NN scattering with N ∆ coupling: Dibaryon resonances without “dibaryons”?

It is known that at their threshold intermediate two-baryon N ∆ states can produce resonance-like structures in some isospin one states, often interpreted as more exotic manifest six-quark states. This paper applies the coupled-channel method to study details of the N ∆ eﬀect in isospin one NN scattering with interactions constrained to exclude the inﬂuence of such extraneous hypothetical particles in two respects. Firstly, the calculated phase parameters are ﬁtted below pion inelasticity, i.e. far below the region of “dibaryons”. Secondly, the strength of the NN → N ∆ transition is constrained to agree with the pion production reaction NN → dπ , fairly independent of the details of the pure “diagonal” NN potential and its eﬀect practically as strong as the NN itself. The results may be considered as a necessary background for searches of additional dibaryon eﬀects. Optimal conditions for resonant N ∆ appearances involve a decrease of the orbital angular momentum in the transition NN → N ∆, as is the case in the 1 D 2 and 3 F 3 waves. Also the eﬀective N ∆ threshold as well as its width are state dependent. Detailed complex phase shift results are presented for isospin one partial waves up to J = 6, where N ∆ excitation is still found to be comparable to or even larger than one pion exchange.


I. INTRODUCTION
Shortly after the appearance of the quark model in 1964 [1] it was suggested [2,3] the possible existence also of states with six quarks (somewhat like six-quark bags).Although this idea concerned only color singlet involving all three quark flavors (known at the time), as a follow-up it was further suggested that this kind of objects would play an important role as resonant s-channel intermediate states in the interaction dynamics of two nucleons (for a balanced review see e.g.Ref. [4]).Searches have since continued for indications in experimental data (for an early review in the NN sector see e.g.Ref. [5]).An interest in such activity has been sustained over decades and got new wind from observations at the WASA@COSY detector of Forschungszentrum Jülich in two-pion production reactions, where a resonant structure with I(J P ) = 0(3 + ), called d ′ (2380), was seen at 2380 MeV with the width of 70 MeV [6].
However, there is another well established possibility for strong resonance-like energy dependencies, namely two-pion exchange with the excitation of an intermediate state with a ∆(1232) resonance.It was realized early in pion physics that pion-nucleon interaction, both elastic scattering and reactions, in the region of a few hundred MeV is dominated by this spin- 3  2 and isospin-3 2 particle [7].Pion exchange can naturally induce such an excitation in isospin one NN scattering [8], transforming one of the nucleons or both into a ∆.Then its consequent decay may end up in pion production reaction or (if reabsorbed by the other nucleon) generate strong attractive two-pion exchange, which is not reducible to two-nucleon intermediate states appearing in iteration of a normal NN potential [9,10].This mechanism acts in the same energy region as several of the conjectured dibaryons and may compete with their effects or be interpreted as those.In the 1970's and 1980's the attractive effect of ∆ excitation was included for example in the so called Bonn boson exchange potentials [11,12], well fitted to phase shift analyses in the elastic energy region.The aim of the present paper is to elaborate the coupled channels model more in the inelastic region.
The ascent of inelasticity above pion threshold was studied long ago by N∆ coupled channels in NN scattering e.g. in Refs.[13] and [14].The positions of possible associated resonant poles were studied by VerWest [15] and Kloet and Tjon [16] in some detail using exactly solvable separable models, which showed the mathematical reality of such threshold effects in N∆ states.Also Betz and Lee had studied pion-nucleon and pion-deuteron dynamics, albeit also with separable interactions [17].Van Faassen and Kloet went further to devise realistic interactions with meson exchanges in Ref. [18] and produced qualitatively successful phases and inelasticities also in the dibaryon regime at and above the N∆ threshold.Likewise, the Bonn potential was successfully extended to include inelasticity for applicability at the dibaryon energies [19].
However, these realistic interactions [18,19] are still mere NN potentials in the sense that they start with NN and end with NN, producing no N∆ wave function necessary to calculate reactions and other probes of the intermediate state.Neither do the others above.This is the crucial difference to the present paper.Here, in scattering calculations the transition NN → N∆ will be given also a strict external constraint arising from the total cross section of the reaction NN → dπ, which requires and is sensitive to such a wave function and, thus, can fix the strength of that transition.
The next section outlines the basic formalism for generating the N∆ configurations in the isovector NN scattering and discusses their main properties.First the coupled Schrödinger equation is given in Subsection II A and then Subsec.II B introduces its most important and essential ingredient, the transition potential.Rise of inelasticity in terms of the effective width of the N∆ channel is discussed in Subsection II C and the potential present in the incident NN channel in II D. Finally some additional comments on the general formalism are given in Subsec.II E. The numerical results for the NN phase shifts are presented and discussed partial wave by partial wave in Sec.III before summarizing in Sec.IV.

II. N ∆ STATES A. Coupled channels system
In this work (as in its predecessors since Ref. [20] and the more mature Ref.[21]) the N∆ components w i (r) are generated from the incident NN wave u(r) by a coupled system of Schrödinger equations involving as the new element the transition potential NN ↔ ∆N due mainly to pion exchange Here k is the centre of mass momentum, L (L ′ i ) the NN (N∆) angular momentum, M and M ′ twice the respective reduced masses.The mass difference ∆M between the ∆ and nucleon defines the threshold of real ∆'s.For the ∆ mass the real part of the position of the pole is used giving ∆M = 274 MeV.
There is another lower important threshold already halfway from real ∆'s: namely pion production threshold.This inelasticity can and will be treated by an imaginary term, halfwidth in the ∆ mass eventually showing up in the imaginary parts of NN phase shifts.
However, for the reasons pointed out later in Subsection 18 this cannot be the free ∆ width.
It will be discussed closer in Subsec.II C.
Eqs. ( 1) and ( 2) give an impression of quite a lot of freedom in the system, involving altogether three interactions and the width, probably too much to fit by NN scattering alone and to make definitive conclusions about dibaryons.However, these are somewhat interrelated and it is possible also to find some external constraints as seen in the next two Subsections.
First, an interesting possibility would be attraction between the nucleon and ∆, conceivably strong enough for a bound state [22].However, the diagonal V N ∆ is somewhat far removed from NN, as a higher order effect than V N N or pure excitation of ∆ by V tr , and it is difficult to reach unambiguously by experiments.Diagonal one pion exchange N∆ → N∆ may be rather weak due to the small pion coupling to ∆ f π∆∆ = f πN N /5 from the quark model [23].In pion production and photoreactions its effect in two models has been modest or small [20,24].Because the aim of this work is not to try to impose "dibaryons" (i.e. more exotic than the present coupling to N∆ excitation) but only to study minimally how the NN phases behave with minimal assumptions, this interaction is neglected and will not be discussed any more.Therefore, the possibly predicted resonant structures could be regarded as threshold cusps.

