Coupled channel approach to strangeness S = -2 baryon-bayron interactions in Lattice QCD

The baryon-baryon interactions with strangeness S = -2 with the flavor SU(3) breaking are calculated for the first time by using the HAL QCD method extended to coupled channel system in lattice QCD. The potential matrices are extracted from the Nambu-Bethe-Salpeter wave functions obtained by the 2+1 flavor gauge configurations of CP-PACS/JLQCD Collaborations with a physical volume of 1.93 fm cubed and with m_pi/m_K = 0.96, 0.90, 0.86. The spatial structure and the quark mass dependence of the potential matrix in the baryon basis and in the SU(3) basis are investigated.


Coupled-channel BB potentials
In this section, we briefly review the coupled-channel HAL QCD method [9,10] applicable to the inelastic scattering in which a 1 + a 2 → b 1 + b 2 , where (a 1 , a 2 ) = (b 1 , b 2 ).

Formalism
We first define the equal-time NBS wave functions with the total energy W i as where B c j ( x, t) with c = a, b and j = 1, 2 denotes a local composite operator for a baryon B c j with its wave-function renormalization factor Z c j . The state |B = 2, W i stands for a QCD asymptotic in-state with baryon number 2 and energy W i . In the present exploratory studies, we assume that √ Z a1 √ Z a2 = √ Z b1 √ Z b2 , which implies that the flavor SU(3) breaking in the wave-function renormalization factor is not sizable in the present setup. The validity of this assumption is left for future studies.
In the asymptotic region at long distance, these NBS wave functions satisfy free Schrödinger-type equations as where the corresponding asymptotic momentum k c i in the center-of-mass (CM) frame is defined through the relation with m c j being the mass of the baryon B c j , and the reduced mass μ c being given by 1/μ c = 1/m c 1 + 1/m c 2 . On the other hand, in the interaction region at short distance, we have from which we define the energy-independent non-local potential matrix as This is an extension of the HAL QCD definition for the potential to the coupled-channel case [9].
To handle the non-locality of the potential, we introduce the derivative expansion as U ( r , r ) = (V LO ( r ) + V NLO ( r ) + · · · )δ( r − r ), where the N n LO term is of O( ∇ n ). At low energies, a good convergence of the derivative expansion has been confirmed for the NN case [17].

Extraction of potential matrix
In the leading order of the derivative expansion of the non-local potential, Eqs. (4) and (5) can be written as a coupled-channel form of the Schrödinger equation for two independent channels a and b, where the kinetic energy and the free Hamiltonian for channels c = a, b are given by 2μ c and H 0 c = − ∇ 2 2μ c , respectively. Two pairs of NBS wave functions, ψ a 2 , are necessary to extract the local potential matrix from the above coupled-channel equation. In the infinite volume, we can have two states, |a, W and |b, W , with a given energy W , which are connected to the asymptotic scattering states if W is larger than m a 1 + m a 2 and m b 1 + m b 2 . This implies that two nearby eigenstates, |B = 2, W 1 and |B = 2, W 2 with W 1 − W 2 = O(L −2 ), exist even for finite volume. Suppose that W 1 < W 2 are the two lowest energies of two baryons in the finite volume. By using the wall-source operators I a (t) = B a 2 B a 1 (t) and I b (t) = B b 2 B b 1 (t), 1 the states |B = 2, W 1 and |B = 2, W 2 are created as where the coefficient matrix C cj can be determined from the two-baryon correlation functions. We then define the optimized source operators as so that the four-point (4-pt) function F c for i = 1, 2 and c = a, b, where W 3 corresponds to the 3rd state satisfying W 1 < W 2 < W 3 < W j≥4 . By using these 4-pt functions, the coupled-channel potential matrix can be determined as for sufficiently large t, where the states with W (>W 1 , W 2 ) can be neglected in the above 4-pt functions. As the volume increases, however, the spectrum becomes denser and the two low-lying states W 1 and W 2 cannot be isolated unless extremely large t is achieved. This is why we need an improved method in practice, as explained in the next subsection.

