Ab initio many-body calculations of nucleon-nucleus scattering

2.5 MB·English

by  Sofia Quaglioni

Ab initio many-body calculations of nucleon-nucleus scattering
Sofia Quaglioni and Petr Navr´atil
Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, CA 94551, USA
(Dated: January 8, 2009)
Wedevelopanewabinitiomany-bodyapproachcapableofdescribingsimultaneouslybothbound
and scattering states in light nuclei, by combining the resonating-group method with the use of
realistic interactions, and a microscopic and consistent description of the nucleon clusters. This
approach preserves translational symmetry and Pauli principle. We outline technical details and
present phase shift results for neutron scattering on 3H, 4He and 10Be and proton scattering on
3,4He, using realistic nucleon-nucleon (NN) potentials. Our A=4 scattering results are compared
to earlier ab initio calculations. We find that the CD-Bonn NN potential in particular provides an
excellentdescriptionofnucleon-4HeS-wavephaseshifts. Onthecontrary,theexperimentalnucleon-
4He P-wave phase shifts are not well reproduced by any NN potential we use. We demonstrate
that a proper treatment of the coupling to the n-10Be continuum is successful in explaining the
parity-inverted ground state in 11Be.
9
0
0
I. INTRODUCTION sis in the NCSM results in an incorrect description of
2
long-range correlations and a lack of coupling to contin-

n uum. The first NCSM applications to nuclear reactions
Nuclei are open quantum systems with bound states,
a
requiredaphenomenologicalcorrectionoftheasymptotic
J unbound resonances, and scattering states. A realistic
8 ab initiodescriptionoflightnucleiwithpredictivepower behavior ofoverlapfunctions [18]. The present approach
isfullyab initio. WecomplementtheabilityoftheRGM
musthavethecapabilitytodescribealltheaboveclasses
to deal with scattering and reactions with the use of re-
] of states within a unified framework. Over the past
h alistic interactions, and a consistent ab initio description
decade, significant progress has been made in our un-
-t derstandingofthepropertiesoftheboundstatesoflight of the nucleon clusters, achieved via the NCSM. Within
l this new approach we studied the n-3H, n-4He, n-10Be,
c nucleistartingfromrealisticnucleon-nucleon(NN)inter-
u actions,seee.g. Ref.[1]andreferencestherein,andmore and p-3,4He scattering processes, and addressed the par-
n ity inversion of the 11Be ground state (g.s.), using real-
recentlyalsofromNN plusthree-nucleon(NNN)inter-
[ istic NN potentials. In this paper, we give the technical
actions [2, 3, 4]. The solution of the nuclear many-body
details of these calculations, discuss results published in
1 problem is even more complex when scattering or nu-
v clear reactions are considered. For A = 3 and 4 nucleon Ref.[17]moreextensivelyandpresentadditionalresults.
0 InSect.II,wepresenttechnicaldetailsofourapproach.
systems, the Faddeev [5] and Faddeev-Yakubovsky [6]
5 WegivetwoindependentderivationsoftheNCSM/RGM
as well as the hyperspherical harmonics (HH) [7] or the
9 kernels, we discuss orthogonalization of the RGM equa-
Alt,GrassbergerandSandhas(AGS)[8]methodsareap-
0
tions and give illustrative examples of the kernels. Re-
. plicable and successful. However, ab initio calculations
1 sults of ab initio NCSM/RGM applications to A = 4,
for scattering processes involving more than four nucle-
0 A=5andA=11systemsaregiveninSect.III.Conclu-
ons overall are challenging and still a rare exception [9].
9
sionsaredrawninSect.IVandsomeofthemostcomplex
0 Thedevelopmentofanab initiotheoryoflow-energynu-
derivations are summarized in Appendix A.
: clear reactions on light nuclei is key to further refining
v
our understanding of the fundamental nuclear interac-
i
X tions among the constituent nucleons and providing, at
II. FORMALISM
r the same time, accurate predictions of crucial reaction
a rates for nuclear astrophysics.
The wave function for a scattering process involving
Recently we combined the resonating-group method
pairs of nuclei can be cast in the form
(RGM) [10, 11, 12, 13, 14, 15] and the ab initio no-
core shell model (NCSM) [16], into a new many-body |ΨJπT(cid:105)=(cid:88)(cid:90) drr2gνJπT(r)Aˆ |ΦJπT(cid:105), (1)
approach [17] (ab initio NCSM/RGM) capable of treat- r ν νr
ingboundandscatteringstatesoflightnucleiinaunified ν
formalism, starting from the fundamental inter-nucleon through an expansion over binary-cluster channel-states
interactions. The RGM is a microscopic cluster tech- of total angular momentum J, parity π, and isospin T,
nfuiqlluyeabnatis-esdymonmethtreicumseaonfyA-b-onduyclweoanveHfaumncilttioonnisabnus,ilwtaitsh- |ΦJνrπT(cid:105) = (cid:104)(cid:0)|A−aα1I1π1T1(cid:105)|aα2I2π2T2(cid:105)(cid:1)(sT)
suming that the nucleons are grouped into clusters. The (cid:105)(JπT) δ(r−r )
NCSM is an ab initio approach to the microscopic cal- ×Y(cid:96)(rˆA−a,a) rr A−a,a . (2)
culation of ground and low-lying excited states of light A−a,a
nuclei with realistic two- and, in general, three-nucleon Theinternalwavefunctionsofthecollidingnuclei(which
forces. The use of the harmonic oscillator (HO) ba- we will often refer to as clusters), contain A−a and a
2
nucleons (a<A), respectively, are antisymmetric under the charge numbers of the clusters in channel ν:
exchange of internal nucleons, and depend on transla-
tionally invariant internal coordinates. They are eigen- V = A(cid:88)−a (cid:88)A V +V3N −V¯ (r)
states of H(A−a) and H(a), the (A−a)- and a-nucleon rel ij (A−a,a) C
intrinsic Hamiltonians, respectively, with angular mo- i=1 j=A−a+1
mentum quantum numbers I and I coupled together
1 2 A−a A
to form channel spin s. For their parity, isospin and (cid:88) (cid:88) (cid:104)
= V ((cid:126)r −(cid:126)r ,σ ,σ ,τ ,τ )
N i j i j i j
additional quantum numbers we use, respectively, the
i=1 j=A−a+1
notations π ,T , and α , with i = 1,2. The channel
i i i
states (2) have relative angular momentum (cid:96). Denoting e2(1+τz)(1+τz) 1 (cid:105)
with {(cid:126)ri,i = 1,2,··· ,A} the A single-particle coordi- + 4|(cid:126)ri−(cid:126)r | j − (A−a)aV¯C(r)
nates, the clusters centers of mass are separated by the i j
relative vector +V3N . (8)
(A−a,a)
A−a A In the above expression we explicitly distinguished
1 (cid:88) 1 (cid:88)
(cid:126)rA−a,a =rA−a,arˆA−a,a = A−a (cid:126)ri− a (cid:126)rj. (bpeotwinetenannduacvleeornag-neu),claenodn,thnruecel-enaurcl(eVoNn)(Vp3luNs Co)ulcoommb-
i=1 j=A−a+1 (A−a,a)
(3) ponentsoftheinter-clusterinteraction. Thecontribution
The symbols Y and δ denote a spherical har- due to the nuclear interaction vanishes exponentially for
(cid:96)
monic and a Dirac delta, respectively. The inter- increasing distances between particles. Thanks to the
cluster anti-symmetrizer for the (A−a,a) partition subtraction of V (r), the overall Coulomb contribution
C
in Eq. (1) can be schematically written as Aˆ = presents a r−2 behavior, as the distance r between the
ν
[(A−a)!a!/A!]1/2(cid:80)P(−)pP, where P are permutations two clusters increases. Therefore, Vrel is localized also
among nucleons pertaining to different clusters, and p in presence of the Coulomb force. In the present paper
the number of interchanges characterizing them. we will consider only the NN part of the inter-cluster
interaction, and disregard, for the time being, the term
The coefficients of the expansion with respect to the
V3N . The inclusion of the three-nucleon force into
channelindexν ={A−aα Iπ1T ; aα Iπ2T ; s(cid:96)}arethe (A−a,a)
relative-motion wave funct1io1ns g1JπT(r2),2wh2ich represent the formalism, although more involved, is straightfor-
ν ward and will be the matter of future investigations. Fi-
the unknowns of the problem. They can be determined
nally, although in Eq. (8) the strong part of the NN
by solving the many-body Schr¨odinger equation in the
Hilbert space spanned by the basis states Aˆ |ΦJπT(cid:105): force (VN) is represented as a local potential, the above
ν νr separation of the Hamiltonian as well as the rest of the
formalismpresentedthroughoutthispaperarevalidalso
(cid:88)(cid:90) drr2(cid:104)HJπT(r(cid:48),r)−ENJπT(r(cid:48),r)(cid:105)gνJπT(r) =0, in the presence of a non-local potential.
