Pairing in high-density neutron matter including short- and long-range correlations

1.3 MB·

by  D. Ding

Pairing in high-density neutron matter including short- and long-range correlations
D. Ding,1 A. Rios,2 H. Dussan,1 W. H. Dickhoff,1 S. J. Witte,1,3 A. Carbone,4 and A. Polls5
1Department of Physics, Washington University, St. Louis, Missouri 63130, USA
2Department of Physics, Faculty of Engineering and Physical Sciences,
University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom
3Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA
4Institut fu¨r Kernphysik, Technische Universit¨at Darmstadt,
64289 Darmstadt, Germany and ExtreMe Matter Institute EMMI,
GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH, 64291 Darmstadt, Germany
5Departament d’Estructura i Constituents de la Mat`eria and Institut de Ci`ences del Cosmos,
Universitat de Barcelona, Avinguda Diagonal 647, E-8028 Barcelona, Spain
(Dated: August 23, 2016)
Pairinggapsinneutronmatterneedtobecomputedinawiderangeofdensitiestoaddressopen
6
questions in neutron star phenomenology. Traditionally, the Bardeen-Cooper-Schrieffer approach
1
has been used to compute gaps from bare nucleon-nucleon interactions. Here, we incorporate the
0
influence of short- and long-range correlations in the pairing gaps. Short-range correlations are
2
treatedincludingtheappropriatefragmentationofsingle-particlestates,andsubstantiallysuppress

g the gaps. Long-range correlations dress the pairing interaction via density and spin modes, and
u providearelativelysmallcorrection. Weusedifferentinteractions,somewiththree-bodyforces,as
A a starting point to control for any systematic effects. Results are relevant for neutron-star cooling
scenarios, in particular in view of the recent observational data on Cassiopeia A.
2
2

I. INTRODUCTION angular momentum via a substantial reconfiguration of
]
h thevortexnetwork. Thisoccursinasingleviolent,short
t rearrangement event, which gives rise to the glitch. The
- Superfluids play an important role in nuclear physics,
l strength and periodicity of these glitches provide con-
c from nuclear structure to neutron-star observations [1].
straints on the amount of the star’s momentum of iner-
u In the latter, superfluidity is a key ingredient in the de-
n scriptionofseveralrelevantastrophysicalphenomena. In tia which is stored in the superfluid [13–15]. Traditional
[ modelsconsiderthesuperfluidaspinnedtothecrust,but
a sense, neutron stars are essential testing grounds for

arecentanalysishassuggestedthatglitchphenomenare-
2 the pairing properties of dense systems. It is important
quiresuperfluidsintheoutercoreofthestar,too[16]. In
v thattheoreticalconsiderationsondensemattersuperflu-
thispicture,thegapclosuredensityofsingletsuperfluids
0 ids are guided by knowledge on another relevant testing
0 ground of superfluidity: finite nuclei. Many-body theory becomesasensitiveprobeofneutronstarinteriorphysics.
6 While the effect of triplet pairing has not been studied
can provide insight on different aspects of singlet and
1 in this context, its strength and density dependence are
triplet pairing in nuclear matter [2, 3].
0 likely to be relevant as well.
. On the one hand, neutron stars cool predominantly
1
by neutrino emission [4]. This process depends sensi- In nuclei, a well-known experimental property of pair-
0
tively on the neutrino weak rates in the dense nuclear ing is its isovector nature. Neutron-neutron and proton-
6
1 medium [5, 6]. The presence of superfluid pairs in the proton pairs dominate to a large extent nuclear pair-
: crust and the core suppresses some of these rates. A ing [17]. The possibility of np pairing is less likely to
v
comparison with astrophysical observations, providing occur in N (cid:54)= Z nuclei, but even in N ≈ Z nuclei
i
X age and temperature estimates for several pulsars, can isoscalar np pairing is elusive [18]. In stark contrast,
r be used to test the validity of some assumptions on the Bardeen–Cooper–Schrieffer(BCS)pairingcalculationsin
a core’s pairing properties. Recently, observations of rapid sub-saturation infinite matter with bare nucleon-nucleon
coolinginCassiopeiaAhavebeeninterpretedasevidence (NN) forces predict a dominant pairing gap, of the or-
of the onset of triplet-pairing-mediated cooling in the der of 10 MeV, in the np sector [1, 19–21]. Mechanisms
core [7]. While there is some debate regarding these ob- have been proposed to explain why spin triplet pairing
servationsontheastrophysicalcommunity[8,9],itseems is particularly suppressed in nuclei [18, 22]. One needs
appropriate to review the status of many-body calcula- tocalluponmany-bodytheory,particularlybeyondBCS
tions of the pairing gaps in infinite neutron matter. pairing, to reconcile empirical observations with theory.
