On the Equivalence of Three-Particle Scattering Formalisms
A. W. Jackura, S. M. Dawid, C. Fernández-Ramírez, V. Mathieu, M. Mikhasenko, A. Pilloni, S. R. Sharpe, A. P. Szczepaniak
JJLAB-THY-19-2947
On the Equivalence of Three-Particle Scattering Formalisms
A. W. Jackura,
1, 2, ∗ S. M. Dawid,
1, 2, † C. Fern´andez-Ram´ırez, V. Mathieu, M. Mikhasenko, A. Pilloni,
6, 7
S. R. Sharpe, and A. P. Szczepaniak
1, 2, 9 (Joint Physics Analysis Center) Physics Department, Indiana University, Bloomington, IN 47405, USA Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Ciudad de M´exico 04510, Mexico Departamento de F´ısica Te´orica, Universidad Complutense de Madrid, 28040 Madrid, Spain CERN, 1211 Geneva 23, Switzerland European Centre for Theoretical Studies in Nuclear Physics and Related areas(ECT ∗ ) and Fondazione Bruno Kessler, Villazzano (Trento), I-38123, Italy INFN Sezione di Genova, Genova, I-16146, Italy Physics Department, University of Washington, Seattle, WA 98195-1560, USA Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA (Dated: May 30, 2019)In recent years, different on-shell → scattering formalisms have been proposed to be appliedto both lattice QCD and infinite volume scattering processes. We prove that the formulation inthe infinite volume presented by Hansen and Sharpe in Phys. Rev. D92, 114509 (2015) and sub-sequently Brice˜no, Hansen, and Sharpe in Phys. Rev. D95, 074510 (2017) can be recovered fromthe B -matrix representation, derived on the basis of S -matrix unitarity, presented by Mai et al. inEur. Phys. J. A53, 177 (2017) and Jackura et al. in Eur. Phys. J. C79, 56 (2019). Therefore, bothformalisms in the infinite volume are equivalent and the physical content is identical. Additionally,the Faddeev equations are recovered in the non-relativistic limit of both representations. I. INTRODUCTION
Considerable progress has been achieved recently in de-termination of the hadron spectrum from first principlesQuantum Chromodynamics (QCD) [1–8]. Comparison ofexperimental data or lattice results with theoretical mod-els involves analysis of partial wave amplitudes in whichresonances appear as pole singularities in the complexenergy and/or angular momentum planes [9]. Thus, aproper description of resonances requires knowledge ofanalytic properties of the scattering amplitude. Specif-ically, the determination of the hadron spectrum fromlattice calculations is done using a quantization condi-tion [10], which relates discrete energy levels in the finitevolume to the infinite-volume, partial waves evaluated atreal energy values and later analytically continued to thecomplex energy plane. The quantization condition hasbeen extensively studied for systems with strong two-particle interactions (see, e.g., Ref. [11] and referencestherein). However, most of resonances of current interestdecay to three and more particles.Quantization conditions for three hadrons have beenderived by various groups using different approaches [12–22], as recently reviewed in Ref. [23]. If differences existbetween formalisms, this could indicate that importantphysical content is missing, and that results based onthem will lead to unknown systematic errors. There-fore, it is important to unify our understanding of these ∗ email: [email protected] † email: [email protected] approaches and establish relationships between all for-malisms. In addition to quantization conditions, analyticrepresentations of the infinite volume → amplitudesare required to be able to identify pole positions of reso-nances. In this context we discuss two seemingly differentapproaches and demonstrate their equivalence.The first approach is referred to as the B -matrix rep-resentation, and was studied in Refs. [24, 25]. Motivatedby unitarity of the S -matrix, the B -matrix refers to akernel in a linear integral equation for an elastic → connected amplitude. The B -matrix contains both theknown long-range one-particle exchange (OPE) contribu-tions and any short range interactions. The latter playsimilar role to the K -matrix in → scattering ampli-tudes [26]. Aspects of its analytic properties were studiedin Ref. [25], showing how, besides the unitarity branchpoint, there are other singularities near the physical re-gion generated by the one-particle exchange, e.g. trianglesingularities. Applying the B -matrix formalism in finitevolume leads to the quantization condition of Ref. [17].The alternative approach was first derived in Ref. [13],and subsequently generalized to allow for ↔ transi-tions in Ref. [15]. Hereafter we refer to it as the HS-BHSapproach (for the authors initials). It is a bottom-up con-struction of the → amplitude starting from a generic,relativistic effective field theory in finite volume [12]. InRef. [13], the corresponding infinite volume limit of theHS-BHS formalism was derived explicitly, providing anexpression for the → scattering amplitude in termsof a → analog of the K -matrix, referred