Single pole dominance in short- and intermediate-range $NN$ interaction

TL;DR

This paper shows that short- and intermediate-range nucleon-nucleon interactions can be effectively modeled by combining long-range one-pion exchange with a short-range dibaryon-induced interaction, explaining experimental phase shifts.

Contribution

It introduces a model incorporating a specific short-range interaction induced by s-channel dibaryon exchange, aligning with experimental dibaryon resonance parameters.

Findings

The model describes elastic and inelastic NN scattering up to 800 MeV.

Resonance parameters match experimental dibaryon data.

The interaction explains the existence of the singlet deuteron and excited dibaryons.

Abstract

It is demonstrated that both elastic and inelastic $NN$ scattering at laboratory energies up to 600--800 MeV, at least in some partial waves characterized by a large inelasticity, can be described by a superposition of the conventional long-range one-pion exchange and a specific short-range interaction induced by the $s$-channel dibaryon exchange. For the $^3P_2$, $^1D_2$ and $^3F_3$ partial waves, the pole parameters giving the best fit of the real and imaginary parts of the $NN$ phase shifts are consistent with the parameters of the respective isovector dibaryon resonances found experimentally. In the $^1S_0$ channel, the suggested interaction gives two poles of the $S$-matrix -- the well-known singlet deuteron and an excited dibaryon. On the basis of the results presented, a conclusion is made about the nature of $NN$ interaction and its strong channel dependence.

Tables

Table 1. Table 1: Parameters of bare and dressed dibaryon resonances (in GeV) for three isovector N ​ N 𝑁 𝑁 NN channels in comparison with experimental values taken from Refs.  Komarov ( P 2 3 superscript subscript 𝑃 2 3 {}^{3}P_{2} ) and Strak91 ( D 2 1 superscript subscript 𝐷 2 1 {}^{1}D_{2} , F 3 3 superscript subscript 𝐹 3 3 {}^{3}F_{3} ).

${}_{}{}^{2S+1}L_J^{}$	$M_0$	${\Gamma }_0$	$M_{\mathrm{th}}$	${\Gamma }_{\mathrm{th}}$	$M_{\exp }$	${\Gamma }_{\exp }$
${}_{}{}^{3}P_2^{}$	2.21	0.096	2.23	0.15	2.197(8)	0.130(21)
${}_{}{}^{1}D_2^{}$	2.168	0.08	2.18	0.11	2.14–2.18	0.05–0.1
${}_{}{}^{3}F_3^{}$	2.23	0.15	2.22	0.17	2.20–2.26	0.1–0.2

Equations

$$H=\left(\begin{array}[]{cc}h_{NN}&\lambda_{1}|\phi\rangle\langle\alpha|\\ \lambda_{1}|\alpha\rangle\langle\phi|&E_{D}|\alpha\rangle\langle\alpha|\\ \end{array}\right),$$

$$V_{NN}=\frac{f_{\pi }^2}{m_{\pi }^2}\frac{1}{q^2+m_{\pi }^2}{\left(\frac{{\Lambda }_{\pi NN}^2-m_{\pi }^2}{{\Lambda }_{\pi NN}^2+q^2}\right)}^2({\mathbf{\sigma }}_1\cdot \mathbf{q})({\mathbf{\sigma }}_2\cdot \mathbf{q})\frac{({\mathbf{\tau }}_1\cdot {\mathbf{\tau }}_2)}{3},$$

$$\varphi (r)=N{r}^{l+1}\exp \left[-\frac{1}{2}{\left(\frac{r}{r_0}\right)}^2\right],$$

$$H_{\mathrm{eff}}(E)=h_{NN}+\frac{{\lambda }_1^2|\varphi ⟩⟨\varphi |}{E-E_D}.$$

$$t(E)=\frac{{\lambda }_1^2|\varphi ⟩⟨\varphi |}{E+i0-E_D-J_1(E+i0)},$$

$$J_1(Z)={\lambda }_1^2⟨\varphi |g_{NN}(Z)|\varphi ⟩.$$

$$\mathrm{Im}{J}_1(E+i0)=-\pi {\lambda }_1^2|⟨\varphi |{\psi }_0(E)⟩{|}^2,$$

$$S(E)=e^{2i{\delta }_0}\frac{E-E_D-J_1^{\ast }(E+i0)}{E-E_D-J_1(E+i0)},$$

$$Z-E_D-J_1(Z)=0.$$

$$\Gamma_{D}(\sqrt{s})=\left\{\begin{array}[]{lr}0,&\sqrt{s}\leq E_{\rm thr};\\ \displaystyle\Gamma_{0}\frac{F(\sqrt{s})}{F(M_{0})},&\sqrt{s}>E_{\rm thr}\\ \end{array}\right.,$$

