Nature of $\bm{S}$-wave $\bm{NN}$ interaction and dibaryon production at nucleonic resonance thresholds
V. I. Kukulin, O. A. Rubtsova, M. N. Platonova, V. N. Pomerantsev, H. Clement, T. Skorodko
aa r X i v : . [ nu c l - t h ] S e p Eur. Phys. J. A manuscript No. (will be inserted by the editor)
Nature of S -wave N N interactionand dibaryon production at nucleonic resonance thresholds
V.I. Kukulin a,1 , O.A. Rubtsova b,1 , M.N. Platonova c,1 , V.N. Pomerantsev d,1 ,H. Clement e,2 , T. Skorodko Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Leninskie Gory 1/2, 119991 Moscow, Russia Physics Institute and Kepler Center for Astro and Particle Physics, Eberhard–Karls–University T¨ubingen, Auf derMorgenstelle 14, D-72076 T¨ubingen, GermanyReceived: September 15, 2020/ Accepted: date
Abstract
Phase shifts and inelasticity parameters for
N N scattering in the partial-wave channels S – D and S at energies T lab from zero to about 1 GeV aredescribed within a unified N N potential model assum-ing the formation of isoscalar and isovector dibaryonresonances near the
N N ∗ (1440) threshold. Evidence forthese near-threshold resonances is actually found in therecent WASA experiments on single- and double-pionproduction in N N collisions. There, the excitation ofthe Roper resonance N ∗ (1440) exhibits a structure inthe energy dependence of the total cross section, whichcorresponds to the formation of dibaryon states with I ( J π ) = 0(1 + ) and 1(0 + ) at the N N ∗ (1440) threshold.These two S -wave dibaryon resonances may provide anew insight into the nature of the strong N N interac-tion at low and intermediate energies.
Keywords
Nucleon-nucleon interaction · Dibaryonresonances · Single- and double-pion production · Roper resonance
The traditional point of view on the strong
N N in-teraction at low energies ( T lab .
350 MeV) is based onthe classic Yukawa concept [1] suggesting t -channel me-son exchanges between nucleons. Later on, this idea ofYukawa has been realized in the so-called realistic N N potentials [2,3,4]. Recently the realistic
N N potentials(of the second generation) have been replaced by theEffective Field Theory (EFT) which can treat single a e-mail:[email protected] b e-mail:[email protected] c e-mail:[email protected] d e-mail:[email protected] e e-mail:[email protected] and multiple meson exchanges more consistently [5,6].However when the energy is rising beyond 350 MeV,the numerous inelastic processes enter the game andthe application of the traditional approach meets manyserious problems. Importantly, most of them are relatedto our poor understanding of the short-range N N in-teraction and the corresponding short-range two- andmany-nucleon correlations in nuclei and nuclear matter[7]. From the general point of view, these problems shouldbe tightly interrelated to the quark structure of nu-cleons and mesons. On the other hand, the consistenttreatment of the intermediate-energy
N N interaction,especially for inelastic processes, within the microscopicquark models is associated to so enormous difficulties[8,9,10], that nowadays we have to limit ourselves withsome phenomenological or semi-phenomenological treat-ment. However it is still possible to use some hybridapproach and to combine the meson-exchange treat-ment for the long-range
N N interaction with the quark-motivated model for the intermediate- and short-rangeinteraction [11]. Such a model can be naturally based onthe assumption about the six-quark bag (or dibaryon)formation at sufficiently short
N N distances, where thethree-quark cores of two nucleons get overlapped witheach other [12]. Implementation of this idea does not re-quire a detailed knowledge of the six-quark dynamics,but only needs the projection of the six-quark wave-functions onto the
N N channel and an operator cou-pling the two channels of different nature, i.e. , nucleon-nucleon and six-quark ones. So, in the
N N channelwe can take into account only the peripheral meson-exchange interaction, while the influence of the internal6 q channel on the N N interaction can be described bya simple mechanism of an intermediate dibaryon reso- nance formation with appropriate
N N ↔ q transitionform factors [13,14,15].The dibaryon-induced mechanism for the short-range N N interaction was initially suggested in Ref. [12] andquite successfully applied to the description of
N N elas-tic scattering phase shifts and the deuteron propertiesin Refs. [13,14]. However, these works did not considerthe inelastic channels and also did not try to identifythe S -matrix poles resulting from the fit of the phaseshifts using the dibaryon resonances found experimen-tally. On the other hand, in recent years a number ofdibaryon resonances have been discovered which aremanifested most clearly in the inelastic processes [16].In Refs. [17,18,19], we elaborated a unified modelthat can describe well both elastic phase shifts andinelasticities in N N scattering in various partial-wavechannels at laboratory energies from zero up to about1 GeV. Thus, it has been shown that one can reproducequite satisfactorily both elastic and inelastic
N N scat-tering phase shifts in a broad energy range using only aone-term separable potential with a pole-like energy de-pendence (with complex energy) for the main part of in-teraction and the one-pion exchange potential (OPEP)for the peripheral part of interaction. The model wasapplied to various partial-wave channels with