After all, the well-known BCS method works best for
Ontheotherhand,theacceptedmodelforglitchingin
uncorrelated systems, and one can put into question its
pulsar requires the presence of superfluids in the crusts
applicability in dense, correlated nuclear systems [23].
of neutron stars [10–12]. Neutron superfluid vortices are
not free to move, but rather remain pinned through the Other than dressing the in-medium interaction, corre-
interaction with lattice nuclei or defects. While the bulk lationsbeyondthetraditionalmean-fieldandBCSframe-
ofthestarslowsdownbytheemissionofelectromagnetic workshavealargeimpactontheredistributionofsingle-
radiation, the superfluid component can only change its particle (sp) strength [24]. Studies in electron knock-
2
out experiments have unambiguously demonstrated that is also able to describe quantitatively the density and
approximately 35% of the sp strength lies beyond the isospin dependence of high-momentum components and
quasi-particle peak [25–27]. About one third of this sp tensor-like correlations [29, 30, 57]. We extrapolate self-
strength depletion can be attributed to short-range cor- energies to zero temperature, and use them to generate
relations (SRC), that empty the nuclear Fermi sea and spectral functions that provide a quantitative account of
promote strength to the high-momentum region [28–30]. the removal of strength close to the Fermi surface [20].
SRCareverysimilarindifferentnuclei,whichbodeswell BasedontheGorkovformalism,ourapproachprovidesa
withtheideathatSRCarecausedtosomeextentbythe quantitativeestimateoftheeffectofSRConpairinggaps.
local short-distance repulsion of the NN force [31, 32]. The feedback from the superfluid phase into the normal
This idea has been exploited to study generic properties Green’s functions properties below the critical tempera-
of short-range correlated pairs, including their spin and ture is missing, but it is expected to be small.
isospin content [33–35]. All theoretical approaches in- WeextendthetreatmentofSRCpresentedinRef.[20]
dicate that SRC are dominated by np pairs in a region in four directions for the astrophysically relevant case
of the order of 50-100 MeV below the mean-field poten- of neutron matter. First, we consider, in addition to
tial [36], in accordance to experimental findings [37, 38]. the singlet 1S case, the case of pairing in the coupled
0
Pairing is active in a momentum region close to the tripletwave,3PF . Thesizeanddensityregimeinwhich
2
Fermisurface. Theremovalofspstrengthinthismomen- this gap operates could have relevance for neutrino cool-
tumregionduetoSRCshouldimpactthecorresponding ing [7] and glitch phenomena in pulsars [16]. Second, we
pairing gap [20, 39, 40]. To go beyond BCS theory, the have implemented a computational method to extrapo-
Gorkov-Green’sfunctionsdiagrammaticapproachcanbe late self-energies, spectral functions and thermodynam-
used to include correlations systematically into the pair- ical properties from finite-temperature calculations into
ing properties [41]. Microscopic calculations taking into zero temperature.
account SRC indicate a reduction of singlet pairing gaps Third, with the aim of quantifying any potential sys-
in nuclear and neutron matter [20, 42]. A qualitatively tematic uncertainties related to the underlying Hamilto-
similar answer is obtained when using a BCS-like ap- nian, we work with three different NN interactions. We
proach that is quenched by Z factors [40, 43], although use two high-quality phase-shift equivalent potentials,
a realistic description requires a full account of the spec- the CDBonn [58] and Argonne v18 (Av18) [59] forces.
