Hyperon-nucleon potentials from lattice QCD

Abstract

We calculate $p\Xi^0$ potentials from the equal-time Bethe-Salpeter amplitude measured in the quenched QCD simulation with the spatial lattice volume, (4.4 fm)$^3$. The standard Wilson gauge action with the gauge coupling $\beta=5.7$ on $32^4$ lattice together with the standard Wilson quark action are used. The hopping parameter $\kappa_{ud}=0.1678$ is chosen for $u$ and $d$ quarks, which corresponds to $m_{\pi}\simeq 0.37$ GeV. The physical strange quark mass is used by taking the parameter $\kappa_s=0.1643$ which is deduced from the physical $K$ meson mass. The lattice spacing $a=0.1420$ fm is determined by the physical $\rho$ meson mass. We find that the $p\Xi^0$ potential has strong spin dependence. Strong repulsive core is found in $^1S_0$ channel while the effective central potential in the $^3S_1$ channel has relatively weak repulsive core. The potentials also have weak attractive parts in the medium to long distance region (0.6 fm $\lsim r \lsim 1.2$ fm) in both of the $^1S_0$ and $^3S_1$ channels.

Full text

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arXiv:0710.3622v1 [hep-lat] 19 Oct 2007
Hyperon-nucleon potentials from lattice QCD
Hidekatsu Nemura∗
Advanced Meson Science Laboratory, Nishina Center for Accelerator-Based Science,
RIKEN,Wako, 351-0198, Japan
E-mail: nemura@riken.jp
Noriyoshi Ishii
Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan
E-mail: ishii@rarfaxp.riken.jp
Sinya Aoki
Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571,
Japan
Riken BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
E-mail: saoki@het.ph.tsukuba.ac.jp
Tetsuo Hatsuda
Department of Physics, University of Tokyo, Tokyo 113-0033,Japan
E-mail: hatsuda@phys.s.u-tokyo.ac.jp
We calculate pΞ0potentials from the equal-time Bethe-Salpeter amplitude measured in the
quenched QCD simulation with the spatial lattice volume, (4.4 fm)3. The standard Wilson gauge
action with the gauge coupling β = 5.7 on 324lattice together with the standard Wilson quark
action are used. The hopping parameter κud= 0.1678 is chosen for u and d quarks, which cor-
responds to mπ≃ 0.37 GeV. The physical strange quark mass is used by taking the parameter
κs= 0.1643 which is deduced from the physical K meson mass. The lattice spacing a = 0.1420
fm is determined by the physical ρ meson mass. We find that the pΞ0potential has strong spin
dependence. Strong repulsive core is found in1S0channel while the effective central potential in
the3S1channel has relatively weak repulsive core. The potentials also have weak attractive parts
in the medium to long distance region (0.6 fm<
∼r<
∼1.2 fm) in both of the1S0and3S1channels.
The XXV International Symposium on Lattice Field Theory
July 30-4 August 2007
Regensburg, Germany
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/

