. From the Tables, one can observe a verygood agreement among all the three sets of numericalresults, but some comments are in order: (i) the com-parison between a F SV and a UNI clearly confirms thatthe uniqueness of the Nakanishi weight function can beassumed with a very high degree of confidence, as wehave quantitatively shown also for the bound-state case[10,11]; (ii) differences between a CK [13] and our calcu-lations are present for µ = 0 . m when the value of α approaches a value which corresponds to a bound stateof zero-energy. In such a case, the scattering length di-verges (let us recall that, for the bound-state case, α is obtained as an eigenvalue of the homogeneous inte-gral equation, in ladder approximation), or there is achange of sign. Indeed, the above mentioned numericaldifferences do not represent a big issue (nonetheless itwill numerically investigated elsewhere), given the com-pletely different theoretical frameworks adopted in Ref.[13] and in our work, and the well-known resonance be-havior of the scattering length, when a bound state isapproaching a zero-energy scattering state. Finally, itis worth noting that the Born approximation a BA rep-resents a quite good approximation only for small α (see also the following Fig. 1). Summarizing, the re-sults shown in Tables 1, 2 and 3, together with thecalculations for the bound states [4,10,11], are a verystrong evidence that the Nakanishi Ansatz, like the onefor scattering states in Eq. (1), represents a reliabletool for solving both homogeneous and inhomogeneousBSE’s in Minkowski space.In Fig. 1, the scattering lengths for the above threevalues of µ/m are presented as a function of the ab-solute value of the scattering length in Born approx-imation | a BA | (see Eq. (55) and the last columns inTables 1, 2 and 3). Interestingly, in the same figure, itis also shown the comparison with the correspondingnon relativistic scattering lengths, evaluated through awell-known expression (see e.g. [24,25]), that exactly Table 1
Comparison, for µ/m = 0 .
15, between the scatter-ing lengths (see Eq. (54)) evaluated in Ref. [13], a CK (sec-ond column), and the ones, a F SV (third column) and a UNI (fourth column), calculated by adopting the Nakanishi weightfunction obtained from Eqs. (56) and (59), respectively. Allthe calculations are in ladder approximation. The first col-umn contains the coupling constant α = g / (16 πm ). Fi-nally, the fifth column shows the scattering length in Bornapproximation, Eq. (55). The scattering lengths are in unit1 /m .( ∗ Private communication by J. Carbonell) α a CK [13] a F SV a UNI a BA ∗ Table 2
The same as in Tab. 1, but for the mass of theexchanged scalar µ/m = 0 . α a CK [13] a F SV a UNI a BA reproduce the second Born approximation, viz m a = m a BA µ m ( ma BA ) . (66)The chosen range of | a BA | is [0 , . m (the mass of the interacting scalars). Be-yond this interval, the scattering lengths can change thesign, as illustrated by the above Tables. Moreover, since m a BA = − α m /µ , after fixing the value of µ/m onecan follows the dependence of the scattering length on Table 3
The same as in Tab. 1, but for the mass of theexchanged scalar µ/m = 1 . α a CK [13] a F SV a UNI a BA the Yukawa coupling constant g . In particular, fromFig. 1, one can see that for increasing values of g and µ/m the relativistic treatment in Minkowski space, be-comes more and more important, as expected, since theeffect of the attractive interaction becomes more andmore large. Notice that the scalar exchange in Eq. (17)leads to a non relativistic attractive Yukawa potential.Summarizing, modulo the adopted ladder approxima-tion, the comparison suggests that some care should betaken when one has to consider the effect of the interac-tion in the description of both hadronic scattering pro-cesses, even in the low-energy regime, and final states,that, e.g., are relevant for hadronic decays.In Fig. 2, the Nakanishi weight functions for (i) µ/m = 0 . , . , .
0, (ii) α = 0 . , .
5, and (iii) z = 0,but running γ/m , are shown. It should be pointed outthat, for each value of µ/m , the two values of the cou-pling constant α are representatives of a weak-interactionregime and a strong one. Moreover, since the Nakan-ishi weight functions obtained from Eq. (56) and Eq.(59) substantially coincide, only the calculations corre-sponding to Eq. (56) are shown. As mentioned at thebeginning of this Section, the step-function behavior forsmall γ has to be present, and the discontinuities areneeded for obtaining the expected singularities in ψ dist ,like the one due to the global propagation. In Fig.2,the transition from the weak regime to the strong oneincreases the discontinuous behavior, that for large µ become more and more smooth. Finally, recalling thatfor a bound state and µ →
0, i.e. the Wick-Cutkoskymodel [20,21], the Nakanishi weight function becomesproportional to δ ( γ ), it is instructive to see the onset ofsuch a behavior in the upper part of Fig.2. | a BA | [1/m] -4-3-2-10 a [ / m ] Fig. 1
The scattering lengths, calculated by using Eq. (54),i.e. corresponding to solutions of the inhomogeneous BSEat zero-energy, vs | a BA | (Eq. (55)). Thick-solid line: a for µ/m = 1. Thick-dashed line: a for µ/m = 0 .
5. Thick-dottedline: a for µ/m = 0 .
15. The non relativistic scattering lengths,represented by the corresponding thin lines, have been cal-culated by using Eq. (66). Notice that for µ/m = 0 .
15 thenon relativistic calculation largely overlaps the relativistic one(thick-dotted line), and therefore it is indistinguishable.
