What is the right formalism to search for resonances? II. The pentaquark chain

We discuss the differences between several partial-wave analysis formalisms used in the construction of three-body decay amplitudes involving fermions. Specifically, we consider the decay Lambda_b ->psi p K- , where the hidden charm pentaquark signal has been reported. We analyze the analytical properties of the amplitudes and separate kinematical and dynamical singularities. The result is an amplitude with the minimal energy dependence compatible with the S-matrix principles.


I. INTRODUCTION
In the recent years experiments such as BaBar, Belle, BESIII, CLAS, COMPASS, GlueX, LHCb, have produced significant amount of high-precision data on three-body hadron decays, garnering information on new hadronic states [1][2][3][4][5]. To put existence of such states on firm theoretical footing and to determine their physical properties rigorous amplitude analysis is needed. There are well established methods based on first principles of reaction theory for construction of reaction amplitudes describing three particle decays of hadrons [6][7][8][9][10][11][12][13][14][15]. It appears, however that there is significant confusion as to the role of various approximations that these methods entail. In an earlier work [16], we pointed out that, contrary to the common wisdom, differences among the various approaches are dynamical rather than kinematical in nature, and we showed that the lore for the LS formalism to be nonrelativistic is unjustified. As an example, we discussed the decay B → ψπK, which shows nontrivial structures appearing in the Belle and LHCb data in ψ(2S) π [17][18][19][20], and J/ψ π channels [21]. In the present paper, we extend the discussion to the more complicated fermion-boson case. Our main goal is to properly separate kinematical from dynamical singularities. In general, the analysis of kinematical singularities of amplitudes with fermions has to be handled with particular care, because of the additional branch point at vanishing value of the Mandelstam variables [22], and because fermions and antifermions have opposite intrinsic parities. Hence, one expects different behavior of the amplitudes at threshold and pseudothreshold. We thus believe that study of such amplitudes deserves an extended discussion. Moreover, because of the possible existence of hidden charm pentaquarks, there is particular interest in final states containing the nucleon, a light meson and a charmonium [1][2][3]. In this paper we thus study the amplitudes for the reaction Λ b → ψpK − in which a prominent pentaquark-like signal in the ψp invariant mass observed at LHCb [23,24].
The paper is organized as follows. In Sec. II we discuss the canonical approach used to analyze the Λ b → ψpK − decay. By relating the helicity partial waves to the Lorentz scalar amplitudes via the partial-wave expansion, we derive constraints on the amplitudes and isolate the kinematical singularities. The results, and the comparison with the LS partial-wave amplitudes, are summarized in Sec. III. In Sec. IV we examine the Covariant Projection Method (CPM) approach and compare it to our results. Conclusions are given in Sec. V. For ease of readability in the main text, most of the technical details are given in the appendices where we also give a practical parameterization of the amplitudes suitable for data analysis.  Reaction diagrams for (a) the Λ b → ψ(→ µ − µ + )pK − decay process, and for (b) the Λ b ψ → pK − s-channel scattering process. Figure 2. Scattering kinematics in the s-channel rest frame. In the decay kinematics, the momentum and the spin of the ψ is reversed to keep the same helicity. Fig. 1 we specify the kinematics for the decay Λ b → ψ(→ µ + µ − )pK − . In the following, we follow the arguments presented in [16]. We will be able to identify and characterize all kinematical singularities as either pertaining to (pseudo)thresholds, or to the vanishing of particles' energies. The particles Λ b , p, and K − are stable against the strong interaction, and the ψ is narrow enough, allowing one to factorize its decay dynamics. Thus, we focus on the amplitude in which ψ is also considered as stable. In the following, we analyze the equivalent scattering problem Λ b ψ → pK − , and we refer to [16] for an extended discussion of crossing symmetry. We use p i , i = 1, 2, 3, and 4 to label the momenta of Λ b , ψ, p, and K − respectively. We callp ψ = −p ψ the momentum of the ψ in the decay kinematics. The helicity amplitude is denoted by A λp,λ b λ ψ (s, t), where λ p , λ b and λ ψ are the helicities of p, Λ b and ψ, respectively. The amplitude depends on the standard Mandelstam variables s = (p p + p K ) 2 , t = (p b − p p ) 2 , and u = (p b − p K ) 2 with s + t + u = i m 2 i . The Λ b baryon decays weakly, so A λp,λ b λ ψ is given by the sum of a parity conserving (PC) and a parity violating (PV) amplitudes.

