Three-nucleon force effects in proton- deuteron scattering Souichi Ishikawa (Hosei University) FB19, Aug Sep. 5, Bonn

1. Introduction The 2  E-3NF is successful in explaining 3N binding energy and N-d differential cross sections. [On the other hand:] It is unsuccessful for some N-d polarization observables such as A y (  ), iT 11 (  ), and T 21 (  ). (see figure) Refs: S. I. et al., PRC 67 (2003) (R), PRC 75 (2007) (R): (1) Tensor component in the 2  E-3NF is responsible for the undesirable effect on T 21 (  ). (2) A phenomenological 3NF (V phe ) is introduced to remedy the disease. (2  E-3NF)

pd T 21 (  10 MeV: Effects of 2  E-3NF and a phenomenological 3NF pd T E p =10.0 MeV (E d =20.0 MeV) Exp. data: M. Sawada et al., PRC27 (1983)  E-3NF V phe

Phenomenological 3NF: V phe S. I., PRC 75 (2007) (R). In this talk: To show “3NF from the exchange of pion and  - boson is a possible candidate to V phe ”.

2.  -  three-nucleon force: V ps 3NF model considered here: - Pion produces “scalar” + ”tensor” forces -  -boson produces “scalar” force Exchange of  and  produces a tensor component, which is different from that of 2  E-3NF. 3NF from  -  Born diagram Ref: J. Adam, Jr. et al., PRC 69 (2004)

General form of  -3NF in Q-space m  (MeV) g  2 /4  V ps 5009 OBEP (BONN ) 584~10 Cutoff masses of the form factors:   =800 MeV,   =1300 MeV The parameter g  is chosen to reproduce the 3 H binding energy  

3. Proton-deuteron calculations Calculations of proton-deuteron scattering are performed including the proton-proton long-range Coulomb force effects based on an integral equation approach to solve Coulomb-modified Faddeev equation in coordinate space. Refs.: S. I., - Mod. Phys. Lett. A 70 (2009) 855 [APFB08] - arXiv: Inputs: -2NF: Argonne V 18 potential (AV18) -3NF: Brazil 2  E-3NF (BR) -2N states : J max = 4 -3N states : J 0,max = 19/2

4. Results T 21 (  ) Polarization transfer coeffients d(p ,p  )d K x x’ (  ), K y y’ (  ), etc  (  ) at forward angle

Effects of V ps on T 21 (  ) at 10 MeV 2  E-3NF V ps Similar effect as V phe is obtained !

Polarization transfer coefficients in d(p ,p  )d reaction at 22.5MeV V phe gives wrong effect !

Effects of V ps on polarization transfer coefficients V ps gives desirable effect !

Effects of 3NFs on  (  ) at E p =10MeV At forward angles: AV18+(2  ) ~ Kyushu data AV18+(2 p )+( ,  ) ~ Koeln data

5. SUMMARY 1. We studied effects of a 3NF from pion-sigma exchange mechanism (V ps ) on some pd observables. 2. V ps may cancel a wrong behavior of the tensor components in 2  E-3NF, and as a result, works to explain the tensor analyzing power T 21 (  ). 3. V ps also successfully explains the polarization transfer coefficients, K x x’ (  ) and K y y’ (  ). 4. V ps gives a large effect in Nd differential cross section at forward angles. (Precise data are required to check.)

Energy dependence of T 21 (  =90 o )

References: S. Ishikawa “Spin-dependent three-nucleon force effects on nucleon-deuteron scattering” Phys. Rev. C 75 (2007) (R) (arXiv: ) S. Ishikawa “Calculation of Proton-Deuteron Scattering with Coulomb Force Effects” Mod. Phys. Lett. A 70 (2009) 855 [APFB08] S. Ishikawa “Coordinate space proton-deuteron scattering calculations including Coulomb force effects” submitted for publication (arXiv: )

General form of  -3NF in R-space  s

Coordinate space integral equation approach - Wave functions expanded by complete sets (plane wave or Coulomb wave) with respect to variable y -Sets of equations for the variable x (2-body equations) -Use of 2-body Coulomb wave functions Modification of Faddeev eq.: T. Sasakawa and T. Sawada, Phys. Rev. C 20, 1954 (1979). Introducing an auxiliary potential Appendix:A formalism for proton-proton- neutron system y x

Consider 1(p)+2(p)+3(n) system [V C (x 3 ) - u C (y 1 )] ~0 for a large x 3 ? Introduction of an auxiliary potential u C (y 1 ) p-p Coulomb potential causes a singularity in Faddeev equation Yes for finite x 1 : bound state & scattering (E< E 3BB ) No, when 3-body breakup occurs (E> E 3BB )

Modified Faddeev eq. u C (y 1 ): Distortion of spectator (proton) Cancellation of the long-range character in p-p Coulomb force R=8fm

Phase shift parameters with AV18 5 MeV 2 S 1/2 4 D 1/2  1/2+ KVP(-41.8, 1.74)(-5.43, 0.004)(1.05, -0.03) Ours(-41.8, 2.15)(-5.45, 0.000)(1.05, -0.04) 10 MeV 2 S 1/2 4 D 1/2  1/2+ KVP(-60.6, 11.7)(-7.30, 0.24)(1.01, 0.06) Ours(-60.8, 11.9)(-7.34, 0.22)(1.01, 0.04) KVP (Kohn variational principle): M. Viviani et al., FBS 30, 39 (2001).

Cancellation of the long-range character in p-p Coulomb force