On kinematical constraints in boson-boson systems
OOn kinematical constraints in boson-boson systems
M.F.M. Lutz ∗ GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,Planck Str. 1, 64291 Darmstadt, Germany
I. Vida˜na † Centro de F´ısica Computacional. Department of Physics,University of Coimbra, PT-3004-516 Coimbra Portugal (Dated: November 3, 2018)We consider the scattering of two-bosons with negative parity and spin 0 or 1. Starting fromhelicity partial-wave scattering amplitudes we derive transformations that eliminate all kinematicalconstraints. Such amplitudes are expected to satisfy partial-wave dispersion relations and thereforeprovide a suitable basis for data analysis and the construction of effective field theories. Our deriva-tion relies on a decomposition of the various scattering amplitudes into suitable sets of invariantfunctions. A novel algebra was developed that permits the efficient computation of such functionsin terms of computer algebra codes.
PACS numbers: 11 . . − m , 13 . .Cs , 11 . . − m I. INTRODUCTION
In a strongly interacting quantum field theory likeQCD an important challenge is the reliably and predic-tive treatment of final-state interactions. Given someeffective degrees of freedom micro-causality and coupled-channel unitarity are crucial constraints that help to es-tablish coupled-channel reaction amplitudes from a suit-able effective Lagrangian (see e.g. [1–3]).Though it is straight forward to introduce partial-wavescattering amplitudes in the helicity formalism of Jacoband Wick [4], it is a nontrivial task to derive transfor-mations that lead to amplitudes that are kinematicallyunconstrained. Such amplitudes are useful for partial-wave analysis or effective field theory approaches whichconsider the consequences of micro causality in terms ofpartial-wave dispersion-integral representations [1, 5–13].It is the purpose of the present work to derive such ampli-tudes by suitable transformations of the helicity partial-wave scattering amplitudes for two-body systems with J P = 0 − or 1 − particles. In a previous work one ofthe authors studied the scattering of 0 − off 1 − parti-cles [14] and fermion-antifermion annihilation processeswith
12 + and
32 + particles [15]. So far reactions involv-ing two body states with two 1 − particles have not beendealt with. The technique applied in this work has beenused previously in studies of two-body scattering systemswith photons, pions and nucleons [5, 16–24]. A possiblyrelated approach is by Chung and Friedrich [25, 26]. Ourresults will be relevant for the PANDA experiment atFAIR, where protons and antiprotons may be annihilatedinto systems of spin 0 or 1 states [27].We consider partial-wave projections of the scattering ∗ Electronic address: [email protected] † Electronic address: ividana@fis.uc.pt amplitude. Our goal is to establish partial-wave ampli-tudes with convenient analytic properties that justify theuse of uncorrelated integral-dispersion relations. We con-sider all two-body reactions possible with spin 0 and 1bosons. In an initial step we decompose the scatter-ing amplitude into invariant functions that are free ofkinematical constraints. Such amplitudes are expectedto satisfy a Mandelstam dispersion-integral representa-tion [16, 28]. A given choice of basis is free of kinemat-ical constraints if any additional structure can be de-composed into the basis with coefficients that are regu-lar. The identification of such a basis is a nontrivial taskas the spins of the involved particles increase. Helicitypartial-wave amplitudes are correlated at various kine-matical conditions. The derivation of such constraints isbased on an application of the previously constructed ba-sis of kinematically unconstrained invariant amplitudes.The kinematical constraints in the helicity partial-waveamplitudes are eliminated by means of non-unitary trans-formation matrices that map the initial, respectively finalhelicity sates to new covariant states.The work is organized as follows. Section II introducesthe conventions used for the kinematics and the spin-1 helicity wave functions. The scattering amplitudes aredecomposed into sets of invariant amplitudes free of kine-matical constraints. In the following section the helicitypartial-wave amplitudes are constructed within the givenconvention. The central results are presented in sectionIV, where the transformation to partial-wave amplitudesfree of kinematical constraints are derived and discussed.
II. ON-SHELL SCATTERING AMPLITUDES
We consider two-body reactions involving pseudo-scalar and vector particles. All derivations will be com-pletely generic. We introduce the 4-momenta p and ¯ p of the incoming and outgoing first particle and those of a r X i v : . [ h e p - ph ] N ov the second particle, p and ¯ p . In the center of massframe we write p µ = ( ω , , , + p ) , ¯ p µ = (¯ ω , +¯ p sin θ, , +¯ p cos θ ) ,ω = (cid:113) m + p , ¯ ω = (cid:113) ¯ m + ¯ p ,p µ = ( ω , , , − p ) , ¯ p µ = (¯ ω , − ¯ p sin θ, , − ¯ p cos θ ) ,ω = (cid:113) m + p , ¯ ω = (cid:113) ¯ m + ¯ p , (1)where θ is the scattering angle, p and ¯ p are the magni-tudes of the initial and final three-momenta. The relativemomenta p and ¯ p can be expressed in terms of the totalenergy √ s of the system p = 14 s (cid:0) s − ( m + m ) (cid:1) (cid:0) s − ( m − m ) (cid:1) w µ = p µ + p µ = ¯ p µ + ¯ p µ , s = w , ¯ p = 14 s (cid:0) s − ( ¯ m + ¯ m ) (cid:1) (cid:0) s − ( ¯ m − ¯ m ) (cid:1) . (2)It is convenient to introduce some further notation k µ = 12 ( p µ − p µ ) , r µ = k µ − p − p s w µ , ¯ k µ = 12 (¯ p µ − ¯ p µ ) , ¯ r µ = ¯ k µ −
12 ¯ p − ¯ p s w µ , ˆ g µν = g µν − s w µ w ν , ¯ r · r = − ¯ p p cos θ , (3)where the two 4-vectors r µ and ¯ r µ have a transparentrelation to the center-of-mass momenta p and ¯ p .We specify the spin-one wave functions (cid:15) µ (¯ p , ±
1) = ∓ cos θ √ − i √ ± sin θ √ , (cid:15) µ (¯ p ,
0) = ¯ p ¯ m ¯ ω ¯ m sin θ ¯ ω ¯ m cos θ ,(cid:15) µ (¯ p , ±
1) = ± cos θ √ − i √ ∓ sin θ √ , (cid:15) µ (¯ p ,
0) = ¯ p ¯ m − ¯ ω ¯ m sin θ − ¯ ω ¯ m cos θ , where the wave function of the corresponding initialstates is recovered with θ = 0.The on-shell production and scattering amplitudes aredefined in terms of plane-wave matrix elements of thescattering operator T . We represent the scattering am-plitudes in terms of a complete set of invariant func-tions F n ( s, t ). The merit of the decomposition lies in thetransparent analytic properties of such functions F n ( s, t ),which are expected to satisfy Mandelstam’s dispersionintegral representation [16, 28]. For reactions involvingspin-one particles it is not straight forward to identifysuch amplitudes.We begin with the elastic scattering of two pseu-doscalar particles T → (¯ k, k, w ) = F ( s, t ) , (4) which is characterized by one scalar function F ( s, t ) de-pending on two Mandelstam variables, e.g. s and t with s + t + u = m + m + ¯ m + ¯ m . (5)We suppress internal degrees of freedom like isospin orstrangeness quantum numbers for simplicity.A slightly more complicated process involves one vec-tor particle in the final state T → (¯ k, k, w ) = F ( s, t ) (cid:104) T (1)¯ ν (cid:105) ¯ ν → , (cid:104) T (1)¯ ν (cid:105) ¯ ν → = (cid:15) † , ¯ ν (¯ p , ¯ λ ) T (1)¯ ν ,T (1)¯ ν = i (cid:15) ¯ νταβ w τ ¯ p α p β , (6)where we use a notation analogous to the one introducedin [15]. For notational simplicity we do not introducedifferent notations for the invariant amplitudes F ( s, t )in the two reactions (4, 6). Further processes related to(6) are obtained by the exchange of the out or ingoingmomenta.The structure of the on-shell reaction amplitudes turnsmore complicated with increasing number of spin-1 par-ticles involved. Consider the production of two vectorparticles T → (¯ k, k, w ) = (cid:88) n =1 F n ( s, t ) (cid:104) T ( n )¯ µ ¯ ν (cid:105) ¯ µ ¯ ν → , (cid:104) T ( n )¯ µ ¯ ν (cid:105) ¯ µ ¯ ν → = (cid:15) † ¯ µ (¯ p , ¯ λ ) (cid:15) † ¯ ν (¯ p , ¯ λ ) T ( n )¯ µ ¯ ν ,T (1)¯ µ ¯ ν = ˆ g ¯ µ ¯ ν , T (2)¯ µ ¯ ν = w ¯ µ w ¯ ν ,T (3)¯ µ ¯ ν = w ¯ µ r ¯ ν , T (4)¯ µ ¯ ν = r ¯ µ w ¯ ν ,T (5)¯ µ ¯ ν = r ¯ µ r ¯ ν , (7)which is characterized by five invariant amplitudes, F n ( s, t ). The choice of Lorentz tensors in (7) is not unam-biguous. Various linear combinations of the given tensorsmay be used. For instance we could have used the 5 ten-sors which follow from (7) by the replacements r µ → k µ and ¯ r µ → ¯ k µ . The suggested form proves most conve-nient when calculating helicity matrix elements.The number of invariant amplitudes is easily deter-mined for the reaction 0 0 → k, k, w and v µ = (cid:15) µατβ ¯ k α w τ k β (8)At first there are 4 × v µ introduced in (8). This elim-inates 6 structures. The transversality of the spin-onewave functions with2 (cid:15) µ (¯ p ) ¯ k µ = + (cid:15) µ (¯ p ) w µ , (cid:15) µ (¯ p ) ¯ k µ = − (cid:15) µ (¯ p ) w µ , (9)eliminates additional 5 structures for on-shell conditions.Altogether there are 6 structures left. The five termsdisplayed in (7) and the Lorentz tensor v ¯ µ v ¯ ν . To showthe on-shell redundance of the extra term requires anexplicit computation of on-shell matrix elements.In a practical application it is important to derive ex-plicit expressions for the invariant amplitudes F n ( s, t )(see e.g. [29]). In the general case this is may be a te-dious exercise, which is considerably streamlined by thederivation and application of a set of projection tensors P ( n ) µν with the following properties P ( n ) µν g ¯ µµ g ¯ νν T ( m )¯ µ ¯ ν = δ nm ,P ( n )¯ µ ¯ ν ¯ p ¯ µ = 0 , P ( n )¯ µ ¯ ν ¯ p ¯ ν = 0 . (10)Given any off-shell production amplitude the invariantfunction F n ( s, t ) is obtained by the contraction with the n th projection tensor. We decompose the projection ten-sors into a basis P ( n )¯ µ ¯ ν = (cid:88) k =1 c ( n ) k Q ( k )¯ µ ¯ ν ,Q ¯ µ ¯ ν = v ¯ µ v ¯ ν /v ,Q ¯ µ ¯ ν = w (cid:99) ¯ µ w (cid:98) ¯ ν , Q ¯ µ ¯ ν = w (cid:99) ¯ µ r (cid:99)(cid:98) ¯ ν ,Q ¯ µ ¯ ν = r (cid:99)(cid:98) ¯ µ w (cid:98) ¯ ν , Q ¯ µ ¯ ν = r (cid:99)(cid:98) ¯ µ r (cid:99)(cid:98) ¯ ν , (11)where the 4-vectors r (cid:99)(cid:98) , w (cid:99) and w (cid:98) are suitable linear com-binations of ¯ r, r and w as to have the convenient proper-ties r (cid:99)(cid:98) · r = 1 , r (cid:99)(cid:98) · ¯ r = 0 = r (cid:99)(cid:98) · w ,w (cid:99) · w = 1 , w (cid:99) · r = 0 = w (cid:99) · ¯ p ,w (cid:98) · w = 1 , w (cid:98) · r = 0 = w (cid:98) · ¯ p . (12)The index (cid:99) and (cid:98) of a vector indicates whether it is or-thogonal to the 4 momentum of the first or second par-ticle respectively. The patched symbol (cid:99)(cid:98) implies the or-thogonality to both 4 momenta. Given such vectors thecoefficients c ( n ) k are readily determined. We find c ( n ) n = 1 , c (2)1 = − w (cid:99) · w (cid:98) + 1 /s , (13) c (3)1 = − w (cid:99) · r (cid:99)(cid:98) , c (4)1 = − r (cid:99)(cid:98) · w (cid:98) , c (5)1 = − r (cid:99)(cid:98) · r (cid:99)(cid:98) , where we display non-vanishing elements only.We construct the auxiliary vectors r (cid:99)(cid:98) , w (cid:99) and w (cid:98) . Forthis purpose we introduce an intermediate notation.Given three 4-vectors a µ , b µ and c µ we introduce a vector, a µb c = a µc b , as follows a µb c a b c · a b c = a µ − a · cc · c c µ − a · ( b − c · bc · c c )( b − c · bc · c c ) (cid:18) b µ − c · bc · c c µ (cid:19) ,a µb c a µ = 1 , a µb c b µ = 0 , a µb c c µ = 0 . (14)In the notation of (14) the desired vectors are identifiedwith r (cid:99)(cid:98) µ = r µ ¯ r w , w (cid:99) µ = w µr ¯ p , w (cid:98) µ = w µr ¯ p , ¯ r (cid:99)(cid:98) µ = ¯ r µr w , ¯ w (cid:99) µ = w µ ¯ r p , ¯ w (cid:98) µ = w µ ¯ r p , (15) where we introduced the additional vectors ¯ r (cid:99)(cid:98) , ¯ w (cid:99) and ¯ w (cid:98) that will turn useful below.It remains the question why did we select the five ten-sors in (7) and did not include the extra structure v ¯ µ v ¯ ν into our basis? The reason is our request that the invari-ant amplitudes should be free of kinematical constraints.The issue is nicely illustrated at hand of the on-shell iden-tity v ¯ µ v ¯ ν = v (cid:104) g ¯ µ ¯ ν − ( w (cid:99) · w (cid:98) ) w ¯ µ w ¯ ν − ( w (cid:99) · r (cid:99)(cid:98) ) w ¯ µ r ¯ ν − ( w (cid:99) · r (cid:99)(cid:98) ) r ¯ µ w ¯ ν − ( r (cid:99)(cid:98) · r (cid:99)(cid:98) ) r ¯ µ r ¯ ν (cid:105) ,v = s (cid:104) (¯ r · r ) − ¯ r r (cid:105) , (16)with the tensor basis introduced in (7). Eliminating anyof the five tensors in favor of the structure v ¯ µ v ¯ ν leads toinvariant functions singular at various kinematical condi-tions. This is evident from the regularity of the expres-sions v ( r (cid:99)(cid:98) · r (cid:99)(cid:98) ) = − s ¯ r ,v ( w (cid:99) · w (cid:98) ) = (¯ r · r ) + r ¯ p · ¯ p ,v ( r (cid:99)(cid:98) · w (cid:99) ) = ( ¯ m − ¯ m − s ) (¯ r · r ) ,v ( r (cid:99)(cid:98) · w (cid:98) ) = ( ¯ m − ¯ m + s ) (¯ r · r ) . (17)We continue with the scattering of pseudo-scalar offvector particles. There are again five invariant ampli-tudes needed to characterize the scattering amplitude T → (¯ k, k, w ) = (cid:88) n =1 F n ( s, t ) (cid:104) T ( n )¯ νν (cid:105) ¯ νν → , (cid:104) T ( n )¯ νν (cid:105) ¯ νν → = (cid:15) † ¯ ν (¯ p , ¯ λ ) T ( n )¯ νν (cid:15) ν ( p , λ ) ,T (1)¯ νν = ˆ g ¯ νν , T (2)¯ νν = w ¯ ν w ν ,T (3)¯ νν = w ¯ ν ¯ r ν , T (4)¯ νν = r ¯ ν w ν ,T (5)¯ νν = r ¯ ν ¯ r ν , (18)where it suffices to assume the second particles with mo-menta p and ¯ p to carry the spin. The type argumentsthat lead to the given choice of tensors in (18) are identi-cal to those given for the two vector production process(7). The construction of the associated projection tensors P ( n )¯ µµ g µν g ¯ µ ¯ ν T ( m )¯ νν = δ nm ,P ( m )¯ νν ¯ p ¯ ν = 0 , P ( m )¯ νν p ν = 0 , (19)is analogous to (11). We find P ( n )¯ νν = (cid:88) k =1 c ( n ) k Q ( k )¯ νν , (20) Q ¯ νν = v ¯ ν v ν /v ,Q ¯ νν = w (cid:98) ¯ ν ¯ w (cid:98) ν , Q ¯ νν = w (cid:98) ¯ ν ¯ r (cid:99)(cid:98) ν ,Q ¯ νν = r (cid:99)(cid:98) ¯ ν ¯ w (cid:98) ν , Q ¯ νν = r (cid:99)(cid:98) ¯ ν ¯ r (cid:99)(cid:98) ν , k n c ( n ) k k n c ( n ) k k n c ( n ) k r s − (¯ r · r ) s r · r ) 2 5 − (¯ r · r ) 2 7 ¯ r − ¯ r α + − ¯ α + ¯ r s ¯ α + (¯ r · r ) s ¯ α − (¯ r · r ) 3 5 ¯ α + (¯ r · r )3 7 ¯ α − ¯ r ¯ α + ¯ r − ¯ α + s − ¯ α − α − ¯ α + ¯ r s α − ¯ α + (¯ r · r ) s − α − ¯ α + (¯ r · r ) 5 5 ¯ α + ( α − (¯ r · r ) + 2 r )5 6 (¯ r · r ) 5 7 − α − ¯ α + ¯ r ¯ α + ( α − ¯ r + 2 (¯ r · r ))5 9 ¯ r ¯ α − α − ¯ α + r s − (¯ r · r ) s r · r ) 6 5 − (¯ r · r )6 7 ¯ r − ¯ r − ¯ α − ¯ α − ¯ r s − ¯ α − (¯ r · r ) s ¯ α − (¯ r · r )7 5 ¯ α + (¯ r · r ) 7 7 ¯ α − ¯ r ¯ α + ¯ r ¯ α − s ¯ α − ¯ α + − (¯ r · r ) s r s − r r − (¯ r · r ) 8 8 (¯ r · r ) 8 12 19 1 (¯ r · r ) − ¯ r r − α − ¯ α + ¯ r s α − ¯ α + (¯ r · r ) s ( − α − ¯ α + (¯ r · r ) − α − r ) 9 5 α − ¯ α + (¯ r · r ) 9 6 (¯ r · r )9 7 ( − α − ¯ α + ¯ r − α − (¯ r · r )) 9 8 α − ¯ α + ¯ r r ¯ α − α − ¯ α +