ν(cid:48)ν ν(cid:48)ν r
ν
(4)
A. Cluster eigenstate calculation
where
We obtain the cluster eigenstates entering Eq. (2)
(cid:68) (cid:12) (cid:12) (cid:69)
HJπT(r(cid:48),r) = ΦJπT(cid:12)Aˆ HAˆ (cid:12)ΦJπT , (5) by diagonalizing H and H in the model space
ν(cid:48)ν ν(cid:48)r(cid:48)(cid:12) ν(cid:48) ν(cid:12) νr (A−a) (a)
spanned by the NCSM basis. This is a complete HO
(cid:68) (cid:12) (cid:12) (cid:69)
NνJ(cid:48)πνT(r(cid:48),r) = ΦJν(cid:48)πrT(cid:48)(cid:12)(cid:12)Aˆν(cid:48)Aˆν(cid:12)(cid:12)ΦJνrπT , (6) basis, the size of which is defined by the maximum num-
ber, N , of HO quanta above the lowest configuration
max
sharedbythenucleons(thedefinitionofthemodel-space
are called the Hamiltonian and norm kernels, respec- size coincides for eigenstates of the same parity, differs
tively. Here E is the total energy in the center-of-mass by one unity for eigenstates of opposite parity; the same
(c.m.) frame, and H is the intrinsic A-nucleon micro- HOfrequencyΩisusedforbothclusters). IftheNN (or
scopic Hamiltonian, for which it is useful to use the de- NNN)potentialusedinthecalculationgeneratesstrong
composition, e.g.: short-rangecorrelations,whichistypicalforstandardac-
curate NN potentials, the H and H Hamiltoni-
(A−a) (a)
H =T (r)+V +V¯ (r)+H +H . (7) ansaretreatedasNCSMeffectiveHamiltonians,tailored
rel rel C (A−a) (a) to the N truncation, obtained employing the usual
max
NCSM effective interaction techniques [4, 16]. The ef-
Further, T (r) is the relative kinetic energy and V fective interactions are derived from the underlying NN
rel rel
is the sum of all interactions between nucleons belong- and, in general, three-nucleon potential models (not in-
ing to different clusters after subtraction of the average cluded in the present investigations) through a unitary
Coulombinteractionbetweenthem,explicitlysingledout transformation in a way that guarantees convergence to
in the term V¯ (r)=Z Z e2/r, where Z and Z are the exact solution as the model-space size increases. On
C 1ν 2ν 1ν 2ν
3
(cid:113) (cid:113)
the other hand, if low-momentum NN potentials, which where (cid:126)x = 1((cid:126)r −(cid:126)r ) and p(cid:126)= 1(p(cid:126) −p(cid:126) ), we intro-
2 1 2 2 1 2
have high-momentum components already transformed
duce here a second, modified two-nucleon Hamiltonian,
awaybyunitarytransformations,areemployedinthecal-
deprived of the nuclear interaction:
culations, the H and H Hamiltonians are taken
(A−a) (a)
unrenormalized or “bare.” p(cid:126)2 1 mΩ2
H(cid:48)Ω =H +V(cid:48) = + mΩ2(cid:126)x2− (cid:126)x2. (10)
Thanks to the unique properties of the HO basis, we 2 02 12 2m 2 A
canmakeuseofJacobi-coordinatewavefunctions[19,20] Themodifiedtwo-bodyeffectiveinteractionisthendeter-
for both nuclei or only for the lightest of the pair (typi- mined from the two-nucleon Hermitian effective Hamil-
cally a≤4) referenced further on as projectile, and still toniansH¯ andH¯(cid:48) ,obtainedviatheLee-Suzukisim-
2eff 2eff
preserve the translational invariance of the problem. In ilaritytransformationmethod[23]startingfromEqs.(9)
the second case we expand the eigenstates of the heav- and (10), respectively:
ier cluster (target) on a Slater-determinant (SD) basis,
V(cid:48) =H¯ −H¯(cid:48) . (11)
and remove completely the spurious c.m. components in 2eff 2eff 2eff
a similar fashion as in Refs. [18, 21, 22]. We exploited We note that i) V(cid:48) →V in the limit N →∞, and
2eff N max
this dual approach to verify our results. The use of the ii) for each model space, the renormalizations related to
SD basis is computationally advantageous and allows us the kinetic energy and the HO potential introduced in
to explore reactions involving p-shell nuclei. H¯ are compensated by the subtraction of H¯(cid:48) .