For γ = 0 and | z | (cid:54) = 1, only the first part of thedecomposition in Eq. (62), i.e. g (+)0 Ld ( γ, z ) ∼ β θ ( µ (1 −| z | )), is dominant, and therefore trivial.In Fig. 3, the same quantities as in Fig. 2, butfor γ/m = 0 . z , are also shown. As il-lustrated by the figure, the Nakanishi weight functionacquaints a quite discontinuos behaviour, as µ/m in-creases.Indeed, it is more profitable to present LF distri-butions, obtained from the distorted part of the zero-energy 3D scattering wave function. In analogy withthe bound-state case (see Refs. [10,11]), one can con-struct transverse and longitudinal LF momentum dis-tributions. In particular, one gets the following expres-sion for ψ ( Ld ) dist (cid:0) z, γ ; κ = z i = 0 (cid:1) ψ ( Ld ) dist (cid:0) z, γ ; κ = z i = 0 (cid:1) == (1 − z )4 (cid:90) ∞ dγ (cid:48) g (+)0 Ld ( γ (cid:48) , z )[ γ (cid:48) + γ + z m ] . (67)It should be noticed that inserting in Eq. (67) only thefirst part of the decomposition (62), one quickly reob-tains the singular behavior due to the global propaga- tion as shown in Eq. (12), viz ψ ( Ld ) dist (cid:0) z, γ ; κ = z i = 0 (cid:1) ∼ β (1 − z )4 × (cid:20) γ + z m − µ (1 − | z | ) + γ + z m (cid:21) = β × (1 − z )4 (cid:20) µ (1 − | z | )( γ + z m ) [ µ (1 − | z | ) + γ + z m ] (cid:21) . (68)Therefore, one has to expect singularities in the LFmomentum distributions, that we would introduce inanalogy with the ones for the bound states [10]. Let usemphasize, that only for the bound states they have aprobabilistic interpretation. One could defines (i) thedistorted transverse LF distribution P dist ( γ ) = 12(2 π ) (cid:90) dξ ξ (1 − ξ ) × (cid:90) π dφ [ ψ ( Ld ) dist (cid:0) z, γ ; κ = z i = 0 (cid:1) ] = 1(16 π ) × (cid:90) − dz (1 − z ) (cid:34)(cid:90) ∞ dγ (cid:48) g (+)0 Ld ( γ (cid:48) , z )[ γ (cid:48) + γ + z m ] (cid:35) , (69)and (ii) the longitudinal one, viz φ dist ( ξ ) = 1(2 π ) ξ (1 − ξ ) × (cid:90) d k ⊥ [ ψ ( Ld ) dist (cid:0) z, γ ; κ = z i = 0 (cid:1) ] = 2 (1 − z )(16 π ) × (cid:90) ∞ dγ (cid:34)(cid:90) ∞ dγ (cid:48) g (+)0 Ld ( γ (cid:48) , z )[ γ (cid:48) + γ + z m ] (cid:35) , (70)with the fraction of longitudinal momentum given by ξ = 1 − z p + (cid:18) p + k + (cid:19) . (71)For the sake of presentation, it is useful to partiallyremoving the singularities affecting the above distribu-tions. Therefore, in Fig. 4, γ P ( γ ) and | − ξ | / φ ( ξ )are shown. Figure 4 illustrates the overall behavior ofthe LF distributions by varying the coupling α and themass of the exchanged scalar µ , as in Figs. 2 and 3.It is worth noting the order-of-magnitude differences,when the coupling α is changed, but the typical fea-tures that one expects are still recognizable. A part thedivergent behavior, already pointed out, that can be as-cribed to the global propagation, the transverse distri-bution shows the ultraviolet tail produced by the dom-inance of a single exchanged scalar, exactly as in thecase of the corresponding distribution for bound states(see Ref. [11]). As to the longitudinal distributions, theexpected peak at ξ = 1 / z = 0 is also seen. γ /m g ( + ) L d ( γ , z = ) / (- π m a ) µ /m= 0.15 γ /m -15-10-5051015 g ( + ) L d ( γ , z = ) / (- π m a ) µ /m= 0.15 γ /m g ( + ) L d ( γ , z = ) / (- π m a ) µ /m= 0.50 γ /m -0.500.511.522.533.5 g ( + ) L d ( γ , z = ) / (- π m a ) µ /m= 0.50 γ /m g ( + ) L d ( γ , z = ) / (- π m a ) µ /m= 1.0 γ /m g ( + ) L d ( γ , z = ) / (- π m a ) µ /m= 1.0 Fig. 2
The Nakanishi weight function g (+)0 Ld ( γ, z ), in the zero-energy limit, vs γ/m , for µ/m = 0 . , . , . z = 0,Left panels: weak-interaction regime with a chosen value α = 0 .
1. Right panels: strong-interaction regime with a chosen value α = 2 . The practical use of ψ dist is given by the evaluationof reactions that need a reliable treatment of the rela-tivistic effects, i.e. when the coupling constant becomeslarger and larger. In the present paper, our approach [9,10,11], basedon the Nakanishi integral representation of the Bethe-Salpeter amplitude, is extended for the first time tothe quantitative investigation of the zero-energy limitof the inhomogeneous Bethe-Salpeter Equation, in lad-der approximation, for an interacting system composed -1 -0.5 0 0.5 1 z -0.0100.010.02 g ( + ) L d ( γ = . m , z ) / (- π m a ) µ /m= 0.15 -1 -0.5 0 0.5 1 z g ( + ) L d ( γ = . m , z ) / (- π m a ) µ /m= 0.15 -1 -0.5 0 0.5 1 z -1-0.50 g ( + ) L d ( γ = . m , z ) / (- π m a ) µ /m= 0.50 -1 -0.5 0 0.5 1 z -3.5-3-2.5-2-1.5-1-0.50 g ( + ) L d ( γ = . m , z ) / (- π m a ) µ /m= 0.50 -1 -0.5 0 0.5 1 z -1-0.50 g ( + ) L d ( γ = . m , z ) / (- π m a ) µ /m= 1.0 -1 -0.5 0 0.5 1 z -1-0.50 g ( + ) L d ( γ = . m , z ) / (- π m a ) µ /m= 1.0 Fig. 3
The same as in Fig. 2 but for running z , and γ = 0 . m . by two massive scalars that exchange a massive scalar.This achievement represents a non trivial task, that hasallowed us to gain a sound confidence in the Nakan-ishi Ansatz, as an effective and workable tool for ob-taining actual solutions of the homogeneous and in-homogeneous BSE’s in Minkowski space. Indeed, thesame approach that leads to a careful description of thebound states also yields a very accurate evaluation ofthe scattering length, as shown in Tables 1, 2 and 3 by the quantitative comparisons with the same observableevaluated within a totally different framework, basedon the direct calculation of the contributions from thesingularities of the inhomogeneous BSE [13,14].As in the bound state case, we have performed thecalculations by using the integral representation of theBS amplitude in terms of the Nakanishi weight func-tion, Eq. (1), that explicitly shows the analytic depen-dence of the BS amplitude upon the invariant kinemat- γ / m γ P d i st ( γ ) µ /m= 0.15 ξ | - ξ | / φ d i s t ( ξ ) µ /m=0.15 γ / m γ P d i st ( γ ) µ /m= 0.50 ξ | - ξ | / φ d i s t ( ξ ) µ /m=0.50 γ / m γ P d i st ( γ ) µ /m= 1.0 ξ | - ξ | / φ d i s t ( ξ ) µ /m=1.0 Fig. 4
The LF distributions obtained from the distorted part of the 3D LF scattering wave function (see Eq. (67) for µ/m = 0 . , . , . γ P ( γ ) vs the γ/m (cf the transverseLF-distribution expression in Eq.(69)). Solid line: strong-interaction regime with α = 2 .