B. N N → N ∆ coupling potential
The familiar one pion exchange (OPE) potential between two nucleons can be naturally and straightforwardly generalized by the transition spin formalism featured e.g. in Refs.[8,23] into a transition potential of the form Here f 2 /4π = 0.076 and f * 2 /4π = 0.35 from the ∆ width [8] are the pion (pseudovector) coupling constants to the nucleon and N ↔ ∆ vertices, and µ is the pion mass.S and S † are the transition spin operators [23] (analogously T and T † for the isospin) defined for example by their reduced matrix elements < ∆||S||N >= 2 in the convention of Refs.[25][26][27].Here the standard tensor operator has been generalized to in the case of the NN → ∆N transition.The radial functions are This procedure yields an admixture of N∆ configurations in principle on the same footing as the NN (apart from the mass difference and asymptotic attenuation) into the wave function, which must necessarily be there due to the nucleon and and pion coupling to the ∆.The method was applied relatively successfully to calculate both total and differential cross sections and analyzing powers [20] and other spin observables [28,29] for the reaction pp → dπ + .Significant improvements and complements have been included in Ref. [21].It may be useful to have in mind that the background of this paper lies in this reaction, and some argumentation to follow largely arises and and is adopted from that work and its later developments.
At this stage it may be worth commenting on the radial dependencies (6) that, following the suggestion by Durso et al. [30], originally two time orderings in OPE taking into account the different ∆ mass was used for the range in Ref. [20].However, inclusion of all time orderings gives back the "normal" static OPE range 1/µ in Eq. ( 6) used later since Ref. [21].
The transition potential presently also includes ρ exchange and dipole form factors with cut-off masses Λ π = 1000 MeV and Λ ρ = 1050 MeV.Interfering destructively with pion exchange, ρ exchange acts in the dominant tensor term V T (r) similarly to a long range cutoff.The spin-spin part V SS (r) of the transition potential (3) is a much weaker effect.Actually for practical purposes, it would be nearly enough to get the strength of the dominant tensor part from the top of the peak of the pp → dπ + total cross section [31] as the only adjusted parameter relevant for N∆.
Of course, the whole transition potential (3) is used with the above form factors, well in accord with the nucleon size and mesonic corrections to the free ∆ width [35].Supplemented by the effective N∆ state width calculated in the next Subsection it reproduces the NN → dπ + cross section excellently from the threshold to 800 MeV and a bit beyond [21].This choice of the form factors is the only fine-tuning of the model done in the "dibaryon" region (i.e.above inelasticity threshold) and it is not sensitive or closely related to NN scattering itself.Therefore, it is reasonable to adopt the transition potential defined above, acceptable for pp → dπ + , also to NN scattering.In this success another essential element is also the treatment of the N∆ channel width discussed in the next Subsection.This is not the free ∆ width but it is not a free parameter either.
One may note that, although, due to the two-body phase space, this cross section is much smaller than the total inelasticity of NN scattering, its data are very good and decisive [31].
Actually, its highest reported point σ prod = 3.148 mb at 575.2 MeV (laboratory energy) is probably the most sensitive probe for the N∆ transition potential strength.(An interpolation of the data would give 3.157 mb at 577 MeV.)In this way the uncertainty of the N∆ vs.
NN effect be reduced from even some tens of percent to few percent, as might be inferred from Figs. 2 and 3 of Ref. [24].The present transition potential defined above yields a flat maximum of about 3.285 mb between 583 and 588 MeV, very close.Then the elastic NN phase shift is fitted by V N N only below inelasticity.No overall phase shift fit extending above pion threshold is attempted, so that the structures in the "dibaryon" region are due to the model, not from fits.
Heavy mesons like the ρ may be frowned in more recent chiral effective field theories (EFT), composed as systematic pion exchange perturbation series including πN and ππ contact terms with few low-energy parameters.(See e.g.[32] for recent progress and extensive bibliography.)However, so far this approach has been limited to lower NN momenta below inelasticity.Therefore, a more phenomenological approach consisting of fewer plausible iterative diagrams, but extending higher in energy, may have value and interest.
Concerning the role of ρ exchange, it might be of some interest in future work to compare spin observables of pp → dπ + including and omitting ρ exchange to test the justification of its inclusion, since, contrary to the tensor part, its spin-spin term (albeit much weaker) adds constructively with pion exchange, which should give differing spin dependence.

C. Rise of inelasticity
The imaginary term from the ∆ width gives naturally also a prediction for NN inelasticity above pion threshold.There are some subtleties required in its inclusion.In Ref. [20] a simple kinematic adjustment is applied on the free ∆ width.(Also in [18,19] different state-independent prescriptions were used.) However, it is important to realize that, because different dynamics in different N∆ channels yield different wave functions (and therefore different momentum distributions), also widths are affected.In particular, the baryon kinetic energy should not be available for the ∆ decay but should be subtracted from the total energy by kinematics.The effective N∆ width taking into account the relative kinetic energy of the nucleon and ∆ is in the momentum representation [21,33] Γ Here Ψ N ∆ (p) is the Fourier transform of the appropriate N∆ channel wave function Ψ N ∆ (r) = w i (r) and Γ(q) the free ∆ → Nπ width [34] Γ(q) = 142 (0.81 q/µ) 3  1 + (0.81 q/µ) 2 MeV (8) with q as the relative Nπ momentum compatible with the total energy and baryon (c.m.) momentum p.The subscript 3 refers to three-body decay N∆ → NNπ, the main inelasticity.
In the calculations of the width and the associated scattering also the calculated two-body inelasticity dπ is taken into account as described in Refs.[21,33].
Because in Eq. ( 7) the wave function influences the channel interaction, which in turn yields the wave function, the procedure requires self-consistent iterative calculations for this inelasticity generating agent.The width decreases quite substantially from its free value, more for higher orbital angular momenta of the N∆ system [33].The main effect to the cross section is increase, because the the absorption acts much like effective repulsion.The result is a significant improvement of the calculated observables [21] and nearly perfect agreement with the total cross section of pp → dπ + .It should be noted that though the success is based on both the transition potential and width, the latter is not an arbitrary or fitted quantity but calculated from the well known ∆ free width (8) and the only free fitted parameters of significance to pp → dπ + are the form factors, basically the height of the peak.
Further, it may also be remarked that the width, a uniform imaginary potential extending spatially to infinity, causes the wave function to fall off making the N∆ channel asymptotically closed also above threshold as well as below, not unlike a bound state.A rotational series of energy states is not far fetched as will be seen later in the next Subsection.

