Sequential single pion production explaning the dibaryon "d^*(2380)" peak
SSequential single pion production explaning the dibaryon “ d ∗ (2380) ” peak R. Molina, ∗ Natsumi Ikeno,
1, 2, † and Eulogio Oset ‡ Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain Department of Agricultural, Life and Environmental Sciences, Tottori University, Tottori 680-8551, Japan
We study the two step sequential one pion production mechanism, np ( I = 0) → π − pp , followedby the fusion reaction pp → π + d , in order to describe the np → π + π − d reaction with π + π − in I = 0, where a narrow peak, so far identified with a “ d (2380)” dibaryon, has been observed. Wefind that the second step pp → π + d is driven by a triangle singularity that determines the positionof the peak of the reaction and the large strength of the cross section. The combined cross sectionof these two mechanisms produce a narrow peak with the position, width and strength compatiblewith the experimental observation within the approximations done. This novel interpretation of thepeak without invoking a dibaryon explains why the peak is not observed in other reactions where ithas been searched for. PACS numbers:
The np → π π d reaction exhibits a sharp peak around2370 MeV with a narrow width of about 70 MeV, whichis also seen in the pp → π + π − d reaction with approxi-mately double strength [1–3]. In the absence of a conven-tional reaction mechanism that can explain these peaks,they have been interpreted as a signal of a dibaryon thathas been named d ∗ (2380). On the base of this hypothesisseveral other features observed in π production experi-ments and N N phase shifts have been interpreted (see[4] for a recent review). Actually, the narrow peak in np → ππd affects the inelasticity of the N N phase shiftsand should have repercussion in
N N phase shifts as em-phasized in [5, 6]. Several mechanisms of two pion pro-duction leading to ππd have been studied in [1–3] basedupon the model of [7] for
N N → N N ππ , which containdouble ∆ production, with subsequent ∆ → πN decayor N ∗ (1440) production with decay of N ∗ to N ππ , or N ∗ → π ∆( N π ). In all these cases the resulting np par-ticles are fused into the deuteron. The results of thesecalculations give rise to cross sections with small strengthcompared to the peak of the np → π π d reaction andno peak at the energy of the observed one. Such conclu-sions were already drawn in an early paper [8] and wehave explicitly recalculated the cross sections from thesemechanisms reconfirming all these earlier findings. Inter-estingly, in the same work [8] a peak with poor statistics,already visible for the np → π + π − d reaction was ex-plained from a different mechanism, two step sequential π production, np → ppπ − followed by pp → π + d . Thecross section for np → π + π − d was evaluated factorizingcross sections for the two latter reactions in an “on-shell”approach that called for further checks concerning its ac-curacy. Such mechanism has no further been invokedconcerning the new improved data on the np → π + π − d , ∗ Electronic address:
[email protected] † Electronic address: [email protected] ‡ Electronic address:
[email protected] dπ + N ′ Nppπ − np ∆ FIG. 1: Two step mechanism for np → π + π − d suggested in[8] with explicit ∆ excitation in the pp → π + d last step asfound in [9–11]. The mechanism with the nn intermediatestate is considered in addition. np → π π d reactions [1–3].On the other hand, the time reversal reaction of pp → π + d , π + absorption in the deuteron, π + d → pp , wasthe subject of study in the past [9–11] and it was shownto have a neat peak corresponding to the ∆ excitation.Combining the work of [9–11] with the idea of [8] on the np → π + π − d reaction, the mechanism for np → π + π − d can be expressed diagrammatically as in Fig. 1.After many years, more refined data and new theo-retical developments make most