2eff 2eff
C. Coordinates and basis states
B. Interaction between nucleons belonging to
different clusters
We neglect the difference between proton and neu-
tron masses, and denote the average nucleon mass with
Incalculating(5,6),all“direct”termsarisingfromthe
m. The formalism presented in this paper is based
identical permutations in both Aˆ and Aˆ are treated
ν ν(cid:48) bothonthesingle-particleCartesiancoordinates,{(cid:126)r ,i=
exactly(withrespecttotheseparationr)withtheexcep- i
tion of (cid:10)ΦJν(cid:48)πrT(cid:48)(cid:12)(cid:12)Vrel(cid:12)(cid:12)ΦJνrπT(cid:11). The latter and all remaining 1n,a2t,e·s:·· ,A}, and on the following set of Jacobi coordi-
terms are localized and can be obtained by expanding
the Dirac δ of Eq. (2) on a set of HO radial wave func- ξ(cid:126) =(cid:114)1 (cid:88)A (cid:126)r , (12)
tions with identical frequency Ω, and model-space size 0 A i
N consistent with those used for the two clusters. i=1
max
The rate of convergence of these terms is closely related the vector proportional to the center of mass (c.m.) co-
to the nuclear force model adopted in the Hamiltonian ordinate of the A-nucleon system (Rc.m. = √1 ξ(cid:126)0);
A
(7). For most nuclear interaction models that generate
(cid:114)
strongshort-rangenucleon-nucleoncorrelationsthelarge 1
ξ(cid:126) = ((cid:126)r −(cid:126)r ),
but finite model spaces computationally achievable are 1 2 1 2
not sufficient to reach the full convergence through a (cid:114) (cid:34) k (cid:35)
“bare” calculation. In these cases it is crucial to uti- ξ(cid:126) = k 1 (cid:88)(cid:126)r −(cid:126)r ,2≤k ≤A−a−1;
lizeeffectiveinteractionstailoredtothetruncatedmodel k k+1 k i k+1
i=1
spaces. In our approach the effective interactions are de-
(13)
rived from the underlying NN potential through a uni-
tary transformation as already pointed out in the pre- the translationally-invariant internal coordinates for the
vious Subsection. While the cluster eigenstates are ob- first A−a nucleons;
tained employing the usual NCSM effective interaction (cid:114)  A−a A 
[t1e6ri]n,ginVplac(e8)ofwteheadboaprte aNmNonduificeledartwpoo-tbeondtiyaleffVeNcteivne- (cid:126)ηA−a = (A−Aa)aA−1 a (cid:88)(cid:126)ri− a1 (cid:88) (cid:126)rj ,
rel i=1 j=A−a+1
interaction, V(cid:48) , which avoids renormalizations related
2eff (14)
tothekineticenergy. Whilethekinetic-energyrenormal-
the vector proportional to the relative position between
izationsareappropriatewithinthestandardNCSM,they (cid:113)
the c.m. of the two clusters ((cid:126)r = A (cid:126)η );
wouldcompromisescatteringresultsobtainedwithinthe A−a,a (A−a)a A−a
NCSM/RGM approach, in which the relative kinetic en- and, finally,
ergy and the average Coulomb interaction between the (cid:114) (cid:34) k (cid:35)
clusters are treated exactly. More specifically, in addi- ϑ(cid:126) = k 1 (cid:88)(cid:126)r −(cid:126)r ,a−1≥k ≥2,
tion to the relevant two-nucleon Hamiltonian (see also A−k k+1 k A−i+1 A−k
i=1
Refs. [16, 20]) (cid:114)
1
ϑ(cid:126) = ((cid:126)r −(cid:126)r ), (15)
p(cid:126)2 1 √ mΩ2 A−1 2 A−1 A
HΩ =H +V = + mΩ2(cid:126)x2+V ( 2(cid:126)x)− (cid:126)x2,
2 02 12 2m 2 N A the translationally-invariant internal coordinates for the
(9) last a nucleons.
4
1. Jacobi basis Note that the HO basis states depending on the Ja-
cobi coordinates introduced in Section IIC are all char-
acterized by the same oscillator-length parameter b =
Nuclei are translationally invariant systems. There- 0
(cid:112)(cid:126)/mΩ. However, the oscillator-length parameter asso-
fore, the use of Jacobi coordinates and translationally-
ciated with the separation r between the centers of mass
invariant basis states represents a “natural” choice for
oftargetandprojectileisdefinedintermsofthereduced
the solution of the many-nucleon problem.
mass µ=[(A−a)am]/A of the channel under considera-
Working with the Jacobi relative coordinates of
Eqs. (13), (14), and (15), it is convenient to introduce tion: b=(cid:112)(cid:126)/µΩ=(cid:112)A/[(A−a)a]b0. Inthefollowingwe
the (translationally-invariant) Jacobi channel states will drop the explicit reference to the HO length param-
eter in the arguments of the HO radial wave functions,
|ΦJπT(cid:105) = (cid:104)(cid:0)|A−aα Iπ1T (cid:105)|aα Iπ2T (cid:105)(cid:1)(sT) and in the HO Jacobi channel states |ΦJνnπT(cid:105).
νη 1 1 1 2 2 2
(cid:105)(JπT) δ(η−η )
×Y (ηˆ ) A−a , (16)
(cid:96) A−a ηηA−a 2. Single-particle Slater-determinant basis
whichareclearlyproportionaltothebinary-clusterbasis
Thanks to the unique properties of the HO basis, we
presented in Eq. (2):
canmakeuseofJacobi-coordinatewavefunctions[19,20]
forbothnucleioronlyforthelighterofthepair(typically
(cid:20)(A−a)a(cid:21)3/2
|ΦJπT(cid:105)= |ΦJπT(cid:105). (17) a ≤ 4), and still preserve the translational invariance of
νr A νη the problem (see also discussions in Refs. [21, 22]). In
the second case we introduce the SD channel states
The clusters intrinsic wave functions depend on their re-
spective set of Jacobi, spin (σ) and isospin (τ) coordi- |ΦJπT(cid:105) = (cid:104)(cid:0)|A−aα I T (cid:105) |aα I T (cid:105)(cid:1)(sT)
nates: νn SD 1 1 1 SD 2 2 2
(cid:105)(JπT)
(cid:104)ξ(cid:126) ···ξ(cid:126) σ ···σ τ ···τ |A−aα Iπ1T (cid:105), (18) ×Y(cid:96)(Rˆc(a.m).) Rn(cid:96)(Rc(a.m).), (24)
1 A−a−1 1 A−a 1 A−a 1 1 1
in which the eigenstates of the (A−a)-nucleon fragment
(cid:104)ϑ(cid:126) ···ϑ(cid:126) σ ···σ τ ···τ |aα Iπ2T (cid:105), are obtained in the SD basis,
A−a+1 A−1 A−a+1 A A−a+1 A 2 2 2
(19)
and are obtained by diagonalizing the H(A−a) and H(a) (cid:104)(cid:126)r1···(cid:126)rA−aσ1···σA−aτ1···τA−a|A−aα1I1π1T1(cid:105)SD, (25)
intrinsic Hamiltonians in the model spaces spanned by
the NCSM Jacobi-coordinate basis [20]. The same HO i.e.,byusingashell-modelcode(suchase.g. Antoine[24]
frequency Ω is used for both clusters. The model-space orMFD[25]), andcontainthereforethespuriousmotion
size coincides for eigenstates of the same parity and dif- of the (A−a)-nucleon cluster c.m. The SD and Jacobi-
fers by one unit for eigenstates of opposite parity. coordinate eigenstates are related by the expression:
In calculating the integral kernels of Eqs. (5) and (6),
(cid:10)ΦJν(cid:48)πrT(cid:48)(cid:12)(cid:12)Vrel(cid:12)(cid:12)ΦJνrπT(cid:11) and all “exchange” terms, arising |A−aα1I1T1(cid:105)SD =|A−aα1I1T1(cid:105) ϕ00(R(cid:126)c(A.m−.a)). (26)
from the permutations in A or A different from the
ν ν(cid:48)
identity,areobtainedbyexpandingtheDiracδofEq.(2) The c.m. coordinates introduced in Eqs. (24) and (26)
on a set of HO radial wave functions with identical fre-
qthuoesnecyusΩed, faonrdthmeotwdeol-cslpuascteerss:ize Nmax consistent with R(cid:126)c(A.m−.a) =(cid:114)A−1 aA(cid:88)−a(cid:126)ri ; R(cid:126)c(a.m). =(cid:114)a1 (cid:88)A (cid:126)ri,
i=1 i=A−a+1
|ΦJνrπT(cid:105) = (cid:20)(A−Aa)a(cid:21)3/2(cid:88)Rn(cid:96)(η,b0)|ΦJνnπ,Tb0(cid:105) (20) areanorthogonaltransformationofthec.m. andrela(t2iv7e)
n coordinates of the A-nucleon system, ξ(cid:126) (12) and (cid:126)η
= (cid:88)R (r,b)|ΦJπT(cid:105), (21) (14), respectively: 0 A−a
n(cid:96) νn,b
n
(cid:114) (cid:114)
a A−a
where the HO Jacobi-channel states are given by (cid:126)η = R(cid:126)(A−a)− R(cid:126)(a) , (28)
A−a A c.m. A c.m.