to here as a r X i v : . [ h e p - ph ] M a y K df . This HS-BHS representation is written in termsof two integral equations, the first summing one-particleexchanges between → subprocesses, and the secondinvolving all orders in K df . Since this approach is basedon Feynman diagrams, one expects that the result is con-sistent with unitarity, and, indeed, very recently this hasbeen shown explicitly [27].In the HS-BHS representation, the kernel K df appearsto play a similar role to that of the short-range part of the B -matrix, but it is actually quite different. It is the mainpurpose of this work to show that, nevertheless, the tworepresentations are equivalent. Specifically, we derive anintegral equation relating the R -matrix of Ref. [25] andthe → K -matrix of Ref. [28]. Furthermore, we showthat the reason for the superficial difference lies in theorganization of the short-range rescattering effects anddifference in the order in which symmetrization of theamplitude is applied.The paper is organized as follows. Section II sum-marizes definitions of on-shell → amplitudes andthe relevant kinematic variables. Section III reviewsthe B -matrix and HS-BHS on-shell representations forthe → amplitude. In Section IV, we derive therelationship between these two representations, provingtheir equivalence. In Section V we show that in thenon-relativistic limit the B -matrix can be reduced tothe Faddeev equations. Our findings and outlook aresummarized in Section VI. We include three technicalAppendices. Appendix A reviews the unitarity relationfor → amplitudes, and Appendix B shows how torewrite the B -matrix representation in a form analogousto that of the HS-BHS representation, which is used inthe demonstration of Section IV. Finally, Appendix Cproves a crucial relation discussed in Section IV. II. 3 → We consider the elastic scattering of three spinlessidentical particles of mass m , e.g., 3 π + → π + scat-tering. Note that Ref. [25] considered distinguishableparticles, while here we consider identical particles tocompare with Refs. [13, 15]. Internal symmetries suchas isospin are not considered, but can be included ina straightforward manner. The initial and final three-particle state have a total energy momentum P = ( E, P )and P (cid:48) = ( E (cid:48) , P (cid:48) ), respectively. This exemplifies a con-vention we use throughout, namely that primed (un-primed) variables denote quantities in the final (initial)state. Total energy-momentum is conserved, as is thethree-particle invariant mass squared s ≡ P = E − P , (1) This quantity is denoted K df , in Ref. [13]. p ?
Our interest is in constructing on-shell representationsfor the connected → scattering amplitude. Here wereview the relevant features of the B -matrix representa-tion discussed in Ref. [25] and the HS-BHS representationof Ref. [13]. A. B-Matrix Representation
As discussed in Refs. [24, 25], the B -matrix is an on-shell representation for the connected → amplitudethat was constructed to satisfy elastic → unitarity.In the p(cid:96)m (cid:96) -basis, the B -matrix representation it leadsto the integral equation A p (cid:48) p = F p (cid:48) B p (cid:48) p F p + (cid:90) k F p (cid:48) B p (cid:48) k A kp (11)where B p (cid:48) p = G p (cid:48) p + R p (cid:48) p is the B -matrix driving term,with G p (cid:48) p being the OPE contribution and R p (cid:48) p a real In Ref. [25] we denoted the OPE contribution by the symbol E ,while here we use G to provide a closer connection to the notationof Ref. [13]. =
We now turn to the definitions of the on-shell → scattering equations of HS-BHS as given in Ref. [13]. Weremind the reader that the unsymmetrized elastic → amplitude, A p (cid:48) p , is a matrix in the angular momentumspace of the pair labeled by the spectator. The unsym-metrized elastic → amplitude in HS-BHS represen-tation is given via the integral equation A p (cid:48) p = D p (cid:48) p + (cid:90) k (cid:48) (cid:90) k L p (cid:48) k (cid:48) T ( k (cid:48) , k ) L (cid:62) k (cid:48) p . (20)The symmetrized amplitude can be recovered as inEq. (7). The ladder series, D p (cid:48) p , is defined exactly likein Eq. (15), and the end cap operators, L p (cid:48) p are definedby L p (cid:48) p = (cid:18)
13 + F p (cid:48) iρ p (cid:48) (cid:19) δ p (cid:48) p + D p (cid:48) p iρ p , (21)with L (cid:62) p (cid:48) p defined with iρ p on the left of F p (cid:48) and D p (cid:48) p .The quantity ρ p is the phase space factor for the twoparticles in the pair, [ ρ p ] (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) = δ (cid:96) (cid:48) (cid:96) δ m (cid:48) (cid:96) m (cid:96)