L > D , P , P , D – G and others, and the theoreticalparameters (mass and width) of dibaryon resonancesfound from the fit of N N scattering in these channelsturned out to be very close to their experimental values[17,18,19].However, the description of just S -wave N N scatter-ing (both elastic and inelastic) at low and intermediateenergies should be especially sensitive to the assump-tions made in the dibaryon-induced model. In fact, inthe case of S -wave N N scattering, absence of a cen-trifugal barrier allows the closest rapprochement of twonucleons to each other, so the short-range interactionin the S -wave channels provides the strongest impactto the phase shifts. This is especially true for inelasticscattering which, in turn, should be governed by thesame mechanism as elastic scattering. Thus, it seemsevident that the quark degrees of freedom should play amajor role in the interaction mechanism. In the presentpaper, we study in detail just S -wave N N scatteringwithin the dibaryon-induced approach.In fact, our model is based mainly on an assumptionabout the formation of dibaryon resonances which canbe coupled to the various
N N channels. Therefore theexistence (or nonexistence) of such states plays a deci-sive role in whole our approach. So, it is worth to brieflydiscuss the current experimental status of the dibaryonresonances before we can proceed further with the the-oretical description of
N N scattering. In recent years, many so-called exotic states havebeen observed in the charmed and beauty meson andbaryon sectors. Common to these X , Y , Z and pen-taquark states is that they appear as narrow resonancesnear particle thresholds constituting weakly bound sys-tems of presumably molecular character [20]. A similarsituation is also present in the dibaryonic sector, whichcan be investigated by elastic and inelastic N N scat-tering.Following the recent observation of the narrow dibaryonresonance d ∗ (2380) with I ( J P ) = 0(3 + ) in two-pionproduction [21,22] and then in N N elastic scattering[23,24], new measurements and investigations revealedand/or reconfirmed evidences for a number of statesnear the
N ∆ threshold. Among these the most pro-nounced resonance is the one with I ( J P ) = 1(2 + ), mass m ≈ Γ ≈
126 MeV. Since its massis close to the nominal
N ∆ threshold of 2.17 GeV andits width is compatible with that of the ∆ itself, its na-ture has been heavily debated in the past, though itspole has been clearly identified in a combined analysis of pp , πd scattering and pp ↔ dπ + reaction [25]. For a re-cent review about this issue see, e.g. , Ref. [16]. Very re-cently also evidence for a resonance with mirrored quan-tum numbers, i.e. , I ( J P ) = 2(1 + ) has been found hav-ing a mass m = 2140(10) MeV and width Γ = 110(10)MeV [26,27]. Remarkably, both these states have beenpredicted already in 1964 by Dyson and Xuong [28]based on SU (6) considerations and more recently cal-culated in a Faddeev treatment by Gal and Garcilazo[29] providing agreement with the experimental findingsboth in mass and in width.Whereas these two states represent weakly boundstates relative to the nominal N ∆ threshold and are ofpresumably molecular character with N and ∆ in rela-tive S wave, new evidence has been presented recentlyalso for two states, where the two baryons are in rela-tive P wave: a state with I ( J P ) = 1(0 − ), m = 2201(5)MeV and Γ = 91(12) MeV as well as a state with I ( J P ) = 1(2 − ), m = 2197(8) MeV and Γ = 130(21)MeV [30]. The values for the latter state agree withthose obtained before in SAID partial-wave analyses[25]. The masses of these p -wave resonances are slightlyabove the nominal N ∆ threshold, which is understoodas being due to the additional orbital motion [30]. Thereis suggestive evidence for the existence of still furtherstates like a P -wave I ( J P ) = 1(3 − ) state, for which,however, the experimental situation is not yet as clear[16]. It is also worth emphasising that the three res-onances 1(2 + ), 1(2 − ) and 1(3 − ) have been shown togive a sizeable contribution to the pp → dπ + cross sec-tions and polarisation observables [31] (the resonance − ) is not allowed in this reaction by the parity andmomentum conservation).In the description of N N scattering within the dibaryon-induced model, we considered first the isovector partialchannels D , P , F and others, where the dibaryonresonances near the N ∆ threshold (respectively, 1(2 + ),1(2 − ), 1(3 − ), etc.) can be formed [17,18]. We have shownthat these resonances determine almost completely N N scattering in the respective partial channels at ener-gies from zero to about 600–800 MeV (lab.). Then inthe work [19]
N N scattering in the isoscalar D – G channels has been shown to be governed by the 0(3 + )dibaryon d ∗ (2380) which is located 80 MeV below the ∆∆ threshold (and thus can be treated not as a molecular-like but as a deeply bound ∆∆ state). By analogy, forthe S -wave partial channels, with which we are con-cerned here, the respective dibaryons could be locatednear the N N ∗ (1440) threshold, since the Roper res-onance N ∗ (1440) has the same quantum numbers asthe nucleon, and an S -wave N N ∗ resonance can eas-ily transform into an S -wave N N state. In compari-son to
N ∆ dibaryons which can couple to the isovector
N N channels only, both isospin assignments I = 0 and I = 1 are allowed for the N N ∗ (1440) resonances. So,these resonances, if they exist, can couple to the S – D (the deuteron) and S (the singlet deuteron) N N channels, respectively.Fortunately, a strong indication of existence of thesetwo dibaryon resonances near the threshold of the Roperresonance excitation have been found in the recent WASAexperiments on single- and double-pion production inisoscalar and isovector