D. Diagonal N N potential
Apart from the explicit generation and use of the N∆ wave function with coupled channels, as iterated pion exchange V tr (r) produces a strong attraction at least below the formal N∆ threshold, which must be accounted for in dealing with the overall effective two-nucleon force V N N (r).Namely, phenomenological data-fitted NN potentials, such as e.g. the Reid potential [36], already by definition include this attraction provided by Nature itself and, therefore, they cannot be V N N (r) in Eq.II A. A theorist working with the equation must then subtract this attraction to avoid doubly counting its influence.
In Ref. [20] this was done by assuming a closure approximation for the iterated transition and using an additive repulsive term with the energy denominator ∆E suitably adjusted to reproduce the phase shifts at each energy.It should be stressed that at that time the purpose was not to calculate or publish NN scattering and its phases but only to generate dependable N∆ wave functions for computing the pion production matrix elements necessary in the reaction pp → dπ + .As discussed before, the knowledge of the N∆ component from this reaction can be used for fixing the transition potential for coupled channels independently of pure NN scattering as presented above.It may be noted that, because the NN potential has to be adjusted anyway (and pp → dπ + is rather insensitive to that), it does not matter much which phenomenological potential is used as a starting point.
In the present context of dibaryons it is also of interest to note that, to produce phase equivalent interactions with and without the N∆ admixture by this procedure, ∆E of Eq. ( 9) predicted a rotational series [37] ∆E ≈ const + 40L N ∆ (L N ∆ + 1) MeV for the effective intermediate state (isospin one "dibaryon") masses, well in accord with the contemporary experimental situation [5].Effectively this result could also be interpreted so that the N∆ state wave function acted as if it were approximately concentrated at the distance of about 1 fm.This is also well commensurate with the OPE range.
Later the double-counting modification has been performed by inclusion of energyindependent short and intermediate range Yukawa terms [33,38] into the Reid potential [36] and its extension to higher partial waves with J = 3 [39].A short digression on these changes is given in the end of this Subsection.From Eq. 9 and second order perturbation arguments it should be clear that the correction should be proportional to some average of the square of the transition strength and from experience that it is strong.And as given, the correction is energy independent.
It is worth reiterating that, as different as the above described methods to deal with double counting may appear, their principal application to the reaction pp → dπ + is rather independent of these details.In particular, its total cross section is practically independent of the NN phase shifts, whereas it is very sensitive to the N∆ component and thus to the An essential issue of this paper is the following: to obtain from a seemingly energy-independent potential strong energy dependencies, often attributed to and parametrized and fitted with explicitly energy-dependent "dibaryons".As discussed in Introduction, arguments to this end have been given long ago e.g. in Refs.[15][16][17][18][19].However, it is hard to say from these works (fixed by NN scattering) which transition strength V tr is correct, since the effect of its changes can be countered in the elastic NN sector.
Because of the interplay between the (complex) N∆ related attraction and the "diagonal" pure NN potential, from the above adjustments it may be seen that NN scattering alone is quite soft for an unambiguous sharing of the contributions from the N∆ and NN parts.
Namely, for example, missing attraction can be provided by either increasing the strength of V tr or inserting it directly in V N N and vice versa for repulsion.Furthermore, increasing the width does not necessarily increase inelasticity linearly, since it also shrinks the N∆ component.Anyway, a weaker transition potential in turn has directly a side effect e.g. as lack of inelasticity in Ref. [19], which uses a weaker πN∆ coupling f *2 /4π = (72/25)f 2 /4π ≈ 0.23 from the quark model. 1 In fact, the difference between the present N∆π coupling f * 2 /4π = 0.35 can be accounted for by vertex corrections [35].
Further, another complication in these analyses with complex potentials is the fact that the absorptive imaginary interaction term itself acts effectively as repulsion [40].Therefore, it is a great boon and benefit to have an additional constraint for the transition strength rather independent of NN scattering as described above.

Modification of V N N
For definiteness here the repulsive modification (in MeV) described above is given to lower partial waves.These were fitted for the Reid potential [36] and its extension to3 F 3 [39], but they should be reasonable at least in pion production for any modern phenomenological potential.The radial variable is x = µr = 0.7r× fm −1 .The potential changes have been adjusted in the elastic region to the data of Ref. [47].They have been very stable below 1000 MeV, can be considered sufficiently precise for the present purpose as will be seen in comparisons later.
For 1 G 4 wave initially the Reid potential for the 1 D 2 state was used modified by