5; dotted line: weak-interaction regimewith α = 0 . | − ξ | / φ ( ξ ), (cf the longitudinal LF-distribution in Eq. 70) ical scalars of the scattering process, under scrutiny.Then, by applying the LF projection onto the null-planeto the inhomogeneous BSE in Minkowski space, Eq. (4)(without self-energy and vertex corrections), one is ableto formally obtain the inhomogeneous integral equationfor the Nakanishi weight function, that depends uponreal variables. Its expression in ladder approximation is given by Eq. (18). Eventually, one can deduce an-other inhomogeneous integral equation for the Nakan-ishi weight function, Eq. (26), by assuming to be validthe uniqueness of the Nakanishi weight function, alsoin the non perturbative regime (recall that the theoremwas demonstrated by Nakanishi [12] in a perturbativeframework, but taking into account the whole set of infinite diagrams contributing to a given n -leg ampli-tude).The numerical comparisons for the scattering lengths,obtained by using our Eqs (56) and (59), and the cor-responding quantities calculated in Ref. [13] are shownin great detail in Tables 1, 2 and 3. It has to be em-phasized that the high accuracy reached by our calcu-lations is due to the new decomposition (62), suitablefor obtaining the numerical solutions of the two inho-mogeneous integral equations, involving the Nakanishiweight function. The comparison with the non rela-tivistic scattering lengths (cf Fig. 1) has illustrated thepotential impact of a proper treatment of the relativis-tic effects in the investigation of hadronic scatteringstates, even in the low-energy regime.For the sake of completeness, the behavior of theNakanishi weight functions, in the zero-energy limit, forweak- and strong interaction regimes have been shownin Figs. 2 and 3. Those figures illustrate the non smoothbehavior of the Nakanishi weight functions for certainranges of the variables, that is an inheritance of the sin-gular behavior of the scattering states. Finally, we havedefined LF momentum distributions, longitudinal andtransverse ones, in analogy with the bound state case,(but without the probabilistic interpretation, entailedfrom the normalization of a bound state). Those distri-butions are shown in Fig. 4, for the sake of illustrationand reference purpose. It should be noticed that thetransverse LF distributions show the expected ultravi-olet behavior, i.e. a power-like one, already found in thebound state case.In conclusion, the Nakanishi Ansatz for the BS am-plitude allows one to numerically solve in a very ac-curate way the inhomogeneous BSE, at least for thezero-energy limit. Such an outcome of our approach,together with the very nice results obtained for thebound-state case, strongly encourages to move to positive-energy scattering states, in order to evaluate the phase-shifts. If the phase-shifts evaluated within our approach(presented elsewhere [18]) will agree with the ones inliterature[13], then the reliability of the Nakanishi Ansatzas a starting guess for obtaining exact solutions of BSE’sin Minkowski space could make a substantial step for-wards, confirming the great potentiality of this method,that can be applied to many other cases, changing di-mensions [26], statistics, kernels, etc. Acknowledgements
We gratefully thank Jaume Carbonelland Vladimir Karmanov for very stimulating discussions. TFacknowledges the warm hospitality of INFN Sezione di Pisaand thanks the partial financial support from the ConselhoNacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq),the Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo(FAPESP). GS thanks the partial support of Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES) ofBrazil. MV and GS acknowledge the warm hospitality of theInstituto Tecnol´ogico de Aeron´autica, S˜ao Jos´e dos Campos,S˜ao Paulo, Brazil, where part of this work was performed.7
Appendix A: Boundary properties of theNakanishi weight function g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ) This Appendix is devoted to the analysis of the relationbetween (i) the Nakanishi weight function g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ),that yields the integral representation of the distortedpart of the 3D LF scattering wave function, and (ii)the weight function (cid:101) G + ( γ (cid:48)(cid:48) , ζ, ζ (cid:48) ), that yields the inte-gral representation of the half-off-shell T-matrix (cf Eq.(58) in Ref. [9]). Notice that the dependence upon κ and z i has been dropped for the sake of a light nota-tion. This analysis allows one to obtain the conditionsfulfilled by g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ) when z (cid:48) = ± z (cid:48)(cid:48) = ± (cid:101) G + ( γ (cid:48)(cid:48) , ζ, ζ (cid:48) ) theconstraint θ (1 − | ζ | − | ζ (cid:48) | ) holds, for the variables z and z (cid:48) an analogous relation does not exist.The above mentioned relation between g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) )and (cid:101) G + ( γ (cid:48)(cid:48) , ζ, ζ (cid:48) ) reads as follows (cf Eq. (63) in Ref. [9],where a factor of two is missing, as well as in Eq. (60),but it was not relevant for the formal discussion, sinceit can be reabsorbed in (cid:101) G + ) g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ) = i (cid:90) dα (cid:90) dα − α − α ) × θ (1 − α − α ) θ (1 − α − α − | z (cid:48) | − | z (cid:48)(cid:48) − α + α | ) × (cid:101) G + (cid:18) γ (cid:48) (1 − α − α ) , z (cid:48) (1 − α − α ) , z (cid:48)(cid:48) − α + α (1 − α − α ) (cid:19) , (A.1)where z (cid:48) = ζ (1 − α − α ) z (cid:48)(cid:48) = ζ (cid:48) (1 − α − α ) + α − α . (A.2)Notice that the constraints θ (1 − α − α ) and θ (1 −| ζ | − | ζ (cid:48) | ) = θ (1 − α − α − | z (cid:48) | − | z (cid:48)(cid:48) − α + α | ) havebeen explicitly written, differently from Eq. (58) in Ref.[9]. In what follows it will be shown that the above thetafunctions lead to a vanishing Nakanishi weight functionat | z (cid:48) | = 1 or | z (cid:48)(cid:48) | = 1.Given the presence of θ (1 − α − α −| z (cid:48) |−| z (cid:48)(cid:48) − α + α | ) and 0 ≤ α i ≤
1, it is easily seen that for | z (cid:48) | = 1one has g (+) ( γ (cid:48) , z (cid:48) = ± , z (cid:48)(cid:48) ) = 0The same holds for | z (cid:48)(cid:48) | = 1. First of all, let us performa change of variables, viz ξ = 1 − ( α + α ) , ∆ = α − α α = 1 − ξ + ∆ , α = 1 − ξ − ∆ . (A.3) then Eq. (A.1) becomes g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) ) = i (cid:90) dξ × (cid:90) − d∆ θ (1 − ξ + ∆ ) θ (1 − ξ − ∆ ) ξ × (cid:101) G + (cid:18) γ (cid:48) ξ , z (cid:48) ξ , z (cid:48)(cid:48) − ∆ξ (cid:19) θ ( ξ − | z (cid:48) | − | z (cid:48)(cid:48) − ∆ | ) . (A.4)For z (cid:48)(cid:48) = 1, one gets z (cid:48)(cid:48) − ∆ ≥
0, since ∆ ∈ [ − , θ (1 − ξ − | ∆ | ) θ ( ξ − | z (cid:48) | − ∆ ) = θ (1 − ξ − | ∆ | ) θ [ ∆ − (1 − ξ ) − | z (cid:48) | ] = 0since ∆ ≥ (1 − ξ )+ | z (cid:48) | ≥ − ξ ≥ | ∆ | . For z (cid:48)(cid:48) = − z (cid:48)(cid:48) − ∆ ≤
0, and then θ (1 − ξ − | ∆ | ) θ [ ξ − | z (cid:48) |− (1 + ∆ )] = θ (1 − ξ − | ∆ | ) θ [ − (1 − ξ ) − | z (cid:48) | − ∆ ] = 0since − ∆ ≥ (1 − ξ )+ | z (cid:48) | ≥ − ξ ≥ | ∆ | . Therefore,from the above results, one gets g (+) ( γ (cid:48) , z (cid:48) , z (cid:48)(cid:48) = ±
1) = 0.
