Post on 15-Jul-2020
Hypernuclei in halo/cluster EFT
Shung-Ichi Ando
Sunmoon University, Asan, Republic of Korea
arXiv:1512.07674
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 1
Outline
• Singular potentials:
Limit cycle and Efimov states in three-body systems
• 4ΛΛH as ΛΛd system in halo EFT
• 6ΛΛHe as ΛΛα system in cluster EFT
• Summary
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 2
Limit cycle
• Three-body systems in unitary (asymptotic) limit
• If an interaction is singular, the system exhibitscyclic singularities, so called limit cycle.
• It is necessary to introduce a counter term forrenormalization.
• Efimov-like bound statesInfinitely many three-body bound states (whose
energies B(n)) appear, for three-boson case,
B(n) =(
e−2π/s0)n−n∗
κ2∗/m ,
where s0 ≃ 1.00624 and eπ/s0 ≃ 22.7.31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 3
Halo/Cluster EFT
• Effective Field Theories (EFTs)
• Model independent approach
• Separation scale
• Counting rules
• Parameters should be fixed by experiments
• For the study of three-body systems the unitarylimit can be chosen as a first approximation.
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 4
4ΛΛ
H at LO
• ΛΛd
• 3ΛH, BΛ = 0.13 MeV
• d, B2 = 2.22 MeV
• S-waves are considered at LO.
• S = 0: no limit cycle, one parameter γΛd, and wefind no bound state for 4
ΛΛH and a0 = 16.0± 3.0 fm
for Λ-3ΛH scattering.
• S = 1: 4ΛΛH shows a limit cycle, three parameters,
aΛΛ, γΛd, g1(Λc), and the three-body interaction isfixed by using the results of the potential models.
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 5
Two-body part: ΛΛ in 1S0 state
• Dressed dibaryon propagator
= + + + ...
• Renormalized dressed dibaryon propagator
Ds(p0, ~p) =4π
mΛy2s
1
1aΛΛ
−√
−mΛp0 + 14~p2 − iǫ
.
aΛΛ = −1.2± 0.6 fm ,
from 12C(K−,K+ΛΛX) reaction [Gasparyan et al., PRC85(2012)015204].
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 6
Two-body part: Λd in 3ΛH channel
• Dressed 3ΛH propagator
= + + + ...
• Renormalized dressed 3ΛH propagator
Dt(p0, ~p) =2π
µΛdy2t
1
γΛd −√
−2µΛd
(
p0 − 12(mΛ+md)
~p2)
,
with
γΛd =√
2µΛdBΛ .
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 7
Three-body part: S = 1 channel
= + + + +
= +
a(p, k;E) = K(a)(p, k;E)−g1(Λc)
Λ2c
−1
2π2
∫ Λc
0dll2
[
K(a)(p, l;E)−g1(Λc)
Λ2c
]
Dt
(
E −1
2mΛl2,~l
)
a(l, k;E)
−1
2π2
∫ Λc
0dll2K(b1)(p, l;E)Ds
(
E −1
2mdl2,~l
)
b(l, k;E) ,
b(p, k;E) = K(b2)(p, k;E)
−1
2π2
∫ Λc
0dll2K(b2)(p, l;E)Dt
(
E −1
2mΛl2,~l
)
a(l, k;E) ,
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 8
where
K(a)(p, l;E) =mdy
2t
6plln
p2 + l2 + 2µΛd
md
− 2µΛdE
p2 + l2 − 2µΛd
md
− 2µΛdE
,
K(b1)(p, l;E) = −√
2
3
mΛysyt
2plln
(
p2 + mΛ
2µΛd
l2 + pl −mΛE
p2 + mΛ
2µΛd
l2 − pl −mΛE
)
,
K(b2)(p, l;E) = −√
2
3
mΛysyt
2plln
( mΛ
2µΛd
p2 + l2 + pl −mΛEmΛ
2µΛd
p2 + l2 − pl −mΛE
)
.
