Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js

Sequential single pion production explaining the dibaryon "d*(2380)" peak

  • In this study, we investigate the two step sequential one pion production mechanism, that is, np(I=0)πpp followed by the fusion reaction ppπ+d, to describe the npπ+πd reaction with π+π in state I=0. In this reaction, a narrow peak identified with a "d(2380)" dibaryon has been previously observed. We discover that the second reaction step ppπ+d is driven by a triangle singularity that determines the position of the peak of the reaction and the high strength of the cross section. The combined cross section of these two mechanisms produces a narrow peak with a position, width, and strength, that are compatible with experimental observations within the applied approximations made. This novel interpretation of the peak accomplished without invoking a dibaryon explains why this peak has remained undetected in other reactions.
  • The npπ0π0d reaction presents a sharp peak around 2370 MeV with a narrow width of approximately 70 MeV; this peak is also observed in the ppπ+πd reaction but with approximately double the strength [13]. In the absence of a conventional reaction mechanism that can explain the occurrence of these peaks, they have been interpreted as signals of a dibaryon labeled as d(2380). Based on this hypothesis, other features observed in π production experiments and NN phase shifts have also been interpreted (see [4] for a recent review). Notably, the narrow peak in npππd affects the inelasticity of the NN phase shifts and is expected to influence these NN phase shifts, as emphasized in [5, 6]. Previously, several mechanisms of the two pion production leading to ππd have been studied [13] based on the model reported in [7] for NNNNππ; this system features double Δ production with subsequent ΔπN decay or N(1440) production with the decay of N to Nππ or NπΔ(Nπ). In all these cases, the resulting np particles are fused into a deuteron. However, the results of these calculations lead to cross sections with smaller strengths compared to the peak of the npπ0π0d reaction and produce no peaks with the energy of the observed peak. Such conclusions have already been drawn in a previous study [8], and we explicitly recalculated the cross sections based on these mechanisms to corroborate all previous findings. Interestingly, in the same study [8], a peak with poor statistics, already visible for the npπ+πd reaction, was explained using a different mechanism, namely two step sequential π production npppπ followed by ppπ+d. The cross section for the npπ+πd reaction was evaluated by factorizing the cross sections for the two latter reactions based on an "on-shell" approach, which, however, required further tests for accuracy. Notably, such a mechanism has not been further invoked considering the availability of new improved data on the npπ+πd and npπ0π0d reactions [13].

    Meanwhile, the time reversal reaction involved in ppπ+d, that is, π+ absorption in a deuteron, π+dpp, has been studied in the past [911]. Such investigations have revealed that this reaction has a neat peak corresponding to the Δ excitation. Combining the results of previous studies [911] with the idea reported in [8] on the npπ+πd reaction, the mechanism underlying the npπ+πd reaction can be expressed diagrammatically, as shown in Fig. 1.

    Figure 1

    Figure 1.  Two step mechanism for npπ+πd, as suggested in [8], with explicit Δ excitation in the last step of ppπ+d, as reported in [911]. The mechanism with the nn intermediate state is also considered.

    After several years, the availability of more refined data and new theoretical developments suggest the need to revisit this issue based on the same idea:

    1) Recent data on npπ+πd and npπ0π0d have excellent precision [14].

    2) The npπ0π0d reaction involves π0π0 in isospin state I=0, and hence, the inital np state must also be in state I=0.

    In [3], the npπ+πd reaction was split into I=0 and I=1 states, and as expected, the same peak visible for the npπ0π0d reaction was observed for the np(I=0)π+πd reaction with approximately double the strength. This implies that in the npπpp reaction, the first step of the sequential single pion production mechanism, i.e., the inital np state, is also in state I=0. Recently, the first step, shown in Fig. 1 as npπpp with np in I=0, has been singled out with a relatively good precision [12] (see revision about normalization in [12, 13]).

    3) New developments concerning triangle singularities [14] facilitate our understanding of the high strength of the ppπ+d reaction with a triangle singularity, as depicted in the triangle diagram shown in the last section of Fig. 1. This corresponds to the simultaneous occurrence of Δ and two nucleons on shell and collinear. A simplification of the formalism on the triangle singularities, as accomplished in [15], allows immediate localization of the peak of the ppπ+d cross section based on Eq. (18) presented in [15]; here, the mass d is considered slightly unbound to determine a solution to the given equation. One predicts that a peak of the cross section should appear around Minv(pp)2179 MeV, which is very close to the location of the peak found in the experiment reported in [16]. The Coleman-Norton theorem [17] clarifies this case to visualize the process. The theorem states that a triangle singularity appears if the process visualized in the triangle diagram can occur at the classical level. In our case, this would occur as follows: the pp system produces ΔN back to back in the pp rest frame; Δ decays into a π+along the direction of Δ and N along the opposite direction, which is along the direction of N. Following this, N moves faster than N (encoded by Eq. (18) in [15]) and approaches N to fuse into a deuteron. The fusion of the two nucleons into a deuteron occurs naturally when the discussed mechanism has a triangle singularity, producing a neat peak and a cross section that are notably large compared with those of typical fusion reactions [18]. We reanalyzed the ppπ+d reaction based on this new perspective [19]; however, corresponding details are nonessential for the derivation of the npπ+πd cross section, which, as reported in [8], relies on experimental cross sections, using the new np(I=0)πpp cross section [12, 13] and the data available for ppπ+d [16]. We also improved the on shell approach adopted in [8]. It is worth mentioning that while the mechanism for ppπ+d described in [911] could not be identified as a triangle singularity, the authors of [11] demonstrated that the cross section blew up when the Δ width was set to zero, a characteristic of the triangle singularity. In [19], the authors demonstrated that the dominant term in ppπ+d is the partial wave 1D2(2S+1LJ), which is in agreement with the experimental observations reported in [20]. Along similar lines, one may trace back JP=1+,3+ for the dπ+π system, with some preference for 3+ and 3D3 in the initial np system, i.e., the preferred quantum numbers associated with the d(2380) peak [4].

    It is also worth mentioning that the dominance of the 1D2 partial wave leads to a structure indicative of a resonance in the ppπd reaction [21]: a dibaryon other than "d(2380)." Theoretical groups also suggest bound states of NΔ, three body πNN [22], or NΔ [23] to explain the peak produced by this reaction; however, in [24], the authors could not prove adequate binding. In fact, as presented in [25], the Argand plots corresponding to a resonance and the triangle singularity are similar. We strictly follow the basic rule stating that if one phenomenon can be explained using conventional, well established facts, this interpretation should be favored over less conventional ones. Previous studies [911, 19] explaining the ppπd reaction on conventional grounds have proved that a new dibaryon is not essential to explain the foregoing reaction.

    The derivation of the npπ+πd cross section follows the steps involved in the optical theorem [26]. Let us denote t as the amplitude for the isoscalar np(I=0)πpp reaction, t as the amplitude for ppπ+d, and tas the amplitude for np(I=0)π+πd. The differential cross section for the isoscalar np(I=0)πpp reaction is given by

    dσInpπppdMinv(p1p1)=14ps(2MN)4116π3pπ˜p1|ˉt|212,

    (1)

    where σI stands for the isoscalar cross section, s is the center-of-mass (CM) energy of the inital np state, Minv(p1p1) is the invariant mass of the final two protons involved in this reaction, p is the CM momentum of the inital n or p particles, pπ is the pion momentum in the np rest frame, and ˜p1 is the momentum of the final protons in the pp rest frame. We use the (2MN)4 factor of the fermion field normalization for nucleons following the formalism proposed by Mandl and Shaw [27]. The magnitude |ˉt|2 indicates the angle averaged value of |t|2, and the factor 12 considers the identity of the two final protons.

    Similarly, the cross section for ppπ+d in the second part of the diagram of Fig. 1 is given by

    σppπ+d=116πM2inv(p1p1)pπ˜p1|ˉt|2(2MN)2(2Md),

    (2)

    where pπ is the π+ momentum in the pp rest frame, and |ˉt|2 stands for angle averaged |t|2. We choose to normalize the deuteron field in the form of nucleons, and we added a factor 2Md (which disappears from the final formulas). Meanwhile, the amplitude for the npππ+d process in Fig. 1 is given by

    it=12d4p1(2π)4(2MN)22EN(p1)2EN(p1)ip01EN(p1)+iϵ×isp01ωπEN(p1)+iϵ(i)t(i)t.

    (3)

    Here, the factor 12 accounts for the intermediate propagator of two identical particles. In the d4p1 integrations, t and t would be off shell. In [8], the pion and two protons in the intermediate state were assumed to be on shell, and t and t were used with the on shell variables. However, theoretical advances [28] allow us to proceed beyond this approximation. Indeed, the chiral unitary approaches reported in [28] for meson-meson interactions or in [29] for meson-baryon interactions factorize the vertices on-shell and perform a loop integral of the two intermediate states. A different justification has been provided in [30], wherein a dispersion relation is derived for the inverse of the hadron-hadron scattering amplitude. Another justification has been provided in [31, 32], wherein with the aid of chiral Lagrangians, the authors demonstrated that the off shell parts of the amplitudes appearing in the approach canceled with counterterms provided by the same theory. This implies that in Eq. (3), we can take tt outside the dp01 integral with the on-shell values and evaluate the remainder of the integral.

    On performing the p01 integration analytically with Cauchy's residues, we obtain

    t=12d3p1(2π)3(2MN)22EN(p1)2EN(p1)×ttsEN(p1)EN(p1)ωπ+iϵ,

    (4)

    where p1, p1 denote the momenta of the intermediate pp particles in Fig. 1, and ωπ is the π energy. The t,t amplitudes are Lorentz invariant, and we choose to evaluate the d3p12E1(p1) integral in the pp rest frame, where |p1|=|p1| and sωπ becomes the invariant mass of the two protons. This integral is logarithmically divergent and requires regularization. The result smoothly depends on a cut off p1,max for |p1|, which is used to regularize the d3p1 integral. We set some values for p1,max within a reasonable range. Despite this, we anticipate that the on shell part given by Eq. (5) below provides the largest contribution to the t amplitude. Given that \tilde{p}_1= 552 MeV/c for M_{{\rm inv}(p_1p_1')}= 2179 MeV, the triangle singularity would appear for t' and Δ with a zero width or a pronounced peak when the width is considered; thus, values of p_{1,\mathrm{max}} around 700-800 MeV seem reasonable.

    The on-shell approximation used in [8] allows one to express the cross section for np\to\pi^+\pi^-d in terms of the cross sections for np {(I=0)}\to \pi^-pp and pp\to\pi^+d . This approximation is obtained in the present formalism by considering the imaginary part of the two nucleon propagator:

    \begin{aligned} & \frac{1}{M_{\mathrm{inv}}(p_1p_1')-2E_N(p_1)+{\rm i}\epsilon}\\\equiv& {\cal P}\left[\frac{1}{M_{\mathrm{inv}}(p_1p_1')-2E_N(p_1)}\right]-{\rm i}\pi\delta(M_{\mathrm{inv}}(p_1p_1')-2E_N(p_1)).\\ \end{aligned}

    (5)

    Following this, we have

    t^{\prime\prime}_\mathrm{on}=-{\rm i}\frac{1}{2}\frac{\tilde{p}_1}{8\pi}\frac{(2M_N)^2}{M_\mathrm{inv}(p_1p_1')}\bar{tt'} ,

    (6)

    where we factorized the angle averaged value of tt' , \bar{tt'} . Using the analogous expression in Eq. (1) for {\rm d}\sigma_{np\to\pi^+\pi^-d}/ {\rm d}M_{\mathrm{inv}}(\pi^+\pi^-) and the on-shell approximation given by Eqs. (6) and (1), we can write the following:

    \begin{aligned}[b] \frac{{\rm d}\sigma_{np\to\pi^+\pi^-d}}{{\rm d}M_{\mathrm{inv}}(\pi^+\pi^-)}=&(2M_N)^2(2M_d)p_d\tilde{p}_\pi\frac{1}{4}\frac{\tilde{p}_1^2}{64\pi^2}\\&\times\frac{1}{M^2_\mathrm{inv}(p_1p_1')}\frac{1}{p_\pi\tilde{p}_1}2|\bar{t}'|^2\frac{{\rm d}\sigma_{np\to\pi^- pp}}{{\rm d}M_\mathrm{inv}(p_1p_1')} , \end{aligned}

    (7)

    where p_d is the deuteron momentum in the original np rest frame, |\bar{t}'|^2 is the angle averaged |t'|^2 , and \tilde{p}_\pi is the pion momentum in the \pi^+\pi^- rest frame. In Eq. (7), we assumed that |\bar{tt'}|^2=|\bar{t}|^2|\bar{t'}|^2. Note that the amplitudes t,t' in [12, 16] have some angular structures, but these are smooth enough to consider this assumption a sensible approximation.

    Next, we use physical arguments to express the np\to \pi^+\pi^-d cross section in the form of an easy compact formula. Note that \pi^0\pi^0 or \pi^+\pi^- in state I=0 , as discussed earlier, require even values for their relative angular momentum l, and when l=0 , \pi^0\pi^0 or the symmetrized (\pi^+\pi^-+\pi^-\pi^+) behave as identical particles, which revert to a Bose enhancement when the two pions move together. Certainly, if these pions are exactly together, we also have the phase space factor \tilde{p}_\pi in the term {\rm d}\sigma/{\rm d} M_\mathrm{inv}(\pi^+\pi^-) of Eq. (7), which makes this distribution null in the two pion threshold; however, some enhancement for small invariant masses is still expected. Our argument is supported by the results reported in [1, 2] for \pi^0\pi^0 (see Fig. 2 in [1] and Fig. 4 in [2]) and in [3] for charged pions, although the natures of I=0 and I=1 in this case slightly distort the mass distribution compared to the clean I=0 \;\pi^0\pi^0 case.

