Ballot, J.L.

da Rocha, C.A.

Robilotta, M.R.

HAL-IN2P3

Two-pion exchange nuclear potential - chiralcancellationsJ.L. Ballot, M.R. Robilotta, C.A. Da RochaTo cite this version:J.L. Ballot, M.R. Robilotta, C.A. Da Rocha. Two-pion exchange nuclear potential - chiralcancellations. International Journal of Modern Physics E, World Scientific Publishing, 1997, 6,pp.83-88. <in2p3-00000295>HAL Id: in2p3-00000295http://hal.in2p3.fr/in2p3-00000295Submitted on 6 Jan 1999HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.HEP-PH-9502369TWO-PION EXCHANGE NUCLEAR POTENTIAL -CHIRAL CANCELLATIONSJ-L. BallotDivision de Physique Theorique, Institut de Physique NucleaireF91406 Orsay Cedex, France, e-mail:ballot@frcpn11.in2p3.frM.R. RobilottaInstituto de Fsica, Universidade de S~ao PauloC.P. 20516, 01452-990, S~ao Paulo, SP, Brazil, e-mail: robilotta@if.usp.brC.A. da RochaInstituto de Fsica Teorica, Universidade Estadual PaulistaR. Pamplona, 145 - 01405-900 - S~ao Paulo, SP, Brazil, e-mail: carocha@vax.ift.unesp.br(February 23, 1995)AbstractWe show that chiral symmetry is responsible for large cancellations in thetwo-pion exchange nucleon-nucleon interaction, which are similar to thoseoccuring in free pion-nucleon scattering.Typeset using REVTEXUnite de Recherche des Universites Paris 6 et Paris 11 associee au CNRS.1The two-pion exchange nucleon-nucleon potential (E-NNP) in the framework of chiralsymmetry has recently attracted considerable attention, especially as far as the restrictedpion-nucleon sector is concerned [1{5]. This process is closely related to the pion-nucleonscattering amplitude, as pointed out many years ago by Brown and Durso [6]. In the non-linear realization of chiral symmetry, one possibility is to write the pion-nucleon interactionLagrangian as a sum of scalar and pseudoscalar terms, as follows:LPS+S=      gNqf2  2+ i 5N +    ; (1)where g is the N coupling constant, fis the pion decay constant and N and  are thenucleon and pion elds, respectively.In the case of pion-nucleon scattering, this Lagrangian yields a tree amplitude whichcontains poles in the s and u channels, as well as a scalar contact interaction, which isthe signature of chiral symmetry. At low energies, this last term cancels a large part ofthe pole amplitude, giving rise to a nal result which is much smaller than the individualcontributions.As far as nucleon-nucleon scattering is concerned, the two-pion exchange amplitude toorder g4is given by ve diagrams, usually named box, crossed box, triangle (twice) andbubble [5]. The rst two diagrams contain only nucleon propagators and are independentof chiral symmetry, whereas the triangles and the bubble involve the scalar interaction andhence are due to the symmetry. When one considers the potential instead of the amplitude,the iterated OPEP has to be subtracted from the box diagram.In this work we show that, as in pion-nucleon scattering, there are large cancellationsamong the various individual contributions to the interaction, that yield a relatively smallnet result, and thus prevent the perturbative explosion of the amplitude.Using the potential in coordinate space produced recently [5] and parametrized in Ref. [7],it is possible to notice two important cancellations within the scalar-isoscalar sector of theE-NNP. The rst of them happens between the triangle and bubble contributions, asshown in Fig. 1. It is worth noting that the scale of this gure is given in thousands of MeV.2The other one occurs when the remainder from the previous cancellation (S) is added tothe sum of the box and crossed box diagrams (PS). In this last case, the direct inspectionof the prole functions for the potential, given in Fig 2, provides just a rough estimateof the importance of the cancellation, since the iterated OPEP is not included. Thereforethe second cancellation can be better studied directly in the NN scattering problem, byconsidering the singlet waves, where the inuence of chiral symmetry is stronger.It is well known that a full