E. Additional comments
Dibaryon fits have often only considered a single reaction (e.g.NN scattering, the topic here) or a single observable (typically a total cross section) without a careful study of N∆ and its ramifications.For example Kamo and Watari [41] fitted the total cross section of pp → dπ + with three dibaryons (i.e. six parameters), assuming a constant background from the ∆.However, the N∆ threshold effect does not behave like a constant background and, in fact, the coupled N∆ model [20] reproduced both the total cross section and also differential and spin observables quite well, even better than Ref. [41], particularly after some improvements in the treatment of pion s-wave rescattering and the N∆ width [21,33].
(To my knowledge dibaryon fits have not often been applied to differential observables in pion production, though Ref. [41] does give predictions or fits with also these.) It might appear that, in second order iteration of pion exchange in NN scattering (and first order in pion production), the effect of the intermediate ∆ could be obtained simply by inserting into the perturbation theory energy denominator the ∆−N mass difference ∆M and width giving E − ∆M + iΓ/2.Although this looks like a trivial way to get a resonant structure, however, it is open to some questions.It was pointed out above that already the extremely phenomenological numerical calculation [37] for phase equivalent potentials indicates some more subtle structure in the energy denominator, e.g. from the centrifugal effects discussed in more detail in Ref. [42].This is due to the fact that in asymptotically closed N∆ channels the expectation value of the centrifugal potential is actually well defined and quantized as argued above in Subsec.II C. Furthermore, the orbital angular momentum of the excitation can be L N N ±2 in addition to L N N , and the N∆ centrifugal barrier obviously favoring L N N − 2 and strongly suppressing L N N + 2.Even the definition of the width in the two-baryon system needs special attention and becomes strongly state dependent.As seen, quite obviously it cannot be the free width [21,33], but consideration of the different parts of the N∆ kinetic energy expectation values should be addressed apart from the internal ∆ particle excitation as in Eq. ( 7).
Moreover, there is a perhaps deeper problem of the applicability of perturbation theory in this context: Perturbation theory needs the matrix element of the perturbation between the initial and intermediate (or final) state to generate its effect.But what is the "unperturbed" wave function of the N∆, which has yet to be computed to get this matrix element?This is a problem in old perturbation calculations such as Ref. [43], which can reproduce the total cross section of pp → dπ + well by straightforward substitution of the above energy denominator, but not the differential one or spin observables [44][45][46].
The question of the applicability of perturbation theory may not be as relevant in NN scattering for the fits with hypothesized dibaryons.The resonant structure as substituted by hand is then trivial and automatic.The positions and widths are free to choose in this procedure.With the N∆ coupled channels there is no such luxury for these.The position and magnitude come directly from the coupled Schrödinger equation as a threshold cusp (with the ∆M and centrifugal forces as necessary and unavoidable parts of the equation), and the width is calculated self-consistently for each channel along the lines of Ref. [33].
Although in the coordinate space equations the resonant structure is not manifest, in the equivalent momentum (or energy) representation, of course, it must be and, after calculation, becomes visible for observables also in the coordinate representation.This is schematically demonstrated below and above the N∆ threshold in of adding the width to curve c (only the possible real part shown).So, from this on we'll stick to this representation, which is also the most transparent framework to incorporate the necessary centrifugal barriers.The dashed curves only incorporate the 5 S 2 (N ∆) states as described in the text.
(the solid curves).Here the data have been fitted to the real parts of phase shifts ℜδ [47] below inelasticity (below 300 MeV), first starting with the phenomenological Reid potential [36] and making the modifications to avoid the double counting of the additional attraction from the explicit N∆ inclusion as explained in the previous section. 2The data of Ref. [47] have been very stable below 1000 MeV, so they have been deemed precise enough especially compared with any meaningful deviations from the calculations.
The agreement with the elastic data [47] is nearly perfect in the fitted region (as also for other partial waves).Then the resonance-like energy dependence will arise above 400 MeV due to the coupling to the isobar excitations, most pronouncedly to 5 S 2 (N∆), not by any further fitting.It can be seen that, compared with the energy-dependent analysis of pp scattering [47], above 300 MeV the results even slightly exaggerate resonant behavior still after the smoothing effect of the width inclusion.The structure follows clearly perturbative argumentation: attraction below the threshold at about 600 MeV and then a change of the sign into repulsion.This behavior is faster above 600 MeV than in Refs.[18,19] perhaps due to the kinematically suppressed N∆ width and, because, to leave space for possible "dibaryons", here no optimization is done above pion threshold.
The inelasticities of Elster et al. [19] are smaller than here, because they use the weaker quark model πN∆ coupling in the NN → N∆ transition.As already emphasized in Sec.
II, this transition strength is essential and is fixed externally from pp → dπ + in this paper.
The quark model coupling gives also 20% underestimate for this reaction even if it is used only for the transition potential (as it would appear in [19], more if it is also used in the final production vertex [24].
In general, the S-wave N∆ component is the most important in reactions in this energy regime and it is favoured also by the decrease of the angular momentum [37] in the D-to S-wave transition as noted in Sec.II.Incidentally, at distances most relevant for pion exchange, here this decrease is associated with an effective decrease in the centrifugal potential comparable to the threshold ∆M, making the N∆ excitation effectively rather degenerate with the initial NN state in this region, and therefore strongly boosting this transition.
More specifically, the p-wave pions associated with the initial 1 D 2 (NN) state account for about 80% of the pp → dπ + total cross section in the peak region around 580 MeV (laboratory energy) [31], and this affluence arises mainly from the 5 S 2 (N∆) admixture.Therefore, it might be of interest to check separately and explicitly the importance of different N∆ configurations for this reaction as well as for scattering.In Fig. 2 the dashed curves show the effect of neglecting the coupling to the higher 5 D 2 (N∆) and 5 G 2 (N∆) channels in the equations (II A).The result is very interesting in that the higher excitations (with even much higher effective thresholds [42]) are, however, still very important and essential in providing attraction to ℜδ 2 .A similar but even more drastic influence of D-wave N∆'s will be seen later in the case of the 1 S 0 (NN) partial wave.
In contrast, the effect of the higher N∆ states to ℑδ 2 is very small except fairly above the N∆ threshold, reflecting the fact of the diminished width due to the smaller free phase space for the ∆ decay [33].Clearly the fast changing resonant behavior has moved to lower energies also with much lesser attraction.It should be noted that, to keep comparison simpler in this calculation of the dashed curves, no further readjustments for the NN potential or the N∆ widths were made.Also a possibly interesting finding for pp → dπ + was that the cross section with 5 S 2 (N∆) alone in this exercise was in fact larger than the original, due to destructive interference between 5 S 2 (N∆) and the smaller component 5 D 2 (N∆), as may be inferred from e.g.Ref. [20].
The missing attraction at the highest energies might possibly be achieved also by inclusion of some higher nucleon resonance (N(1520) or N(1535)) or double ∆ excitation, beyond the scope of this paper.
Fig. 3 shows the second important dibaryon candidate 3 F 3 starting from the NN potential extension [39] of the Reid potential.Here the included N∆ configurations are 5 P 3 (N∆), 5 F 3 (N∆) and 3 F 3 (N∆) with the first one being dominant due to the decrease of the orbital angular momentum and consequently the smaller centrifugal potential in the N∆ system favouring it as argued already in Ref. [37].This plot is very similar to Fig. 6 of Ref. [33] where, however, only the "dibaryonic" state 5 P 3 (N∆) was included to emphasize its special influence and nature as a "dibaryon".The higher angular momentum states give minor attraction of 0.5-1.0degrees above 500 MeV and, in comparison with Ref. [33], perhaps one might also argue a very slight broadening upwards in energy due to the stronger centrifugal effect in them.The "knee" at 600 MeV is better reproduced here than in Refs.[18,19] probably because of the dynamically calculated and reduced N∆ state width and strong enough NN → N∆ transition potential.
This wave was found absolutely essential in successful production of differential observables in pp → dπ + [20,21] and it also accounts for 10-20% of its total cross section in the peak region.Probably the two dibaryonic states discussed so far dominate also its isospin-cousin component in pp → npπ + at intermediate and high energies.Furthermore, 3 F 3 appears to give the largest individual contribution at least to the total cross section of the reaction pp → ppπ 0 , with the final isotriplet NN state, although overall this is significantly smaller than the isosinglet.The next potential N∆ dibaryon-like state, as argued in Refs.[37,42] and also experimentally suggested in Yokosawa's review [5], 1 G 4 (NN) is discussed in Fig. 4 with the 5 D 4 (N∆), 5 G 4 (N∆) and 5 I 4 (N∆) admixtures.Again, computationally a clear broad resonance-like maximum is seen in the phase shift at about 800 MeV laboratory energy, while in the data the peak is missing.Also inelasticity is overestimated in the calculated scattering.crosses the imaginary axis slightly above 700 MeV (lab energy) but with rather strong inelasticity [33].On the other hand, the 3 F 3 amplitude remains on the left-hand side of the imaginary axis at all energies [33].The 1 G 4 state has a calculated bump of ℜδ not seen in the data.However, although the latter NN amplitudes may not show particular drastic features in actual pure phase shift Argand diagrams, interestingly the N∆ wave functions do suggest resonant structures, both in magnitude and phase displayed for example in two Argand-like diagrams designed as probes for the 5 D 4 (N∆) component in Ref. [42], which may show up in reactions sensitive to N∆ and involving overlaps with N∆ configurations.
Apparently the maximum in the phase shift here is due to constructive interference of the N∆ effect with the attractive background from the NN diagonal potential (a modified 1 D 2 Reid potential very similar to that used for Fig. 2 was employed).
To study a little bit further this interplay in the 1 G 4 wave, the dashed curves utilize instead of the modified Reid potential as the diagonal NN interaction the one pion exchange potential (OPE), expected to dominate peripheral waves.Although the fit (solid) is perfect, also OPE as the diagonal potential is acceptable below 300-400 MeV (dashed).The overall phase ℜδ for this is slightly more attractive, increasingly so for higher energies, and the maximum moves to higher energy.This probably follows from the lack of the short range repulsion present in the phenomenological potential.
But more importantly, it is very interesting and intriguing that the N∆ excitation is still very active and alive at such a high value of L, as seen in comparison of dashed curves vs.
dash-dot (pure OPE), where the latter does not have this excitation at all.This is probably due to the long (OPE) range of the transition potential V tr (r) itself and the favourable decrease of the centrifugal repulsion in the change 1 G 4 (NN) → 5 D 4 (N∆).
It might be argued that in high-L peripheral waves only the lightest meson π could be exchanged and so only long range OPE could have any effect, and excitation of ∆ would be of shorter range.However, the transition potential has the same range and once within that range the centrifugal NN repulsion can be much larger than in the N∆ system for L ′ = L − 2. Therefore, the N∆ generation is not energetically an excitation at all, i.e. not like a heavier meson exchange.So once OPE is possible, so, to some extent is N∆.
The OPE range is also essentially the range of the N∆ wave function [42].Again, due to the difference in the centrifugal potentials the N∆ admixture may not be practically an excitation at all.The coupling effect looks even more drastic when comparing with the pure OPE in the NN wave without the ∆-resonance generation and the associated strong attraction (dash-dot curve).Clearly, the G-wave is not peripheral enough for the pure OPE to dominate alone at least above 300 MeV.Quite evidently this raises the need to go further to even more peripheral waves.Below the inelastic threshold 300 MeV, the result here (dashed vs. dash-dot) is numerically in perfect agreement with the EFT calculation [32] in next-to-next-to-leading order (NNLO) vs. leading order (LO).
The dipole form factor with the range parameter Λ π = 1000 MeV has been included in the OPE's all the time.The dotted curve shows the 1 S 0 phase calculated with the double-counting correction but without the N ∆ coupling.

