Paper
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Nucleon-Nucleon Scattering in a Three-Dimensional Approach
Authors
I. Fachruddin,Ch. Elster
W. Glöckle,Hamid Ronagh,Anna IANNONE,RAFAEL PEREZ DEL REAL,bruce cornell,Marcelo Tolmasky,Edward Gregorich,François Molle,Signorino Galvagno,Vladimir Kibirev,Claudia Benitez-Nelson,Roger Slatt,John Sauk,Wilbur Campbell,Francisco J. Gonzalez-Vila,Maria Benedetta Donati,Luis Ortiz ,Manuel Delfino,Shyi-Kaan Wu,Ramon Miquel,Brigitte Bloch-Devaux,Alexander Finch,Pier Simone Marrocchesi,Isidoro Ferrante,Marco Cattaneo,Egidio Longo,Lluís Garrido Beltran,Vitaliano Chiarella,Mokhtar Chmeissani,Anuj Jain,Anna Westermair,Hanneke Muthert,Juan Ayala,Ndue KANARI,Richard Christen,Robert Dantzer,Eugene Flynn,Larry Claxton,Isao Yamamoto,Judith Smith,Olivier Hantz,Ernst P. Stephan,KarolÃna KubÃÄková,Mats Halvarsson,DANIELE NANNI,Abdelaziz Souizi,David Kilcrease,ANDREA GHISELLI,A. Trabelsi,Roger Stoller,Stanislaw Krukowski,Magnus Gottfredsson,Mohamed El Ansari,Federico Sanchez Nieto,Sutthisak Phongthanapanich,Donald Olson,Ruth Timme,Magued Iskander,Simon Cross,Antonella POGGI,Ryszard Piramidowicz,Vladimir Evenko,Victor Dilman,Igor Solodov,Victor Krylov,Atef Othman,Anna Glushakova,Milo Koretsky,Donald Macfarlane,Silvio Garattini,Dwight Deugo,Pietro GIUSTi,W. D. Walls,Maria Malysheva,Li Li,jamilah bakar,Arun Wagh,David Montgomery,Chandrasekhar Kambhampati,Antonio Lanzirotti,Henryk Kozłowski,Gian Luigi de'Angelis,Domingo Dominguez,Margarita Martínez-Gómez,Marion Scheepers,Erkki Mäkinen,Stuart Reid,Antonio Mur,Juan M. D. Tascon,Oleg Krishtal,Kari-Jouko Räihä,Gregory Tucker,Soldatkin Alexei,Haydar Demirel,Arseniy Martinchik,Dr Claudio Brito,Edömér Tassonyi,Kevin Roth,Helmut Fritzsche,Joseph Newkirk,Eric Devaux,Vijey Aanandhi M,EMILIO FRANZONI,Kevin Ronan,Hong-Guang Ma,Mark Thursz,Michael Page,Dietrich RT Zahn,Alan Litke,Cristobal Padilla,Petar Lukic,Gabriel Martínez-Pinedo,Леонид Шемчук,Shiva Ghimire,steven allin,Neil Brockdorff,Giovanna BLANDINO,Rauf Izmailov,James Morris,Hyo-Sok Ahn,Jose C R Silva,Laurence Eaves,Merja Itävaara,R Rengarajan,Richard Buckingham,Jean-Michel ESCOUBAS,Peter Charles Doherty,Luigi Giusto Spagnoli,Lewis Joel Greene,Giuseppe CIMINO,Ali Ghanbari,Bohdan Skalski,Julio Ly,Leon Gorris,Aránzazu Peña,Giovanni Franchin,Miroslav Karabaliev,Stephen Gourley,Željko Grabarević,Walter Gottlieb Land,Iñigo Fernandez de Larrinoa,Vedat Sansoy,EMMANUEL GANOTAKIS,Flavio Tosello,nabil el-sayed,Evgeny Kravtsov,Marco Raffaelli,Emmanuel Balanzat,Fabio CATANI,Larissa Leites,Anne Laure Fauchais,Herbert Dawid,Prasanta Tripathy,José D. Méndez,Klaus Benndorf,Lisa Klein,CESIRA FOPPOLI,Ana P. Tedim,Vasileios Basios,pithi chanvorachote,Waleed Shaheen,Peter Münger,Mark Hahn,Sohan Hayreh,A. S. 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,George A. BrooksJournal
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Apr 7, 2001
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arXiv:nucl-th/0104027v1 7 Apr 2001
1
Nucleon-Nucleon Scattering in a Three-Dimensional Approach
I. Fachruddina, Ch. Elsterb, W. Gl¨ocklea
aInstitut f¨ur Theoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum
bInsitut f¨ur Kernphysik, Forschungszentrum J¨ulich, and Institute for Nuclear and
Particle Physics, Ohio University, Athens, OH 45701
Two-nucleon scattering at intermediate energies of a few hundred MeV requires quite
a few angular momentum states in order to achieve convergence of e.g. scattering ob-
servables. This is even more true for the scattering of three or more nucleons upon each
other. An alternative approach to the conventional one, which is based on angular mo-
mentum decomposition, is to work directly with momentum vectors, specifically with the
magnitudes of momenta and the angles between them. We formulate and numerically
illustrate [1] this alternative approach for the case of NN scattering using two realistic
interaction models, the Argonne AV18 [2] and the Bonn-B [3] potentials. The momentum
vectors enter directly into the scattering equation, and the total spin of the two nucleons
is treated in a helicity representation with respect to the relative momenta q of the two
nucleons.
