$d^\ast (2380)$ dibaryon from lattice QCD

The $\Delta\Delta$ dibaryon resonance $d^\ast (2380)$ with $(J^P, I)=(3^+, 0)$ is studied theoretically on the basis of the 3-flavor lattice QCD simulation with heavy pion masses ($m_\pi =679, 841$ and $1018$ MeV). By using the HAL QCD method, the central $\Delta$-$\Delta$ potential in the ${}^7S_3$ channel is obtained from the lattice data with the lattice spacing $a\simeq 0.121$ fm and the lattice size $L\simeq 3.87$ fm. The resultant potential shows a strong short-range attraction, so that a quasi-bound state corresponding to $d^\ast (2380)$ is formed with the binding energy $25$-$40$ MeV below the $\Delta\Delta$ threshold for the heavy pion masses. The tensor part of the transition potential from $\Delta\Delta$ to $NN$ is also extracted to investigate the coupling strength between the $S$-wave $\Delta\Delta$ system with $J^P=3^+$ and the $D$-wave $NN$ system. Although the transition potential is strong at short distances, the decay width of $d^\ast (2380)$ to $NN$ in the $D$-wave is kinematically suppressed, which justifies our single-channel analysis at the range of the pion mass explored in this study.


Introduction
Recently much interest has been attracted to decuplet-decuplet dibaryons as well as to octetoctet and octet-decuplet dibaryons [1,2,3,4,5,6,7,8]. Theoretically, the quark Pauli principle provides an important guideline to identify possible dibaryon channels [9,10]: If the overlap of the quark wave functions is forbidden by the quark Pauli principle, it is difficult to form dibaryons, while if the overlap is allowed or only partially forbidden, there is a chance.
Email address: shinya.gongyo@riken.jp (Shinya Gongyo) To see the role of quark Pauli principle more explicitly in the decuplet-decuplet system, let us consider its irreducible representation of the SU(3) flavor symmetry, where "sym." and "anti-sym." stand for the flavor symmetry under the exchange of two baryons. Then one finds that there are two Pauli-allowed Swave states: Spin 0 in symmetric 28 representation and spin 3 in anti-symmetric 10 * representation. The ΩΩ system in the spin-0 channel belongs to the former, while the ∆∆ system in the spin-3 and isospin-0 channel belongs to the latter [11]. In fact, these two systems have been studied exten-sively by using phenomenological models (see e.g. [12,13,14,15] for the ΩΩ, and [16,17] for the ∆∆). Only recently, the first principle lattice QCD simulation of the baryon-baryon interactions near the physical point became possible thanks to the HAL QCD method, and it was shown that the ΩΩ interaction in the spin-0 channel supports a shallow dibaryon state, the di-Omega, near unitarity [3]. It is also proposed to search for such a state by the momentum correlation of Ω-pairs in future heavy-ion collision experiments [6].
As for the ∆∆ system, a dibaryon with spin-3 and isospin-0 has been reported experimentally [18,19]. It is now called d * (2380) and has a resonance peak about 80 MeV below the ∆∆ threshold with the total width Γ 70MeV. The recent exclusive experiment has revealed its detailed properties such as the branching ratios into N N , N N π, and N N ππ [2,20]. Thus it is highly desirable to make a first principle lattice QCD calculation of d * (2380). However, it is a much involved task in comparison to di-Omega primarily because d * (2380) is a resonance above multi-particle thresholds such as N N π and N N ππ. Instead of studying the problem with extensive coupled-channel approach on the lattice, we take heavy quark masses to capture the essential mechanism of the formation of d * (2380) from two ∆s. In such a lattice setup, ∆ becomes a stable particle without decaying into N π and d * (2380) may appear as a spin-3 and S-wave quasi-bound state of ∆∆ which can decay to N N only through the D-wave and the G-wave. Such a lattice result not only reveals the physics behind d * (2380) but also provides useful input to the effective field theory approach toward the physical point [21]. This paper is organized as follows. In Sec.2, we introduce the HAL QCD method to extract the ∆-∆ central potential from lattice QCD. In Sec. 3, we summarize setup of our lattice QCD simulations. In Sec. 4, we show the numerical results of ∆-∆ central potential in 7 S 3 channel. Sec. 5 is devoted to summary. In Appendix A, we show the transition potential from ∆∆ to N N and estimate the decay rate to be small, which justifies the single-channel approach.