$$F(\sqrt{s})=\frac{1}{s}{\int }_{2m}^{\sqrt{s}-m_{\pi }}d{M}_{NN}\frac{q^{2l_{\pi }+1}{k}^{2L_{NN}+1}}{(q^2+{\Lambda }^2{)}^{l_{\pi }+1}(k^2+{\Lambda }^2{)}^{L_{NN}+1}},$$

$$K=\tan \delta +i{\tan }^2\rho ,$$

$$V_{\mathrm{rep}}=\lambda |{\varphi }_0⟩⟨{\varphi }_0|,$$

Full text

nucleon-nucleon interaction, dibaryon resonances

I Introduction

Nowadays the Effective Field Theory (EFT), or Chiral Perturbation Theory (ChPT), is a dominating framework for the quantitative description of the $NN$ interaction at low and moderate energies Weinberg ; Machl ; Eppelbaum ; Ordones . In this approach, a peripheral part of the $NN$ interaction is described via a superposition of terms of the perturbation theory series corresponding to the subsequent orders, i.e., the leading order (LO), the next-to-leading order (NLO), the next-to-next-to-leading order (N2LO), etc., while the short-range contributions are parameterized through the so-called contact terms which, according to the general concept, should be independent of energy and the expansion order. By construction, this general approach is valid until the collision energies $T_{\rm lab}\simeq 350$ MeV, while at higher energies it should be supplemented by an appropriate theoretical model to describe the short-range components of the $NN$ interaction and the short-range $NN$ correlations in nuclei. For this purpose, one can consider the well-known quark model in its various versions (see, e.g., Stancu ). However such a hybrid approach inevitably leads to serious difficulties with double counting because in quark models the gluon exchanges between quarks are usually supplemented by meson ($\pi$ and $\sigma$) exchanges, which immediately results in appearance of not only short-range but also long-range meson-exchange forces between nucleons.

At the moment in nuclear physics there is a wide class of phenomena where one observes a close connection between short-range correlations of nucleons in nuclei and some distortions of quark momentum distributions in the deuteron, 3He and other nuclei Zhihong ; Weinstein . These phenomena include the EMC effect, DIS observations, cumulative effects, etc. It is evident that neither the traditional meson-exchange nor the modern EFT approaches are relevant for a description of such effects occurring at very high momentum transfers. On the other hand, since all these phenomena are closely related to nuclear force and nuclear structure at short distances, they should be described within some general scheme which includes the correct treatment of the $2N$ and $3N$ forces.

Therefore, for further progress in this area it would be highly desirable to treat the short- and intermediate-range $NN$ interactions by using some QCD-motivated model which could reproduce correctly the basic effects of the nucleon quark structure in different $NN$ partial waves, however without addressing all complexity of the multi-quark dynamics. At the same time, such a model description of the short-range $NN$ dynamics should not modify the known interaction at long distances.

In our opinion, suitable objects, which, on the one hand, can reproduce the main features of six-quark dynamics and, on the other hand, are strongly coupled to the hadronic ($NN$, $N\Delta$ and $\Delta\Delta$) channels at low and moderate energies are dibaryon resonances which were predicted by Dyson and Xuong in 1964, at the very beginning of quark era Dyson . Just recently, after many years of rejection, doubt, and contradictory findings, a number of dibaryon resonances have been eventually confirmed in the modern high-precision experiments Adlarson11 ; Adlarson14 ; Komarov ; Adlarson18 (see also the recent review Clement ).

Dibaryon resonances are very attractive to describe the short-range $NN$, $N\Delta$ and $\Delta\Delta$ forces not only due to their six-quark structure but, first of all, because they are specific relatively long-lived states in which six-quark dynamics should be manifested most clearly. So, the goal of the present paper is to demonstrate that the $NN$ interaction at least in particular partial-wave channels can be described properly by a superposition of the long-range meson-exchange potentials and one simple $s$-channel exchange potential driven by a dibaryon intermediate state.