tral function width [20, 44]. At finite temperature, SRC In terms of SRC, the latter is traditionally considered
alsocorrectthecorrespondingcriticaltemperatureofthe to be harder than the former, in the sense that it in-
pairingtransition. WeemphasizethataT-matrixresum- duces larger high-momentum components in the many-
mation that fully accounts for SRC is needed to describe body wave function [29, 30]. Moreover, we also employ
pseudo-gapphenomenainthevicinityofthephasetransi- the next-to-next-to-next-to-leading-order (N3LO) Idaho
tion[45]. Moreover, andaccordingtotheThoulesscrite- potential, that has been derived in the context of chiral
rion,theonsetofpairingisinone-to-onecorrespondence perturbation theory [60]. The cut-off associated to the
with the appearance of a pole in the T-matrix [46–48]. chiral expansion is implemented in the form of a regula-
In finite nuclei, long range correlations (LRC) are re- torfunctioninrelativemomentumwhichsharplycutsthe
sponsible for about 20% of the sp fragmentation via the potentialfromΛ=500MeVon. Asaconsequence,pair-
coupling of sp states to low-lying resonances and collec- ing calculations, which are directly sensitive to the rela-
tive modes [24]. For pairing properties, polarization tive momentum dependence of the matrix elements due
effects should mostly renormalize the effective interac- to the BCS kinematics, become sensitive to the artificial
tion [49], even at very low densities [50]. Density and regulator function for Fermi momenta above k ≈ 2.5
F
spin collective motion is expected to dress the interac- fm−1. We also present results which are computed by
tion of paired particles, which does not necessarily re- supplementing the Idaho NN potential with an N2LO
semble the free space NN force [40]. While the proper- three-neutron force (3NF), following an uncorrelated av-
ties of LRC can be different in finite and infinite nuclear erage over the third particle that is consistent with the
matter, a screening mechanism that dresses the pairing use of the Green’s function formalism [61–64]. 3NFs are
interaction is expected to affect both singlet and triplet includedbothattheleveloftheeffectiveinteractionand
pairing in neutron-star matter [40, 51–53]. In particular, onthetreatmentofSRC,inthisinitialexploratorystudy.
correctionsattheleveloftheeffectivepairinginteraction More sophisticated calculations including 3NFs are our
occur at low energies and Fermi liquid theory (FLT) can priority for the near future.
be used to provide a phenomenological, but systematic, Finally, we supplement the calculations of SRC in
understanding of screening [49, 54–56]. the gap equation with a physically relevant screening
In the following, we use a theoretical approach that of the pairing matrix elements. To account for LRC
combines self-consistent Green’s functions (SCGF) tech- in our infinite matter calculations, we screen the effec-
niquesandFLTtoincludeconsistentlytheeffectofSRC tive interaction using FLT as a starting point. Following
and LRC in the singlet and triplet gaps of neutron mat- Refs.[40,52,53],wedresstheinteractionwithsuccessive
ter. SRC are included by means of a well established particle-holeexcitations. Thesearecoupledtothepaired
finite-temperature ladder resummation scheme, which nucleonsbyverticesthatareself-consistentlydetermined
3
usingtheconceptoftheinducedinteraction[65]. Collec- this formulation goes beyond the BCS approach in that
tivemodesontopofthisaredescribedusingFermiliquid the fragmentation of states at the normal level is de-
theory, withLandauparametersobtainedfromRef.[52]. scribed in terms of a fully dressed sp normal propagator,
As a general conclusion, we find that the effect of LRC GN.
on triplet pairing gaps is smaller than that of SRC. Working in a partial wave basis, and after a suitable
This paper is organized as follows. Section II provides angle-averageprocedurehasbeenconsidered,theexpres-
an overview of the method, with specific subsections de- sion for the lowest-order dressed anomalous self-energy
voted to the discussion of our treatment of SRC, tem- leads to the generalized gap equation,
perature extrapolations and LRC. Results are summa-
rized in Sec. III. Singlet pairing gaps are discussed in (cid:88)(cid:90) ∞ dk(cid:48)k(cid:48)2(cid:104)k|VJST|k(cid:48)(cid:105)
∆JST(k)=− LL(cid:48) ∆JST(k(cid:48)).