Appendix B: The distorted part of the 3D LFscattering wave function
In this Appendix, it will be shown how the expectedglobal free propagation of the constituents can be fac-torized out in the expression of ψ (+) dist , as in the nonrelativistic case. This result is relevant in two respects.On one side, it emphasizes the analogy with the nonrelativistic approach, and on the other side it allowsone to understand the support of the Nakanishi weightfunction g (+) ( γ (cid:48) , z (cid:48) , z ; γ i , z i ), when γ (cid:48) runs.In the CM frame ( p ⊥ = 0 and p ± = M ), assumingwithout loss of generality a head-on scattering, i.e. γ i =0, the 3D LF scattering wave function, is given by [9] ψ (+) (cid:0) z, γ ; κ , z i (cid:1) = p + (1 − z )4 (cid:90) dk − π Φ (+) ( k, p ) == p + (1 − z )4 (2 π ) δ (3) (˜ k − ˜ k i ) + ψ dist ( z, γ ; κ , z i ) , (B.5)where ˜ k ≡ { k + , k ⊥ } and ψ dist is ψ dist ( z, γ ; κ , z i ) = (1 − z )4 (cid:90) − dz (cid:48) (cid:90) ∞−∞ dγ (cid:48) g (+) ( γ (cid:48) , z (cid:48) , z ; κ , z i )[ γ (cid:48) + γ + z m + (1 − z ) κ + M z z (cid:48) z i − i(cid:15) ] , (B.6) By using the Nakanishi weight function for the half-off-shell T-matrix, one gets the following expression [9] ψ dist ( z, γ ; κ , z i ) == p + (1 − z )4 (cid:90) dk − π (cid:104) k µ | G ( p ) T ( p ) | k µi (cid:105) == p + (1 − z )4 (cid:90) dk − π i (cid:0) p + k (cid:1) − m + i(cid:15) × i (cid:0) p − k (cid:1) − m + i(cid:15) (cid:90) − dz (cid:48) (cid:90) − dζ (cid:48) (cid:90) ∞−∞ dγ (cid:48) × (cid:101) G + ( γ (cid:48) , z (cid:48) , ζ (cid:48) ; κ , z i ) θ (1 − | z (cid:48) | − | ζ (cid:48) | ) k + p − m + ζ (cid:48) p · k + z (cid:48) k · k i − γ (cid:48) + i(cid:15) . (B.7)Then, one can write ψ dist ( z, γ ; κ , z i ) == − p + (1 − z )4 (cid:90) − dz (cid:48) (cid:90) − dζ (cid:48) (cid:90) ∞−∞ dγ (cid:48) × (cid:101) G + ( γ (cid:48) , z (cid:48) , ζ (cid:48) ; κ , z i ) θ (1 − | z (cid:48) | − | ζ (cid:48) | ) (cid:90) dk − π × M/ k + ) ( M/ − k + ) 1( k + + ζ (cid:48) M + z (cid:48) k + i ) × p + k ) − − ( p + k ) − on + i(cid:15)/ ( M/ k + ) × p − k ) − − ( p − k ) − on + i(cid:15)/ ( M/ − k + ) × k − + k + ( ζ (cid:48) M + z (cid:48) k − i ) − γ − γ (cid:48) − κ + i(cid:15) ( k + + ζ (cid:48) M + z (cid:48) k + i ) , (B.8)where( p k ) − on = 2( m + γ ) M (1 − z )( p − k ) − on = 2( m + γ ) M (1 + z ) . (B.9)with k + = − zM/
2. Since k + i = − k − i = − z i M/
2, onegets ψ dist ( z, γ ; κ , z i ) = − M (cid:90) − dz (cid:48) (cid:90) − dζ (cid:48) (cid:90) ∞−∞ dγ (cid:48) × (cid:101) G + ( γ (cid:48) , z (cid:48) , ζ (cid:48) ; κ , z i ) θ (1 − | z (cid:48) | − | ζ (cid:48) | )( ζ (cid:48) − z − z (cid:48) z i ) × (cid:90) dk − π M + k − − ( p + k ) − on + i (cid:15)/ [ M/ (1 − z )] × M − k − − ( p − k ) − on + i (cid:15)/ [ M (1 + z )] × k − − k − Na + i (cid:15)/ [ M ( ζ (cid:48) − z − z (cid:48) z i )] , (B.10)with k − Na = 2 M ( ζ (cid:48) − z − z (cid:48) z i ) × (cid:20) M z ( ζ (cid:48) + z (cid:48) z i ) + γ + γ (cid:48) + κ (cid:21) . (B.11) One has the following poles (recall that 1 > z > − k L = ( p k ) − on − M − i(cid:15)k U = − ( p − k ) − on + M i(cid:15)k LU = k − Na − i (cid:15)M ( ζ (cid:48) − z − z (cid:48) z i ) . (B.12)In order to evaluate the analytic integration on k − , onecan consider the following two cases.If ζ (cid:48) > z + z (cid:48) z i , one can close the integration contourinto the upper plane, taking the residue at k U , i.e. (cid:90) ∞−∞ dk − π k − − k L ] 1[ − k − + k U ] 1[ k − − k LU ] == − i (cid:2) M − ( p − k ) − on − ( p + k ) − on + i(cid:15) (cid:3) × (cid:2) M − ( p − k ) − on − k − Na + i(cid:15) (cid:3) == − i M ζ (cid:48) − z − z (cid:48) z i ) × (1 − z )[ κ (1 − z ) + m z + γ − i(cid:15) ] (1 + z )(1 + ζ (cid:48) − z (cid:48) z i ) × κ (1 − z ) + m z + γ + (1+ z ) (cid:16) M zz (cid:48) z i + γ (cid:48) (cid:17) (1+ ζ (cid:48) − z (cid:48) z i ) − i(cid:15) . (B.13)If z + z (cid:48) z i > ζ (cid:48) , one can close the integration contourinto the lower plane, taking the residue at k L , i.e. (cid:90) ∞−∞ dk − π k − − k L ] 1[ − k − + k U ] 1[ k − − k LU ] == − i M ζ (cid:48) − z − z (cid:48) z i ) × (1 − z )(1 − ζ (cid:48) + z (cid:48) z i ) (1 − z )[ κ (1 − z ) + m z + γ − i(cid:15) ] × κ (1 − z ) + m z + γ + (1 − z ) (cid:16) M zz (cid:48) z i + γ (cid:48) (cid:17) (1 − ζ (cid:48) + z (cid:48) z i ) − i(cid:15) , (B.14)where ( ζ (cid:48) − z − z (cid:48) z i ) (cid:15) → − (cid:15) , since ( ζ (cid:48) − z − z (cid:48) z i ) < Collecting all the above results, one gets the follow-ing expression for ψ dist ψ dist ( z, γ ; κ , z i ) == i (1 − z )4 1[ κ (1 − z ) + m z + γ − i(cid:15) ] (cid:90) − dz (cid:48) (cid:90) − dζ (cid:48) (cid:90) ∞−∞ dγ (cid:48) (cid:101) G + ( γ (cid:48) , z (cid:48) , ζ (cid:48) ; κ , z i ) × (cid:104) (1 + z )(1 + ζ (cid:48) − z (cid:48) z i ) × θ ( ζ (cid:48) − z − z (cid:48) z i ) θ (1 − | z (cid:48) | − | ζ (cid:48) | ) κ (1 − z ) + m z + γ + (1+ z ) (cid:16) M zz (cid:48) z i + γ (cid:48) (cid:17) (1+ ζ (cid:48) − z (cid:48) z i ) − i(cid:15) + (1 − z )(1 − ζ (cid:48) + z (cid:48) z i ) × θ ( z + z i z (cid:48) − ζ (cid:48) ) θ (1 − | z (cid:48) | − | ζ (cid:48) | ) κ (1 − z ) + m z + γ + (1 − z ) (cid:16) M zz (cid:48) z i + γ (cid:48) (cid:17) (1 − ζ (cid:48) + z (cid:48) z i ) − i(cid:15) (cid:105) . (B.15)One can reobtain the expression in Eq. (B.6) by apply-ing the Feynman trick to Eq. (B.15). For instance, onehas θ ( ζ (cid:48) − z − z i z (cid:48) )[ κ (1 − z ) + m z + γ − i(cid:15) ] × κ (1 − z ) + m z + γ + (1+ z ) (cid:16) M zz (cid:48) z i + γ (cid:48) (cid:17) (1+ ζ (cid:48) − z (cid:48) z i ) − i(cid:15) == θ ( ζ (cid:48) − z − z i z (cid:48) ) (cid:90) dξ × (cid:20) γ + m z + κ (1 − z ) + ξ (1+ z ) (cid:16) M zz (cid:48) z i + γ (cid:48) (cid:17) (1+ ζ (cid:48) − z (cid:48) z i ) − i(cid:15) (cid:21) = (1 + ζ (cid:48) − z (cid:48) z i )(1 + z ) (cid:90) dα θ (cid:20) (1 + z )(1 + ζ (cid:48) − z (cid:48) z i ) − α (cid:21) × θ ( ζ (cid:48) − z − z i z (cid:48) ) (cid:2) γ + m z + κ (1 − z ) + α (cid:0) M zz (cid:48) z i + γ (cid:48) (cid:1) − i(cid:15) (cid:3) , (B.16)with1 ≥ (1 + z )(1 + ζ (cid:48) − z (cid:48) z i ) = (1 + z )[1 + z + ( ζ (cid:48) − z − z (cid:48) z i )]since θ ( ζ (cid:48) − z − z i z (cid:48) ). Inserting the above expression,together with the one containing θ ( z + z i z (cid:48) − ζ (cid:48) ), inEq. (B.15), one gets the following expression for the distorted term ψ dist ( z, γ ; z i , κ ) == i (1 − z )4 (cid:90) − dζ (cid:48)(cid:48) (cid:90) − dζ (cid:48) (cid:90) ∞−∞ dγ (cid:48)(cid:48) (cid:90) dαα × θ ( α − | ζ (cid:48)(cid:48) | ) θ (1 − | ζ (cid:48)(cid:48) α | − | ζ (cid:48) | ) × (cid:101) G + ( γ (cid:48)(cid:48) α , ζ (cid:48)(cid:48) α , ζ (cid:48) ) (cid:2) γ + m z + κ (1 − z ) + γ (cid:48)(cid:48) + M zζ (cid:48)(cid:48) z i − i(cid:15) (cid:3) { θ [(1 + z + ζ (cid:48)(cid:48) z i ) − α (1 + ζ (cid:48) )] θ [ α ( ζ (cid:48) − z ) − z i ζ (cid:48)(cid:48) ]+ θ [(1 − z − ζ (cid:48)(cid:48) z i ) − α (1 − ζ (cid:48) ))] θ [ − α ( ζ (cid:48) − z ) + z i ζ (cid:48)(cid:48) ] } = i (1 − z )4 (cid:90) − dζ (cid:48)(cid:48) (cid:90) ∞−∞ dγ (cid:48)(cid:48) (cid:90) dαα (cid:90) − dy × θ ( α − | y | ) θ ( α − | ζ (cid:48)(cid:48) | ) θ ( α − | ζ (cid:48)(cid:48) | − | y | ) × (cid:101) G + ( γ (cid:48)(cid:48) α , ζ (cid:48)(cid:48) α , yα ) (cid:2) γ + m z + κ (1 − z ) + γ (cid:48)(cid:48) + M zζ (cid:48)(cid:48) z i − i(cid:15) (cid:3) { θ (1 + z + ζ (cid:48)(cid:48) z i − α − y ) θ ( y − αz − z i ζ (cid:48)(cid:48) )+ θ (1 − z − ζ (cid:48)(cid:48) z i − α + y ] θ [ − y + αz + z i ζ (cid:48)(cid:48) ) } , (B.17)where γ (cid:48)(cid:48) = α γ (cid:48) and ζ (cid:48)(cid:48) = α z (cid:48) .The theta functions between curly brackets singleout the following integration regions – − α + z + ζ (cid:48)(cid:48) z i ≥ y ≥ αz + z i ζ (cid:48)(cid:48) – αz + z i ζ (cid:48)(cid:48) ≥ y ≥ − (1 − α ) + z + ζ (cid:48)(cid:48) z i The above intervals lead to the following constraint1 − α ≥ y − z − ζ (cid:48)(cid:48) z i ≥ − (1 − α )namely θ (1 − α − | y − z − ζ (cid:48)(cid:48) z i | ) . Then one gets ψ dist ( z, γ ; z i , κ ) = i (1 − z )4 (cid:90) − dζ (cid:48)(cid:48) (cid:90) ∞−∞ dγ (cid:48)(cid:48) × (cid:90) dαα (cid:90) − dy (cid:101) G + ( γ (cid:48)(cid:48) α , ζ (cid:48)(cid:48) α , yα ) × θ ( α − | ζ (cid:48)(cid:48) | − | y | ) θ (1 − α − | y − z − ζ (cid:48)(cid:48) z i | ) (cid:2) γ + m z + κ (1 − z ) + γ (cid:48)(cid:48) + M zζ (cid:48)(cid:48) z i − i(cid:15) (cid:3) . (B.18)The above expression of ψ dist allows one to write thefollowing relation between the Nakanishi weight func-tion g + ( γ (cid:48) , z (cid:48) , z ; κ , z i ), that appears in Eq. (B.6), and (cid:101) G + , namely the Nakanishi weight function involved inthe description the half-off-shell T-matrix, g + ( γ (cid:48) , z (cid:48) , z ; κ , z i ) == i (cid:90) dαα (cid:90) − dy (cid:101) G + ( γ (cid:48) α , z (cid:48) α , yα ; κ , z i ) × θ ( α − | z (cid:48) | − | y | ) θ (1 − α − | y − z − z (cid:48) z i | ) . (B.19)Notice that Eq. (B.19) can be transformed into Eq.