    We could consider certain M_\mathrm{inv}(\pi^+\pi^-) distributions as inputs. However, to make the results as model independent as possible, we set \bar{M}_\mathrm{inv}(\pi^+\pi^-)\sim 2m_\pi+60 MeV, which is not far from the threshold; however, we varied this value to evaluate the dependence on \bar{M}_\mathrm{inv} . The stability of the results with varying values of \bar{M}_{\mathrm{inv}}(\pi\pi) justifies this approximation a posteriori. Thus, we can write

    \frac{{\rm d}\sigma_{np\to\pi^+\pi^-d}}{{\rm d}M_{\mathrm{inv}(\pi^+\pi^-)}}=\sigma_{np\to\pi^+\pi^-d}\delta(M_{\mathrm{inv}}(\pi^+\pi^-)-\bar{M}_{\pi\pi})\ .

    (8)

    The approximation given by Eq. (8) is sufficiently good and provides a more transparent picture of the reason responsible for the appearance of the peak in the np\to \pi^+\pi^-d reaction. In this case, the energy of the two pions is obtained as

    \begin{equation} E_{2\pi}=\frac{s+M^2_\mathrm{inv}(\pi\pi)-M^2_d}{2\sqrt{s}}\ , \end{equation}

    (9)

    and given that both pions move relatively together, we set E_\pi=E_{2\pi}/2 , which allows relating M_\mathrm{inv}(p_1p_1') with \sqrt{s} via

    \begin{array}{*{20}{l}} M^2_\mathrm{inv}(p_1p_1')=(P(np)-p_{\pi^-})^2=s+m^2_\pi-2\sqrt{s}E_\pi \end{array}

    (10)

    and formally

    \begin{aligned}[b] & 2M_{\mathrm{inv}}(p_1p_1'){\rm d} M_\mathrm{inv}(p_1p_1')\\=&-2\sqrt{s}{\rm d} E_\pi=-M_\mathrm{inv}(\pi\pi){\rm d} M_\mathrm{inv}(\pi\pi)\ . \end{aligned}

    (11)

    Using this relationship, we can integrate Eq. (8) with respect to M_\mathrm{inv}(\pi\pi) , and using Eqs. (2) and (7), we obtain

    \begin{equation} \sigma_{np\to\pi^+\pi^-d}=\frac{M_\mathrm{inv}(p_1p_1')}{4\pi}\frac{\sigma_{np\to\pi^-pp}\sigma_{pp\to\pi^+d}}{M_\mathrm{inv}(\pi\pi)}\frac{\tilde{p}_1^2}{p_\pi p'_{\pi}}p_d\tilde{p}_\pi . \end{equation}

    (12)

    Here, one last detail is required. As stated, we considered the two step np {(I=0)}\to\pi^-pp reaction followed by pp\to\pi^+d . An appropriately symmetrized t^{\prime\prime} amplitude requires the addition of np {(I=0)}\to\pi^+nn followed by nn\to \pi^-d . Evidently, considering the isospin, the amplitudes np {(I=0)}\to \pi^-pp and np {(I=0)}\to \pi^+nn are identical up to the phase of \pi^+ ( -1 in our formalism), and the same is true for pp\to\pi^+d and nn\to \pi^-d for the same configuration of the particles. Hence, the product of the amplitudes is the same. If \pi^+ and \pi^- move exactly together, both amplitudes will be identical and will add coherently. However, we observe that the phase space factor \tilde{p}_\pi in Eq. (12) eliminates this contribution. When considering integration over the five degrees of freedom of the three body phase space, the terms are expected to mostly sum incoherently. Consequently, we must multiply Eq. (12) by 2 . Similar arguments can be made with respect to the spin sums and averages. Investigations on the pp\to \pi^+d reaction conducted in [19] indicate the presence of a certain angular dependence on the different spin transitions, and we should expect an incoherent sum over spins. Subsequently, by including the average over initial spins and the sum over final spins in |\bar{t}|^2 , we would be considering the average over spins of the initial np and the sum over spins of the deuteron in our formula, in addition to the intermediate sum over the pp and nn spins.

    Equation (12) still relies on the on-shell approximation expressed by Eq. (6). To consider the off shell effects discussed above, we realized that by factorizing the angular averaged t,t' amplitudes in the term t" of Eq. (4) while maintaining their energy dependence as a function of M_\mathrm{inv}(p_1p_1') , the on shell energy of the intermediate two nucleons (Eq. (4)) presented a remaining structure as the G function of two protons,

    G=\int\frac{{\rm d}^3p_1}{(2\pi)^3}\frac{1}{E_N(p_1)E_N(p_1)}\frac{1}{M_\mathrm{inv}(p_1p'_1)-2E_N(p_1)+{\rm i}\epsilon}\ .\\

    (13)

    Following this, we obtain

    \begin{equation} \mathrm{Im}G=-\frac{1}{2\pi}\frac{\tilde{p}_1}{M_{\mathrm{inv}}(p_1p_1')} \ , \end{equation}

    (14)

    where, as mentioned after Eq. (1), \tilde{p}_1 is the momentum of both protons in their rest frame. The on shell factorization of the terms t and t' outside the G function has been justified in the discussion following Eq. (3) from different perspectives. We proceed along a different direction with respect to the derivation of Eq. (6): we now retain the two terms in Eq. (5) rather than retaining only the imaginary part. In Eq. (6), we considered the imaginary part of the integral of Eq. (4). This is equivalent to taking \mathrm{Im}\,G of Eq. (14) instead of G in the integral of Eq. (13). To revert this approximation and approximately determine the effects of the off shell part of the integral, we replace the following in Eq. (12):

    \left(\frac{\tilde{p}_1}{2\pi M_\mathrm{inv}(p_1p_1')}\right)^2\to |G(M_\mathrm{inv})|^2\ .

    (15)

    The last step in the evaluation of \sigma_{np\to\pi^+\pi^-d} requires the use of experimental data for np(I=0)\to\pi^-pp and pp\to \pi^+d . We directly extracted \sigma_{pp\to \pi^+d} from the experiments reported in [16]. For \sigma_{np\to\pi^-pp} in I=0 , additional analysis is however required. In [33, 34], the isoscalar NN\to\pi NN amplitude was obtained via isospin symmetry from \sigma_{np\to pp\pi^-} and \sigma_{pp\to pp\pi^0} , and relatively precise results were reported in [12] based on improved measurements of these cross sections. In the erratum of [12] and in [13], it has been clarified that the actual value \sigma_{pn(I=0)\to NN\pi} is one half that of the value of \sigma_{NN(I=0)\to NN\pi} reported in [12]. The required cross section is \sigma_{pn(I=0)\to pp\pi^-} . Thus, using isospin symmetry, we can observe that \sigma_{pn(I=0)\to pp\pi^-} , \sigma_{pn(I=0)\to nn\pi^+} , and (\sigma_{pn(I=0) \to pn\pi^0}+\sigma_{pn(I=0)\to np\pi^0 }) are all equal. Following this, we express the relationship among the results of [12, 33, 34] as

    \begin{aligned}[b] \sigma_{np(I=0)\to pp\pi^-}= &\frac{1}{3}\sigma_{np(I=0)\to NN\pi} = \frac{1}{6}\sigma_{NN(I=0)\to NN\pi}\\=&\frac{1}{6}3(2\sigma_{np\to pp \pi^-}-\sigma_{pp\to pp\pi^0}) \end{aligned} .

    (16)

    For the above, we consider the data for \sigma_{np(I=0)\to NN\pi} from Fig. 1 in [13]. Notably, statistical and systematic errors have been considered in [12, 13]. To ensure a realistic fit for the data, we also included systematic errors from the uncertainty given in Eq. (16) when using isospin symmetry. We assumed a typical 5% violation of the isospin in each of the last two terms of Eq. (16) and summed the errors in quadrature. The systematic errors obtained were of the order of 0.5 mb in \sigma_{np(I=0)\to NN\pi} , which were also added in quadrature to the errors reported in [13]. With these errors, we obtained several good fits with reduced \chi^2 , ( \chi^2_r ), smaller than 1 . We selected two of these, one peaking on the lower side of \sqrt{s} and the other on the upper side for the np\mathrm{(I=0)}\to NN\pi cross section, and we parameterized the cross section as given below.

    \sigma_i=\left|\frac{\alpha_i}{\sqrt{s}-\tilde{M}_i+{\rm i}\dfrac{\tilde{\Gamma}}{2}}\right|^2

    (17)

    Here, set I has the following set of parameters: \tilde{M}_1=2326 MeV, \tilde{\Gamma}_1=70 MeV, and \alpha^2_1=2.6\left(\frac{\tilde{\Gamma}_1}{2}\right)^2 mb MeV ^2 ( \chi^2_r= 0.50); set II has the following set of parameters: \tilde{M}_2= 2335 MeV, \tilde{\Gamma}_2=80 MeV, and \alpha^2_2=2.5\left(\frac{\tilde{\Gamma}_2}{2}\right)^2 mb MeV ^2 ( \chi^2_r=0.52 ). The pp\to\pi^+d cross section has accurate data, and we parameterize it as

    \sigma_3=\left|\frac{\alpha_3}{M_\mathrm{inv}(p_1p'_1)-\tilde{M}_3+{\rm i}\frac{\tilde{\Gamma}_3}{2}}\right|^2 ,

    (18)

    with \tilde{M}_3=2165\; {\rm MeV} , \tilde{\Gamma}_3=123.27\; {\rm MeV} , and \alpha^2_3= 3.186\times \left(\frac{\tilde{\Gamma}_3}{2}\right)^2 mb MeV ^2 .

    Based on the above discussions, our final formula on shell can be expressed as

    \sigma_{np\to\pi^+\pi^-d}=\frac{M_{\mathrm{inv}}(p_1p'_1)}{6\pi}\frac{\sigma^I_{np\to NN\pi}\sigma_{pp\to\pi^+d}}{M_{\mathrm{inv}}(\pi\pi)}\frac{\tilde{p}_1^2}{p_\pi p'_\pi}p_d\tilde{p}_\pi,

    (19)

    with \sigma^I_{np\to NN\pi}=\sigma_{np(I=0)\to NN\pi} from [12, 13]. The corresponding results are depicted in Fig. 2.

    Figure 2

    Figure 2.  (color online) Plots of \sigma_{np\to\pi^-pp}(I=0) and \sigma_{pp\to\pi^+d} as functions of \sqrt{s} and M_{\mathrm{inv}}(p_1p_1') , respectively, where M_{\mathrm{inv}}(p_1p_1') is evaluated using Eq. (10). The results with \sigma_{np\to\pi^+\pi^-d} in I=0 of Eq. (19) are multiplied by 10 for better comparison. Left: Results for set I. Right: Results for set II. \bar{M}_{\pi\pi}=2m_\pi+60 MeV. Inset: \sigma_{pp\to\pi^+d} as a function of M_{\mathrm{inv}}(p_1p'_1) . Data for pp\to\pi^+d extracted from [16]. Data for np\mathrm{(I=0)}\to\pi NN extracted from Dakhno et al. [33] and WASA-at-COSY ^{(*)} [12, 13], including systematic errors from isospin violation.

    We can observe that the cross sections of pp\to \pi^+d and np (I=0) \to NN\pi overlap around the center of their energy distributions such that their product in Eq. (19) produces a narrow peak around \sqrt{s}=2340 MeV, which is close to the position of the experimental np\to\pi^+\pi^-d peak around 2365 MeV.

    Table 1 summarizes the results obtained using sets I and II for the strength of \sigma_{np\to\pi^+\pi^-d} at the peak, peak position, peak width, varying \bar{M}_{\pi\pi} , and p_{1,\mathrm{max}} for the off shell calculations. The results appear stable with changes in \bar{M}_{\pi\pi} , justifying the use of Eq. (8). Note that the off shell effects resulting from Eq. (15) are small, justifying the on shell approximation used in [8]. The strength at the peak between 0.72-0.96 mb can be considered good compared to the experimental one around 0.5 mb, given the different applied approximations (note: the fits to the np {(I=0)}\to \pi^-pp cross section with systematic errors presenting 20\%-30\% smaller strengths at the peak are still acceptable; hence, such uncertainties in the resulting np\to\pi^+\pi^-d cross section are expected). The peak position from 2332-2345 MeV can also be considered good compared to the approximate peak positions at 2365 MeV, as experimentally reported previously [13, 12]. The narrow width observed in the experiment with 70-75 MeV is also well reproduced by our results in the range of 75–88 MeV.

