Hypernuclei in halo/cluster EFT Shung-Ichi Ando Sunmoon University, - - PowerPoint PPT Presentation

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

• 4

ΛΛH as ΛΛd system in halo EFT

• 6

ΛΛHe 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 exhibits

cyclic singularities, so called limit cycle.

• It is necessary to introduce a counter term for

renormalization.

• Efimov-like bound states

Infinitely many three-body bound states (whose energies B(n)) appear, for three-boson case,

B(n) =

• e−2π/s0n−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 unitary

limit can be chosen as a first approximation.

31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 4

4 ΛΛH at LO

• ΛΛd
• 3

ΛH, BΛ = 0.13 MeV

• d, B2 = 2.22 MeV
• S-waves are considered at LO.
• S = 0: no limit cycle, one parameter γΛd, and we

find no bound state for

4 ΛΛH and a0 = 16.0 ± 3.0 fm

for Λ-3

ΛH scattering.

• S = 1:

4 ΛΛH shows a limit cycle, three parameters,

aΛΛ, γΛd, g1(Λc), and the three-body interaction is

fixed 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 mΛy2

s

1

1 aΛΛ −

• −mΛp0 + 1

4

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 3

ΛH channel

• Dressed 3

ΛH propagator

= + + + ...

• Renormalized dressed 3

ΛH propagator Dt(p0, p) = 2π µΛdy2

t

1 γΛd −

• −2µΛd
• p0 −

1 2(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) Λ2

c

− 1 2π2 Λc dll2

• K(a)(p, l; E) − g1(Λc)

Λ2

c

• Dt
• E −

1 2mΛ l2, l

• a(l, k; E)

− 1 2π2 Λc dll2K(b1)(p, l; E)Ds

• E −

1 2md l2, l

• b(l, k; E) ,

b(p, k; E) = K(b2)(p, k; E) − 1 2π2 Λc dll2K(b2)(p, l; E)Dt

• E −

1 2mΛ l2, l

• a(l, k; E) ,

31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 8

where K(a)(p, l; E) = mdy2

t

6pl ln   p2 + l2 + 2µΛd

md

− 2µΛdE p2 + l2 − 2µΛd

md

− 2µΛdE   , K(b1)(p, l; E) = −

• 2

3 mΛysyt 2pl ln p2 +

mΛ 2µΛd l2 + pl − mΛE

p2 +

mΛ 2µΛd l2 − pl − mΛE

• ,

K(b2)(p, l; E) = −

• 2

3 mΛysyt 2pl ln

• mΛ

2µΛd p2 + l2 + pl − mΛE mΛ 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) µ3/2

Λd

, where µd(ΛΛ) = 2mΛmd/(2mΛ + md), and I1(s) = 2π s sin[s sin−1 1

2 a

• ]

cos π

2 s

• ,

I2(s) = 2π s 1 bs/2 sin[s cot−1 √4b − 1

• ]

cos π

2 s

• ,

I3(s) = 2π s bs/2 sin[s cot−1 √4b − 1] cos π

2 s

• ,

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

5 10 15 20 101 102 103 104 105 106 g1(Λc) 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 (MeV)

• aΛΛ (fm)

Λc = 300 MeV = 150 MeV = 50 MeV (II), (III)

31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 13

6 ΛΛHe at LO

• ΛΛα (S = 0)
• 5

ΛHe, 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 is

fixed 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

5 10 100 1000 10000 100000 g(Λc) 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 (MeV)

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

B

ΛΛ (MeV)

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 hypernuclei

are constructed.

• Those three-body systems described by means of

EFTs at LO exhibit a limit cycle in the asymptotic limit which implies the formation of bound states.

• For more conclusive results, we need to have the
• exp. data and include higher order corrections.
• We have applied the present approach to the

study of nnΛ system [SIA, Raha, Oh, PRC92(2015)024325].

31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 18