II. ANALYTICITY CONSTRAINTS FOR
We discuss here the PC amplitude in the s-channel, and we refer to Appendix D for the summary of the PV amplitude. The s-channel resonances correspond to the Λ * 's and dominate the reaction [25]. As discussed in the previous section, the analysis of the experimental data indicates a possible signal of resonances in the exotic ψp spectrum, which in our notation correspond to the u-channel.
In the center of mass of the s-channel scattering process, the momentum p b defines the z-axis, the momenta p p and p K lie in the xz-plane, p and q denote magnitudes of relative momenta in the incoming (Λ b , ψ) and the outgoing (p, K − ) states. The scattering angle θ s is the polar angle of the proton (see Fig. 2). The quantities are expressed through the Mandelstam invariants, The function 4s n(s, t) is a polynomial in s and t. 1 To incorporate resonances in the pK − system with a certain spin j, we expand the amplitude in partial waves, where A j λp,λ b λ ψ (s) are the helicity partial-wave amplitudes in the s-channel, λ = λ b − λ ψ , λ = λ p , and M = max(|λ|, |λ |) [8]. We use the definition of the Wigner d function as in [26], i.e. d j λλ (cos θ) = j, λ| exp (−iJ y θ) |j, λ , that differs from the one in [8] by θ → −θ. This results in a difference in our definition of the parity conserving helicity amplitudes given in Eq. (7) below.
Instead of working with an infinite number of helicity partial waves, we will consider the isobar model, customarily used in data analysis. 2 The dynamical singularities in s, t and u are taken into account explicitly by a sum of different terms, with and J max < ∞. In this model, it is assumed that the entire dynamical information is expressed by the isobar amplitudes, which are functions of a single Mandelstam variable: , with x = s, t, u. The expressions for the (t) and (u) isobars are similar to Eq. (4). In the following we focus on the s-channel isobars, and drop the (s) superscript. The u-channel isobars are described in the appendices E, F). In Eqs. (2),(4) the entire t dependence originates from the d functions. Specifically, the d functions have singularities in z s which lead to kinematical singularities in t. We define the functionŝ with being the so-called half angle factor that contains all the kinematic singularities in t. The reduced rotational function d j λλ (z s ) is a polynomial of n(s, t)/pq of order j − M , see Eq. (1). For λ, λ = 0, the functions d j λλ (z s ) have no definite parity. This means that the product (pq) j−Md j λλ (z s ) contains terms with odd powers of pq that still have kinematic branch-point singularities in s. To be able to remove these singularities from the amplitude, we need to define the so-called parity-conserving helicity amplitudes (PCHAs), where η ψ = η b = + are the naturality of the ψ and Λ b , respectively. These functions are free of kinematical singularities in t. Similarly, we can split the isobars into natural and unnatural ones where we defined the definite-parity partial-wave/isobar amplitudes, We introduce the definite-parity Wigner d functions by, One can check that the functiond j+ λλ (z s ) is a definite-parity polynomial of order j − M , i.e.d j+ λλ (−z s ) = (−1) j−Md j+ λλ (z s ). 3 Similarly,d j− λλ (z s ) is a definite parity polynomial of order j − M − 1, and therefore subleading in the z s → ∞ limit. We refer the reader to Appendix B for a more detailed discussion on the (sub)leading behavior of thed j(η) λλ (z s ). In terms of these isobars, the PCHAs read where we applied the parity relations in Eq. (9) in the transition from the first to second line. We note that, for given η, isobars with both naturalities contribute to the A η λp,λ b λ ψ (s, t). 4 The helicity isobars A j λp,λ b λ ψ (s) have singularities in s, which have both dynamical and kinematical origin. The kinematical singularities in s, just like the t-dependent kinematical singularities, arise because of particle spin. We explicitly isolate the kinematic factors in s, and denote the kinematical singularity-free helicity isobar amplitudes byÂ j λp,λ b λ ψ (s). First, we take out the factor (pq) j−M from the A j λp,λ b λ ψ (s). This factor cancels the threshold and pseudothreshold singularities in s that appear ind j λλ (z s ). Second, we follow [8] and introduce the additional kinematic factor K η M N . These factors are required to account for the mismatch between the j and L dependence in the angular momentum barrier factors in the presence of particles with spin. Specifically, it is expected that where L 1 and L 2 are the lowest possible orbital angular momenta in the given helicity and parity combination. The definite-parity, kinematical-singularity-free helicity isobar amplitudesÂ jη λp,λ b λ ψ (s) are defined by In addition, the K-factors have powers of √ s as required to ensure factorization of the isobar amplitude into contributions from distinct vertices [8]. The isobar amplitudesÂ j,η λp,λ b λ ψ (s) contain the dynamical information of the model. Often they are parameterized in terms of a sum of Breit-Wigner amplitudes with Blatt-Weisskopf barrier factors.