    Table 1

    Table 1.  Values of the peak strength ("strength"), peak position ("position"), and width for intermediate particles on shell (columns with \delta \bar{M}_{\pi\pi} , \bar{M}_{\pi\pi}=2m_\pi+\delta \bar{M}_{\pi\pi} ) and off-shell ("o.s."), where we set p_{1,\mathrm{max}}=700 and 800 MeV and \delta\bar{M}_{\pi\pi}=60 MeV.
    \delta\bar{M}_{\pi\pi}/{\rm MeV} p_{1,\mathrm{max}}^{\mathrm{o.s.}}/{\rm MeV}
    Set I 40 60 80 700 800
    strength/mb 0.72 0.76 0.75 0.82 0.95
    position/MeV 2332 2332 2332 2332 2332
    width/MeV 76 76 81 75 75
    Set II
    strength/mb 0.75 0.80 0.80 0.85 0.96
    position/MeV 2342 2345 2345 2343 2342
    width/MeV 86 87 88 87 84
    DownLoad: CSV
    Show Table

    The appeareance of the peak at approximately 25 MeV below the experimental one is not significant given that as discussed in [12], the authors achieved a resolution in \sqrt{s} with a value of approximately 20 MeV; in addition, the pp\to pp\pi^0 and pn\to pp\pi^- cross sections, based on which \sigma_{np\mathrm{(I=0)}\to pp\pi^-} is obtained using Eq. (16) with large cancellations, were measured using data bins of 50 MeV in T_p .

    The derivation involves the use of basic ingredients of dynamics in a skilled manner, resulting in some approximations that are reliant on experimental cross sections. It is remarkable that a narrow peak at approximately the correct position, with a strength and width comparable to those of the experimental peak of np\to\pi^+\pi^-d , appears despite the considered approximations, and the stability of the results allows us to conclude that a peak with the properties of the experimental one associated with the " d^*(2380) " dibaryon is unavoidable according to the evaluated mechanism.

    Based on the fact that the np\to\pi^+\pi^-d reaction is involved in the particular reaction mechanism shown in Fig. 1, with a two step sequential one pion production, it is easy to understand why the narrow peak of the np\to \pi^+\pi^-d reaction cannot be observed for the \gamma d\to \pi^0\pi^0d reaction, despite the same final state [35]. The first reaction is a fusion reaction, with the last step connected to a triangle singularity. The \gamma d\to \pi^0\pi^0d reaction is a coherent reaction; here, d is already present in the initial state, and the reaction mechanisms are drastically different.

    In summary, we investigated the reaction mechanism producing a narrow peak in the np\to \pi^+\pi^-d reaction cross sections without invoking the "dibaryon" resonance. Based on this perspective, it is easy to understand why the peak is not observed in other investigated reactions, despite the fact that the peak contributing to the inelastic channels of pn\to all presents traces in the NN phase shifts, as anticipated in [5, 6] and discussed in [4].

    Previous studies have claimed the occurrence of the d^*(2380) state in other reactions (see review in [4]). Moreover, the results presented in the previous section, posted on arXiv (arXiv: 2102.05575), have been further analyzed in [36], where apparent contradictions with experiments have been reported. In this section, we address the different points raised in [36] to indicate that our approach does not contradict previous experimental results and that the "proofs" presented in favor of the dibaryon hypothesis are unsupported. We follow the points presented in [36] in the discussion below.

    As mentioned in the previous section, studies on the np\to \pi^0\pi^0d and np\to \pi^+\pi^-d reactions [2, 3] have revealed an unexpected narrow peak in the cross section around 2380 MeV. This peak has been attributed to a dibaryon, labeled as d^*(2380) , by an experimental team. It is interesting to note that, based on earlier data pertaining to the reaction, the peak was attributed to a reaction mechanism based on sequential one pion production, that is, np\to \pi^-pp\to \pi^-\pi^+d together with np\to\pi^+nn\to \pi^+\pi^-d , in a study conducted by Bar-Nir et al. [8]. The second step, pp\to\pi^+d , was the object of theoretical investigation in [911], and it has been demonstrated to be driven by \Delta(1232) excitation. We propose a reformulation of the idea of these former works, which has been done in [19] from a Feynman diagrammatic point of view, indicating that the process develops a triangle singularity (TS) [9, 14, 15]. This finding is relevant for the present discussion because it is well known that a TS produces an Argand plot that is similar to the one of an ordinary resonance, even if the origin is a kinematical singularity rather than a genuine physical state [25, 37].

    The idea adopted by Bar-Nir et al. has been revisited in the former reaction, and with some reasonable approximations and experimental data on the np(I=0)\to pp \pi^- and pp\to \pi^+d reactions, a peak could be obtained for the pp\to\pi^+\pi^-d reaction; this peak was in qualitative agreement with previous experimental results with regard to its position, width, and strength. Even with the approximations involved and the resulting qualitative agreement, it is important to note that such an agreement, together with the results reported by Bar-Nir et al., is extremely unlikely to be a mere coincidence and hence offers an alternative explanation of the experimentally observed peak.

    In [36], the authors present a few arguments. We address these arguments as follows.

    i) In point 1), the authors note that the error resulting from the proposed approach increases in the pn (I=0) \to \pi^- pp reaction. The reason for this increase is that the cross section for this reaction is obtained using isospin symmetry with the following relation:

    \qquad\qquad \sigma_{np(I=0) \to pp \pi^-} = ( \sigma_{np \to pp \pi^-} - \sigma_{pp \to pp \pi^0}/2)\ .

    (20)

    However, note that this formula involves extensive cancelations, and the result is ten times smaller than each individual term. Thus, we assumed a 5 % uncertainty in the terms obtained from isospin violation and determined the errors in the results. These correspond to systematic uncertainties that should have been considered by the experimentalists; however, they were ignored, which is why we accounted for these errors in our analysis. In the high energy region of the spectrum, where the cross section decreases and produces the shape of the cross section, the systematic errors are much greater than the statistical ones. Therefore, the manner in which the statistical errors are summed is inconsequential. The magnitude depends on the systematic errors. In any case, the obtained cross sections are not influenced by these errors.

    ii) In point 2), the authors claim that our analysis lacked a precise description of the data. With the approximations mentioned above, we cannot possibly pretend to have achieved precise agreements. However, in our opinion, the fact that for such a complicated reaction, we could qualitatively obtain a peak with the correct energy, width, and strength is an accomplishment.

    iii) For this argument, first, let us note that in a previous study [19] on the pp\to \pi^+d reaction, we proved the dominance of ^1D_2 , as determined experimentally in [38, 39]. Second, the argument states that because in the np(I=0)\to\pi^-pp reaction, the invariant mass of \pi^-p is large, the mass of pp is small, and it only accommodates L=0,L=1 waves and not D-waves that are necessary for the overlap of the two-step mechanism. It is interesting to make this argument more quantitative. Let us consider two situations wherein M(pp) can be easily evaluated. They correspond to the case of M(\pi^-p)\vert_{\mathrm{min}}=m_p+m_{\pi^-} and M(\pi^-p)\vert_{\mathrm{max}}=\sqrt{s}-m_p . In the first case, p and \pi^- move together along a direction opposite to that of the other proton. In the second case, one proton is produced at rest. Note that M(pp) is trivially evaluated in the foregoing two cases, and for an energy of T_p=1200 MeV in Fig. 6 of [12], we find the following:

    a) M(pp) (at M(\pi^-p)\vert_{\mathrm{min}} ) =2239.47 MeV, with an excess energy of the two protons = 362.9 MeV.

    b) M(pp) (at M(\pi^-p)\vert_{\mathrm{max}}) =1920.2 MeV, with an excess energy of 43.65 MeV.

    c) Another situation allowing such easy evaluation also exists. This is at the peak of the distribution around M(\pi^-p)=1370 MeV, where M(\pi^-p) from either of the protons is approximately the same and enhances the contribution in this region. Here, we have

    2M^2(\pi^-p)+M^2(pp) = s+2m_p^2+m_{\pi^-}^2;

    from this, we obtain M(pp)\simeq 1951 MeV corresponding to an excess energy of approximately 75 MeV for the two protons. Assuming a relative distance of r\simeq 2.13 fm for the produced two protons, the same as the radius of the deuteron, which corresponds to the range of pion exchange, we determined that the angular momentum, L\sim r\times p , could reach up to L=6 in case a), L=2 in case b), and L=3 in the most favorable case c).

    In [36], L=2 was already ruled out, and with no calculations, the sequential pion production cross section was deemed to be very small, contradicting the conclusions made in [8]. However, with the dominance of the Roper excitation, as claimed in [36], we could observe that S=0 for both protons is the dominant mode in np(I=0)\to\pi^-pp , as in the second step of [19] for the pp\to \pi^+d reaction, and several L values are allowed. Notably, research along these lines is still underway, given that Roper excitation is not the only ingredient of the np(I=0)\to\pi^-pp reaction.

    iv) We would like to state that point 4) is illustrative. Two independent studies, [5] and [6], have proved the existence of a relationship between the pn (I=0) \to \pi^+ \pi^- d reaction and the one where the np of the deuteron emerge as free states. The existence of the peak for the pn (I=0) \to \pi^+ \pi^- d reaction also consequently produces a peak in pn \to \pi^+ \pi^- n p and related reactions at the same energy. However, this is the case regardless of the reason responsible for the appearance of the peak in the fusion reaction. This is a key point. Notably, to calculate the cross section of the open reactions, the authors of [36] used the results reported in these references and added the contribution to the results of the standard model, which were also obtained from [7]. Despite the fact that the authors of [5, 6] indicated that the new contribution was necessary regardless of the reason for the fusion reaction, the authors of [36] considered the above as an evidence of a dibaryon.

    v) In point 5), the authors state that we do not to calculate differential distributions. This is true; however, we did not need the above to prove our points. At the considered qualitative level, we demonstrated that the distribution had to peak at small invariant masses of the two pions because we had two contributions: pn (I=0) \to \pi^- pp followed by pp \to \pi^+ d , together with pn (I=0) \to \pi^+ nn followed by nn \to \pi^- d . We could prove that when the momenta of the two pions were equal, the two amplitudes were identical, and they summed to produce a Bose enhancement. In this case, the invariant mass of the two pions has the smallest value. This is why the cross section peaks at a low \pi \pi invariant mass.

    vi) Next, the authors claim that the picture proposed by Bar-Nir [8] presented in the former section is not adequate to explain the observed pole in ^3D_3-^3G_3 np partial waves. This statement is incorrect. The NN , I=0 phase shifts are affected by the peak in the pn (I=0) \to \pi^+ \pi^- d reaction because one can have pn(I=0) \to \pi^+ \pi^- d \to pn(I=0) , where \pi^+ \pi^- d is in an intermediate state and contributes to the inelasticities. This will be particularly the case in the quantum numbers preferred by the pn (I=0) \to \pi^+ \pi^- d reaction discussed in our previous papers, in particular the ^3D_3 partial wave (see the discussion at the end of [19]). At the energy of the peak of the pn(I=0) \to \pi^+ \pi^- d reaction, the np \to np amplitude will have an enhanced imaginary part according to the optical theorem. This has an impact on the phase shift at this energy. This can also be stated for most of the reactions claiming to observe the dibaryon. In these reactions, what is actually observed is a consequence of the peak observed in the pn(I=0) \to \pi^+ \pi^- d reaction, regardless of reason for this peak. This is an important point. The peak of the pn(I=0) \to \pi^+ \pi^- d reaction will have effects on numerous observables; however, this does not indicate that the reason for the peak is a dibaryon. Nevertheless, it will have consequences. In fact, the effect of the peak of the pn(I=0)\to \pi^+\pi^-d reaction on the ^3D_3 and ^3G_3 partial waves has already been discussed in [5, 6]. Although it appears that the pn\to \pi^+\pi^-d\to pn process would provide a small contribution to the pn \to pn amplitude, the relatively large strength of the pn \to \pi^+\pi^-d peak makes this two step process not too small, as demonstrated in [5, 6]. It is also worth mentioning that small terms in an amplitude can often emerge more clearly in polarization observables than in direct cross sections, as shown in [40].

    The pole or resonant structure is guaranteed by the triangle singularity of the last step pp\to\pi^+d , as shown in [19]. It is well-known that a triangle singularity produces an Argand plot similar to that of a resonance [25, 37].

    vii) The argument made here is rather weak. The authors mention that " d^*(2380) " has been observed in the \gamma d\to d\pi^0\pi^0 and \gamma d\to pn reactions. In fact, in [41, 42], one can observe a deviation of the experimental cross section from the theoretical results reported by [43, 44] at low photon energies. These calculations are based on the impulse approximation, and the π rescattering terms are neglected. However, the rescattering contributions of pions are important, particularly at low energies because the momentum transfer is shared between two nucleons, and one picks up smaller deuteron momentum components, where the wave function is larger. Thus, concluding that the discrepancies observed between experiments and calculations based on the impulse approximation are attributable to the dibaryon is incorrect.

    Let us further extend this discussion. The authors of [41, 42] also studied the \gamma d\to \pi^0 \eta d reaction [45, 46]. This reaction was throughly studied theoretically in [47], and the authors discovered that the pion rescattering mechanisms were particularly important, and the most striking feature of the reaction, i.e., the shift of the shape of the invariant mass distributions, was well reproduced. In the \gamma d\to \pi^0\eta d reaction, η rescattering had a small effect, and only the π rescattering was relevant. In the \gamma d\to \pi^0\pi^0 d reaction, the two pions could rescatter, making the rescattering mechanism in \gamma d\to \pi^0\pi^0 d even more important than that in \gamma d\to \pi^0\eta d .

    The \gamma d\to \pi^0\pi^0 d reaction was measured more accurately in [48]. The same comments can be made with regard to this analysis, given that the comparison with the data was conducted with the impulse approximation in [43, 44]. Notably, in the foregoing study, three dibaryons were claimed to be observed. While it is not the purpose of our discussion to criticize these conclusions, we must point out that a fit to the data with a straight line provides a better \chi^2 value than the one obtained with three dibaryons.