10 2 ¯ r s
10 3 − (¯ r · r ) s
10 4 (¯ r · r ) 10 5 − (¯ r · r ) 10 7 ¯ r
10 8 − ¯ r
10 10 s
10 13 ¯ α +
11 2 α − ¯ r s
11 3 − α − (¯ r · r ) s
11 4 α − (¯ r · r )11 5 − α − (¯ r · r ) − r
11 7 α − ¯ r
11 8 − α − ¯ r − (¯ r · r )11 13 − ¯ α − α −
12 2 α − ¯ r s
12 3 − α − (¯ r · r ) s
12 4 α − (¯ r · r ) − r
12 5 − α − (¯ r · r ) 12 7 α − ¯ r − (¯ r · r )12 8 − α − ¯ r
12 13 α − ¯ α +
13 6 − r
13 9 − (¯ r · r ) 13 12 α − TABLE I: The non-vanishing coefficients c ( n ) k in the expansion (24). Here is α ± = 1 ± m − m s and ¯ α ± = 1 ± ¯ m − ¯ m s . c ( n ) n = 1 , c (2)1 = − w (cid:98) · ¯ w (cid:98) + 1 /s ,c (3)1 = − w (cid:98) · ¯ r (cid:99)(cid:98) , c (4)1 = − r (cid:99)(cid:98) · ¯ w (cid:98) , c (5)1 = − r (cid:99)(cid:98) · ¯ r (cid:99)(cid:98) , where we display non-vanishing elements only and usethe 4-vectors introduced in (15).While the construction of a suitable basis was almosttrivially implied for the reactions 0 0 → → g µν and the three 4 momenta ¯ r µ , r µ and w µ . Owing to therelation (16) any structure involving an even number of v µ vectors is redundant. Altogether there are 61 Lorentzstructures with the proper parity transformation. Due tothe Schouten identity [30, 31] g στ (cid:15) αβγδ = g ατ (cid:15) σβγδ + g βτ (cid:15) ασγδ + g γτ (cid:15) αβσδ + g δτ (cid:15) αβγσ , (21)only a subset of 28 structures are off-shell independent.This is in contrast to the number of independent helic-ity amplitudes, which there are 13. Thus using on-shell conditions out of the 28 tensors only 13 are linear inde-pendent. The task is to find a subset which is free ofkinematical constraints. The construction of such a setis quite tedious and to the best knowledge of the authorssuch amplitudes did not exist for the considered reaction.The on-shell scattering amplitude may be parameterizedin terms of the following 13 scalar amplitudes F ,..., with T → (¯ k, k, w ) = (cid:88) n =1 F n ( s, t ) (cid:104) i T ( n )¯ µ ¯ ν,ν (cid:105) ¯ µ ¯ ν,ν → , (22) (cid:104) T ( n )¯ µ ¯ ν,ν (cid:105) ¯ µ ¯ ν,ν → = (cid:15) † ¯ µ (¯ p , ¯ λ ) (cid:15) † ¯ ν (¯ p , ¯ λ ) × T ( n )¯ µ ¯ ν,ν (cid:15) ν ( p , λ ) , T (1)¯ µ ¯ ν,ν = (cid:15) ¯ µ ¯ ννα w α , T (2)¯ µ ¯ ν,ν = (cid:15) ¯ µ ¯ ννα r α ,T (3)¯ µ ¯ ν,ν = (cid:15) ¯ µ ¯ ννα ¯ r α , T (4)¯ µ ¯ ν,ν = w ¯ ν (cid:15) ¯ µναβ ¯ r α w β ,T (5)¯ µ ¯ ν,ν = w ¯ µ (cid:15) ¯ νναβ ¯ r α w β , T (6)¯ µ ¯ ν,ν = r ¯ ν (cid:15) ¯ µναβ ¯ r α w β ,T (7)¯ µ ¯ ν,ν = w ¯ ν (cid:15) ¯ µναβ w α r β , T (8)¯ µ ¯ ν,ν = w ¯ µ (cid:15) ¯ νναβ w α r β ,T (9)¯ µ ¯ ν,ν = r ¯ ν (cid:15) ¯ µναβ w α r β , T (10)¯ µ ¯ ν,ν = w ¯ ν (cid:15) ¯ µναβ ¯ r α r β ,T (11)¯ µ ¯ ν,ν = ˆ g ¯ νν (cid:15) ¯ µατβ ¯ r α w τ r β ,T (12)¯ µ ¯ ν,ν = r ¯ ν w ν (cid:15) ¯ µατβ ¯ r α w τ r β ,T (13)¯ µ ¯ ν,ν = (cid:0) w ¯ ν w ν (cid:15) ¯ µατβ − w ¯ µ w ν (cid:15) ¯ νατβ (cid:1) ¯ r α w τ r β . We assure that our choice of amplitudes in (22) excludesthe occurrence of kinematical constraints with the pos-sible exception at s = 0. Using slightly modified am-plitudes as implied by the replacement r µ → k µ and¯ r µ → ¯ k µ removes the constraints at s = 0.Again we provide the convenient projection tensorsthat streamline the computation of the invariant ampli-tudes F n ( s, t ) by means of algebraic computer codes. Wefind P ( n ) αβ,τ g α ¯ µ g β ¯ ν g τν T ( m )¯ µ ¯ ν,ν = δ nm , (23) P ( n )¯ µ ¯ ν,ν ¯ p ¯ µ = 0 , P ( n )¯ µ ¯ ν,ν ¯ p ¯ ν = 0 , P ( n )¯ µ ¯ ν,ν p ν = 0 .P ( n )¯ µ ¯ ν,ν = (cid:88) k =1 c ( n ) k Q ( k )¯ µ ¯ ν,ν , (24) Q ¯ µ ¯ ν,ν = v ¯ µ v ¯ ν v ν /v /v ,Q ¯ µ ¯ ν,ν = v ¯ µ w (cid:98) ¯ ν ¯ w (cid:98) ν /v − ( w (cid:98) · ¯ w (cid:98) ) Q ¯ µ ¯ ν,ν ,Q ¯ µ ¯ ν,ν = v ¯ µ w (cid:98) ¯ ν ¯ r (cid:99)(cid:98) ν /v − ( w (cid:98) · ¯ r (cid:99)(cid:98) ) Q ¯ µ ¯ ν,ν ,Q ¯ µ ¯ ν,ν = v ¯ µ r (cid:99)(cid:98) ¯ ν ¯ w (cid:98) ν /v − ( r (cid:99)(cid:98) · ¯ w (cid:98) ) Q ¯ µ ¯ ν,ν ,Q ¯ µ ¯ ν,ν = v ¯ µ r (cid:99)(cid:98) ¯ ν ¯ r (cid:99)(cid:98) ν /v − ( r (cid:99)(cid:98) · ¯ r (cid:99)(cid:98) ) Q ¯ µ ¯ ν,ν ,Q ¯ µ ¯ ν,ν = w (cid:99) ¯ µ v ¯ ν ¯ w (cid:98) ν /v , Q ¯ µ ¯ ν,ν = w (cid:99) ¯ µ v ¯ µ ¯ r (cid:99)(cid:98) ν /v ,Q ¯ µ ¯ ν,ν = r (cid:99)(cid:98) ¯ µ v ¯ ν ¯ w (cid:98) ν /v , Q ¯ µ ¯ ν,ν = r (cid:99)(cid:98) ¯ µ v ¯ ν ¯ r (cid:99)(cid:98) ν /v ,Q ¯ µ ¯ ν,ν = w (cid:99) ¯ µ w (cid:98) ¯ ν v ν /v , Q ¯ µ ¯ ν,ν = w (cid:99) ¯ µ r (cid:99)(cid:98) ¯ ν v ν /v ,Q ¯ µ ¯ ν,ν = r (cid:99)(cid:98) ¯ µ w (cid:98) ¯ ν v ν /v , Q ¯ µ ¯ ν,ν = r (cid:99)(cid:98) ¯ µ r (cid:99)(cid:98) ¯ ν v ν /v , where the explicit form of the coefficients c ( n ) k can befound in Tab. I.We turn to the most complicated reaction 1 1 → v µ vectors there are 138 structures with the properparity transformation that one may write down. A subsetof 136 structures are off-shell independent. Using on-shell conditions out of the 136 tensors only 41 are linearindependent. We constructed a subset which is free ofkinematical constraints with the possible exception of s =0. To the best knowledge of the authors such amplitudesdid not exist for the considered scattering process. Theon-shell scattering amplitude may be parameterized interms of the following 41 scalar amplitudes F ,..., with T → (¯ k, k, w ) = (cid:88) n =1 F n ( s, t ) (cid:104) T ( n )¯ µ ¯ ν,µν (cid:105) ¯ µ ¯ ν,µν → , (25) (cid:104) T ( n )¯ µ ¯ ν,µν (cid:105) ¯ µ ¯ ν,µν → = (cid:15) † ¯ µ (¯ p , ¯ λ ) (cid:15) † ¯ ν (¯ p , ¯ λ ) × T ( n )¯ µ ¯ ν,µν (cid:15) ν ( p , λ ) (cid:15) ν ( p , λ ) , T (1)¯ µ ¯ ν,µν = ˆ g ¯ µµ ˆ g ¯ νν , T (2)¯ µ ¯ ν,µν = ˆ g ¯ µν ˆ g ¯ νµ ,T (3)¯ µ ¯ ν,µν = ˆ g ¯ µ ¯ ν ˆ g µν , T (4)¯ µ ¯ ν,µν = ˆ g ¯ νν w ¯ µ w µ ,T (5)¯ µ ¯ ν,µν = ˆ g ¯ νν r ¯ µ w µ , T (6)¯ µ ¯ ν,µν = ˆ g ¯ νν w ¯ µ ¯ r µ ,T (7)¯ µ ¯ ν,µν = ˆ g ¯ νν r ¯ µ ¯ r µ , T (8)¯ µ ¯ ν,µν = ˆ g ¯ µµ w ¯ ν w ν ,T (9)¯ µ ¯ ν,µν = ˆ g ¯ µµ r ¯ ν w ν , T (10)¯ µ ¯ ν,µν = ˆ g ¯ µµ w ¯ ν ¯ r ν ,T (11)¯ µ ¯ ν,µν = ˆ g ¯ µµ r ¯ ν ¯ r ν , T (12)¯ µ ¯ ν,µν = ˆ g ¯ νµ w ¯ µ w ν ,T (13)¯ µ ¯ ν,µν = ˆ g ¯ νµ r ¯ µ w ν , T (14)¯ µ ¯ ν,µν = ˆ g ¯ νµ w ¯ µ ¯ r ν ,T (15)¯ µ ¯ ν,µν = ˆ g ¯ νµ r ¯ µ ¯ r ν , T (16)¯ µ ¯ ν,µν = ˆ g ¯ µν w ¯ ν w µ ,T (17)¯ µ ¯ ν,µν = ˆ g ¯ µν r ¯ ν w µ , T (18)¯ µ ¯ ν,µν = ˆ g ¯ µν w ¯ ν ¯ r µ ,T (19)¯ µ ¯ ν,µν = ˆ g ¯ µν r ¯ ν ¯ r µ , T (20)¯ µ ¯ ν,µν = ˆ g ¯ µ ¯ ν w µ w ν ,T (21)¯ µ ¯ ν,µν = ˆ g ¯ µ ¯ ν ¯ r µ w ν , T (22)¯ µ ¯ ν,µν = ˆ g ¯ µ ¯ ν w µ ¯ r ν ,T (23)¯ µ ¯ ν,µν = ˆ g ¯ µ ¯ ν ¯ r µ ¯ r ν , T (24)¯ µ ¯ ν,µν = ˆ g µν w ¯ µ w ¯ ν ,T (25)¯ µ ¯ ν,µν = ˆ g µν r ¯ µ w ¯ ν , T (26)¯ µ ¯ ν,µν = ˆ g µν w ¯ µ r ¯ ν ,T (27)¯ µ ¯ ν,µν = ˆ g µν r ¯ µ r ¯ ν , T (28)¯ µ ¯ ν,µν = w ¯ µ w ¯ ν w µ w ν ,T (29)¯ µ ¯ ν,µν = r ¯ µ r ¯ ν w µ w ν , T (30)¯ µ ¯ ν,µν = w ¯ µ w ¯ ν ¯ r µ ¯ r ν ,T (31)¯ µ ¯ ν,µν = r ¯ µ w ¯ ν w µ w ν , T (32)¯ µ ¯ ν,µν = w ¯ µ r ¯ ν w µ w ν ,T (33)¯ µ ¯ ν,µν = r ¯ µ w ¯ ν ¯ r µ ¯ r ν , T (34)¯ µ ¯ ν,µν = w ¯ µ r ¯ ν ¯ r µ ¯ r ν ,T (35)¯ µ ¯ ν,µν = w ¯ µ w ¯ ν ¯ r µ w ν , T (36)¯ µ ¯ ν,µν = r ¯ µ r ¯ ν ¯ r µ w ν ,T (37)¯ µ ¯ ν,µν = w ¯ µ w ¯ ν w µ ¯ r ν , T (38)¯ µ ¯ ν,µν = r ¯ µ r ¯ ν w µ ¯ r ν ,T (39)¯ µ ¯ ν,µν = (cid:0) r ¯ µ w ¯ ν + w ¯ µ r ¯ ν (cid:1) (cid:0) ¯ r µ w ν − w µ ¯ r ν (cid:1) ,T (40)¯ µ ¯ ν,µν = (cid:0) r ¯ µ w ¯ ν − w ¯ µ r ¯ ν (cid:1) (cid:0) ¯ r µ w ν + w µ ¯ r ν (cid:1) ,T (41)¯ µ ¯ ν,µν = (cid:0) r ¯ µ w ¯ ν − w ¯ µ r ¯ ν (cid:1) (cid:0) ¯ r µ w ν − w µ ¯ r ν (cid:1) , where our invariant functions are kinematically corre-lated at s = 0 only. The latter constraint can be elim-inated by the use of the modified vectors r µ → k µ and¯ r µ → ¯ k µ in (25). An algebra to project onto the invariantamplitudes F n ( s, t ) is developed in Appendix A.We emphasize that all amplitudes F n ( s, t ) introducedin this section are truly uncorrelated and satisfy Mandel-stam’s dispersion integral representation [16, 28]. III. PARTIAL-WAVE DECOMPOSITION
