We report on an ab initio calculation of the {sup 4}He total photo-absorption cross section using two- and three-nucleon interactions based upon chiral effective field theory. The microscopic treatment of the continuum problem is achieved using the Lorentz integral transform method, applied within the no-core shell model approach

Navratil, P

Quaglioni, S

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LLNL-PROC-401226Application Of Chiral Two- AndThree-Nucleon Interactions To The 4HePhoto-DisintegrationS. Quaglioni, P. NavratilFebruary 11, 2008International Symposium on New Facet of Three-NucleonForce - 50 years of Fujita-Miyazawa Three-Nucleon ForceTokyo, JapanOctober 29, 2007 through October 31, 2007Disclaimer  This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.  Application of chiral two- and three-nucleoninteractions to the 4He photo-disintegrationSofia Quaglioni and Petr NavrátilLawrence Livermore National Laboratory, L-414, P.O. Box 808, Livermore, CA 94551, USAAbstract. We report on an ab initio calculation of the 4He total photo-absorption cross sectionusing two- and three-nucleon interactions based upon chiral effective field theory. The microscopictreatment of the continuum problem is achieved using the Lorentz integral transform method,applied within the no-core shell model approach.PACS: 25.20.Dc; 21.30.-x; 21.60.Cs, 27.10.+hINTRODUCTIONChiral effective filed theory (χEFT) [1, 2] represents our best opportunity to reach aconsistent picture of the interaction among nucleons, that is based on the underlyingand fundamental theory of quantum chromo-dynamics (QCD). Indeed, χEFT may bethought as an effective theory for QCD in the low energy regime relevant for describingthe properties of nuclei. In the framework of χEFT the nucleon-nucleon (NN) interac-tion is predicted at the leading order, together with a three-nucleon (NNN) interactionat the next-to-next-to-leading order or N2LO [2, 3, 4], and even a four-nucleon (NNNN)interaction at the fourth order (N3LO) [5]. The details of QCD are contained in param-eters, the so-called low-energy constants (LECs), that are not fixed by the symmetry,but can be constrained by experiment. Therefore, it is of the utmost importance to applyχEFT to nuclei in an ab initio framework.We performed an ab initio calculation [6] of the 4He total photo absorption crosssection in unretarded dipole approximation, using the high quality NN potential at thefourth order (N3LO) in the χEFT expansion of Ref. [7], and the NNN interaction atthe highest order presently available (N2LO) [3, 5]. The two low-energy constants thatparameterize the short-range NNN interaction were selected as discussed in Ref. [8].The microscopic treatment of the continuum problem was achieved by means of theLorentz integral transform (LIT) method [9], applied within the ab initio no-core shellmodel (NCSM) approach [10].THE AB INITIO NO-CORE SHELL MODELThe NCSM is a technique for the solution of the A-nucleon bound-state problem. Start-ing from an Hamiltonian containing realistic NN and NNN forces (both coordinate-and momentum-space interactions can be equally handled), the Schrrödinger equationis solved by expanding the wave functions in terms of a complete set of A-nucleonharmonic oscillator (HO) basis states up to a maximum excitation Nmaxh¯Ω above theminimum energy configuration, with Ω the HO frequency. Both Jacobi relative coordi-nates or Cartesian single-particle coordinates can be used. Indeed, in a complete Nmaxh¯Ωspace translational invariance is preserved even in the Slater-determinant basis. The con-vergence to the exact result with increasing Nmax is accelerated by the use of an effectiveinteraction tailored to the model-space truncation. The effective interaction is obtainedusing a unitary transformation in a n−body cluster approximation, where n is typically2 or 3 [11].TOTAL PHOTO-ABSORPTION CROSS SECTION VIA THELORENTZ INTEGRAL TRANSFORM METHODIn the long wave-length approximation, the total photo-absorption cross sectionσγ(ω) = 4pi2e2h¯cωR(ω) , (1)is proportional to the inclusive dipole response function:R(ω) =∫dΨ f∣∣〈Ψ f ∣∣ ˆD |Ψ0〉∣∣2 δ (E f −E0−ω) . (2)This is the sum of all the transitions from the ground state |Ψ0〉 to the various allowedfinal states |Ψ f 〉 induced by the dipole operator:ˆD =√4pi3A∑i=1τzi2riY10(rˆi) . (3)In the above equations ground- and final-state energies are denoted by E0 and E f ,respectively, whereas τzi and~r i = rirˆi represent the isospin third component and centerof mass frame coordinate of the ith nucleon. The direct calculation of such responsefunction is extremely difficult, especially for energies beyond the three-body breakupthreshold. However, it is possible to obtain the response function by following a smalldetour, the LIT method [12].While in conventional approaches one usually starts from Eq. (2), in the LIT methodone obtains R(ω) after the inversion of its integral transform with a Lorentzian kernelL(σR,σI) =∫dω R(ω)(ω−σR)2+σ2I= 〈Ψ˜|Ψ˜〉 . (4)Indeed, in order to calculate such a transform it is sufficient to solve the in-homogeneous“Schrödinger-like” equation(H−E0−σR+ iσI)|Ψ˜〉= ˆD|Ψ0〉. (5)Because of the presence of an imaginary part σI in Eq. (5) and the fact that the right-hand side of this same equation is localized (〈Ψ0 ∣∣ ˆD† ˆD∣∣Ψ0〉<∞), one has an asymptotic-31.0-30.0-29.0-28.0-27.0-26.0-25.01.351.401.451.501.550 2 4 6 8 10 12 14 16 18 20 220.750.800.850.900.95E0[MeV]〈r2 p〉1 2[fm]〈Ψ0|Dˆ†Dˆ|Ψ0〉[fm2]Nmax(a)(b)(c)NNNN+NNNh¯Ω = 22 MeVh¯Ω = 28 MeVFIGURE 1. The 4He ground-state (a) energy, (b) point-proton root-mean-square radius 