Most Strange Dibaryon from Lattice QCD
Abstract
The system in the channel (the most strange dibaryon) is studied on the basis of the (2+1)flavor lattice QCD simulations with a large volume (8.1 fm) and nearly physical pion mass MeV at a lattice spacing fm. We show that lattice QCD data analysis by the HAL QCD method leads to the scattering length , the effective range and the binding energy . These results indicate that the system has an overall attraction and is located near the unitary regime. Such a system can be best searched experimentally by the pairmomentum correlation in relativistic heavyion collisions at LHC.
Introduction: Dibaryon is defined as a baryon number = 2 system (equivalently a 6quark system) in quantum chromodynamics Mulders:1980vx ; Oka:1988yq ; Gal:2015rev . So far, only one stable dibaryon, the deuteron, has been observed: It is a loosely bound system of the proton and the neutron in spintriplet and isospinsinglet channel. In recent years, there are renewed experimental interests in the dibaryons due to exclusive measurements in hadron reactions Clement:2016vnl as well as the direct measurement in relativistic heavyion collisions Cho:2017dcy . Also from the theoretical side, (2+1)flavor lattice QCD simulations of the 6quark system in a large box with nearly the physical quark masses became possible recently Doi:2017cfx . The main aim of this Letter is to report the first result and physical implication of , the strangeness dibaryon (“most strange dibaryon”), in full QCD simulations with the lattice volume (8.1 fm) and the pion mass MeV at a lattice spacing fm Ishikawa:2015rho .
Before entering the detailed discussions of our study, we first introduce the reason why such an exotic channel () is of interest in QCD. Let us consider octet and decuplet baryons in the flavor SU(3) classification. All the members of are stable under strong decay. This is why the forces between octet baryons in are most relevant in the physics of hypernuclei and of neutron stars. Also, the elusive dibaryon (a combination of , and ) is in this representation Jaffe1977 ; Inoue:2010es ; Beane:2010hg and does not suffer from the Pauli exclusion principle in the flavorSU(3) limit.
On the other hand, only in is stable under strong decay. Therefore, in the representation, the most promising candidate of stable dibaryon is Goldman:1987ma . The Pauli exclusion principle does not work in this case too, so that there is a possibility to have a bound state in the Swave and totalspin 2 channel Etminan:2014tya . Such a system is indeed studied by the twoparticle momentum correlation in highenergy heavyion collisions both theoretically and experimentally Morita:2016auo .
In the decupletdecuplet channnel, we have
where ”sym.” and ”antisym.” stand for the flavor symmetry under the exchange of two baryons. Only possible stable state under strong decay is the system in the symmetric representation. Again, the quark Pauli principle does not operate in this channel Oka:2000wj . Note here that the celebrated ABC resonance ( in the spin3 and isospin0 channel) Dyson:1964xwa ; Clement:2016vnl belongs to the antisymmetric representation, while in the spin0 and isospin3 channel is in the same multiplet with .
The interaction at low energies has been investigated so far by using phenomenological quark models or by using lattice QCD simulations with heavy quark masses. Very recently, the chiral effective field theory has also been applied to the scattering of the baryons Haidenbauer:2017sws . In some quark models, strong attraction is reported zhang:1997 ; zhang:2000 , while other models show weak repulsion wang:1992 ; wang:1997 . A (2+1)flavor lattice QCD study with MeV by using the finite volume method shows weak repulsion Buchoff:2012prd , while a study with MeV by using the HAL QCD method shows moderate attraction Yamada:2015cra . Under such a controversial situation, it is most important to carry out firstprinciples lattice QCD simulations in a large volume with the pion mass close to the physical point.
HAL QCD method: In the HAL QCD method Ishii:2006ec ; Aoki:2010ptp ; Ishii:2012plb ; Aoki:2012tk , the observables such as the binding energy and phase shifts are obtained from the equaltime NambuBetheSalpeter (NBS) wave function and associated twobaryon irreducible kernel . It was recently demonstrated that the traditional finite volume method with the plateau fitting luscher:1991 has a fatal problem for systems in large volumes because of its inability to differentiate each scattering state Lepage:1990 ; Iritani:2016jie ; Iritani:2017rlk ; Aoki:2017byw . On the other hand, the timedependent HAL QCD method Ishii:2012plb can avoid such a problem since all the elastic scattering states are dictated by the same kernel and there is no need to identify each scattering state in a finite box.