HAL QCD method for ∆∆ interaction
In QCD, the ∆∆ potential in the 7 S 3 channel is obtained from the equal-time Nambu-Bethe-Salpeter (NBS) wave function defined by where |W n ; J = 3, I = 0 stands for a QCD eigenstate which has the total energy W n = 2 k 2 n + m 2 ∆ with m ∆ being ∆-baryon's mass, the total spin J = 3 and the isospin α,β,l,m,A,B being the projection operator onto the internal spin s = 3 and I = 0. The ∆-baryon operator ∆ A α,l ( x + r) with the charge index A, the spinor index α, and the Lorentz index l is constructed from the liner combinations of interpolating operators, abc q T a (x)Cγ l q b (x)q cα (x) with q = u, d and C ≡ γ 4 γ 2 .
We first assume that the couplings of ∆∆( 7 S 3 ) to the D-wave and the G-wave N N states below the ∆∆ threshold is small and consider the single channel analysis between ∆s. Justification of this assumption will be discussed in Appendix A.
Since the NBS wave function in the asymptotically large distance is identical to that of the scattering state or bound state in 2-body quantum mechanics [22,23], one can define the ∆∆ potential via the Schrödinger-type equation obtained from the equal-time NBS equation as [24]: with m ∆ being the mass of ∆ and E n = k 2 n /m ∆ . Note that the non-local potential U ∆∆ ( r, r ) is energy-independent. The NBS wave function is related to the reduced four-point function, with a n = W n ; J = 3, I = 0|J J=3 ∆∆ (0) |0 , δW n = W n − 2m ∆ , ∆E * (> 0) being the energy difference between the inelastic threshold and 2m ∆ , and J J=3 ∆∆ (0) being a source operator with J = 3. Below the inelastic threshold, R ∆∆ J=3 ( r, t) satisfies 2 the time-dependent HAL QCD equation [25], Using the derivative expansion of the non-local potential, , the leading-order (LO) local potential can be obtained as The conventional finite volume method with the naive plateaux fitting to obtain E n turns out to have difficulty in disentangling the ground state from other excited states in large lattice volumes as demonstrated in [26,27,28]. On the other hand, the present HAL QCD method provides a potential from the information of both ground state and excited states below the inelastic threshold without such disentanglement. The resultant potential can then be used to calculate the observables such as the binding energy and the phase shift in the infinite volume.
The systematic error in Eq.(5) originating from the LO truncation of the derivative expansion can be estimated from the residual time-dependence of V ∆∆ ( r). Also, the higher-order terms can be determined by using the multiple source functions for J J=3 ∆∆ . It was shown in [28,29] that the next-to-LO potential obtained by combining a wall source and a smeared source for a two-octet baryon system gives negligible effects to physical observable at low energies for heavy pion masses.

Simulation setup
We employ the full QCD gauge configurations in the flavor-SU (3) limit with the renormalizationgroup improved gauge action and the nonperturbatively O(a) improved Wilson quark action at β = 1.83 and κ uds = 0.13710, 0.13760, 0.13800 for 32 3 × 32 lattice. The lattice spacing a and the physical volume corresponds to 0.121fm and (3.87fm) 3 , respectively. We have used 360 configurations for κ uds = 0.13710, 0.13800 and 480 configurations for κ uds = 0.13800 given in Ref. [10]. The wall-type quark source with the Coulomb gauge fixing is employed.