Time-dependent method
An improved method to extract the potentials without using ground-state saturation has been proposed in Ref. [18] in the case of the single channel. In this subsection, we extend this method to the coupled-channel case. We first introduce the normalized 4-pt correlation function R, defined as where here is defined through the original wall-source operator I d (0) instead of I W i (0). The ellipses in Eq. (11) denote inelastic contributions from channels other than a and b.
In the non-relativistic approximation valid at low energies, W c j E c j , we can replace the kinetic energy term in the equation with the time derivative as with which we obtain the Schrödinger-type equation where If we go beyond the non-relativistic approximation, higher-order time derivatives appear, which we will not consider in this paper. Expanding U in terms of derivatives again, the leading-order coupled-channel potentials can be obtained as  2 27 Extension of this formula to three channels is straightforward. For Eq. (14) to work, two independent source operators I a and I b are needed, while no optimization is required. Note that isolation of each eigenstate is not necessary in this method [18]. The only constraint is to keep moderately large t so that other channels with larger threshold energies than a and b can be suppressed. In the following, we employ this improved method in our numerical calculations.

Strangeness S = −2 two-baryon system
We emply the following interpolating operator for octet baryons: with the Dirac index α, which represents the spin of the octet baryons. Denoting the quark flavors as q = u, d, s for "up", "down", and "strange", respectively, the flavor structures of the baryons are given in terms of the isospin multiplets as  Table 1, where the I z = 0 components are given as

Numerical simulations
We   Wall-source operators that generate positive-parity two-baryon states with flavor structures h 1 and h 2 are given by whereQ = xq ( x) is the quark wall source. Projection operators for spin-singlet and spin-triplet states are given by Quark propagators are calculated for the wall source at t 0 with the Dirichlet boundary condition in the temporal direction at t = 16 + t 0 . The wall source is placed at 32 different values of t 0 on each gauge configuration, in order to increase the statistics, in addition to the average over forward and backward propagations in time. The A + 1 projection of the cubic group is taken for the sink operator to obtain the relative S-wave in the BB wave function. 2 Numerical computations have been carried out using the KEK supercomputer system, Blue Gene/L, and the kaon and jpsi clusters at Fermilab. The hadron masses obtained in our calculation are given in Table 2. The thresholds of two-baryons with strangeness S = −2 for each set of gauge configurations are plotted in Fig. 1.

Numerical results
We now present our results of coupled-channel BB potentials in the strangeness S = −2 sector.

Time dependence
We first show how the time-dependent method extended to the coupled-channel system works in our calculation. For this purpose, we investigate the time dependences of the diagonal potentials. Figure 2 shows V in the 1 S 0 (I = 2) channel (5th line in Table 1) and V N N in the 3 S 1 (I = 0) channel (2nd line in Table 1) at three values of t − t 0 (= 8, 9, 10) with Set 3, which has the lightest pion mass in our calculation. Within statistical errors, no significant t − t 0 dependence is observed for these PTEP 2015, 113B01 K. Sasaki et al. single-channel potentials with Set 3, showing that t − t 0 = 8 is large enough to suppress inelastic contributions and that higher-order contributions in the derivative expansion are negligible.
In Fig. 3, two diagonal potentials in the 1 S 0 (I = 1) channel (3rd line in Table 1) calculated with Set 1 are shown at t − t 0 = 8-10. Again, no significant t − t 0 dependence is observed at this quark mass and this is true at other quark masses. Similarly, three diagonal potentials in the 3 S 1 (I = 1) channel (4th line in Table 1) and those in the 1 S 0 (I = 0) channel (1st line in Table 1) show no significant t − t 0 dependence at all quark masses, as seen in Fig. 4 for Set 2.
Since no significant t − t 0 dependences are observed for all diagonal potentials at t − t 0 = 8, 9, 10, we hereafter consider the results at t − t 0 = 8, where statistical errors are smallest.

Hermiticity
Hermiticity of the potential matrix is a sufficient condition for the probability conservation, though it is not a necessary condition. In this subsection, we investigate the hermiticity of the potential matrix, V a b = V b a , since it is not automatically guaranteed in the definition of the coupled-channel potential matrix in Eq. (5). As in the case of the diagonal parts, we confirm that the off-diagonal parts of the potential matrix show no significant t − t 0 dependence, so we take the results at t − t 0 = 8 in our analysis.
We introduce a hermiticity measure to see the relative magnitude of the hermiticiy violation of the potential matrix.  (I = 1) channel with Set 1 (red), Set 2 (blue), and Set 3 (green). It satisfies the hermiticity well within the statistical errors. Figure 6 shows δV a−b for a, b = N , , in the 3 S 1 (I = 1) channel. Some violations of hermiticity can be seen in δV N − and δV − in the r < 0.5 fm region. Those for a, b = , N , in 1 S 0 (I = 0) are given in Fig. 7. Hermiticity is more or less satisfied within the statistical errors. It is a problem for future work to check whether the possible hermiticity breaking for small r in Fig. 6 and large r in Fig. 7 disappears or not by removing our assumption on the wave-function renormalization factor introduced after Eq. (1).