subsection IIIA and triplet gaps, in subsection IIIB. A L π ξ(k(cid:48)) L(cid:48)
preliminary discussion on the effect of 3NF is provided L(cid:48) 0
(3)
in subsection IIIC. We draw conclusions and provide an
outlook of potential future work in Sec. IV. The appen-
∆JST(k) is the pairing gap for a given partial wave, L,
dicesprovideadiscussionofnumericalaspectsrelatedto L
in the channel of total angular momentum J, pair spin
zero-temperature extrapolations and fits.
S and pair isospin T. Pairs are in a BCS-like state of
opposite sp momenta, k = −k . The effective pairing
1 2
interaction, V , is therefore a function of relative mo-
II. METHODS LL(cid:48)
mentum,k =k ,andweworkatzeropaircentre-of-mass
1
momentum. InBCStheory,V wouldsimplybeabare
A. Short-range correlations LL(cid:48)
NN interaction. However, the effect of the medium is
important for the pairing interaction, even at very low
Inthefollowing, wedescribeamethodtoincludeSRC densities [50, 53, 66]. We will therefore introduce a de-
intothepairingpropertiesofdensenuclearmatter. More scriptionofthescreeningofV withpolarizationeffects
LL(cid:48)
specifically, we look at the inclusion of fragmented sp inSec.IID.Inournomenclature,thepolarizationeffects
states into the gap equation. Technically, our method is in the effective interaction are equivalent to LRC effects.
founded in the Gorkov-Green’s functions theory for the
Our main emphasis is on quantifying the effect of cor-
descriptionofcondensedfermionicsystems[23,41,66]. A
relations in pairing properties, and particularly in find-
diagrammaticexpansionintermsofself-consistentprop-
ing behaviours that are generic to all nuclear forces. We
agators exists in this case, and involves, in addition to
will therefore focus on calculations involving NN inter-
theusualGreen’sfunctions, anomalouspropagators. We
actions with different short-range and tensor structures.
donotprovidedetailsonthederivationsoftheequations
The results that contain 3NF have been obtained with
here: they have been presented elsewhere in the nuclear
a suitable non-correlated average over the third particle.
physics literature [20, 23, 42]. We also note that simi-
Attheleveloftheeffectivetwo-bodyinteraction,thisin-
lar approaches exist in condensed matter, particularly in
volvesanintegrationoverthethirdparticlethatincludes
thecontextoftheBCS-BECcrossover[67]. Inthatfield,
thefullanti-symmetrizationofthethree-bodymatrixel-
our approach is reminiscent to the fluctuation-exchange
ement [61, 62]. At the self-energy level, there is another,
(FLEX) scheme [68].
differently weighted, one-body contribution of 3NFs to
A specific formulation of Gorkov’s theory allows for
the Hartree-Fock term.
the resummation of correlations on the normal compo-
Inadditiontotheeffectiveinteraction,thekernelofthe
nent of the propagator using the normal Dyson equa-
gap equation is determined by an energy denominator,
tion [20, 23, 54, 69]. This is particularly useful for
ξ(k). In the Gorkov approach, the denominator is the
strongly correlated nuclear systems, in which there is al-
double energy convolution [20]:
ready a substantial fragmentation in the normal state
sp propagator, GN. Below the critical temperature, the 1 (cid:90) dωdω(cid:48)
Gorkovformalismcouplesthefullsuperfluidsppropaga- = AN(k,ω)A(k,ω(cid:48))×
2ξ(k) 2π 2π
tor, G, to its normal component via an anomalous self-
1−f(ω)−f(ω(cid:48))
energy, . (4)
ω+ω(cid:48)
G(k,ω)=GN(k,ω)−GN(k,ω)∆(k,ω)G(k,ω). (1)
The temperature, T, is included in the Fermi-Dirac dis-
In turn, the anomalous propagator, F, and self-energy tribution, f(ω) = [1+exp(ω−µ)/T)]−1. The spectral
(or superfluid gap), ∆, are related to both the normal function AN(k,ω) is related to the normal component of
and the full propagators: the self-energy,
F(k,ω)=GN(−k,−ω)G(k,ω)∆(k,ω). (2) −2ImΣN(k,ω)
AN(k,ω)= ,
(cid:2)ω− k2 −ReΣN(k,ω)(cid:3)2+ImΣN(k,ω)2
In the lowest order diagrammatic approximation, ∆ is 2m
(5)
an energy-independent quantity. We note, however, that
4
and includes information related to sp fragmenta- Consequently,AN andAonlydifferfromeachotherclose
tion in the normal phase [23, 29]. We work with totheFermimomentumandenergy,wherepairingeffects
self-energies that have been obtained within a finite- are more prominent [20, 39].