opportune to revise thisissue along the same idea. We can quote:1) The data on np → π + π − d and np → π π d havenowadays excellent precision [1–4].2) The np → π π d reaction has π π in isospin I = 0,and hence the inital np state must also be in I = 0.The work of [3] splits the np → π + π − d reactioninto I = 0 and I = 1, and, as expected, the samepeak visible in the np → π π d reaction is seen inthe np (I = 0) → π + π − d reaction with about dou-ble strength. This means that in the np → π − pp reaction, the first step of the sequential single pionproduction mechanism, the inital np state is alsoin I = 0. Only very recently the first step in Fig.1, np → π − pp with np in I = 0 has been singledout with relatively good precission [12] (see revisionabout normalization in [12, 13]) a r X i v : . [ nu c l - t h ] M a r
3) New developments about triangle singularities [14]allow us to identify the large strength of the pp → π + d reaction with the presence of a triangle sin-gularity in the triangle diagram shown in the lastpart of Fig. 1. This corresponds to having si-multaneously the ∆ and the two nucleons on shelland collinear. The simplification of the formal-ism on the triangle singularities done in [15] al-lows us to see immediately where the peak of the pp → π + d cross section should appear, using Eq.(18) of [15] with the d mass slightly unbound tofind a solution of that equation. One predicts thata peak of the cross section should appear around M inv ( pp ) ∼ pp system produces ∆ N back to back in the pp rest frame; the ∆ decays intoa π + in the direction of the ∆ and N (cid:48) in opposite di-rection, which is the direction of N . The N (cid:48) movesfaster than N (encoded in Eq. (18) of [15]) andcatches up with N to fuse into the deuteron. Thefusion of the two nucleons into the deuteron comesout naturally when the mechanism discussed has atriangle singularity, giving rise to a neat peak anda cross section rather large compared with typicalfusion reactions [18]. We have done a recalculationof the pp → π + d reaction from this new perspective[19], but the details are unnecessary in the deriva-tion done here for the np → π + π − d cross sectionwhich, as in [8], relies on experimental cross sec-tions, using the new np (I = 0) → π − pp cross sec-tion [12, 13] and the data for pp → π + d [16]. Wealso improve on the on shell approach used in [8].It is also worth mentioning that while the mech-anism for pp → π + d in [9–11] was not identifiedas a triangle singularity, it was shown in [10] thatthe cross section was blowing up when the ∆ widthwas set to zero, a characteristic of the triangle sin-gularity. In Ref. [19] it is shown that the dominantterm in pp → π + d is the partial wave D ( S +1 L J ),in agreement with the experimental observation in[20], and from there one traces back J P = 1 + , + for the dπ + π − system, with some preference for3 + , and D for the initial np system, the preferredquantum numbers associated to the d ∗ (2380) peak[4].The derivation of the np → π + π − d cross sectionthat we do follows the steps of the derivation of theoptical theorem [21]. We call t the amplitude forisoscalar np (I = 0) → π − pp , t (cid:48) for pp → π + d and t (cid:48)(cid:48) for np (I = 0) → π + π − d . The differential cross section for the isoscalar np (I = 0) → π − pp reaction is given by dσ Inp → π − pp dM inv ( p p (cid:48) ) = 14 ps (2 M N ) π p π ˜ p | ¯ t |
