Modern Physics Letters A Vol. 24, Nos (2009) c World Scientific Publishing Company

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1 Modern Physics Letters A Vol. 24, Nos (2009) c World cientific Publishing Copany TOWARD A THREE-DIMENIONAL OLUTION FOR 3N BOUND TATE WITH 3NFs M. R. HADIZADEH and. BAYEGAN Departent of Physics, University of Tehran, P. O. Box , Tehran, Iran hadizade@khaya.ut.ac.ir bayegan@khaya.ut.ac.ir After a brief discussion about the necessity of using the 3D approach, we present the non partial wave (PW) foralis for 3N bound state with the inclusion of 3N force (3NF). As an exaple the evaluation of 3NF atrix eleents, which appear in the obtained coupled three-diensional integral equations, for 2π-exchange Tucson Melbourne 3NF show how would be this foralis efficient and less cubersoe in coparison with the PW foralis. Keywords: Three-diensional approach; 3N bound state; Tucson Melbourne 3N force. PAC Nos.: v, Ff, Dr, Hw 1. Why Do We Use 3D Approach Instead of PW Approach? The answer to this ain question indicates the otivation for using this approach. Few-body calculations are traditionally carried out by solving the relevant equations in a PW basis which after truncation they lead to coupled equations on angular oentu quantu nubers. A few PWs often provide qualitative insight, but odern calculations need any different spin, isospin and angular oentu cobinations. It is clear that in PW approach one should su all PWs to infinite order, but in practice one truncates the su to a finite angular oentu nuber which is dependent to the energy that one is working. It eans that in higher energies one will need ore PWs to obtain a convergence. In contrast to the traditional PW representation, the novel 3D approach replaces the discrete angular oentu quantu nubers with continuous angle variables and consequently it considers autoatically all PW coponents to infinite order o the nuber of equations in 3D foralis is energy independent, also this foralis avoids the very involved angular oentu algebra occurring for the perutations, transforations and especially for the 3NFs. 2. 3D Representation of Faddeev Equations with 3NF We have recently applied the 3D approach to the 3N bound state, where the Faddeev equations with NN interactions are successfully solved with Bonn-B potential

2 Towards a Three-Diensional olution for 3N Bound tates with 3NFs 817 In this article we extend this foralis by considering the 3NF, which helps us to reach to the full solution of the 3N bound state in a straightforward anner. The 3N bound state in the presence of the 3NF is described by the Faddeev equation: ψ = G 0 tp ψ + (1 + G 0 t)g 0 V (3) Ψ, (1) where the quantity V (3) defines a part of the 3NF which is syetric under the exchange of the particle and 2. In order to solve the Eq. (1) in oentu space we introduce the 3N basis states in a 3D foralis as: p q α p q α α T p q (2 1 2 ) M (2 1 2 )T M T. (2) Evaluation of the transition and perutation operators need to the free 3N basis states p q γ, where the spin-isospin parts γ are given as: γ γ γ T s1 s2 s3 t1 t2 t3. To this ai by changing the 3N basis states α to the free 3N basis states γ one needs to calculate the usual Clebsch-Gordan coefficients γ α = g γα gγα gt γα, see appendix (A) of Ref. 10. The evaluation of Faddeev equation with the inclusion of 3NF will be exactly the sae as Eq. (19) of Ref. 10 except that an extra ter with V (3). This is } p q α (1+G 0 t)g 0 V (3) Ψ = 1 E p2 p q α V (3) 3q2 Ψ + p q α tg 0V (3) Ψ. (3) The atrix eleents of the second ter can be evaluated as: Ψ = d 3 p d 3 q g γ α g γ α 3q 2 p q γ t p q γ p q α V (3) Ψ, (4) after evaluating the atrix eleents of the NN t-atrix and integrating over q vector, one obtains: Ψ = d 3 p g γ α g γ α γ,γ,α δ 3q2 s3 δ t3 p t(ɛ) p p q α V (3) Ψ. (5) By using the syetry property of 3NF and the anti-syetry property of the total wave function under exchange of nucleon and 2, one can rewrite Eq. (5) as: Ψ = d 3 p g γ α g γ α δ δ t3 3q2 ) ( ( ) s 12 +t 12 p t(ɛ)p 12 p p q α V (3) Ψ, (6)