Potential matrices and their quark mass dependence
We here discuss the properties of potentials in three cases, single channel, two channels, and three channels, separately.     Figure 8 shows the quark mass dependences of the potential in the 1 S 0 (I = 2) channel (left) and the N potential in the 3 S 1 (I = 0) channel (right). We first notice non-smooth behaviors as a function of r at large r for both cases, which indicates that the spatial volume is not sufficiently large. In addition, the non-smooth behavior at short distance observed in the N 3 S 1 (I = 0) channel may be caused by the finite-lattice-spacing effect. With these systematics, we discuss only the qualitative features of the potentials in this report, and leave quantitative analysis such as the extraction of scattering phase shifts for future studies with larger and finer lattices.

Single channel
The potential in the 1 S 0 (I = 2) channel (left), which belongs to the 27-plet irreducible representation in the flavor SU(3), has repulsion at short distance and attraction at long distance.  Also, the magnitude of these two components increases as the light (ud) quark mass decreases, as in the case of the NN potential in the 1 S 0 sector belonging to the 27-plet. An increase of attraction at long distance, r > 0.8 fm, may be related to the decrease of the mass of the pion exchanged between the two .
Similarly, the N potential in the 3 S 1 (I = 0) channel (right) has both repulsion at short distance and attraction at long distance. The magnitude of these two components is enhanced as the light quark mass decreases. It should be remarked that the repulsion at short distance here is weaker than that of the potential in the 1 S 0 (I = 2) channel. This difference may be related to the fact that the Pauli blocking in the quark level for N in the 3 S 1 (I = 0) channel is weaker than in the 1 S 0 (I = 2) channel.

Two channels
A potential matrix in the 1 S 0 (I = 1) channel, which has N and components, is given in Fig. 9, which shows that the diagonal elements of the potential matrix in this channel, V N N and V , are both strongly repulsive and the off-diagonal element, V N , is comparable to or even larger than the diagonal elements. These features have already been observed in the flavor SU(3) symmetric limit [8]. Figure 10 shows a potential matrix in the 3 S 1 (I = 1) channel, which has N , , and components. All diagonal elements of the potential matrix, middle), and V (left lower), have an attraction at long distance and a repulsive core at short distance. The largest attraction in this channel appears in V , whose maximum depth is about −60 MeV at around r ∼ 0.6 fm. All the diagonal potentials show a tendency for the magnitudes of both repulsion at short distance and attraction at long distance to increase as the light quark masses decrease.

Three channels
For the off-diagonal elements of the potential matrix, V N and V are much smaller than V N . These off-diagonal potentials, V N and V , almost vanish at r > 1.2 fm and have a small quark mass dependence, while V N increases as the light quark masses decrease. Figure 11 shows the potential matrix in the 1 S 0 (I = 0) channel, where the H dibaryon state may appear. This potential matrix is also important for the study of double-hypernuclei [23][24][25]. All diagonal elements of the potential matrix have a repulsive core at short distance, whose strength, however, depends strongly on the state. An attractive pocket, on the other hand, appears only in two diagonal elements, V and V N N , where V N N has a much deeper attractive pocket than V does, while V is totally repulsive in the whole range of r . The off-diagonal element, V N , is smaller than the other two, so that the decay rate from N to may be relatively suppressed. The diagonal elements of the potential matrix generated with the configuration Set 2 are most attractive, while the off-diagonal potentials with the configuration Set 3 are the strongest in magnitude for r > 0.5 fm.
We have estimated the volume integral of the diagonal N -potential with spin and isospin averages. The result shows that there is an overall attraction in all sets. This may have some relevance to the first hypernucleus recently reported in Ref. [26].