|ΦJπT(cid:105) = (cid:104)(cid:0)|A−aα Iπ1T (cid:105)|aα Iπ2T (cid:105)(cid:1)(sT)
νn,b 1 1 1 2 2 2
(cid:114) (cid:114)
×Y(cid:96)(ηˆA−a)(cid:105)(JπT)Rn(cid:96)(rA−a,a,b) (22) ξ(cid:126)0 = AA−aR(cid:126)c(A.m−.a)+ AaR(cid:126)c(a.m).. (29)
(cid:34)(cid:114) (cid:35)3/2
(A−a)a
= |ΦJπT (cid:105). (23) Therefore,intheSDbasisofEq.(24),theHOwavefunc-
A νn,b0 tions depending on these coordinates transform accord-
5
ing to ν0,r0 1 2···A-2A A-1
(cid:0)ϕ (R(cid:126)(A−a))ϕ (R(cid:126)(a) )(cid:1)((cid:96)) =
00 c.m. n(cid:96) c.m.
(cid:88) (cid:104)00n(cid:96)(cid:96)|nr(cid:96)rNL(cid:96)(cid:105)A−aa(cid:0)ϕnr(cid:96)r((cid:126)ηA−a)ϕNL(ξ(cid:126)0)(cid:1)((cid:96)), ··· ···
nr(cid:96)r,NL
(30)
where the coefficients of the expansion are generalized
HO brackets for two particles with mass ratio d = a ν,r 1 2···A-2A-1 A
A−a
that can be calculated as described e.g. in Ref. [26]. As
(a) (b)
a result the SD and Jacobi channel states are related by:
|ΦJπT(cid:105) = (cid:88) (cid:96)ˆJˆ (−1)(s+(cid:96)r+L+J) FIG. 1: (Color online.) Diagrammatic representation of the
νn SD r “direct”(a)and“exchange”(b)componentsofthenormker-
nr(cid:96)r,NL,Jr nel. The first group of circled black lines represents the first
(cid:26) (cid:27)
s (cid:96) J cluster, the bound state of A−1 nucleons. The separate red
× L Jr (cid:96)r (cid:104)nr(cid:96)rNL(cid:96)|00n(cid:96)(cid:96)(cid:105)A−aa linerepresentsthesecondcluster,inthespecificcaseasingle
nucleon. Bottom and upper part of the diagram represent
×(cid:104)|ΦJνrrπnrrT(cid:105)ϕNL(ξ(cid:126)0)(cid:105)(JπT), (31) initial and final states, respectively.
where ν ={A−aα I T ; aα I T ; s(cid:96) }. It is therefore
r 1 1 1 2 2 2 r In calculating (5) and (6), it is convenient to isolate
possible to extract the translationally-invariant matrix
the “direct” terms arising from the identical permuta-
elements from those calculated in the SD basis, which tion in Aˆ. Considering that the full A-nucleon Hamil-
contain the spurious c.m. motion, by inverting the fol-
toniancommuteswiththeinter-clusteranti-symmetrizer
lowing expression: ([Aˆ,H]=0), and that
SD(cid:68)ΦJν(cid:48)πnT(cid:48)(cid:12)(cid:12)(cid:12)Oˆt.i.(cid:12)(cid:12)(cid:12)ΦJνnπT(cid:69)SD = Aˆ2(cid:12)(cid:12)(cid:12)ΦJνrπT(cid:69)=(cid:104)1−A(cid:88)−1PˆiA(cid:105)(cid:12)(cid:12)(cid:12)ΦJνrπT(cid:69), (34)
i=1
(cid:88) (cid:68)ΦJνrr(cid:48)πnr(cid:48)rT(cid:12)(cid:12)(cid:12)Oˆt.i.(cid:12)(cid:12)(cid:12)ΦJνrrπnrrT(cid:69) winetchaenSwNrPitebtahseis:followingexpressionforthenormkernel
n(cid:48)r(cid:96)(cid:48)r,nr(cid:96)r,Jr
×(cid:88)(cid:96)ˆ(cid:96)ˆ(cid:48)Jˆr2(−1)(s+(cid:96)−s(cid:48)−(cid:96)(cid:48))(cid:26)Ls (cid:96)Jr J(cid:96)r (cid:27)(cid:26)sL(cid:48) (cid:96)J(cid:48)r J(cid:96)r(cid:48) (cid:27) NνJ(cid:48)πνT(r(cid:48),r)=δν(cid:48)ν δ(rr(cid:48)(cid:48)−r r) +Nνe(cid:48)xν(r(cid:48),r). (35)
NL
Here, we have singled out the non-local exchange part of
×(cid:104)nr(cid:96)rNL(cid:96)|00n(cid:96)(cid:96)(cid:105) a (cid:104)n(cid:48)r(cid:96)(cid:48)rNL(cid:96)|00n(cid:48)(cid:96)(cid:48)(cid:96)(cid:48)(cid:105) a , (32) the matrix elements in the term (we drop for simplicity
A−a A−a the JπT superscript)
wtrhaenrselatiOoˆnt.ai.l-inivsariaannyt opsecraaltaorr (aOˆntd.i. =paAˆr,itAyˆ-HcoAnˆs,eertvci.n)g. Nνe(cid:48)xν(r(cid:48),r) = −(cid:68)ΦJν(cid:48)πrT(cid:48)(cid:12)(cid:12)(cid:12)A(cid:88)−1PˆiA(cid:12)(cid:12)(cid:12)ΦJνrπT(cid:69) (36)
We exploited this dual approach to verify our results.
i=1
TheuseoftheSDbasisiscomputationallyadvantageous (cid:88)
= −(A−1) R (r(cid:48))R (r)
andallowsustoexplorereactionsinvolvingp-shellnuclei. n(cid:48)(cid:96)(cid:48) n(cid:96)
n(cid:48)n
(cid:68) (cid:12) (cid:12) (cid:69)
× ΦJπT(cid:12)Pˆ (cid:12)ΦJπT . (37)
ν(cid:48)n(cid:48)(cid:12) A−1,A(cid:12) νn
D. Translational invariant kernels in the
single-nucleon-projectile basis In deriving Eq. (37) we used the expansion (21), and
tookadvantageoftheinternalsymmetrypropertiesofthe
All calculations in the present paper were carried out (A−1)-cluster wave function. A similar decomposition
in the single-nucleon projectile (SNP) basis, i.e., using can be performed also for the Hamiltonian kernel,
binary-cluster channels (2) with a = 1. In this case, the
ϑ coordinates are not defined, the channel index reduces HJπT(r(cid:48),r) = (cid:68)ΦJπT(cid:12)(cid:12)H(cid:104)1−A(cid:88)−1Pˆ (cid:105)(cid:12)(cid:12)ΦJπT(cid:69) (38)
to ν = {A−1α Iπ1T ; 111; s(cid:96)}, and the inter-cluster ν(cid:48)ν ν(cid:48)r(cid:48)(cid:12) iA (cid:12) νr
1 1 1 22 i=1
anti-symmetrizer is simply given by
= (cid:104)Tˆ (r(cid:48))+V¯ (r(cid:48))+EI1(cid:48)π1(cid:48)T1(cid:48)(cid:105)NJπT(r(cid:48),r)
(cid:34) A−1 (cid:35) rel C α(cid:48)1 ν(cid:48)ν
Aˆ ≡Aˆ= √1 1− (cid:88)Pˆ . (33)
ν iA
A +VD (r(cid:48),r)+Vex(r(cid:48),r), (39)
i=1 ν(cid:48)ν ν(cid:48)ν
6
FIG.2: (Coloronline.)