3 818 M. R. Hadizadeh &. Bayegan under the exchange of the labels, to,, reverse of it and changing p to p one finds: Ψ = d 3 p g γ α g γ α 3q2 δ s3 δ t3 p t(ɛ)p 12 p p q α V (3) Ψ. (7) Now one can consider Eqs. (5) and (7) together to achieve: Ψ = 1 2 d 3 p g γ α g γ α 3q2 δ s3 δ t3 p t(ɛ)(1 P 12 ) p p q α V (3) Ψ, (8) by applying the introduction of the anti-syetrized NN t-atrix, i.e. a t a = t(1 P 12 ), one can obtain the coupled three-diensional Faddeev integral equations as: 1 p q α ψ = E p2 [ d 3 q 3q2 γ,γ,α g αγ g γ α δ δ a p t(ɛ) 1 2 q q a q q q α ψ + p q α V (3) Ψ g αγ g γ α d 3 p δ δ t3 E p 2 3q2 }] a p t(ɛ) p a p q α V (3) Ψ. (9) To represent the generality of our 3D foralis we can siplify the Eq. (9) to the bosonic case by switching off the spin-isospin quantu nubers, see Ref. 3. In order to show the efficiency of the presented foralis the realistic 2π-exchange TM 3NF 15 has been used to evaluate the atrix eleents of p q α V (3) Ψ. 3. The Evaluation of pqα V (3) Ψ for the TM 3NF For evaluation of the coupled three-diensional Faddeev equations, the atrix eleents p q α V (3) Ψ need to be calculated. In this section these atrix eleents have been evaluated for the TM 2π-exchange 3NF. To this ai one should first

4 Towards a Three-Diensional olution for 3N Bound tates with 3NFs 819 prepare this force in the for that can be evaluated easily in 3D representation as: V (3) = V 0Q Q σ 1 Q σ 2 Q F (Q 2 ) F (Q 2 ) Q π Q π ( ) [τ 1 τ 2 A + B QQ γ + C (Q 2 + Q 2 ) 3ia + D QQ a τ 3 τ 1 τ 2 4γ i 2γa (σ 3 Q ) 2 + i 2a σ 3 Q σ 3 Q + i 8γa (σ 3 Q ) 2 (σ 3 Q) 2 i } ] 2γa (σ 3 Q) 2, (10) where γ = Q Q and a = 1 γ 2. The Q and Q are oentu transfers, σ i s are Pauli spin atrices and F (Q 2 ) are for factors. One distinguishes four A-, B-, C- and D-ters in the TM force. The scalar product of the spin-oentu vectors can be evaluated as: ẑ s σ Q ẑ s = s i=a i=a s D 1 2 s s ( Q) D 1 2 s s( Q) = Ø Q s s, (11) where D 1 2 s s is Wigner D-function which is defined generally as D s s(ˆq) = s ẑs s ˆqs s. The application of the TM 3NF to the total wave function Ψ can be considered as su of the four independent ters: D V (3) Ψ = D V (i) 31 I(i) V (i) 23 Ψ = ψ i, I (i) τ1 τ = 2 i = A, B, C. (12) τ 3 τ 1 τ 2 i = D In the following the atrix eleents of p q α ψ i ters have been evaluated. It is clear that V (3) can be splitted into two parts and each part contains a eson exchange, the esons are exchanged in the subsystes (31) and (23), which are called for convenience subsyste 2 and 1 correspondingly. o it is convenient to insert a coplete set of states of type 2 between V (3) and Ψ and another coplete set of states of type 1 between the two eson exchanges. Then the atrix eleents of ψ i can be written as: 3 p q α ψ i = d 3 p d 3 q 3 p q α p q α 1 α α α d 3 p d 3 p d 3 q 1 p q α V (i) 31 p q α 1 d 3 q 1 p q α I (i) p q α 2 α d 3 p d 3 q 2 p q α V (i) 23 p q α 2 2 p q α Ψ. (13)

5 820 M. R. Hadizadeh &. Bayegan Here the subscript, 2, 3 of the bra and ket vectors stand for differenn chains. The coordinate transforation fro the syste of type 1 to one of type 3 can be evaluated explicitly as: 3 p q α p q α 1 = g α3α δ3 (p p q)δ3 (q p + 1 q). (14) 2 The atrix eleents of the isospin coordinate transforations can be evaluated as: 1 p q α I (i) p q α 2 = 1 p q α p q α 2 1 α T I(i) α T 2, (15) 1 p q α p q α 2 = g α 1 α 2 δ3 (p p q )δ 3 (q p q ). (16) The atrix eleents of 1 α T I(i) α T 2 have been derived in Ref. 14. In the following the atrix eleents for the different V s are evaluated. In the first step the atrix eleents of V31 i are evaluated. ince both pion-exchange propagators in the 3NF ter depend only on the oentu transfer in a twonucleon subsyste and also because of the separation of the isospin parts of 3NF, the atrix eleents of V31 i can be evaluated as: 1 p q α V (i) 31 p q α 1 = δ 3 (p p ) δ α T α T 1 q α (i) V 31 q α 1, (17) and the spin-space parts can be ore siplified for the A-,B-,C- and D-ters separately as: 1 q α 31 q α 1 = γ,γ 1 q α 31 q α 1 = g α γ g α γ δ δ s3 1 q 31 q 1, (18) γ,γ g α γ The A-, B- and C-ters can be evaluated as: 1 q 31 q 1 = F ((q q ) 2 ) (q q ) π g α γ δ 1 q 31 q 1. (19) (q q ) q q N Ø, (20) where the value of N i, 2 and 3 corresponding to A-, B- and C-ters. For the different parts, five parts, of the D-ter one obtains: } M 1 q 31 q 1 = 1 q V (B) 31 q (q 1 Ø q ), (21) where the power value of M is 0 for the first and second parts of this ter, it i for the third part and it i for the fourth and fifth parts in Eq. (10). The atrix