The equaltime NBS wave function has the property that its asymptotic behavior at large distance reproduces the scattering phase shifts, which can be proven from the unitarity of matrix in quantum field theories Ishii:2006ec . Moreover, it is related to the following reduced fourpoint (4pt) function,
(1)  
Here is a source operator creating the system at Euclidean time 0, and is the matrix element defined by with representing the elastic scattering states in a finite volume. The scattering energy is represented as with the baryon mass and the relative momentum . Typical excitation energy of a single baryon is denoted by , so that the last term in Eq.(1) is exponentially suppressed as long as Ishii:2012plb with MeV being the QCD scale parameter. A local interpolating operator for the baryon has a general form
(2) 
with , and being color indices, being the Dirac matrix, being the spinor index, and being charge conjugation. An appropriate spin projection is necessary from this operator to single out a particular spin state as mentioned later.
The reduced 4pt function has been shown to satisfy the following master equation Ishii:2012plb ,
which is valid as long as . We emphasize here that we do not need to isolate each scattering state with the energy , so that only the moderate values of up to 1.52 fm are sufficient for a reliable extraction of the kernel . (This can be quantified by the stability of as a function of .) This is a crucial difference from the finite volume method which requires fm (for the lattice volume as large as 8 fm) to identify each . (See a recent summary Aoki:2017byw and references therein.) At low energies, one can use the derivative expansion with respect to the nonlocality of the kernel Murano:2011nz , , so that the leadingorder local potential reads
(4) 
Interpolating operator: The interpolating operator for the baryon with spin and the component can be readily constructed by the appropriate spin projection of the upper two components of Eq.(2) as shown in Yamada:2015cra . The asymptotic system can now be characterized by with the total spin , the orbital angular momentum and the total angular momentum . We consider where maximum attraction is expected at low energies. Then, the Fermi statistics leads to be even (either =0 or 2). Here we consider an system with the interpolating operator
For , we use the walltype quark source with the projection given above. With this source, the total momentum of the system is automatically zero. Also, it has good overlap with the scattering states where in Eq.(LABEL:eq:tdep) is larger than the typical baryon size. To extract the and state at on the lattice, we employ Eq.(Most Strange Dibaryon from Lattice QCD) for the sink operator together with the projection to the representation of the cubicgroup,
(6) 
where is an element of the cubic group acting on the relative distance .
Note here that and depend on the choice of interpolating operators, while observables calculated from these quantities are independent of the choice thanks to the NishijimaZimmermannHaag theorem Aoki:2010ptp .
Lattice Setup: By using the 11PFlops supercomputer K at RIKEN Advanced Institute for Computational Science, flavor gauge configurations on the lattice are generated with the Iwasaki gauge action at and nonperturbatively improved Wilson quark action with stout smearing. The lattice spacing is fm ( GeV) Ishikawa:2015rho and the pion mass, the kaon mass and the nucleon masses are MeV, MeV and MeV, respectively. (These masses are higher than the physical values by about 8%, 6 % and 3 %, respectively, due to slightly larger quark masses at the simulation point.) The lattice size, fm, is sufficiently large to accommodate two baryons in a box.
We employ the wall quark source with the Coulomb gauge fixing, and the periodic (Dirichlet) boundary condition is used for spatial (temporal) directions. Forward and backward propagations are averaged to reduce the statistical fluctuations. We pick 1 configuration per each 5 trajectories, and make use of the rotation symmetry and the translational invariance for the source position to increase the statistics. The total statistics in this Letter amounts to 400 configurations 2 (forward/backward) 4 rotations 48 source positions. The quark propagators are obtained by the domaindecomposed solver Boku:2012zi ; Terai:2013 ; Nakamura:2011my ; Osaki:2010vj and the correlation functions are calculated using the unified contraction algorithm Doi:2012xd .
The baryon mass extracted from the effective mass with being the baryonic twopoint function is MeV (from the plateau in ) and MeV (from ) with the statistical errors. These numbers are about 2% higher than the physical value, 1672 MeV. We take the former number in the following analysis.