The helicity matrix elements of the scattering operator, T , are decomposed into partial-wave amplitudes charac-terized by the total angular momentum J . Given a spe-cific process together with our convention of the helicitywave functions it suffices to specify the helicity projection λ , λ and ¯ λ , ¯ λ as introduced in the previous section.We write (cid:10) ¯ λ ¯ λ (cid:12)(cid:12) T | λ λ (cid:105) = (cid:88) J (2 J +1) (cid:104) ¯ λ ¯ λ | T J | λ λ (cid:105) d ( J ) λ, ¯ λ ( θ ) ,d ( J ) λ, ¯ λ ( θ ) = ( − ) λ − ¯ λ d ( J ) − λ, − ¯ λ ( θ )= ( − ) λ − ¯ λ d ( J )¯ λ,λ ( θ ) = d ( J ) − ¯ λ, − λ ( θ ) , (26) (cid:104) ¯ λ ¯ λ | T J | λ λ (cid:105) = (cid:90) − − d cos θ (cid:104) ¯ λ ¯ λ | T | λ λ (cid:105) d ( J ) λ, ¯ λ ( θ ) , with λ = λ − λ and ¯ λ = ¯ λ − ¯ λ . Wigner’s rotationfunctions, d ( J ) λ, ¯ λ ( θ ), are used in a convention as charac-terized by (26). The phase conventions assumed in this a b (cid:2) U J + , (cid:3) ab a b (cid:2) U J + , (cid:3) ab a b (cid:2) U J + , (cid:3) ab M + − M − ) √ s p α − s p (cid:113) JJ +1 − pa b (cid:2) U J + , (cid:3) ab a b (cid:2) U J + , (cid:3) ab a b (cid:2) U J + , (cid:3) ab −√ sp − ( M + + M − ) √ sp (cid:113) JJ +1 α − ( M + + M − ) √ s p M + + M − ) p √ s − ( M + − M − ) √ sp (cid:113) JJ +1 − α + ( M + − M − ) √ s p M + − M − ) p √ s M + M − p (cid:113) J − J ( J +1) ( J +2) − α − α + s p (cid:113) J − J +2 − M + M − p s (cid:113) J − J +2 p √ a b (cid:2) U J − , (cid:3) ab a b (cid:2) U J − , (cid:3) ab a b (cid:2) U J − , (cid:3) ab M − − M ) s p (cid:113) J +1 J α − α + s p (cid:113) J +1) J − s (cid:113) J +1) J − α − ( M − + M + ) √ s s p M − + M + ) √ s α + ( M + − M − ) √ s s p M + − M − ) √ s √ s ( M − − M + ) 5 1 α − α + s p (cid:113) J − J +2 s (cid:113) J − J +2 − M + M − (cid:113) J − J +2 − α − s (cid:113) J − J +2 − p √ J +1) √ J − √ J +2 TABLE II: Non-zero elements of the transformation matrices U J + , and U J ± , . We use the notation of (35). work imply the parity relations (cid:104)− ¯ λ , − ¯ λ | T | − λ , − λ (cid:105) = ( − ) ∆ (cid:104) ¯ λ , ¯ λ | T | λ , λ (cid:105) , ∆ = S − S + ¯ S − ¯ S + λ − λ − ¯ λ + ¯ λ . (27)It is useful to decouple the two parity sectors by intro-ducing parity eigenstates of good total angular momen- tum J , formed in terms of the helicity states [4]. Follow-ing (26) we introduce the angular momentum projection, | λ , λ (cid:105) J , of the helicity state | λ , λ (cid:105) . We write | λ , λ (cid:105) J , with T | λ , λ (cid:105) J = T J | λ , λ (cid:105) . (28)We introduce parity eigenstates states, | n ± , J (cid:105) , thatare eigenstates of the total angular momentum. We willbe applying the following state convention | − , J (cid:105) = | , (cid:105) J , (29) | − , J (cid:105) = √ (cid:16) | , −(cid:105) J − | , + (cid:105) J (cid:17) , | − , J (cid:105) = | , (cid:105) J , | − , J (cid:105) = √ (cid:16) | + , + (cid:105) J + |− , −(cid:105) J (cid:17) , | − , J (cid:105) = √ (cid:16) | , −(cid:105) J + | , + (cid:105) J (cid:17) , | − , J (cid:105) = √ (cid:16) | + , (cid:105) J + |− , (cid:105) J (cid:17) , | − , J (cid:105) = √ (cid:16) | + , −(cid:105) J + |− , + (cid:105) J (cid:17) , and | + , J (cid:105) = | , (cid:105) J , (30) | + , J (cid:105) = √ (cid:16) | , −(cid:105) J + | , + (cid:105) J (cid:17) , | + , J (cid:105) = √ (cid:16) | + , + (cid:105) J − |− , −(cid:105) J (cid:17) , | + , J (cid:105) = √ (cid:16) | , −(cid:105) J − | , + (cid:105) J (cid:17) , | + , J (cid:105) = √ (cid:16) | + , (cid:105) J − |− , (cid:105) J (cid:17) , | + , J (cid:105) = √ (cid:16) | + , −(cid:105) J − |− , + (cid:105) J (cid:17) , where we suppress the sector index 0 0, 0 1 or 1 1 on theright hand sides and use the short-hand notation ± ≡ ± P | n ± , J (cid:105) = ± ( − J +1 | n ± , J (cid:105) . (31)The partial-wave helicity amplitudes t J ± ,ab that carrygood angular momentum J and good parity are definedwith t J ± ,ab = (cid:104) a ± , J | T | b ± , J (cid:105) , (32)where a and b label the states. For sufficiently large s the unitarity condition takes the simple form (cid:61) (cid:2) t ( J ) (cid:3) − ab = − p a π √ s δ ab , (33)where the index a and b spans the basis of two-particlehelicity states in the (0 , , (0 ,
1) and (1 ,
1) sectors.Helicity-partial-wave amplitudes are correlated at spe-cific kinematical conditions. This is seen once the am-plitudes t J ± ,ab ( s ) are expressed in terms of the invariantfunctions F n ( s, t ). This is a well know problem relatedto the use of the helicity basis in covariant models, seefor example the review [23]. In contrast covariant-partialwave amplitudes T J ± ( s ) are free of kinematical constraintsand can therefore be used efficiently in partial-wave dis-persion relation. They are associated to covariant statesand a covariant projector algebra which diagonalizes theBethe-Salpeter two-body scattering equation for local in-teractions [9, 10, 14, 15]. We introduce T J ± ( s ) = (cid:18) s ¯ p p (cid:19) J (cid:2) ¯ U J ± ( s ) (cid:3) T t J ± ( s ) U J ± ( s ) , (34)with nontrivial matrices U J ± ( s ) and ¯ U J ± ( s ) characterizingthe transformation for the initial and final states fromthe helicity basis to the new kinematic-free basis. Thematrices U J ± ( s ) are given in Tab. III, where we use theconventions M ± = m ± m , ¯ M ± = ¯ m ± ¯ m ,δ = M + M − s , ¯ δ = ¯ M + ¯ M − s ,α ± = 1 ± δ , ¯ α ± = 1 ± ¯ δ . (35)The transformation (34) implies a change in the phase-space distribution: ρ J ± ( s ) = −(cid:61) (cid:104) T J ± ( s ) (cid:105) − = 18 π (cid:18) p √ s (cid:19) J +1 (cid:104) U J ± ( s ) (cid:105) − (cid:104) U J ± ( s ) (cid:105) T, − . (36)We adapt a convention for the transformation matricesthat lead to an asymptotically bounded phase-space ma-trix, i.e. we requirelim s →∞ det ρ J ± ( s ) = const (cid:54) = 0 . (37)In contrast to the helicity states the phase-space matrixin the covariant states does have off-diagonal elements.The quest for the elimination of kinematical constraintsleads necessarily to off-diagonal elements. To the bestknowledge of the authors transformation matrices for the’1 1’ case in Tab. III are novel and not presented in