Once we have removed the kinematic singularities from the isobar amplitudes and the corresponding angular functions, we are now in a position to remove the singularities from the full amplitude. Therefore, we take out the 4 The name "Parity-Conserving Helicity Amplitudes" arises from Regge theory, where in the limit t → ∞ (which implies zs → ∞) the contribution from the opposite naturality, −η, is proportional tod j− λλ (zs) which is negligible compared tod j+ λλ (zs). In the case at hand, however, we consider the limits q → 0 or p → 0, where the kinematic factors of the partial-wave amplitudes are also relevant to determine the leading behavior of the two contributions in Eq. (11). factor K η M N and define the amplitudes F which are kinematic singularity-free PCHAs (KSF-PCHAs), where the ratio K −η M N /K η M N = pm p qm ψ η . While the KSF-PCHAs are free of kinematical singularities, they are not necessarily independent for all kinematics. Indeed, we will illustrate below that additional constraints must be fulfilled by the isobar amplitudes for certain kinematics. Therefore, as in [16], we seek a representation of A λp,λ b λ ψ (s, t) in terms of a set of covariant structures that explicitly account for the kinematic part of the amplitude. For the PC amplitude, the basis with minimal energy dependence is given by with and the scalar functions C i (s, t) are free from kinematical singularities. There are six independent M µ 1...6 tensors, and any other possible combination can be reduced to these using the Dirac equation for the spinors, or the orthogonality relation µ (p ψ , λ ψ )p µ ψ = 0. Alternatively, one can use the CGLN basis defined in [27], for pseudoscalar-meson electroproduction. However, these covariant structures enforce a gauge-invariance principle which does not apply here since we ψ is a massive vector particle. Had we used the CGLN basis there would be unnecessary kinematic zeros. The PC amplitude requires a γ 5 because of the unnatural K − parity. The explicit expressions for the polarization vectors and spinors are given in Appendix A. We can match Eq. (15) and (16), and express the scalar functions as a sum over kinematical singularity free helicity isobars. This yields with M a 6 × 6 matrix that encodes all the kinematic factors and is provided in Appendix C. 5 The factors E p + m p and √ E b + m b are factored out to simplify the expression for M. We stress that they have only singularities at s = 0. For example, and the physical region of √ s corresponds to Re √ s > 0. For m b > m ψ , which is the case here, the first factor in Eq. (19) is always positive, and the only singularity is due to the branch point at s = 0. This would be different if the fermion was lighter than the boson. In that case, the factor will have a singularity at pseudothreshold that has , and so on.