Diagrammatic repre-
··· ··· ···
sentation of “direct” (c
and d) and “exchange”
(e) components of the
potential kernel (see
alsoCaptionofFig.1).
(c) (d) (e)
where we divided (cid:10)ΦJν(cid:48)πrT(cid:48)(cid:12)(cid:12)VrelAˆ2(cid:12)(cid:12)ΦJνrπT(cid:11) into “direct” Jacobi-coordinateformalismwithoutrequiringoverlyte-
and “exchange” potential kernels according to: dious manipulations. Interested readers can find a com-
pilation of all Jacobi-coordinate formulae, along with an
(cid:88)
VD (r(cid:48),r) = (A−1) R (r(cid:48))R (r) outline of their derivation, in Appendix A.
ν(cid:48)ν n(cid:48)(cid:96)(cid:48) n(cid:96)
The HO Jacobi channel state of Eq. (22) for the (2,1)
n(cid:48)n
×(cid:68)ΦJπT(cid:12)(cid:12)V (cid:0)1−Pˆ (cid:1)(cid:12)(cid:12)ΦJπT(cid:69) partition can be written as
ν(cid:48)n(cid:48)(cid:12) A−1,A A−1,A (cid:12) νn
(40) |ΦJν,πnT(cid:105) = (cid:88) (cid:10)n1(cid:96)1s1I1T1(cid:12)(cid:12)2α1I1π1T1(cid:11)
Vex(r(cid:48),r) = −(A−1)(A−2)(cid:88)R (r(cid:48))R (r) n1(cid:96)1s1
ν(cid:48)ν n(cid:48)n n(cid:48)(cid:96)(cid:48) n(cid:96) ×(cid:12)(cid:12)(cid:2)(n (cid:96) s I T ;11)sT;n(cid:96)(cid:3)JπT(cid:69), (44)
(cid:68) (cid:12) (cid:12) (cid:69) (cid:12) 1 1 1 1 1 22
× ΦJπT(cid:12)Pˆ V (cid:12)ΦJπT .(41)
ν(cid:48)n(cid:48)(cid:12) A−1,A A−2,A−1(cid:12) νn
where we have expanded the two-nucleon target wave
As pointed out in Sec. II, the channel states (2) are not function onto HO basis states depending on the Jacobi
anti-symmetric with respect to the exchange of nucle- coordinate ξ(cid:126)1 defined in Eq. (13)
ons pertaining to different clusters (fully anti-symmetric
states are recovered through the action of the opera- (cid:104)ξ(cid:126) σ σ τ τ |n (cid:96) s I T (cid:105), (45)
1 1 2 1 2 1 1 1 1 1
tor Aˆ ). As a consequence, the Hamiltonian kernel as
defineνd in Eq. (38) is explicitly non Hermitian. Using and(cid:10)n1(cid:96)1s1I1T1(cid:12)(cid:12)2α1I1π1T1(cid:11)arethecoefficientsoftheex-
AˆHAˆ = 1(Aˆ2H +HAˆ2), we introduce the Hermitized pansion. Here n1,(cid:96)1 are the HO quantum numbers cor-
Hamiltoni2an kernel H¯JπT in the form respondingtotheharmonicoscillatorassociatedwithξ(cid:126)1,
ν(cid:48)ν while s ,I , and T are the spin, total angular momen-
1 1 1
tum, and isospin of the two-nucleon channel formed by
H¯JπT(r(cid:48),r)=(cid:68)ΦJπT(cid:12)(cid:12)H−1A(cid:88)−1(cid:0)HPˆ +Pˆ H(cid:1)(cid:12)(cid:12)ΦJπT(cid:69). nucleons1and2,respectively. Notethatthebasis(45)is
ν(cid:48)ν ν(cid:48)r(cid:48)(cid:12) 2 iA iA (cid:12) νr anti-symmetric with respect to the exchange of the two
i=1
(42) nucleons, (−)(cid:96)1+s1+T1 =−1.
Finally, we note that, according to Eqs. (7,8) and AccordingtoEq.(37),inordertoobtaintheexchange
Eqs.(35,37),thecontributionoftheaverageCoulombpo- part of the norm kernel we need to evaluate matrix ele-
tentialtotheHermitianHamiltoniankernel(42)amounts mentsof thepermutationcorrespondingtotheexchange
overall to: of the last two particles, in this case Pˆ . This task can
23
be accomplished by, e.g., switching to a more convenient
1δ (cid:2)V¯ (r(cid:48))+V¯ (r)(cid:3)(cid:104)δ(r(cid:48)−r)−(cid:88)R (r(cid:48))R (r)(cid:105). coupling of the three-nucleon quantum numbers
2 ν(cid:48)ν C C r(cid:48)r n(cid:96) n(cid:96)
n (43) (cid:12)(cid:12)(cid:2)(n (cid:96) s I T ;11)sT;n(cid:96)(cid:3)JπT(cid:69)
(cid:12) 1 1 1 1 1 22
(cid:26) (cid:27)
1. Jacobi-coordinate derivation =(cid:88)ZˆIˆ1(−)(cid:96)1+s1+21+s (cid:96)11 ss1 IZ1
Z 2
(cid:26) (cid:27)
The main technical as well as computational challenge ×(cid:88)Λˆsˆ(−)Z+(cid:96)+s+Λ Z (cid:96)1 s
(cid:96) J Λ
of the NCSM/RGM approach lies in the evaluation of Λ
norm and Hamiltonian kernels. The analytical expres- ×(cid:12)(cid:12)(cid:2)(n (cid:96) ,n(cid:96))Λ;(cid:0)s 1(cid:1)Z(cid:3)Jπ(cid:69)(cid:12)(cid:12)(cid:0)T 1(cid:1)T(cid:69), (46)
sions for the integral kernels of Eqs. (37), (40), and (41) (cid:12) 1 1 12 (cid:12) 12
assumeaparticularlyinvolvedaspectinthemodelspace
spanned by the HO Jacobi channel states of Eq. (22). and observing that, as a result of the action of
Herewediscusstheexchange-partofthenormkernelfor Pˆ23, the HO state (cid:104)ξ(cid:126)1(cid:126)η2|(n1(cid:96)1,n(cid:96))Λ(cid:105) is changed into
the A=3 system (a=1), which is representative of the (cid:104)ξ(cid:126)(cid:48)(cid:126)η(cid:48)|(n (cid:96) ,n(cid:96))Λ(cid:105). The new set of Jacobi coordinates
1 2 1 1
7
ξ(cid:126)(cid:48) and (cid:126)η(cid:48) (obtained from ξ(cid:126) and (cid:126)η , respectively, by ex- ones. Consequently, the HO states depending on them
1 2 1 2
changing the single-nucleon indexes 2 and 3) can be ex- are related by the orthogonal transformation
pressedasanorthogonaltransformationoftheunprimed
(cid:104)ξ(cid:126)(cid:48)(cid:126)η(cid:48)|(n (cid:96) ,n(cid:96))Λ(cid:105)= (cid:88) (−)L+L1−Λ(cid:104)NL,N L ,Λ|n (cid:96) ,n(cid:96),Λ(cid:105) (cid:104)ξ(cid:126) (cid:126)η |(N L ,NL)Λ(cid:105), (47)
1 2 1 1 1 1 1 1 3 1 2 1 1
NL,N1L1
where the elements of the transformation are the general HO brackets for two particles with mass ratio d=3.