For the particular $NN$ partial waves considered below (and for many others) it is crucial to include the $NN\to N\Delta$ (and/or $NN\to\Delta\Delta$) coupling. In turn, the $NN\to N\Delta$ transition may or may not be accompanied by a dibaryon formation. While the latter type of coupling has been effectively included into our model consideration through the resonance parameters (in fact, the isovector dibaryon width is mainly due to the $D\to N\Delta$ decay), the former (pure $t$-channel) coupling is actually a background to the $s$-channel dibaryon generation which should be included explicitly into the $NN$ interaction potential. Though the background processes may affect significantly the description of $NN$ inelasticities above the resonance energy, they will not change our main results concerning the impact of the basic dibaryon mechanism at lower energies. Thus we postpone the consistent treatment of the $NN\to N\Delta$ coupling to the future work.

It should be noted that the initial version of the dibaryon model for $NN$ and $3N$ interactions developed in JPhys2001 (see also AnnPhys2010 and references therein) provided quite encouraging results in description of $NN$ scattering in various partial waves. In particular, it allowed to reproduce real parts of phase shifts in different $NN$ channels up to energies $T_{\rm lab}=500$–$600$ MeV. However, since then dibaryons have achieved a more reliable experimental status, while the parameters of the real dibaryon poles found in the work JPhys2001 were never compared to those of the physical dibaryon resonances. So, in the present paper we tried to incorporate into the initial model the experimentally found dibaryon resonances together with their empirical parameters to describe simultaneously the real and imaginary parts of the $NN$ phase shifts in a broad energy range. The results of this study would allow to judge about the true applicability and merit of such a non-conventional description of $NN$ interaction.

The structure of the paper is as follows. In Sec. II, we introduce the two-channel formalism (with one external and one internal channel) which is used further to describe the $NN$ partial phase shifts. In Sec. III and IV, we present the results for the particular isovector $NN$ partial waves and illustrate effectiveness of the dibaryon mechanism. We summarize the results and conclude in Sec. V.

II $\bm{NN}$ scattering driven by a single state with a complex energy

in the internal channel

We consider below a two-channel model with one complex pole in the effective interaction potential. This model corresponds to the physical pattern of the $NN$ scattering driven by the traditional one-pion exchange in the external channel and by one state with a complex eigenvalue (i.e., the “bare” dibaryon state) in the internal channel. The complex energy of this state can be interpreted as a consequence of different modes of its decay which do not include the $NN$ mode.111The initial dibaryon state with the real energy can be treated in terms of the field theory as a “bare” dibaryon, while the coupled-channel dibaryon which is able to decay into the $NN$ continuous spectrum can be identified as a “dressed” dibaryon. In this sense, the dibaryon with the complex energy can be called as a “semi-dressed” one. However, to retain the unified notation, we will refer to the initial dibaryon with the complex energy as a “bare” one.

The total matrix Hamiltonian for such a two-channel problem has the form:

[TABLE]

where the external-channel Hamiltonian $h_{NN}=h_{NN}^{(0)}+V_{NN}$ acts in the space of $NN$ variables and includes the peripheral $NN$ interaction $V_{NN}$ which is exhausted by the one-pion exchange potential (OPEP):

[TABLE]

where $m_{\pi}=(m_{\pi^{0}}+2m_{\pi^{\pm}})/3$ is the averaged pion mass and $\Lambda_{\pi NN}$ — the high-momentum cutoff parameter222In our calculations, the averaged pion-nucleon constant $f_{\pi}^{2}/(4\pi)=0.075$ and the soft cutoff $\Lambda_{\pi NN}=0.65$ GeV/$c$ are used..

In case of the single-pole model, the Hilbert space of the internal channel is one-dimensional and therefore the internal Hamiltonian is reduced to a single term with a complex eigenvalue $E_{D}=E_{0}-i\Gamma_{\rm inel}/2$, the imaginary part of which is defined by the decay width of the dibaryon into all inelastic channels.

To determine the form factor $|\phi\rangle$ of the transition between the external ($NN$) and the internal (dibaryon) channels, it is necessary to use some microscopic model that describes both channels within a unified approach. In our previous works JPhys2001 ; AnnPhys2010 , a dibaryon model for nuclear forces was developed, in which a microscopic six-quark shell model in a combination with the ${}^{3}P_{0}$ quark mechanism of pion production was used to determine the transition amplitude between two channels. Note that the transition form factor $|\phi\rangle$ is a function of the relative coordinate $r$ (or the relative momentum) in the $NN$ channel, and also depends on the spin, isospin, orbital and total angular momenta of the two-nucleon state. Here we use the same form of this function as in Ref. JPhys2001 , i.e., the harmonic oscillator term:

[TABLE]

where $l$ is the orbital angular momentum in the two-nucleon system, $N$ — the normalization factor, and $r_{0}$ — the scale parameter.