12 (1)where σ I stands for the isoscalar cross section, √ s isthe center-of-mass (CM) energy of the inital np state, M inv ( p p (cid:48) ) the invariant mass of the final two protonsin this reaction, p the CM momentum of the inital n or p particles, p π the pion momentum in the np rest frameand ˜ p the momentum of the final protons in the pp restframe. We use the (2 M N ) factor of fermion field nor-malization for the nucleons following the formalism ofMandl and Shaw [22]. The magnitude | ¯ t | stands for theangle averaged | t | and the factor takes into accountthe identity of the two final protons.Similarly, the cross section for pp → π + d in the secondpart of the diagram of Fig. 1 is given by σ pp → π + d = 116 πM ( p p (cid:48) ) p (cid:48) π ˜ p | ¯ t (cid:48) | (2 M N ) (2 M d ) (2)where p (cid:48) π is the π + momentum in the pp rest frame and | ¯ t (cid:48) | stands for the angle averaged | t (cid:48) | . We choose tonormalize the deuteron field as the nucleons and add thefactor 2 M d (it disappears from the final formulas). Onthe other hand the amplitude for the np → π − π + d pro-cess in Fig. 1 is given by − it (cid:48)(cid:48) = 12 (cid:90) d p (2 π ) (2 M N ) E N ( p )2 E N ( p (cid:48) ) ip − E N ( p ) + i(cid:15) × i √ s − p − ω π − E N ( p (cid:48) ) + i(cid:15) ( − i ) t ( − i ) t (cid:48) (3)The factor is to account for the intermediate propaga-tor of two identical particles. In the d p integrations t and t (cid:48) would be off shell. In Ref. [8] the pion and thetwo protons of the intermediate state were taken on shelland t and t (cid:48) were used with the on shell variables. The-oretical advances done after [8] allow us to go beyondthis approximation. Indeed, the chiral unitary approachof [23] for meson-meson interaction, or [24] for meson-baryon interaction, factorizes the vertices on-shell andperforms the loop integral of the two intermediate states.A different justification is given in [25] writing a disper-sion relation for the inverse of the hadron-hadron scatter-ing amplitude, and it also finds a justification in [26, 27]showing with chiral lagrangians that off shell parts of theamplitudes appearing in the approach get cancelled withcounterterms provided with the same theory. This meansthat in Eq. (3) we can take tt (cid:48) outside the dp integrationwith their on-shell values and evaluate the remaining ofthe integral of Eq. (3).Performing the p integration analytically withCauchy’s residues we get t (cid:48)(cid:48) = 12 (cid:90) d p (2 π ) (2 M N ) E N ( p )2 E N ( p (cid:48) ) × tt (cid:48) √ s − E N ( p ) − E N ( p (cid:48) ) − ω π + i(cid:15) (4)where (cid:126)p , (cid:126)p (cid:48) are the momenta of the intermediate pp particles in Fig. 1, and ω π the π − energy. The t, t (cid:48) ampli-tudes are Lorentz invariant and we choose to evaluate the (cid:82) d p E ( p ) integral in the pp rest frame, where | (cid:126)p (cid:48) | = | (cid:126)p | and √ s − ω π becomes the invariant mass of the two pro-tons. This integral is logarithmically divergent and re-quires regularization. The result depends smoothly on acut off p , max for | (cid:126)p | that we use to regularize the d p integration, and we shall take some values for p , max ina reasonable range. Yet, we anticipate that the on shellpart given by Eq. (5), below, gives the largest contri-bution to the t (cid:48)(cid:48) amplitude. Since ˜ p = 552 MeV/c for M inv ( p p (cid:48) ) = 2179 MeV, where the triangle singularitywould appear for t (cid:48) for a ∆ with zero width, or a pro-nounced peak when the width is considered, values of p , max around 700 −
800 MeV seem reasonable.The on-shell approximation used in [8] that allows oneto write the cross section for np → π + π − d in terms ofthe np (I = 0) → π − pp and pp → π + d ones is obtainedin the present formalism by taking the imaginary part ofthe two nucleon propagator1 M inv ( p p (cid:48) ) − E N ( p ) + i(cid:15) ≡P (cid:20) M inv ( p p (cid:48) ) − E N ( p ) (cid:21) − iπδ ( M inv ( p p (cid:48) ) − E N ( p ))(5)We have then t (cid:48)(cid:48) on = − i
12 ˜ p π (2 M N ) M inv ( p p (cid:48) ) ¯ tt (cid:48) (6)where we have factorized the angle averaged value of tt (cid:48) , ¯ tt (cid:48) . Using the analogous equation of Eq. (1) for dσ np → π + π − d /dM inv ( π + π − ), the on-shell approximationof Eq. (6) and Eq. (1) we can already write dσ np → π + π − d dM inv ( π + π − ) = (2 M N ) (2 M d ) p d ˜ p π