N N collisions [32,33]. It will bedemonstrated below that the scenario of dibaryonic res-onances near the
N ∆ threshold is not unique, but isrepeated at the
N N ∗ (1440) threshold. And just theseresonances determine the S -wave N N scattering at lowand intermediate energies.The paper is organised as follows. In Sec. 2 we brieflyoutline the theoretical formalism of the dibaryon-inducedmodel for the
N N interaction [17,18,19] with somemodifications necessary to apply it to S -wave scatter-ing. Then in Sec. 3 we derive the dibaryon parametersfrom the fit to the phase shifts and inelasticities in the S – D and S channels and compare them to theexperimental data which are discussed in Sec. 4. Weconclude in Sec. 5. S -wave N N interaction
As is well known, the effective range approximation forthe low-energy
N N -scattering leads to the S -matrix poles near zero energy for the triplet S – D and sin-glet S channels. According to the Wigner’s idea, onecan treat these S -matrix poles as a result of an s -channel exchange by the deuteron or singlet deuteron(see Fig. 1). NNNN d Fig. 1
Diagram illustrating the low-energy NN interac-tion due to s -channel exchange by the deuteron or singletdeuteron. Then the question arises: whether such s -channelmechanism can provide description not only of low-energy but also intermediate-energy N N scattering?The answer is: surely, if instead of the deuteron polein Fig. 1 one will imply a corresponding dibaryon poleat intermediate energy. The first attempt to treat the S -wave N N scattering at intermediate energies by the s -channel exchange by the dibaryon pole was under-taken at the beginning of 2000s within the framework ofthe dibaryon concept for the nuclear force [13]. The pe-ripheral meson-exchange N N interaction was describedvia the so-called external space (or channel) where onedeals with nucleonic and mesonic degrees of freedom.The main short-range
N N attraction is caused by acoupling between the external and internal channels,where the latter is treated by means of quark-gluon (orstring) degrees of freedom. The rigorous mathematicalformalism to describe such quantum systems combiningtwo Hilbert spaces (or channels) with completely differ-ent degrees of freedom was developed in the numerouspapers of the Leningrad group [34]. We refer the readerto Refs. [13,14] where this approach was used to developa dibaryon-induced
N N -interaction model (referred toas a “dressed bag model”) based on a microscopic six-quark shell model in a combination with the well-known P mechanism of pion production.A deeper insight into the structure of the six-quarksystem in the internal channel may be gained fromthe quark-cluster picture [35,36], where two separatedquark clusters, a tetraquark 4 q and a diquark 2 q areconnected by a color string which can vibrate and ro-tate. In the quark shell-model language, such a statecorresponds to the six-quark configuration | s p [42] x ; L =0 , ST i with two quarks in the p -shell [12,13]. Beingtransformed into the 4 q –2 q two-cluster state, it corre-sponds to the 2 ~ ω excitation of the color string con-necting two clusters. So, coupling between the exter-nal and internal channels corresponds to passing from a bag-like 2 ~ ω -excited six-quark state to N N loops inthe external channel (see Fig. 2). Of course, the inter-mediate dibaryon can decay also into inelastic channels(other than
N N ). In our model, such decays are effec-tively taken into account through the width Γ D (seeEq. (6) below). So that, in Fig. 2, dibaryon decays into N N ∗ channel are implicitly included in the dibaryonpropagator as well. + N N NNNN N NN + NNNN NNND D D D D NN + ... D Fig. 2
Graphical representation of the NN scattering am-plitude driven by the intermediate dibaryon D formation inthe NN system. In Refs. [17,18,19], the dibaryon-induced model hasbeen generalised further to effectively include the in-elastic processes. Below, for the readers’ convenience,we briefly outline the basic formalism of the dibaryon-induced model with a special emphasis on S -wave N N scattering. As has been mentioned above, the total Hilbertspace of the model includes the external and internalchannels. The external channel corresponds to the rel-ative motion of two nucleons, while the internal chan-nel corresponds to the formation of the six-quark (ordibaryon) state. In the simplest case, the internal spaceis one-dimensional, and a single internal state | α i is as-sociated with the “bare dibaryon” having the complexenergy E D . So, the total Hamiltonian has the matrixform: H = (cid:18) h NN λ | Φ ih α | λ | α ih Φ | E D | α ih α | (cid:19) , (1)where the transition form factor | Φ i is defined in theexternal space and represents a projection of the total6 q wavefunction onto the N N channel. In particular, incase of the coupled spin-triplet
N N partial waves S – D , | Φ i is a two-component column (see Ref. [19]).The external Hamiltonian h NN includes the periph-eral interaction of two nucleons which is given by theone-pion exchange potential V OPEP . Here we use thesame form and the same parameters of V OPEP as inRef. [19] . For S -wave N N scattering, one should alsotake into account the six-quark symmetry aspects lead-ing to an additional repulsive term V orth in the N N potential [18]. Thus, the external Hamiltonian is repre-sented as a sum of three terms: h NN = h NN + V OPEP + V orth , (2) For the coupled spin-triplet channels S – D , we use a bitlower cutoff parameter Λ πNN = 0 .
