Triton-like molecules Li Ma 13/01/2020 Introduction and Motivation - PowerPoint PPT Presentation

Triton-like molecules Li Ma 13/01/2020

Introduction and Motivation 1 2 Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ 3 Application to the NNN system 4 Numerical results for the ΛΛΛ, ΞΞΞ and ΣΣΣ Some other numerical results 5 Summary 6 13/01/2020 2 / 20

Introduction and Motivation The deuteron-like molecules � � � � � � � � π � η � σ π � η � σ π � η � σ ρ � ω ρ � ω ρ � ω � � � � � � � � Deuteron Di-meson molecule Di-baryon molecule X (3872) → D ¯ Z b (10610) → B ¯ D ∗ , D s 1 (2460) → D ∗ K , B ∗ Z c (4020) → D ∗ ¯ Y (4260) → D 1 ¯ Z c (3900) → D ¯ D , D ∗ , D ∗ P c (4380) → Σ c ¯ c ¯ D ∗ , P c (4450) → Σ ∗ D ∗ 13/01/2020 3 / 20

Introduction and Motivation The triton-like molecules � � � � � � � � � π � η � σ � ρ � ω � � � � � � � � � � Benzene ring Triton NDK , ¯ KDN , ND ¯ D , N ¯ KK , DKK , DK ¯ K , BBB ∗ , DDK , DD ∗ K , Ω NN and ΩΩ N Faddeev equation, or FCA Dimer or isobar formalism Gaussian expansion method (GEM) 13/01/2020 4 / 20

Introduction and Motivation Efimov effect � 2 ma 2 (1 + O ( r 0 B 2 = a )) B 3 (1 + O ( r 0 a )) = 1 s 0 κ 2 − � 2 ma 2 + [ e − 2 π n f ( ξ )] ∗ m s 0 = 1 . 00624 ... , ξ is defined by tan ξ = − ( mB 3 ) 1 / 2 a / � , f ( ξ )is a universal function with f ( − π/ 2) = 1, � K = sgn ( E ) m | E | . Figure: Efimov plot 13/01/2020 5 / 20

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ The OBE interaction indicates that there is only one virtual boson exchanged by any two constituents as shown in the following. � � � Ξ Λ Σ π π π Ξ Ξ Λ Λ Σ Σ � � � � � � � � � Figure: Dynamical illustration of the ΛΛΛ, ΞΞΞ and ΣΣΣ systems with a circle describing the delocalized π bond inside. Monopole form factor F ( q ) = Λ 2 − m 2 Λ 2 − q 2 , ( α = π, η, ρ, ω, σ, φ ) with q the α four-momentum of the pion and Λ the cutoff parameter. 13/01/2020 6 / 20

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ Born-Oppenheimer potential Considering that the particle b and c are static with the separation r bc , one can separate the degree of freedom of a from the three-body system. We assume the distance r bc is a parameter. The mesons b and c are static, and have one-pion interactions with meson a , which can be viewed as two static sources. We explore the dynamics for the meson a in the limit r bc → ∞ , and subtract the binding energy for the break-up state which is trivial for the three-body bound state. � � Ξ Ξ  ��  �� � � ��� π π  bc ) V BO ( r Ξ Ξ Ξ Ξ � �  �� �  �� � � � � � fixed fixed ( � ) ( � ) 13/01/2020 7 / 20 Figure: Illustration of the BO potential. (a) illustrates the calculation procedure

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ Interpolating wave function of meson a � Ξ  �� )  �� ) ψ ( � ψ ( � Ξ Ξ � �  �� � � fixed fixed The zero order of the final wave function for the meson a could be the superposition of these two components ψ ( � r ab ,� C [ ψ ( � r ab ) ± ψ ( � r ac ) = r ac )] | ΞΞΞ � Accordingly, one can obtain the energy eigenvalue of the meson a E a (Λ , � r bc ) = � ψ ( � r ab ,� r ac ) | H a | ψ ( � r ab ,� r ac ) � 13/01/2020 8 / 20

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ BO potential and Its physical meaning (Intensity of ”glue”) We define the BO potential as � Ξ V BO (Λ , � r bc ) = E a (Λ , � r bc ) − E 2 (Λ) . Ξ Ξ The BO potential can describe the contribution for the � � � one meson on the dynamics of the two remaining mesons. The meson a here works like a mass of ”glue”. V ( r )/ MeV 2 4 6 8 10 V tot V π ss - 1 40 V η ss V σ ss - 2 V ρ ss 20 V ω ss V ϕ ss - 3 r / fm 0.5 1.0 1.5 2.0 2.5 3.0 - 4 - 5 - 20 - 6 - 40 - 7 Figure: Here we chose the parameter Λ = 900 MeV. E ΞΞ I =0 = − 3 . 09 MeV. Ψ T = α Φ( � r bc ) ψ ( � r ab , � r ac ) . 13/01/2020 9 / 20

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ The configurations of the three-body systems Ξ � Ξ � � Ξ Ξ Ξ � � Ξ Ξ � � � Ξ Ξ � � � Figure: Every meson can be considered to be a lighter one and separated from the three-body system. Each of them can generate the ”glue” for the remaining mesons. � � � Ξ Ξ Ξ  ��  �� � �  ac )  ab ) V BO ( r V BO ( r  bc ) V BO ( r � � � � � � Ξ � � � Ξ Ξ Ξ Ξ Ξ  �� � ( � ) ( � ) ( � ) Figure: (a), (b) and (c) correspond to the wave functions ψ / a = Φ( � r bc ) ψ ( � r ab , � r ac ), ψ / b = Φ( � r ac ) ψ ( � r ab , � r bc ) and ψ / c = Φ( � r ab ) ψ ( � r bc , � r ac ), respectively. 13/01/2020 10 / 20 The basis constitute a configuration space { ψ , ψ , ψ } .