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Hyperon-nucleon potentials from lattice QCD
Hidekatsu Nemura
1. Introduction
Study of hypernuclei is one of the frontiers in nuclear physics. The strange degree of freedom
gives a new dimension to the description of nuclear structure. A hyperon (or a strange quark) em-
bedded in nuclei plays a characteristic role as an “impurity” or a probe in the many body system[1].
Modern nucleon-nucleon (NN) potentials give a successful description of the NN scattering
data and have been used to make precise calculations of light nuclei[2, 3]. In contrast, hyperon-
nucleon (YN) and hyperon-hyperon (YY) interactions have large uncertainties, because the scat-
tering experiments are either difficult or impossible due to the short life-time of hyperons. The
full phase shift analysis at the same level of the NN case is not available yet. Although a lot of
theoretical models which describe YN interaction together with the NN interaction have been pub-
lished, the different model predicts different phase shifts and scattering parameters, e.g., of the ΛN
potential in spite of the nice description of the NN sector[4, 5, 6, 7, 8, ?].
Recently, a lattice QCD study of the NN potential has been performed[10]. This approach may
lead to a new paradigm to study the YN and YY interactions too, since the lattice QCD is an ab
initio method of treating the fundamental theory of strong interaction. (See also the conventional
approach to the YN phase shifts using the Lüscher’s finite volume formula[11].)
The purpose of the present report is to explore theYN andYY potentials from the lattice QCD
simulation on the basis of the methodology developed in Refs. [10, 12]. The extension from the NN
potential toYN andYY potentials isrelatively straightforward. Inthe case ofthe NN potential, there
are only two representations in the isospin channel, i.e., 2⊗2 = 3⊕1 in SU(2), which correspond
to isovector and isoscalar channels. Including the strange degrees of freedom extends the arithmetic
into flavor SU(3), 8⊗8=27⊕10⊕1⊕8⊕10⊕8. Here the isovector (isoscalar) channel of the NN
sector is assigned to be a subset in the 27-plet (10-plet) representation. The potentials for newly
arising channels are hardly determined from the real experiment so far. The lattice QCD simulation
with physical strange quark mass provides new numerical “data” for these strange channels.
In this report, we focus on the NΞ potential in the isovector (I = 1) channel as a first step.
We keep away, at present, from the isoscalar (I = 0) channel of the NΞ potential, since it is not
the lowest state of the isoscalar1S0channel and ΛΛ strong decay mode may open below the NΞ.
There are almost no experimental information on the NΞ interaction, although a few experimental
data[13, 14, 15] suggest that the Ξ-nucleus potential would be weakly attractive. Moreover, the
Ξ-nucleus interaction will be studied as a day one experiment in the near future at J-PARC[16]
through (K−,K+) reaction on the nuclear target such as12C.
2. Formulation
As we mentioned, the methodology to obtain the potential is along the lines of Refs. [12]
and [10]. The latter describes, successfully for the first time, the NN potentials from lattice QCD
simulation. We start from an effective Schrödinger equation for NΞ system at low energies:
−1
2µ∇2φ(? r)+
where µ = mNmΞ/(mN+mΞ) and E are the reduced mass of the NΞ system and the nonrelativistic
energy in the center-of-mass frame, respectively. The nonlocal potential can be represented by the
?
d3r′U(? r,? r′)φ(? r′) = Eφ(? r),
(2.1)
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Hyperon-nucleon potentials from lattice QCD
Hidekatsu Nemura
derivative expansion as
U(? r,? r′) =VNΞ(? r,∇)δ(? r−? r′).
(2.2)
The general expression of the potential VNΞwould be
VNΞ= V0(r)+Vσ(r)(? σN·? σΞ)+Vτ(r)(? τN·? τΞ)+Vστ(r)(? σN·? σΞ)(? τN·? τΞ)
+VT(r)S12+VTτ(r)S12(? τN·? τΞ)+VLS(r)(?L·?S+)+VLSτ(r)(?L·?S+)(? τN·? τΞ)
+VALS(r)(?L·?S−)+VALSτ(r)(?L·?S−)(? τN·? τΞ)+O(∇2).
(2.3)
Here S12= 3(? σ1·? n)(? σ2·? n)−? σ1·? σ2is the tensor operator with? n =? r/|? r|,?S±= (? σ1±? σ2)/2 sym-
metric (+) and antisymmetric (−) spin operators,?L = −i? r×?∇ the relative angular momentum
operator, and ? τN (? τΞ) is isospin operator for N (Ξ). We note that the antisymmetric spin-orbit
forces (VALSand VALSτ) newly come up because the constituents (N and Ξ) are not identical.
According to the above expansion of the potential, the wave function should be classified by
the total isospin I, the total angular momentum and parity Jπwith?J =?L+?S+. Particular spin
(isospin) projection can be made in terms of ? σN·? σΞ(? τN·? τΞ), e.g., for the isospin projection we
have P(I=0)= (1−? τN·? τΞ)/4 and P(I=1)= (3+? τN·? τΞ)/4. In this work, we focus only the isospin
I = 1, S-wave component of the wave function, so as to obtain the (effective) central potential
through
Vcentral(r) = E +1
2µ
?∇2φ(r)
φ(r)
.
(2.4)
The S-wave wave function is measured from the equal-time Bethe-Salpeter (BS) amplitude
φ(? r;k) as
φ(? r;k) =
1
24∑
R∈O
1
L3∑
? x
Pσ
αβ
?
0
???pα(R[? r]+? x)Ξ0
β(? x)
??? pΞ0;k
?
,
(2.5)
pα(x) = εabc(ua(x)Cγ5db(x))ucα(x),
Ξ0
(2.6)
β(y) = εabc(ua(y)Cγ5sb(y))scβ(y),
(2.7)
where α and β denote the Dirac indices, a, b and c the color indices, and C = γ4γ2the charge
conjugation matrix. The summation over R ∈ O is taken for cubic transformation group to project