temperatureT−matrixSCGFapproach,discussedinde- A fully self-consistent description of pairing requires
tail in Refs. [70–72]. At temperatures close to the pair- an explicit iterative calculation of both AN and A [39].
ing phase transition, AN develops a characteristic two- Here, we take a different approach, which is an initial
peak structure as a function of energy, which is an in- step towards a fully self-consistent solution, including all
dication of a pseudo-gap phase [45]. At and below the the relevant correlations. First, because the normal and
critical temperature, the method is not valid anymore, superfluid spectral functions are very similar in a wide
as evidenced by the appearance of the Thouless pole in energy and momentum domain, we compute the contri-
the T−matrix [46–48]. We therefore obtain the normal butiontotheeffectivedenominatorfromthedoublecon-
spectral function at zero temperature by extrapolating volution of normal spectral functions,
finite temperature results down to zero temperature, as
1 (cid:90) dωdω(cid:48)
explained in the following subsection. This is in agree- = AN(k,ω)AN(k,ω(cid:48))×
ment with the physical interpretation of AN as a normal 2χ(k) 2π 2π
state spectral function. 1−f(ω)−f(ω(cid:48))
. (10)
Inthelowest-orderBCSapproachatzerotemperature, ω+ω(cid:48)
the spectral functions in the convolution of Eq. (4) be-
This will account explicitly for fragmentation effects on
come delta functions in energy. The normal self-energy
the gap equation. The superfluid gap is then generated
has a single peak at the normal quasi-particle energy,
by the gap equation, Eq. (3), with an energy denomina-
whereas the superfluid spectral function shows two un-
tor that has the same structure as in the BCS expres-
equallyweightedsolutionsforagivenmomentum[20,42].
sion, Eqs. (6) and (7). One can indeed generate a su-
The energy denominator is a function of momentum,
perfluidspectralfunctionfromtheresultinggap, seee.g.
Ref. [20]. Further calculations of the normal self-energy
ξ(k)=|E(k)| (6)
then in principle require the effect of the gap to be in-
with the effective sp energy, cluded in the normal propagators [39]. We expect such
feedback effects from the superfluid phase to be small
E2(k)=χ2(k)+∆2(k). (7) in comparison to the relatively large fragmentation of
strength associated with SRC which is captured effec-
To obtain this result, we have also assumed that there tively by Eq. (10).
is no renormalisation of the sp peaks, Z(k) ≈ 1. In a This approach was first exploited in Ref. [20] to study
low-density BCS approximation, the effective sp energy the singlet pairing properties of neutron matter and
corresponds to a kinetic spectrum, χ(k)= k2 −µ, with triplet pairing in symmetric nuclear matter. A major
2m conclusion of that study was the large impact of sp frag-
µthechemicalpotentialofthesystem. Onecanalsoadd
mentation on pairing properties. Gaps in infinite matter
a mean-field potential contribution to the spectrum, or
are substantially quenched by the removal of strength
describe its effect by means of an effective mass [23, 66].
mediated by SRC, generally by a factor of ≈ 20%. In
The averaged gap, ∆ is associated with the partial wave
nuclear matter, the SRC effect alone precludes the for-
that is active for a given Fermi momentum. In practice,
mation of a 3SD pairing gap in nuclear matter at sat-
different partial waves are active in different regions of 1
uration density. This provides support for the lack of
Fermi momentum. Consequently, we consider ∆(k) ≈
experimental evidence of isoscalar np pairing from an
∆JST(k) in the solution of the gap equation in a given
L infinite-matter calculation.