(A.4) by applying a suitable change of variables. Appendix C: Zero-energy limit
The zero-energy limit of the relevant integral equationsfulfilled by the Nakanishi weight function amounts toconsider the case κ = 0, namely M = 4 m . This en-tails γ i = z i = 0 through M = 4( m + γ i ) / (1 − z i ).In this Appendix, the integral equations obtained bothwithout applying the uniqueness theorem [12] and byexploiting it, are obtained following a simpler proce-dure than the one adopted in Ref. [9] (notice that amistyping present in Eq. (103) of [9] has been fixed inthis Appendix, as explained in what follows).The Nakanishi integral equation, involving ψ dist , for κ ≤ (cid:90) − dz (cid:48) (cid:90) ∞−∞ dγ (cid:48)(cid:48) g (+)( Ld ) ( γ (cid:48)(cid:48) , z (cid:48) , z ; γ i , z i ) × γ + γ (cid:48)(cid:48) + z m + (1 − z ) κ + z (cid:48) M zz i − i(cid:15) ] == g (cid:90) − dz (cid:48) (cid:90) ∞−∞ dγ (cid:48)(cid:48) θ ( − z (cid:48) ) δ ( γ (cid:48)(cid:48) − γ a ( z (cid:48) )) × (cid:2) γ + γ (cid:48)(cid:48) + (1 − z ) κ + z m + z (cid:48) M zz i − i(cid:15) (cid:3) (cid:110) θ ( z − z i ) θ [1 − z + z (cid:48) (1 − z i )]+ θ ( z i − z ) θ [1 + z + z (cid:48) (1 + z i )] (cid:111) − g π ) (cid:90) ∞−∞ dγ (cid:48)(cid:48) (cid:90) − dz (cid:48) × (cid:2) γ + γ (cid:48)(cid:48) + z m + κ (1 − z ) + z (cid:48) M zz i − i(cid:15) (cid:3) (cid:90) ∞−∞ dγ (cid:48) (cid:90) − dζ (cid:90) − dζ (cid:48) g (+)( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ , z i ) × (cid:104) (1 + z )(1 + ζ (cid:48) − z i ζ ) × θ ( ζ (cid:48) − z − z i ζ ) h (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ )+ (1 − z )(1 − ζ (cid:48) + z i ζ ) × θ ( z − ζ (cid:48) + z i ζ ) h (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , − z, − z i ; γ (cid:48) , ζ, − ζ (cid:48) , µ ) (cid:105) , (C.20)with h (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , z, z i ; γ (cid:48) , ζ, ζ (cid:48) , µ ) = (1 + z )(1 + ζ (cid:48) − z i ζ ) × (cid:26) ∂∂λ (cid:90) ∞ dy (cid:90) dξ δ [ z (cid:48) − ξZ ( z, ζ, ζ (cid:48) ; z i )] × δ (cid:2) F ( λ, y, ξ ; γ (cid:48)(cid:48) , z, ζ, ζ (cid:48) , γ (cid:48) ; z i , κ , µ ) (cid:3)(cid:9) λ =0 . (C.21) where F ( λ, y, ξ ; γ (cid:48)(cid:48) , z, ζ, ζ (cid:48) , γ (cid:48) ; z i , κ , µ ) == γ (cid:48)(cid:48) − ξ (1 + z )(1 + ζ (cid:48) − z i ζ ) × (cid:18) y A ( ζ, ζ (cid:48) , γ (cid:48) , κ ) + y ( µ + γ (cid:48) ) + µ y (cid:19) − ξλ (C.22)For κ = 0, it follows that γ i = z i = 0 since κ = m − M (cid:16) p ± k i (cid:17) − M − γ i − z i M g (+)( Ld ) ( γ, z (cid:48) , z ; κ = 0)has support only for positive γ , one can write (cf Eq.(C.20) and subsec. 2.2) (cid:90) − dz (cid:48) (cid:90) ∞ dγ (cid:48)(cid:48) g (+)( Ld ) ( γ (cid:48)(cid:48) , z (cid:48) , z ; κ = 0)[ γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] == g (cid:90) ∞ dγ (cid:48)(cid:48) (cid:90) − dz (cid:48) θ ( − z (cid:48) ) δ ( γ (cid:48)(cid:48) + z (cid:48) µ )[ γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] × (cid:104) θ ( z ) θ (1 − z + z (cid:48) ) + θ ( − z ) θ (1 + z + z (cid:48) ) (cid:105) − g π ) (cid:90) ∞ dγ (cid:48) (cid:90) − dζ × (cid:90) − dζ (cid:48) g (+)( Ld ) ( γ (cid:48) , ζ, ζ (cid:48) ; κ = 0) × (cid:90) ∞−∞ dγ (cid:48)(cid:48) γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] (cid:90) − dz (cid:48) × (cid:104) (1 + z )(1 + ζ (cid:48) ) θ ( ζ (cid:48) − z ) Z (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ )+ (1 − z )(1 − ζ (cid:48) ) θ ( z − ζ (cid:48) ) Z (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , − z ; γ (cid:48) , − ζ (cid:48) , µ ) (cid:105) , (C.23)where the kernel Z (cid:48) is given by Z (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ) = (1 + z )(1 + ζ (cid:48) ) × (cid:110) ∂∂λ (cid:90) ∞ dy (cid:90) dξ δ [ γ (cid:48)(cid:48) − ξΓ ( y, z, ζ (cid:48) , γ (cid:48) ) − ξλ ] × δ (cid:20) z (cid:48) − ξ (1 + z )(1 + ζ (cid:48) ) ζ (cid:21)(cid:111) λ =0 (C.24)with Γ ( y, z, ζ (cid:48) , γ (cid:48) ) = (1 + z )(1 + ζ (cid:48) ) 1 y × (cid:8) y A ( ζ (cid:48) , γ (cid:48) ) + y ( µ + γ (cid:48) ) + µ (cid:9) (C.25)and A ( ζ (cid:48) , γ (cid:48) ) = ζ (cid:48) M γ (cid:48) = ζ (cid:48) m + γ (cid:48) ≥ , (C.26)Notice that γ (cid:48) is positive and therefore also Γ has tobe positive. Finally, γ (cid:48)(cid:48) in Eq. (C.24) has to be positivefor getting a non vanishing Z (cid:48) ( γ (cid:48)(cid:48) , z (cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ). Performing (i) the integration on z (cid:48) in both sides ofEq. (C.23) (recall that 1 > | ξζ (1 ± z ) / (1 ± ζ (cid:48) ) | ) and (ii)the integration on ζ in the rhs, one gets (cid:90) ∞ dγ (cid:48)(cid:48) g (+)0 Ld ( γ (cid:48)(cid:48) , z )[ γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] == g µ (cid:90) ∞−∞ dγ (cid:48)(cid:48) θ ( γ (cid:48)(cid:48) )[ γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] × (cid:110) θ ( z ) θ (cid:2) − z − γ (cid:48)(cid:48) /µ (cid:3) + θ ( − z ) θ (cid:2) z − γ (cid:48)(cid:48) /µ (cid:3)(cid:111) + − g π ) (cid:90) ∞ dγ (cid:48) (cid:90) − dζ (cid:48) g (+)0 Ld ( γ (cid:48) , ζ (cid:48) ) × (cid:90) ∞−∞ dγ (cid:48)(cid:48) θ ( γ (cid:48)(cid:48) ) 1[ γ + γ (cid:48)(cid:48) + z m − i(cid:15) ] × (cid:104) (1 + z )(1 + ζ (cid:48) ) θ ( ζ (cid:48) − z ) h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ )+ (1 − z )(1 − ζ (cid:48) ) θ ( z − ζ (cid:48) ) h (cid:48) ( γ (cid:48)(cid:48) , − z ; γ (cid:48) , − ζ (cid:48) , µ ) (cid:105) , (C.27)where1. g (+)0 Ld ( γ (cid:48)(cid:48) , z ) = (cid:90) − dz (cid:48) g (+)( Ld ) ( γ (cid:48)(cid:48) , z (cid:48) , z ; κ = 0) . (C.28)2. (cid:90) − dz (cid:48) θ ( − z (cid:48) ) δ ( γ (cid:48)(cid:48) + z (cid:48) µ ) == 1 µ θ ( γ (cid:48)(cid:48) ) θ ( µ − γ (cid:48)(cid:48) ) . (C.29)3. h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ) = (1 + z )(1 + ζ (cid:48) ) ∂∂λ (cid:90) ∞ dy (cid:90) dξ δ [ γ (cid:48)(cid:48) − ξΓ ( y, z, ζ (cid:48) , γ (cid:48) ) − ξλ ] (cid:12)(cid:12)(cid:12)(cid:12) λ =0 . (C.30)From Ref. [10], one recognizes that h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ) isthe suitable kernel for a bound state with zero energy.Therefore one can write h (cid:48) ( γ (cid:48)(cid:48) , z ; γ (cid:48) , ζ (cid:48) , µ ) == θ (cid:34) − B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) − µ (cid:114) ζ (cid:48) M γ (cid:48) (cid:35) × (cid:104) − B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) A ( ζ (cid:48) , γ (cid:48) ) ∆ ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) 1 γ (cid:48)(cid:48) + (1 + ζ (cid:48) )(1 + z ) × (cid:90) y + y − dy y [ y A ( ζ (cid:48) , γ (cid:48) ) + y ( µ + γ (cid:48) ) + µ ] (cid:105) − (1 + ζ (cid:48) )(1 + z ) (cid:90) ∞ dy y [ y A ( ζ (cid:48) , γ (cid:48) ) + y ( µ + γ (cid:48) ) + µ ] , (C.31) with (see also Eq. (C.26)) B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) = µ + γ (cid:48) − γ (cid:48)(cid:48) (1 + ζ (cid:48) )(1 + z ) ≤ ,∆ ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) == B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) − µ A ( ζ (cid:48) , γ (cid:48) ) ≥ ,y ± = 12 A ( ζ (cid:48) , γ (cid:48) ) × (cid:2) −B ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) ± ∆ ( z, ζ (cid:48) , γ (cid:48) , γ (cid:48)(cid:48) , µ ) (cid:3) . (C.32)Notice that in the inhomogeneous term in Eq. (C.27)the factor θ ( µ − γ (cid:48)(cid:48) ) has been dropped, given the pres-ence of the step functions θ (cid:2) ± z − γ (cid:48)(cid:48) /µ (cid:3) . In Eq.