12! 116 π (cid:115) − m σ p (22) L is the same as the quantity L ( u,u ) of Ref. [13]. To see thisrequires accounting for the different integration measures usedin the two works: our measure includes a factor of 1 / (2 ω p ) thatis not present in the measure of Ref. [13]. In Ref. [13], the two-body phase space is defined slightly dif-ferently, ρ p (Ref.[13]) = − iρ p H ( p ). Thus there are no explicitfactors of i in the expression for L in Ref. [13], whereas we preferhere to keep such factors explicit. The object H ( p ) is a cutofffunction, absent here because our momentum integrals implicitlyinclude an ultraviolet cutoff, as discussed above. where the 2! is the symmetry factor. Finally, T ( p (cid:48) , p ) isdefined via the integral equation T ( p (cid:48) , p ) = K df ( p (cid:48) , p )+ (cid:90) k (cid:48) (cid:90) k K df ( p (cid:48) , k (cid:48) ) iρ k (cid:48) L k (cid:48) k T ( k , p ) , (23)where K df ( p (cid:48) , p ) is the three-particle K -matrix. The am-plitudes T ( p (cid:48) , p ) and K df ( p (cid:48) , p ) are matrices in angularmomenta,[ K df ( p (cid:48) , p ) ] (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) = K df ,(cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , s, p ) , (24)and similarly for T ( p (cid:48) , p ), however we denote them dif-ferently than all other amplitudes due to their symme-try properties. As defined in Ref. [13], T ( p (cid:48) , p ) and K df ( p (cid:48) , p ) are symmetric under interchange of any pair ofinitial or final state particles, after we sum over the prod-uct of the amplitude and its spherical harmonics of thepair orientations. Thus, the symmetric divergence-free K matrix is given by K df = 4 π (cid:88) (cid:96) (cid:48) ,m (cid:48) (cid:96) (cid:96),m (cid:96) Y ∗ (cid:96) (cid:48) m (cid:48) (cid:96) ( ˆq (cid:63) p (cid:48) ) K df ,(cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , s, p ) Y (cid:96)m (cid:96) ( ˆq (cid:63) p ) , (25)with a similar expression for T . Note that Eq. (25) is dif-ferent from Eq. (7) since the latter requires a further sym-metrization operation. The K matrix on the left handside in Eq. (25) is fully symmetric under interchange ofany pair of particles in either the initial or final state. T ( p (cid:48) , p ) is viewed as an amputated amplitude forwhich, in addition to all → rescatterings being re-moved, the possibility of no rescattering in either ini-tial or final is included. This possibility is allowed bythe term involving the constant 1 / / T ( p (cid:48) , p ) and K df ( p (cid:48) , p ), Eq. (25), which is different thanEq. (7). Therefore, when we symmetrize the amplitudein Eq. (20), we would overcount the terms with no rescat-terings if the 1 / IV. EQUIVALENCE OF THE B -MATRIX ANDHS-BHS REPRESENTATIONS Having established the B -matrix and HS-BHS equa-tions, we now show that they are equivalent. To do sowe assume that the → amplitudes in both represen-tations are equal, and search for a relation between R and K df . We first express the HS-BHS end caps, L p (cid:48) p , interms of the B -matrix rescattering functions, (cid:101) L p (cid:48) p , L p (cid:48) p = 13 δ p (cid:48) p + (cid:101) L p (cid:48) p iρ p . (26)The result for L (cid:62) p (cid:48) p simply has iρ p (cid:48) and (cid:101) L p (cid:48) p inter-changed. We will find that the first term of Eq. (26) can be traced to the differences in symmetrization and re-moval of → rescatterings between (cid:101) T p (cid:48) p and T ( p (cid:48) , p ),while the iρ p factor in the second term is due to a differ-ence in the definition of on-shell amputation.To proceed, we rewrite Eq. (26) as L p (cid:48) p = (cid:90) k (cid:101) L p (cid:48) k U kp , (27)where U p is the “conversion factor” U p (cid:48) p = iρ p (cid:48) δ p (cid:48) p + 13 (cid:101) L − p (cid:48) p (28)= iρ p (cid:48) δ p (cid:48) p + 13 F − p (cid:48) δ p (cid:48) p − G p (cid:48) p , (29)where the second line follows from the inverse of (cid:101) L p (cid:48) p obtained from Eq. (19). In a similar manner, the trans-pose is given by L (cid:62) p (cid:48) p = (cid:82) k U p (cid:48) k (cid:101) L kp . Now, equating theexpressions for A in the two formalisms, Eqs. (16) and(20), and using Eq. (27), we find the equivalence if thefollowing relation holds, (cid:101) T p (cid:48) p = (cid:90) k (cid:48) (cid:90) k U p (cid:48) k (cid:48) T ( k (cid:48) , k ) U k , p . (30)The amplitudes (cid:101) T p (cid:48) p and T ( p (cid:48) , p ) can be formally solvedin terms of R p (cid:48) p and K df ( p (cid:48) , p ), respectively, as one doesin matrix equations, e.g. (cid:101) T = (cid:104) − R (cid:101) L (cid:105) − R , which isa matrix in both angular and spectator momenta. Com-bining the formal solutions for (cid:101) T p (cid:48) p and T ( p (cid:48) , p ), therelation Eq. (30), and using the definition of U p (cid:48) p inEq. (28), we arrive at an integral equation relating R p (cid:48) p and K df ( p (cid:48) , p ) R p (cid:48) p = (cid:90) k (cid:48) (cid:90) k U p (cid:48) k (cid:48) K df ( k (cid:48) , k ) U kp − (cid:90) k (cid:48) (cid:90) k U p (cid:48) k (cid:48) K df ( k (cid:48) , k ) R kp (31)If this equation holds, then the two representations of theamplitude A are equivalent.The final step is to show that Eq. (31) is consistentwith the reality