Since the total Hamiltonian (1) couples the internal channel to the $NN$ relative-motion channel, it is convenient to exclude the internal channel by the commonly-used method Feshb and to deal with the $NN$ relative motion only. Then the resulted effective energy-dependent Hamiltonian in the $NN$ channel takes the form:

[TABLE]

Due to the fact that the basic term of the effective Hamiltonian (4) has a separable form, it is convenient to define explicitly an additional $t$-matrix in the distorted-wave representation:

[TABLE]

where $J_{1}(Z)$ is the matrix element of the resolvent of the external $NN$ Hamiltonian $g_{NN}(Z)\equiv[Z-h_{NN}]^{-1}$:

[TABLE]

Note that the imaginary part of this function at a real positive energy can be found in an explicit form as

[TABLE]

where $|\psi_{0}(E)\rangle$ is the scattering function for the external Hamiltonian $h_{NN}$. Using the formula for the transition operator (5), one can easily obtain the expression for the total $S$-matrix:

[TABLE]

where $\delta_{0}(E)$ is the phase shift for the external Hamiltonian $h_{NN}$. Thus, the pole of the total $S$-matrix can be found in a complex energy $Z$ plane from the condition:

[TABLE]

The condition (9) makes it possible to find the renormalized position of the dressed dibaryon resonance $Z_{R}=E_{R}-i\Gamma_{\rm th}/2$ (relatively to the initial “bare” value of $E_{D}$), i.e., the complex function $J_{1}(Z)$ gives shifts of the real and imaginary parts of the dibaryon pole. These shifts arise due to coupling of the initial dibaryon to the external $NN$ channel. The invariant mass of the dressed dibaryon can be calculated from the real part of the energy $E_{R}$ by the relation $M_{\rm th}=2\sqrt{m(E_{R}+m)}$, where $m$ is the nucleon mass333Here we use the “minimal” account of the relativistic effects (see, e.g., Ref. Geramb ) by keeping the relation $E=k^{2}/m$ between the energy $E$ and the relative momentum $k$ in the Lippmann–Schwinger equation, while calculating $k$ from the laboratory energy $T_{\rm lab}$ via the relation $k=\sqrt{mT_{\rm lab}/2}$. Finally, the invariant energy is calculated from $E$ by using the formula $\sqrt{s}=2\sqrt{m(E+m)}$..

For the effective account of inelastic processes and description of the threshold behavior of the reaction cross section, one should introduce the energy dependence of the bare dibaryon width $\Gamma_{\rm inel}$. The main inelastic process for the isovector $NN$ channels considered here is the one-pion production. The three-body mode $D\to\pi NN$ dominates in the decay width of dibaryons in these partial-wave channels, while the two-body mode $D\to\pi d$ takes $\lesssim 30$% Strak91 of the total dibaryon width and has the similar threshold behavior. Thus, the respective inelastic width can be represented as follows:

[TABLE]

where $\sqrt{s}$ is the total invariant energy of the decaying resonance, $M_{0}$ — the bare dibaryon mass, $E_{\rm thr}=2m+m_{\pi}$ — the threshold energy, and $\Gamma_{0}$ defines the decay width at the resonance energy.

The function $F(\sqrt{s})$ should take into account the dibaryon decay into the channel $\pi NN$. So that, for the given values of the orbital angular momenta of the pion $l_{\pi}$ and $NN$ pair $L_{NN}$, this function can be parameterized in the form:

[TABLE]

where \displaystyle q={\sqrt{(s-m^{2}_{\pi}-M^{2}_{NN})^{2}-4m_{\pi}^{2}M_{NN}^{2}}}\Big{/}{2\sqrt{s}} is the pion momentum in the total center-of-mass frame, $\displaystyle k={\frac{1}{2}}\sqrt{M_{NN}^{2}-4m^{2}}$ — the momentum of the nucleon in the center-of-mass frame of the final $NN$ subsystem with the invariant mass $M_{NN}$, and $\Lambda$ — the high-momentum cutoff parameter which prevents an unphysical rise of the width $\Gamma_{\rm inel}$ at high energies. The orbital momenta $l_{\pi}$ and $L_{NN}$ may take different values however their sum is restricted by the total angular momentum and parity conservation. In practical calculations, the values of $l_{\pi}$, $L_{NN}$ and $\Lambda$ were adjusted to get the best description of the phase shifts in the given $NN$ partial wave. It should be emphasized that the values of these parameters affect mainly the threshold behavior of the inelastic partial phase shifts. They are used for the “fine-tuning” of the model while the resulting phase shifts in a broad energy range are much more sensitive to the mass and width of the bare resonance.