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 9
Evaluation formula for the limit cycle
• In the asymmetric limit, there is no scale in the integral
equations. The scale invariance suggests that the
power-law behavior for the amplitude
a(p) ∼ p−1+s .
• After Mellin transformations we have
1 = C1I1(s) + C2I2(s)I3(s) .
• It has imaginary solutions for s, s = ±is0,
s0 = 0.4492 · · · ,
and thus eπ/s0 ≃ 1.09× 103.31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 10
• Evaluation formula for the limit cycle (for the ΛΛd system)
1 = C1I1(s) + C2I2(s)I3(s) ,
with
C1 =1
6π
md
µΛd
√
µΛ(Λd)
µΛd, C2 =
√2
3π2
√mΛµd(ΛΛ)µΛ(Λd)
µ3/2Λd
,
where µd(ΛΛ) = 2mΛmd/(2mΛ +md), and
I1(s) =2π
s
sin[s sin−1(
12a)
]
cos(
π2s) ,
I2(s) =2π
s
1
bs/2sin[s cot−1
(√4b− 1
)
]
cos(
π2s) ,
I3(s) =2π
sbs/2
sin[s cot−1(√
4b− 1)
]
cos(
π2s) ,
and a = 2µΛd
md
and b = mΛ
2µΛd
.31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 11
Numerical results: S = 1 channel
• With g1(Λc),
(BΛΛ, aΛΛ) = (I) (0.2 MeV, −0.5 fm), (II) (0.6, −1.5), (III) (1.0, −2.5).
-20
-15
-10
-5
0
5
10
15
20
101 102 103 104 105 106
g 1(Λ
c)
Λc (MeV)
(I)(II)(III)
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 12
Numerical results: S = 1 channel
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 1.1 1.2
0.5 1 1.5 2 2.5 3 3.5
B ΛΛ
(M
eV)
-aΛΛ (fm)
Λc = 300 MeV= 150 MeV= 50 MeV
(II), (III)
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 13
6ΛΛ
He at LO
• ΛΛα (S = 0)
• 5ΛHe, BΛ ≃ 3 MeV
• First excited energy of α, B1 ≃ 20 MeV
• The limit cycle appears, three parameters, aΛΛ,γΛα, g(Λc), at LO, and the three-body interaction isfixed by using the Nagara event, BΛΛ ≃ 6.93 MeV
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 14
Numerical results:
• With g(Λc) (Input: BΛΛ =6.93MeV)
-10
-5
0
5
10
100 1000 10000 100000
g(Λ
c)
Λc (MeV)
aΛΛ = -1.8 fm = -1.2 fm = -0.6 fm
Λn = Λ0 exp(nπ/s0) , s0 ≃ 1.05 , exp(π/s0) ≃ 19.9 .
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 15
Numerical results:
• Without g(Λc)
4
5
6
7
8
9
10
11
12
300 350 400 450 500 550 600 650
B ΛΛ
(M
eV)
Λc (MeV)
aΛΛ = - 1.8 fm = - 1.2 fm = - 0.6 fm
• rc = Λ−1 ≃ 0.35 to 0.5 fm.31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 16
Numerical results:
• With g(Λc) (Input: BΛΛ =6.93MeV, aΛΛ = −0.5fm)
5
6
7
8
9
10
11
12
13
14
-3 -2.5 -2 -1.5 -1 -0.5 0
B ΛΛ
(M
eV)
1/aΛΛ (fm-1)
Λc = 430 MeV = 300 MeV = 170 MeV
Potential models
[Filikhin and Gal, NPA707,491(2002)]31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 17
Summary
• Halo/cluster EFTs at LO for the light hypernucleiare constructed.
• Those three-body systems described by means ofEFTs at LO exhibit a limit cycle in the asymptoticlimit which implies the formation of bound states.
• For more conclusive results, we need to have theexp. data and include higher order corrections.
• We have applied the present approach to thestudy of nnΛ system [SIA, Raha, Oh, PRC92(2015)024325].
31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 18