of both R and K df . This result is notmanifest, as U is complex. Its imaginary part is readilyfound to be Im U p (cid:48) p = 13 (cid:16) ρ p δ p (cid:48) p − C p (cid:48) p (cid:17) , (32)where we have used the → unitarity relation for theinverse amplitude,Im F − p = − ¯ ρ p = − ρ p Θ( σ p − m ) , (33)which follows from Eq. (A2), as well as the result C p (cid:48) p =Im G p (cid:48) p . To proceed we need the important results (cid:90) k Im U p (cid:48) k K df ( k , p ) = (cid:90) k K df ( p (cid:48) , k ) Im U kp = 0 . (34)which are demonstrated in Appendix C. Essentially, theaction of C p (cid:48) p on an object with the symmetry propertiesof Eq. (25) yields a phase-space factor that cancels thefirst term of Eq. (32). Combining these results, we find that R is real if K df is,Im R p (cid:48) p = (cid:90) k (cid:48) (cid:90) k U p (cid:48) k (cid:48) Im K df ( k (cid:48) , k ) U kp − (cid:90) k (cid:48) (cid:90) k U p (cid:48) k (cid:48) Im K df ( k (cid:48) , k ) R kp − (cid:90) k (cid:48) (cid:90) k U p (cid:48) k (cid:48) K df ( k (cid:48) , k ) Im R kp = 0 . (35)The inverse result can be shown similarly. We con-clude that the B -matrix and HS-BHS representations areequivalent.The relationship between R p (cid:48) p and K df ( p (cid:48) , p ) in Eq. (31) can be better understood if rewritten as (cid:90) k (cid:48) (cid:90) k (cid:101) L p (cid:48) k (cid:48) R k (cid:48) k (cid:101) L kp = (cid:90) k (cid:48) (cid:90) k (cid:18) δ p (cid:48) k (cid:48) + (cid:101) L p (cid:48) k (cid:48) iρ k (cid:48) (cid:19) K df ( k (cid:48) , k ) (cid:18) δ kp + iρ k (cid:101) L kp (cid:19) − (cid:90) k (cid:48) (cid:90) k (cid:90) k (cid:48)(cid:48) (cid:18) δ p (cid:48) k (cid:48) + (cid:101) L p (cid:48) k (cid:48) iρ k (cid:48) (cid:19) K df ( k (cid:48) , k ) R kk (cid:48)(cid:48) (cid:101) L k (cid:48)(cid:48) p . (36)We now assume that K df ( p (cid:48) , p ) is momentum-independent, K df = λ , with λ a small constant. This isotropic form isthe leading contribution in an expansion about threshold [29]. Truncating the series solution of Eq. (31) at leadingorder in K df , we obtain (cid:90) k (cid:48) (cid:90) k (cid:101) L p (cid:48) k (cid:48) R k (cid:48) k (cid:101) L kp = 19 K df ( p (cid:48) p ) + 13 (cid:90) k (cid:48) (cid:101) L p (cid:48) k (cid:48) iρ k (cid:48) K df ( k (cid:48) , p ) + 13 (cid:90) k K df ( p (cid:48) , k ) iρ k (cid:101) L kp + (cid:90) k (cid:48) (cid:90) k (cid:101) L p (cid:48) k (cid:48) iρ k (cid:48) K df ( k (cid:48) , k ) iρ k (cid:101) L kp + O ( K ) , (37)which is represented diagrammatically in Fig. 4. Since K df ( p (cid:48) , p ) represents three-body interactions such as contactinteractions, the right hand side shows that there is a possibility that the interaction is not dressed by → rescatterings on either the initial or final state (or both). Contrarily, the R p (cid:48) p matrix is always dressed by → interactions in both the initial and final state. Thus, the R p (cid:48) p matrix represents a different organization of amplitudes.The factors 1 / / K df ( p (cid:48) , p ) has a decomposition given by Eq. (25), which does not includesumming over all spectator momenta. Thus, the factors are needed to remove the overcounting when we sum overall spectator momenta in the initial or final state. Finally, the left hand side has no iρ p factors, whereas the righthand side does. This is due to the differences in how the amplitudes are amputated. For the B -matrix, which isbased on satisfying the unitarity relations, the amputation was made by removing the partial wave amplitudes via A p (cid:48) p = F p (cid:48) (cid:101) A p (cid:48) p F p where the (cid:101) A p (cid:48) p is the amputated partial wave amplitude. This was convenient as it simplifiedthe unitarity relation (see Appendix A and Ref. [25] for details). This amputation is not unique, as we can freelyremove any real quantity from (cid:101) A p (cid:48) p , including ( iρ p ) . The HS-BHS equations involve an all-orders summation ofamplitudes in an effective field theory, which includes loop integrals over four-momenta of intermediate states. Whenthe two-particle loop integral is put on-shell, the iρ p factors naturally emerge. V. NON-RELATIVISTIC LIMIT AND FADDEEVEQUATIONS
In the non-relativistic limit, which is relevant for nearthreshold processes, we can investigate the relation be-tween these representations to the Faddeev equations.If we assume that the three-body interactions are neg-ligible compared to the two-body, then we can set the K df -matrix (or equivalently the R -matrix by Eq. (31)) tozero at leading order, leaving only the ladder rescattering solutions, A p (cid:48) p = D p (cid:48) p + O ( K df ) . (38)The three-body amplitudes are dominated by exchangesbetween → processes, as in typical Faddeev-typeapproximations. In that case, as can be seen from Eqs.(8) and (19), the unsymmetrized scattering amplitude M becomes (cid:101) L , which gives M p (cid:48) p = F p δ p (cid:48) p + (cid:90) k F p (cid:48) G p (cid:48) k M kp . (39) =