14 ˜ p π × M ( p p (cid:48) ) 1 p π ˜ p | ¯ t (cid:48) | dσ np → π − pp dM inv ( p p (cid:48) )(7)where p d is the deuteron momentum in the original np rest frame, | ¯ t (cid:48) | the angle averaged | t (cid:48) | , and ˜ p π the pionmomentum in the π + π − rest frame. In Eq. (7) we haveassumed that | ¯ tt (cid:48) | = | ¯ t | | ¯ t (cid:48) | . The amplitudes t, t (cid:48) inRefs. [12, 16] have some angular structure, but theseare smooth enough to make this assumption a sensibleapproximation.Next we use physical arguments to write the np → π + π − d cross section with an easy compact formula. Wenote that π π or π + π − in I = 0, as we discussed earlier,require an even value of their relative angular momen-tum l , and when l = 0 the π π , or the symmetrized( π + π − + π − π + ), behave as identical particles, which re-verts into a Bose enhancement when the two pions go together. Certainly if they are exactly together we shallalso have the phase space factor ˜ p π in dσ/dM inv ( π + π − )of Eq. (7) which makes null this distribution in the twopion threshold, but some enhancement for small invari-ant masses is expected. Our argumentation is supportedby the results of [1, 2] for π π (see Fig. 2 of [1] and Fig.4 of [2]) and also in [3] for charged pions, although thenature of I = 0 and I = 1 in this case distorts a bit themass distribution compared to the clean I = 0 π π case.We could take some M inv ( π + π − ) distribution as in-put, but to make the results as model independent aspossible we take the ¯ M inv ( π + π − ) ∼ m π + 60 MeV, notfar from threshold but we change it to see how the re-sults depend on ¯ M inv . The stability of the results thatwe find by changing the value of ¯ M inv ( ππ ) justifies thisapproximation a posteriori. Then, we can write dσ np → π + π − d dM inv( π + π − ) = σ np → π + π − d δ ( M inv ( π + π − ) − ¯ M ππ ) . (8)The approximation of Eq. (8) is sufficiently good andallows us to get a more transparent picture of what is thereason for the appearance of the peak in the np → π + π − d reaction. Note now that the energy of the two pions isobtained as E π = s + M ( ππ ) − M d √ s , (9)and since the two pions go relatively together, we take E π = E π /
2, which allows to relate M inv ( p p (cid:48) ) with √ s via M ( p p (cid:48) ) = ( P ( np ) − p π − ) = s + m π − √ sE π (10)and formally2 M inv ( p p (cid:48) ) dM inv ( p p (cid:48) )= − √ sdE π = − M inv ( ππ ) dM inv ( ππ ) . (11)Using this relationship we can integrate Eq. (8) with re-spect to M inv ( ππ ) and using Eqs. (2) and (7) we obtain, σ np → π + π − d = M inv ( p p (cid:48) )4 π σ np → π − pp σ pp → π + d M inv ( ππ ) ˜ p p π p (cid:48) π p d ˜ p π (12)One last detail is needed. We have considered the twostep np (I = 0) → π − pp followed by pp → π + d . Aproperly symmetrized t (cid:48)(cid:48) amplitude requires the addi-tion of np (I = 0) → π + nn followed by nn → π − d . Itis trivial to see considering isospin that the amplitudes np (I = 0) → π − pp and np (I = 0) → π + nn are identicalup to the phase of π + ( − pp → π + d and nn → π − d for the sameconfiguration of the particles. Hence, the product of theamplitudes is the same. In the case that the π + and π − go exactly together, the two amplitudes will be identi-cal and add coherently, but we saw that the phase spacefactor ˜ p π of Eq. (12) kills this contributions. When con-sidering the integration over the five degrees of freedomof the three body phase space the terms are expected tosum mostly incoherently and we must multiply by 2 Eq.