D. J = 0 waves
After these somewhat peripheral waves it may be in order to study similarly the opposite situation with the J = 0 states, where the transition into dibaryonic states with decreasing L is not possible.Quite contrary, the 1 S 0 state is coupled only to the D-wave 5 D 0 (N∆), which should involve some 300 MeV extra centrifugal energy (plus some radial addition, too [42]), bringing its effective excitation energy very high.
Fig. 5 presents the results for this wave by circles and solid curves and for 3 P 0 by squares and dash-dot curves.A good agreement can be found for the real part in both waves up to 800 MeV (well, up to 500-600 MeV for the 3 P 0 wave).In the S-wave the N∆ coupling is complemented with the repulsive double counting correction 770 exp(−4µr)/(µr) MeV to the Reid potential [36] -the phase has been fitted between 50 and 300 MeV (the solid curve vs. filled circles).However, consistently with the arguments given earlier, the isobar excitation produces very little inelasticity -hardly distinguishable from the energy axis even after multiplying by ten.This is due to the minuteness of the 5 D 0 (N∆) width calculated with the prescription of Refs.[21,33].Likewise, van Faassen and Tjon [18] get also consistently small inelasticity in this wave, whereas Elster et al. [19] have this rising with energy.The latter may be, because in an optical NN potential it is difficult to include this state dependence on the N∆ effect.In fact, also the 1 S 0 contribution to pp → dπ + cross section is very small, in part due to destructive interference between the NN and N∆ components.The same seems to be true for pp → ppπ 0 s-wave production (with 3 P 0 final state nucleons).To make the imaginary parts of the NN phase shifts comparable in the scale, all of them have been multiplied by 10.
As a sideline, it is worth noting that, as a check, it was verified that in the low-energy limit the phase turns down at about 1 MeV laboratory energy yielding for the singlet scattering For the 3 P 0 (NN) wave (coupled to 3 P 0 (N∆)), also with an attempted fit up to 300 MeV, ℜδ agrees well with data up to about 600 MeV.Its inelasticity agrees roughly with data in magnitude but would peak at higher energy.As noted before, this paper includes the inelasticity as the imaginary part ℑδ of the phase shift, which causes a different energy dependence from the ρ parameter of [47] for small values.It may be of some interest to note that, especially for the 3 P 0 state, also the large real part of the phase shift can have a strong effect in the inelasticity (with the notation of Ref. [47]) bringing it down in spite of the seemingly very large values of ρ.
The J = 0 state pion reactions are somewhat superficially related by their possibility to s-wave pion production by transitions 1 S 0 → 3 P 0 s and 3 P 0 → 1 S 0 s.However, an explicit calculation suggests these reaction channels to be significantly smaller than production in other NN states discussed here.Further, in their comparison, the former is nearly an order of magnitude smaller than the latter.