chiral calculation of the potential in momentum space requiresthe inclusion of undetermined counterterms in the Lagrangian, involving higher orders ofthe relevant momenta [1]. In conguration space, on the other hand, these countertermsbecome delta functions which aect only the origin and hence are eective just for waveswith low orbital momentum. Therefore, in order to avoid these undetermined short rangeeects, we consider the behaviour of the1D2;1G4;1F3, and1H5waves.For each channel, we decompose the full NN potential V asV = U+ UPS+ US+ UC; (2)where Uis the OPEP, UCrepresents the short ranged core contributions, UPSis due to thebox and crossed box diagrams whereas USis associated with the chiral triangle and bubbleinteractions. Using the variable phase method, it is possible to write the phase shift forangular momentum L as [8,9]L=  mkZ10dr V P2L: (3)In this expression, the structure function PLis given byPL= |^LcosDL  n^LsinDL; (4)where |^Land n^Lare the usual Bessel and Neumann functions multiplied by their argumentsand DLis the variable phase. Using the decomposition of the potential given in Eq. 2, onewrites the perturbative resultL=  mkZ10drnU|^2L+hUP2L  |^2l+ UPSP2l+ USP2Li+ UCP2Lo L)L+ [L)I+ L)PS+ L)S] + L)C: (5)3In this expression, the rst term represents the perturbative long range OPEP (L), thesecond the iterated OPEP (I), the third the part due to the box and crossed-box diagrams(PS), the fourth the contribution from chiral symmetry (S). The last one is due to the coreand vanishes for waves with L 6= 0. In Fig. 3 we show the partial contributions to the1D2and1F3phase shifts as functions of energy. There it is possible to see large cancellations ofthe medium range contributions, represented by the terms within square brackets in Eq. 5.In the case of the1F3wave one notes a contribution from the iterated OPEP, the cancellationis almost complete and the total phase shift is very close to that due to the long-OPEP term.The same patterns also holds for the1G4and1H5waves.These results show that chiral symmetry is responsible for large cancellations in the two-pion exchange interaction. This process is therefore similar to threshold pion-nucleon orpion-deuteron [10] scattering amplitudes, where the main role of the symmetry is to set thescale to the problem to be small.ACKNOWLEDGMENTSM.R.R. would like to acknowledge the kind hospitality of the Division de PhysiqueTheorique of the Institut de Physique Nucleaire in Orsay, where part of this work wasperformed. The work of C.A.R. was supported by FAPESP, Brazilian Agency.4REFERENCES[1] C. Ordo~nez and U. Van Kolk, Phys. Lett. B291, 459 (1992).[2] L.S. Celenza, A. Pantziris and C.M. Shakin, Phys. Rev. C 46, 2213 (1992).[3] M.C. Birse, Phys. Rev. C 49, 2212 (1994).[4] J.L. Friar and S.A. Coon, Phys. Rev. C 49, 1272 (1994).[5] C.A. da Rocha and M.R. Robilotta, Phys. Rev. C, 49, 1818 (1994).[6] G.E. Brown and J.W. Durso, Phys. Lett. 35B, 120 (1971).[7] C.A. da Rocha and M.R. Robilotta, submitted to Phys. Rev. C.[8] F. Calogero, N. Cim. 27, 261 (1963); Variable Phase Approach to Potential Scattering,Academic Press, New York (1967).[9] J.L. Ballot and M.R. Robilotta, J. Phys. G 20, 1595 (1994).[10] M.R. Robilotta and C. Wilkin, J. Phys. G 4, L115 (1978).5FIGURESFIG. 1. Prole functions for the bubble (()) and triangle (r) scalar-isoscalar potentials and fortheir chiral sum (S), showing a strong cancellation between these two contributions.FIG. 2. Prole functions for the chiral sum (S - same as in Fig. 1) and pseudoscalar (PS)scalar-isoscalar potentials and for their sum, representing the full medium ranged potential (M).One sees another strong cancellation between these two contributions.FIG. 3. Contributions for the long-OPEP (L), iterated OPEP (I), pseudoscalar (PS) andchiral (S) terms of the potential to the phase shifts for1D2and1F3waves. The total phase shiftsare indicated by (T).6

Two-pion exchange nuclear potential - chiral cancellations

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