Low Momentum Nucleon-Nucleon Interactions and Shell-Model Calculations

0.28 MB·English

by  L. Coraggio

Low Momentum Nucleon-Nucleon Interactions and Shell-Model Calculations
L. Coraggio,1 A. Covello,1 A. Gargano,1 N. Itaco,1 D. R. Entem,2 T. T. S. Kuo,3 and R. Machleidt4
1Dipartimento di Scienze Fisiche, Universit`a di Napoli Federico II,
and Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
2Grupo de F´ısica Nuclear, IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain
3Department of Physics, SUNY, Stony Brook, New York 11794
4Department of Physics, University of Idaho, Moscow, Idaho 83844
(Dated: February 6, 2008)
In the last few years, thelow-momentum nucleon-nucleon (NN) interaction Vlow−k derived from
free-spaceNN potentialshasbeensuccessfullyusedinshell-modelcalculations. Vlow−k isasmooth
potential which preserves the deuteron binding energy as well as the half-on-shell T-matrix of the
original NN potential up to a momentum cutoff Λ. In this paper we put to the test a new low-
7 momentumNN potentialderivedfromchiralperturbationtheoryatnext-to-next-to-next-to-leading
0 order with a sharp low-momentum cutoff at 2.1 fm−1. Shell-model calculations for the oxygen
0
isotopes using effective hamiltonians derived from both types of low-momentum potential are per-
2
formed. Wefindthatthetwopotentialsshowthesameperturbativebehaviorandyieldverysimilar

n results.
a
J PACSnumbers: 21.30.Fe,21.60.Cs,27.20.+n,27.30.+t

3
2
I. INTRODUCTION Inspired by EFT a new approach based on the renor-

1 malization group (RG) has been recently introduced to
v A challenging goal of nuclear structure is to perform derive a low-momentum NN interaction Vlow−k [13, 14,
5 shell-model calculations with single-particle (SP) ener- 15]. The starting point in the construction of Vlow−k
6 is a realistic model for V such as the CD-Bonn [16],
giesandresidualtwo-bodyinteractionbothderivedfrom NN
0 Nijmegen [17], Argonne V18 [18], or N3LO [11] poten-
1 a realistic nucleon-nucleon (NN) potential VNN. To
tials. AcutoffmomentumΛthatseparatesfastandslow
0 tackle this problem, a well-established framework is the
modes is then introduced and from the original V an
7 time-dependent degenerate linked-diagram perturbation NN
0 theory as formulated by Kuo, Lee and Ratcliff [1, 2], effective potential, satisfying a decoupling condition be-
/ tween the low-andhigh-momentum spaces, is derivedby
h which enables the derivation of a shell-model effective
integrating out the high-momentum components. The
l-t haAmsiltisonwiaenllHkneoffwsnta,ratinmgafinromfeaVtuNrNe.of V is the pres- main result is that Vlow−k is a smooth potential which
c NN preserves exactly the onshell properties of the original
u ence of a built-in strong short-range repulsion dealing
V ,andissuitabletobe useddirectlyinnuclearstruc-
n with high-momentum components of the potential. This NN
: hindersanorder-by-orderperturbativecalculationofH ture calculations. In the pastfew years,Vlow−k has been
v eff fruitfullyemployedinmicroscopiccalculationswithindif-
interms ofV , asthe matrixelements ofthe latterare
i NN ferentperturbativeframeworkssuchasthe realisticshell
X generally very large. However, in nuclear physics there
model[19,20,21,22,23,24],theGoldstoneexpansionfor
r is a naturalseparationofenergyscales,thatcanbe used
a doubly closed-shell nuclei [25, 26, 27], and the Hartree-
to formulate an advantageous theoretical approach for
Fock theory for nuclear matter calculations [28, 29, 30].
V . As a matter of fact, the characteristic quantum
NN
chromodynamics(QCD)energyscaleis M ∼1GeV,
QCD The success of Vlow−k suggests that there may be a
while for nuclear systems we have M ∼100 MeV [3].
nuc way to construct a low-momentum NN potential which
This consideration was at the origin of the seminal is more deeply rooted in EFT and can be used in a
work of Weinberg [4, 5], who introduced into nuclear perturbative approach. Namely, instead of taking the
physics the method of effective field theory (EFT) to detour through a NN potential with high-momentum
studytheS-matrixforaprocessinvolvingarbitrarynum- components,onemayaswellconstructalow-momentum
bers of low-momentum pions and nucleons. This ap- potential from scratch using chiral perturbation theory.