    With regard to the \gamma d\to p n (or pn\to \gamma d ) signals observed in polarization observables in [4951], the following considerations are in order. The cross section for the \gamma d\to pn (pn\to \gamma d) reaction has a clear peak attributed to the \Delta(1232) excitation around E_\gamma=260 MeV [52, 53]. This reaction is similar to the pp\to \pi^+d reaction studied in [19], which develops a triangle singularity. It is easy to conclude based on the procedure adopted in [19] that the \gamma d\to pn reaction is also driven by the same triangle singularity. In the cross section, no trace of " d^*(2380) " can be observed. However, it is well known that polarization observables are sensitive to small terms of the amplitude, which are not present in integrated cross sections [40]. Thus, the combined reaction np\to \pi^+\pi^-d\to \gamma d provides a contribution to the np\to \gamma d reaction through an intermediate state with a peak in the " d^*(2380) " region. As already reported in [40], this small amplitude can emerge in polarization observables, justifying the observation reported in [4951]. However, this cannot be considered a proof of the existence of a dibaryon, given that it will occur regardless of the reason for the pn(I=0)\to\pi^+\pi^-d peak.

    viii) The next point is also somewhat illustrative. The authors claim that the cross section for the pn (I=I) \to \pi^+ \pi^- d reaction obtained based on our approach should be approximately 4 times larger than the one for pn(I=0) \to \pi^+ \pi^- d. However, it is approximately 10 times smaller according to the experiments. This can be rather easily explained. As mentioned before, the two step process has two amplitudes: pn (I=0) \to \pi^- pp followed by pp \to \pi^+ d and pn (I=0) \to \pi^+ nn followed by nn \to \pi^- d . For equal momenta of the pions, both amplitudes sum and produce an enhancement in the cross section. By contrast, for I=1 , we have pn (I=1) \to \pi^- pp followed by pp \to \pi^+ d and pn (I=1) \to \pi^+ nn followed by nn \to \pi^- d . However, in this case, both amplitudes cancel exactly. We have proved this analytically; however, such an explanation is the only possible one because the two pions are in state I=1 , implying a p-wave, and therefore, they cannot move together.

    ix) Finally, the comment regarding the Argand plot has an easy explanation. Given that the final step of our mechanism contains a triangle singularity, it creates a structure similar to that of a normal resonance, as discussed in a paper by the COMPASS collaboration [25].

    We thank Luis Alvarez Ruso for a careful reading of the paper and useful suggestions. R. M. acknowledges support from the CIDEGENT program with Ref. CIDEGENT/2019/015 and from the spanish national grant PID2019-106080GB-C21. The work of N. I. was partly supported by JSPS Overseas Research Fellowships and JSPS KAKENHI Grant Number JP19K14709. This work is also partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P B and No. FIS2017-84038-C2-2-P B. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 824093 for the STRONG-2020 project.

    One should not attribute this shape to the Roper excitation as assumed in [12, 13]. We have seen that the Roper excitation grows smoothly monotonically around this energy region (see also Fig. 1 of Ref. [13]) and there are many other mechanisms contributing to the amplitude with cancellations among them.

    [1] M. Bashkanov et al., Phys. Rev. Lett. 102, 052301 (2009), arXiv:0806.4942 doi: 10.1103/PhysRevLett.102.052301
    [2] P. Adlarson et al. (WASA-at-COSY), Phys. Rev. Lett. 106, 242302 (2011), arXiv:1104.0123 doi: 10.1103/PhysRevLett.106.242302
    [3] P. Adlarson et al. (WASA-at-COSY), Phys. Lett. B 721, 229 (2013), arXiv:1212.2881 doi: 10.1016/j.physletb.2013.03.019
    [4] H. Clement and T. Skorodko, Chin. Phys. C 45, 022001 (2021), arXiv:2008.07200 doi: 10.1088/1674-1137/abcd8e
    [5] G. Faldt and C. Wilkin, Phys. Lett. B 701, 619 (2011), arXiv:1105.4142 doi: 10.1016/j.physletb.2011.06.054
    [6] M. Albaladejo and E. Oset, Phys. Rev. C 88, 014006 (2013), arXiv:1304.7698 doi: 10.1103/PhysRevC.88.014006
    [7] L. Alvarez-Ruso, E. Oset, and E. Hernandez, Nucl. Phys. A 633, 519 (1998), arXiv:nucl-th/9706046 doi: 10.1016/S0375-9474(98)00126-2
    [8] I. Bar-Nir et al., Nucl. Phys. B 54, 17 (1973) doi: 10.1016/0550-3213(73)90062-X
    [9] D. O. Riska, M. Brack, and W. Weise, Phys. Lett. B 61, 41 (1976) doi: 10.1016/0370-2693(76)90556-6
    [10] A. M. Green and J. A. Niskanen, Nucl. Phys. A 271, 503 (1976) doi: 10.1016/0375-9474(76)90258-X
    [11] M. Brack, D. O. Riska, and W. Weise, Nucl. Phys. A 287, 425 (1977) doi: 10.1016/0375-9474(77)90055-0
    [12] P. Adlarson et al. (WASA-at-COSY), Phys. Lett. B 774, 599 (2017), [Erratum: Phys. Lett. B 806, 135555 (2020)], arXiv: 1702.07212
    [13] H. Clement and T. Skorodko (2020), 2010.09217
    [14] L. D. Landau, Nucl. Phys. 13, 181 (1960), [Sov. Phys. JETP 10(1), 45 (1959); Zh. Eksp. Teor. Fiz. 37(1), 62 (1959)]
    [15] M. Bayar, F. Aceti, F.-K. Guo et al., Phys. Rev. D 94, 074039 (2016), arXiv:1609.04133 doi: 10.1103/PhysRevD.94.074039
    [16] C. Richard-Serre, W. Hirt, D. F. Measday et al., Nucl. Phys. B 20, 413 (1970) doi: 10.1016/0550-3213(70)90329-9
    [17] S. Coleman and R. E. Norton, Nuovo Cim. 38, 438 (1965) doi: 10.1007/BF02750472
    [18] P. W. F. Alons, R. D. Bent, J. S. Conte et al., Nucl. Phys. A 480, 413 (1988) doi: 10.1016/0375-9474(88)90458-7
    [19] N. Ikeno, R. Molina, and E. Oset, Phys. Rev. C 104, 014614 (2021), arXiv:2103.01712 doi: 10.1103/PhysRevC.104.014614
    [20] M. G. Albrow, S. Andersson-Almehed, B. Bosnjakovic et al., Phys. Lett. B 34, 337 (1971) doi: 10.1016/0370-2693(71)90619-8
    [21] T. Ueda, Y. Ikegami, and K. Tada, Few Body Syst. 18, 133 (1995) doi: 10.1007/s006010050007
    [22] A. Gal and H. Garcilazo, Nucl. Phys. A 928, 73 (2014), arXiv:1402.3171 doi: 10.1016/j.nuclphysa.2014.02.019
    [23] Z.-T. Lu, H.-Y. Jiang, and J. He, Phys. Rev. C 102, 045202 (2020), arXiv:2007.15878 doi: 10.1103/PhysRevC.102.045202
    [24] H. Huang, X. Zhu, J. Ping et al., Phys. Rev. C 98, 034001 (2018), arXiv:1805.04873 doi: 10.1103/PhysRevC.98.034001
    [25] G. D. Alexeev et al. (COMPASS), Phys. Rev. Lett. 127, 082501 (2021), arXiv:2006.05342 doi: 10.1103/PhysRevLett.127.082501
    [26] C. Itzykson and J. B. Zuber, Quantum Field Theory, International Series In Pure and Applied Physics (McGrawHill, New York, 1980), ISBN 978-0-486-44568-7
    [27] F. Mandl and G. Shaw, Quantum Field Theory (1985)
    [28] J. A. Oller and E. Oset, Nucl. Phys. A 620, 438 (1997), [Erratum: Nucl. Phys. A 652, 407-409 (1999)], arXiv: hep-ph/9702314
    [29] E. Oset and A. Ramos, Nucl. Phys. A 635, 99 (1998), arXiv:nucl-th/9711022 doi: 10.1016/S0375-9474(98)00170-5
    [30] J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263 (2001), arXiv:hep-ph/0011146 doi: 10.1016/S0370-2693(01)00078-8
    [31] A. Martinez Torres, K. P. Khemchandani, and E. Oset, Phys. Rev. C 77, 042203 (2008), arXiv:0706.2330 doi: 10.1103/PhysRevC.77.042203
    [32] A. Martinez Torres, K. P. Khemchandani, L. S. Geng et al., Phys. Rev. D 78, 074031 (2008), arXiv:0801.3635 doi: 10.1103/PhysRevD.78.074031
    [33] L. G. Dakhno et al., Phys. Lett. B 114, 409 (1982) doi: 10.1016/0370-2693(82)90081-8
    [34] J. Bystricky, J.Phys. France 48, 1901 (1987) doi: 10.1051/jphys:0198700480110190100
    [35] M. S. Guenther (A2), PoS Hadron 2017, 051 (2018)
    [36] M. Bashkanov, H. Clement, and T. Skorodko (2021), arXiv: 2106.00494
    [37] F.-K. Guo, X.-H. Liu, and S. Sakai, Prog. Part. Nucl. Phys. 112, 103757 (2020), arXiv:1912.07030 doi: 10.1016/j.ppnp.2020.103757
    [38] R. A. Arndt, I. I. Strakovsky, R. L. Workman et al., Phys. Rev. C 48, 1926 (1993) doi: 10.1103/PhysRevC.48.1926
    [39] C.-H. Oh, R. A. Arndt, I. I. Strakovsky et al., Phys. Rev. C 56, 635 (1997), arXiv:nucl-th/9702006 doi: 10.1103/PhysRevC.56.635
    [40] L. Roca, Nucl. Phys. A 748, 192 (2005), arXiv:nuclth/0407049 doi: 10.1016/j.nuclphysa.2004.10.028
    [41] T. Ishikawa et al., Phys. Lett. B 772, 398 (2017), arXiv:1610.05532 doi: 10.1016/j.physletb.2017.04.010
    [42] T. Ishikawa et al., Phys. Lett. B 789, 413 (2019), arXiv:1805.08928 doi: 10.1016/j.physletb.2018.12.050
    [43] A. Fix and H. Arenhoevel, Eur. Phys. J. A 25(1), 115 (2005), arXiv:nucl-th/0503042 doi: 10.1140/epja/i2005-10067-5
    [44] M. Egorov and A. Fix, Nucl. Phys. A 933, 104 (2015), arXiv:1412.0089 doi: 10.1016/j.nuclphysa.2014.10.002
    [45] T. Ishikawa, H. Fujimura, H. Fukasawa et al., Phys. Rev. C 104, L052201 (2021) doi: 10.1103/PhysRevC.104.L052201
    [46] T. Ishikawa et al., Phys. Rev. C 105, 045201 (2022) doi: https://journals.aps.org/prc/abstract/10.1103/PhysRevC.105.045201
    [47] A. Martinez Torres, K. P. Khemchandani, and E. Oset, Phys. Rev. C 107, 025202 (2023) doi: 10.1103/PhysRevC.107.025202
    [48] T. C. Jude et al., Phys. Lett. B 832, 137277 (2022), arXiv:2202.08594 doi: 10.1016/j.physletb.2022.137277
    [49] H. Ikeda et al., Phys. Rev. Lett. 42, 1321 (1979) doi: 10.1103/PhysRevLett.42.1321
    [50] M. Bashkanov et al., Phys. Lett. B 789, 7 (2019) doi: 10.1016/j.physletb.2018.12.026
    [51] M. Bashkanov et al. (A2), Phys. Rev. Lett. 124, 132001 (2020), arXiv:1911.08309 doi: 10.1103/PhysRevLett.124.132001
    [52] P. Rossi, E. De Sanctis, P. Levi Sandri et al., Phys. Rev. C 40, 2412 (1989) doi: 10.1103/PhysRevC.40.2412
    [53] C. S. Whisnant, Phys. Rev. C 73, 044005 (2006) doi: 10.1103/PhysRevC.73.044005
  • [1] M. Bashkanov et al., Phys. Rev. Lett. 102, 052301 (2009), arXiv:0806.4942 doi: 10.1103/PhysRevLett.102.052301
    [2] P. Adlarson et al. (WASA-at-COSY), Phys. Rev. Lett. 106, 242302 (2011), arXiv:1104.0123 doi: 10.1103/PhysRevLett.106.242302
    [3] P. Adlarson et al. (WASA-at-COSY), Phys. Lett. B 721, 229 (2013), arXiv:1212.2881 doi: 10.1016/j.physletb.2013.03.019
    [4] H. Clement and T. Skorodko, Chin. Phys. C 45, 022001 (2021), arXiv:2008.07200 doi: 10.1088/1674-1137/abcd8e
    [5] G. Faldt and C. Wilkin, Phys. Lett. B 701, 619 (2011), arXiv:1105.4142 doi: 10.1016/j.physletb.2011.06.054
    [6] M. Albaladejo and E. Oset, Phys. Rev. C 88, 014006 (2013), arXiv:1304.7698 doi: 10.1103/PhysRevC.88.014006
    [7] L. Alvarez-Ruso, E. Oset, and E. Hernandez, Nucl. Phys. A 633, 519 (1998), arXiv:nucl-th/9706046 doi: 10.1016/S0375-9474(98)00126-2
    [8] I. Bar-Nir et al., Nucl. Phys. B 54, 17 (1973) doi: 10.1016/0550-3213(73)90062-X
    [9] D. O. Riska, M. Brack, and W. Weise, Phys. Lett. B 61, 41 (1976) doi: 10.1016/0370-2693(76)90556-6
    [10] A. M. Green and J. A. Niskanen, Nucl. Phys. A 271, 503 (1976) doi: 10.1016/0375-9474(76)90258-X
    [11] M. Brack, D. O. Riska, and W. Weise, Nucl. Phys. A 287, 425 (1977) doi: 10.1016/0375-9474(77)90055-0
    [12] P. Adlarson et al. (WASA-at-COSY), Phys. Lett. B 774, 599 (2017), [Erratum: Phys. Lett. B 806, 135555 (2020)], arXiv: 1702.07212
    [13] H. Clement and T. Skorodko (2020), 2010.09217
    [14] L. D. Landau, Nucl. Phys. 13, 181 (1960), [Sov. Phys. JETP 10(1), 45 (1959); Zh. Eksp. Teor. Fiz. 37(1), 62 (1959)]
    [15] M. Bayar, F. Aceti, F.-K. Guo et al., Phys. Rev. D 94, 074039 (2016), arXiv:1609.04133 doi: 10.1103/PhysRevD.94.074039
    [16] C. Richard-Serre, W. Hirt, D. F. Measday et al., Nucl. Phys. B 20, 413 (1970) doi: 10.1016/0550-3213(70)90329-9
    [17] S. Coleman and R. E. Norton, Nuovo Cim. 38, 438 (1965) doi: 10.1007/BF02750472
    [18] P. W. F. Alons, R. D. Bent, J. S. Conte et al., Nucl. Phys. A 480, 413 (1988) doi: 10.1016/0375-9474(88)90458-7
    [19] N. Ikeno, R. Molina, and E. Oset, Phys. Rev. C 104, 014614 (2021), arXiv:2103.01712 doi: 10.1103/PhysRevC.104.014614
    [20] M. G. Albrow, S. Andersson-Almehed, B. Bosnjakovic et al., Phys. Lett. B 34, 337 (1971) doi: 10.1016/0370-2693(71)90619-8
    [21] T. Ueda, Y. Ikegami, and K. Tada, Few Body Syst. 18, 133 (1995) doi: 10.1007/s006010050007
    [22] A. Gal and H. Garcilazo, Nucl. Phys. A 928, 73 (2014), arXiv:1402.3171 doi: 10.1016/j.nuclphysa.2014.02.019
    [23] Z.-T. Lu, H.-Y. Jiang, and J. He, Phys. Rev. C 102, 045202 (2020), arXiv:2007.15878 doi: 10.1103/PhysRevC.102.045202
    [24] H. Huang, X. Zhu, J. Ping et al., Phys. Rev. C 98, 034001 (2018), arXiv:1805.04873 doi: 10.1103/PhysRevC.98.034001
    [25] G. D. Alexeev et al. (COMPASS), Phys. Rev. Lett. 127, 082501 (2021), arXiv:2006.05342 doi: 10.1103/PhysRevLett.127.082501
    [26] C. Itzykson and J. B. Zuber, Quantum Field Theory, International Series In Pure and Applied Physics (McGrawHill, New York, 1980), ISBN 978-0-486-44568-7
    [27] F. Mandl and G. Shaw, Quantum Field Theory (1985)
    [28] J. A. Oller and E. Oset, Nucl. Phys. A 620, 438 (1997), [Erratum: Nucl. Phys. A 652, 407-409 (1999)], arXiv: hep-ph/9702314
    [29] E. Oset and A. Ramos, Nucl. Phys. A 635, 99 (1998), arXiv:nucl-th/9711022 doi: 10.1016/S0375-9474(98)00170-5
    [30] J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263 (2001), arXiv:hep-ph/0011146 doi: 10.1016/S0370-2693(01)00078-8
    [31] A. Martinez Torres, K. P. Khemchandani, and E. Oset, Phys. Rev. C 77, 042203 (2008), arXiv:0706.2330 doi: 10.1103/PhysRevC.77.042203
    [32] A. Martinez Torres, K. P. Khemchandani, L. S. Geng et al., Phys. Rev. D 78, 074031 (2008), arXiv:0801.3635 doi: 10.1103/PhysRevD.78.074031
    [33] L. G. Dakhno et al., Phys. Lett. B 114, 409 (1982) doi: 10.1016/0370-2693(82)90081-8
    [34] J. Bystricky, J.Phys. France 48, 1901 (1987) doi: 10.1051/jphys:0198700480110190100
    [35] M. S. Guenther (A2), PoS Hadron 2017, 051 (2018)
    [36] M. Bashkanov, H. Clement, and T. Skorodko (2021), arXiv: 2106.00494
    [37] F.-K. Guo, X.-H. Liu, and S. Sakai, Prog. Part. Nucl. Phys. 112, 103757 (2020), arXiv:1912.07030 doi: 10.1016/j.ppnp.2020.103757
    [38] R. A. Arndt, I. I. Strakovsky, R. L. Workman et al., Phys. Rev. C 48, 1926 (1993) doi: 10.1103/PhysRevC.48.1926
    [39] C.-H. Oh, R. A. Arndt, I. I. Strakovsky et al., Phys. Rev. C 56, 635 (1997), arXiv:nucl-th/9702006 doi: 10.1103/PhysRevC.56.635
    [40] L. Roca, Nucl. Phys. A 748, 192 (2005), arXiv:nuclth/0407049 doi: 10.1016/j.nuclphysa.2004.10.028
    [41] T. Ishikawa et al., Phys. Lett. B 772, 398 (2017), arXiv:1610.05532 doi: 10.1016/j.physletb.2017.04.010
    [42] T. Ishikawa et al., Phys. Lett. B 789, 413 (2019), arXiv:1805.08928 doi: 10.1016/j.physletb.2018.12.050
    [43] A. Fix and H. Arenhoevel, Eur. Phys. J. A 25(1), 115 (2005), arXiv:nucl-th/0503042 doi: 10.1140/epja/i2005-10067-5
    [44] M. Egorov and A. Fix, Nucl. Phys. A 933, 104 (2015), arXiv:1412.0089 doi: 10.1016/j.nuclphysa.2014.10.002
    [45] T. Ishikawa, H. Fujimura, H. Fukasawa et al., Phys. Rev. C 104, L052201 (2021) doi: 10.1103/PhysRevC.104.L052201
    [46] T. Ishikawa et al., Phys. Rev. C 105, 045201 (2022) doi: https://journals.aps.org/prc/abstract/10.1103/PhysRevC.105.045201
    [47] A. Martinez Torres, K. P. Khemchandani, and E. Oset, Phys. Rev. C 107, 025202 (2023) doi: 10.1103/PhysRevC.107.025202
    [48] T. C. Jude et al., Phys. Lett. B 832, 137277 (2022), arXiv:2202.08594 doi: 10.1016/j.physletb.2022.137277
    [49] H. Ikeda et al., Phys. Rev. Lett. 42, 1321 (1979) doi: 10.1103/PhysRevLett.42.1321
    [50] M. Bashkanov et al., Phys. Lett. B 789, 7 (2019) doi: 10.1016/j.physletb.2018.12.026
    [51] M. Bashkanov et al. (A2), Phys. Rev. Lett. 124, 132001 (2020), arXiv:1911.08309 doi: 10.1103/PhysRevLett.124.132001
    [52] P. Rossi, E. De Sanctis, P. Levi Sandri et al., Phys. Rev. C 40, 2412 (1989) doi: 10.1103/PhysRevC.40.2412
    [53] C. S. Whisnant, Phys. Rev. C 73, 044005 (2006) doi: 10.1103/PhysRevC.73.044005
  • 加载中