Potential matrix in the SU(3) irreducible representation basis
We here present potential matrices in the SU(3) irreducible representation basis (SU(3) basis for short hereafter) such as 1, 8 s , 8 a , 10, 10, 27, obtained from the particle basis by using Clebsch-Gordan coefficients. This makes it possible for us to compare the results with those in the flavor SU(3) symmetric limit [8].
The transformation from particle basis to SU(3) basis is defined as for the 3 S 1 (I = 1) channel. Figure 12 shows the potential matrix in the SU(3) basis for 3 S 1 (I = 1), which is composed of 8 a , 10, and 10. While all the diagonal elements of the potential matrix have a repulsive core, the height of the repulsive core in V 8 a 8 a is much lower than that for the other two and its attractive-pocket depth is the deepest of the three. On the other hand, V 10 10 is strongly repulsive and has quite a shallow attractive pocket at all quark masses, though the height of the repulsive core and the range of the attractive pocket increase as the ud quark mass decreases. As far as the off-diagonal elements are concerned, they are very small. In particular, V 10 10 vanishes at all quark masses including Set 3, where the SU(3) breaking by the difference between the ud and s quark masses is maximal in our calculation. The other two off-diagonal elements, V 8 a 10 and V 8 a 10 , have small non-zero values in the short-distance region (r < 0.6 fm), which gradually increase as the ud quark masses decrease. Figure 13 shows the potential matrix in the SU(3) basis for 1 S 0 (I = 0). As in the case of the SU(3) limit [8], the diagonal element for the flavor singlet state, V 1 1 , is strongly attractive, while V 8 s 8 s is repulsive, at all distances. The absence of a repulsive core in V 1 1 is consistent with the absence of the quark Pauli blocking effect. 3 The shape of V 27 27 is similar to the 1 S 0 nuclear force, which also belongs to the 27-plet.
The quark mass dependences of the diagonal potentials can be clearly seen in the flavor basis. As the light quark mass decreases, the attraction in V 1 1 gradually increases, while both repulsive core and attraction at long distance in V 27 27 are enhanced. The off-diagonal elements of the potential matrix in the SU(3) basis are presented in the right-hand three panels in Fig. 13, which give effective measures of the flavor SU(3) breaking effects since they are absent in the flavor SU(3) symmetric limit. Figure 13 shows that distances, r < 0.2 fm, where the cutoff effects could be sizable. These results tell us that the flavor SU(3) breaking effects in the off-diagonal parts are much smaller than those in the diagonal part within the quark masses adopted in this paper. The 27-plet state is almost uncoupled even if the ud quark mass is different from the s quark mass.    Figure 14 shows the diagonal and off-diagonal parts of the potential matrix in the SU(3) basis for the 1 S 0 (I = 1) channel. We find that the diagonal elements, V 8 s 8 s and V 27 27 , have similar behaviors to those obtained from 1 S 0 (I = 0), and the transition potential between the 8 s -plet and the 27-plet is quite small for all sets.
To see the effects of SU(3) breakings, it is interesting to compare the potentials in the SU(3) basis extracted from two different channels, 1 S 0 (I = 0) and 1 S 0 (I = 1). In Fig. 15, we show the measure of SU(3) breaking defined as . We find that, for V (8 s ) , there are no significant deviations from zero within statistical errors for all sets. Similar behavior to the V (8 s ) case can be seen again for the V (27) case, except for a singular behavior at r 0.4 fm where the 27-plet potentials themselves almost vanish. 15 (27) (right) as a function of r . Red, blue, and green symbols stand for results with Sets 1, 2, and 3, respectively.

Summary and conclusions
In this paper, we have investigated the S = −2 BB potentials from 2 + 1-flavor lattice QCD by using the HAL QCD method extended to coupled-channel systems in Ref. [9]. Combining the coupled-channel formalism with the time-dependent Schrödinger equation [18], we could extract potential matrices for the first time without ground-state saturation and without diagonalization of the source operators.
By considering two-baryon systems with S = −2, , N , , and , which are mutually coupled, we successfully extracted potential matrices. They are approximately hermitician within the statistical errors, which is not guaranteed from the definition. A small violation of hermiticity may be removed at least partly by proper treatment of the renormalization factors, which is left for future studies.
We have discussed the properties of the potential matrices for all S = −2 two-baryon systems. We found that all diagonal elements of the potential matrix have a repulsive core, while their heights largely depend on their flavor structure. Our previous works show that a decreasing ud quark mass leads to an enhancement in the short-range repulsion and the long-range attraction. Although such quark mass dependence was clearly seen in the single-channel case, it becomes less pronounced in the two-and three-channel cases.
The potentials in the SU(3) basis have also been investigated, where we could see clear quark mass dependence. We found a strongly attractive potential for V 1 PTEP 2015, 113B01 K. Sasaki et al.
breaking. In the 3 S 1 (I = 1) channel, except for V 10 10 , only a small transition potential between irreducible representations could be seen at short distances. In the 1 S 0 (I = 0) channel, we found a clear mixture of the flavor singlet state and the octet state. The other off-diagonal potentials have only a small magnitude at short distances. Note, however, that the SU(3) breaking introduced in this paper (m π /m K = 0.96, 0.90, 0.86) is still small compared to the realistic magnitude of the breaking (m π /m K = 0.27). Nevertheless, the present paper provides a first theoretical and numerical step toward a realistic BB potential matrix at physical quark masses.