E. 3 P 1 wave
Another NN wave without a possibility of the favorable decrease of L in the N∆ transition is 3 P 1 , which has particular interest because of its significance in pion production.Especially, due to s-wave pion rescattering in s-wave pion production, it dominates pp → dπ + and pp → npπ + at threshold.Fig. 6 shows the results for this wave (solid line) with a global fit to the phases of Ref. [47] deep into pion production region.Generally there is a fair agreement of ℜδ with the smooth data behavior, quite enough for the originally intended calculation of pion production reactions and isospin mixings.However, one may envision a tendency to slight similarity with the 3 F 3 wave (Fig. 3).To emphasize this tendency one might try to improve the already good fit by mimicking the curvature in the elastic region below 300 MeV (dash-dot line).Rather clearly now the latter curve has as well a wide shoulder in the energy region 600-800 MeV, similar to the case of 3 F 3 also with P -wave N∆'s, but not really visible in the data for this wave.
The data indicate small and remarkably constant inelasticity ℑδ ≈ 4 • − 5 • above 600 MeV, whereas the calculated results increase rather linearly with energy exaggerating it above 600 MeV.It is probably useful to remember the behavior of s-wave pion production in pp → dπ + [20].At threshold this NN wave dominates, the cross section being linear in the pion c.m. momentum q π , that is square root of the excess energy as is usual for the two-body final state phase space.However, this two-body reaction dominates the total inelasticity of 3 P 1 (NN) only up to 420 MeV.Aut above this energy the scattering inelasticity calculated from the absorptive width quickly takes over so that already by 500 MeV twobody pion production pp → dπ + is less than 10% of the total.Moreover, due to destructive interference between direct production and pion s-wave rescattering the cross section from this amplitude virtually vanishes around 600 MeV [20].Above that energy direct production from the axial charge part of the pion-baryon vertex wins and the amplitude changes sign, but the s-wave pion cross section remains still very small as seen already in Ref. [20].This wave is illustrated by the dotted curve in Fig. 6 (multiplied by the factor of 100, including also d-wave pions).In the width prescription of Ref. [33] the phase space integral over intermediate N∆ momenta is augmented also by the pp → dπ + cross section.
To check the validity of this procedure and to avoid problems with possible double counting in explicit consideration of this reaction, the dash-dot curve now includes only the phase-space integrated width in the calculation of the scattering inelasticity (and the elastic part of the phase shift).The pp → dπ + cross section is then added explicitly by hand to yield the total absorption cross section and finally to calculate the inelastic phase shift for the dash-dot curve from σ abs (18) with k the nucleon c.m. momentum.In spite of the dominance of the dπ + final state at threshold (due to different phase space), its overall effect is very little for the present purpose, since actually also the two-body reaction cross section is very small there.Above 600 MeV the inelasticity from this pion production is less than 1% of the scattering inelasticity so that its effect is very small whether it is included in the width (imaginary potential) or explicitly added afterwards by Eq. ( 18) into the total absorption cross section to calculate ℑδ.Namely the dotted curve shows 100 times its contribution.It might still be noted that except for the minimum around 600 MeV this 3 P 1 -wave contribution to pp → dπ + is still definitively dominated by s-wave pions over the also possible d-wave.
Is there a way to obtain the observed constancy of the inelasticity showing in the data?
The structure of the 3 P 1 contribution to pp → dπ + (dotted curve) is intriguing as a total change from monotonously growing cross section at threshold (solid and dash-dot curves).
In spite of this, the view of the pp → dπ + possibly changing the trend of ℑδ towards a more constant value seems hopeless due to the smallness of its 3 P 1 -wave contribution.However, quite probably the importance of s-wave pion rescattering at threshold also holds in the isospin related three-particle final state in pp → npπ + .Furthermore, also the destructive competition between the axial charge contributions could remain true producing a similar strong pion minimum for wave functions at any given final NN energy.This might reduce the smeared final state phase space integral of the cross section to resemble better the constant inelastic data.
True enough, also in the three-body pp → ppπ 0 reaction there is a strong minimum in s-wave production around 550 MeV (however originating from the initial 3 P 0 (NN) state), giving considerable constancy of the cross section up to near 600 MeV [48][49][50] (though the cross section in this final state isospin arrangement is much smaller).The present phase calculation, taking the width from integration over intermediate N∆ momenta (and using as input for this the free ∆ width) to simulate the three-body reaction [33], does not have these effects in.It does not require much imagination to consider how much the general outlook and the average of the dotted curve (Fig. 6) would differ, if the minimum caused by the destructive interference would be smeared over in the phase space integral.However, it would require a totally independent and different calculation of the three-body reaction.
Some hints in this direction may be found in Ref. [51].
The tensor coupled case 3 P 2 − 3 F 2 is significantly more involved [47].In the elastic region it is standard to express the 2 × 2 S-matrix in the Stapp form [52] where, in addition to the phase shifts δ ± in the respective ± = J ± 1 NN channels treatable as before in this paper, the mixing parameter ǫ opens the possibility of their mixing.In this paper inelasticity has been implemented by complex phase shifts, which automatically arise for complex potentials.For convenience in analyses Ref. [47] accommodates inelasticity instead by adding a symmetric imaginary 2 × 2 component into the otherwise real elastic Thus, keeping this as the real part of the K-matrix, it is generalized to the complex form However, I proceed in the present work by forcing inelasticity into the Stapp form.Then, even the mixing parameter may become complex, in contrast to the choice of [47].
In the phases of Fig. 7 the elastic region fit of ℜδ continues as essentially perfect somewhat above 400 MeV, but above 500 MeV deviates strongly especially for the too attractive (solid curve vs. full circles).The same behavior at high energies occurs also in Refs.[18,19] (essentially constant ℜδ − ≈ 20 • from 200 MeV to 1000 MeV).However, inelasticity in this incident wave is excellent (the imaginary part of the phase, solid curve vs. hollow circles).The experimental phase parameters are extracted from the data of Ref.
[47] using their prescription for transforming the K-matrix to the S-matrix elements.In contrast, for the incident 3 F 2 -wave the elastic phase remains reasonable, but the imaginary part is tenfold as compared with the nearly elastic data.However, one should note the earlier warning about the different sensitivity of the two representations for small parameters (ℑδ vs. ρ) in this case.Fig. 8 shows the J = 2 mixing parameter ǫ.In form ℜǫ is very reasonable in comparison with data (the lower curves vs. circles).However, the imaginary part (squares) may be surprising, not present in the original definition (21) of Ref. [47].Nevertheless, from the inverse of Eq. ( 20) one can get the off-diagonal elements related by (Eq.( 7) of Ref. [47]) where S 0 (resp.K 0 ) is the off-diagonal element of the S-and K-matrix and D K is the determinant of (1 − iK).By definition (5) of Ref. [47] above inelastic threshold K acquires both real and imaginary parts, and it is perfectly plausible that the mixing parameter ǫ is complex.It is just a consequence of the different definitions, and ℑǫ may be rather directly (and simply) linked to ρ at least for small mixing angles.In comparisons between theory and experiment, the difference between the upper curves and the squares is about the most relevant at this point in consideration of how inelasticity is dynamically embedded in theory and shows up in ℑǫ.It is probably worth noting that also van Faassen and Tjon get an imaginary component in ǫ 2 remarkably the same size as here [18].The real part is somewhat better here, though qualitative trends are the same.
There is another perhaps strange point of interest and suspect.Numerically it was found that the exact values of ǫ depend on which of the two waves is the incident state (see solid vs. dashed curves in Fig. 8).This means that the S-matrix is not perfectly symmetric as normally expected. 4Apparently this feature has to do with the numerical finding reported earlier in Ref. [33] that, in addition to the rather natural dependence on the relative orbital angular momentum between the ∆ and the nucleon themselves in the intermediate state, also the external incident NN angular momentum matters in the calculation and the value of the effective N∆ width.As noted already in Ref. [33], the treatment of inelasticity did not start from explicitly Hermitian first principles, but used existing inelasticity (width) of the πN resonance, the ∆, as the starting assumption.However, this may be a practicable procedure for implementing intuitively apparently physical effects in the system.Thus there would be somewhat different absorption for the two different initial waves.If this difference was taken off, the S-matrix came out as symmetric (as in the formulation of Ref. [47]).
Also, for the applicability of the theory it may be happy that this effect is actually small as compared even with the (already rather small) total values of ǫ.
Finally, just for completeness and easier comparisons Fig. 9 presents the inelasticity parameter ρ as defined by Arndt et al. [47] easily obtainable by its definition (20) from the calculated S-matrix as Here, in an awkward notation S − 0 is the off-diagonal S-matrix element originating from the initial − channel (respectively + ↔ −) and D S is the determinant of (1 + S). parameters ℜδ ± are practically the same in both representations, Stapp or Arndt.