To increase the statistics, the forward and backward propagations are averaged and the rotational symmetry on the lattice (4 rotations) and the translational invariance for the source position (32 temporal positions) are utilized for each configuration. The hadron masses obtained by the single exponential fit are summarized in Table I. The statistical errors are estimated by the Jackknife method with 18 samples for κ uds = 0.13710, 0.13800 and 24 samples for κ uds = 0.13760. The fit results are slightly different from Ref. [10], because we use more statistics and different fit ranges. In all cases, m ∆ is below the threshold, m π + m N , so that ∆ is a stable baryon.  Fig.1 are the central potentials in the 7 S 3 channel, V ∆∆ (r) as a function of r in the range, t/a = 9, 10, 11 and m π = 679, 841, 1018 MeV. As seen from Fig.1 (a), V ∆∆ (r) for different t are nearly identical within the statistical errors indicating that the contribution from higher-order potential is not relevant. We also find that V ∆∆ (r) is attractive in the whole range of r. Moreover, the long-range part of the attraction becomes stronger as m π decreases as seen from Fig.1(b). These features can be understood by (i) the absence of Pauli exclusion effect for quarks in this channel, (ii) the absence of the color magnetic effect in one-gluon exchange at short distance [9], and (iii) the attractive one-pion exchange at long distance. We fit the lattice data of the ∆∆ potential in the range r = 0 − 1.5 fm by two Gaussians plus one Yukawa  form with a form factor as The binding energy B ∆∆ can be also obtained from the Schrödinger equation. The results of the bound state energy E 0 = −B ∆∆ for different t/a and m π are shown in Fig.3(a). Also shown in Fig.3(b) are the bound state energy E 0 and the root-mean-square distance with the statistical errors (first) and systematic errors from the t dependence (second).

Summary
We have studied the ∆∆ system in the J = 3 channel, where the resonant dibaryon d * (2380) was observed, from the lattice QCD simulation with heavy quark masses in the flavor-SU (3) limit. The ∆-∆ central potential in the 7 S 3 channel calculated by the HAL QCD method is found to be attractive in all distance. The phase shifts obtained by solving the Schrödinger equation using the potential show the presence of the deep quasi-bound state below the ∆∆ threshold. The energy below the threshold is estimated from t/a = 10 to be about 30MeV in the case of the lightest pion mass m π = 679MeV.
The lattice simulation of the ∆∆ system near the physical point is left for future studies. Since ∆ baryon can decay into N π, the ∆∆ system can also Inner bars correspond to the statistical errors, while the outer bars are obtained by the quadrature of the statistical and systematic errors estimated from the central values for t/a = 9, 11. decay into N N π and N N ππ as well as N N . Therefore, the coupled channel equations associated with three and four hadron systems, which is challenging not only in the simulation but also in the formulation on the lattice [30], are needed to extract potentials. Appendix A. Transition from ∆∆ to N N The threshold of the N N system (J = 3) in higher partial waves, 3 D 3 and 3 G 3 , are below the quasi-bound state of ∆∆ system. In the main text, we have neglected such transition and derived the single-channel ∆∆ potential in the S-wave. To estimate the magnitude of the the decay rate from the quasi-bound state to the N N scattering states, let us calculate the transition potential V N N ;∆∆ ( r) by using the general operator form in I = 0 [31,32], where S i (i = 1, 2) is the transition operator from the spin-3/2 state to the spin-1/2 state 1 , and S A 12 (A = σ, S) is the tensor operator associated with σ and S, respectively: For N N system with s = 1 and I = 0, we have σ 1 · σ 2 = 1, so that V N N ;N N 0 (r) and V N N ;N N σ (r) σ 1 · σ 2 are combined into The potentials, V N N ;N N (r) and V N N ;∆∆ (r), appear in the coupled channel equations between N N and ∆∆ [30] ∇ 2 m N − ∂ ∂t + 1 4m N ( r, t) being the N N operator with s = 1, I = 0, and J = 1, 3, andJ (s ,I=0) ∆∆ (0) being the ∆∆ source operator constructed from wall-type quark source with internal spin s = J. R ∆∆ J ( r, t) is defined to include the wave function renormalization factor (Z-factor) and the kinetic correction factor to compensate the threshold energy difference between ∆∆ and N N [33,5].
