Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 47 (2014)

Cutoﬀ eﬀects on lattice nuclear forces†
T. Doi∗1†∗1

Condensed from the article in PoS LAT2013, 226 (2013)
RIKEN Nishina Center

1400

a=0.1076fm
a=0.1555fm
a=0.2150fm

V(r) [MeV]

1200
1000

0800
600
400

0200

0.5

111.5
r [fm]

2000.5Fig. 1. Central potentials VC (r) in 1 S0 channel. Inset
shows r2 VC (r) to include phase space factor. Solid lines
correspond the fit for the potentials.
3scattering length [fm]Nuclear forces, the interactions among nucleons,serve as the cornerstone in nuclear physics. While they
have been traditionally determined through the scattering experiments, their theoretical understanding byusing the fundamental theory, quantum chromodynamics (QCD), has not been established yet. Recently,
a novel approach was proposed to determine nuclear
forces on a lattice1,2) . In this approach, now called
the HAL QCD method, nuclear forces are directly obtained from Nambu-Bethe-Salpeter wave functions calculated on the lattice. The method has been successfully extended to general hadron interactions such as
three-nucleon forces3) . See Ref.4) for a recent review.
For the quantitative determination of nuclear forces,
systematic uncertainties in lattice simulations should
be carefully examined, such as the eﬀect of discretization artifacts. There have been, however, no work that
performs the continuum extrapolation on nuclear interactions. The aim of this work is to perform the ﬁrst
systematic study for the lattice cutoﬀ dependence ofnuclear interactions.
We employ Nf = 2 conﬁgurations with clover
fermion generated by CP-PACS collaboration5) . The
measurements are performed at three lattice spacings,
a = 0.2150, 0.1555, 0.1076 fm. The physical lattice sizeis L3 ×T  (2.5 fm)3 ×5 fm, and the hadron masses are(mπ , mN )  (1.1, 2.2) GeV. The computational cost in
the Wick and color/spinor contractions is reduced by
the uniﬁed contraction algorithm6) . For details about
the simulation parameters, see Doi (2013)7) .
In Fig. 1, we plot the nuclear central potential in 1 S0channel for each lattice cutoﬀ. We observe that cutoﬀ
dependence is nonnegligible at short distances, while
it is suppressed at long distances. This is a naturalconsequence of the discretization being an intrinsicallyshort-range eﬀect. It is also interesting that repulsive
core is enhanced on a ﬁner lattice, which is consistent
with the study on the operator product expansion8) .
Although the lattice cutoﬀ dependence on potentials
looks sizable at short distances, such eﬀect is expected
to be suppressed in physical observables such as phaseshifts and scattering length, because of the phase spacefactor of ∝ r2 , as is shown in the inset of Fig. 1.
In order to quantitatively study the cutoﬀ eﬀect on
physical observables, we ﬁt the potential and solve theSchrödinger equation in inﬁnite volume. In Fig. 2, weshow the preliminary results for the scattering length
against the lattice spacing a, with only a statistical error. Because the scattering length represents the lowenergy phenomena, the cutoﬀ dependence is found to2
10-1-200.05

0.1

0.15
a [fm]

0.2

0.25

0.3

Fig. 2. Scattering length in 1 S0 channel against latticespacing a. Blue point corresponds to the result in
the continuum limit obtained by linear extrapolation
against a.

be negligible compared to the statistical errors. Detailed studies for the systematic uncertainties on phaseshifts and scattering length are in progress.
References
1) N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99(2007) 022001 [nucl-th/0611096].
2) S. Aoki, T. Hatsuda and N. Ishii, Prog. Theor. Phys.
123 (2010) 89 [arXiv:0909.5585 [hep-lat]].
3) T. Doi et al. [HAL QCD Coll.], Prog. Theor. Phys. 127(2012) 723 [arXiv:1106.2276 [hep-lat]].
4) S. Aoki et al. [HAL QCD Coll.], Prog. Theor. Exp. Phys.
2012 (2012) 01A105 [arXiv:1206.5088 [hep-lat]].
5) A. Ali Khan et al. [CP-PACS Coll.], Phys. Rev. D 65
(2002) 054505 [E: D 67 (2003) 059901].
6) T. Doi and M. G. Endres, Comput. Phys. Commun.
184 (2013) 117 [arXiv:1205.0585 [hep-lat]].
7) T. Doi [HAL QCD Collaboration], PoS LAT2013, 226(2013), arXiv:1311.2697 [hep-lat].
8) S. Aoki, J. Balog and P. Weisz, JHEP 1005 (2010) 008[arXiv:1002.0977 [hep-lat]].

- 113 -