We have shown that the relativistic on-shell represen-tation of the → scattering amplitude of Hansen andSharpe [13] and Brice˜no, Hansen, and Sharpe [15], andthe B -matrix representation presented by Mai et al. [17]and Jackura et al. [25] are equivalent, and their physicalcontent is identical. The results of the present work areconsistent with the conclusions of Ref. [27] that the HS-BHS approach is unitary. The difference in these repre-sentations is how the formalism incorporates rescatteringeffects. In the B -matrix representation, the → ampli-tude is always dressed by → rescatterings in both theinitial and final states, as shown by Eq. (16). Contrar-ily, the representation by Brice˜no, Hansen, and Sharpeallows the possibility of no initial/ final state rescatter-ings. It was shown in Section IV that the differences be-tween these rescattering functions manifest themselves asdifferences in the real part of the on-shell equations, giv-ing the integral equation (31). The non-relativistic limitof both formalisms reproduced the Faddeev equations,providing a consistency check to well known low-energyapproaches. As was discussed in Ref. [25], the B -matrixrepresentations of Refs. [24] and [25] differ only in thereal part as a result of the latter approach using a cut-off on the integration range that eliminated unphysicalmodes.All of the proposed formalisms require regulation ofthe high-energy modes in order to arrive at a convergentsolution to the integral equations. Regulating the di-vergent behavior introduces additional cutoff dependencein the equations. Physical quantities must, however, becutoff-independent, and this is achieved by introducingcutoff dependence into the real, K -matrix-like quantitiesin the formalisms (i.e. R and K df ). For example, as wasdiscussed in Ref. [25], the B -matrix representations ofRef. [24] and [25] differ only in their real parts, as a re-sult of the latter using a cutoff in the integration rangewhich eliminated unphysical modes.It remains to be seen if the quantization conditions cor-responding to the different formalisms are also identical.Naively, one might assume that, since the infinite volumeequations are identical, the quantization conditions mustalso be, at least up to exponentially suppressed correc-tions. However, the details of transitioning from infiniteto finite volume, e.g., the handling of angular momen-tum mixing, are nontrivial and have not yet been workedout. This is an interesting area of study and must becompleted to ensure consistency.An interesting direction for future studies is com-paring numerical results from each representation. Al-though equivalent, parametrizations using the R -matrixof Ref. [25] or the K -matrix of Ref. [13] may turn out tobe advantageous for particular numerical analyses. ACKNOWLEDGMENTS
We thank Ra´ul Brice˜no and Maxwell Hansen formany useful discussions. This work was supportedby the U.S. Department of Energy under grantsNo. DE-SC0011637 (SRS), No. DE-AC05-06OR23177,and No. DE-FG02-87ER40365, U.S. National Sci-ence Foundation under award number PHY-1415459,PAPIIT-DGAPA (UNAM, Mexico) grant No. IA101819,CONACYT (Mexico) grants No. 251817 and No. A1-S-21389. VM acknowledges support from ComunidadAut´onoma de Madrid through Programa de Atracci´on deTalento Investigador 2018 (Modalidad 1). The work ofSRS was partly supported by the International ResearchUnit of Advanced Future Studies at Kyoto University.
Appendix A: Unitarity Relations
Unitarity of the S -matrix constrains the imaginarypart of on-shell scattering amplitudes. Given the unitar-ity constraints, one can construct an on-shell representa-tion for scattering amplitudes in terms of real quantitiesand kinematic functions. We present here a brief sum-mary of the unitarity relations for identical particles. Therelations for distinguishable particles have been discussedin detail in Ref. [25]. Elastic three-particle scattering sat-isfies the unitarity relation2 Im M = 13! (cid:89) j =1 (cid:90) k j (2 π ) δ (4) (cid:32) (cid:88) j =1 k j − P (cid:33) M ∗ M , (A1)where the integration is over the on-shell intermediatestate momenta. Writing M in terms of the unsymmeter-ized amplitudes, Eq. (7), and separating the disconnected → amplitude from the connected → amplitudevia Eq. (8), we arrive at two unitarity equations. Thefirst is the well-known → unitarity relation in angu-lar momentum space,Im F p = F † p ¯ ρ p F p , (A2)where ¯ ρ is the two-body phase space defined in Eq. (33).Equation (A2) admits the on-shell K -matrix representa-tion for F p , F p = K p + K p i ¯ ρ p F p = [1 − K p i ¯ ρ p ] − K p , (A3)where [ K p ] (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) = δ (cid:96) (cid:48) (cid:96) δ m (cid:48) (cid:96) m (cid:96) K (cid:96) ( σ p ) is the → K -matrix, which is a real function of σ p in the elas-tic kinematic