To extract the potentials from Eq. (A.4), we have to utilize R N N J ( r, t) and R ∆∆ J ( r, t) with given J. Since our ∆∆ source operator with internal spin s is invariant under the A + 1 projection, it contains not only l = 0 but also l ≥ 4. Therefore, it couples to the multiple total angular momenta, J = s , |s − 4|, |s − 4| + 1, . . . . To construct the N N -∆∆ correlation with given J, we employ the Misner's projection, where each (l, l z ) contribution can be obtained separately by using points inside the shell that are not connected with each other under the cubic transformation [34,35]. For the sink operator with the internal spin s, we perform the (l, l z ) projection by Misner's method and have constructed J-projection using appropriate Clebsch-Gordan coefficients.
In Alternatively, by assuming that the spin-spin part of the transition potential, V N N ;∆∆ S (r) S 1 · S 2 , which cannot make the transition from S-wave to higher partial waves, is negligibly small, we have extracted the remnant three potentials from the l = 0 and l = 2 components in J = 1 and the l = 2 component in J = 3. Again, we have used Misner's projection.
Shown in Fig (r) show the qualitatively similar behavior of the phenomenologically well-known potential in the spin-triplet channel of N N system: the short-range repulsion and the intermediate-range and long-range attraction for the central potential and the all-range negative tensor potential. Furthermore, we find that all the results obtained by the coupled channel equations using ∆∆ sources in J = 1 and J = 3 are nearly identical with the previous results obtained by the single-channel equation using N N source in J = 1 [36,10]. In Fig. A.4(a) -(b), we also show the results from the single-channel equation at κ uds = 0.13800 corresponding to m π = 679MeV, taken from Ref. [10] (where m π = 672MeV is quoted due to the different statistics and fit-range). This good agreement implies that the analysis of the three potentials by neglecting the spin-spin part of the transition potential works well 2 .
In Fig.A.4(c), we find that the tensor part of the transition potential increases significantly as r decreases for all the quark masses, while it has relatively large statistical errors compared with the other potentials. Using the transition potential, we then have estimated the decay rate at J = 3 from the quasi-bound state of the ∆∆ system in the Swave to N N in the D-wave given by with K µ (2m ∆ − B ∆∆ , 0) andψ N N 3 D3 (r) and ψ ∆∆ 7 S3 (r) being radial wave function of N N scattering state in 3 D 3 channel and that of the ∆∆ quasi-bound state in 7 S 3 channel, respectively. Here, we have used the transition potential at t/a = 10 by fitting the two r-Gaussian form, with fitting parameters p i , q i (i = 1, 2), and the ∆∆ wave function by solving the Schrödinger equation using the central potential at t/a = 10. For the 2 Having neglected the tensor part instead of the spin-spin part, the obtained central potential and tensor potential in N N system are completely different from the previous results in Ref. [10]. Even the short-range repulsion cannot be found. sake of simplicity, we have employed the free radial wave function for the N N scattering state, ψ N N 3 D3 (r) = − √ 10πj 2 (kr), with j 2 (kr) being the spherical Bessel function of order two. This results in Γ = (1.6(6)MeV, 5.6(1.7)MeV, 6.4(1.8)MeV) for m π = (1018MeV, 841MeV, 679MeV). Due to the repulsive interaction, the wave function for the N N scattering state in higher partial waves becomes smaller at short distances, only where the transition potential becomes non-negligible. Therefore, the decay rate is further reduced if more realistic wave function is employed. (r) of the diagonal potential in N N system and the tensor part of the transition potential V N N ;∆∆ T (r) from ∆∆ to N N at κ uds = 0.13710, 0.13760, 0.13800 corresponding to mπ = 1018MeV, 841MeV, 679MeV, and t/a = 10. The single channel results for the central potential and the tensor potential at κ uds = 0.13800 obtained by using N N source in the conventional method are taken from Ref. [10].