onto the S-wave, and the summation over? x for zero total spatial momentum. pα(x) and Ξ0
the local field operators for the proton and Ξ0. We take the upper components of the Dirac indices
α and β to construct the spin singlet (triplet) channel by Pσ
The φ(? r) with ? r =? x−? y is understood as the probability amplitude to find “nucleon-like”
three quarks located at point? x and “Ξ-like” three quarks located at point? y. The φ(? r) includes not
only the elastic amplitude NΞ → NΞ but also the inelastic amplitudes such as NΞ → πNΞ and
NΞ → ΛΣ, and so on. Note that, at low energies below the thresholds, however, the asymptotic
behavior of φ(? r) is not affected by the inelastic contributions, since they decrease exponentially in
the asymptotic region. (In the present calculation with the isospin I = 1, the ΛΛ channel is closed
due to the isospin conservation.) On the other hand, φ(? r) and hence the potential may depend on
the interpolating fields in the interaction region. Further study on this issue is found in Ref.[17] for
the NN potential.
β(y) are
αβ= (σ2)αβ(Pσ
αβ= (σ1)αβ).
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Hyperon-nucleon potentials from lattice QCD
Hidekatsu Nemura
In the actual simulations, the BS amplitude is obtained through the four-point correlator,
FpΞ0(? x,? y,t;t0) =
?
0
???pα(? x,t)Ξ0
An
?
β(? y,t)JpΞ0(t0)
???pα(? x)Ξ0
???0
?
(2.8)
=∑
n
0
β(? y)
???n
?
e−En(t−t0).
(2.9)
Here JpΞ0(t0) is a source term located at t = t0. We utilize the wall source for JpΞ0in this work
in order to enhance the lowest scattering state of the pΞ0system. Enis the energy of the pΞ0state,
|n?, and An(t0) = ?n|JpΞ0(t0)|0?.
We also calculate the two-point correlator, C(t;t0) =∑? x?0|Bα(? x,t)Bα(? x′,t0)|0?, for the octet
baryons (B = N,Ξ,Λ,Σ), in order to check whether various two baryon (ΛΛ,NΞ,ΛΣ and ΣΣ)
thresholds are reproduced in the correct order. The interpolating fields for Λ and Σ+, employed in
this work, are given by
Λα(x) = εabc{(da(x)Cγ5sb(x))ucα(x)+(sa(x)Cγ5ub(x))dcα(x)−2(ua(x)Cγ5db(x))scα(x)},
(2.10)
(2.11)
Σ+
β(y) = −εabc(ua(y)Cγ5sb(y))ucβ(y).
3. Numerical calculation
We use the standard Wilson gauge action at the gauge coupling β = 5.7 on the 323×32
lattice together with the standard Wilson quark action. See Ref.[17] for details. The hopping
parameter of κud= 0.1678 is chosen for the u and d quarks, which corresponds to mπ≃ 0.37 GeV,
mρ≃0.81 GeV,and mN≃1.16 GeV. In order to determine the parameter for the strange quark mass
(κs), we first measure the correlators of pseudo scalar and vector mesons by using the interpolating
fields given by
Mps(x) = q1,cα(x)γ5,αβq2,cβ(x)
Mv,k(x) = q1,cα(x)γk,αβq2,cβ(x)
for pseudo scalar meson,
for vector meson,
(3.1)
(3.2)
with applying six set of hopping parameters; first three sets for κ1= κ2(π and ρ mesons) with tak-
ing a number from {0.1678,0.1665,0.1640}, and another three sets for κ1> κ2(K and K∗mesons)
with taking different two numbers from {0.1678,0.1665,0.1640}. Assuming the following func-
tional forms for the pseudo scalar meson mass squared and for the vector meson mass,
(mpsa)2=B
2
?1
κ1−1
?1
κc
?
+B
2
+D
?1
κ2−1
?1
κc
?
,
(3.3)
(mva) =C+D
2
κ1−1
κc
?
2
κ2−1
κc
?
,
(3.4)
we obtained critical hopping parameter κc= 0.1693, and the physical parameter, κphys= 0.1691,
from (mπa/mρa) = (135/770). The lattice scale is determined so as to be a = 0.1420 fm from the
physical ρ meson mass. The parameter for the strange quark mass is determined as κs= 0.1643
from the physical K meson mass (494 MeV).
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Hyperon-nucleon potentials from lattice QCD
Hidekatsu Nemura
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.00.51.01.5 2.0
r (fm)
t−t0=6
1
3S0
S1
Figure 1: The radial wave function of pΞ0, in1S0(circle) and3S1(triangle) channels, obtained from the
lattice QCD at t −t0= 6.
4. Results and Discussion
Table 1 lists the hadron masses measured from the present lattice QCDsimulation. 1283 gauge
configurations are used to calculate the hadron masses and wave functions. (17 exceptional config-
urations are not used out of totally 1300 gauge configurations.) We note that the present results for
the baryon masses provide the correct order of particular threshold energy of two baryon states in
the strangeness S=−2 sector; Eth(ΛΛ) =2525(11)MeV (this channel is not allowed in the present
case because of isospin conservation), Eth(NΞ) = 2544(12)MeV, Eth(ΛΣ) = 2575(11)MeV, and
Eth(ΣΣ) = 2624(11)MeV. This warrants the desirable asymptotic behavior of the wave function.
Figure 1 shows the wave function obtained at the time slice t −t0= 6. The1S0(3S1) channel
is plotted by circles (triangles), which are normalized at the spatial boundary? r = (32/2,0,0). All
the data are taken into account for r<
∼0.7 fm, while only the data on the x-, y-, and z-axis and
their nearest neighbors are used to plot for the outer region. As seen in the Figure, the wave
functions are suppressed in the short distance region, and a slight enhancement is found in the
medium range region for both1S0and3S1channels. There is sizable difference between the1S0and
3S1, particularly of suppression in the short distance, suggesting that the1S0channel has stronger
repulsive core.
mπ
mρ
mK
mK∗
882(2)
mp
mΞ0
mΛ
mΣ+
1312(6) 367(1)811(4)552.6(5)1164(7) 1379(6)1263(6)
Table 1: Hadron masses, given in units of MeV, measured from the lattice QCD simulation. The number in
the parenthesis shows the errorbar in the last digit.
5