AftertakingcareoftheactionofPˆ alsoonthespinandisospinstates, onecancompletethederivationandwrite
23
the following expression for the A=3 exchange part of the norm kernel in the SNP basis:
Nνe(cid:48)xν(r(cid:48),r) = −2(cid:88)Rn(cid:48)(cid:96)(cid:48)(r(cid:48))Rn(cid:96)(r) (cid:88) (cid:10)n(cid:48)1(cid:96)(cid:48)1s(cid:48)1I1(cid:48)T1(cid:48)(cid:12)(cid:12)2α1(cid:48)I1(cid:48)π1(cid:48)T1(cid:48)(cid:11) (cid:88) (cid:10)n1(cid:96)1s1I1T1(cid:12)(cid:12)2α1I1π1T1(cid:11)
n(cid:48)n n(cid:48)1(cid:96)(cid:48)1s(cid:48)1 n1(cid:96)1s1
(cid:40) 1 1 T (cid:41) (cid:40) 1 1 s (cid:41)(cid:40)(cid:96)(cid:48) Z s(cid:48) (cid:41)
×Tˆ1(cid:48)Tˆ1(−)T1(cid:48)+T1 21 T2 T1(cid:48) sˆ(cid:48)1sˆ1Iˆ1(cid:48)Iˆ1sˆ(cid:48)sˆ(−)(cid:96)1+(cid:96)(cid:88)Λˆ2Zˆ2(−)Λ 21 Z2 s(cid:48)1 J1 (cid:96)(cid:48) Λ
2 1 Λ,Z 2 1
(cid:40)(cid:96)(cid:48) Z s(cid:48) (cid:41)(cid:40)(cid:96) Z s (cid:41)(cid:40)(cid:96) Z s (cid:41)
× 1 1 1 (cid:104)n(cid:48)(cid:96)(cid:48),n(cid:48)(cid:96)(cid:48),Λ|n (cid:96) ,n(cid:96),Λ(cid:105) . (48)
1 I(cid:48) s(cid:48) J (cid:96) Λ 1 I s 1 1 1 1 3
2 1 1 2 1 1
Hereweremindthattheindexν standsforthecollection 2. Single-particle Slater determinant derivation
of quantum numbers {A−1α Iπ1T ; 111; s(cid:96)}, while ν(cid:48)
1 1 1 22
is an analogous index containing the primed quantum
numbers.
The matrix elements of the operators Pˆ ,
The derivation of “direct”- and “exchange”-potential A−1,A
V (1−Pˆ ), and Pˆ V can be more
kernels, although complicated by the need for additional A−1,A A−1,A A−1,A A−2,A−1
intuitively derived working within the SD basis of
orthogonal transformations and the presence of the two-
Eq. (24). Using the second-quantization formalism, they
body matrix elements of the interaction, proceeds along
can be related to linear combinations of matrix elements
the same lines presented here (see Appendix A1). As
of creation and annihilation operators between (A−1)-
final remark, we note that while the exchange part of
nucleons SD states. These quantities can be easily cal-
thenormkernel(48)andthedirectpotentialkernel(A4)
culated by shell model codes. Here we outline the main
aresymmetricunderexchangeofprimeandunprimedin-
stages of the derivation.
dexes,andprimedandunprimedcoordinates,thesameis
nottrueoftheexchangepartofthepotentialkernel(A5).
Indeed, as anticipated in Sec. IID, the Hamiltonian ker- The SD basis (24) simplifies in the case of a single-
nel defined in Eq. (38) is explicitly non Hermitian. nucleon projectile to
|ΦJνnπT(cid:105)SD = (cid:104)(cid:0)|A−1α1I1T1(cid:105)SD(cid:12)(cid:12)(cid:12)(cid:12)12112(cid:29)(cid:1)(sT)Y(cid:96)(rˆA)(cid:105)(JπT)Rn(cid:96)(rA)
= (cid:88)(−1)I1+J+j(cid:40)I1 12s(cid:41)sˆˆj(cid:104)|A−1α1I1T1(cid:105)SDϕn(cid:96)j1((cid:126)rAσAτA)(cid:3)(JπT), (49)
(cid:96) Jj 2
j
with ν = {A−1α Iπ1T ; 111; s(cid:96)} and the HO calculate the permutation operator matrix elements
1 1 1 22
single-particle wave function ϕ ((cid:126)r σ τ ) = within the basis (49). By expressing the position
R (r )(cid:0)Y (rˆ )χ (σ )(cid:1)(j)χ (τn)(cid:96).jm12mTto AobAtaiAn the state of the nucleon (A−1) as |(cid:126)rA−1σA−1τA−1(cid:105) =
exnc(cid:96)hanAge (cid:96)paArt o12f Athemno21rmmt kAernel (37) we first (cid:80)n(cid:96)jm21mtϕ∗n(cid:96)jm21mt((cid:126)rA−1σA−1τA−1)a†n(cid:96)jm12mt|0(cid:105) we ar-
8
rive at
(cid:104)ΦJπT|Pˆ |ΦJπT(cid:105)
SD ν(cid:48)n(cid:48) A,A−1 νn SD
= 1 (cid:88) sˆsˆ(cid:48)ˆjˆj(cid:48)Kˆτˆ(−1)I1(cid:48)+j(cid:48)+J(−1)T1+12+T (cid:40)I1 12s(cid:41)(cid:40)I1(cid:48) 12s(cid:48)(cid:41)(cid:40)I1KI1(cid:48) (cid:41)(cid:40)T1 τ T1(cid:48)(cid:41)
A−1 (cid:96) Jj (cid:96)(cid:48) Jj(cid:48) j(cid:48) J j 1 T 1
jj(cid:48)Kτ 2 2
× (cid:104)A−1α(cid:48)I(cid:48)T(cid:48)|||(a† a˜ )(Kτ)|||A−1αI T (cid:105) . (50)
SD 1 1 n(cid:96)j12 n(cid:48)(cid:96)(cid:48)j(cid:48)12 1 1 SD
Here, (cid:104)A−1α(cid:48)I(cid:48)T(cid:48)|||(a† a˜ )(Kτ)|||A−1αI T (cid:105) are one-body density matrix elements (OBDME) of the
SD 1 1 n(cid:96)j12 n(cid:48)(cid:96)(cid:48)j(cid:48)12 1 1 SD
targetnucleusanda˜n(cid:48)(cid:96)(cid:48)j(cid:48)m(cid:48)1m(cid:48) =(−1)j(cid:48)−m(cid:48)+12−m(cid:48)t an(cid:48)(cid:96)(cid:48)j(cid:48)−m(cid:48)1−m(cid:48). Nextweextractthecorrespondingtranslationally-
2 t 2 t
invariant matrix elements, (cid:104)Φ(νAr(cid:48)−n(cid:48)r1,1)JrπrT|PˆA,A−1|Φν(Ar−nr1,1)JrπrT(cid:105), by inverting Eq. (32) for a = 1 and Oˆt.i. = PˆA−1,A.