region, and diagonal in angular momentaspace. Since the phase space factor contains the kine-matic information of two on-shell propagating particles,the K -matrix represents all the dynamical content of thetwo-particle system, e.g. such as the short range forcesbetween pions in elastic ππ scattering. This can in prin-ciple include virtual exchanges leading to left hand cuts0or higher multiparticle thresholds, e.g. four particle pro-duction, which do not give singular contributions in theelastic domain. Since → amplitudes are diagonalin angular momentum space, Eq. (A3) reduces to a sim-ple algebraic relation. It is straightforward to verify thatEq. (A3) satisfies Eq. (A2).The second unitarity relation is for the connected → amplitude, which in the p(cid:96)m (cid:96) -basis isIm A p (cid:48) p = (cid:90) k A † p (cid:48) k ¯ ρ k A kp + (cid:90) k (cid:48) (cid:90) k A † p (cid:48) k (cid:48) C k (cid:48) k A kp + F † p (cid:48) ¯ ρ p (cid:48) A p (cid:48) p + (cid:90) k F † p (cid:48) C p (cid:48) k A kp + A † p (cid:48) p ¯ ρ p F p + (cid:90) k A † p (cid:48) k C kp F p + F † p (cid:48) C p (cid:48) p F p , (A4)where C p (cid:48) p is the recoupling coefficient between a pair inone state to a different pair in the same state, e.g., froman angular momentum coupling (12)3 to (23)1, whichis defined as the imaginary part of the amputated OPEamplitude, Eq. (13),[ C p (cid:48) p ] (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) ≡ Im [ G p (cid:48) p ] (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) = π δ (cid:0) ( P p − p (cid:48) ) − m (cid:1) × π Y (cid:96) (cid:48) m (cid:48) (cid:96) ( ˆp (cid:63) p (cid:48) ) Y ∗ (cid:96)m (cid:96) ( ˆp (cid:48) (cid:63) p ) . (A5)The recoupling coefficients are an additional feature ofthree-body scattering that can be seen in Fig. 5 whena diagram with a crossed exchange in the intermediatestate is cut. Diagrams that are cut where no exchange oc-curs gives rise to the conventional two-body phase space.One may be concerned that the complexity of spheri-cal harmonics is not taken into account. The phases inthe unitarity relation cancel since the intermediate statesums over all possibilities. To avoid this bookkeepingduring intermediate calculations, one can use real spher-ical harmonics, which have the same completeness andorthonormality relations as the usual ones, to formallymanipulate the expressions. Since the final results do notdepend on the choice of harmonics, we are guaranteed thevalidity of the unitarity relations and the solutions.Equation (A4) admits the on-shell representation givenby Eq. (11), which we now verify. We find the follow-ing demonstration more direct than the one presentedin Ref. [25]. First, let us introduce amplitudes whichhave the final state → amplitudes amputated, i.e., A p (cid:48) p = F p (cid:48) (cid:101) A p (cid:48) p F p . Equation (A4) then simplifies toIm (cid:101) A p (cid:48) p = (cid:90) k (cid:101) A † p (cid:48) k F † k ¯ ρ k F k (cid:101) A kp + (cid:90) k (cid:48) (cid:90) k (cid:101) A † p (cid:48) k (cid:48) F † k (cid:48) C k (cid:48) k F k (cid:101) A kp + (cid:90) k C p (cid:48) k F k (cid:101) A kp + (cid:90) k (cid:101) A † p (cid:48) k F † k C kp + C p (cid:48) p , (A6) and the corresponding amputated B -matrix representa-tion is (cid:101) A p (cid:48) p = B p (cid:48) p + (cid:90) k B p (cid:48) k F k (cid:101) A kp = (cid:90) k B p (cid:48) k (cid:16) δ kp + F k (cid:101) A kp (cid:17) , (A7)where we remind the reader that B p (cid:48) p = G p (cid:48) p + R p (cid:48) p , where G p (cid:48) p is given in Eq. (13) and R p (cid:48) p isa real function that contains the unconstrained three-body dynamics. It is straightforward to verify thatEq. (A7) satisfies Eq. (A6) directly by taking the dif-ference between the amplitude and its Hermitian con-jugate. Note that if the matrix elements of A p (cid:48) p are A (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , s, p ), then the Hermitian conjugate, A † p (cid:48) p ,has elements A ∗ (cid:96)m (cid:96) ; (cid:96) (cid:48) m (cid:48) (cid:96) ( p , s, p (cid:48) ), since it acts on both theangular momentum space and the spectator space. TheHermitian analytic properties [33] of amplitudes thenstate A ∗ (cid:96)m (cid:96) ; (cid:96) (cid:48) m (cid:48) (cid:96) ( p , s, p (cid:48) ) = A ∗ (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , s, p ), so that A p (cid:48) p − A † p (cid:48) p = A p (cid:48) p − A ∗ p (cid:48) p = 2 i Im A p (cid:48) p .We begin by rewriting the difference by adding andsubtracting a judiciously chosen term, leading to (cid:101) A p (cid:48) p − (cid:101) A † p (cid:48) p = (cid:90) k (cid:101) A † p (cid:48) k (cid:16) F k − F † k (cid:17) (cid:101) A kp + (cid:90) k (cid:16) δ p (cid:48) k + (cid:101) A † p (cid:48) k F † k (cid:17) (cid:101) A kp − (cid:90) k (cid:101) A † p (cid:48) k (cid:16) δ kp + F k (cid:101) A kp (cid:17) . (A8)Next we insert Eq. (A7) into (cid:101) A kp on