Numerical results: The potential obtained from Eq.(4) with the lattice measurement of is shown in Fig. 1 for and 18. Here Laplacian and the timederivative in Eq.(4) are approximated by the central (symmetric) difference. The statistical errors for at each are estimated by the jackknife method with the bin size of 40 configurations. A comparison with the bin size of 20 configurations shows that the bin size dependence is small. The particular region in Fig. 1 is chosen to suppress contamination from excited states in the single propagator at smaller and simultaneously to avoid large statistical errors at larger . We observe that the potentials at and 18 are nearly identical within statistical errors as expected from the timedependent HAL QCD method Ishii:2012plb .
The potential has qualitative features similar to the central potential of the nucleonnucleon () interaction, i.e., the short range repulsion and the intermediate range attraction Doi:2017cfx . There are, however, some quantitative differences: First of all, the short range repulsion is much weaker in the case possibly due to the absence of quark Pauli exclusion effect. Secondly, the attractive part is much shortranged due to the absence of pion exchanges.
For the purpose of converting the potential to the physical observables such as the scattering phase shifts and the binding energy, we fit in Fig. 1 in the range by three Gaussians, . For example, the uncorrelated fit in the case of gives the following parameters: in MeV and in fm with . Another functional form such as two Gaussians + (Yukawa function) provides equally well fit, and the results are not affected within errors. The finite volume effect on the potential is expected to be small due to the large lattice size. The naive order estimate of the finite effect for the physical observables is also small ( %) thanks to the nonperturbative improvement, but an explicit confirmation would be desirable in the future.
The scattering phase shifts in the channel obtained from are shown in Fig.2 for and as a function of the kinetic energy in the center of mass frame, . The error bands reflect the statistical uncertainty of the potential in Fig. 1. All three cases show that starts from 180, which indicates the existence of a bound system.
The scattering length and the effective range for the system in the channel can be extracted from through the effective range expansion (ERE), , with the sign convention of nuclear and atomic physics:
(7)  
(8) 
The central values and the statistical errors in the first parentheses are extracted from at , and the systematic errors in the second parentheses are estimated from the results at and 18. To get a feel for the magnitude of these values, we recapitulate here the experimental values of and in the systems; and Hackenburg:2006qd . There exists no symmetry reason that the scattering parameters in the systems and those in the system should be similar. Nevertheless, it is remarkable that they are all close to the unitary region where is substantially smaller than 1 as shown in Fig.3.
Shown in Fig.4 are the bound state energy given by the opposite sign of the binding energy, , and the rootmeansquare distance () of the bound state obtained from the potential. The blue diamond is taken from the data at without the Coulomb repulsion. The blue solid and dashed lines are the statistical error at and the systematic error estimated from the data at , respectively:
(9) 
This is consistent with the value obtained from the general formula for loosely bound states Naidon:2016dpf with (7) and (8); MeV. Associated with this small binding energy, is as large as 34 fm which is consistent with the expectation, , for loosely bound states. The Coulomb repulsion reduces the above binding energy by a factor of two, as shown in Fig.4 by the red triangle.
It is in order here to remark that there are three energy scales in the present problem: 3400 MeV 50 MeV MeV. Since only the relative difference between the interacting and noninteracting two systems matters, the absolute magnitude of the uncertainty of is not reflected directly in . This is why we could extract rather accurately as shown in Fig.1 despite of the large scale difference between and . Then the small binding energy as well as the large scattering length are the natural consequence of the large cancellation between the longrange attraction and the shortrange repulsion of , a situation common in nuclear and atomic physics. Although is not a direct observable, it provides an essential intermediate step to link the QCD scale (GeV) to nuclear physics scale (MeV) since attempts to measure the binding energy directly from lattice QCD using the finite volume method do not work for large lattice volumes and physical quark masses (see the critical review Aoki:2017byw ).
Finally let us estimate the effect of slightly heavy quark masses in our simulation. First of all, the attractive part of the potential would be slightly longranged at the physical point due to the kaon mass, MeV MeV. On the other hand, the effect of the mass, MeV MeV, would lead to lessbinding due to the larger kinetic energy. Therefore, conservative estimate is obtained by keeping the same in Fig.1 and to adopt to calculate the phase shifts and the binding energy. This results in and for the potential at . These numbers are well within errors of the results of the present simulation shown in Figs. 3 and 4.