Figures(2) / Tables(1)

Get Citation
R. Molina, Natsumi Ikeno and Eulogio Oset. Sequential single pion production explaining the dibaryon "d*(2380)" peak[J]. Chinese Physics C. doi: 10.1088/1674-1137/acb0b7
R. Molina, Natsumi Ikeno and Eulogio Oset. Sequential single pion production explaining the dibaryon "d*(2380)" peak[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acb0b7 shu
Milestone
Received: 2022-11-08
Article Metric

Article Views(1316)
PDF Downloads(32)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Sequential single pion production explaining the dibaryon "d*(2380)" peak

  • 1. Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, Spain
  • 2. Department of Agricultural, Life and Environmental Sciences, Tottori University, Tottori 680-8551, Japan

Abstract: In this study, we investigate the two step sequential one pion production mechanism, that is, np(I=0)\to \pi^-pp followed by the fusion reaction pp\to \pi^+d , to describe the np\to \pi^+\pi^-d reaction with \pi^+\pi^- in state I=0 . In this reaction, a narrow peak identified with a " d(2380) " dibaryon has been previously observed. We discover that the second reaction step pp\to \pi^+d is driven by a triangle singularity that determines the position of the peak of the reaction and the high strength of the cross section. The combined cross section of these two mechanisms produces a narrow peak with a position, width, and strength, that are compatible with experimental observations within the applied approximations made. This novel interpretation of the peak accomplished without invoking a dibaryon explains why this peak has remained undetected in other reactions.

    HTML

    I.   INTRODUCTION
    • The np\to \pi^0\pi^0d reaction presents a sharp peak around 2370 MeV with a narrow width of approximately 70 MeV; this peak is also observed in the pp\to \pi^+\pi^-d reaction but with approximately double the strength [13]. In the absence of a conventional reaction mechanism that can explain the occurrence of these peaks, they have been interpreted as signals of a dibaryon labeled as d^*(2380) . Based on this hypothesis, other features observed in π production experiments and NN phase shifts have also been interpreted (see [4] for a recent review). Notably, the narrow peak in np\to \pi\pi d affects the inelasticity of the NN phase shifts and is expected to influence these NN phase shifts, as emphasized in [5, 6]. Previously, several mechanisms of the two pion production leading to \pi\pi d have been studied [13] based on the model reported in [7] for NN\to NN\pi\pi ; this system features double Δ production with subsequent \Delta\to \pi N decay or N^*(1440) production with the decay of N^* to N\pi\pi or N^*\to \pi\Delta(N\pi) . In all these cases, the resulting np particles are fused into a deuteron. However, the results of these calculations lead to cross sections with smaller strengths compared to the peak of the np\to \pi^0\pi^0d reaction and produce no peaks with the energy of the observed peak. Such conclusions have already been drawn in a previous study [8], and we explicitly recalculated the cross sections based on these mechanisms to corroborate all previous findings. Interestingly, in the same study [8], a peak with poor statistics, already visible for the np\to \pi^+\pi^-d reaction, was explained using a different mechanism, namely two step sequential π production np\to pp\pi^- followed by pp\to \pi^+d . The cross section for the np\to \pi^+\pi^-d reaction was evaluated by factorizing the cross sections for the two latter reactions based on an "on-shell" approach, which, however, required further tests for accuracy. Notably, such a mechanism has not been further invoked considering the availability of new improved data on the np\to \pi^+\pi^-d and np\to \pi^0\pi^0d reactions [13].

      Meanwhile, the time reversal reaction involved in pp\to \pi^+d , that is, \pi^+ absorption in a deuteron, \pi^+d\to pp , has been studied in the past [911]. Such investigations have revealed that this reaction has a neat peak corresponding to the Δ excitation. Combining the results of previous studies [911] with the idea reported in [8] on the np\to \pi^+\pi^-d reaction, the mechanism underlying the np\to \pi^+\pi^-d reaction can be expressed diagrammatically, as shown in Fig. 1.

      Figure 1.  Two step mechanism for np\to \pi^+\pi^-d , as suggested in [8], with explicit Δ excitation in the last step of pp\to \pi^+d , as reported in [911]. The mechanism with the nn intermediate state is also considered.

      After several years, the availability of more refined data and new theoretical developments suggest the need to revisit this issue based on the same idea:

      1) Recent data on np\to \pi^+\pi^-d and np\to \pi^0\pi^0 d have excellent precision [14].

      2) The np\to \pi^0\pi^0d reaction involves \pi^0\pi^0 in isospin state I=0 , and hence, the inital np state must also be in state I=0 .

      In [3], the np\to \pi^+\pi^-d reaction was split into I=0 and I=1 states, and as expected, the same peak visible for the np\to \pi^0\pi^0 d reaction was observed for the np\mathrm{(I=0)}\to \pi^+\pi^-d reaction with approximately double the strength. This implies that in the np\to \pi^-pp reaction, the first step of the sequential single pion production mechanism, i.e., the inital np state, is also in state I=0 . Recently, the first step, shown in Fig. 1 as np\to \pi^-pp with np in I=0 , has been singled out with a relatively good precision [12] (see revision about normalization in [12, 13]).