to be considered separately. The relation in Eq. (18) can be inverted, leading to where the matrices B and Reg are regular at p = 0. The explicit expression for the M −1 and the B matrices are in Appendix C. We just report a few terms here to ease the discussion, Since the C i functions must be regular at (20) must conspire to cancel the 1/p 2 pole. This translates into a relation between the various isobar amplitudesÂ jη λp,λ b λ ψ (s). As an example, let us consider the last two rows in Eq. (20). Inspecting the matrix elements in Eq. (21) one finds that two emerging conditions are not independent and lead to, where we mean here is that this combination must vanish as p 2 for p → 0. The conspiracy relation can be written in terms of the isobar amplitudes by inserting the expression for the F 's in terms of the isobars, given in Eq. (15). Since the isobars of different spin are independent, we can consider each j individually. For j ≥ 3 2 we obtain When p → 0, z s → ∞ and the leadingd j+ −1/2,1/2 (z s ) andd j+ 3/2,1/2 (z s ) diverge as 1/p j−1/2 and 1/p j−3/2 , respectively. This divergence is canceled by the threshold factor (pq) j−1/2 , but an additional relation betweenÂ j+ +,++ (s) and A j+ +,+− (s) is needed to cancel the additional 1/p 2 pole appearing in Eq. (20). On the other hand, the subleadinĝ d j− −1/2,1/2 (z s ) andd j− 3/2,1/2 (z s ) diverge as 1/p j−3/2 and 1/p j−5/2 only, and together with the additional factor of p coming from the mismatch factors K η M N and the threshold factor, vanish as p 2 to cancel the 1/p 2 pole. Therefore, the opposite-naturality waves do not contribute to this type of conspiracy relations. It is also straightforward to check that the expressions are regular when q → 0. One can use the asymptotic expansion of the Wigner d functions (the full expressions are in Appendix B), We remark that these relations hold for the j = 1 2 case as well, and we do not need any separate consideration for it. There is only one LS coupling with nonminimal L, which calls for one conspiracy equation only. However, the equations obtained from the first and third line in Eq. (21) give and it is easy to check the two equations to be independent out of (pseudo)threshold. We evaluate the constraints at both threshold and pseudothreshold, By restoring the kinematic factors, At threshold, this matches with the LS constraint. To interpolate with the pseudothreshold result, we replace To summarize, we used analyticity constraints to derive relations between the different helicity isobars. At threshold, these relations are in agreement with the expectations derived from the LS decomposition. Similar constraints are derived at pseudothreshold. To interpolate between the two constraints, we add an energy dependent factor E ψ /m ψ in the λ ψ = 0 amplitude. This results in the minimal kinematic dependence as required by analyticity.

III. THE GENERIC PARAMETERIZATION FOR THE s-CHANNEL ISOBARS
In this section we derive a general parametrization for the isobar amplitude which takes into account the conspiracy relations derive in the preceding section. A generic parameterization for the natural isobars which fullfills Eqs. (25) and (30) is given by where g ( , ) j+ (s) and f ( , ) j+ (s) are regular functions at s = s ± , and g j+ (s ± ) = g j+ (s ± ) = g j+ (s ± ). The branch points at s = 0 are not constrained by the threshold relations and a germane to the fermion-boson system. Their origin is dynamical and has to be addressed in the context of analyticity in j. We decided to include appropriate factors of √ s to make these formulae more similar to the LS ones. For example, the additional factor of m p m ψ / √ s in front of f j+ (s) in Eq. (37c) is unconstrained by these relations, but it has been inserted by analogy with LS. Similarly, we decided the subleading f ( , ) j+ (s) functions to appear with a factor p 2 insted of λ bψ , unlike in [16]. Upon restoration of the kinematic factors, the original helicity isobars amplitudes read (j ≥ 3 2 ) A particular choice of the functions g j+ (s) constitutes a given hadronic model. For j = 1 2 , no conspiracy constraint is needed. Indeed, the isobarsÂ j+ λp,λ b λ ψ (s) always appears with an additional factor ∝ p 2 , as shown in Eq. (12b), and the pole is automatically canceled.