(103) of Ref. [9] the step function θ ( µ − γ (cid:48)(cid:48) ) has beenaccidentally overlooked.In conclusion, by applying the Nakanishi theoremon the uniqueness of the weight function [12], one hasthe following integral equation g (+)0 Ld ( γ, z ) = g µ θ ( γ ) θ (cid:2) µ (1 − | z | ) − γ (cid:3) − g π ) θ ( γ ) (cid:90) ∞ dγ (cid:48) (cid:90) − dz (cid:48) g (+)0 Ld ( γ (cid:48) , z (cid:48) ) × (cid:104) (1 + z )(1 + z (cid:48) ) θ ( z (cid:48) − z ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ )+ (1 − z )(1 − z (cid:48) ) θ ( z − z (cid:48) ) h (cid:48) ( γ, − z ; γ (cid:48) , − z (cid:48) , µ ) (cid:105) . (C.33) Appendix D: An effective decomposition of g (+)0 Ld ( γ, z ) In this Appendix, the decomposition of g (+)0 Ld ( γ, z ) shownin Eq. (62) is applied to the simple case of Eq. (59),based on the Nakanishi uniqueness theorem [12], in or-der to give the explicit representation of the numericalsystem to be solved.Inserting the decomposition (62), g (+)0 Ld ( γ, z ) = β θ ( − t ) + θ ( t ) N z (cid:88) (cid:96) =0 N g (cid:88) j =0 A (cid:96)j G (cid:96) ( z ) L j ( t ) , (D.34)with t = γ − µ (1 − | z | ), in Eq. (59), given by (noticethat in the following expression, the symmetry proper-ties of both the weight function and the kernel h (cid:48) areexploited), g (+)0 Ld ( γ, z ) = g µ θ ( γ ) θ (cid:2) µ (1 − | z | ) − γ (cid:3) − g (4 π ) θ ( γ ) (cid:90) ∞ dγ (cid:48) (cid:90) − dz (cid:48) g (+)0 Ld ( γ (cid:48) , z (cid:48) ) × (1 + z )(1 + z (cid:48) ) θ ( z (cid:48) − z ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ ) , (D.35) one can quickly obtain the following coupled system β θ ( − t ) = g µ θ ( γ ) θ ( − t ) − g (4 π ) θ ( γ ) θ ( − t ) β × (cid:90) − dz (cid:48) (cid:90) µ (1 −| z (cid:48) | )0 dγ (cid:48) × (1 + z )(1 + z (cid:48) ) θ ( z (cid:48) − z ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ ) − g (4 π ) θ ( γ ) θ ( − t ) N z (cid:88) (cid:96) =0 N g (cid:88) j =0 A (cid:96)j (cid:90) − dz (cid:48) × (cid:90) ∞ µ (1 −| z (cid:48) | ) dγ (cid:48) G (cid:96) ( z (cid:48) ) L j (cid:2) γ (cid:48) − µ (1 − | z (cid:48) | ) (cid:3) × (1 + z )(1 + z (cid:48) ) θ ( z (cid:48) − z ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ ) , (D.36)and θ ( t ) N z (cid:88) (cid:96) =0 N g (cid:88) j =0 A (cid:96)j G (cid:96) ( z ) L j ( t ) == − g (4 π ) θ ( γ ) θ ( t ) β (cid:90) − dz (cid:48) (cid:90) µ (1 −| z (cid:48) | )0 dγ (cid:48) × (1 + z )(1 + z (cid:48) ) θ ( z (cid:48) − z ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ ) + − g (4 π ) θ ( γ ) θ ( t ) N z (cid:88) (cid:96) =0 N g (cid:88) j =0 A (cid:96)j (cid:90) − dz (cid:48) (cid:90) ∞ µ (1 −| z (cid:48) | ) dγ (cid:48) G (cid:96) ( z (cid:48) ) L j (cid:2) γ (cid:48) − µ (1 − | z (cid:48) | ) (cid:3) × (1 + z )(1 + z (cid:48) ) θ ( z (cid:48) − z ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ ) . (D.37)For γ = 0 and z = 0, Eq. (D.36) reduces to β = g µ + β I β,β + N z (cid:88) (cid:96) =0 N g (cid:88) j =0 I β,(cid:96)j A (cid:96)j , (D.38)where I ββ = − g (4 π ) (cid:90) dz (cid:48) (cid:90) µ (1 − z (cid:48) )0 dγ (cid:48) × z (cid:48) ) h (cid:48) ( γ = 0 , z = 0; γ (cid:48) , z (cid:48) , µ ) (D.39)and I β,(cid:96)j = − g (4 π ) × (cid:90) dz (cid:48) (cid:90) ∞ µ (1 − z (cid:48) ) dγ (cid:48) G (cid:96) ( z (cid:48) ) L j (cid:2) γ (cid:48) − µ (1 − z (cid:48) ) (cid:3) × z (cid:48) ) h (cid:48) ( γ = 0 , z = 0; γ (cid:48) , z (cid:48) , µ ) , (D.40) with (cf Eq. (57)) h (cid:48) ( γ = 0 , z = 0; γ (cid:48) , z (cid:48) , µ ) = − (1 + z (cid:48) ) × (cid:90) ∞ dy y [ y ( γ (cid:48) + m z (cid:48) ) + y ( γ (cid:48) + µ ) + µ ] . (D.41)A matrix representation can be obtained for Eq. (D.37)by multiplying both sides by G (cid:96) (cid:48) ( z ) L j (cid:48) ( t ) and integrat-ing, viz A (cid:96) (cid:48) j (cid:48) = β I (cid:96) (cid:48) j (cid:48) ,β + N z (cid:88) (cid:96) =0 N g (cid:88) j =0 I (cid:96) (cid:48) j (cid:48) ,(cid:96)j A (cid:96)j (D.42)where (cf Eq. (57)) I (cid:96) (cid:48) j (cid:48) ,β = − g (4 π ) (cid:90) − dz (cid:90) ∞ dt G (cid:96) (cid:48) ( z ) L j (cid:48) ( t ) (cid:90) z dz (cid:48) (cid:90) µ (1 −| z (cid:48) | )0 dγ (cid:48) (1 + z )(1 + z (cid:48) ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ ) (D.43)and I (cid:96) (cid:48) j (cid:48) ,(cid:96)j = − g (4 π ) (cid:90) − dz (cid:90) ∞ dt G (cid:96) (cid:48) ( z ) L j (cid:48) ( t ) (cid:90) z dz (cid:48) (cid:90) ∞ µ (1 −| z (cid:48) | ) dγ (cid:48) G (cid:96) ( z (cid:48) ) ×L j (cid:2) γ (cid:48) − µ (1 − | z (cid:48) | ) (cid:3) (1 + z )(1 + z (cid:48) ) h (cid:48) ( γ, z ; γ (cid:48) , z (cid:48) , µ ) . (D.44) References
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