The momentum-helicity states are given as
|q; ˆqSΛ〉 ≡ |q〉 |ˆqSΛ〉 = |q〉 R(ˆq) ∑
m1m2
C(1
2
1
2 S; m1m2Λ)
∣
∣
∣
∣ˆz 1
2 m1
〉 ∣
∣
∣
∣ˆz 1
2m2
〉
, (1)
where R(ˆq) = exp(−iSz φ) exp(−iSy θ) is the rotation operator, Sz, Sy the components of
S = 1
2 (σ1 + σ2), and Λ the eigenvalue of the helicity operator S · ˆq. Introducing parity
and two-body isospin states |tmt〉, the antisymmetrized two-nucleon state is given by
|q; ˆqSΛ; t〉πa = 1
√2 (1 − ηπ(−)S+t) |t〉 |q; ˆqSΛ〉π , (2)
with the parity eigenvalues ηπ = ±1 and |q; ˆqSΛ〉π ≡ 1√2 (|q〉 + ηπ |−q〉) |ˆqSΛ〉.
With these basis states we formulate the Lippmann-Schwinger (LS) equations for NN
scattering. In the singlet case, S = 0, there is one single equation for each parity,
T πSt
00 (q′, q) = V πSt
00 (q′, q) + 1
4
∫
d3q′′ V πSt
00 (q′, q′′)G0(q′′)T πSt
00 (q′′, q) (3)
Using rotational and parity invariance one finds in the triplet case, S = 1, a set of two
coupled LS equations for each parity and each initial helicity state
T πSt
Λ′Λ (q′, q) = V πSt
Λ′Λ (q′, q) + 1
2
∫
d3q′′ V πSt
Λ′1 (q′, q′′)G0(q′′)T πSt
1Λ (q′′, q)
+ 1
4
∫
d3q′′ V πSt
Λ′0 (q′, q′′)G0(q′′)T πSt
0Λ (q′′, q). (4)
1
Nucleon-Nucleon Scattering in a Three-Dimensional Approach
I. Fachruddina, Ch. Elsterb, W. Gl¨ocklea
aInstitut f¨ur Theoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum
bInsitut f¨ur Kernphysik, Forschungszentrum J¨ulich, and Institute for Nuclear and
Particle Physics, Ohio University, Athens, OH 45701
Two-nucleon scattering at intermediate energies of a few hundred MeV requires quite
a few angular momentum states in order to achieve convergence of e.g. scattering ob-
servables. This is even more true for the scattering of three or more nucleons upon each
other. An alternative approach to the conventional one, which is based on angular mo-
mentum decomposition, is to work directly with momentum vectors, specifically with the
magnitudes of momenta and the angles between them. We formulate and numerically
illustrate [1] this alternative approach for the case of NN scattering using two realistic
interaction models, the Argonne AV18 [2] and the Bonn-B [3] potentials. The momentum
vectors enter directly into the scattering equation, and the total spin of the two nucleons
is treated in a helicity representation with respect to the relative momenta q of the two
nucleons.
The momentum-helicity states are given as
|q; ˆqSΛ〉 ≡ |q〉 |ˆqSΛ〉 = |q〉 R(ˆq) ∑
m1m2
C(1
2
1
2 S; m1m2Λ)
∣
∣
∣
∣ˆz 1
2 m1
〉 ∣
∣
∣
∣ˆz 1
2m2
〉
, (1)
where R(ˆq) = exp(−iSz φ) exp(−iSy θ) is the rotation operator, Sz, Sy the components of
S = 1
2 (σ1 + σ2), and Λ the eigenvalue of the helicity operator S · ˆq. Introducing parity
and two-body isospin states |tmt〉, the antisymmetrized two-nucleon state is given by
|q; ˆqSΛ; t〉πa = 1
√2 (1 − ηπ(−)S+t) |t〉 |q; ˆqSΛ〉π , (2)
with the parity eigenvalues ηπ = ±1 and |q; ˆqSΛ〉π ≡ 1√2 (|q〉 + ηπ |−q〉) |ˆqSΛ〉.
With these basis states we formulate the Lippmann-Schwinger (LS) equations for NN
scattering. In the singlet case, S = 0, there is one single equation for each parity,
T πSt
00 (q′, q) = V πSt
00 (q′, q) + 1
4
∫
d3q′′ V πSt
00 (q′, q′′)G0(q′′)T πSt
00 (q′′, q) (3)
Using rotational and parity invariance one finds in the triplet case, S = 1, a set of two
coupled LS equations for each parity and each initial helicity state
T πSt
Λ′Λ (q′, q) = V πSt
Λ′Λ (q′, q) + 1
2
∫
d3q′′ V πSt
Λ′1 (q′, q′′)G0(q′′)T πSt
1Λ (q′′, q)
+ 1
4
∫
d3q′′ V πSt
Λ′0 (q′, q′′)G0(q′′)T πSt
0Λ (q′′, q). (4)
100%