theliterature before.We provide particularly detailed results for the 0 1 → → → U J − , = 1 , U J − , = 1 . (38) It follows T J − , → ( s ) = A J ( s ) ,T J − , → ( s ) = (cid:112) J ( J + 1)(2 J + 1) √ s (cid:104) s A J − ( s ) − ¯ p p A J +11 ( s ) (cid:105) ,T J − , → ( s ) = − A J ( s ) + ¯ p p (2 J + 1) s A J +15 ( s ) − s J + 1 A J − ( s ) , (39)with A Jn ( s ) = (cid:18) s ¯ p p (cid:19) J (cid:90) − d cos θ F n ( s, t ) P J (cos θ ) . (40)The important merit of (39) is the absence of kinemati-cal constraints, with the repeatedly discussed exceptionat s = 0. A potential singularity at ¯ p p = 0 in (40) is notrealized due to the properties of the Legendre polynomi-als P J (cos θ ).The corresponding decomposition of the partial-wavescattering amplitudes T J ( s ) into integrals over the in-variant amplitude F n ( s, t ) for the 0 1 → → → → T J ± ( s ) = (cid:88) k,n a J + k ± n ( s ) A J + kn ( s ) , (41)with coefficients a J + k ± n ( s ).In Tab. III we detail those results for the coefficients a J + k ± n ( s ) which demonstrate the absence of kinematicalsingularities in all partial-wave amplitudes considered inthis work. We note that the transformation matrices inTab. III are derived from a study of the three reactions0 1 → → → a J + k ± n ( s )in Tab. III it follows the absence of kinematical sin-gularities. Additional and tedious computations revealthat there are also no correlations of the various partial-wave amplitudes T J ± ( s ) at any kinematical point but at s = 0. The covariant amplitudes are associated to a pro-jector algebra for the Bethe-Salpeter scattering equation[9, 10, 14, 15]. A consistency check was performed thatconfirm the claimed properties for the remaining cases inparticular the most tedious 1 1 → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → s α − α − s J +1)(2 J +3) − ¯ α − α − s
14 2 J +1 J +1 α − ¯ p s α − p s α − α − ¯ p p J +2)(2 J +1)( J +1)(2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − s √ J √ J +12 J +1 α − p s √ J ( J +2) √ J +1 (2 J +3) α − p √ J √ J +1 − ¯ p p √ J √ J +12 J +1 − ¯ α − ¯ p p s √ J ( J +2) √ J +1 (2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − s √ J √ J +12 J +1 α − ¯ p s √ J ( J +2) √ J +1 (2 J +3) α − ¯ p √ J √ J +1 − ¯ p p √ J √ J +12 J +1 − α − ¯ p p s √ J ( J +2) √ J +1 (2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − s J − J − − − s J +12 J +1 − ¯ p p J ( J +1) − J ( J +1) − − ¯ p p s J J +1 p p s J ( J +2)4 J ( J +2)+3 k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − √ s s J J +1 − − √ s s J J +1 − √ s s α − √ s s α + J +1( J +1)(2 J +3) α − √ s s JJ +1 α − √ s s JJ +1 α − √ s s J +1)(¯ α − J +2)( J +1)(2 J +3) − α − √ s s J +1 J +1 − p √ s − α − ¯ p √ s s JJ +1 − α − ¯ p √ s s JJ +1 α − α − p √ s s JJ +1 − p p √ s J J +1 p p √ s J J +1 − α − ¯ p p √ s
12 (2 J +1)(¯ α − J +2)( J +1)(2 J +3) − α − ¯ p p √ s
12 (2 J +1)(¯ α − J +2)( J +1)(2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − √ s s √ J √ J +12 J +1 − √ s s √ J √ J +12 J +1 p √ s s √ J (¯ α − J − δ +1) √ J +1 (2 J +3) − p √ s s √ J √ J +1 − p √ s s √ J √ J +1 − p √ s s √ J (¯ α − J +2) √ J +1 (2 J +3) p √ s √ J √ J +1 p p √ s √ J J √ J +1 (2 J +1) p p √ s √ J J √ J +1 (2 J +1) − ¯ α − p √ s √ J √ J +1 p p √ s √ J (¯ α − J +2) √ J +1 (2 J +3) p p √ s √ J (¯ α − J +2) √ J +1 (2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − α + √ s s √ J √ J +12 J +1 − − ¯ α + √ s s √ J √ J +12 J +1 − α − √ s s √ J √ J +1 − ¯ α − ¯ α + α − √ s s √ J √ J +1 (2 J +3) α + α − √ s s √ J √ J +1 − ¯ α − α − √ s s √ J √ J +1 α − ¯ α + α − √ s s √ J J (2 J +1) √ J +1 (2 J +3) − α − ¯ p √ s √ J √ J +1 − ¯ α + α − ¯ p √ s s √ J √ J +1 α − α − ¯ p √ s s √ J √ J +1 α − ¯ α + α − p √ s s √ J J √ J +1 − ¯ α + ¯ p p √ s √ J √ J +12 J +1 α + ¯ p p √ s √ J √ J +12 J +1 − ¯ α − ¯ α + α − ¯ p p √ s √ J (2 J +1) √ J +1 (2 J +3) − ¯ α − ¯ α + α − ¯ p p √ s √ J (2 J +1) √ J +1 (2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − √ s s J +12 J +1 − α + √ s s J +12 J +1 − − ¯ α − √ s s J +12 J +1 p √ s α − ¯ α + p √ s s J +32 J +3 − ¯ α + p √ s s α − p √ s s − ¯ α − ¯ α + p √ s s J J +3 p p √ s J J +1 α + ¯ p p √ s J J +1 − ¯ α − ¯ p p √ s J J +1 − ¯ α − ¯ α + p √ s α − ¯ α + ¯ p p √ s J J +3 α − ¯ α + ¯ p p √ s J J +3 TABLE III: Non-vanishing coefficients a J + k ± n in the expansion (41) for the reactions 0 1 → → → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − −√ s s √ J √ J +12 J +1 − − α − √ s s √ J √ J +12 J +1 − − ¯ p √ s s √ J √ J +12 J +1 − p √ s s √ J √ J +12 J +1 − α − ¯ p √ s s − √ J ( − δ + J +1) √ J +1 (2 J +3) − α − ¯ p √ s s √ J √ J +1 − α − ¯ p √ s s √ J √ J +1 α − ¯ p √ s s √ J (2 ¯ δ J +¯ δ − J − √ J +1 (2 J +3) α − ¯ p √ s √ J √ J +1 p p √ s √ J √ J +12 J +1 α − ¯ p √ s √ J √ J +1 α − ¯ p √ s √ J √ J +1 α − ¯ p p √ s √ J (2 ¯ δ J +¯ δ − J ) √ J +1 (2 J +1) p p √ s √ J √ J +12 J +1 − ¯ p p √ s √ J √ J +12 J +1 − α − ¯ p p √ s √ J (2 ¯ δ J +¯ δ − J − √ J +1 (2 J +3) − α − ¯ p p √ s √ J (2 ¯ δ J +¯ δ − J − √ J +1 (2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − −√ s s J − J − − −√ s s J − J − − −√ s s J +12 J +1 − − ¯ p √ s s J +12 J +1 − − ¯ p √ s s J +12 J +1 − p √ s s J +12 J +1 p p √ s
12 ¯ δ ( J +3)(2 J − − J ( J +1)4 J ( J +1) − p p √ s p p √ s − ¯ p p √ s
12 2 (¯ δ − J − (¯ δ +2) J +34 J ( J +1) − − ¯ p p √ s J J +1 − ¯ p p √ s J J +1 − ¯ p p √ s J J +1 − ¯ p p √ s δ J +¯ δ − J J +1 p p √ s s J (2 ¯ δ J +¯ δ − J − J ( J +2)+3 p p √ s s J (2 ¯ δ J +¯ δ − J − J ( J +2)+3 k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − α − √ s s √ J √ J ( J +1) − √ J +1 (2 J +1) − α − ¯ p √ s s √ J √ J ( J +1) − √ J +1 (2 J +3) − α − ¯ p √ s s √ J √ J ( J +1) − √ J +1 (2 J +3) − α − ¯ p p √ s √ J √ J ( J +1) − √ J +1 (2 J +1) α − ¯ p p √ s √ J √ J ( J +1) − √ J +1 (2 J +3) α − ¯ p p √ s √ J √ J ( J +1) − √ J +1 (2 J +3) k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → k n (cid:2) a J + k + n (cid:3) → − √ s s J +1) √ J ( J +1) − J − J +1) − √ s s J +1) √ J ( J +1) − J − J +1) − − p √ s s √ J ( J +1) − J +1 − p p √ s √ J ( J +1) − J − J +3) − p p √ s √ J ( J +1) − J − J +3) p p √ s √ J ( J +1) − J +1 − ¯ p p √ s s J √ J ( J +1) − J +1)(2 J +3) − ¯ p p √ s s J √ J ( J +1) − J +1)(2 J +3) k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → − s √ J +1 √ J − ¯ α − p s √ J +1 √ J α + p s √ J +1 √ J α − ¯ α + p √ J +1 √ J k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → − s √ J +1 √ J p s √ J +1 √ J (2 J +3) − p p s √ J +1 √ J (2 J +3) k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → − − s √ J √ J +1 (2 J +3)4 J ( J +2)+3 − − s √ J √ J +1 (2 J +3)4 J ( J +2)+3 M − ¯ M + p √ J √ J +1 (2 J +1)4 J ( J +2)+3 p p √ J √ J +1 (2 J +3)4 J ( J +2)+3 p p √ J √ J +1 (2 J +3)4 J ( J +2)+3 − ¯ M − ¯ M + ¯ p p s √ J √ J +1 (2 J +1)4 J ( J +2)+3 k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → − s √ J √ J +1 (2 J +3)4 J ( J +2)+3 α − p s √ J √ J +1 (2 J +1)4 J ( J +2)+3 − p p √ J √ J +1 (2 J +3)4 J ( J +2)+3 − ¯ α − ¯ p p s √ J √ J +1 (2 J +1)4 J ( J +2)+3 k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → k n (cid:2) a J + k − n (cid:3) → − − s √ J √ J +1 (2 J +3)4 J ( J +1) − p p √ J √ J +1 (2 J +1)4 J ( J +1) − − ¯ p p s √ J √ J +1 (2 J − J ( J +1) − TABLE IV: Continuation of Tab. III. We use the notation of (35). IV. CONCLUSIONS