G. More peripheral waves
In Subsec.III C the persistent prominence of the attractive N∆ contribution even in the high 1 G 4 partial wave is very suggestive that this behavior may extend even higher.Fig. 10 presents results for two more peripheral waves, making possible a comparison of 3 H 5 and 1 I 6 waves including OPE together with N∆ or OPE alone.The dashed curves can be contrasted with the data of Arndt et al. [47] in the case of 3 H 5 .The model gives quite similar behavior as seen for 3 F 3 , only at about 300 MeV higher energy.The trend may also be seen in the data (negative triangles), though the rise of the data at high energy is more pronounced and pointing towards even higher energies.The predicted inelasticity is also tolerable.The solid curve vs. circles, displaying the real part of the phase shift for the 1 I 6 wave, is perfectly satisfactory with coupled channels.Also the minor imaginary part ℑδ is acceptable, though the data are negligibly small and not shown.
One can make a very interesting comparison with the pure OPE calculation (dotted curves).Clearly at and above the nominal ∆ threshold isobar excitation is still well comparable with OPE in such high angular momentum states.The effect is especially outstanding and manifest in the destructive interference with the repulsive triplet OPE both in 3 F 3 and Similarly to the 1 G 4 wave, below 300 MeV the deviation from OPE to full coupled channels (solid vs. dense dots for 1 I 6 and dashed vs. sparse dots for 3 H 5 ) is similar to the EFT results of Ref. [32] from LO to NNLO.

IV. CONCLUSION
The aim of the present paper has been to study NN + N∆ dynamics in the framework of coupled channels to produce resonant structures in NN scattering without explicit input of energy dependencies, for reference and benchmarks to confront and compare with explicitly energy dependent fits, such as those with hypothesized dibaryons.Much of the necessary physics can be incorporated by a formulation of coupled channels in terms of excitation of intermediate N∆ components.As already seen at several points, this is not a new idea [8] and, in principle, it has strong foundations in pion exchange: the pion coupling to nucleons and ∆'s is well established and should be reconciled parallel with NN scattering dynamics.
Only after considering this is it meaningful to involve other dibaryons.
The model, shortly outlined in Sec.II. is substantiated by references to older work, in particular on pion production, which is a natural association and a relative to the present work.The relation to "dibaryon" fits is first to obtain as good a fit to the data as possible below inelasticity and then leave the rest to depend on N∆ dynamics above the pion threshold with no supplemental degrees of freedom.
The desired features in the ∆ excitation energy region are successfully predicted for the NN waves which have also the most important resonant amplitudes in pion production in pp → dπ + ( 1 D 2 and 3 F 2 , Figs. 2 and 3).The results overall agree quite well with earlier work on NN scattering with N∆'s [18,19].Some of the differences may be identified in the state and energy dependence of the effective N∆ width and the NN → N∆ transition strength.This strength seems to be too weak in Ref. [19].Also the restriction of the phase shift fit only to the elastic region below 300 MeV may be reflected into the resonant region as some deviation from models using global optimization.This is a calculated choice to allow space for possible extraneous dibaryons.
The more peripheral wave 1 G 4 is seen to obtain still a major contribution from the N∆ coupling, even overpowering OPE, normally expected to overwhelmingly dominate (Fig. 4).
This may be understood by the interplay with centrifugal effects: effectively the intermediate 5 D 4 (N∆) state is not energetically higher than the NN centrifugal barrier within the OPE range and thus is not particularly an excitation.The calculated phase shift ℜδ 4 displays rather a clear and wide maximum not seen in the data.The inelasticity ℑδ 4 is somewhat overestimated in comparison with Ref. [47], but this may be exaggerated for small phases by the different parametrization.The importance of the N∆ coupling prompted further study of the peripheral waves in Subsec.III G.
In the J = 0 states the real phases come out rather good and also ℑδ for the 3 P 0 is reasonable though not perfect (Fig. 5).The 1 S 0 inelasticity does not agree with data, being much smaller, essentially negligible.Calculationally the reason is the very small calculated N∆ width and the large centrifugal barrier in the intermediate 5 D 0 (N∆) state essentially doubling the effective N∆ threshold [42] (both are interrelated with each other).Also the phase shift analysis result [47] for ℑδ is exceedingly small for such a low partial wave, and one might again remind on the sensitivity of the different parametrizations for small phases.
The calculated 1 S 0 contribution to pp → dπ + is also very small, practically negligible in its total cross section, well in line with the present results.Theoretically it is interesting that the attractive strength of the explicit N∆ is huge, 30-40 degrees from Fig. 5, or seen in another way alone by itself enough to bind the NN.
In the 3 P 1 incident wave a slight shoulder is predicted in ℜδ 1 , not seen in the smooth data (Fig. 6).Also the present calculation overestimates the small and constant ℑδ 1 .
However, possible improvements are suggested in the text from explicit experience in pion production (e.g.destructively interfering pion s-wave rescattering [20] missing in the present calculation).After all, the inelasticities are predominantly pion production.
In the tensor-coupled system 3 P 2 − 3 F 2 the real phases are reasonable, except for the Pwave ℜδ above 600 MeV, which is much too attractive (Fig. 7).The real part of the mixing angle ℜǫ 2 compares reasonably with the data (which only do give the real part) (Fig. 8).
However, the dynamic coupled channels model calculation also predicts an imaginary part for this.An interesting peculiarity (due to the different definitions) is getting an imaginary part for the Stapp mixing angle ǫ 1 from the real Arndt parameter ρ (therefore only about half of it being of dynamical origin, the other half coming from the transformation between the two representations).Another interesting, perhaps troublesome, result was that, parallel to earlier findings [21,33], also the angular momentum of the incident external NN state influences the widths calculated in the N∆ intermediate states and the S-matrix becomes slightly asymmetric.
Furthermore, even in higher peripheral partial waves 3 H 5 and 1 I 6 the N∆ was seen to contribute to the elastic parts of the phase shifts comparably with OPE and to inelasticity perhaps alone by construction.One might raise the question whether this behavior continues without end, thus challenging the idea of neglecting anything else but OPE in peripheral waves.Below pion threshold the results for these peripheral waves agree numerically with those of Ref. [32] using EFT.
Obviously much of resonances or resonance-like structures can be obtained by coupled channels involving N∆ excitations, even slight indications arose in waves where they are not seen in data.The method appears to be a very reasonable, even successful starting point for producing realistic wave functions to use in further calculations.The main shortcoming may be that some additional interactions in strongly interacting medium are not automatically included which affect inelasticity, such as e.g.s-wave rescattering of pions in the 3 P 1 wave.
But this is just what the "further calculations" mean, with some references pointed out in the text.However, most of the p-wave pion effects come already in with the dominant ∆.In spite of the aforementioned shortcomings, apart possibly from excessive attraction in the 3 P 2 wave above 500 MeV, the remaining deviations from data do not show particularly systematic and pressing need for extraneous dibaryons.
also familiar from pion exchange between two nucleons.The NN equation is then coupled by V tr to the N∆ equations including (in addition to their possible mutual interactions not discussed here) the ∆ − N mass difference ∆M = 274 MeV as given in the previous Subsection and (above the inelastic threshold) the effect of the width −iΓ i /2 in the N∆ channel i.