(12). Similar arguments can be done with respect to thespin sums and averages. The study of the pp → π + d reaction in [19] indicates that there is a certain angu-lar dependence on the different spin transitions and weshould expect an incoherent sum over spins. Then by in-cluding also in | ¯ t | the average over initial spins and sumover final spins we would be considering in our formulathe average over spins of the initial np and the sum overspins of the deuteron, plus the intermediate sum over the pp and nn spins.Eq. (12) still relies on the on-shell approximation ofEq. (6). To take into account the off shell effects dis-cussed above we replace (cid:18) ˜ p πM inv ( p p (cid:48) ) (cid:19) → | G ( M inv ) | (13)with G the loop function with two protons G = (cid:90) d p (2 π ) E N ( p ) E N ( p ) 1 M Inv ( p p (cid:48) ) − E N ( p ) + i(cid:15) The last step in the evaluation of σ np → π + π − d requiresto use the experimental data for np ( I = 0) → π − pp and pp → π + d . We get σ pp → π + d directly from experiment[16]. The σ np → π − pp in I = 0 requires some thoughts. In[28, 29] the isoscalar N N → πN N amplitude is obtainedvia isospin symmetry from σ np → ppπ − and σ pp → ppπ , andrelatively precise results are obtained in [12] from im-proved measurements of these cross sections. In the er-ratum of [12] and in [13] it is clarified that the actual σ pn ( I =0) → NNπ is one half of σ NN ( I =0) → NNπ of [12]. Thecross section that we need is σ pn ( I =0) → ppπ − . It is triv-ial to see using isospin symmetry that σ pn ( I =0) → ppπ − , σ pn ( I =0) → nnπ + and ( σ pn ( I =0) → pnπ + σ pn ( I =0) → npπ ) areall equal. Then we write the relationship of [12, 28, 29]as σ np ( I =0) → ppπ − = 13 σ np ( I =0) → NNπ (14)= 16 σ NN ( I =0) → NNπ = 16 3(2 σ np → ppπ − − σ pp → ppπ )and we take data for σ np ( I =0) → NNπ from Fig. 1 of [13].Statistical and some systematic errors are considered in[12, 13]. With the only purpose of making a realistic fitto the data we include also systematic errors from theuncertainty in Eq. (14) when using isospin symmetry.We assume a typical 5% violation of isospin in each ofthe last two terms of Eq. (14) and sum the errors inquadrature. The systematic errors obtained are of theorder of 0 . σ np ( I =0) → NNπ , which we also add inquadrature to the former ones of [13]. With those errorswe have many good fits with reduced χ , ( χ r ), smallerthan 1. We take two of them, one peaking on the lower side of √ s and the other one on the upper side for the np (I = 0) → N N π cross section, parameterized as, σ i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i √ s − ˜ M i + i ˜Γ2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (15)and call set I the one with the parameters: ˜ M = 2326MeV, ˜Γ = 70 MeV, α = 2 . (cid:16) ˜Γ (cid:17) mb MeV ( χ r =0 . M = 2335 MeV, ˜Γ = 80 MeV, α = 2 . (cid:16) ˜Γ (cid:17) mb MeV ( χ r = 0 . . The pp → π + d cross section has accurate data and we parameterize it as σ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α M inv ( p p (cid:48) ) − ˜ M + i ˜Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (16)with ˜ M = 2165 MeV, ˜Γ = 123 .
27 MeV, α =3 . (cid:16) ˜Γ (cid:17) mb MeV .With the former discussions our final formula on shellis given by σ np → π + π − d = M inv ( p p (cid:48) )6 π σ Inp → NNπ σ pp → π + d M inv ( ππ ) ˜ p p π p (cid:48) π p d ˜ p π (17)with σ Inp → NNπ = σ np ( I =0) → NNπ of [12, 13] and we showthe results in Fig. 2.We can see that the cross section of pp → π + d and np (I = 0) → N N π overlap around the middle of theirenergy distributions such that their product in Eq. (17)gives rise to a narrow peak around √ s = 2340 MeV,close to the position of the experimental np → π + π − d peak around 2365 MeV.In table I we show the results obtained with set I andset II for the strength of σ np → π + π − d at the peak, thepeak position and the width of the peak, varying ¯ M ππ ,and p , max for the off shell calculations. What one sees isa stability of the results upon changes of ¯ M ππ , which jus-tifies the use of Eq. (8). We also find that off shell effectsusing Eq. (13) are small, justifying the on shell approx-imation used in [8]. The strength at the peak between0 . − .