We can immediately cast this expression in the LS basis. For the natural isobars considered, this reads with G j+ LS (s) = p L q j+1/2Ĝ j+ LS (s), and the dependence on λ p is understood. We remark that there are two LS couplings having nonminimal L in Eq. (39), and at p = 0 the three helicity couplings will depend on one independent LS coupling only. This will require indeed two equations to be satisfied, i.e. Eqs. (25) and (30). This expression matches Eq. (37) by identifying f j (s) = 2j + 2 2j + 1 For the unnatural isobars, the minimal parameterization fulfilling Eq. (36) is with g j− (s ± ) − g j− (s ± ) = g j− (s ± ) and h j− (s ± ) = h j− (s ± ) = h j− (s ± ). The identification with the LS couplings is straightforward, As shown in Eq. (40) and (42) (42), instead of the LS amplitudes helps prevent unwanted singularities. A practical use of these formulae, and the analogous forms for the PV amplitude and for the u-channel exchanges, are given in Appendix G.

IV. COMPARISON WITH THE COVARIANT PROJECTION METHOD
The LS and helicity partial waves can now be compared to the CPM formalism. The latter builds LS-like partialwave amplitudes, based on covariant structures that are interpreted as spin (S) and orbital-momentum (L) covariant tensors. We follow the methodology outlined by the Bonn-Gatchina partial-wave analysis group in [28]. We consider the example of a Λ * resonance with J P = 3 2 − in the s-channel. First, we consider the interaction in the scattering regime. In this case, all structures must be orthogonalized to the center-of-mass momentum P = p b + p ψ . We define the relative four-momenta in the intial and final state p = (p b − p ψ )/2 and q = (p p − p K )/2, respectively. The orbital momentum component of the vertex Λ * → pK − is described by the D-wave tensor with q µ ⊥ = q µ − P µ P · q/s, and g ρµ ⊥ = g ρµ − P ρ P µ /s, such that q µ ⊥ P µ = q ρ g ρµ ⊥ P µ = 0. Furthermore, we define γ µ ⊥ = g µν ⊥ γ ν . The initial state can be S-wave or D-wave. The orbital tensor structure for the latter reads with p µ ⊥ = p µ − P µ P · p/s. The 3 2 − contribution to the helicity amplitudes is therefore fully determined by the expression 7 7 We neglect overall factors of i.
where we introduced the spin-3 2 projector Explicitly, the corresponding isobar amplitudes read Notice that the expression in Eq. (47c) indeed does not contain a contribution from the D 1 2 component, as expected from the LS in Eq. (31). As discussed in Sec. II, the square roots have no singularities at (pseudo)threshold. The role of Clebsch-Gordan coefficients can be enlighted by writing It is worth noticing that the Clebsch-Gordan multiplying the g D 3 2 (s) coupling is not the one expected according to the LS construction: the ψ spin is coupled with the orbital angular momentum first, and only after with the spin of the the Λ b , while the canonical LS construction would couple the two spins first, and the angular momentum after. This is also evident by looking at the covariant structures the second line of Eq. (45), and explains why the various tensors are not orthogonal.
The same framework can be applied to the decay chain, where the tensor structures of the initial Λ b → ψΛ * decay must be orthogonalized with the respect to the Λ b momentum p b , rather than the isobar momentum P . The ψ is now in the final state with momentump ψ = −p ψ and polarization * µ (p ψ , λ ψ ). We therefore obtain In the above, the γ ⊥ (p b ) and γ ⊥ (P ) are orthogonalized with respect to p b and P respectively. We will show the results for the g D 3 2 only (equating the other couplings to zero), in the decay chain and scattering regime. In the isobar rest frame, the contribution in the decay chain reads To summarize, the amplitudes that follow from the CPM method contain the factor E ψ /m ψ in the λ ψ = 0 isobar. This factor asserts the fulfillment of the conspiracy relation in Eq. (30). This factors was included in the canonical helicity amplitudes, but does not follow from the LS method. Additional energy dependent factors E p + m p √ E b + m b are found, which are not required by analyticity since they are smooth. As already pointed out in [16], the CPM formalism was shown to violate crossing symmetry, since the amplitudes in the decay and scattering kinematics differ. Also, the coupling of the external particle spins and orbital momentum occur in a different way than in the LS for the D 3 2 component.