proach is based on the known symmetries of QCD and Wehaveconstructedsuchapotentialatnext-to-next-to-
parametrizestheunknowndynamicaldetailsintroducing next-to-leading order (N3LO) using a sharpcutoff at 2.1
a number of constants to be determined. fm−1. ThispotentialreproducestheNN phaseshifts up
Since then much work has been carried out on this to 200 MeV laboratoryenergy and the deuteron binding
subject (see for instance [3, 6, 7, 8, 9, 10]), leading to energy. While Vlow−k allows only for a numerical rep-
the construction of V ’s based on chiral perturbation resentation, the new low-momentum potential (dubbed
NN
theory that are able to reproduce accurately the NN N3LOW) is given in analytic form. Moreover, the low-
data [11, 12]. However, also these potentials cannot be energy constants are explicitly known so that the chi-
used in a perturbative nuclear structure calculation. ral three-nucleon forces consistent with N3LOW can be
2
properly defined. Qˆ is an irreducible vertex function whose intermediate
In order to investigate the perturbative properties of states are all beyond Λ and Qˆ′ is obtained by removing
this new chiral low-momentum potential, we perform from Qˆ its terms first order in the interaction V . In
NN
shell-model calculations for even oxygen isotopes. We additiontothepreservationofthehalf-on-shellT matrix,
employ twodifferent effective hamiltonians: one is based whichimpliespreservationofthephaseshifts,thisVlow−k
on the Vlow−k derived numerically from a ‘hard’ N3LO preserves the deuteron binding energy, since eigenvalues
potential [11] and the other on N3LOW. are preserved by the KLR effective interaction. For any
The paper is organizedas follows. In Sec. II we give a value of Λ, the low-momentum potential of Eq. (4) can
short description of how Vlow−k is derived as well as an be calculated very accurately using iteration methods.
outline of the construction of the N3LOW potential. In Our calculationof Vlow−k is performedby employing the
Sec. III a summary of the derivation of the shell-model iteration method proposedin [31], which is based on the
H is presented with some details of our calculations. Lee-Suzuki similarity transformation [32].
eff
In Sec. IV we present and discuss our results. Some The Vlow−k given by the T-matrix equivalence ap-
concluding remarks are given in Sec. V. proach mentioned above is not hermitian. Therefore, an
additional transformation is needed to make it hermi-
tian. To this end, we resort to the hermitization proce-
II. LOW-MOMENTUM NUCLEON-NUCLEON dure suggested in [31], which makes use of the Cholesky
POTENTIALS decomposition of symmetric positive definite matrices.
A. Potential model Vlow−k
B. Potential model N3LOW
First, we outline the derivation of Vlow−k [13, 14, 15].
As pointed out in the Introduction, the repulsive core The general and fundamental reason why the Vlow−k
contained in VNN is smoothed by integrating out the approach to nuclear structure physics works is that the
high-momentummodesofVNN downtoacutoffmomen- dynamicswhichrulesnuclearphysicscanbedescribedin
tum Λ. This integration is carried out with the require- the framework of a low-energy EFT. This nuclear EFT
ment that the deuteron binding energy and low-energy is characterized by the symmetries of low-energy QCD,
phase shifts of VNN are preserved by Vlow−k. This is inparticular,spontaneouslybrokenchiralsymmetry,and
achievedbythefollowingT-matrixequivalenceapproach. the degrees of freedom relevant for nuclear physics, nu-
We start from the half-on-shell T matrix for VNN cleons and pions. The expansion based upon this EFT
T(k′,k,k2)=V (k′,k)+ has become known as chiral perturbation theory (χPT),
NN which is an expansion in terms of (Q/M )ν where Q
∞ QCD
1
+P q2dqV (k′,q) T(q,k,k2) , (1) denotes the magnitude of a nucleon three-momentum or
Z0 NN k2−q2 a pion four-momentum, and MQCD is the QCD energy
where P denotes the principal value and k, k′, and scale [4, 5]. For this expansion to converge at a proper
rate, we have to have Q ≪ M . To enforce this, chi-
q stand for the relative momenta. The effective low- QCD
ralNN potentials are multiplied by a regulatorfunction
momentum T matrix is then defined by
thatsuppressesthepotentialfornucleonmomentaQ>Λ
Tlow−k(p′,p,p2)=Vlow−k(p′,p)+ with Λ ≪ MQCD. Present chiral NN potentials [11, 12]
Λ 1 typically apply values for Λ around 2.5 fm−1.
+PZ q2dqVlow−k(p′,q)p2−q2Tlow−k(q,p,p2) , (2) It is of course not an accident that the latter cutoff
0 value is not too different from what is typically used for
where the intermediate state momentum q is integrated Vlow−k, namely, Λ = 2.1 fm−1. This fact stimulates an
from 0 to the momentum space cutoff Λ and (p′,p)≤Λ. obvious question, namely: Is it possible to construct a
The above T matrices are required to satisfy the condi- chiral NN potential with Λ = 2.1 fm−1? Not surpris-
tion ingly, the answer is in the affirmative.