Summary and Discussions: In this Letter, we presented a first realistic calculation on the most strange dibaryon, , in the channel on the basis of the (2+1)flavor lattice QCD simulations with a large volume and nearly physical quark masses. The scattering length, effective range and the binding energy obtained by the HAL QCD method strongly indicate that the system in the channel has an overall attraction and is located in the vicinity of the unitary regime. From the phenomenological point of view, such a system can be best searched by the measurement of pairmomentum correlation with being the relative momentum between two baryons produced in relativistic heavyion collisions at LHC Cho:2017dcy . Experimentally, each can be identified through a successive weak decay, . Note that a large scattering length (not the existence of a bound state) is the important element for to have characteristic enhancement at small relative momentum . Moreover, the effect of the Coulomb interaction can be effectively eliminated by taking a ratio of between small and large collision systems Morita:2016auo .
We are currently underway to increase the statistics of the lattice data with the same lattice setup. These results together with the detailed examination of the spectrum analysis in a finite lattice volume and the effective range expansion will be reported elsewhere.
Acknowledgements.
We thank members of PACS Collaboration for the gauge configuration generation. The lattice QCD calculations have been performed on the K computer at RIKEN, AICS (hp120281, hp130023, hp140209, hp150223, hp150262, hp160211, hp170230), HOKUSAI FX100 computer at RIKEN, Wako (G15023, G16030, G17002) and HAPACS at University of Tsukuba (14a20, 15a30). We thank ILDG/JLDG conf:ildg/jldg which serves as an essential infrastructure in this study. We thank the authors of cuLGT code Schrock:2012fj used for the gauge fixing. This study is supported in part by GrantinAid for Scientific Research on Innovative Areas(No.2004:20105001, 20105003) and for Scientific Research(Nos. 25800170, 26400281), SPIRE (Strategic Program for Innovative REsearch), MEXT GrantinAid for Scientific Research (JP15K17667, JP16H03978, JP16K05340), ”Priority Issue on PostK computer” (Elucidation of the Fundamental Laws and Evolution of the Universe), and by Joint Institute for Computational Fundamental Science (JICFuS). S.G. is supported by the Special Postdoctoral Researchers Program of RIKEN. T.D. and T.H. are partially supported by RIKEN iTHES Project and iTHEMS Program. T.H. is grateful to the Aspen Center for Physics, supported in part by NSF Grants PHY1066292 and PHY1607611. The authors thank C. M. Ko for drawing out attention to the system, and K. Yazaki for fruitful discussions on the short range part of baryonbaryon interactions, and Y. Namekawa for his careful reading of the manuscript.References
 (1) P. J. Mulders, A. T. M. Aerts and J. J. De Swart, Phys. Rev. D 21, 2653 (1980).
 (2) M. Oka, Phys. Rev. D 38, 298 (1988).
 (3) A. Gal, Acta Phys. Polon. B 47, 471 (2016) [arXiv:1511.06605 [nuclth]].
 (4) H. Clement, Prog. Part. Nucl. Phys. 93, 195 (2017) [arXiv:1610.05591 [nuclex]].
 (5) S. Cho et al. [ExHIC Collaboration], Prog. Part. Nucl. Phys. 95, 279 (2017) [arXiv:1702.00486 [nuclth]].
 (6) T. Doi et al., PoS LATTICE 2016, 110 (2017) [arXiv:1702.01600 [heplat]]. H. Nemura et al., PoS LATTICE 2016, 101 (2017) [arXiv:1702.00734 [heplat]]. K. Sasaki et al., PoS LATTICE 2016, 116 (2017) [arXiv:1702.06241 [heplat]]. N. Ishii et al., PoS LATTICE 2016, 127 (2017) [arXiv:1702.03495 [heplat]].
 (7) K.I. Ishikawa et al. [PACS Coll.], PoS LATTICE 2015, 075 (2016) [arXiv:1511.09222 [heplat]].
 (8) R.L. Jaffe, Phys. Rev. Lett. 38, 195 (1977).
 (9) T. Inoue et al. [HAL QCD Collaboration], Phys. Rev. Lett. 106, 162002 (2011) [arXiv:1012.5928 [heplat]].
 (10) S. R. Beane et al. [NPLQCD Collaboration], Phys. Rev. Lett. 106, 162001 (2011) [arXiv:1012.3812 [heplat]].
 (11) J. T. Goldman, K. Maltman, G. J. Stephenson, Jr., K. E. Schmidt and F. Wang, Phys. Rev. Lett. 59, 627 (1987).
 (12) F. Etminan et al. [HAL QCD Collaboration], Nucl. Phys. A 928, 89 (2014) [arXiv:1403.7284 [heplat]].