We have constructed partial-wave amplitudes for two-body reactions involving J P = 0 − and J P = 1 − particleswhich are free from kinematical constraints and frame in-dependent. Those covariant partial-wave amplitudes arewell suited to be used in partial-wave dispersion rela-tions and data analysis. Explicit transformations fromthe conventional helicity states to the covariant stateswere derived and presented in this work. In an initialstep we identified complete sets of invariant functionsthat parameterize the scattering amplitudes of the vari-ous processes and are kinematically unconstrained. Thelatter are expected to satisfy Mandelstam’s dispersionintegral representation. Explicit expressions for the co-variant partial-wave scattering amplitudes in terms of in-tegrals over the invariant amplitudes were derived andpartially presented in this work. A convenient projectionalgebra was constructed that streamlines the derivationof the invariant amplitudes by means of computer algebracodes significantly.The present paper thus offers an efficient startingpoint for analyzing boson-boson scattering in a covari-ant coupled-channel approach that takes into account theconstraints set by micro-causality and coupled-channelunitarity. Acknowledgments
M.F.M.L. thanks the Centro de F´ısica Computacionalat the University of Coimbra for kind hospitalilty.
Appendix A
We provide projection tensors for the 1 1 → F n ( s, t ) introduced in (25) by means ofalgebraic computer codes. We find P ( n ) αβ,τσ g α ¯ µ g β ¯ ν g τµ g σν T ( m )¯ µ ¯ ν,µν = δ nm ,P ( n )¯ µ ¯ ν,µν ¯ p ¯ µ = 0 , P ( n )¯ µ ¯ ν,µν ¯ p ¯ ν = 0 ,P ( n )¯ µ ¯ ν,µν p µ = 0 , P ( n )¯ µ ¯ ν,µν p ν = 0 .P ( n )¯ µ ¯ ν,µν = (cid:88) k =1 c ( n ) k Q ( k )¯ µ ¯ ν,νµ , (A1) Q ¯ µ ¯ ν,µνn = ¯ L ¯ µ ¯ ν (cid:100) n/ (cid:101) L µνn +5 − (cid:100) n/ (cid:101) for n ≤ ,Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v µ w (cid:98) ¯ ν ¯ w (cid:98) ν , Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v µ w (cid:98) ¯ ν ¯ r (cid:99)(cid:98) ν ,Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v µ r (cid:99)(cid:98) ¯ ν ¯ w (cid:98) ν , Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v µ r (cid:99)(cid:98) ¯ ν ¯ r (cid:99)(cid:98) ν ,Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v ν w (cid:99) ¯ µ ¯ w (cid:99) µ , Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v ν w (cid:99) ¯ µ ¯ r (cid:99)(cid:98) µ ,Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v ν r (cid:99)(cid:98) ¯ µ ¯ w (cid:99) µ , Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v ν r (cid:99)(cid:98) ¯ µ ¯ r (cid:99)(cid:98) µ , Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v µ w (cid:99) ¯ µ ¯ w (cid:98) ν , Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v µ w (cid:99) ¯ µ ¯ r (cid:99)(cid:98) ν ,Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v µ r (cid:99)(cid:98) ¯ µ ¯ w (cid:98) ν , Q ¯ µ ¯ ν,µν = ˆ v ¯ ν ˆ v µ r (cid:99)(cid:98) ¯ µ ¯ r (cid:99)(cid:98) ν ,Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v ν w (cid:98) ¯ ν ¯ w (cid:99) µ , Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v ν w (cid:98) ¯ ν ¯ r (cid:99)(cid:98) µ ,Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v ν r (cid:99)(cid:98) ¯ ν ¯ w (cid:99) µ , Q ¯ µ ¯ ν,µν = ˆ v ¯ µ ˆ v ν r (cid:99)(cid:98) ¯ ν ¯ r (cid:99)(cid:98) µ , with ˆ v µ = v µ /v . The ceiling function, (cid:100) x (cid:101) − ≤ x ≤ (cid:100) x (cid:101) , (A2)maps a real number x onto an integer number (cid:100) x (cid:101) asdefined by (A2). The tensors L µνn and ¯ L ¯ µ ¯ νn used in (A1)are L µν = v µ v ν /v ,L µν = ¯ w (cid:99) µ ¯ w (cid:98) ν − (cid:2) ( ¯ w (cid:99) · ¯ w (cid:98) ) − ( ¯ w (cid:99) · w ) ( w · ¯ w (cid:98) ) /w (cid:3) L µν ,L µν = ¯ w (cid:99) µ ¯ r (cid:99)(cid:98) ν − ( ¯ w (cid:99) · ¯ r (cid:99)(cid:98) ) L µν ,L µν = ¯ r (cid:99)(cid:98) µ ¯ w (cid:98) ν − (¯ r (cid:99)(cid:98) · ¯ w (cid:98) ) L µν ,L µν = ¯ r (cid:99)(cid:98) µ ¯ r (cid:99)(cid:98) ν − (¯ r (cid:99)(cid:98) · ¯ r (cid:99)(cid:98) ) L µν , ¯ L ¯ µ ¯ ν = v ¯ µ v ¯ ν /v , ¯ L ¯ µ ¯ ν = w (cid:99) ¯ µ w (cid:98) ¯ ν − (cid:2) ( w (cid:99) · w (cid:98) ) − ( w (cid:99) · w ) ( w · w (cid:98) ) /w (cid:3) ¯ L ¯ µ ¯ ν , ¯ L ¯ µ ¯ ν = w (cid:99) ¯ µ r (cid:99)(cid:98) ¯ ν − ( w (cid:99) · r (cid:99)(cid:98) ) ¯ L ¯ µ ¯ ν , ¯ L ¯ µ ¯ ν = r (cid:99)(cid:98) ¯ µ w (cid:98) ¯ ν − ( r (cid:99)(cid:98) · w (cid:98) ) ¯ L ¯ µ ¯ ν , ¯ L ¯ µ ¯ ν = r (cid:99)(cid:98) ¯ µ r (cid:99)(cid:98) ¯ ν − ( r (cid:99)(cid:98) · r (cid:99)(cid:98) ) ¯ L ¯ µ ¯ ν . (A3)The derivation of the coefficient matrix c ( n ) k appearsprohibitively cumbersome at first. A 41 ×
41 matrix needsto be inverted. The merit of the algebra developed in thiswork are concise and manageable expressions for the coef-ficients c ( n ) k . They are presented in terms of the buildingblocks δ = m − m s , α ± = 1 ± δ , (A4)¯ δ = ¯ m − ¯ m s , ¯ α ± = 1 ± ¯ δ ,β ± = ± δ δ + ¯ δ − δ , γ ± = ± δ δ + ¯ δ + δ . In Tab. V we specify k and n for all coefficients with c ( n ) k = 1. In Tab. VI we detail all remaining non-vanishing elements in the expansion (A1). k n k
11 12 15 16 17 20 21 22 23 24 n
26 32 34 25 31 33 27 29 38 36TABLE V: The non-vanishing coefficients c ( n ) k = 1. k n c ( n ) k k n c ( n ) k k n c ( n ) k
13 1 − v /s
13 2 v /s
13 4 ¯ α + α + (¯ r · r )13 5 α + ¯ r
13 6 ¯ α + r
13 7 (¯ r · r )13 8 ¯ α − α − (¯ r · r ) 13 9 − α − ¯ r
13 10 − ¯ α − r
13 11 (¯ r · r ) 13 12 α − ¯ α + (¯ r · r ) 13 13 α − ¯ r
13 14 − ¯ α + r
13 15 − (¯ r · r ) 13 16 ¯ α − α + (¯ r · r )13 17 − α + ¯ r
13 18 ¯ α − r
13 19 − (¯ r · r )13 39 − α −
13 40 − ¯ α −
13 41 1 − ¯ δ δ
14 1 − v /s
14 2 v /s
14 4 ¯ α + α + (¯ r · r )14 5 α + ¯ r
14 6 ¯ α + r
14 7 (¯ r · r )14 8 ¯ α − α − (¯ r · r ) 14 9 − α − ¯ r
14 10 − ¯ α − r
14 11 (¯ r · r ) 14 12 α − ¯ α + (¯ r · r ) 14 13 α − ¯ r
14 14 − ¯ α + r
14 15 − (¯ r · r ) 14 16 ¯ α − α + (¯ r · r )14 17 − α + ¯ r
14 18 ¯ α − r
14 19 − (¯ r · r )14 39 α +
14 40 − ¯ α −
14 41 − ¯ δ δ −
118 1 − v /s
18 2 v /s
18 4 ¯ α + α + (¯ r · r )18 5 α + ¯ r
18 6 ¯ α + r
18 7 (¯ r · r )18 8 ¯ α − α − (¯ r · r ) 18 9 − α − ¯ r
18 10 − ¯ α − r
18 11 (¯ r · r ) 18 12 α − ¯ α + (¯ r · r ) 18 13 α − ¯ r
18 14 − ¯ α + r
18 15 − (¯ r · r ) 18 16 ¯ α − α + (¯ r · r )18 17 − α + ¯ r
18 18 ¯ α − r
18 19 − (¯ r · r )18 39 − α −
18 40 ¯ α +
18 41 − ¯ δ δ −
119 1 − v /s
19 2 v /s
19 4 ¯ α + α + (¯ r · r )19 5 α + ¯ r
19 6 ¯ α + r
19 7 (¯ r · r )19 8 ¯ α − α − (¯ r · r ) 19 9 − α − ¯ r
19 10 − ¯ α − r
19 11 (¯ r · r ) 19 12 α − ¯ α + (¯ r · r ) 19 13 α − ¯ r
19 14 − ¯ α + r
19 15 − (¯ r · r ) 19 16 ¯ α − α + (¯ r · r )19 17 − α + ¯ r
19 18 ¯ α − r
19 19 − (¯ r · r )19 39 α +
19 40 ¯ α +
19 41 1 − ¯ δ δ
25 1 − (¯ δ δ + 1) v /s
25 2 (¯ δ δ − v /s
25 3 v /s
25 4 ¯ α + α + (¯ δ δ + 1) (¯ r · r ) 25 5 α + (¯ δ δ + 1) ¯ r
25 6 ¯ α + (¯ δ δ + 1) r
25 7 (¯ δ δ + 1) (¯ r · r ) 25 8 ¯ α − α − (¯ δ δ + 1) (¯ r · r ) 25 9 − α − (¯ δ δ + 1) ¯ r
25 10 − ¯ α − (¯ δ δ + 1) r
25 11 (¯ δ δ + 1) (¯ r · r ) 25 12 α − ¯ α + (¯ δ δ −
1) (¯ r · r )25 13 α − (¯ δ δ −
1) ¯ r
25 14 − ¯ α + (¯ δ δ − r
25 15 − (¯ δ δ −
1) (¯ r · r )25 16 ¯ α − α + (¯ δ δ −
1) (¯ r · r ) 25 17 − α + (¯ δ δ −
1) ¯ r