A. pK − mass distribution in different approaches
We explore the difference between the various approaches and consider two intermediate natural parity, spin- 3 2 Λ * resonances in the s-channel (pK − ): the Λ(1520) with mass M Λ * = 1519.5 MeV and width Γ Λ * = 156 MeV (artificially increased by a factor of 10 for illustration purposes), and the Λ(1690) with M Λ * = 1690 MeV and width Γ Λ * = 60 MeV. We denote the dynamical part of the amplitude as T Λ * . We consider the CPM formalism discussed in Eq. (47) and Eq. (50) (for scattering and decay respectively), setting g S 3 2 (s) = g D 1 2 (s) = 0 and g S 1 2 (s) = g D 5 2 (s) = 0 respectively. We assume g D 3 2 (s) = T Λ * (s) to be identical in the scattering and decay kinematics, with For the LS formalism, we choose the couplings in Eq. (31) to beĜ = T Λ * . The LS amplitude in the decay kinematics differs from the one in the scattering kinematics only because of the breakup momentum of Λ b → ψΛ * , calculated in the Λ b rest frame or in the Λ * rest frame, respectively. Finally, we show the results for our proposed amplitude given in Appendix G. The model is obtained by taking g Λ(1520) + g Λ(1690) = T Λ * in Eq. (G3).
As in [16], we illustrate the effect of including Blatt-Weisskopf factors in the dynamic part of the amplitude. In the case at hand, this amounts to multiplying the dynamic amplitude T Λ * by a factor B 2 (p)B 2 (q), where B 2 is defined as (x = p, q) and assume R = 3 GeV −1 as in [23]. The differential width is given by where ρ(s) = λ 1/2 bψ λ 1/2 pK /s and N j is a normalization constant. The effect of the different kinematic structures is clearly observed in the invariant mass distributions in Fig. 3. Our proposed amplitudes from Appendix G, referred to as the JPAC amplitudes, differ from the LS amplitudes given in Eq. (40) by the factor of E ψ /m ψ in the λ ψ = 0 helicity partial-wave amplitude (cf. Eq. (G12)). This factor also follows naturally from the CPM formalism in the scattering kinematics. The CPM amplitudes in the scattering and decay frame (see Eq. (47) and Eq. (50) respectively) both include an additional factor of √ E b + m b E p + m p compared to the JPAC and LS formalism. In addition the CPM formalism applied to the decay kinematics introduces redundant kinematic factors of s in all partial-wave amplitudes. Additionally, the λ ψ = 0 amplitude has a factor of (s − m 2 b − m 2 ψ )E b /E ψ in the decay kinematics. The differences shown in Fig. 3, particularly between the LS decay and the CPM scattering, are enough to significantly impact the extraction of the couplings.

V. CONCLUSIONS
We have expanded the discussion in [16] about the different approaches for constructing amplitudes for scattering and decay processes to the fermion-boson case. In particular, we have studied the Λ b → ψpK − decay, which is of interest for hidden charm pentaquark searches. The inclusion of fermion spins in the helicity formalism introduces mismatches between threshold and pseudothreshold, which makes the discussion more complicated and the equations cumbersome. We used analyticity as a guiding principle to examine the canonical helicity formalism [6][7][8][9] and the covariant projection method [10][11][12][13]. We have shown how the latter violates crossing symmetry. The matching of the helicity amplitudes to the most general covariant expression allows us to identify the kinematical singularities, and to determine the minimal energy dependence required, summarized in Sec. III. In order to factor out the kinematical singularities we need to build the hadronic part of the amplitudes with a well defined parity. A √ s singularity cannot be removed with these considerations and needs to be taken care of through the corresponding dynamical model. As in our previous work [16], we find meaningful differences among the amplitude building methodologies which do affect the resonance pole position extraction, and consequently to the data analysis oriented to determine the existence and properties of the resonances. Although there is no way to escape all model dependence, our analysis maximizes the consistency of a given model with the S-matrix principles. A practical formulation of the amplitudes, both in the s and u channels, and both for the parity-conserving and parity-violating case, can be found in Appendix G. In the s-channel center of mass frame the spinors are given by and the ψ polarization by We apply the 'particle 2' convention for a particle going into the −z direction, as in [6]. The energies E i are calculated from the momenta and are fully determined by s.