 (13) K. Morita, A. Ohnishi, F. Etminan and T. Hatsuda, Phys. Rev. C 94, 031901 (2016) [arXiv:1605.06765 [hepph]].
 (14) M. Oka, K. Shimizu and K. Yazaki, Prog. Theor. Phys. Suppl. 137, 1 (2000).
 (15) F. Dyson and N. H. Xuong, Phys. Rev. Lett. 13, 815 (1964).
 (16) J. Haidenbauer, S. Petschauer, N. Kaiser, U. G. Mei ner and W. Weise, arXiv:1708.08071 [nuclth].
 (17) Z.Y. Zhang, Y.W. Yu, P.N. Shen, L.R. Dai, A. Faessler, and U. Straub, Nucl. Phys. A 625, 59 (1997).
 (18) Z.Y. Zhang, Y.W.Yu, C.R.Ching, T.H.Ho, and Z.D.Lu, Phys. Rev. C 61, 065204 (2000).
 (19) F. Wang, J Ping, G Wu, L Teng, and T Goldman, Phys. Rev. C 51, 3411 (1995).
 (20) F. Wang, G. Wu, L. Teng, and T. Goldman, Phys. Rev. Lett. 69, 2901 (1992) .
 (21) M. Buchoff et al. Phys. Rev. D85, 09451 (2012) [arXiv:1201.3596[heplat]] .
 (22) M. Yamada et al. [HAL QCD Collaboration], PTEP 2015, 071B01 (2015) [arXiv:1503.03189 [heplat]].
 (23) N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99, 022001 (2007).
 (24) S. Aoki, T. Hatsuda and N. Ishii, Prog. Theor. Phys. 23, 89 (2010) [arXiv:0909.5585 [heplat]].
 (25) N. Ishii et al. [HAL QCD Collaboration], Phys. Lett. B 712, 437 (2012) [arXiv:1203.3642 [heplat]].
 (26) S. Aoki et al. [HAL QCD Collaboration], PTEP 2012, 01A105 (2012) [arXiv:1206.5088 [heplat]].
 (27) M. Lüscher, Nucl. Phys. B354, 531 (1991).
 (28) G. P. Lepage, in From Actions to Answers: Proceedings of the TASI 1989, edited by T. Degrand and D. Toussaint (World Scientific, Singapore, 1990).
 (29) T. Iritani et al., JHEP 1610, 101 (2016) [arXiv:1607.06371 [heplat]].
 (30) T. Iritani et al., Phys. Rev. D 96, no. 3, 034521 (2017) [arXiv:1703.07210 [heplat]].
 (31) S. Aoki, T. Doi and T. Iritani, arXiv:1707.08800 [heplat].
 (32) K. Murano, N. Ishii, S. Aoki and T. Hatsuda, Prog. Theor. Phys. 125, 1225 (2011) [arXiv:1103.0619 [heplat]].
 (33) T. Boku et al., PoS LATTICE 2012, 188 (2012) [arXiv:1210.7398 [heplat]].
 (34) M. Terai, K. I. Ishikawa, Y. Sugisaki, K. Minami, F. Shoji, Y. Nakamura, Y. Kuramashi, M. Yokokawa, ”Performance Tuning of a Lattice QCD code on a node of the K computer,” IPSJ Transactions on Advanced Computing Systems, Vol.6 No.3 4357 (Sep. 2013) (in Japanese).
 (35) Y. Nakamura, K.I. Ishikawa, Y. Kuramashi, T. Sakurai and H. Tadano, Comput. Phys. Commun. 183, 34 (2012) doi:10.1016/j.cpc.2011.08.010 [arXiv:1104.0737 [heplat]].
 (36) Y. Osaki and K. I. Ishikawa, PoS LATTICE 2010, 036 (2010) [arXiv:1011.3318 [heplat]].
 (37) T. Doi and M. G. Endres, Comput. Phys. Commun. 184, 117 (2013) [arXiv:1205.0585 [heplat]].
 (38) R. W. Hackenburg, Phys. Rev. C 73, 044002 (2006).
 (39) P. Naidon and S. Endo, Rept. Prog. Phys. 80, 056001 (2017)

(40)
"http://www.lqcd.org/ildg"
,"http://www.jldg.org"
 (41) M. Schröck and H. Vogt, Comput. Phys. Commun. 184 (2013) 1907 [arXiv:1212.5221 [heplat]].