62 GeV (instead of 0 . D phase shift. where h NN is the two-nucleon kinetic energy operator(which may include the Coulomb interaction for the pp case) and V orth has a separable form V orth = λ | φ ih φ | . (3)The symmetry-induced operator V orth was introducedfor the first time in Ref. [37]. It corresponds to thefull or partial exclusion of the space symmetric six-quark component | s [6] i from the total N N wavefunc-tion and is needed to fulfil the orthogonality conditionbetween the small | s [6] i and the dominating mixed-symmetry | s p [42] i components in the N N system. Ithas been shown [38] that the operator V orth plays therole of the traditional N N repulsive core. In fact, this s -eliminating potential provides a stationary node inthe N N wavefunctions at different energies, and theposition of the node corresponds to the radius of therepulsive core [39].To satisfy the orthogonality condition strictly, onehas to take the limit λ → ∞ in V orth . However, sincethe S -wave N N channels have a strong coupling to the S -wave N N ∗ (1440) channels near the Roper resonanceexcitation threshold, the 2 ~ ω excitation in N N rela-tive motion can pass into the 2 ~ ω inner monopole ex-citation of the Roper resonance N ∗ (1440) . Thus, forsuch a strong coupling, there should not be a strict or-thogonality condition for the symmetric configuration | s [6] i at energies near the resonance, and the value of λ should be finite. There is another good reasoningto this point. The S -wave dibaryon state located nearthe N N ∗ (1440) threshold can decay into both N N and
N N ∗ channels. While the relative-motion wavefunctionin the N N channel has a stationary node at r c = 0 . N N scattering the
N N ∗ wave function has not got a node because the 2 ~ ω exci-tation in the initial six-quark wave function passes into2 ~ ω inner excitation in the Roper state itself. Hence,for the channel N N ∗ , the projection operator V orth isnot needed. The especially strong mixing of the N N and
N N ∗ channels happens just in the near-thresholdarea where the effect of V orth almost disappears.So that, we use here the orthogonalising term V orth with the finite values of λ . It provides a node in the N N relative motion wavefunctions at small energies,but at intermediate energies,
N N scattering states havesome admixture of the nodeless state | φ i . So, in thisapproach, the finite properly chosen value of λ provides In the quark shell-model language, the N ∗ (1440) structurecorresponds to the mixture of the 3 q configurations 0 s − (1 p ) and (0 s ) − s , both carrying 2 ~ ω excitation. In particular, a resonance state in the NN channel may havea noticeable overlap with the state | φ i . The detailed studyof this formalism will be published elsewhere. an effective account of the strong coupling between the N N and
N N ∗ (1440) channels.After excluding the internal channel, one gets theeffective Hamiltonian in which the main attraction isgiven by the energy-dependent pole-like interaction: H eff ( E ) = h NN + λ E − E D | Φ ih Φ | . (4)By using the separable form for the energy-dependentpart of interaction, one can find explicitly an equa-tion for the poles of the total S -matrix (see details inRefs. [18,19]): Z − E D − J ( Z ) = 0 , (5)where the function J ( Z ) is determined from the matrixelement of the external Hamiltonian resolvent g NN ( Z ) =[ Z − h NN ] − , i.e. , J ( Z ) = λ h Φ | g NN ( Z ) | Φ i .Finally, for the effective account of inelastic pro-cesses, we introduce the imaginary part of the inter-nal pole position E D = E − iΓ D /
2, which is energy-dependent and describes the possible decays of the “bare”dibaryon into all inelastic channels ( i.e. , except for the
N N one). For a single decay channel, the width Γ D canbe represented as follows: Γ D ( √ s ) = , √ s ≤ E thr ; Γ F ( √ s ) F ( M ) , √ s > E thr , (6)where √ s is the total invariant energy of the decay-ing resonance, M is the bare dibaryon mass, E thr isthe threshold energy, and Γ defines the partial decaywidth at √ s = M . For the S -wave dibaryon resonancelocated near the N N ∗ (1440) threshold, the dominantdecay channel is D → N N ∗ (1440). Here we take intoaccount only the main decay mode of the Roper reso-nance N ∗ (1440) → πN , and thus the main decay chan-nel of the dibaryon is D → πN N . For such a case, theparametrization of the function F ( √ s ) in Eq. (6) hasbeen introduced in Ref. [18]: F ( √ s ) = 1 s Z √ s − m π m dM NN q l π +1 k L NN +1 ( q + Λ ) l π +1 ( k + Λ ) L NN +1 . (7)Here q = q ( s − m π − M NN ) − m π M NN (cid:14) √ s and k = 12 q M NN − m are the pion momentum in thetotal center-of-mass frame and the momentum of thenucleon in the center-of-mass frame of the final N N subsystem with the invariant mass M NN , respectively.In Eq. (7), the high momentum cutoff parameter Λ isused to prevent an unphysical growth of the width Γ inel at high energies. The possible values of the pion orbital angular momentum l π with respect to the N N subsys-tem and the orbital angular momentum of two nucleons L NN are restricted by the total angular momentum andparity conservation. Below, the particular values of l π , L NN and Λ are adjusted to get the best fit of inelastic-ity parameters in the partial N N channels in question. S – D and S partial-wavechannels In this section, we present the results of calculations forthe