96 mb should be considered quite good comparedto the experimental one around 0 . np (I = 0) → π − pp cross section with the systematic errors with 20 −
30 %smaller strength at the peak are still acceptable, hencesuch uncertainties in the resulting np → π + π − d cross sec-tion are expected). The peak position from 2332 − One should not attribute this shape to the Roper excitation asassumed in [12, 13]. We have seen that the Roper excitationgrows smoothly monotonically around this energy region (seealso Fig. 1 of Ref. [13]) and there are many other mechanismscontributing to the amplitude with cancellations among them. σ pn → NN π I σ pp → d π + σ pn → d π + π - Dakhno et al.WASA - at - COSY (*) s ( MeV ) σ ( m b ) M inv ( p p ' )( MeV ) σ pp → π + d ( m b ) σ pn → NN π I σ pp → d π + σ pn → d π + π - Dakhno et al.WASA - at - COSY (*) s ( MeV ) σ ( m b ) FIG. 2: Plots of σ np → π − pp ( I = 0) and σ pp → π + d , as a function of √ s and M inv ( p p (cid:48) ), respectively, where M inv ( p p (cid:48) ) is evaluatedby means of Eq. (10). The results with σ np → π + π − d in I = 0 of Eq. (17) are multiplied by 10 for a better comparison. Left:Results with set I; Right: Results for set II. ¯ M ππ = 2 m π + 60 MeV. Inset: σ pp → π + d as a function of M inv ( p p (cid:48) ). Data for pp → π + d from [16]. Data for np (I = 0) → πNN are taken from Dakhno et al. [28], and WASA-at-COSY ( ∗ ) [12, 13], includingsystematic errors from isospin violation.TABLE I: Values of the peak strength (“strength”), peak po-sition (“position”), and width, for intermediate particles onshell (columns with δ ¯ M ππ , ¯ M ππ = 2 m π + δ ¯ M ππ ), and off-shell(“o.s.”), where we have taken p , max = 700 and 800 MeV and δ ¯ M ππ = 60 MeV. δ ¯ M ππ (MeV) p o . s . , max (MeV)Set I 40 60 80 700 800strength (mb) 0 .
72 0 .
76 0 .
75 0 .
82 0 . .
75 0 .
80 0 .
80 0 .
85 0 . MeV should also be considered rather good compared tothe about 2365 MeV of the experiment [1–3, 12]. Thenarrow width observed in the experiment of 70 −
75 MeVis also well reproduced by our results in the range of[75 −
88] MeV.The appeareance of the peak about 25 MeV below theexperimental one is not significant with the perspectivethat, as discussed in [12], the authors achieve a resolu-tion in √ s of about 20 MeV and the pp → ppπ and pn → ppπ − cross sections, from where σ np (I=0) → ppπ − isobtained via Eq. (14) with large cancellations, are mea-sured using data bins of 50 MeV in T p .The derivation done contains the basic dynamical in- gredients in a skilled way, making some approximationsto rely upon experimental cross sections. We think thatit is remarkable that a narrow peak, at about the rightposition, with strength and width comparable to the ex-perimental peak of np → π + π − d , appears in spite ofthe approximations done, and the stability of the re-sults allows us to conclude that a peak with the prop-erties of the experimental one associated so far to the“ d ∗ (2380)” dibaryon is unavoidable from the mechanismthat we have studied.From the perspective of the np → π + π − d reaction be-ing tied to the particular reaction mechanism of Fig. 1,with a two step sequential one pion production, it is easyto understand why the narrow peak of the np → π + π − d reaction is not seen in γd → π π d in spite of havingthe same final state [30]. The first reaction is a fusionreaction, with the last step tied to a triangle singular-ity. The γd → π π d reaction is a coherent reaction, the d is already present in the initial state and the reactionmechanisms are drastically different.In summary, we have identified the reaction mecha-nism that produces a narrow peak in the np → π + π − d cross sections without having to invoke a “dibaryon” res-onance. From this perspective it is also easy to under-stand why the peak is not seen in other reactions whereit has been searched for, although the peak contributingto the inelastic channels of pn → all can show traces in N N phase shifts, as anticipated in [5, 6] and discussed in[4].
I. ACKNOWLEDGMENTS