25 18 ¯ α − (¯ δ δ − r
25 19 − (¯ δ δ −
1) (¯ r · r ) 25 20 − α − α + ¯ r
25 21 − α − (¯ r · r )25 22 α + (¯ r · r ) 25 23 r
25 24 − ¯ α − ¯ α + r
25 25 − ¯ α − (¯ r · r ) 25 26 ¯ α + (¯ r · r ) 25 27 ¯ r
25 28 − ¯ α − α − ¯ α + α +
25 39 − α − α + ¯ δ
25 40 − ¯ α − ¯ α + δ
25 41 − (¯ δ δ −
1) 26 1 (¯ r · r ) v
26 2 − (¯ r · r ) v
26 4 − ¯ α + α + (¯ r · r ) s
26 5 − α + ¯ r (¯ r · r ) s
26 6 − ¯ α + (¯ r · r ) r s
26 7 − (¯ r · r ) s
26 8 (4 v − ¯ α − α − (¯ r · r ) s ) 26 9 α − ¯ r (¯ r · r ) s
26 10 ¯ α − (¯ r · r ) r s
26 11 ( − (¯ r · r ) ) s
26 12 − α − ¯ α + (¯ r · r ) s
26 13 − α − ¯ r (¯ r · r ) s
26 14 ¯ α + (¯ r · r ) r s
26 15 (¯ r · r ) s
26 16 − ¯ α − α + (¯ r · r ) s
26 17 α + ¯ r (¯ r · r ) s
26 18 − ¯ α − (¯ r · r ) r s
26 19 (¯ r · r ) s
26 28 − ¯ α + α + (¯ r · r ) s
26 31 − α + ¯ r s
26 35 − ¯ α + r s
26 39 − α + (¯ r · r ) s
26 40 − ¯ α + (¯ r · r ) s
26 41 (¯ δ δ −
1) (¯ r · r ) s
27 1 α + ¯ r v ) 27 2 − α + ¯ r v
27 4 − ¯ α + α ¯ r (¯ r · r ) s
27 5 − α (¯ r ) s
27 6 − ¯ α + α + ¯ r r s
27 7 − α + ¯ r (¯ r · r ) s
27 8 − ¯ α − α − α + ¯ r (¯ r · r ) s
27 9 α − α + (¯ r ) s
27 10 (¯ α − α + ¯ r r s + 4 v ) 27 11 − α + ¯ r (¯ r · r ) s
27 12 − α − ¯ α + α + ¯ r (¯ r · r ) s
27 13 − α − α + (¯ r ) s
27 14 ¯ α + α + ¯ r r s
27 15 α + ¯ r (¯ r · r ) s
27 16 − ¯ α − α ¯ r (¯ r · r ) s
27 17 α (¯ r ) s
27 18 − ¯ α − α + ¯ r r s
27 19 α + ¯ r (¯ r · r ) s
27 30 − ¯ α + r s
27 33 − (¯ r · r ) s
27 37 − ¯ α + α + (¯ r · r ) s
27 39 α − α + ¯ r s
27 40 − ¯ α + α + ¯ r s
27 41 α + (¯ δ δ + 1) ¯ r s
28 1 ¯ α + r v
28 2 − ¯ α + r v
28 4 − ¯ α α + (¯ r · r ) r s
28 5 − ¯ α + α + ¯ r r s
28 6 − ¯ α ( r ) s
28 7 − ¯ α + (¯ r · r ) r s
28 8 − ¯ α − α − ¯ α + (¯ r · r ) r s
28 9 ( α − ¯ α + ¯ r r s + 4 v )28 10 ¯ α − ¯ α + ( r ) s
28 11 − ¯ α + (¯ r · r ) r s
28 12 − α − ¯ α (¯ r · r ) r s
28 13 − α − ¯ α + ¯ r r s
28 14 ¯ α ( r ) s
28 15 ¯ α + (¯ r · r ) r s TABLE VI: The non-vanishing coefficients c ( n ) k in the expansion (A1). k n c ( n ) k k n c ( n ) k k n c ( n ) k
28 16 − ¯ α − ¯ α + α + (¯ r · r ) r s
28 17 ¯ α + α + ¯ r r s
28 18 − ¯ α − ¯ α + ( r ) s
28 19 ¯ α + (¯ r · r ) r s
28 29 − α + ¯ r s
28 32 − ¯ α + α + (¯ r · r ) s
28 36 − (¯ r · r ) s
28 39 − ¯ α + α + r s
28 40 ¯ α − ¯ α + r s
28 41 ¯ α + (¯ δ δ + 1) r s
29 1 ( γ + + 3) (¯ r · r ) v
29 2 − ( γ + −
1) (¯ r · r ) v
29 3 − (¯ r · r ) v
29 4 − ¯ α + α + ( γ + + 3) (¯ r · r ) s
29 5 − α + ( γ + + 3) ¯ r (¯ r · r ) s
29 6 − ¯ α + ( γ + + 3) (¯ r · r ) r s
29 7 − ( γ + + 3) (¯ r · r ) s
29 8 − ¯ α − α − ( γ + + 3) (¯ r · r ) s
29 9 α − ( γ + + 3) ¯ r (¯ r · r ) s
29 10 ¯ α − ( γ + + 3) (¯ r · r ) r s
29 11 − (( γ + −
1) (¯ r · r ) + 4 ¯ r r ) s
29 12 − α − ¯ α + ( γ + −
1) (¯ r · r ) s
29 13 − α − ( γ + −
1) ¯ r (¯ r · r ) s
29 14 ¯ α + ( γ + −
1) (¯ r · r ) r s
29 15 ( γ + −
1) (¯ r · r ) s
29 16 − ¯ α − α + ( γ + −
1) (¯ r · r ) s
29 17 α + ( γ + −
1) ¯ r (¯ r · r ) s
29 18 − ¯ α − ( γ + −
1) (¯ r · r ) r s
29 19 ( γ + −
1) (¯ r · r ) s
29 20 α − α + ¯ r (¯ r · r ) s
29 21 α − (¯ r · r ) s
29 22 − α + (¯ r · r ) s
29 23 − (¯ r · r ) r s
29 24 ¯ α − ¯ α + (¯ r · r ) r s
29 25 ¯ α − (¯ r · r ) s
29 26 − ¯ α + (¯ r · r ) s
29 27 − ¯ r (¯ r · r ) s
29 28 ¯ α − α − ¯ α + α + (¯ r · r ) s
29 34 − ¯ α + r s
29 38 − α + ¯ r s
29 39 α − α + (3 ¯ δ + 1) (¯ r · r ) s
29 40 ¯ α − ¯ α + (3 δ + 1) (¯ r · r ) s
29 41 (¯ δ δ −
1) ( γ + + 3) (¯ r · r ) s
30 1 (¯ r · r ) v
30 2 − (¯ r · r ) v
30 4 (4 v − ¯ α + α + (¯ r · r ) s ) 30 5 − α + ¯ r (¯ r · r ) s
30 6 − ¯ α + (¯ r · r ) r s
30 7 − (¯ r · r ) s
30 8 − ¯ α − α − (¯ r · r ) s
30 9 α − ¯ r (¯ r · r ) s
30 10 ¯ α − (¯ r · r ) r s
30 11 − (¯ r · r ) s
30 12 − α − ¯ α + (¯ r · r ) s
30 13 − α − ¯ r (¯ r · r ) s
30 14 ¯ α + (¯ r · r ) r s
30 15 (¯ r · r ) s
30 16 − ¯ α − α + (¯ r · r ) s
30 17 α + ¯ r (¯ r · r ) s
30 18 − ¯ α − (¯ r · r ) r s
30 19 (¯ r · r ) s
30 28 − ¯ α − α − (¯ r · r ) s
30 32 α − ¯ r s
30 37 ¯ α − r s
30 39 α − (¯ r · r ) s
30 40 ¯ α − (¯ r · r ) s
30 41 (¯ δ δ −
1) (¯ r · r ) s
31 1 − α − ¯ r v
31 2 α − ¯ r v
31 4 α − ¯ α + α + ¯ r (¯ r · r ) s
31 5 α − α + (¯ r ) s
31 6 ( α − ¯ α + ¯ r r s + 4 v )31 7 α − ¯ r (¯ r · r ) s
31 8 ¯ α − α − ¯ r (¯ r · r ) s
31 9 − α − (¯ r ) s
31 10 − ¯ α − α − ¯ r r s
31 11 α − ¯ r (¯ r · r ) s
31 12 α − ¯ α + ¯ r (¯ r · r ) s
31 13 α − (¯ r ) s
31 14 − α − ¯ α + ¯ r r s
31 15 − α − ¯ r (¯ r · r ) s
31 16 ¯ α − α − α + ¯ r (¯ r · r ) s
31 17 − α − α + (¯ r ) s
31 18 ¯ α − α − ¯ r r s
31 19 − α − ¯ r (¯ r · r ) s
31 30 (¯ α − r s/ − (¯ r · r ) s
31 35 − ¯ α − α − (¯ r · r ) s
31 39 α − α + ¯ r s
31 40 − ¯ α − α − ¯ r s
31 41 − α − (¯ δ δ + 1) ¯ r s
32 1 − ¯ α − r v
32 2 (¯ α − r v /
232 4 ¯ α − ¯ α + α + (¯ r · r ) r s
32 5 (¯ α − α + ¯ r r s + 4 v ) 32 6 ¯ α − ¯ α + ( r ) s
32 7 ¯ α − (¯ r · r ) r s
32 8 ¯ α − α − (¯ r · r ) r s
32 9 − ¯ α − α − ¯ r r s
32 10 − ¯ α − ( r ) s
32 11 ¯ α − (¯ r · r ) r s
32 12 ¯ α − α − ¯ α + (¯ r · r ) r s
32 13 ¯ α − α − ¯ r r s
32 14 − ¯ α − ¯ α + ( r ) s
32 15 − ¯ α − (¯ r · r ) r s
32 16 ¯ α − α + (¯ r · r ) r s
32 17 − ¯ α − α + ¯ r r s
32 18 ¯ α − ( r ) s
32 19 − ¯ α − (¯ r · r ) r s
32 29 ( α − ¯ r s/ − ¯ α − α − (¯ r · r ) s
32 38 − (¯ r · r ) s
32 39 − ¯ α − α − r s
32 40 ¯ α − ¯ α + r s
32 41 − ¯ α − (¯ δ δ + 1) r s
33 1 − ( γ − −
3) (¯ r · r ) v
33 2 ( γ − + 1) (¯ r · r ) v
33 3 − (¯ r · r ) v
33 4 ¯ α + α + ( γ − −
3) (¯ r · r ) s
33 5 α + ( γ − −
3) ¯ r (¯ r · r ) s
33 6 ¯ α + ( γ − −
3) (¯ r · r ) r s
33 7 (( γ − + 1) (¯ r · r ) − r r ) s
33 8 ¯ α − α − ( γ − −
3) (¯ r · r ) s
33 9 − α − ( γ − −
3) ¯ r (¯ r · r ) s
33 10 − ¯ α − ( γ − −
3) (¯ r · r ) r s
33 11 ( γ − −
3) (¯ r · r ) s
33 12 α − ¯ α + ( γ − + 1) (¯ r · r ) s
33 13 α − ( γ − + 1) ¯ r (¯ r · r ) s
33 14 − ¯ α + ( γ − + 1) (¯ r · r ) r s
33 15 − ( γ − + 1) (¯ r · r ) s
33 16 ¯ α − α + ( γ − + 1) (¯ r · r ) s
33 17 − α + ( γ − + 1) ¯ r (¯ r · r ) s
33 18 ¯ α − ( γ − + 1) (¯ r · r ) r s
33 19 − ( γ − + 1) (¯ r · r ) s
33 20 α − α + ¯ r (¯ r · r ) s
33 21 α − (¯ r · r ) s
33 22 − α + (¯ r · r ) s
33 23 − (¯ r · r ) r s
33 24 ¯ α − ¯ α + (¯ r · r ) r s
33 25 ¯ α − (¯ r · r ) s
33 26 − ¯ α + (¯ r · r ) s
33 27 − ¯ r (¯ r · r ) s
33 28 ¯ α − α − ¯ α + α + (¯ r · r ) s
33 33 (¯ α − r s/
233 36 α − ¯ r s
33 39 α − α + (3 ¯ δ −
1) (¯ r · r ) s
33 40 ¯ α − ¯ α + (3 δ −
1) (¯ r · r ) s
33 41 − (¯ δ δ −
1) ( γ − −
3) (¯ r · r ) s
34 1 (¯ r · r ) v
34 2 − (¯ r · r ) v
34 4 − ¯ α + α + (¯ r · r ) s
34 5 − α + ¯ r (¯ r · r ) s
34 6 − ¯ α + (¯ r · r ) r s
34 7 − (¯ r · r ) s
34 8 − ¯ α − α − (¯ r · r ) s
34 9 α − ¯ r (¯ r · r ) s
34 10 ¯ α − (¯ r · r ) r s
34 11 − (¯ r · r ) s
34 12 (4 v − α − ¯ α + (¯ r · r ) s )34 13 − α − ¯ r (¯ r · r ) s
34 14 ¯ α + (¯ r · r ) r s
34 15 (¯ r · r ) s
34 16 − ¯ α − α + (¯ r · r ) s
34 17 α + ¯ r (¯ r · r ) s
34 18 − ¯ α − (¯ r · r ) r s
34 19 (¯ r · r ) s
34 28 ¯ α − α + (¯ r · r ) s
34 32 − α + ¯ r s
34 35 ¯ α − r s
34 39 − α + (¯ r · r ) s
34 40 ¯ α − (¯ r · r ) s
34 41 (¯ δ δ + 1) (¯ r · r ) s
35 1 α + ¯ r v / − α + ¯ r v
35 4 − ¯ α + α ¯ r (¯ r · r ) s
35 5 − α (¯ r ) s
35 6 − ¯ α + α + ¯ r r s
35 7 − α + ¯ r (¯ r · r ) s
35 8 − ¯ α − α − α + ¯ r (¯ r · r ) s
35 9 α − α + (¯ r ) s TABLE VII: Continuation of Tab. V. k n c ( n ) k k n c ( n ) k k n c ( n ) k
35 10 ¯ α − α + ¯ r r s