2
JST channel[1]. Forcoupledchannels, wetake∆ (k)=
Finally,wewouldliketocommentonquasi-particleap-
(cid:0)∆JST(k)(cid:1)2+(cid:0)∆JST(k)(cid:1)2. proximations to the pairing problem. In a quasi-particle
L L(cid:48)
The gap equation is an integral non-linear equation limit,thespectralfunctionsinEq.(4)andEq.(10)arere-
for∆JST(k),whichappearsexplicitlyinthedenominator placed by delta functions centered around quasi-particle
L
viaEq.(7). Inthefullycorrelatedtheory,incontrast,the energies. In both the normal and the superfluid case,
gap appears indirectly in the definition of the superfluid these are weighted by the corresponding renormaliza-
spectral function, tionsofthequasi-particlepoles,orZ-factors[42,43,73].
Thesewilleffectivelyaccountforaremovalofstrengthin
A(k,ω)=−2ImG(k,ω). (8) differentregionsofmomentumspace,includingtheFermi
surface. In general, one finds that the quasi-particle en-
The full sp propagator, G(k,ω), computed in the super- ergydenominatorofEq.(6)isdividedbyafactorZ2(k).
fluid phase, differs from the normal one by a factor that
These quasi-particle approximations, however, include
is proportional to the square of the gap,
fragmentation in a very crude way and cannot reliably
G(k,ω)=GN(k,ω)(cid:2)1−GN(−k,−ω)∆2(k,ω)G(k,ω)(cid:3) . predicttheeffectofquenchingduetoSRCobtainedwith
realistic spectral functions in the double convolution of
(9)
Eq. (10) [20].
5
k=0 k=kF k=2kF
Im(k,) [MeV]Σω---- 08642 (a) ----- 543210 (b) ------110000- ......10428642 TTTTTT======11864020 MMMM MMeeeeeeVVVVVV (c) mentum distribution, n(k) 0000 ....12468 TTTTTTTTTT==========2111118640086420 MMMM MMMMMM(eeeeaeeeeeeVVVV)VVVVVV χenominator, (k) [MeV] 1 2468000000 CkFD=B1o.3n3n fm-1 (b)
-10 -6 CDBonn -1.6 TT==1164 MMeeVV Mo D
kF=1.33 fm-1 -1.8 TT==2108 MMeeVV 0 0
-12 -7 -2 0 0.5 1 1.5 2 0 0.5 1 1.5 2
-150-100-50 0 50 100 150 -150-100-50 0 50 100 150 -150-100-50 0 50 100 150
ω−µ [MeV] ω−µ [MeV] ω−µ [MeV] Momentum, k/kF Momentum, k/kF
FIG. 1. Imaginary part of the self-energy around the Fermi FIG. 2. Panel (a): momentum distribution at k = 1.33
F
energy at k = 1.33 fm−1 for the CDBonn interaction for fm−1 for the CDBonn interaction at different temperatures,
F
several temperatures. Panels (a), (b) and (c) correspond to including the T = 0 extrapolation (solid line). Panel (b):
momentak=0,k and2k ,respectively. TheT =0extrap- effective sp denominator, χ(k), in the same conditions.
F F
olation is shown in a solid line. Note the different vertical
scales of each panel.