Thus, we have formulated a simple model for coupling between the external $NN$ (driven by OPEP) channel and the internal (“bare” dibaryon) channel which leads to a renormalization of the complex energy of the initial “bare” dibaryon and its transformation to the real mass and width of the “dressed” dibaryon. The simple mechanism for coupling between the external and internal channels can be represented graphically by the diagram series shown in Fig. 1. This series corresponds to the well-known Dyson equation for a dressed particle in the quantum field theory.

III $\bm{NN}$ scattering in isovector channels

In this section, we consider as particular examples the isovector partial waves (${}^{3}P_{2}$, ${}^{1}D_{2}$ and ${}^{3}F_{3}$) of the $NN$ scattering, where the empirical phase shifts do not manifest explicitly the behavior inherent to the repulsive core, at least until relatively high collision energies $T_{\rm lab}\simeq 800$ MeV.444The fact that at higher energies $T_{\rm lab}>800$ MeV the real phase shifts in some channels become negative can be explained by strong absorption at these energies, i.e., by appearance of a large imaginary part of phase shifts that can be interpreted as a consequence of strong repulsion. The resonance behavior in these partial waves near the $N\Delta$ threshold was established long ago in experiments Meshcher ; Auer1 ; Auer2 and was interpreted either by the threshold effect or the true dibaryon formation. The dibaryon interpretation was then supported by the partial-wave analyses of different groups which found the $S$-matrix poles corresponding to dibaryon resonances Hoshiz ; Arndt ; Strak (see also the review paper Strak91 ). In our recent works NPA2016 ; PRD2016 the importance of these three resonances was established for description of the basic pion-production reaction $pp\to d\pi^{+}$, and the ${}^{3}P_{2}$ dibaryon was shown to play a crucial role in reproducing the polarization observables. Simultaneously, the ${}^{3}P_{2}$ dibaryon was observed in the recent high-precision experiment on the reaction $pp\to pp(^{1}S_{0})\pi^{0}$ Komarov .

It should be emphasized that previous analyses of $NN$ scattering which considered dibaryon resonances generally suggested that the dibaryon pole can give a significant contribution to the $NN$ phase shifts in a respective partial wave only near the resonance energy. Contrary to this, we will show below that the single pole (combined with a long-range OPEP contribution) can explain the phase shifts behaviour in the isovector channels in a broad energy range from zero up to the resonance energy. The isoscalar channels of $NN$ scattering will be studied in our subsequent paper.

For description of the partial phase shifts, the Arndt parametrization for the $K$-matrix SAID2007 ; Geramb is used. For the uncoupled $NN$ channels, it has a simple form:

[TABLE]

where $\delta$ is the real phase shift and $\rho$ is a parameter related to inelasticity. For the sake of simplicity, below we will refer to the parameter $\rho$ as the imaginary phase shift.

III.1 Channel $\bm{{}^{3}P_{2}}$

The empirical $NN$ phase shifts in the triplet channel ${}^{3}P_{2}$ as found by the George Washington University group (SAID) SAID2007 do not display any sign of the repulsive core and remain to be positive at least up to energies of $T_{\rm lab}\simeq 1$ GeV.555Such a behavior can be explained in the conventional $NN$ potential models by a complete compensation of the short-range repulsive core by the very strong attractive spin-orbital potential, so that, the resulting potential in this channel turns out to be attractive Bohr . This can be interpreted as a fact that the traditional repulsive core does not play a crucial role in this channel, and thus $NN$ interaction here is managed by a rather strong attraction, which is likely due to generation of a dibaryon resonance.

Below we show that such an attraction can be reproduced by a single dibaryon pole in the effective Hamiltonian (4) via varying its position $E_{D}=E_{0}-i\Gamma_{D}/2$ and the coupling constant $\lambda_{1}$ for the external and internal channels.