the second line ofEq. (A8), and its Hermitian conjugate, (cid:101) A † p (cid:48) p = B † p (cid:48) p + (cid:90) k (cid:101) A † p (cid:48) k F † k B † kp = (cid:90) k (cid:16) δ p (cid:48) k + (cid:101) A † p (cid:48) k F † k (cid:17) B † kp , (A9)into the (cid:101) A † p (cid:48) k on the third line of Eq. (A8). This gives (cid:101) A p (cid:48) p − (cid:101) A † pp (cid:48) = (cid:90) k (cid:101) A † p (cid:48) k (cid:16) F k − F † k (cid:17) (cid:101) A kp + (cid:90) k (cid:48) (cid:90) k (cid:16) δ p (cid:48) k (cid:48) + (cid:101) A † p (cid:48) k F † k (cid:48) (cid:17) × (cid:16) B k (cid:48) k − B † k (cid:48) k (cid:17) × (cid:16) δ kp + F k (cid:101) A kp (cid:17) , (A10)which can then be simplified using F p − F † p = 2 i Im F p and Eq. (A2), as well as the result that B p (cid:48) p − B † p (cid:48) p =2 i Im G p (cid:48) p since R p (cid:48) p is real. Then, since the recouplingcoefficients are C p (cid:48) p = Im G p (cid:48) p , we arrive at Eq. (A6),thus proving that the B -matrix representation satisfiesthe unitary condition.1 =
In this appendix, we show how the B -matrix represen-tation can be expressed in terms of the full OPE lad-der summation and a remaining piece containing gen-uine three-body interactions (see also Ref. [30]). The B -matrix representation for the full amplitude is givenin Eq. (11). In the limit that the scattering is dominatedby → interactions, and three-body interactions arenegligible ( R p (cid:48) p → → ampli-tudes. We defined this process as the ladder amplitude, D p (cid:48) p , which satisfies Eq. (15). We now want to removethe ladder solution from the general three-body system.In the same vein as HS-BHS, we define the divergence-free amplitude, A df , p (cid:48) p ≡ A p (cid:48) p − D p (cid:48) p , which the → amplitude free from the ladder diagram and its singular-ities. We can then separate the ladder solution from the B -matrix representation and are left with an equation for A df , p (cid:48) p , A df , p (cid:48) p = (cid:90) k F p (cid:48) R p (cid:48) k ( F k δ kp + D kp )+ (cid:90) k F p (cid:48) ( R p (cid:48) k + G p (cid:48) k ) A df , kp . (B1) Now define the → rescattering function Eq. (19), andamputate the end caps from the divergent free amplitude, A df , p (cid:48) p = (cid:90) k (cid:48) (cid:90) k (cid:101) L pk (cid:101) T k (cid:48) k (cid:101) L kp . (B2)Substituting Eq. (B2) into Eq. (B1) and collecting terms,we arrive at (cid:90) k (cid:48) (cid:90) k [ δ p (cid:48) k (cid:48) − F p (cid:48) G p (cid:48) k (cid:48) ] (cid:101) L k (cid:48) k (cid:101) T kp = F p (cid:48) R p (cid:48) p + (cid:90) k (cid:48) (cid:90) k F p (cid:48) R p (cid:48) k (cid:48) (cid:101) L k (cid:48) k (cid:101) T kp , (B3)where we have removed the right-most rescattering func-tion, and collected all terms with R p (cid:48) p on the right handside. Finally, the combination on the left hand side sim-plifies to (cid:90) k (cid:48) [ δ p (cid:48) k (cid:48) − F p (cid:48) G p (cid:48) k (cid:48) ] (cid:101) L k (cid:48) k = F p (cid:48) δ p (cid:48) k + (cid:32) D p (cid:48) k − F p (cid:48) G p (cid:48) k F k − (cid:90) k (cid:48) F p (cid:48) G p (cid:48) k (cid:48) D k (cid:48) k (cid:33) , (B4)where the term inside the parenthesis is zero fromEq. (15). Factorizing the final → amplitude fromthe left hand side, we arrive at the resummed → amplitude, A p (cid:48) p = D p (cid:48) p + (cid:90) k (cid:48) (cid:90) k (cid:101) L p (cid:48) k (cid:48) (cid:101) T k (cid:48) k (cid:101) L kp , (B5)with the new amputated amplitude satisfying (cid:101) T p (cid:48) p = R p (cid:48) p + (cid:90) k (cid:48) (cid:90) k R p (cid:48) k (cid:48) (cid:101) L k (cid:48) k (cid:101) T kp . (B6)Equation (B5), along with Eqs. (15) and (B6), is an al-ternative on-shell representation for the → scatteringamplitude that satisfies unitarity. We now proceed withsimilar manipulations on the unitarity relation, Eq. (A4),allowing one to derive Eq. (B6) directly from unitarity.It is clear from the demonstration in Appendix A that D p (cid:48) p satisfies the same unitarity relation as Eq. (A4).Therefore, the unitarity relation for the divergence-freeamplitude statesIm A df , p (cid:48) p = (cid:90) k (cid:90) q A † df , p (cid:48) q (cid:16) ¯ ρ q δ qk + C qk (cid:17) A df , kp + (cid:90) k (cid:90) q A † df , p (cid:48) q (cid:16) ¯ ρ q δ qk + C qk (cid:17) (cid:101) L kp + (cid:90) k (cid:90) q (cid:101) L † p (cid:48) q (cid:16) ¯ ρ q δ qk + C qk (cid:17) A df , kp . (B7)Since F p obeys the → unitarity relation, Eq. (A2),and D p (cid:48) p satisfies Eq. (A4), we can see that the rescat-tering function (cid:101) L p (cid:48) p satisfies the relationIm (cid:101) L p (cid:48) p = (cid:90) k (cid:101) L † p (cid:48) k ¯ ρ k (cid:101) L kp + (cid:90) k (cid:48) (cid:90) k (cid:101) L † p (cid:48) k (cid:48) C k (cid:48) k (cid:101) L kp . (B8)2The amputated divergence-free amplitude can be definedas in Eq. (B2), so that the unitarity relation becomesIm (cid:101) T p (cid:48) p = (cid:90) k (cid:48) (cid:90) k (cid:90) q (cid:101) T † p (cid:48) k (cid:48) (cid:101) L † k (cid:48) q ¯ ρ q (cid:101) L qk (cid:101) T kp + (cid:90) k (cid:48) (cid:90) k (cid:90) q (cid:48) (cid:90) q (cid:101) T † p (cid:48) k (cid:48) (cid:101) L † k (cid:48) q (cid:48) C q (cid:48) q (cid:101) L qk (cid:101) T kp = (cid:90) k (cid:48) (cid:90) k (cid:101) T † p (cid:48) k (cid:48) Im (cid:101) L k (cid:48) k (cid:101) T kp . (B9) Using similar manipulations as in Appendix A, it isstraightforward to verify that Eq. (B6) satisfies the uni-tarity relation Eq. (B9). Appendix C: Proof of Eq. (34)