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ Interpolating wave functions The basis constitute a configuration space { ψ / a , ψ / b , ψ / c } . Ψ T = α Φ( � r bc ) ψ ( � r ab , � r ac ) + β Φ( � r ac ) ψ ( � r ab , � r bc ) + γ Φ( � r ab ) ψ ( � r bc , � r ac )   α  , = αψ / a + βψ / b + γψ / c = β  γ Expand Φ( � r bc ), Φ( � r ac ) and Φ( � r ab ) as a set of Laguerre polynomials � (2 λ ) 2 l +3 n ! Γ(2 l + 3 + n ) r l e − λ r L 2 l +2 χ nl ( r ) = (2 λ r ) , n = 1 , 2 , 3 ... n � � � ψ / a = φ i ( � r bc ) ψ ( � r ab , � r ac ) , ψ / b = φ i ( � r ac ) ψ ( � r ab , � r bc ) , ψ / c = φ i ( � r ab ) ψ ( � r bc , � r ac ) . i i i 13/01/2020 11 / 20

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ Orthonormalization We orthonormalize the { ψ / a , ψ / b , ψ / c } into a new � + ψ � � + ψ � � basis { ˜ a , ˜ b , ˜ ψ � ψ / ψ / ψ / c } . 1 ˜ � x ij ψ j ψ i ( ψ i a + ψ i b + ψ i � � = c ) − , a / / / / / a N i � i ψ � 1 ˜ �  ψ i ( ψ i a + ψ i b + ψ i x ij ψ j � � � ψ � = c ) − , � ψ � � ψ � / / / / b b / N i i   1 � � ψ � ψ � ψ i ˜ ( ψ i a + ψ i b + ψ i � x ij ψ j � � = c ) − , / c / / / c / N i i where the x ij is a parameter matrix which will be determined later. The N i are normalization coefficients. Then the eigenvector for the three-body system B ( ∗ ) a B ( ∗ ) b B ( ∗ ) can be written as a c vector in the configuration space { ˜ a , ˜ b , ˜ ψ / ψ / ψ / c } . Therefore, we have � α i ˜ � β j ˜ ˜ ψ j � γ k ˜ ψ i ψ k Ψ T = ˜ a + b + ˜ c , / / / i j k 13/01/2020 12 / 20

Tri-baryon systems ΛΛΛ, ΞΞΞ and ΣΣΣ First order correction � Ξ  ( �  ( �  �� )  �� ) ψ * ψ * Ξ Ξ � �  �� � � fixed fixed The first order of the final wave function for the meson a could be the superposition of these two components ˜ C ∗ [ ˜ r ab ) ± ˜ ψ ∗ ( � r ab ,� ψ ∗ ( � ψ ∗ ( � r ac ) = r ac )] | ΞΞΞ � Accordingly, one can obtain the energy eigenvalue of the meson a � ˜ r ac ) | H a | ˜ E ∗ a (Λ , � r bc ) = ψ ( � r ab ,� ψ ( � r ab ,� r ac ) � We define the BO potential as V ∗ BO (Λ , � r bc ) = E ∗ a (Λ , � r bc ) − E 2 (Λ) . 13/01/2020 13 / 20

Application to the NNN system Application to the NNN system (Triton or Helium-3 nucleus) � �� ���� � ��� � ��� [ ��� ] �� NNN = 5.38 MeV E I = 1 / 2 ���� � �� ���� � �� NNN = 1.71 MeV E I = 1 / 2 �� � � = � / � π � � � � � � � ���� � �� - � � � � � � �� �� [ ��� ] � � = � Figure: Dependence of the reduced three-body binding energy on the binding energy of its two-body subsystem (the deuteron). The result is comparable with the empirical binding energies of the triton (8.48 MeV) and helium-3 (7.80 MeV) nuclei. 13/01/2020 14 / 20

Application to the NNN system Numerical results for the NNN system (Triton or Helium-3 nucleus) Λ(MeV) E 2 (MeV) E 3 (MeV) E T (MeV) V BO (0)(MeV) S wave(%) D wave(%) r rms (fm) 830.00 -0.18 -1.93 -2.11 -4.54 94.01 5.99 4.21 850.00 -0.67 -2.71 -3.38 -5.36 93.36 6.64 4.00 870.00 -1.23 -3.65 -4.88 -6.32 92.68 7.32 3.78 890.00 -1.88 -4.77 -6.66 -7.42 91.99 8.01 3.54 899.60 -2.23 -5.38 -7.62 -8.00 91.66 8.34 3.42 900.00 -2.25 -5.41 -7.66 -8.03 91.64 8.36 3.42 920.00 -3.05 -6.85 -9.90 -9.35 90.97 9.03 3.18 940.00 -3.98 -8.51 -12.49 -10.83 90.35 9.65 2.95 960.00 -5.03 -10.42 -15.45 -12.46 89.76 10.24 2.74 980.00 -6.21 -12.57 -18.78 -14.23 89.23 10.77 2.54 1000.00 -7.55 -14.97 -22.51 -16.14 88.73 11.27 2.37 1020.00 -9.04 -17.61 -26.65 -18.19 88.27 11.73 2.23 1040.00 -10.69 -20.51 -31.20 -20.37 87.84 12.16 2.10 Table: Bound state solutions for the NNN system with isospin I 3 = 1 / 2. E 2 is the energy eigenvalue of its subsystem. E 3 is the reduced three-body energy eigenvalue relative to the break-up state of the NNN system. E T is the total three-body energy eigenvalue relative to the NNN threshold. 13/01/2020 15 / 20