N N scattering phase shifts and inelasticity param-eters in the lowest partial-wave channels, viz., S – D and S , within the dibaryon-induced model.For the model form factors entering Eqs. (1) and (3),we have employed the harmonic oscillator functions withthe orbital momentum L and the radial quantum num-ber n equal to the number of nodes in the N N relativemotion wavefunction. In particular, | φ i has the formof the S -wave state with n = 0 and an effective range r . The “dibaryon” form factor | Φ i has two componentscorresponding to S and D waves, i.e. , | Φ i = (cid:18) α | φ S i β | φ D i (cid:19) ,where α + β = 1. Here | φ S i has the same effectiverange r as | φ i , but n = 1, so, these two functions areorthogonal to each other. The D -wave part of | Φ i is anodeless function with the effective range r D . The po-tential parameters used for both spin-triplet and spin-singlet partial-wave channels are listed in Tab. 1. Here λ S ≡ αλ and λ D ≡ βλ for the spin-triplet channel. Forthe dibaryon width defined by Eqs. (6) and (7), we usedthe values l π = 0, L NN = 1 and Λ = 0 . c for the S channel and l π = 1, L NN = 2 and Λ = 1 . c for the coupled S – D channels. The concrete valuesof these parameters are important mainly for the fit ofinelasticities in the near-threshold region and have alittle impact on the overall fit quality. Table 1
Parameters of the dibaryon model potential for thelowest spin-triplet and spin-singlet NN partial-wave channels. λ r λ S λ D r D M Γ MeV fm MeV MeV fm MeV MeV SD
165 0.475 248.1 65.9 0.6 2275 80 S
165 0.48 274.2 - - 2300 40
The partial phase shifts and mixing angle for thecoupled channels S – D are shown in Fig. 3 in com-parison with the single-energy (SE) solution of the SAIDpartial-wave analysis (PWA) [40]. It is seen from Fig. 3that within the dibaryon model we can reproduce thePWA data on the S – D partial phase shifts and mix- ( d e g ) T lab (GeV)(a) S D ( d e g ) T lab (GeV)(b) 0.0 0.2 0.4 0.6 0.8 1.0 1.2-10-5051015 ( d e g ) T lab (GeV)(c) Fig. 3 (Color online) Partial phase shifts (a), (b) and mixing angle (c) for the coupled NN channels S – D found withinthe dibaryon model (solid curves) in comparison with the single-energy SAID PWA [40] (filled circles) and results for the pureOPEP (dash-dotted curves). ( d e g ) T lab (GeV) S Fig. 4 (Color online) Partial phase shifts for the NN channel S found within the dibaryon model (solid curves) in com-parison with the single-energy SAID PWA [40] (filled circles)and results for the pure OPEP (dash-dotted curves). ing angle in a broad energy range from zero up to about1.2 GeV.The partial phase shifts for the spin-singlet channel S calculated with the model parameters which arerather close to those used for the spin-triplet case (seeTab. 1) are shown in Fig. 4 in comparison with theSAID SE data. Again the dibaryon model allows forthe very good description of the partial phase shifts atenergies from zero up to 1.2 GeV.The comparison of inelasticities for the S -wave chan-nels with the SAID single-energy data is presented inFig. 5 (a) and (b). Here we see reasonable agreementfor the S -wave inelasticity parameters with the PWAdata up to the energies corresponding to the resonanceposition ( T lab ≃ . N N scattering in S waves in a broad energy range from zero up to about 1GeV.In Figs. 3–5 the contribution of the pure OPEP isshown by dash-dotted curves. It is clearly seen that just the dibaryon excitation mechanism allows for a reason-able description of both partial phase shifts and inelas-ticities for S -wave N N scattering. The coupling with adibaryon in the D -wave component of the spin-tripletchannel S – D is weaker, so the dibaryon mechanismmakes some important contribution here only above theinelastic threshold (see Fig. 3 (b)). The situation hereis very similar to that for the D – G partial-wavechannels studied in Ref. [19].It is extremely interesting that the S -matrices forthe model N N potentials in both singlet and tripletpartial channels have two poles. For the S – D case,the first pole corresponds to the bound state, i.e. , thedeuteron, which is reproduced rather accurately. Thesecond pole here corresponds to the dibaryon resonancewith the parameters: M th ( SD ) = 2310 MeV , Γ th ( SD ) = 157 MeV . (8)For the S channel, the first pole is the well-knownsinglet deuteron state, while the position of the secondone is: M th ( S ) = 2330 MeV , Γ th ( S ) = 51 MeV . (9)Both these resonance positions are rather close tothe N N ∗ (1440) threshold . As will be shown below,the resonance parameters given in Eqs. (8) and (9) alsoturn out to be very close to the values derived from therecent single- and double-pion production experiments(see Sec. 4). However, the inaccuracy in description ofinelasticity parameters at energies above the resonanceposition in the considered N N partial-wave channels aswell as a too narrow width of the dibaryon resonance in The difference between the resonance parameters found herefor the S channel from the preliminary ones obtained inRef. [18] is due to the use of the finite λ in the orthogonal-ising potential V orth . S ( d e g ) T lab (GeV)(a) 0.0 0.2 0.4 0.6 0.8 1.001020 S ( d e g ) T lab (GeV)(b) Fig. 5 (Color online) Inelasticity parameters for the NN channel S (a) and S (b) found within the dibaryon model(solid curves) in comparison with the single-energy SAID PWA [40] (filled circles) and results for the pure OPEP (dash-dottedcurves). the S channel show that a more detailed treatmentof inelastic processes is required within the dibaryonmodel.As s -channel resonances, the two predicted dibaryonstates have to