Thus,wehaveconstructedaNN potentialatN3LOof
T(p′,p,p2)=Tlow−k(p′,p,p2); (p′,p)≤Λ. (3) chiral perturbation theory that carries a sharp momen-
tum cutoff at 2.1 fm−1. We have dubbed this potential
The above equations define the effective low-
N3LOW.
momentum interaction Vlow−k, and it has been shown
One advantage of this potential is that it is given in
[14] that they are satisfied by the solution:
analytic form. The analytic expressions are the same as
Vlow−k =Qˆ−Qˆ′Z Qˆ+Qˆ′Z QˆZ Qˆ−Qˆ′Z QˆZ QˆZ Qˆ+... , foofr2t0h0e3“[h1a1r]da”nNd3aLrOe gpiovteenntiinalRbeyf.En[3t3em]. aNnodteMtahcahtletihdet
(4) procedure that needs to be followed to construct Vlow−k
which is the well known Kuo-Lee-Ratcliff (KLR) folded- from a free NN potential (cf. Sec. II.A) can be carried
diagram expansion [1, 2], originally designed for con- out only numerically.
structing shell-model effective interactions. In Eq. (4) Another important issue are many-body forces. As
3
dtbmntaiueoaaimcdnlt.lyteoneonOfrunosctndrlfrceoeoaeaertsgrscere(dmne3iaoiNsanttnFrRasye)adp-ebtmfvru.eoaarsd[nya3etytn0abet]sg,eeywewdscithortbeuhfeymcnoχiuasaaP,tlpl.TtopahwFleyi3o-siNmrinntFegochxmlaauawtemsVhnilioepottnwnuleg−m,otekhfnnpeiuetnohrcttacrlweeetenaeoersr---- Mixing Parameter (deg) 0246 ε1 Phase Shift (deg) 27..0555 1D2
nuclear two- and many-body forces on an equal footing 0 50 100 150 200 0 50 100 150 200
Lab. Energy (MeV) Lab. Energy (MeV)
(for an overview of this aspect, see Ref. [34]). Most
30
interaction vertices that appear in the 3NF and in the g) 3D g) 16 3P
four-nucleon force (4NF) also occur in the two-nucleon de 2 de 2
force(2NF).Theparameterscarriedbytheseverticesare hift ( 20 hift ( 12
fixed in the construction of the chiral 2NF. Consistency se S 10 se S 8
requires that for the same vertices the same parameter ha ha 4
P P
values are used in the 2NF, 3NF, 4NF, .... If the 2NF 0 0
0 50 100 150 200 0 50 100 150 200
is analytic, these parameters are known, and there is no
Lab. Energy (MeV) Lab. Energy (MeV)
problem with their consistent proliferation to the many-
g) ε
bii4fintNrolymdnFyuk,cmno.foo.newr.rscmiincestufeatsonentrrcdmmyb,tesbh.tebehtaewHpsnaeeoderwtanhemuovtpseeweortoe,np-risaf‘arentadaodmupcbemaoettetaeerundnssyte-aigdbaruoeliednesnysxoteithfsso’tee.rsxc3epoTNsnlihFlciyes-, Phase Shift (deg) 01..1255 3F2 xing Parameter (de --021 2
lost. Moreover, if a potential is given only in numeric 0 Mi -3
0 50 100 150 200 0 50 100 150 200
form, one does not known what order of χPT it belongs Lab. Energy (MeV) Lab. Energy (MeV)
to. Thus, it is also not clear which orders of 3NF and
4NF to include to be consistent with the order of the FIG. 2: Same as Fig. 1.
2NF.
75 15 OurnewlyconstructedN3LOWreproducesaccurately
eg) 1S0 eg) 3P0 the empirical deuteron binding energy, the experimen-
hift (d 50 hift (d 10 tal low-energy scattering parameters, and the empirical
ase S 25 ase S 5 plahbaosrea-tsohrifytsenoefrgNy,NsesecFatigtesr.in1g,2.upMotoreadtetleaailssta2b0o0utMtheVis
Ph Ph 0 potential will be published elsewhere [35]. It is the main
0
0 50 100 150 200 0 50 100 150 200 purposeofthispapertotestiftheperturbativeproperties
Lab. Energy (MeV) Lab. Energy (MeV) ofN3LOWwhenappliedinmicroscopicnuclearstructure
eg) 0 1P1 eg) 0 3P1 are as good as the ones of typical Vlow−k potentials.