      3) New developments concerning triangle singularities [14] facilitate our understanding of the high strength of the pp\to \pi^+d reaction with a triangle singularity, as depicted in the triangle diagram shown in the last section of Fig. 1. This corresponds to the simultaneous occurrence of Δ and two nucleons on shell and collinear. A simplification of the formalism on the triangle singularities, as accomplished in [15], allows immediate localization of the peak of the pp\to \pi^+d cross section based on Eq. (18) presented in [15]; here, the mass d is considered slightly unbound to determine a solution to the given equation. One predicts that a peak of the cross section should appear around M_{\mathrm{inv}}(pp)\sim 2179 MeV, which is very close to the location of the peak found in the experiment reported in [16]. The Coleman-Norton theorem [17] clarifies this case to visualize the process. The theorem states that a triangle singularity appears if the process visualized in the triangle diagram can occur at the classical level. In our case, this would occur as follows: the pp system produces \Delta N back to back in the pp rest frame; Δ decays into a \pi^+ along the direction of Δ and N' along the opposite direction, which is along the direction of N. Following this, N' moves faster than N (encoded by Eq. (18) in [15]) and approaches N to fuse into a deuteron. The fusion of the two nucleons into a deuteron occurs naturally when the discussed mechanism has a triangle singularity, producing a neat peak and a cross section that are notably large compared with those of typical fusion reactions [18]. We reanalyzed the pp\to \pi^+d reaction based on this new perspective [19]; however, corresponding details are nonessential for the derivation of the np\to\pi^+\pi^-d cross section, which, as reported in [8], relies on experimental cross sections, using the new np {(I=0)}\to \pi^-pp cross section [12, 13] and the data available for pp\to \pi^+d [16]. We also improved the on shell approach adopted in [8]. It is worth mentioning that while the mechanism for pp\to\pi^+d described in [911] could not be identified as a triangle singularity, the authors of [11] demonstrated that the cross section blew up when the Δ width was set to zero, a characteristic of the triangle singularity. In [19], the authors demonstrated that the dominant term in pp\to \pi^+d is the partial wave ^1D_2\,(^{2S+1}L_J) , which is in agreement with the experimental observations reported in [20]. Along similar lines, one may trace back J^P=1^+,3^+ for the d\pi^+\pi^- system, with some preference for 3^+ and ^3D_3 in the initial np system, i.e., the preferred quantum numbers associated with the d^*(2380) peak [4].

      It is also worth mentioning that the dominance of the ^1D_2 partial wave leads to a structure indicative of a resonance in the pp\to \pi d reaction [21]: a dibaryon other than " d^*(2380) ." Theoretical groups also suggest bound states of N\Delta , three body \pi NN [22], or N\Delta [23] to explain the peak produced by this reaction; however, in [24], the authors could not prove adequate binding. In fact, as presented in [25], the Argand plots corresponding to a resonance and the triangle singularity are similar. We strictly follow the basic rule stating that if one phenomenon can be explained using conventional, well established facts, this interpretation should be favored over less conventional ones. Previous studies [911, 19] explaining the pp\to \pi d reaction on conventional grounds have proved that a new dibaryon is not essential to explain the foregoing reaction.

    II.   FORMALISM
    • The derivation of the np\to \pi^+\pi^-d cross section follows the steps involved in the optical theorem [26]. Let us denote t as the amplitude for the isoscalar np {(I=0)}\to \pi^-pp reaction, t' as the amplitude for pp\to\pi^+d , and t^{\prime\prime} as the amplitude for np {(I=0)}\to \pi^+\pi^-d . The differential cross section for the isoscalar np {(I=0)}\to \pi^-pp reaction is given by

      \frac{{\rm d}\sigma_{np\to \pi^-pp}^I}{{\rm d} M_\mathrm{inv}(p_1p_1')}=\frac{1}{4ps}(2 M_N)^4\frac{1}{16\pi^3}p_\pi\tilde{p}_1|\bar{t}|^2\frac{1}{2},

      (1)

      where \sigma^I stands for the isoscalar cross section, \sqrt{s} is the center-of-mass (CM) energy of the inital np state, M_{\mathrm{inv}}(p_1p'_1) is the invariant mass of the final two protons involved in this reaction, p is the CM momentum of the inital n or p particles, p_\pi is the pion momentum in the np rest frame, and \tilde{p}_1 is the momentum of the final protons in the pp rest frame. We use the (2M_N)^4 factor of the fermion field normalization for nucleons following the formalism proposed by Mandl and Shaw [27]. The magnitude |\bar{t}|^2 indicates the angle averaged value of |t|^2 , and the factor \dfrac{1}{2} considers the identity of the two final protons.

      Similarly, the cross section for pp\to \pi^+d in the second part of the diagram of Fig. 1 is given by

      \sigma_{pp\to \pi^+d}=\frac{1}{16\pi M_{\mathrm{inv}}^2(p_1p_1')}\frac{p_\pi'}{\tilde{p}_1}|\bar{t}\,'|^2 (2M_N)^2(2M_d) ,

      (2)

      where p'_\pi is the \pi^+ momentum in the pp rest frame, and |\bar{t}'|^2 stands for angle averaged |t'|^2 . We choose to normalize the deuteron field in the form of nucleons, and we added a factor 2M_d (which disappears from the final formulas). Meanwhile, the amplitude for the np\to\pi^-\pi^+d process in Fig. 1 is given by

      \begin{aligned}[b] -{\rm i} t^{\prime\prime}=&\frac{1}{2}\int\frac{{\rm d}^4p_1}{(2\pi)^4}\frac{(2M_N)^2}{2E_N(p_1)2E_N(p'_1)}\frac{\rm i}{p_1^0-E_N(p_1)+{\rm i}\epsilon}\\&\times\frac{\rm i}{\sqrt{s}-p_1^0-\omega_\pi-E_N(p'_1)+{\rm i}\epsilon}\,(-{\rm i})t\,(-{\rm i})t' . \end{aligned}

      (3)

      Here, the factor \dfrac{1}{2} accounts for the intermediate propagator of two identical particles. In the d^4p_1 integrations, t and t' would be off shell. In [8], the pion and two protons in the intermediate state were assumed to be on shell, and t and t' were used with the on shell variables. However, theoretical advances [28] allow us to proceed beyond this approximation. Indeed, the chiral unitary approaches reported in [28] for meson-meson interactions or in [29] for meson-baryon interactions factorize the vertices on-shell and perform a loop integral of the two intermediate states. A different justification has been provided in [30], wherein a dispersion relation is derived for the inverse of the hadron-hadron scattering amplitude. Another justification has been provided in [31, 32], wherein with the aid of chiral Lagrangians, the authors demonstrated that the off shell parts of the amplitudes appearing in the approach canceled with counterterms provided by the same theory. This implies that in Eq. (3), we can take tt' outside the dp_1^0 integral with the on-shell values and evaluate the remainder of the integral.

      On performing the p_1^0 integration analytically with Cauchy's residues, we obtain

      \begin{aligned}[b] t^{\prime\prime}=&\frac{1}{2}\int\frac{{\rm d}^3p_1}{(2\pi)^3}\frac{(2M_N)^2}{2E_N(p_1)2E_N(p'_1)}\\&\times\frac{tt'}{\sqrt{s}-E_N(p_1)-E_N(p'_1)-\omega_\pi+{\rm i}\epsilon} , \end{aligned}

      (4)

      where \vec{p}_1 , \vec{p}_1\,' denote the momenta of the intermediate pp particles in Fig. 1, and \omega_\pi is the \pi^- energy. The t,t' amplitudes are Lorentz invariant, and we choose to evaluate the \int \dfrac{{\rm d}^3p_1}{2E_1(p_1)} integral in the pp rest frame, where |\vec{p}\,'_1|= |\vec{p}_1| and \sqrt{s}-\omega_\pi becomes the invariant mass of the two protons. This integral is logarithmically divergent and requires regularization. The result smoothly depends on a cut off p_{1,\mathrm{max}} for |\vec{p}_1| , which is used to regularize the d^3p_1 integral. We set some values for p_{1,\mathrm{max}} within a reasonable range. Despite this, we anticipate that the on shell part given by Eq. (5) below provides the largest contribution to the t'' amplitude. Given that \tilde{p}_1= 552 MeV/c for M_{{\rm inv}(p_1p_1')}= 2179 MeV, the triangle singularity would appear for t' and Δ with a zero width or a pronounced peak when the width is considered; thus, values of p_{1,\mathrm{max}} around 700-800 MeV seem reasonable.

      The on-shell approximation used in [8] allows one to express the cross section for np\to\pi^+\pi^-d in terms of the cross sections for np {(I=0)}\to \pi^-pp and pp\to\pi^+d . This approximation is obtained in the present formalism by considering the imaginary part of the two nucleon propagator:

      \begin{aligned} & \frac{1}{M_{\mathrm{inv}}(p_1p_1')-2E_N(p_1)+{\rm i}\epsilon}\\\equiv& {\cal P}\left[\frac{1}{M_{\mathrm{inv}}(p_1p_1')-2E_N(p_1)}\right]-{\rm i}\pi\delta(M_{\mathrm{inv}}(p_1p_1')-2E_N(p_1)).\\ \end{aligned}

      (5)

      Following this, we have

      t^{\prime\prime}_\mathrm{on}=-{\rm i}\frac{1}{2}\frac{\tilde{p}_1}{8\pi}\frac{(2M_N)^2}{M_\mathrm{inv}(p_1p_1')}\bar{tt'} ,

      (6)

      where we factorized the angle averaged value of tt' , \bar{tt'} . Using the analogous expression in Eq. (1) for {\rm d}\sigma_{np\to\pi^+\pi^-d}/ {\rm d}M_{\mathrm{inv}}(\pi^+\pi^-) and the on-shell approximation given by Eqs. (6) and (1), we can write the following:

      \begin{aligned}[b] \frac{{\rm d}\sigma_{np\to\pi^+\pi^-d}}{{\rm d}M_{\mathrm{inv}}(\pi^+\pi^-)}=&(2M_N)^2(2M_d)p_d\tilde{p}_\pi\frac{1}{4}\frac{\tilde{p}_1^2}{64\pi^2}\\&\times\frac{1}{M^2_\mathrm{inv}(p_1p_1')}\frac{1}{p_\pi\tilde{p}_1}2|\bar{t}'|^2\frac{{\rm d}\sigma_{np\to\pi^- pp}}{{\rm d}M_\mathrm{inv}(p_1p_1')} , \end{aligned}

      (7)

      where p_d is the deuteron momentum in the original np rest frame, |\bar{t}'|^2 is the angle averaged |t'|^2 , and \tilde{p}_\pi is the pion momentum in the \pi^+\pi^- rest frame. In Eq. (7), we assumed that |\bar{tt'}|^2=|\bar{t}|^2|\bar{t'}|^2. Note that the amplitudes t,t' in [12, 16] have some angular structures, but these are smooth enough to consider this assumption a sensible approximation.

      Next, we use physical arguments to express the np\to \pi^+\pi^-d cross section in the form of an easy compact formula. Note that \pi^0\pi^0 or \pi^+\pi^- in state I=0 , as discussed earlier, require even values for their relative angular momentum l, and when l=0 , \pi^0\pi^0 or the symmetrized (\pi^+\pi^-+\pi^-\pi^+) behave as identical particles, which revert to a Bose enhancement when the two pions move together. Certainly, if these pions are exactly together, we also have the phase space factor \tilde{p}_\pi in the term {\rm d}\sigma/{\rm d} M_\mathrm{inv}(\pi^+\pi^-) of Eq. (7), which makes this distribution null in the two pion threshold; however, some enhancement for small invariant masses is still expected. Our argument is supported by the results reported in [1, 2] for \pi^0\pi^0 (see Fig. 2 in [1] and Fig. 4 in [2]) and in [3] for charged pions, although the natures of I=0 and I=1 in this case slightly distort the mass distribution compared to the clean I=0 \;\pi^0\pi^0 case.

      We could consider certain M_\mathrm{inv}(\pi^+\pi^-) distributions as inputs. However, to make the results as model independent as possible, we set \bar{M}_\mathrm{inv}(\pi^+\pi^-)\sim 2m_\pi+60 MeV, which is not far from the threshold; however, we varied this value to evaluate the dependence on \bar{M}_\mathrm{inv} . The stability of the results with varying values of \bar{M}_{\mathrm{inv}}(\pi\pi) justifies this approximation a posteriori. Thus, we can write

      \frac{{\rm d}\sigma_{np\to\pi^+\pi^-d}}{{\rm d}M_{\mathrm{inv}(\pi^+\pi^-)}}=\sigma_{np\to\pi^+\pi^-d}\delta(M_{\mathrm{inv}}(\pi^+\pi^-)-\bar{M}_{\pi\pi})\ .

      (8)

      The approximation given by Eq. (8) is sufficiently good and provides a more transparent picture of the reason responsible for the appearance of the peak in the np\to \pi^+\pi^-d reaction. In this case, the energy of the two pions is obtained as

      \begin{equation} E_{2\pi}=\frac{s+M^2_\mathrm{inv}(\pi\pi)-M^2_d}{2\sqrt{s}}\ , \end{equation}

      (9)

      and given that both pions move relatively together, we set E_\pi=E_{2\pi}/2 , which allows relating M_\mathrm{inv}(p_1p_1') with \sqrt{s} via

      \begin{array}{*{20}{l}} M^2_\mathrm{inv}(p_1p_1')=(P(np)-p_{\pi^-})^2=s+m^2_\pi-2\sqrt{s}E_\pi \end{array}

      (10)

      and formally

      \begin{aligned}[b] & 2M_{\mathrm{inv}}(p_1p_1'){\rm d} M_\mathrm{inv}(p_1p_1')\\=&-2\sqrt{s}{\rm d} E_\pi=-M_\mathrm{inv}(\pi\pi){\rm d} M_\mathrm{inv}(\pi\pi)\ . \end{aligned}

      (11)

      Using this relationship, we can integrate Eq. (8) with respect to M_\mathrm{inv}(\pi\pi) , and using Eqs. (2) and (7), we obtain

      \begin{equation} \sigma_{np\to\pi^+\pi^-d}=\frac{M_\mathrm{inv}(p_1p_1')}{4\pi}\frac{\sigma_{np\to\pi^-pp}\sigma_{pp\to\pi^+d}}{M_\mathrm{inv}(\pi\pi)}\frac{\tilde{p}_1^2}{p_\pi p'_{\pi}}p_d\tilde{p}_\pi . \end{equation}

      (12)

      Here, one last detail is required. As stated, we considered the two step np {(I=0)}\to\pi^-pp reaction followed by pp\to\pi^+d . An appropriately symmetrized t^{\prime\prime} amplitude requires the addition of np {(I=0)}\to\pi^+nn followed by nn\to \pi^-d . Evidently, considering the isospin, the amplitudes np {(I=0)}\to \pi^-pp and np {(I=0)}\to \pi^+nn are identical up to the phase of \pi^+ ( -1 in our formalism), and the same is true for pp\to\pi^+d and nn\to \pi^-d for the same configuration of the particles. Hence, the product of the amplitudes is the same. If \pi^+ and \pi^- move exactly together, both amplitudes will be identical and will add coherently. However, we observe that the phase space factor \tilde{p}_\pi in Eq. (12) eliminates this contribution. When considering integration over the five degrees of freedom of the three body phase space, the terms are expected to mostly sum incoherently. Consequently, we must multiply Eq. (12) by 2 . Similar arguments can be made with respect to the spin sums and averages. Investigations on the pp\to \pi^+d reaction conducted in [19] indicate the presence of a certain angular dependence on the different spin transitions, and we should expect an incoherent sum over spins. Subsequently, by including the average over initial spins and the sum over final spins in |\bar{t}|^2 , we would be considering the average over spins of the initial np and the sum over spins of the deuteron in our formula, in addition to the intermediate sum over the pp and nn spins.