Fig. 1 .FIG. 1 .
FIG. 1. Schematic representation of the N ∆ wave function around the channel threshold.Curves a, b and c below, at and above threshold without width.Curve d depicts the attenuation due to the width in the oscillatory case.

FIG. 2 .
FIG. 2. The 1 D 2 phase shift.The solid lines are the calculation with 5 S 2 (N ∆), 5 D 2 (N ∆) and 5 G 2 (N ∆) admixtures and the filled circles and triangles are the experimental real and imaginary component of the phase shift extracted from the energy-dependent fit to pp data [47], respectively.

FIG. 5 .
FIG.5.J = 0 phase shifts: solid curves 1 S 0 and dash-dot 3 P 0 .The circles show the analysis of Arndt et al.[47] for 1 S 0 and the squares for 3 P 0 .The filled symbols present the real parts and the hollow ones the imaginary parts.For clarity, all imaginary parts ℑδ have been multiplied by 10.
length a negative value a S ≈ −38 fm.(Of course, in this context only the sign matters.)Therefore, the possible presence of a bound state and the ensuing extra node in the NN wave function should not be the source for the smallness of the width.As another sideline one might also note that, in spite of the 5 D 0 (N∆) state being unfavoured in reactions such as pp → dπ + (due to destructive NN and N∆ interference), its attractive effect may, however, be surprisingly strong.A quick calculation indicated that, with the direct NN interaction totally discarded, the iteration of just the N∆ transition predicts a positive scattering length a S ≈ 2.3 fm suggesting that the N∆ component alone could support a bound state with a binding energy of E B ≈ 8 MeV.To further illustrate the strength and importance of the N∆ configurations even in elastic scattering, the dotted curve still shows the 1 S 0 phase shift using the Reid potential with the above double-counting repulsion correction but without including the 5 D 0 (N∆) coupled channel.The gap of[30][31][32][33][34][35][36][37][38][39][40] degrees between the two results (solid vs. dotted) is massive and is one way of viewing the significance of the N∆ coupling effect.

FIG. 6 . 3 P 1
FIG.6.3P 1 phase shift: solid curves as described in Ref.[33] and earlier figures.The circles and triangles show the analysis of Arndt et al.[47].The dot-dash curve shows the modification described in the text, and the dotted one is 100 times the sole contribution from pp → dπ + to inelasticity, (s + d)-wave pions.

3 P 2 -FIG. 7 . 3 P 2 − 3 F 2
FIG. 7.3 P 2 − 3 F 2 phase shifts: solid curves the real and imaginary parts of the P -wave shift, the dashed curves for the F wave.Respectively, the circles and squares show the analysis of Arndt et al.[47] transformed to the Stapp parametrization (19) (filled symbols for ℜδ, hollow symbols for ℑδ).The F -wave experimental inelasticity is multiplied by ten.

FIG. 8 .
FIG.8.The mixing angle ǫ as defined by Stapp et al.[52] extended to inelastic region.The circles (ℜǫ) and squares (ℑǫ) show the analysis of Arndt et al.[47] forced into the Stapp form.The solid curves show the results originating from the incident P -wave and dashed curves from the F -wave.

FIG. 9 .
FIG.9.The inelasticity parameter ρ ± as defined by Arndt et al.[47].The P -wave solid vs. circles, the F -wave dashed vs. squares.This may be compared with the ℑδ's of Fig.7in the Stapp representation.

FIG. 10 .
FIG.10.Solid curves and circles show the 1 I 6 partial wave phase from the coupled channels calculation and from[47] (only ℜδ given for data).The dashed curves and triangles present the 5 H 5 wave, and the dotted curves display the pure OPE result for both.