The final step follows easily from Eq. (37).
The same procedure is applied also for calculating “direct”- and “exchange”-potential kernels. In this case the
transition matrix elements on the SD basis are respectively:
(cid:104)ΦJπT|V (1−Pˆ )|ΦJπT(cid:105)
SD ν(cid:48)n(cid:48) A−1,A A,A−1 νn SD
= A−1 1 (cid:88) (cid:88) (cid:88) (cid:88)sˆsˆ(cid:48)ˆjˆj(cid:48)KˆτˆJˆ02Tˆ02(−1)I1(cid:48)+j(cid:48)+J(−1)T1−12+T (cid:40)I(cid:96)1J12js(cid:41)(cid:40)I(cid:96)1(cid:48)(cid:48) J21js(cid:48)(cid:48)(cid:41)
jj(cid:48)KτnalajanblbjbJ0T0
×(cid:40)I1KI1(cid:48) (cid:41)(cid:40)jbjaK(cid:41)(cid:40)T1 τ T1(cid:48)(cid:41)(cid:40) τ 12 12(cid:41)(cid:113)1+δ (cid:113)1+δ
j(cid:48) J j j(cid:48) j J 1 T 1 T 1 1 (nalaja),(n(cid:48)(cid:96)(cid:48)j(cid:48)) (nblbjb),(n(cid:96)j)
0 2 2 0 2 2
1 1 1 1
×(cid:104)(n l j )(n(cid:48)(cid:96)(cid:48)j(cid:48) )J T |V|(n(cid:96)j )(n l j )J T (cid:105)
a a a2 2 0 0 2 b b b2 0 0
× (cid:104)A−1α(cid:48)I(cid:48)T(cid:48)|||(a† a˜ )(Kτ)|||A−1αI T (cid:105) , (51)
SD 1 1 nalaja12 nblbjb21 1 1 SD
and
(cid:104)ΦJπT|Pˆ V |ΦJπT(cid:105)
SD ν(cid:48)n(cid:48) A,A−1 A−2,A−1 νn SD
= 1 (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) sˆsˆ(cid:48)ˆjˆj(cid:48)KˆτˆKˆ τˆ Kˆ τˆ
2(A−1)(A−2) a a cd cd
jj(cid:48)KτnalajanblbjbnclcjcndldjdKaτaKcdτcd
×(−1)I1(cid:48)+j(cid:48)+J+K+j+ja+jc+jd(−1)T1+21+τ+T
(cid:40)I 1s(cid:41)(cid:40)I(cid:48) 1s(cid:48)(cid:41)(cid:40)I KI(cid:48) (cid:41)(cid:40)K K K(cid:41)(cid:40)T τ T(cid:48)(cid:41)(cid:40)τ τ τ (cid:41)
× 1 2 1 2 1 1 a cd 1 1 a cd
(cid:96) Jj (cid:96)(cid:48) Jj(cid:48) j(cid:48) J j j(cid:48) j j 1 T 1 1 1 1
a 2 2 2 2 2
(cid:113) (cid:113) 1 1 1 1
× 1+δ 1+δ (cid:104)(n(cid:48)(cid:96)(cid:48)j(cid:48) )(n l j )K τ |V|(n l j )(n l j )K τ (cid:105)
(nalaja),(n(cid:48)(cid:96)(cid:48)j(cid:48)) (nclcjc),(ndldjd) 2 a a a2 cd cd d d d2 c c c2 cd cd
×SD(cid:104)A−1α(cid:48)I1(cid:48)T1(cid:48)|||((a†n(cid:96)j21a†nalaja12)(Kaτa)(a˜nclcjc12a˜ndldjd21)(Kcdτcd))(Kτ)|||A−1αI1T1(cid:105)SD. (52)
While the “direct” matrix element (51) depends on the merinvolvesonlyasinglenucleonofthetarget,whilethe
OBDME, the “exchange” matrix element (52) depends latterinvolvestwonucleonsofthetarget, seealsoFig.2.
on two-body density matrix elements (TBDME) of the We note that the two-body matrix elements of the inter-
target nucleus. This is easily understandable as the for- action V are evaluated using just the first two terms of
9
e2(1+τz)(1+τz)
Eq. (8), i.e. V = V (ij)+ i j as the aver-
ij N 4|(cid:126)ri−(cid:126)rj| 0.03 n+α(g.s.)
age Coulomb interaction is taken care of with the help
of Eq. (43). We also note that as a consistency check, it Nmax
11
is possible to recover the expression (50) from either the
0.02 13
expression(51)ortheexpression(52)bysettingtheNN
15
interaction operator V to identity. −3] 32−21 (2P3/2) 1179
m0.01 21
f
[
3. Illustrative examples m) h¯Ω=19MeV
f
1 00
ploTrheethne-αchsayrsatecmteripsrtoicvidfeeastuarecsonovfenthieentingteroguranldkteorneexls- 0r=--00..11 N3LO 21+12 (2S1/2)
obtainedapplyingtheNCSM/RGMapproachwithinthe r,( Nmax
SNP formalism. Thanks to the tightly-bound structure exN``--00..22 -240 4 8 12 16 20
of 4He, an expansion in n-α channel states allows us to V]
describe fairly well the low-energy properties of the 5He --00..33 Me-26
system. The latter (likewise 5Li) is an unbound system, [-28
its ground state being a narrow P-wave resonance in the --00..44 E..gs-30 α
3− 1 channel. --00..55 -32
2 2
Figures 3 to 7, and Table I present results of single-
0 1 2 3 4 5 6 7 8
channel calculations carried out using n-α cluster chan- r [fm]
nels with the α particle in its g.s. (note that through-
out this Section the index ν ={4g.s.0+0;11+1;1(cid:96)} can FIG. 3: (Color online.) Dependence on N of the “ex-
2 2 2 max
and will be simply replaced by the quantum number change” part of the diagonal norm kernel for the n-4He(g.s.)
(cid:96)). The interaction models adopted are the N3LO NN 1+1 (2S ), and 3−1 (2P ) channels as a function of the
2 2 1/2 2 2 3/2
potential [27] derived within chiral effective-field theory relative coordinate r at r(cid:48) = 1 fm, using the N3LO NN po-
(χEFT)atthenext-to-next-to-next-to-leadingorder,and tential[27]at(cid:126)Ω=19MeV.Intheinset,convergencepattern
theV NN potential[28]derivedfromAV18withcut- oftheenergyofthe4Heg.s.,usedtobuildthebinary-cluster
lowk
off Λ = 2.1 fm−1. Although χEFT forces are known to basis. The green dashed line indicates the previous NCSM
present a relatively soft core, the large but finite model evaluation of Eg.s. =−25.39(1) MeV [29].
spaces computationally achievable are still not sufficient
to reach a full convergence through a “bare” calculation.