The comparison of empirical (SAID) and theoretical ${}^{3}P_{2}$ $NN$ phase shifts in the energy interval $T_{\rm lab}=0$–$800$ MeV is presented in Fig. 2.666Though the SAID results for $np$ scattering are plotted in Fig. 2 and subsequent figures, the $pp$ scattering phase shifts are indistinguishable from them in the scale of the picture. Here the following potential parameters were used: $\lambda_{1}=0.065$ GeV and $r_{0}=0.71$ fm. The initial dibaryon mass was taken to be $M_{0}=2.21$ GeV, and the inelastic width had the form (10), (11) with the parameters $\Gamma_{0}=0.096$ GeV, $l_{\pi}=2$, $L_{NN}=0$ and $\Lambda=0.3$ GeV/$c$.

It is seen from Fig. 2 that the single-pole model in a combination with a simple OPEP provides almost quantitative agreement with the empirical data for the real part and reasonable agreement for the imaginary part of the ${}^{3}P_{2}$ phase shifts up to energies $T_{\rm lab}\simeq 700$ MeV. Needless to say that the above agreement for both real and imaginary parts of the phase shifts, i.e., for elastic $NN$ scattering and meson production simultaneously, was attained with the same parameters for the bare dibaryon. Some discrepancy observed for the imaginary part of the phase shift, starting just above the pion-production threshold, is likely related to our simplified description of the energy dependence of the dibaryon width, and can probably be decreased by a more sophisticated treatment of $\Gamma_{D}(E)$. The proper way to proceed here is to include explicitly in the model the $N\Delta$ $P$-wave channels which couple to the $NN(^{3}P_{2})$ channel.

Now, using Eq. (9), one can find easily the renormalized position of the resonance pole in this channel. We found the following parameters for the dressed dibaryon in the channel ${}^{3}P_{2}$: $M_{\rm th}(^{3}P_{2})=2.23$ GeV and $\Gamma_{\rm th}(^{3}P_{2})=0.15$ GeV. These parameters should be compared with the respective experimental values found recently by the ANKE-COSY Collaboration Komarov : $M_{\rm exp}(^{3}P_{2})=2.197(8)$ GeV and $\Gamma_{\rm exp}(^{3}P_{2})=0.130(21)$ GeV (the numbers in parentheses denote the uncertainty in the last figure). It is worth noting that the authors of Komarov used two different fitting procedures, the first leading to the above quoted values and the second (the global fit) — to somewhat larger values $M_{\rm exp}(^{3}P_{2})=2.207(12)$ GeV and $\Gamma_{\rm exp}(^{3}P_{2})=0.170(32)$ GeV still consistent with the above ones within errors. Thus, the mass and width obtained in this work for the ${}^{3}P_{2}$ dibaryon turn out to be rather close to their experimental values (with account of the experimental uncertainties).

Since our model does not include any other assumptions, except the OPEP and presence of a single dibaryon pole, we owe to conclude that the results obtained point directly to the dominance of the intermediate dibaryon production in the $NN$ interaction in this channel.

III.2 Channel $\bm{{}^{1}D_{2}}$

A fully similar consideration of the $NN$ phase shifts in the singlet channel ${}^{1}D_{2}$ leads to theoretical predictions shown in comparison with the respective empirical data (SAID) in Fig. 3. The potential parameters for this channel have been taken to be $\lambda_{1}=0.048$ GeV and $r_{0}=0.82$ fm. The initial dibaryon mass is $M_{0}=2.168$ GeV, and the width parameters are $\Gamma_{0}=0.08$ GeV, $l_{\pi}=0$, $L_{NN}=1$ and $\Lambda=0.2$ GeV/$c$.

Here we see again almost perfect agreement for the real and reasonable agreement for the imaginary phase at energies up to 600 MeV. The parameters of the dressed dibaryon in the ${}^{1}D_{2}$ channel found in our calculations are $M_{\rm th}(^{1}D_{2})=2.18$ GeV and $\Gamma_{\rm th}(^{1}D_{2})=0.11$ GeV. These parameters are consistent with those found previously in experiments Meshcher ; Auer2 and partial-wave analyses Hoshiz ; Arndt ; Strak and turn out to be rather close to the values obtained in the early experiment of 1955 Meshcher where the ${}^{1}D_{2}$ resonance was first observed: $M_{\rm exp}(^{1}D_{2})\simeq 2.16$ GeV and $\Gamma_{\rm exp}(^{1}D_{2})\simeq 0.12$ GeV.

It is important to emphasize that the $NN$ channel ${}^{1}D_{2}$ strongly couples to the $N\Delta$ channel ${}^{5}S_{2}$, and just this strong coupling with a subsequent decay of the $\Delta$-isobar determines a large portion of inelasticity in the ${}^{1}D_{2}$ partial wave. Since we did not take into account the $t$-channel $\Delta$ excitation in our model, the description of inelastic phase shifts turns out to be not perfect. However, almost quantitative description of the real part of the phase shifts until $T_{\rm lab}=600$ MeV and reasonable description of their imaginary part (with the same model parameters) point to the dominant role of the dibaryon resonance (including its $N\Delta$ component) in this channel as well.