d d
hift ( -10 hift ( -10
S S III. DERIVATION OF THE SHELL-MODEL
e e
as as EFFECTIVE HAMILTONIAN
h -20 h -20
P P
0 50 100 150 200 0 50 100 150 200
In the framework of the shell model, an auxiliary one-
Lab. Energy (MeV) Lab. Energy (MeV)
body potential U is introduced in order to break up the
g) 200 3S g) 0 3D nuclear hamiltonian as the sum of a one-body compo-
hift (de 112600 1 hift (de -1-05 1 nnuenctleHon0s,,wanhdichadreessicdruibaelsinttheerainctdieopneHnd1e:nt motion of the
S S
se 80 se -15
a a
h h
P 40 P -20 A p2
0 50 100 150 200 0 50 100 150 200 H = i + V =T +V =
ij
Lab. Energy (MeV) Lab. Energy (MeV) X2m X
i=1 i<j
FIG.1: Phaseparametersofneutron-protonscatteringupto = (T +U)+(V −U)=H0+H1 . (5)
200 MeV laboratory energy for partial waves with total an-
gular momentumJ ≤2. Thesolid lines showthepredictions Once H0 has been introduced, a reduced model space
bythenewlow-momentumN3LOWpotential. Thesoliddots isdefinedinterms ofafinite subsetofH0’s eigenvectors.
andopencirclesrepresenttheNijmegenmultienergynpphase The diagonalization of the many-body hamiltonian (5)
shift analysis [51] and the GWU/VPI single-energy np anal- in an infinite Hilbert space, that is obviously unfeasible,
ysis SM99 [52], respectively. is then reduced to the solution of an eigenvalue problem
for an effective hamiltonian H in a finite space.
eff
4
In this paper we derive H by way of the time- rules the convergencerate of the diagrammaticseries for
eff
dependent perturbation theory [1, 2]. Namely, H is H which has to deal with the convergence of both the
eff eff
expressedthroughthe KLRfolded-diagramexpansionin order-by-orderperturbative expansion and the sum over
terms of the vertex function Qˆ-box, which is composed the intermediate states in the Goldstone diagrams. In
of irreducible valence-linked diagrams. We take the Qˆ- this context, we have found it interesting to study and
box to be composed of one- and two-body Goldstone di- compare the convergenceproperties of H derived from
eff
agrams through third order in V [36]. Once the Qˆ-box both the Vlow−k and the N3LOW potentials.
has been calculatedat this perturbative order,the series
of the folded diagrams is summed up to all orders using
the Lee-Suzuki iteration method [32]. TABLE I: Ground state energies (in MeV) of 18O relative to
ThehamiltonianHeff containsone-bodycontributions, 16OcalculatedwithHeff derivedfromtheVlow−kofthe“hard”
which represent the effective SP energies. In realistic N3LO potential as a function of themaximum numberNmax
of the HO quanta (see text for details). Results obtained at
shell-modelcalculationsitiscustomarytouseasubtrac-
second and third order in perturbation theory are reported.
tion procedure [37] so that only the two-body terms of
Heff are retained- the effective interactionVeff - and the Nmax 4 6 8 10 12 14 16
SP energies are taken from the experimental data. 2nd -8.191 -10.615 -12.748 -14.318 -15.037 -15.142 -15.168
In this work, we have followed a different approach 3rd -6.617 -8.987 -11.344 -13.277 -14.294 -14.487 -14.523
and employed the theoretical SP energies obtained from
the calculation of H . This allows to make a consistent
eff
study of the perturbative properties of the input poten-
tial V (Vlow−k or N3LOW) in the shell-model approach. TABLE II: Same as Table I, but with Heff derived from
In this regard, it is worth to point out that, owing to N3LOW.
thepresenceofthe−U terminH ,U-insertiondiagrams
ariseintheQˆ-box. Inourcalcula1tionweuseasauxiliary Nmax 4 6 8 10 12 14 16
2nd -4.806 -6.789 -9.085 -11.759 -13.671 -14.108 -14.162
potentialtheharmonicoscillator(HO),U = 1mω2r2+∆,
2 3rd -3.547 -5.366 -7.683 -10.789 -13.339 -13.992 -14.049
and take into account only the first-order U-insertion
that appears in the collection of the one-body contribu-
tions, which is the dominant one [38]. In Table I we present the Vlow−k g.s. energies of 18O
relative to 16O obtained at second- and third-order per-
turbative expansion of H . In both cases the en-
eff
IV. RESULTS ergies are reported as a function of the maximum al-
lowed excitation energy of the intermediate states, ex-
We have performed shell-model calculations for even- pressed in terms of the oscillator quanta Nmax. It can
massoxygenisotopesbeyondthedoubly-closedcore16O. be seen that the g.s. energy is practically convergent
Oxygen isotopes constitute a quite interesting nuclear at Nmax = 12. As regards the order-by-order conver-
chain both theoretically and experimentally, and they gence, it may be considered quite satisfactory, the dif-
have been considered a testing ground for realistic shell- ference between second- and third-order results being
modelcalculationssincethepioneeringworkbyKuoand around 4 % for Nmax = 16, which is the largest Nmax
Brown [39]. Our calculations have been carried out by wehaveemployed. Similarresultsareobtainedusingthe
using the Oslo shell-model code [40] N3LOW potential, as shown in Table II. In this case
for N =16 the difference between second- and third-
Two H ’s have been obtained using, respectively, a max
eff
Vlow−k with a cutoff momentum Λ = 2.1 fm−1 derived order results is less than 1 %.
numerically from the ‘hard’ N3LO chiral NN potential
[11], and the new N3LOW potential described in Sec.