35 11 − α + ¯ r (¯ r · r ) s
35 12 − α − ¯ α + α + ¯ r (¯ r · r ) s
35 13 − α − α + (¯ r ) s
35 14 (¯ α + α + ¯ r r s + 4 v ) 35 15 α + ¯ r (¯ r · r ) s
35 16 − ¯ α − α ¯ r (¯ r · r ) s
35 17 α (¯ r ) s
35 18 − ¯ α − α + ¯ r r s
35 19 α + ¯ r (¯ r · r ) s
35 30 ¯ α − r s ) 35 34 − (¯ r · r ) s
35 37 ¯ α − α + (¯ r · r ) s
35 39 α − α + ¯ r s
35 40 ¯ α − α + ¯ r s
35 41 α + (¯ δ δ −
1) ¯ r s
36 1 − ¯ α − r v
36 2 ¯ α − r v
36 4 ¯ α − ¯ α + α + (¯ r · r ) r s
36 5 ¯ α − α + ¯ r r s
36 6 ¯ α − ¯ α + ( r ) s
36 7 ¯ α − (¯ r · r ) r s
36 8 ¯ α − α − (¯ r · r ) r s
36 9 − ¯ α − α − ¯ r r s
36 10 − ¯ α − ( r ) s
36 11 ¯ α − (¯ r · r ) r s
36 12 ¯ α − α − ¯ α + (¯ r · r ) r s
36 13 (¯ α − α − ¯ r r s + 4 v ) 36 14 − ¯ α − ¯ α + ( r ) s
36 15 − ¯ α − (¯ r · r ) r s
36 16 ¯ α − α + (¯ r · r ) r s
36 17 − ¯ α − α + ¯ r r s
36 18 ¯ α − ( r ) s
36 19 − ¯ α − (¯ r · r ) r s
36 29 − α + ¯ r s
36 31 ¯ α − α + (¯ r · r ) s
36 36 − (¯ r · r ) s
36 39 ¯ α − α + r s
36 40 ¯ α − ¯ α + r s
36 41 − ¯ α − (¯ δ δ − r s
37 1 ( β + + 1) (¯ r · r ) v
37 2 − ( β + −
3) (¯ r · r ) v
37 3 − (¯ r · r ) v
37 4 − ¯ α + α + ( β + + 1) (¯ r · r ) s
37 5 − α + ( β + + 1) ¯ r (¯ r · r ) s
37 6 − ¯ α + ( β + + 1) (¯ r · r ) r s
37 7 − ( β + + 1) (¯ r · r ) s
37 8 − ¯ α − α − ( β + + 1) (¯ r · r ) s
37 9 α − ( β + + 1) ¯ r (¯ r · r ) s
37 10 ¯ α − ( β + + 1) (¯ r · r ) r s
37 11 − ( β + + 1) (¯ r · r ) s
37 12 − α − ¯ α + ( β + −
3) (¯ r · r ) s
37 13 − α − ( β + −
3) ¯ r (¯ r · r ) s
37 14 ¯ α + ( β + −
3) (¯ r · r ) r s
37 15 (( β + + 1) (¯ r · r ) − r r ) s
37 16 − ¯ α − α + ( β + −
3) (¯ r · r ) s
37 17 α + ( β + −
3) ¯ r (¯ r · r ) s
37 18 − ¯ α − ( β + −
3) (¯ r · r ) r s
37 19 ( β + −
3) (¯ r · r ) s
37 20 α − α + ¯ r (¯ r · r ) s
37 21 α − (¯ r · r ) s
37 22 − α + (¯ r · r ) s
37 23 − (¯ r · r ) r s
37 24 ¯ α − ¯ α + (¯ r · r ) r s
37 25 ¯ α − (¯ r · r ) s
37 26 − ¯ α + (¯ r · r ) s
37 27 − ¯ r (¯ r · r ) s
37 28 ¯ α − α − ¯ α + α + (¯ r · r ) s
37 33 ¯ α − r s
37 38 − α + ¯ r s
37 39 α − α + (3 ¯ δ −
1) (¯ r · r ) s
37 40 ¯ α − ¯ α + (3 δ + 1) (¯ r · r ) s
37 41 ( β + −
3) (¯ δ δ + 1) (¯ r · r ) s
38 1 (¯ r · r ) v
38 2 − (¯ r · r ) v
38 4 − ¯ α + α + (¯ r · r ) s
38 5 − α + ¯ r (¯ r · r ) s
38 6 − ¯ α + (¯ r · r ) r s
38 7 − (¯ r · r ) s
38 8 − ¯ α − α − (¯ r · r ) s
38 9 α − ¯ r (¯ r · r ) s
38 10 ¯ α − (¯ r · r ) r s
38 11 ( − (¯ r · r ) ) s
38 12 − α − ¯ α + (¯ r · r ) s
38 13 − α − ¯ r (¯ r · r ) s
38 14 ¯ α + (¯ r · r ) r s
38 15 (¯ r · r ) s
38 16 (4 v − ¯ α − α + (¯ r · r ) s ) 38 17 α + ¯ r (¯ r · r ) s
38 18 − ¯ α − (¯ r · r ) r s
38 19 (¯ r · r ) s
38 28 α − ¯ α + (¯ r · r ) s
38 31 α − ¯ r s )38 37 − ¯ α + r s
38 39 α − (¯ r · r ) s
38 40 − ¯ α + (¯ r · r ) s
38 41 (¯ δ δ + 1) (¯ r · r ) s
39 1 − α − ¯ r v
39 2 α − ¯ r v )39 4 α − ¯ α + α + ¯ r (¯ r · r ) s
39 5 α − α + (¯ r ) s
39 6 α − ¯ α + ¯ r r s
39 7 α − ¯ r (¯ r · r ) s
39 8 ¯ α − α − ¯ r (¯ r · r ) s
39 9 − α − (¯ r ) s
39 10 − ¯ α − α − ¯ r r s
39 11 α − ¯ r (¯ r · r ) s
39 12 α − ¯ α + ¯ r (¯ r · r ) s
39 13 α − (¯ r ) s
39 14 − α − ¯ α + ¯ r r s
39 15 − α − ¯ r (¯ r · r ) s
39 16 ¯ α − α − α + ¯ r (¯ r · r ) s
39 17 − α − α + (¯ r ) s
39 18 (¯ α − α − ¯ r r s + 4 v )39 19 − α − ¯ r (¯ r · r ) s
39 30 − ¯ α + r s
39 33 − (¯ r · r ) s
39 35 α − ¯ α + (¯ r · r ) s
39 39 α − α + ¯ r s
39 40 α − ¯ α + ¯ r s
39 41 − α − (¯ δ δ −
1) ¯ r s
40 1 ¯ α + r v
40 2 − ¯ α + r v
40 4 − ¯ α α + (¯ r · r ) r s
40 5 − ¯ α + α + ¯ r r s
40 6 − ¯ α ( r ) s
40 7 − ¯ α + (¯ r · r ) r s
40 8 − ¯ α − α − ¯ α + (¯ r · r ) r s
40 9 α − ¯ α + ¯ r r s
40 10 ¯ α − ¯ α + ( r ) s
40 11 − ¯ α + (¯ r · r ) r s
40 12 − α − ¯ α (¯ r · r ) r s
40 13 − α − ¯ α + ¯ r r s
40 14 ¯ α ( r ) s
40 15 ¯ α + (¯ r · r ) r s
40 16 − ¯ α − ¯ α + α + (¯ r · r ) r s
40 17 (¯ α + α + ¯ r r s + 4 v ) 40 18 − ¯ α − ¯ α + ( r ) s
40 19 ¯ α + (¯ r · r ) r s
40 29 α − ¯ r s
40 32 α − ¯ α + (¯ r · r ) s
40 38 − (¯ r · r ) s
40 39 α − ¯ α + r s
40 40 ¯ α − ¯ α + r s
40 41 ¯ α + (¯ δ δ − r s
41 1 − ( β − −
1) (¯ r · r ) v
41 2 ( β − + 3) (¯ r · r ) v
41 3 − (¯ r · r ) v
41 4 ¯ α + α + ( β − −
1) (¯ r · r ) s
41 5 α + ( β − −
1) ¯ r (¯ r · r ) s
41 6 ¯ α + ( β − −
1) (¯ r · r ) r s
41 7 ( β − −
1) (¯ r · r ) s
41 8 ¯ α − α − ( β − −
1) (¯ r · r ) s
41 9 − α − ( β − −
1) ¯ r (¯ r · r ) s
41 10 − ¯ α − ( β − −
1) (¯ r · r ) r s
41 11 ( β − −
1) (¯ r · r ) s
41 12 α − ¯ α + ( β − + 3) (¯ r · r ) s
41 13 α − ( β − + 3) ¯ r (¯ r · r ) s
41 14 − ¯ α + ( β − + 3) (¯ r · r ) r s
41 15 − ( β − + 3) (¯ r · r ) s
41 16 ¯ α − α + ( β − + 3) (¯ r · r ) s
41 17 − α + ( β − + 3) ¯ r (¯ r · r ) s
41 18 ¯ α − ( β − + 3) (¯ r · r ) r s
41 19 − (( β − −
1) (¯ r · r ) + 4 ¯ r r ) s
41 20 α − α + ¯ r (¯ r · r ) s
41 21 α − (¯ r · r ) s
41 22 − α + (¯ r · r ) s
41 23 − (¯ r · r ) r s
41 24 ¯ α − ¯ α + (¯ r · r ) r s
41 25 ¯ α − (¯ r · r ) s
41 26 − ¯ α + (¯ r · r ) s
41 27 − ¯ r (¯ r · r ) s
41 28 ¯ α − α − ¯ α + α + (¯ r · r ) s
41 34 − ¯ α + r s
41 36 α − ¯ r s ) 41 39 α − α + (3 ¯ δ + 1) (¯ r · r ) s
41 40 ¯ α − ¯ α + (3 δ −
1) (¯ r · r ) s
41 41 − ( β − + 3) (¯ δ δ + 1) (¯ r · r ) s TABLE VIII: Continuation of Tab. V. [1] A. Gasparyan and M. F. M. Lutz, Nucl. Phys. A848 ,126 (2010), 1003.3426.[2] I. V. Danilkin, A. M. Gasparyan, and M. F. M. Lutz,Phys. Lett.
B697 , 147 (2011), 1009.5928.[3] I. V. Danilkin, L. I. R. Gil, and M. F. M. Lutz, Phys.Lett. B, in print (2011), 1106.2230.[4] M. Jacob and G. C. Wick, Ann. Phys. , 404 (1959).[5] G. F. Chew, M. L. Goldberger, F. E. Low, and Y. Nambu,Phys. Rev. , 1345 (1957).[6] N. Nakanishi, Phys. Rev. , 1225 (1962).[7] F. A. Berends, A. Donnachie, and D. L. Weaver, Nucl.Phys. B4 , 1 (1967).[8] M. Lutz, Nucl. Phys. A677 , 241 (2000), nucl-th/9906028.[9] M. F. M. Lutz, G. Wolf, and B. Friman, Nucl.Phys.
A706 , 431 (2002), nucl-th/0112052, [Erratum-ibid.A765:431-496,2006].[10] M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys.
A700 ,193 (2002), nucl-th/0105042.[11] M. F. M. Lutz and C. L. Korpa, Nucl. Phys.
A700 , 309(2002), nucl-th/0105067.[12] M. F. M. Lutz, C. L. Korpa, and M. Moller, Nucl. Phys.
A808 , 124 (2008), 0707.1283.[13] C. L. Korpa, M. F. M. Lutz, and F. Riek, Phys. Rev.
C80 , 024901 (2009), 0809.4604.[14] M. Lutz and E. Kolomeitsev, Nucl.Phys.
A730 , 392(2004), nucl-th/0307039. [15] S. Stoica, M. F. M. Lutz, and O. Scholten, Phys. Rev.D, in print (2011), 1106.5619.[16] J. S. Ball, Phys. Rev. , 2014 (1961).[17] A. O. Barut, I. Muzinich, and D. N. Williams, Phys.Rev. , 442 (1963).[18] Y. Hara, Phys. Rev. , B507 (1964).[19] J. D. Jackson and G. E. Hite, Phys. Rev. , 1248(1968).[20] L.-L. C. Wang, Phys. Rev. , 1187 (1966).[21] J. Franklin, Phys. Rev. , 1606 (1968).[22] M. D. Scadron and H. F. Jones, Phys. Rev. , 1734(1968).[23] G. Cohen-Tannoudji, A. Morel, and H. Navelet, Annalsof Physics , 239 (1968).[24] W. A. Bardeen and W. K. Tung, Phys. Rev. , 1423(1968).[25] S. U. Chung, Phys. Rev. D , 1225 (1993).[26] S.-U. Chung and J. Friedrich, Phys. Rev. D78 , 074027(2008), 0711.3143.[27] PANDA, M. F. M. Lutz et al. , (2009), 0903.3905.[28] S. Mandelstam, Phys. Rev. , 1344 (1958).[29] M. F. M. Lutz and M. Soyeur, Nucl. Phys.
A813 , 14(2008), 0710.1545.[30] T. H. West, Comput. Phys. Commun. , 286 (1993).[31] A. L. Bondarev, Theor. Math. Phys.101