display a counter clockwise looping in theArgand diagrams of amplitudes in S and S partialwaves. For these partial waves, two S -wave trajectories are shown by the solid lines in Fig. 6 in comparison withthe different SAID PWA solutions [40]. In fact, we ob-serve the counter clockwise loopings for the amplitudesfound within the dibaryon model indicating the reso-nance presence in both cases.Since these resonances are highly inelastic, the reso-nance loops are rather tiny. The theoretical predictionsshould be compared with three PWA solutions of theSAID group, viz., the single-energy as well as the globalsolutions AD14 and SM16 [40]. The scatter within thesingle-energy data as well as the differences among thevarious SAID solutions may serve as an indication forinherent ambiguities in the partial-wave analysis, es-pecially for the S channel. In fact, the differencesbetween two recent solutions SM16 and AD14 are ofthe same order as those differences between theoreticalloops and each of the above SAID solutions. Hence, theabsence of the loops in the current SAID solutions can-not argue against the suggested dibaryon resonances.We note that the situation here is much differentfrom that for the dibaryon resonance d*(2380). In caseof the latter, the partial waves D and G were in-volved, which both carry large orbital angular momen-tum and hence have a large impact on the analyz-ing power. Since this observable is the only one, whichsolely consists of interference terms, it is predestinatedto exhibit substantial effects even from tiny resonance Here the partial amplitude A is defined as A = ( S L − / i ,where S L is the S -matrix for the given orbital angular mo-mentum L . admixtures in partial waves. Unfortunately, we deal herewith S -wave resonances, which make no contribution tothe analyzing power due to the missing orbital angu-lar momentum. Hence this key observable for revealingloops and resonances is not working here. The only wayout of this dilemma is to look into reactions, where thesehighly inelastic resonances decay to, namely single- anddouble-pion production. We discuss these processes inthe next Section. N N induced single-and double-pion production and thenear-threshold dibaryon resonances
The Roper resonance N ∗ (1440) excitation appears usu-ally quite hidden in the observables and in most casescan be extracted from the data only by sophisticatedanalysis tools like partial-wave decomposition. By con-trast, it can be observed free of background in N N -induced isoscalar single-pion production, where the over-whelming isovector ∆ excitation is filtered out by isospinselection as demonstrated by recent WASA-at-COSYresults [32] for the N N → [ N N π ] I =0 reaction. Thoughthe primary aim of this experiment was the search for adecay d ∗ (2380) → [ N N π ] I =0 , it also covers the regionof the Roper excitation, which is discussed here.Since the ∆ excitation is filtered out by the isospincondition, there is only a single pronounced structureleft in the isoscalar nucleon-pion invariant mass spec-trum as seen in Fig. 6 of Ref. [32], which peaks at m ≈ Γ ≈
150 MeV. Thesevalues are compatible with the pole values for the Roperresonance deduced in diverse πN and γN studies [41].Our values for the Roper peak also are in good agree-ment with earlier findings from hadronic J/Ψ → ¯ NN π decay [42] and αN scattering [43,44]. -0.48 -0.45 -0.42 -0.39 -0.360.250.300.350.400.450.500.550.60 I m A Re A (a) S -0.50 -0.45 -0.40 -0.350.40.50.60.70.8 I m A Re A (b) S Fig. 6 (Color online) Argand diagrams for the NN channels S (a) and S (b) found within the dibaryon model (solidcurves) in comparison with different solutions of the SAID PWA [40]: single-energy (filled circles), SM16 (dashed curves) andAD14 (dash-dotted curves). The numbers near the single-energy points reflect the corresponding values of the lab. energy T lab in MeV. The energy excitation function of the measured
N N -induced isoscalar single-pion production cross sectionis displayed in Fig. 7. Near threshold the Roper res-onance is produced in S wave in relation to the othernucleon, whereas the pion from the Roper decay is emit-ted in relative p wave. Hence we expect for the energydependence of the total cross section in the isoscalar N N → [ N N π ] I =0 channel a threshold behavior likefor pion p waves — as is actually born out by the ex-plicit calculations for the t -channel Roper excitation inthe framework of the modified Valencia model [32,46].These calculations are displayed in Fig. 7 by the dashedline, which is arbitrarily adjusted in height to the data.The data [32,45] presented in Fig. 7 follow this expec-tation by exhibiting an increasing cross section withincreasing energy up to about √ s ≈ t -channel productionprocess. The observed behavior is rather in agreementwith a s -channel resonance process as expected for theformation of a dibaryonic state at the N N ∗ threshold.Due to the relative S wave between N and N ∗ as well asdue to the isoscalar nature of this system, it must havethe unique quantum numbers I ( J P ) = 0(1 + ). From afit of a simple Lorentzian to the data we obtain m =2315(10) MeV and Γ = 150(30) MeV. The large uncer-tainty on the latter results from the large uncertaintiesof the data at lower energies (for a fit with a Gaussian,which leads to a width of 170 MeV, see Ref. [32]). Fora more detailed treatment of the resonance structureone would need to use a momentum-dependent width,which takes into account the nearby pion