TABLEIII:SameasTableI,butwithexperimentalSPener-
II.B.TheCoulombforcebetweenproton-protoninterme-
gies.
diate states has been explicitly taken into account. For
theHOparameter~ω wehaveusedthevalueof14MeV, Nmax 4 6 8 10 12 14 16
asobtainedfromtheexpression~ω =45A−1/3−25A−2/3 2nd -12.201 -12.000 -11.847 -11.771 -11.750 -11.748 -11.749
3rd -12.373 -12.328 -12.286 -12.281 -12.290 -12.294 -12.296
[41] for A = 16. The value of ∆ has been chosen so
thattheself-energyandtheU-insertiondiagramsalmost
cancel each other [38]. The numerical value is -54 MeV
yieldinganunperturbedenergyofthe0s1dshellequalto
-5 MeV, which is not far the experimental value of the TABLE IV: Same as Table II, but with experimental SP en-
ground state (g.s.) energy of 17O relative to 16O (-4.144 ergies.
MeV [42]). However, it is worth to point out that, when
Nmax 4 6 8 10 12 14 16
including only first-order U-insertions, our H matrix
eff 2nd -12.386 -12.202 -12.072 -12.002 -11.973 -11.968 -11.967
elements do not depend on the choice of ∆.
3rd -12.663 -12.637 -12.612 -12.656 -12.722 -12.737 -12.739
The perturbative behavior of the input potential V
5
It is now worth to comment on the dimension of the USDA hamiltonian, see Ref. [43] for details. It is inter-
intermediate-state space. We have found that, when in- estingtonotethattheTBMEofUSDAareclosertoours
creasing N , the matrix elements of the two-body ef- thanthose ofthe originalUSD [44, 45], whichwas based
max
fective interaction V (TBME) are much more stable on a smaller set of data. Regarding the SP energies, we
eff
than the calculated SP energies. In this connection, an seethatthe USDAabsoluteenergyofthe d stateisin
5/2
inspection of Tables III and IV, where we present re- good agreement with the experimental g.s. state energy
sults obtainedusing experimentalSPenergies,evidences of 17O, while both our calculations give about one MeV
the rapid convergence of the g.s. energy with N . morebinding. Ontheotherhand,ourcalculatedandthe
max
This supports the choice of moderately large interme- USDA excitationenergies of the s state come close to
1/2
diate state spaces in standard realistic shell-model cal- eachotheranddonotdiffersignificantlyfromthe exper-
culations, where in general experimental SP energies are imental value. As regards the relative energy of the d
3/2
employed. state, the USDA value is larger than the experimental
A comparison between the results obtained from the one, in line with our findings.
Vlow−k and the N3LOW potentials (see Tables I and II,
respectively)showsthatatthirdorder,withasufficiently –3.0
largenumberofintermediatestates,the twointeractions V
predict very close values for the relative binding energy low–k
–4.0
of18O.Notethattheexperimentalvalueis−12.187MeV V]
e Expt.
[42]. From now on we refer to calculations with Nmax = M
16andQˆ-boxdiagramsuptothirdorderinperturbation [al –5.0 Theor.
Nv
theory. y/
g
er –6.0
n
E
TABLE V: Jπ = 0+ TBME of Heff (in MeV) obtained from s.
theVlow−kofthe“hard”N3LOpotentialandN3LOWatthird g. –7.0
orderinperturbationtheorywithNmax =16. Theyarecom-
pared with the USDATBME of Brown and Richter[43].
–8.0
0 2 4 6 8 10 12 14
configuration Vlow−k N3LOW USDA
(0d5/2)2 (0d5/2)2 -2.435 -2.689 -2.480 N
val
(0d5/2)2 (0d3/2)2 -3.464 -3.741 -3.569
(0d5/2)2 (1s1/2)2 -1.269 -1.337 -1.157 FIG. 3: (Color online) Experimental and calculated ground-
(0d3/2)2 (0d3/2)2 -0.845 -1.147 -1.505 state energy per valence neutron for even oxygen isotopes
(0d3/2)2 (1s1/2)2 -0.862 -0.967 -0.983 from A = 18 to 28. Nval is the number of valence neutrons.
(1s1/2)2 (1s1/2)2 -2.385 -2.563 -1.846 The calculated values are obtained with Heff derived from
the Vlow−k of the “hard” N3LO potential at third order in
perturbation theory,with Nmax =16.
