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

Are two nucleons bound in lattice QCD for heavy quark masses?†T. Iritani∗1 for HAL QCD Collaboration
The interactions between hadrons are important forunderstanding the origin of the matter. These interactions are described by quantum chromodynamics.
There are two approaches to study two-hadron systems
based on lattice QCD. In the direct method, one extracts the eigenenergy of a two-particle system throughtemporal correlation. In the HAL QCD method, potential is determined by spatial correlation.
These two methods should be equivalent. However, in the previous studies on heavy quarks masses,
deuterons and dineutrons have been found to be boundin the direct method and unbound in the HAL QCD
method. In a series of papers,1,2) we reported that thedirect method experiences systematic uncertainties resulting from elastic scattering states.
In the direct method, the energy shift is measured
using the temporal correlation where the ground stateis dominant. For example, inelastic excitation can beneglected around 1 fm. However, in two-particle systems, there are elastic scattering states, whose gap
is considerably smaller than that of inelastic scattering states. Typically, ground state saturation requires
O(10) fm. Owing to its slow convergence, one might
misidentify the ground state energy around 1.5 fm.
To resolve such a fake measurement problem, we propose a simple check of the energy shift (∆EL ) using thescattering phase shift, δ0 (k), which is given by
11 ∑k cot δ0 (k) =(1)2 − |kL/(2π)|2πL|⃗n|3Fig. 1. Scattering phase shift (Upper) NN(1 S0 ) in Yamazaki et al.3) (Lower) NN(3 S1 ) in NPLQCD Coll.4)
The red (blue) band denotes the fitted ERE for k2 < 0(k 2 > 0).

⃗n∈Zeach other. Moreover, ERE (NPL2015) violates thephysical pole condition. These EREs suggest that theground state is not measured correctly. We have concluded that there are serious uncertainties about theexistence of two-nucleon bound states for heavy quark
masses in all previous studies.1)
While the temporal correlation in the direct method
experiences elastic scattering states, we deﬁne potential through spatial correlation in the HAL QCD
method, which does not encounter this problem. Furtheremore, we show the reliability of the HAL QCD
method.5)

according to Lüscher’s ﬁnite√ volume formula, wherek is given by ∆EL = 2 m2B + k 2 − 2mB in a ﬁnite box of volume L3 . The bound state corresponds
to the pole of the S-matrix at k cot δ0 (k)|k2 =−κ20 =√− −k 2 |k2 =−κ20 , which is determined by the extrapolation of k cot δ0 (k) using eﬀective range expansion(ERE), i.e., k cot δ0 (k) ≃ 1/a0 + (1/2)reﬀ k 2 + · · ·. Inaddition, the pole satisﬁes the physical residue condition by�dd [ √ 2 ]��
�− −k � 2<. (2)[kcotδ
(k)]
�0
dk 2
dk 2
k2 =−κ20k =−κ20

However, none of the previous works analyze the boundstate correctly based on the ﬁnite volume method. We
re-examine their results carefully. For example, Fig. 1
shows the scattering phase shift obtained by Yamazakiet al.3) for NN(1 S0 ) and NPLQCD Coll.4) for NN(3 S1 ).
The ERE in the upper panel shows singular behavior,
while the EREs in the lower panel are inconsistent with†∗1References
1) T. Iritani for HAL QCD Coll., Phys. Rev. D 96, 034521(2017).
2) T. Iritani for HAL QCD Coll., J. High Energy Phys.
1610, 101 (2016).
3) T. Yamazaki et al., Phys. Rev. D 86, 074514 (2012).
4) S. R. Beane et al. [NPLQCD Coll.], Phys. Rev. D 87,034506 (2013).
5) T. Iritani for HAL QCD Coll., arXiv:1710.06147.

Condensed from the article in Physical Review D 96, 034521(2017)1)
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