      Equation (12) still relies on the on-shell approximation expressed by Eq. (6). To consider the off shell effects discussed above, we realized that by factorizing the angular averaged t,t' amplitudes in the term t" of Eq. (4) while maintaining their energy dependence as a function of M_\mathrm{inv}(p_1p_1') , the on shell energy of the intermediate two nucleons (Eq. (4)) presented a remaining structure as the G function of two protons,

      G=\int\frac{{\rm d}^3p_1}{(2\pi)^3}\frac{1}{E_N(p_1)E_N(p_1)}\frac{1}{M_\mathrm{inv}(p_1p'_1)-2E_N(p_1)+{\rm i}\epsilon}\ .\\

      (13)

      Following this, we obtain

      \begin{equation} \mathrm{Im}G=-\frac{1}{2\pi}\frac{\tilde{p}_1}{M_{\mathrm{inv}}(p_1p_1')} \ , \end{equation}

      (14)

      where, as mentioned after Eq. (1), \tilde{p}_1 is the momentum of both protons in their rest frame. The on shell factorization of the terms t and t' outside the G function has been justified in the discussion following Eq. (3) from different perspectives. We proceed along a different direction with respect to the derivation of Eq. (6): we now retain the two terms in Eq. (5) rather than retaining only the imaginary part. In Eq. (6), we considered the imaginary part of the integral of Eq. (4). This is equivalent to taking \mathrm{Im}\,G of Eq. (14) instead of G in the integral of Eq. (13). To revert this approximation and approximately determine the effects of the off shell part of the integral, we replace the following in Eq. (12):

      \left(\frac{\tilde{p}_1}{2\pi M_\mathrm{inv}(p_1p_1')}\right)^2\to |G(M_\mathrm{inv})|^2\ .

      (15)

      The last step in the evaluation of \sigma_{np\to\pi^+\pi^-d} requires the use of experimental data for np(I=0)\to\pi^-pp and pp\to \pi^+d . We directly extracted \sigma_{pp\to \pi^+d} from the experiments reported in [16]. For \sigma_{np\to\pi^-pp} in I=0 , additional analysis is however required. In [33, 34], the isoscalar NN\to\pi NN amplitude was obtained via isospin symmetry from \sigma_{np\to pp\pi^-} and \sigma_{pp\to pp\pi^0} , and relatively precise results were reported in [12] based on improved measurements of these cross sections. In the erratum of [12] and in [13], it has been clarified that the actual value \sigma_{pn(I=0)\to NN\pi} is one half that of the value of \sigma_{NN(I=0)\to NN\pi} reported in [12]. The required cross section is \sigma_{pn(I=0)\to pp\pi^-} . Thus, using isospin symmetry, we can observe that \sigma_{pn(I=0)\to pp\pi^-} , \sigma_{pn(I=0)\to nn\pi^+} , and (\sigma_{pn(I=0) \to pn\pi^0}+\sigma_{pn(I=0)\to np\pi^0 }) are all equal. Following this, we express the relationship among the results of [12, 33, 34] as

      \begin{aligned}[b] \sigma_{np(I=0)\to pp\pi^-}= &\frac{1}{3}\sigma_{np(I=0)\to NN\pi} = \frac{1}{6}\sigma_{NN(I=0)\to NN\pi}\\=&\frac{1}{6}3(2\sigma_{np\to pp \pi^-}-\sigma_{pp\to pp\pi^0}) \end{aligned} .

      (16)

      For the above, we consider the data for \sigma_{np(I=0)\to NN\pi} from Fig. 1 in [13]. Notably, statistical and systematic errors have been considered in [12, 13]. To ensure a realistic fit for the data, we also included systematic errors from the uncertainty given in Eq. (16) when using isospin symmetry. We assumed a typical 5% violation of the isospin in each of the last two terms of Eq. (16) and summed the errors in quadrature. The systematic errors obtained were of the order of 0.5 mb in \sigma_{np(I=0)\to NN\pi} , which were also added in quadrature to the errors reported in [13]. With these errors, we obtained several good fits with reduced \chi^2 , ( \chi^2_r ), smaller than 1 . We selected two of these, one peaking on the lower side of \sqrt{s} and the other on the upper side for the np\mathrm{(I=0)}\to NN\pi cross section, and we parameterized the cross section as given below.

      \sigma_i=\left|\frac{\alpha_i}{\sqrt{s}-\tilde{M}_i+{\rm i}\dfrac{\tilde{\Gamma}}{2}}\right|^2

      (17)

      Here, set I has the following set of parameters: \tilde{M}_1=2326 MeV, \tilde{\Gamma}_1=70 MeV, and \alpha^2_1=2.6\left(\frac{\tilde{\Gamma}_1}{2}\right)^2 mb MeV ^2 ( \chi^2_r= 0.50); set II has the following set of parameters: \tilde{M}_2= 2335 MeV, \tilde{\Gamma}_2=80 MeV, and \alpha^2_2=2.5\left(\frac{\tilde{\Gamma}_2}{2}\right)^2 mb MeV ^2 ( \chi^2_r=0.52 ). The pp\to\pi^+d cross section has accurate data, and we parameterize it as

      \sigma_3=\left|\frac{\alpha_3}{M_\mathrm{inv}(p_1p'_1)-\tilde{M}_3+{\rm i}\frac{\tilde{\Gamma}_3}{2}}\right|^2 ,

      (18)

      with \tilde{M}_3=2165\; {\rm MeV} , \tilde{\Gamma}_3=123.27\; {\rm MeV} , and \alpha^2_3= 3.186\times \left(\frac{\tilde{\Gamma}_3}{2}\right)^2 mb MeV ^2 .

      Based on the above discussions, our final formula on shell can be expressed as

      \sigma_{np\to\pi^+\pi^-d}=\frac{M_{\mathrm{inv}}(p_1p'_1)}{6\pi}\frac{\sigma^I_{np\to NN\pi}\sigma_{pp\to\pi^+d}}{M_{\mathrm{inv}}(\pi\pi)}\frac{\tilde{p}_1^2}{p_\pi p'_\pi}p_d\tilde{p}_\pi,

      (19)

      with \sigma^I_{np\to NN\pi}=\sigma_{np(I=0)\to NN\pi} from [12, 13]. The corresponding results are depicted in Fig. 2.

      Figure 2.  (color online) Plots of \sigma_{np\to\pi^-pp}(I=0) and \sigma_{pp\to\pi^+d} as functions of \sqrt{s} and M_{\mathrm{inv}}(p_1p_1') , respectively, where M_{\mathrm{inv}}(p_1p_1') is evaluated using Eq. (10). The results with \sigma_{np\to\pi^+\pi^-d} in I=0 of Eq. (19) are multiplied by 10 for better comparison. Left: Results for set I. Right: Results for set II. \bar{M}_{\pi\pi}=2m_\pi+60 MeV. Inset: \sigma_{pp\to\pi^+d} as a function of M_{\mathrm{inv}}(p_1p'_1) . Data for pp\to\pi^+d extracted from [16]. Data for np\mathrm{(I=0)}\to\pi NN extracted from Dakhno et al. [33] and WASA-at-COSY ^{(*)} [12, 13], including systematic errors from isospin violation.

      We can observe that the cross sections of pp\to \pi^+d and np (I=0) \to NN\pi overlap around the center of their energy distributions such that their product in Eq. (19) produces a narrow peak around \sqrt{s}=2340 MeV, which is close to the position of the experimental np\to\pi^+\pi^-d peak around 2365 MeV.

      Table 1 summarizes the results obtained using sets I and II for the strength of \sigma_{np\to\pi^+\pi^-d} at the peak, peak position, peak width, varying \bar{M}_{\pi\pi} , and p_{1,\mathrm{max}} for the off shell calculations. The results appear stable with changes in \bar{M}_{\pi\pi} , justifying the use of Eq. (8). Note that the off shell effects resulting from Eq. (15) are small, justifying the on shell approximation used in [8]. The strength at the peak between 0.72-0.96 mb can be considered good compared to the experimental one around 0.5 mb, given the different applied approximations (note: the fits to the np {(I=0)}\to \pi^-pp cross section with systematic errors presenting 20\%-30\% smaller strengths at the peak are still acceptable; hence, such uncertainties in the resulting np\to\pi^+\pi^-d cross section are expected). The peak position from 2332-2345 MeV can also be considered good compared to the approximate peak positions at 2365 MeV, as experimentally reported previously [13, 12]. The narrow width observed in the experiment with 70-75 MeV is also well reproduced by our results in the range of 75–88 MeV.

      \delta\bar{M}_{\pi\pi}/{\rm MeV} p_{1,\mathrm{max}}^{\mathrm{o.s.}}/{\rm MeV}
      Set I 40 60 80 700 800
      strength/mb 0.72 0.76 0.75 0.82 0.95
      position/MeV 2332 2332 2332 2332 2332
      width/MeV 76 76 81 75 75
      Set II
      strength/mb 0.75 0.80 0.80 0.85 0.96
      position/MeV 2342 2345 2345 2343 2342
      width/MeV 86 87 88 87 84

      Table 1.  Values of the peak strength ("strength"), peak position ("position"), and width for intermediate particles on shell (columns with \delta \bar{M}_{\pi\pi} , \bar{M}_{\pi\pi}=2m_\pi+\delta \bar{M}_{\pi\pi} ) and off-shell ("o.s."), where we set p_{1,\mathrm{max}}=700 and 800 MeV and \delta\bar{M}_{\pi\pi}=60 MeV.

      The appeareance of the peak at approximately 25 MeV below the experimental one is not significant given that as discussed in [12], the authors achieved a resolution in \sqrt{s} with a value of approximately 20 MeV; in addition, the pp\to pp\pi^0 and pn\to pp\pi^- cross sections, based on which \sigma_{np\mathrm{(I=0)}\to pp\pi^-} is obtained using Eq. (16) with large cancellations, were measured using data bins of 50 MeV in T_p .

      The derivation involves the use of basic ingredients of dynamics in a skilled manner, resulting in some approximations that are reliant on experimental cross sections. It is remarkable that a narrow peak at approximately the correct position, with a strength and width comparable to those of the experimental peak of np\to\pi^+\pi^-d , appears despite the considered approximations, and the stability of the results allows us to conclude that a peak with the properties of the experimental one associated with the " d^*(2380) " dibaryon is unavoidable according to the evaluated mechanism.

      Based on the fact that the np\to\pi^+\pi^-d reaction is involved in the particular reaction mechanism shown in Fig. 1, with a two step sequential one pion production, it is easy to understand why the narrow peak of the np\to \pi^+\pi^-d reaction cannot be observed for the \gamma d\to \pi^0\pi^0d reaction, despite the same final state [35]. The first reaction is a fusion reaction, with the last step connected to a triangle singularity. The \gamma d\to \pi^0\pi^0d reaction is a coherent reaction; here, d is already present in the initial state, and the reaction mechanisms are drastically different.

    III.   CONCLUSIONS
    • In summary, we investigated the reaction mechanism producing a narrow peak in the np\to \pi^+\pi^-d reaction cross sections without invoking the "dibaryon" resonance. Based on this perspective, it is easy to understand why the peak is not observed in other investigated reactions, despite the fact that the peak contributing to the inelastic channels of pn\to all presents traces in the NN phase shifts, as anticipated in [5, 6] and discussed in [4].

    IV.   DISCUSSION ON OTHER REACTIONS THAT HAVE BEEN CLAIMED TO WITNESS THE " {\boldsymbol{d^*}} \boldsymbol{(2380)} " STATE
    • Previous studies have claimed the occurrence of the d^*(2380) state in other reactions (see review in [4]). Moreover, the results presented in the previous section, posted on arXiv (arXiv: 2102.05575), have been further analyzed in [36], where apparent contradictions with experiments have been reported. In this section, we address the different points raised in [36] to indicate that our approach does not contradict previous experimental results and that the "proofs" presented in favor of the dibaryon hypothesis are unsupported. We follow the points presented in [36] in the discussion below.