6 Towards a Three-Diensional olution for 3N Bound tates with 3NFs 821 eleents of V23 i can be evaluated by following the sae algorith as above. o one can obtain: 2 p q α V (i) 23 p q α 2 = δ 3 (p p ) δ α 2 q α T α T V (i) 23 q α 2, (22) 2 q α (A,B,C) V 23 q α 2 = γ,γ 2 q α (D) V 23 q α 2 = g α γ g α γ δ δ 2 q 23 q 2, (23) γ,γ g α γ g α γ δ 2 q 23 q 2. (24) The A-, B- and C-ters which have been considered in the right side of Eq. (23) can be evaluated as: 2 q 23 q 2 = F ((q q ) 2 ) (q q ) π (q q ) q q N Ø, (25) where the value of N i, 2 and 3 corresponding to A-, B- and C-ters. Also the different parts of the D-ter which have been considered in the right side of Eq. (24) can be evaluated in the following: 2 q 23 q 2 = 2 q V (B) 23 q 2 Ø (q q ) } M, (26) where the power value of M is 0 for the first and fifth parts of this ter, it i for the third part and it i for the second and fourth parts in Eq. (10). 4. uary A new foralis of three-diensional Faddeev integral equations for the 3N bound state including the 3NF is proposed. This forulation leads to a coupled set of a strictly finite nuber of equations in two vector variables for the aplitudes. The coparison of 3D and PW foraliss shows that this non PW foralis avoids the very involved angular oentu algebra occurring for the perutations and transforations and it is ore efficient and less cubersoe for considering the 3NF. This foralis enables us to handle the realistic NN and 3N forces with all their coplexity in 3N bound state calculations.

7 822 M. R. Hadizadeh &. Bayegan Acknowledgents M. R. H. would like to thank the organizers of the APFB08 conference for their kind invitation, war hospitality and support, and the very pleasant and stiulating atosphere at the conference. We would like to express our acknowledgents to our colleagues M. Harzchi and M. A. halchi for the enriching discussions. This work was supported by the research council of the University of Tehran. References 1. I. Fachruddin, Ch. Elster, and W. Glöckle, Phys. Rev. C 62, (2000). 2. I. Fachruddin, Ch. Elster, and W. Glöckle, Phys. Rev. C 63, (2001). 3. H. Liu, Ch. Elster, and W. Glöckle, Few Body yst. 33, 241 (2003). 4. I. Fachruddin, Ch. Elster, and W. Glöckle, Phys. Rev. C 68, (2003). 5. I. Fachruddin, W. Glöckle, Ch. Elster, and A. Nogga, Phys. Rev. C 69, (2004). 6. H. Liu, Ch. Elster, and W. Glöckle, Phys. Rev. C 72, (2005). 7. T. Lin, Ch. Elster, W. N. Polyzou, and W. Glöckle, Phys. Rev. C 76, (2007). 8. M. R. Hadizadeh and. Bayegan, Few Body yst. 40, 171 (2007). 9. M. R. Hadizadeh and. Bayegan, Eur. Phys. J. A 36, 201 (2008) Bayegan, M. R. Hadizadeh, and M. Harzchi, Phys. Rev. C 77, (2008). 11. Ch. Elster, T. Lin, W. Glöckle, and. Jeschonnek, Phys. Rev. C 78, (2008) Bayegan, M. Harzchi, and M. R. Hadizadeh, Nucl. Phys. A 814, 21 (2008) Bayegan, M. R. Hadizadeh, and W. Glöckle, Prog. Theor. Phys. 120, 887 (2008). 14. D. Hüber, H. Witala, A. Nogga, W. Glöckle, and H. Kaada, Few-Body yst. 22, 107 (1997) A. Coon and W. Glöckle, Phys. Rev. C 23, 1790 (1981).