TABLEVI:Calculated SPrelative energies of Heff (in MeV)
obtained from the Vlow−k of the “hard” N3LO potential and
N3LOW at third order in perturbation theory with Nmax = –3.0
16. TheyarecomparedwiththeUSDASPenergiesofBrown N3LOW
and Richter [43] and the experimental ones. The values in
–4.0
parenthesis are theabsolute SPenergies. V]
e Expt.
M
oνr0bdi5t/a2l 0.0V(lo-w5−.4k25) 0.N03(L-4O.9W09) 0.0U(S-3D.9A44) 0.0E(-x4p.1t44) N [val –5.0 Theor.
ν0d3/2 7.323 7.117 5.924 5.085 y/
g
ν1s1/2 1.257 0.818 0.883 0.871 er –6.0
n
E
s.
In Table V and VI, we present the J = 0+ TBME g. –7.0
and the SP energies obtained from the Vlow−k and the
N3LOW potentials. It turns out that the results from –8.0
0 2 4 6 8 10 12 14
thetwopotentialsarequitesimilar,thelargestdifference
regardingtheabsoluteenergyofthed5/2 SPlevel. Inthe Nval
lastcolumnofTableVandVIwealsoshowtheJπ =0+
TBME and SP energies obtained by Brown and Richter FIG. 4: (Color online) Same as Fig. 3,but with Heff derived
[43] from a least-squares fit of a large set of energy data from N3LOW.
for the sd-shell nuclei. The reported values refer to the
6
–2.0 mental 18O binding energy, then we obtain a very good
V agreement, as shown in Figs. 5 and 6.
low–k
–3.0 This result is very interesting: our calculations fail to
V] predictthat26Oand28Oareunboundto2-neutrondecay
e Expt.
M as indicated by experimental studies [46, 47]. However,
N [val –4.0 Theor. whentheSPenergiesareshiftedupward,thenweobtain
y/ the correct binding properties, as it may be seen from
g
er –5.0 Figs. 5 and 6. In some recent papers [48, 49] it has
En been argued that with a realistic shell-model interaction
s. itisnotpossibletoreproducethecorrectbehaviorofthe
g. –6.0 binding energy of oxygen isotopes. In Refs. [49, 50] this
hasbeenascribedtothelackofbothgenuineandeffective
–7.0 three-bodycorrelations. Ourresultssuggestthatamajor
0 2 4 6 8 10 12 14
factor is the lack of a real three-body force. In fact, as
N discussed above, we have obtained a good description
val
of the binding energy of oxygen isotopes by modifying
FIG. 5: (Color online) Same as Fig. 3, but with SP energies the SP spectrum so as to reproduce the g.s. energy of
shifted upward by 1.1 MeV. two-valence nucleon 18O. The last quantity is obviously
notaffectedbyeffectivethree-bodycorrelations,butitis
sensitive to genuine three-body forces.
In Figs. 7 and 8 we compare the experimental and
–2.0 calculated excitation energies of the first 2+ states for
N3LOW the Vlow−k and the N3LOW potential, respectively. In
both cases the experimental behavior as a function of A
–3.0
V] is well reproduced.
e Expt.
M
[al –4.0 Theor. 7.0
Nv 6.0 Vlow–k
y/
erg –5.0 5.0 Expt.
En eV] 4.0 Theor.
g.s. –6.0 E [M 3.0
2.0
1.0
–7.0
0 2 4 6 8 10 12 14 0.0
16 18 20 22 24 26
N A
val
FIG.7: (Coloronline)Experimentalandcalculatedexcitation
FIG. 6: (Color online) Same as Fig. 4, but with SP energies
energyofthefirst2+stateforoxygenisotopes. Thecalculated
shifted upward by 1 MeV.
values are obtained with Heff derived from the Vlow−k of the
“hard” N3LO potential. Experimental data are taken from
[53].
It is worth recalling that H , which has been derived
eff
intheframeworkoftheKLRapproach,isdefinedforthe
two valence-nucleon problem. It is of interest, however,
to test the theoretical SP energies, as given in Table VI, 7.0
N3LOW
and the TBME in systems with many valence nucleons. 6.0
To this end, we have calculated for evenoxygenisotopes 5.0
Expt.
the g.s. energies relative to 16O core up to A=28, and eV] 4.0 Theor.
the excitation energies of the first 2+ state up to A=24. M
In Figs. 3 and 4 the experimental g.s. energies per va- E [ 3.0
2.0
lence nucleon are compared with the calculated ones for
the Vlow−k and the N3LOW potential, respectively. The 1.0
predictions from both potentials yield binding energies 0.0
16 18 20 22 24 26
which are larger than the experimental ones, with the
A
differenceremainingalmostconstantwhenincreasingthe
number of valence neutrons Nval. In fact, if we shift the FIG. 8: (Color online) Same as in Fig. 7, but with Heff
Vlow−k and N3LOW SP spectra upwardby 1.1 MeV and derived from N3LOW.