productionthreshold and lowers the resonance cross section at thelow-energy side.A very similar situation is also observed in N N -induced two-pion production. The situation is particu- [ m b ] σ − WASA-at-COSY Dakhno et al. (I=0) σ s [MeV] Fig. 7 (Color online) The NN -induced isoscalar single-pionproduction cross section in dependence of the total c.m. en-ergy √ s . Shown are the recent results from WASA-at-COSY[32] (solid circles) together with earlier results [45] at lowerenergies. The dashed line shows the expected energy depen-dence based on t -channel Roper excitation [32,46], the solidline a Lorentzian fit to the data with m = 2315 MeV and Γ =150 MeV. larly clear in the pp → ppπ π reaction, the total crosssection of which is plotted in Fig. 8. Since ordinarysingle- ∆ excitation is excluded here due to the nec-essary production of two pions, the Roper excitationis the only resonance process at low energies. Hencewe would again expect a phase-space-like (dotted line)growth of the cross section, which is also born out bydetailed model calculations (dash-dotted line) [46,52].But the data follow this trend up to T p ≈ √ s ≈ t -channel ∆∆ process with double- p -wave emission(dashed line) starts. Isospin decomposition of the data in the various N N ππ channels tells us that the energydependence of the Roper excitation is experimentallygiven by the filled star symbols in Fig. 8 [33]. Againwe see a resonance-like energy dependence, which indi-cates a
N N ∗ (1440) molecular system also in this case,but now with quantum numbers I ( J P ) = 1(0 + ), m ≈ Γ ≈
150 MeV. We note that the fadingaway of the Roper excitation at energies beyond T p ≈ b ] µ [ σ − − CELSIUS/WASA WASA-at-COSY Shimizu et al. PROMICE/WASA Brunt at al. π π pp → pp s [MeV] σ N* σ ValenciaN* phase space σ ∆∆ Fig. 8 (Color online) Energy dependence of the total pp → ppπ π cross section. Shown are the data from CEL-SIUS/WASA [33] as well as WASA-at-COSY [47] (filledcircles), PROMICE/WASA [48] (filled squares), and ear-lier work [49,50] (open symbols). The dotted and dash-dotted lines show the expected energy dependence of simplephase-space and modelled Roper excitation [46], respectively.The dashed line shows the t -channel ∆∆ excitation [46,51],whereas the filled stars display the result of the isospin de-composition for N ∗ excitations [33]. Here, the first structureis due to the Roper N ∗ (1440) excitation. The rerise at higherenergies signals higher-lying N ∗ excitations. We have shown within the dibaryon-induced model for
N N scattering that the
N N interaction in the basicspin-singlet and spin-triplet S -wave partial channels atenergies T lab from zero up to about 1 GeV is gov-erned by the formation of the I ( J π ) = 0(1 + ) and 1(0 + )dibaryon resonances near the N N ∗ (1440) threshold. Thiswork continues a series of the previously published pa-pers [17,18,19] where the N N interaction in higherpartial waves was shown to be dominated by the in-termediate dibaryon excitation (supplemented by theperipheral one-pion exchange) in the respective partialchannels near the
N ∆ or ∆∆ thresholds.From the energy dependence of N N -induced isoscalarsingle-pion and isovector double-pion production we see also that both isospin-spin combinations in the
N N ∗ (1440)system lead obviously to dibaryonic threshold statesat the Roper excitation threshold — analogous to thesituation at the ∆ threshold. However, compared tothe situation there the Roper excitation cross sectionsdiscussed here are small. Since these structures decaymainly into inelastic channels, their partial decay widthinto the elastic ( N N ) channel should be only a smallfraction of the total width, similarly to the respectivebranching ratio for the
N ∆ near-threshold states [16].Despite this fact, our results show that the contribu-tions of these dibaryon states to the low- and intermediate-energy
N N elastic scattering are dominating.On the other hand, at energies below the inelasticthresholds
N N π and
N N ππ , decay of the dibaryonsinto these channels is forbidden. However, a strong cou-pling between the
N N and the closed (virtual) channelslike
N ∆ , N N ∗ (1440), N N π and
N N ππ is still possi-ble. So, at low energies ( T lab .
350 MeV) the couplingof the dibaryon to these closed channels appears to bestrong and thus the whole picture of the
N N interac-tion at these energies is dominated just by this cou-pling. This explains how the intermediate dibaryon for-mation near the nucleonic resonance threshold can bethe leading mechanism for the
N N interaction at lowenergies. When the collision energy is rising and the in-elastic channels open, the same intermediate dibaryonsprovide single- and double-pion production. Thus, inthe dibaryon-induced approach to the
N N interaction,the elastic and inelastic
N N collision processes havea common origin and can be described via a commonmechanism. These results may provide a novel insightinto the nature of the
N N interaction at low and in-termediate energies and should be confirmed by furtherexperimental and theoretical research.
Acknowledgments.
We are indebted to L. Alvarez-Ruso for using his code and to I.T. Obukhovsky forfruitful discussions of the microscopic quark model. Thework has been supported by DFG (grants CL 214/3-2 and 3-3) and the Russian Foundation for Basic Re-search, grants Nos. 19-02-00011 and 19-02-00014. M.N.P.also appreciates support from the Foundation for theAdvancement of Theoretical Physics and Mathematics“BASIS”.