Appendix B: Leading and next-to-leading expansion of Wigner d-functions
Thed j λλ (z s ) Wigner functions are polynomials in z s of order j − M , with the usual definition of M = max(|λ|, |λ |) and N = min(|λ|, |λ |). We use the relation between the Wigner d-functions and the Jacobi polynomials P (a,b) n given by equation (3.74) in [29] d j λλ (z s ) = Two useful relations to compare the above to the literature are |λ + λ | + |λ − λ | = 2M and |λ| + |λ | = M + N . The leading and next-to-leading terms of the polynomial are given bŷ As we noted, this polynomial has no definite parity. We defined the parity-conserving Wigner d-functions aŝ With the substitution λ → −λ in Eq. (B2), the second term between brackets obtains a minus sign, since (|λ − λ | − M ) = −(|λ + λ | − M ), while the first term is unaffected. This illustrates the more general fact that even and odd powers of z s obtain opposite sign under λ → −λ. In particular, the leading power in z s obtains an extra phase (−1) M −λ−|λ−λ | from the prefactor in front of the brackets. This phase factor is identically equal to (−1) M −λ used in the definition B3. One can show thatd j λλ (z s ) = (−1) j−λ d j −λλ (−z s ). It follows that thed jη λλ (z s ) have definite parity, and asymptotic expressionsd where depends only on j.
For the u-channel process, we have the initial helicity fixed to 1/2, and the final one running. We can use the same formulae, upond with The inverse matrix is calculated using Mathematica [30]. It is given by with The matching can be performed in the same way, giving a matching equation analogous to Eq. (20), where D = − √ s(1 + E p /m p ), the elements of the B matrix are defined in Appendix C, and B ij,k stands for the term in B ij of order [n(s, t)] k . Up to irrelevant factors which do not enter the equations (as the multiplicative factors of D, or the terms R 13,0 = B 13,0 and R 33,0 = B 33,0 ), the conspiracy relations are going to be the same as in Sec. II, upon swapping the natural and unnatural partial waves. For completeness, we report the matrix elements of Reg : and the proton. The kinematics is summarized in Fig. 4. The quantities are expressed through the Mandelstam invariants, As before, the function 4u n(u, t) is a polynomial in u, t. For the covariant amplitude, we have with the same covariant basis M i as in Eq. (17); the polarization spinors are the same as in Eq. (A1), with the obvious replacements θ s → θ u , E i (s) → E i (u), with i = ψ, b, p. The ψ polarization is given by We remark that, since in the final state the fermion (p) is lighter than the boson (ψ), the factor E p − m p will have the threshold singularity only, while E p + m p the pseudothreshold singularity only. 8 Therefore, the factorized kinematic factors will be different for the threshold and pseudothreshold. The latter is accounted for by the explicit factors of √ u − u − below. The PCHAs read where the matrices B, P, and Reg are regular at (pseudo)threshold: with with g j+ (u ± ) = g j+ (u ± ) = g j+ (u ± ), g j− (u ± ) − g j− (u ± ) = g j− (u ± ) and h j− (u ± ) = h j− (u ± ) = h j− (u ± ).
Appendix F: The u-channel parity-violating amplitude To carry out the analysis for the PV part of the u-channel amplitude, we remark that changing the spin-parity of the Λ b from J P = 1 2 + to J P = 1 2 − only affects the arguments related to the initial state Λ b K + . Since the arguments in the previous channel were based on the (pseudo)threshold of the final state pψ, the derivation of the conspiracy relations is unaffected. Therefore, the kinematical factors are identical to the ones for the PC u-channel amplitudes, and will not be discussed any further. For completeness, we report the matching matrices, where where λ is the Källén triangular function, and Mat(s) is given as M −1 in App. C, Mat (s) is given as (B/p 2 s + Reg) in App. D, Mat(u) is given as (B/q 2 u + P/(u − u − ) + Reg) in App. E, Mat (u) is given as (B/q 2 u + P/(u − u − ) + Reg) in App. F.
The matrices Mat ( ) (x) will be available for download on the JPAC website [32]. The functions F R i contain the kinematical dependence of the KSF-PCHAs,