Ⅰ. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 51 (2018)

Origin of the fake eigen energy of the two-baryon system
in lattice QCD†
T. Iritani∗1 for HAL QCD Collaboration
Both the direct and HAL QCD methods are used
to study two-hadron systems in lattice QCD. In previous studies for large pion masses,2) the direct method
showed that both dineutron and deuteron are bound.
However, the HAL QCD method suggests that theseare unbound. In the series of papers,3,4) we pointed outthat these discrepancies originate from the misidentiﬁcation of the ground state in the direct method due tothe scattering states,3) which can be revealed by some
simple tests using Lüscher’s ﬁnite volume formula.4)
In the direct method, one measures the energy eigenvalue. It is estimated by the plateau value of the eﬀective energy shift, which is given by[] []/ ∑∑1∆Eeﬀ (t) ≡ log
R(⃗r, t)R(⃗r, t + a) (1)a⃗r

Fig. 1. The eﬀective energy shift using the wall and thesmeared source for ΞΞ(1 S0 ) at mπ = 0.51 GeV. The
lattice size L = 48 with the lattice spacing a ≃ 0.09 fm.

⃗rusing the R-correlatorR(⃗r, t) ≡⟨0|T {B(⃗x + ⃗r, t)B(⃗x, t)}J (0)|0⟩,{CB (t)}2(2)where J (B) is a source(sink) operator and the baryon
propagator CB (t) ≡ ⟨B(t)B̄(0)⟩. It converges to theground state energy at a large time, where the groundstate is saturated. For example, the inelastic state becomes negligible around 1 fm, while the elastic excitation in the two-baryon system remains even aroundO(10) fm, which causes a fake plateau-like structurearound 1.5 fm in the actual calculations.
Such a fake plateau problem can be checked by thesource dependence.3) Figure 1 shows the eﬀective energy shift of ΞΞ(1 S0 ) at mπ = 0.51 GeV using the walland the smeared sources. There is a plateau-like structure around t ∼ 15a ≃ 1.5 fm, but it depends on thesource, which means either (or both) of the results isfake.
Since the time-dependent HAL QCD method uses
both the ground and the scattering states simultaneously to extract the interaction, it does not require theground-state saturation. In this method, the potential
is deﬁned from the R-correlator, and some systematic
uncertainties are shown to be under control.1)
Using the correct eigen energies ∆En and eigenfunction Ψn (r), which are obtained by solving H ≡H0 + V (r) with the HAL QCD potential V (r) in theﬁnite box, the R-correlator is expanded by∑∑∑R(⃗
p = 0, t) ≃bn e−∆En t (3)an Ψn (⃗r)e−∆En t =⃗r†∗1nFig. 2. Reconstructed ∆Eeﬀ (t) and its convergence.

The contamination coeﬃcients bn are determined fromthe orthogonality of Ψn (⃗r).
Figure 2 shows the ∆Eeﬀ (t) reconstructed using alow-lying bn and ∆En , which well reproduces the fakeplateau. The ground-state saturation of the smeared
source is estimated to be around t ∼ 100a ∼ 10 fm
at L = 48. This result proves the advantages of theHAL QCD method, and the direct measurement of thetwo-baryon system is not practical.
References
1) T. Iritani for HAL QCD Coll., EPS Web Conf. 165,05008 (2018).
2) T. Yamazaki, PoS LATTICE 2014, 009 (2015).
3) T. Iritani for HAL QCD Coll., J. High Energy Phys.
1610, 101 (2016).
4) T. Iritani for HAL QCD Coll., Phys. Rev. D 96, 034521(2017).

nCondensed from the article in Proceedings for the 35th International Symposium on Lattice Field Theory1)
RIKEN Nishina Center- 13 -