III.3 Channel $\bm{{}^{3}F_{3}}$

First experimental evidence of the ${}^{3}F_{3}$ dibaryon resonance with the mass $M_{D}\simeq 2.26$ GeV was obtained in 1977 Auer1 . It was also established that there is a very large inelasticity in the ${}^{3}F_{3}$ $NN$ partial wave. So, an interesting and challenging question arises: to what degree the ${}^{3}F_{3}$ resonance contributes to the real and imaginary parts of the $NN$ phase shifts in this channel? From the first glance, the possible impact of the dibaryon can be seen mainly at the energies close to the resonance position. However, in general, the $s$-channel dibaryon exchange in $NN$ scattering must be retraced also far from the resonance energy and give an evident impact to the off-shell $t$-matrix.

In Fig. 4 the partial phase shifts in the ${}^{3}F_{3}$ channel are compared with the empirical (SAID) data. As is clearly seen from the figure, the single ${}^{3}F_{3}$ dibaryon pole (in a combination with the peripheral OPEP) can reproduce very well the real $NN$ phase shifts from zero energy up to 700 MeV (see Fig. 4). The parameters of the coupling potential in this case are $\lambda_{1}=0.06$ GeV and $r_{0}=0.5$ fm, while the pole parameters are $M_{0}=2.23$ GeV and $\Gamma_{0}=0.15$ GeV with $l_{\pi}=0$, $L_{NN}=2$ and $\Lambda=0.1$ GeV/$c$.

The parameters of the dressed dibaryon found for the ${}^{3}F_{3}$ channel, i.e., $M_{\rm th}(^{3}F_{3})=2.22$ GeV and $\Gamma_{\rm th}(^{3}F_{3})=0.17$ GeV, are also consistent with the previous findings Auer1 ; Auer2 ; Hoshiz ; Arndt ; Strak and occur to be rather close to the values obtained in the first experimental observation of the ${}^{3}F_{3}$ resonance Auer1 : $M_{\rm exp}(^{3}F_{3})\simeq 2.26$ GeV and $\Gamma_{\rm exp}(^{3}F_{3})\simeq 0.2$ GeV.

The results presented in this section for the isovector dibaryon resonances together with experimental data are summarized in Table 1.

Thus, in this section we have got a very good quantitative approximation for the real parts of the partial phase shifts in the $NN$ channels considered and a rather good qualitative approximation for the imaginary parts of these phase shifts in a broad energy range starting from zero energy which is very far from the position of the “bare” dibaryon. The crucial point of the results presented above is an agreement of the masses and widths of the dressed dibaryons obtained by fitting the $NN$ phase shifts in our model with the parameters of experimentally found dibaryons.

IV Channel $\bm{{}^{1}S_{0}}$. Description of the repulsive core effects

One of the main ingredients of the conventional models for $NN$ interaction is a well-known short-range repulsion induced by vector-meson exchange. However, from the modern point of view such a mechanism looks doubtful.777See, e.g., Ref. Barnes where the authors claim: “A literal attribution of the short-range repulsive core to vector meson exchange, as opposed to a phenomenological parametrization, of course involves a non sequitur: since the nucleons have radii $\approx$ 0.8 fm and the range of the vector exchange force is $\hbar/m_{\omega}c\approx 0.2$ fm one would have to superimpose the nucleon wavefunctions to reach the appropriate internucleon separations. The picture of distinct nucleons exchanging a physical $\omega$-meson at such a small separation is clearly a fiction…” We will demonstrate below how the repulsive core effects can be reproduced within the framework of the quark and dibaryon models (see also Refs. Faess ; Sazonov ; YAF2013 ).

As is well known, the repulsive core effects are manifested very clearly in the channel ${}^{1}S_{0}$, where the phase shifts become negative already at collision energies $T_{\rm lab}\simeq 250$ MeV. The dibaryon model JPhys2001 ; AnnPhys2010 predicts for the $S$-wave $NN$ interaction the dominating six-quark configuration of the type $|s^{4}p^{2}[42]_{x}LST\rangle$ with a two-quantum ($2\hbar\omega$) excitation. When projecting this $2\hbar\omega$-excited configuration onto the $NN$ channel, the $NN$ relative-motion wavefunction $\psi(r)$ automatically acquires an internal node. This node is rather stable and its position $r_{c}$ moves only weakly with increase of collision energy (see Fig. 5). Moreover, the node position ($r_{c}\sim 0.5$ fm) turns out to be very close to that of the traditional repulsive core.