References
1. H. Yukawa, Proc. Phys. Math. Soc. Jpn. , 48 (1935).2. V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen, and J.J.de Swart, Phys. Rev. C , 2950 (1994).3. R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev.C , 38 (1995).4. R. Machleidt, Phys. Rev. C , 024001 (2001).5. E. Eppelbaum, J. Gegelia, Eur. Phys. J. A , 341 (2009).06. R. Machleidt, D.R. Entem, Phys. Rept. , 1 (2011).7. M. Baldo, O. Elgaroey, L. Engvik, M. Hjorth-Jensen, andH.-J. Schulze, Phys. Rev. C , 1921 (1998).8. A. Faessler, F. Fernandes, G. L¨ubeck, and K. Shimizu,Phys. Lett. B , 201 (1982); Nucl. Phys. A , 555(1983); K. Shimizu, Rep. Prog. Phys. , 1 (1989).9. Y. Yamauchi, A. Buchmann, A. Faessler, A. Arima, Nucl.Phys. A , 495 (1991).10. Fl. Stancu, S. Pepin, and L.Ya. Glozman, Phys. Rev. C , 2779 (1997); Erratum ibid. , 1219 (1999).11. M. Beyer and H.J. Weber, Phys. Lett. B , 383 (1984).12. V.I. Kukulin, Proceedings of the XXXIII Winter SchoolPIYaF (Gatchina, 1998), Saint-Petersburg, 1999, p. 207.13. V.I. Kukulin, I.T. Obukhovsky, V.N. Pomerantsev, andA. Faessler, J. Phys. G ,1851 (2001).14. V.I. Kukulin, I.T. Obukhovsky, V.N. Pomerantsev, andA. Faessler, Int. J. Mod. Phys. E , 1 (2002).15. V.I. Kukulin et al. , Ann. Phys. , 1173 (2010).16. H. Clement, Prog. Part. Nucl. Phys. , 195 (2017).17. V.I. Kukulin, V.N. Pomerantsev, O.A. Rubtsova, Few-Body Syst. , 48 (2019).18. V.I. Kukulin, V.N. Pomerantsev, O.A. Rubtsova, M.N.Platonova, Phys. At. Nucl. , 934 (2019).19. V.I. Kukulin et al. , Phys. Lett. B , 135146 (2020).20. R. Aaij et al. , Phys. Rev. Lett. , 222001 (2019).21. M. Bashkanov et al. , Phys. Rev. Lett. , 052301 (2009).22. P. Adlarson et al. Phys. Rev. Lett. , 242302 (2011).23. P. Adlarson et al. , Phys. Rev. Lett. , 202301 (2014).24. P. Adlarson et al. , Phys. Rev. C , 035204 (2014).25. Ch.H. Oh, R.A. Arndt, I.I. Strakovsky, and R.L. Work-man, Phys. Rev. C , 635 (1997) and references therein.26. P. Adlarson et al. , Phys. Rev. Lett. , 052001 (2018).27. P. Adlarson et al. , Phys. Rev. C , 025201 (2019).28. F.J. Dyson and N.-H. Xuong, Phys. Rev. Lett. , 815(1964); Erratum ibid. , 339 (1965).29. A. Gal and H. Garcilazo, Nucl. Phys. A , 73 (2014).30. V. Komarov et al. , Phys. Rev. C , 065206 (2016).31. M.N. Platonova and V.I. Kukulin, Phys. Rev. D ,054039 (2016).32. P. Adlarson et al. , Phys. Lett. B , 599 (2017).33. T. Skorodko et al. , Phys. Lett. B , 30 (2009).34. Yu.A. Kuperin, K.A. Makarov, S.P. Merkuriev, A.K. Mo-tovilov, B.S. Pavlov, J. Math. Phys. , 1681 (1990); Sov.J. Theor. Math. Phys. , 431 (1988); ibid. , 242 (1988);Yu.A. Kuperin, K.A. Makarov, S.P. Merkuriev, A.K. Mo-tovilov, Sov. J. Nucl. Phys. , 358 (1988).35. P.J. Mulders, A.T.M. Aerts, and J.J. de Swart, Phys.Rev. D , 2653 (1980).36. L.A. Kondratyuk, B.V. Martemyanov, M.G. Shchepkin,Sov. J. Nucl. Phys. , 776 (1987).37. V.M. Krasnopolsky, V.I. Kukulin, Sov. J. Nucl. Phys. ,470 (1975).38. V.I. Kukulin et al. , J. Phys. G , 330(1978).39. V.I. Kukulin, M.N. Platonova, Phys. At. Nucl. , 1465(2013).40. All SAID PWA solutions can be accessed via the officialSAID website: http://gwdac.phys.gwu.edu41. M. Tanabashi et al. (Particle Data Group), Phys. Rev. D , 030001 (2018).42. M. Ablikim et al. (BES Collaboration), Phys. Rev. Lett. , 062001 (2006).43. H.P. Morsch et al. , Phys. Rev. Lett. , 1336 (1992).44. H.P. Morsch and P. Zupranski, Phys. Rev. C , 024002(1999).45. L.G. Dakhno et al. , Phys. Lett. B , 409 (1982). 46. L. Alvarez-Ruso, E. Oset, E. Hernandez, Nucl. Phys. A , 519 (1998) and priv. comm.47. P. Adlarson et al. , Phys. Lett. B , 256 (2012).48. J. Johanson et al. , Nucl. Phys. A , 75 (2002).49. F. Shimizu et al. , Nucl. Phys. A , 571 (1982).50. C.D. Brunt et al. , Phys. Rev. , 1856 (1969).51. T. Skorodko et al. , Phys. Lett. B , 115 (2011).52. X. Cao, B.-S. Zou, H.-S. Xu, Phys. Rev. C , 065201(2010).53. T. Skorodko et al. , Eur. Phys. J. A35