1 MeV, respectively, such as to reproduce the experi-
7
V. SUMMARY AND CONCLUSIONS The main purpose of this work was to test the new
chiral low-momentum NN potential N3LOW. Unlike
In this paper, we have performed shell-model calcula- Vlow−k, this potential has the desirable feature that it
tionsemployinganeffectivehamiltonianobtainedfroma is given in analytic form. We have shown here that it
newlow-momentumpotential,dubbedN3LOW,whichis is suitable to be applied perturbatively in microscopic
derivedintheframeworkofchiralperturbationtheoryat nuclear structure calculations and yields results which
next-to-next-to-next-to-leadingorderwithasharpcutoff come quite close to those obtained from the Vlow−k de-
at 2.1 fm−1. rived from the “hard” N3LO potential.
Wehavestudiedtheconvergencepropertiesofthispo- Afterthis firstsuccessfulapplication,thereisnowmo-
tential by calculating the ground-state energy of 18O, tivation to pursue further nuclear structure calculations
andhavecomparedtheresultswiththoseobtainedusing with this new low-momentum potential.
the Vlow−k derived from the “hard” N3LO potential of
Entem and Machleidt [11]. It has turned out that the
two low-momentum potentials show the same perturba- Acknowledgments
tive behavior. We have also performed calculations for
the heavier oxygen isotopes, obtaining binding energies This work was supported in part by the Italian
which are very similar for the two potentials, but larger Ministero dell’Istruzione, dell’Universit`a e della Ricerca
thantheexperimentalones. However,wehavefoundout (MIUR), by the U.S. DOE Grant No. DE-FG02-
thatbyintroducinganempiricalshiftoftheSPspectrum 88ER40388, by the U.S. NSF Grant No. PHY-0099444,
the ground-stateenergies arewellreproducedup to 24O, by the Ministerio de Ciencia y Tecnonolog´ıaunder Con-
while26Oand28Oareunboundtotwo-neutrondecay,in tract No. FPA2004-05616,and by the Junta de Castilla
agreement with the experimental findings. y Le´on under Contract No. SA-104/04.
[1] T. T. S. Kuo, S. Y. Lee, and K. F. Ratcliff, Nucl. Phys. Rev. C 51, 38 (1995).
A 176, 65 (1971). [19] L.Coraggio,A.Covello,A.Gargano,andN.Itaco,Phys.
[2] T. T. S. Kuo and E. Osnes, Lecture Notes in Physics, Rev. C 66, 064311 (2002).
vol. 364 (Springer-Verlag, Berlin, 1990). [20] L. Coraggio, A. Covello, A. Gargano, N. Itaco, T. T. S.
[3] U.van Kolck, Prog. Part. Nucl. Phys. 43, 337 (1999). Kuo, D. R. Entem, and R. Machleidt, Phys. Rev. C 66,
[4] S.Weinberg, Phys. Lett.B 251, 288 (1990). 021303(R) (2002).
[5] S.Weinberg, Nucl. Phys.B 363, 3 (1991). [21] L.Coraggio,A.Covello,A.Gargano,andN.Itaco,Phys.
[6] G.P.Lepage,inNuclearPhysics: ProceedingsoftheVIII Rev. C 70, 034310 (2004).
Jorge Andre’ Swieca Summer School, edited by C. A. [22] L.Coraggio,A.Covello,A.Gargano,andN.Itaco,Phys.
Bertulani, M. E. Bracco, B. V. Carlson, and M. Nielsen Rev. C 72, 057302 (2005).
(World Scientific, Singapore, 1997), p. 135. [23] L. Coraggio and N. Itaco, Phys. Lett.B 616, 43 (2005).
[7] D.B.Kaplan,M.J.Savage,andM.B.Wise,Phys.Lett. [24] L.Coraggio,A.Covello,A.Gargano,andN.Itaco,Phys.
B 424, 390 (1998). Rev. C 73, 031302(R) (2006).
[8] E. Epelbaoum, W. Gl¨ockle, and U.-G. Meissner, Nucl. [25] L. Coraggio, N. Itaco, A. Covello, A. Gargano, and
Phys.A 637, 107 (1998). T. T. S. Kuo, Phys.Rev.C 68, 034320 (2003).
[9] P.F. Bedaque,M.J. Savage,R.Seki,and U.vanKolck, [26] L. Coraggio, A. Covello, A. Gargano, N. Itaco, T. T. S.
eds., Nuclear Physics with Effective Field Theory II, Kuo,andR.Machleidt,Phys.Rev.C71,014307(2005).
vol. 9 of Proceedings from Institute for Nuclear Theory [27] L. Coraggio, A. Covello, A. Gargano, N. Itaco, and
(World Scientific, Singapore, 1999). T. T. S. Kuo, Phys.Rev.C 73, 014304 (2006).