〈r2p〉1/2, and (c)total dipole strength 〈Ψ0| ˆD† ˆD|Ψ0〉 as functions of the model-space truncation Nmax. Present results withχEFT interactions for h¯Ω = 22 and 28 MeV.boundary condition similar to a bound state. Thus, one can apply bound-state techniquesfor its solution, and, in particular, expansions over basis sets of localized functionssuch the NCSM basis. Moreover, the solution of Eq. (5) is unique. Once calculated〈Ψ˜|Ψ˜〉 the response function can be obtained by numerical inversion of the integralequation (4) [13]. During this procedure all the final-state interaction of the problem isfully taken into account.RESULTSWe start by discussing our results for the ground state of the α particle, which enters thedriving term of the LIT equation (5). In Fig. 1 we show the convergence patterns for threedifferent ground-state observables, calculated with and without inclusion of the NNNterms of the interaction: binding energy, point-proton radius and total dipole strength.For all of them we obtain very similar and smooth convergence patterns, using effectiveinteractions at the three-body cluster level. In particular, already for Nmax = 18 we findindependence from both model space and harmonic oscillator parameter. At the ground-state level, the inclusion of the NNN force affects mostly the energy, providing 3.21 MeVadditional binding, while the point-proton radius undergoes only a weak reduction. Thetotal dipole strength follows the same pattern as the radius. Indeed, these two observables20 25 30 350.01.02.03.018/1916/1714/1520 40 60 80 1000.01.02.03.0ω [MeV]σγ(ω)[mb]Nmax= 18/19Nmaxh¯Ω = 22 MeVh¯Ω = 22 MeVh¯Ω = 28 MeVh¯Ω = 28 MeVh¯Ω = 28 MeVNNNN+NNNNN+NNNFIGURE 2. The 4He total photo-absorption cross section as a function of the excitation energy ω .Present results with χEFT interactions, and in particular: (upper panel) convergence pattern of theNN+NNN calculation with respect to the model-space truncation Nmax for h¯Ω = 28 Me; (lower panel)frequency dependence of the best (Nmax = 18/19) results with and without inclusion of the NNN force.are correlated [14]:〈Ψ0| ˆD† ˆD|Ψ0〉 ' ZN3(A−1)〈r2p〉 . (6)This expression, which is exact for deuteron and triton, and for spatially symmetricsystems, is violated of about 9% for the 4He calculated both with and without NNNinteractions terms.By applying the LIT method we have obtained the 4He total photo-absorption crosssection shown in Fig. 2. Also for this observable we find a stable and accurate con-vergence thanks to the use of three-body effective interactions. From the bottom panelof the figure we also see that for the biggest model space used, Nmax = 18/19, the de-pendence on the HO frequency is weak. The inclusion of the NNN interaction termsinduces a suppression of the peak and an enhancement of the tail of the cross section. Inparticular, the reduction of the low-energy cross section is related, through the inverseenergy-weighted sum rule to the the reduction found for the dipole strength∫∞Ethσγ(ω)ωdω = 4pi2 e2h¯c〈Ψ0|ˆD† ˆD|Ψ0〉 . (7)20 25 30 350.01.02.03.04He+γ →Xω [MeV]σγ(ω)[mb]χEFTNNNN+NNNNilsson (‘07)Shima (‘05)Nakayama (‘07)FIGURE 3. The 4He photo-absorption cross section as a function of the excitation energy ω . Presentresults with χEFT interactions compared to the most recent experiments [15, 16, 17].Our final results, presented in Fig. 3 together with the most recent experiments,show a peak around the excitation energy of ω = 27.8 MeV, with a pick height mildlysensitive to the NNN force. The experimental situation in the near-threshold regionis controversial: two direct measurements performed using quasi-mono-energetic pho-tons [15, 16] show discrepancies up to a factor of two on the absolute height of thecross-section peak. We find an overall good agreement with the photo-disintegrationdata from bremsstrahlung photons of Nilsson et al. [16], while we reach only the last ofthe experimental points of Ref. [15]. In particular, the confused experimental situationdrawn by these two data sets does not allow to asses the role of the NNN force effect.Recently Nakayama et al. performed an indirect measurements of the α-particle totalphoto-absorption cross section [17] by observing its analog via the 4He(7Li,7Be) reac-tion at an incident energy of 455 MeV and at forward scattering angles. Although theuncertainty on this extracted absolute cross section is 20% or more, the inclusion of theNNN terms of the interaction appear to improve the agreement of the calculated crosssection with the latter indirect measurement.ACKNOWLEDGMENTSThis work was performed under the auspices of the U.S. Department of Energy byLawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Sup-port from U.S. DOE/SC/NP (Work Proposal Number SCW0498) and the Department ofEnergy under Grant DE-FC02-07ER41457 is acknowledged.REFERENCES1. S. Weinberg, Physica A 96, 327 (1979); J. Gasser, and H. Leutwyler, Ann. Phys. 158, 142 (1984);Nucl. Phys. B250, 465 (1985).2. S. Weinberg, Phys. Lett. B251, 288 (1990); Nucl. Phys. B363, 3 (1991).3. U. van Kolck, Phys. Rev. C 49, 2932 (1994).4. E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner, and H. Witala, Phys. Rev. C 66,064001 (2002).5. E. Epelbaum, Phys. Lett. B639, 456 (2006).6. S. Quaglioni, and P. Navrátil, Phys. Lett. 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Application Of Chiral Two- And Three-Nucleon Interactions To The 4He Photo-Disintegration

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Application Of Chiral Two- And Three-Nucleon Interactions To The 4He Photo-Disintegration

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