RIKEN Accel. Prog. Rep. 50 (2017)

Ⅰ. HIGHLIGHTS OF THE YEAR

Baryon interaction in lattice QCD†
T. Iritani∗1 for HAL QCD Collaboration
Both Lüscher’s ﬁnite volume method (directmethod) and the HAL QCD method are used tostudy hadron interactions in lattice QCD. Theoretically, these approaches are equivalent. However, boththe deuteron and dineutron are considered boundstates in the direct method, while both are consideredscattering states in the HAL QCD method for heavier pion mass.2) To understand the origin of these deviations, we systematically study two-baryon systems
using both the HAL QCD and ﬁnite volume methods.
In Lüscher’s ﬁnite volume method, the phase shift
δ(k) is obtained through11 k cot δ(k) =,(1)
2 − |kL/(2π)|2πL|
n|3Fig. 1. Mock-up data with ﬂuctuations.

n∈Z

where k is given by the energy shift of two-baryon systems ΔEBB = 2 m2B + k 2 − 2mB in the ﬁnite box
L3 . In lattice simulations, ΔEBB is estimated fromthe plateau of the eﬀective energy shift
1eﬀ(t) ≡ − log (RBB (t + a)/RBB (t)) ,ΔEBB
acauses serious concerns on previous studies using thedirect method.2,4)(2)

2
where RBB (t) ≡ CBB (t)/{Cwith
 the two-baryon
 B (t)}22and the baryon
B(t)propagator CBB (t) ≡B̄(0)propagator CB (t) ≡ B(t)B̄(0) .
In lattice QCD, the signal-to-noise ratio of the multibaryon system becomes exponentially worse for A
baryons as S(t)/N (t) ∼ exp[−A(mB − (3/2)mM )t]with the baryon number A and meson mass mM . Inaddition, the energy gap of the elastic scattering statesscales as O(1/L2 ), which is about 50 MeV for a typical case. For ground-state saturation, t ∼ O(10) fm isrequired.
To demonstrate the ground-state saturation problem, we consider the mock-up data as R(t) =b1 e−ΔEBB t + b2 e−(δEel +ΔEBB )t + c1 e−(δEinel +ΔEBB )t ,where δEel(inel) is the gap of the (in)elastic state. Fig. 1
shows the eﬀective energy shift of the mock-up datawith typical values of the energy gaps δEel = 50
MeV and δEinel = 500 MeV. Without the elastic state
b2 /b1 = 0, the system converges to the ground statearound t ∼ 1 fm. However, a contamination of only
10% (b2 /b1 = ±0.1) creates a fake plateau around t ∼ 1fm.
To check such fake plateaux, we study the quark
source dependence of the eﬀective energy shift. Fig. 2
shows the eﬀective energy shift using a smeared source
and wall source. Both show clear a plateau-like structure around t ∼ 15a ∼ 1.5fm. However, their energyshift values are diﬀerent. This fake plateau problem†∗1Fig. 2. Example of the eﬀective energy shift ΔE eﬀ (t).

On the other hand, in the HAL QCD method, we deﬁne the potential through the Nambu-Bethe-Salpeter
function and study interactions based on this potential. In this approach, the ground-state saturation is
not required, and we demonstrate the reliability of the
HAL QCD method in the following papers.3,4)References
1) T. Iritani for HAL QCD Collaboration, J. of High Energy Phys. 1610, 101 (2016).
2) T. Yamazaki, Proceedings of Science (LATTICE2014)009 (2015), and references therein.
3) T. Iritani for HAL QCD Collaboration, Proceedings ofScience (LATTICE2016) 107 (2016).
4) S. Aoki, T. Doi, T. Iritani, Proceedings of Science (LATTICE2016) 109 (2016).

Condensed from the article in Journal of High Energy
Physics 1610, 101 (2016)1)
RIKEN Nishina Center- 16 -