      As mentioned in the previous section, studies on the np\to \pi^0\pi^0d and np\to \pi^+\pi^-d reactions [2, 3] have revealed an unexpected narrow peak in the cross section around 2380 MeV. This peak has been attributed to a dibaryon, labeled as d^*(2380) , by an experimental team. It is interesting to note that, based on earlier data pertaining to the reaction, the peak was attributed to a reaction mechanism based on sequential one pion production, that is, np\to \pi^-pp\to \pi^-\pi^+d together with np\to\pi^+nn\to \pi^+\pi^-d , in a study conducted by Bar-Nir et al. [8]. The second step, pp\to\pi^+d , was the object of theoretical investigation in [911], and it has been demonstrated to be driven by \Delta(1232) excitation. We propose a reformulation of the idea of these former works, which has been done in [19] from a Feynman diagrammatic point of view, indicating that the process develops a triangle singularity (TS) [9, 14, 15]. This finding is relevant for the present discussion because it is well known that a TS produces an Argand plot that is similar to the one of an ordinary resonance, even if the origin is a kinematical singularity rather than a genuine physical state [25, 37].

      The idea adopted by Bar-Nir et al. has been revisited in the former reaction, and with some reasonable approximations and experimental data on the np(I=0)\to pp \pi^- and pp\to \pi^+d reactions, a peak could be obtained for the pp\to\pi^+\pi^-d reaction; this peak was in qualitative agreement with previous experimental results with regard to its position, width, and strength. Even with the approximations involved and the resulting qualitative agreement, it is important to note that such an agreement, together with the results reported by Bar-Nir et al., is extremely unlikely to be a mere coincidence and hence offers an alternative explanation of the experimentally observed peak.

      In [36], the authors present a few arguments. We address these arguments as follows.

      i) In point 1), the authors note that the error resulting from the proposed approach increases in the pn (I=0) \to \pi^- pp reaction. The reason for this increase is that the cross section for this reaction is obtained using isospin symmetry with the following relation:

      \qquad\qquad \sigma_{np(I=0) \to pp \pi^-} = ( \sigma_{np \to pp \pi^-} - \sigma_{pp \to pp \pi^0}/2)\ .

      (20)

      However, note that this formula involves extensive cancelations, and the result is ten times smaller than each individual term. Thus, we assumed a 5 % uncertainty in the terms obtained from isospin violation and determined the errors in the results. These correspond to systematic uncertainties that should have been considered by the experimentalists; however, they were ignored, which is why we accounted for these errors in our analysis. In the high energy region of the spectrum, where the cross section decreases and produces the shape of the cross section, the systematic errors are much greater than the statistical ones. Therefore, the manner in which the statistical errors are summed is inconsequential. The magnitude depends on the systematic errors. In any case, the obtained cross sections are not influenced by these errors.

      ii) In point 2), the authors claim that our analysis lacked a precise description of the data. With the approximations mentioned above, we cannot possibly pretend to have achieved precise agreements. However, in our opinion, the fact that for such a complicated reaction, we could qualitatively obtain a peak with the correct energy, width, and strength is an accomplishment.

      iii) For this argument, first, let us note that in a previous study [19] on the pp\to \pi^+d reaction, we proved the dominance of ^1D_2 , as determined experimentally in [38, 39]. Second, the argument states that because in the np(I=0)\to\pi^-pp reaction, the invariant mass of \pi^-p is large, the mass of pp is small, and it only accommodates L=0,L=1 waves and not D-waves that are necessary for the overlap of the two-step mechanism. It is interesting to make this argument more quantitative. Let us consider two situations wherein M(pp) can be easily evaluated. They correspond to the case of M(\pi^-p)\vert_{\mathrm{min}}=m_p+m_{\pi^-} and M(\pi^-p)\vert_{\mathrm{max}}=\sqrt{s}-m_p . In the first case, p and \pi^- move together along a direction opposite to that of the other proton. In the second case, one proton is produced at rest. Note that M(pp) is trivially evaluated in the foregoing two cases, and for an energy of T_p=1200 MeV in Fig. 6 of [12], we find the following:

      a) M(pp) (at M(\pi^-p)\vert_{\mathrm{min}} ) =2239.47 MeV, with an excess energy of the two protons = 362.9 MeV.

      b) M(pp) (at M(\pi^-p)\vert_{\mathrm{max}}) =1920.2 MeV, with an excess energy of 43.65 MeV.

      c) Another situation allowing such easy evaluation also exists. This is at the peak of the distribution around M(\pi^-p)=1370 MeV, where M(\pi^-p) from either of the protons is approximately the same and enhances the contribution in this region. Here, we have

      2M^2(\pi^-p)+M^2(pp) = s+2m_p^2+m_{\pi^-}^2;

      from this, we obtain M(pp)\simeq 1951 MeV corresponding to an excess energy of approximately 75 MeV for the two protons. Assuming a relative distance of r\simeq 2.13 fm for the produced two protons, the same as the radius of the deuteron, which corresponds to the range of pion exchange, we determined that the angular momentum, L\sim r\times p , could reach up to L=6 in case a), L=2 in case b), and L=3 in the most favorable case c).

      In [36], L=2 was already ruled out, and with no calculations, the sequential pion production cross section was deemed to be very small, contradicting the conclusions made in [8]. However, with the dominance of the Roper excitation, as claimed in [36], we could observe that S=0 for both protons is the dominant mode in np(I=0)\to\pi^-pp , as in the second step of [19] for the pp\to \pi^+d reaction, and several L values are allowed. Notably, research along these lines is still underway, given that Roper excitation is not the only ingredient of the np(I=0)\to\pi^-pp reaction.

      iv) We would like to state that point 4) is illustrative. Two independent studies, [5] and [6], have proved the existence of a relationship between the pn (I=0) \to \pi^+ \pi^- d reaction and the one where the np of the deuteron emerge as free states. The existence of the peak for the pn (I=0) \to \pi^+ \pi^- d reaction also consequently produces a peak in pn \to \pi^+ \pi^- n p and related reactions at the same energy. However, this is the case regardless of the reason responsible for the appearance of the peak in the fusion reaction. This is a key point. Notably, to calculate the cross section of the open reactions, the authors of [36] used the results reported in these references and added the contribution to the results of the standard model, which were also obtained from [7]. Despite the fact that the authors of [5, 6] indicated that the new contribution was necessary regardless of the reason for the fusion reaction, the authors of [36] considered the above as an evidence of a dibaryon.

      v) In point 5), the authors state that we do not to calculate differential distributions. This is true; however, we did not need the above to prove our points. At the considered qualitative level, we demonstrated that the distribution had to peak at small invariant masses of the two pions because we had two contributions: pn (I=0) \to \pi^- pp followed by pp \to \pi^+ d , together with pn (I=0) \to \pi^+ nn followed by nn \to \pi^- d . We could prove that when the momenta of the two pions were equal, the two amplitudes were identical, and they summed to produce a Bose enhancement. In this case, the invariant mass of the two pions has the smallest value. This is why the cross section peaks at a low \pi \pi invariant mass.

      vi) Next, the authors claim that the picture proposed by Bar-Nir [8] presented in the former section is not adequate to explain the observed pole in ^3D_3-^3G_3 np partial waves. This statement is incorrect. The NN , I=0 phase shifts are affected by the peak in the pn (I=0) \to \pi^+ \pi^- d reaction because one can have pn(I=0) \to \pi^+ \pi^- d \to pn(I=0) , where \pi^+ \pi^- d is in an intermediate state and contributes to the inelasticities. This will be particularly the case in the quantum numbers preferred by the pn (I=0) \to \pi^+ \pi^- d reaction discussed in our previous papers, in particular the ^3D_3 partial wave (see the discussion at the end of [19]). At the energy of the peak of the pn(I=0) \to \pi^+ \pi^- d reaction, the np \to np amplitude will have an enhanced imaginary part according to the optical theorem. This has an impact on the phase shift at this energy. This can also be stated for most of the reactions claiming to observe the dibaryon. In these reactions, what is actually observed is a consequence of the peak observed in the pn(I=0) \to \pi^+ \pi^- d reaction, regardless of reason for this peak. This is an important point. The peak of the pn(I=0) \to \pi^+ \pi^- d reaction will have effects on numerous observables; however, this does not indicate that the reason for the peak is a dibaryon. Nevertheless, it will have consequences. In fact, the effect of the peak of the pn(I=0)\to \pi^+\pi^-d reaction on the ^3D_3 and ^3G_3 partial waves has already been discussed in [5, 6]. Although it appears that the pn\to \pi^+\pi^-d\to pn process would provide a small contribution to the pn \to pn amplitude, the relatively large strength of the pn \to \pi^+\pi^-d peak makes this two step process not too small, as demonstrated in [5, 6]. It is also worth mentioning that small terms in an amplitude can often emerge more clearly in polarization observables than in direct cross sections, as shown in [40].

      The pole or resonant structure is guaranteed by the triangle singularity of the last step pp\to\pi^+d , as shown in [19]. It is well-known that a triangle singularity produces an Argand plot similar to that of a resonance [25, 37].

      vii) The argument made here is rather weak. The authors mention that " d^*(2380) " has been observed in the \gamma d\to d\pi^0\pi^0 and \gamma d\to pn reactions. In fact, in [41, 42], one can observe a deviation of the experimental cross section from the theoretical results reported by [43, 44] at low photon energies. These calculations are based on the impulse approximation, and the π rescattering terms are neglected. However, the rescattering contributions of pions are important, particularly at low energies because the momentum transfer is shared between two nucleons, and one picks up smaller deuteron momentum components, where the wave function is larger. Thus, concluding that the discrepancies observed between experiments and calculations based on the impulse approximation are attributable to the dibaryon is incorrect.

      Let us further extend this discussion. The authors of [41, 42] also studied the \gamma d\to \pi^0 \eta d reaction [45, 46]. This reaction was throughly studied theoretically in [47], and the authors discovered that the pion rescattering mechanisms were particularly important, and the most striking feature of the reaction, i.e., the shift of the shape of the invariant mass distributions, was well reproduced. In the \gamma d\to \pi^0\eta d reaction, η rescattering had a small effect, and only the π rescattering was relevant. In the \gamma d\to \pi^0\pi^0 d reaction, the two pions could rescatter, making the rescattering mechanism in \gamma d\to \pi^0\pi^0 d even more important than that in \gamma d\to \pi^0\eta d .

      The \gamma d\to \pi^0\pi^0 d reaction was measured more accurately in [48]. The same comments can be made with regard to this analysis, given that the comparison with the data was conducted with the impulse approximation in [43, 44]. Notably, in the foregoing study, three dibaryons were claimed to be observed. While it is not the purpose of our discussion to criticize these conclusions, we must point out that a fit to the data with a straight line provides a better \chi^2 value than the one obtained with three dibaryons.

      With regard to the \gamma d\to p n (or pn\to \gamma d ) signals observed in polarization observables in [4951], the following considerations are in order. The cross section for the \gamma d\to pn (pn\to \gamma d) reaction has a clear peak attributed to the \Delta(1232) excitation around E_\gamma=260 MeV [52, 53]. This reaction is similar to the pp\to \pi^+d reaction studied in [19], which develops a triangle singularity. It is easy to conclude based on the procedure adopted in [19] that the \gamma d\to pn reaction is also driven by the same triangle singularity. In the cross section, no trace of " d^*(2380) " can be observed. However, it is well known that polarization observables are sensitive to small terms of the amplitude, which are not present in integrated cross sections [40]. Thus, the combined reaction np\to \pi^+\pi^-d\to \gamma d provides a contribution to the np\to \gamma d reaction through an intermediate state with a peak in the " d^*(2380) " region. As already reported in [40], this small amplitude can emerge in polarization observables, justifying the observation reported in [4951]. However, this cannot be considered a proof of the existence of a dibaryon, given that it will occur regardless of the reason for the pn(I=0)\to\pi^+\pi^-d peak.

      viii) The next point is also somewhat illustrative. The authors claim that the cross section for the pn (I=I) \to \pi^+ \pi^- d reaction obtained based on our approach should be approximately 4 times larger than the one for pn(I=0) \to \pi^+ \pi^- d. However, it is approximately 10 times smaller according to the experiments. This can be rather easily explained. As mentioned before, the two step process has two amplitudes: pn (I=0) \to \pi^- pp followed by pp \to \pi^+ d and pn (I=0) \to \pi^+ nn followed by nn \to \pi^- d . For equal momenta of the pions, both amplitudes sum and produce an enhancement in the cross section. By contrast, for I=1 , we have pn (I=1) \to \pi^- pp followed by pp \to \pi^+ d and pn (I=1) \to \pi^+ nn followed by nn \to \pi^- d . However, in this case, both amplitudes cancel exactly. We have proved this analytically; however, such an explanation is the only possible one because the two pions are in state I=1 , implying a p-wave, and therefore, they cannot move together.

      ix) Finally, the comment regarding the Argand plot has an easy explanation. Given that the final step of our mechanism contains a triangle singularity, it creates a structure similar to that of a normal resonance, as discussed in a paper by the COMPASS collaboration [25].

    V.   ACKNOWLEDGMENTS
    • We thank Luis Alvarez Ruso for a careful reading of the paper and useful suggestions. R. M. acknowledges support from the CIDEGENT program with Ref. CIDEGENT/2019/015 and from the spanish national grant PID2019-106080GB-C21. The work of N. I. was partly supported by JSPS Overseas Research Fellowships and JSPS KAKENHI Grant Number JP19K14709. This work is also partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P B and No. FIS2017-84038-C2-2-P B. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 824093 for the STRONG-2020 project.

Reference (53)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return