In the dibaryon model JPhys2001 ; AnnPhys2010 appearance of a stationary node in the $S$-wave $NN$ interaction is provided by an orthogonality condition between the fully symmetric and the mixed-symmetry six-quark configurations, which is realized by adding a projecting operator to the $NN$ potential:

[TABLE]

where $\lambda\to\infty$, and $|\phi_{0}\rangle$ is a projection of the fully symmetric wavefunction $|s^{6}[6]\rangle$ onto the $NN$ channel. The idea behind inclusion of this term into the $NN$ potential is that the six-quark configuration $|s^{6}[6]\rangle$ gives a much smaller contribution to the $S$-wave $NN$ interaction than the mixed-symmetry component. Hence, in case of $S$- and $P$-wave channels which exhibit strong repulsive core effects (${}^{3}P_{0}$, ${}^{3}P_{1}$ and ${}^{1}P_{1}$), the effective $NN$ Hamiltonian (4) should be supplemented with the orthogonalizing pseudopotential (13) with a large positive coupling constant $\lambda$.888For the $P$ waves, $|\phi_{0}\rangle$ is a projection of the six-quark wavefunction $|s^{5}p[51]_{x}LST\rangle$ onto the $NN$ channel.

Fortunately, this modification does not add to our model any free parameters, so that, the number of adjustable parameters for the bare dibaryon remains the same. Thus, varying the position of only one complex pole in the ${}^{1}S_{0}$ channel, one can achieve a very reasonable description of the phase shifts in a wide energy range. However, the resonance in this channel occurs to be very broad, hence, few different sets of parameters can be used to fit the phase shifts. In Fig. 5 we show the results for the potential parameters $\lambda_{1}=1.184$ GeV, $r_{0}=0.51$ fm, the initial dibaryon mass $M_{0}=2.364$ GeV, and the initial width $\Gamma_{0}=0.044$ GeV with $l_{\pi}=1$, $l_{NN}=0$ and $\Lambda=0.4$ GeV/$c$.

It should be stressed that the total $S$-matrix in the ${}^{1}S_{0}$ channel has two poles: the first corresponding to the well-known singlet deuteron just above the $NN$ threshold (at $E=-0.067$ MeV on the unphysical complex energy sheet) and the second being a broad high-lying resonance with $M_{\rm th}=2.59$ GeV and $\Gamma_{\rm th}=0.63$ GeV. We emphasize here once again that the mass and width of the bare ${}^{1}S_{0}$ dibaryon were taken to be the same for description of both real and imaginary phase shifts.

These results give a strong indication for existence of a second high-lying dibaryon in the ${}^{1}S_{0}$ channel in addition to the near-threshold state (the singlet deuteron) predicted by Dyson and Xuong Dyson many years ago. It is interesting to note that while the second pole for the bare dibaryon has been postulated when constructing the effective potential (4), the first one appears in the course of dressing, i.e., when the coupling between the $NN$ and dibaryon channels gets switched on.

V Conclusion

We have demonstrated in this work that the hypothesis about the dibaryon origin of the basic nuclear force makes it possible to describe properly the behavior of both real and imaginary parts of $NN$ phase shifts (at least in some partial-wave channels characterized by a large inelasticity) in a rather wide energy range 0–600 MeV with only a few basic parameters for the initial “bare” dibaryon. This result should be compared to the fact that the traditional realistic $NN$ potentials can describe only the real $NN$ phase shifts in the energy range 0–350 MeV.

What is even more interesting, the results of the present study can also explain the strong partial-wave dependence of the $NN$ phase shifts, which must depend upon the dynamics of dibaryon states in the given partial channel, their spin, isospin, parity, etc. In turn, the dibaryon dynamics is determined completely by the QCD degrees of freedom. Without taking into account the intermediate dibaryons, the $NN$ interaction potential must have very complicated and non-transparent structure and adopt numerous terms with ${\bf S\cdot L}$, $L^{2}$, etc., operators to describe the empirical channel dependence of the $NN$ phase shifts.

Acknowledgment. The work has been partially supported by RFBR, grants Nos. 19-02-00011 and 19-02-00014.