In Sec. IV, we showed that the R -matrix and three-body K -matrix are related by an integral equation, Eq. (31).Proving the reality of the Eq. (31) relied on the claim Eq. (34), which we now prove.From the definition, Eq. (28), we find that3 (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) (cid:90) k Im [ U p (cid:48) k ] (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) [ K df ( k , p ) ] (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) = (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) (cid:90) k (cid:104) ρ p (cid:48) δ p (cid:48) k − C p (cid:48) k (cid:105) (cid:96) (cid:48) m (cid:48) (cid:96) ; (cid:96)m (cid:96) [ K df ( k , p ) ] (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) = (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) δ (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) δ m (cid:48) (cid:96) m (cid:48)(cid:48) (cid:96) ρ p (cid:48) K df ,(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , p ) − (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) (cid:90) k πδ (cid:16) ( P − k − p (cid:48) ) − m (cid:17) πY (cid:96) (cid:48) m (cid:48) (cid:96) ( ˆk (cid:63) ) Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ( ˆp (cid:48) (cid:63) ) K df ,(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) ( k , p ) , (C1)where in the second term, Eq. (A5) was used. We leave the first term as is, and focus on the second term. Accordingto Ref. [13], K df ( k , p ) is defined as a symmetric object after acting with spherical harmonics of the pair orientations on K df ( k , p ), and summing over all angular momenta. We use this property to combine the product of the final sphericalharmonic Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ( ˆp (cid:48) (cid:63) ) and K df ( k , p ), and then switch the role of p (cid:48) and k , finally expanding in spherical harmonics of ˆk (cid:63) . This allows us to write (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ( ˆp (cid:48) (cid:63) ) K df ,(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) ( k , p ) = (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ( ˆk (cid:63) ) K df ,(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , p ) , (C2)Now, K df ( p (cid:48) , p ) is independent of k , thus we can perform the integrations (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) (cid:90) k πδ (cid:16) ( P − k − p (cid:48) ) − m (cid:17) πY (cid:96) (cid:48) m (cid:48) (cid:96) ( ˆk (cid:63) ) Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ( ˆk (cid:63) ) K df ,(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , p )= 14 π (cid:90) ∞ d k (cid:63) k (cid:63) ω k (cid:63) δ ( ω k (cid:63) − E (cid:63) p (cid:48) / (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) (cid:90) d ˆk (cid:63) Y (cid:96) (cid:48) m (cid:48) (cid:96) ( ˆk (cid:63) ) Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ( ˆk (cid:63) ) K df ,(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , p )= 2¯ ρ p (cid:48) (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) (cid:96) δ (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) δ m (cid:48) (cid:96) m (cid:48)(cid:48) (cid:96) K df ,(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) (cid:96) ; (cid:96)m (cid:96) ( p (cid:48) , p ) , (C3)where we converted to spherical coordinates in the finalpair rest frame, and used the composition properties ofDirac delta functions to convert to the on-shell energy ω k (cid:63) . Orthogonality properties of spherical harmonics al-lows the angular integration to be done, showing that the second term is identical to the first of Eq. (C1). Thus,we conclude that (cid:90) k Im U p (cid:48) k K df ( k , p ) = 0 , (C4)3as claimed. The relation, (cid:82) k K df ( p (cid:48) , k ) Im U kp = 0, is verified in an identical manner. [1] S. Durr et al. , Science , 1224 (2008), arXiv:0906.3599[hep-lat].[2] S. R. Beane, E. Chang, W. Detmold, H. W. Lin,T. C. Luu, K. Orginos, A. Parreno, M. J. Savage,A. Torok, and A. Walker-Loud (NPLQCD), Phys. Rev. D85 , 054511 (2012), arXiv:1109.2889 [hep-lat].[3] J. J. Dudek, R. G. Edwards, C. E. Thomas, andD. J. Wilson (Hadron Spectrum), Phys. Rev. Lett. ,182001 (2014), arXiv:1406.4158 [hep-ph].[4] R. Williams, C. S. Fischer, and W. Heupel, Phys. Rev.
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