[10] W.C.HaxtonandC.L.Song,Phys.Rev.Lett.84,5484 [28] J. Kuckei, F. Montani, H. Mu¨ther, and A. Sedrakian,
(2000). Nucl. Phys. A 723, 32 (2003).
[11] D. R. Entem and R. Machleidt, Phys. Rev. C 68, [29] A. Sedrakian, T. T. S. Kuo, H. Mu¨ther, and P. Schuck,
041001(R) (2003). Phys. Lett. B 576, 68 (2003).
[12] E. Epelbaum, W. Gl¨ockle, and U.-G. Meissner, Nucl. [30] S.K.Bogner,A.Schwenk,R.J.Furnstahl,andA.Nogga,
Phys.A 747, 362 (2005). Nucl. Phys. A 763, 59 (2005).
[13] S.Bogner,T.T.S.Kuo,andL.Coraggio, Nucl.Phys.A [31] F. Andreozzi, Phys.Rev.C 54, 684 (1996).
684, 432c (2001). [32] K. Suzuki and S. Y. Lee, Prog. Theor. Phys. 64, 2091
[14] S. Bogner, T. T. S. Kuo, L. Coraggio, A. Covello, and (1980).
N.Itaco, Phys.Rev.C 65, 051301(R) (2002). [33] D.R.EntemandR.Machleidt,Phys.Rev.C66,014002
[15] S. Bogner, T. T. S. Kuo, and A. Schwenk, Phys. Rep. (2002).
386, 1 (2003). [34] R. Machleidt and D. R. Entem, J. Phys. G 31, S1235
[16] R.Machleidt, Phys.Rev. C 63, 024001 (2001). (2005).
[17] V.G.J.Stoks,R.A.M.Klomp,C.P.F.Terheggen,and [35] D. R.Entem and R. Machleidt, to be published.
J. J. D. Swart, Phys. Rev.C 49, 2950 (1994). [36] T.T.S.Kuo,J.Shurpin,K.C.Tam,E.Osnes,andP.J.
[18] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Ellis, Ann.Phys.(NY) 132, 237 (1981).
8
[37] J.Shurpin,T.T.S.Kuo,andD.Strottman,Nucl.Phys. R. Pfaff, C. F. Powell, and B. M. Sherrill, Phys. Rev.
A 408, 310 (1983). C 53, 647 (1996).
[38] J. Shurpin, D. Strottman, T. T. S. Kuo, M. Conze, and [47] O. Tarasov, R. Allatt, J. C. Ang´elique, R. Anne,
P.Manakos, Phys Lett.B 69, 395 (1977). C. Borcea, Z. Dlouhy, C. Donzaud, S. Gr´evy,
[39] T.T.S.KuoandG.E.Brown,Nucl.Phys.85,40(1966). D. Guillemaud-Mueller, M. Lewitowicz, et al., Phys.
[40] T. Engeland, the Oslo shell-model code 1991-2006, un- Lett. B 409, 64 (1997).
published. [48] B. A.Brown, Prog. Part. Nucl. Phys. 47, 517 (2001).
[41] J. Blomqvist and A. Molinari, Nucl. Phys. A 106, 545 [49] P. J. Ellis, T. Engeland, M. Hjorth-Jensen, M. P. Kar-
(1968). tamyshev,andE.Osnes,Phys.Rev.C71,034401(2005).
[42] G.Audi,A.H.Wapstra,andC.Thibault,Nucl.Phys.A [50] A. P. Zuker,Phys.Rev. Lett. 90, 042502 (2003).
729, 337 (2003). [51] V.G.J.Stoks,R.A.M.Klomp,M. C.M.Rentmeester,
[43] B.A.BrownandW.A.Richter,Phys.Rev.C74,034315 and J. J. deSwart, Phys. Rev.C 48, 792 (1993).
(2006). [52] R.A.Arndt,I.I.Strakowsky,andR.L.Workman,George
[44] B. H. Wildenthal, Prog. Part. Nucl. Phys. 11, 5 (1984). Washington University Data Analysis Center (formerly
[45] B.A.BrownandB.H.Wildenthal,Ann.Rev.Nucl.Part. VPI SAIDfacility), solution of summer 1999 (SM99).
Sci. 38, 29 (1988). [53] Data extracted using the NNDC On-line Data Service
[46] M. Fauerbach, D. J. Morrissey, W. Benenson, B. A. from the ENSDF database, file revised as of May 30,
Brown, M. Hellstr¨om, J. H. Kelley, R. A. Kryger, 2006.