Therefore, for this potential we utilize two-body effec- 3% or less off the converged (Nmax = 21) curve in the
tive interactions tailored to the truncated model spaces wholer-rangeupto4.5fm. Ananalogousanalysisofthe
as outlined in Sec. IIB. Results for the Vlowk potential 2P3/2 kernelsyieldsasomewhatlargerrelativedifference
are obtained using the “bare” interaction. (less than 10%) between Nmax =17 and 21 in the range
The overall convergence behavior of the integral ker- between 1 and 4 fm, while the discrepancy increases to-
nels is influenced by both the convergence of the eigen- wards the origin. In this regard, we note that the 3−1
2 2
states entering the binary-cluster basis, in the specific kernel overall is an order of magnitude smaller than the
case the 4He g.s., and the convergence of the radial ex- 1+1 one.
2 2
pansion of Eq. (21). As an example, Fig. 3 presents the The convergence rate for V (see upper panel of
lowk
behavioroftheexchangepartofthenormkernelwithre- Fig. 4) is clearly much faster. Here the 2P results
3/2
specttotheincreaseofthemodel-spacesizeobtainedfor for the two largest model spaces (N =15 and 17) are
max
the JπT = 1+1, and 3−1 five-nucleon channels, using within 0.5% or less in the whole region up to 5 fm.
2 2 2 2
theN3LOpotential. Thecorrespondingconvergencepat- Despitethemilddifferencesinmagnitudeandstrength
tern for the α-particle g.s. energy is shown in the inset. distributionforsmallr,r(cid:48) values,the2S and2P re-
1/2 3/2
Inordertoallowforthecalculationofbothpositive-and sultsofFigs.3and 4presentessentiallythesameshape,
negative-parity five-nucleon channels, for a given trun- andsamerangeofabout5fm. Thiscanbeobservedalso
cation Nmax in the I1π1T1 = 0+0 model space used to inFig.5, whichshowsonceagainthe2S1/2 partialwave,
expand the g.s., a complete calculation of Eq. (37) re- in terms of contour plots (note that the 2S curves of
1/2
quiresanexpansionovern-αJπT statesuptoN +1. Figs. 3 and 4 correspond to slices of the current plot
max
ThisistheoriginoftheoddN valuesinthelegendof along the r(cid:48) = 1 fm line). In particular it is clear that
max
Fig.3(andfollowing). Aswecanseefromthefigure,the the 2S kernels for the two different NN potentials as-
1/2
HO frequency (cid:126)Ω=19 MeV enables a quite satisfactory sume almost-identical values starting from r,r(cid:48) = 2 fm,
convergence of both 4He g.s. and n-α radial expansion, the N3LO results being much shallower near origin and
and hence of the integral kernel. As an example, for the overall less symmetric than those obtained with V .
lowk
2S channel the N = 17 result is already within The latter features reveal differences in the structure of
1/2 max
10
0.05 tions of higherrelativeangular momenta are of the same
n+α(g.s.) orderorsmallerthanthe2D3/2 partialwave. Itisappar-
0.04 ent that the 2S channel dominates all over the others
1/2
Nmax and is negative. This is an effect of the Pauli exclu-
0.03 23−21 (2P3/2) 911 sthioensp-srhinecllipolfe,awnhuicclheafrorsbyisdtsemm.oTrehtehfaonurfonuurcnleuocnlseofnosrmin-
3]0.02 13 ing the 4He g.s. sit mostly in the 0(cid:126)Ω shell. Accord-
− 15
m ingly, in the 2S channel the “exchange”-part of the
17 1/2
[f0.01 normkernelsuppressesthe(dominant)0(cid:126)Ωcontribution
)
m h¯Ω=18MeV totheδfunctionofEq.(35)(and,consequently,totheS-
f
1 0 1+1
wave relative-motion wave function g2 2) coming from
= 0 (cid:96)=0
the fifth nucleon in s-shell configuration. More precisely,
0r V
r,-0.1 lowk the diagonalization of the “exchange” part of the norm
exN(``-0.2 21+12 (2S1/2) kernel reveals the presence of an eigenvector g021,+Γ12 with
-0.3 21−21 (2P1/2) eigenvalue γΓ (cid:39)−1, i.e. a Pauli-forbidden state:
--00..45 Nmax=17 2233+−1221 ((22DP33//22)) (cid:90) drN0e0x(r(cid:48),r)g012,+Γ12(r)=γΓg021,+Γ12(r(cid:48)). (53)
-0.6
TableIpresentsthethreelargest-negativeeigenvaluesfor
0 1 2 3 4 5 6 7 8 the adopted NN potentials along with their dependence
r [fm] upon the model space size. For both interactions the
first eigenvalue clearly corresponds to a Pauli-forbidden
FIG.4: (Coloronline.) “Exchange”partofthediagonalnorm state. Once again, the rate of convergence for V is
lowk
kernel for the n-4He(g.s.) 1±1 and 3±1 channels as a func- visiblyfasterthanforN3LO,and, despitethedifferences
2 2 2 2
tionoftherelativecoordinater atr(cid:48) =1fm,usingtheVlowk noted in the integral kernels, the overall results for the
NN potential [28] at (cid:126)Ω=18 MeV. The upper panel shows
the model-space dependence of the 2P component.
3/2
γ γ γ
1 2 3
N V
the α particle obtained within the N3LO and V NN max lowk
lowk
interactions. We note that g.s. energy and point-proton 9 −0.9547 −0.06609 −0.00310
root-mean-square radius of the α particle are −25.39(1) 11 −0.9539 −0.06600 −0.00288
MeV,1.515(2)fmand−27.77(1)MeV,1.4239(2)fmwith 13 −0.9530 −0.06616 −0.00290
the N3LO and Vlowk potentials, respectively. 15 −0.9526 −0.06617 −0.00292
In Fig. 4, bottom panel, we compare the components 17 −0.9524 −0.06616 −0.00293
of the “exchange”-norm kernel up to (cid:96) = 2. Contribu-
N N3LO
max
9 −0.954 −0.0633 −0.00346
Nex(r,r0) [fm−3] 11 −0.945 −0.0641 −0.00452
``
13 −0.938 −0.0643 −0.00524
N3LO -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 Vlowk 15 −0.933 −0.0646 −0.00599
4 4
3.5 3.5 17 −0.929 −0.0645 −0.00636
3 --00..0155 3 19 −0.927 −0.0644 −0.00661
m] 2.5 --00..2355 2.5 m] 21 −0.926 −0.0645 −0.00684
[f 2 -0.45 2 [f
0r 1.5 --00..5655 1.50r AV14
1 -0.75 1
-0.85
0.5 0.5 FY [30] −0.937 −0.0663 −0.00753
0 0
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4
r [fm] r [fm]
n+α(g.s.) 1+1 (2S ) TABLE I: The three largest negative eigenvalues of the “ex-
2 2 1/2 change” part of the norm kernel (37) for the n-4He(g.s.)
JπT = 1+1 channel. Convergencewithrespecttothemodel-
FIG.5: (Coloronline.) “Exchange”partofthediagonalnorm space si2ze 2N of the NCSM/RGM results obtained using
max
kernel for the n-4He(g.s.) 1+1 channel as a function of the the V [28] and N3LO NN potentials at (cid:126)Ω = 18 and 19
2 2 lowk
relative coordinates r and r(cid:48), using the N3LO [27] (left) and MeV, respectively. The calculated values for the AV14 NN
V [28] (right) NN potentials at (cid:126)Ω = 19 and 18 MeV, potential of Ref. [30] are multiplied by -1 to adhere to the
lowk
respectively. definition of the norm kernel adopted in the present paper.