Fromtheself-energy,onecanobtainotherrelevantmi-
croscopic properties. Panel (a) in Fig. 2 shows an exam-
B. Zero temperature extrapolation ple of the temperature extrapolation of the momentum
distribution. Again, while this specific example is for
the CDBonn interaction at k =1.33 fm−1, very similar
The normal spectral function at zero temperature, F
AN(k,ω),hasbeencomputedasanextrapolationoffinite results are obtained with other forces in a wide density
regime. Astemperaturedecreases,onefindstheexpected
temperature self-energies. Numerical results of SCGF
behavior for the correlated momentum distribution: the
ladder approximation self-energies with microscopic NN
Fermi surface becomes increasingly sharp, and low and
interactions have been available in the literature for the
high-momentum features build up. As expected, the
last decade [57, 71, 72]. The pairing instability, however,
zero temperature n(k) has a sharp discontinuity across
precludes a direct calculation within the ladder approx-
the Fermi surface. The exact shape of the momentum
imation and normal propagators below the critical pair-
distribution for momenta within a few percent of k is
ing temperature, T [20, 48]. Consequently, for a fixed F
c
sensitive to the extrapolation procedure, particularly to
density, we perform a series of finite temperature cal-
the order of the extrapolating polynomial. However, the
culations to determine the real and imaginary parts of
implementation of the thermodynamically consistent ex-
the self-energy, and use these as input for an extrapola-
trapolation procedure guarantees that, on average, the
tion to zero temperature. The fit is further constrained
discontinuityoftheFermisurfaceiswithinafewpercent
by the requirement that the macroscopic properties that
of the derivative of the self-energy (i.e. the Z-factor at
are computed at T = 0 provide a thermodynamically
k =k ). Wenotethatwehavecorrectedthemomentum
consistent description of the system. Numerical details F
distribution for missing strength effects, as discussed in
areprovidedinAppendixA.Below,wediscusstheprop-
Appendix A.
erties of the extrapolated self-energies.
The three panels of Fig. 1 show the imaginary part of
the self-energy as a function of energy, ω, for three char-
acteristicmomenta. Panels(a)and(c)showself-energies C. Pairing kernel with short range correlations
wellbelowandabovetheFermimomentum,respectively.
Panel(b),incontrast,showsthek =k case. Resultsare Pairing calculations require as input the double en-
F
displayedfortheCDBonninteractionatk =1.33fm−1, ergyconvolutionofEq.(10). Thisconvolutionisformally
F
but equivalent conclusions are found with other NN in- equivalent to the dressed but non-interacting two-body
teractions and 3NFs in this density regime. At large propagator,G0 ,inthein-mediumT−matrixequationat
II
temperatures, there is little (or no) distinction between zero energy and centre-of-mass momentum [23, 57, 70].
the hole, ω < µ, and the particle ω > µ, parts of ImΣ. This is in correspondence to the well-known fact that
As temperature decreases, however, a structure develops CooperpairingappearsasapoleinthenormalT−matrix
close to ω ≈ µ, with ImΣ approaching zero in absolute intheseconditions. Theexperiencegatheredinperform-
value. Thisistheareawheretemperatureplaysthemost ing the double convolution in SCGF finite temperature
important role, and where the extrapolation procedure calculationsisusefulincomputingtheenergydenomina-
is most critical. A momentum- and energy-dependent tor [70]. In particular, it is useful to keep track of the
polynomial fit, described in Appendix A, captures this quasi-particle energies for each given sp momentum, so
temperature dependence, and provides an extrapolated thatthequasi-particlepeakiswellsampledinthedouble
self-energy which provides consistent results. folding integrals [71].
6
V] 250 10 spectral functions. The spectral functions of these three
e CDBonn
χ(k) [M 125000 1 ignrtaetreadctcioonnvsolaurteiornelsamtievaerlysoduitsstihmeildairffe[r2e9n],cebsuttotahceeritnatien-
nator, 100 111...350336 222...012220 0.1 eitxstdenent.siCtyondseepqeunednetnlyc,e)thaerererseulalttsivoeblytacilnoesdeffoorraχll(kth)e(aNnNd
Denomi 500 (a) 111...689812 22..2482 (d) 0.01 forIcnesg.eneral,wefindthatthedoubleconvolutiondenom-
V] 250 10 inator increases with density. This density dependence
e Av18
M
χ(k) [ 125000 1 oFcecrmurissautrfaalclemtohmatenistad,isipnlcalyueddin,ginthaelongeaarritvhimcinicitsycaolfe,thine
or, panels (d) to (f) of Fig. 3. The gap equation is mostly
minat 15000 0.1 determinedbytheenergydenominatorclosetotheFermi
no (b) (e) surface, andhenceitisimportantthatthisregioniswell
De 0 0.01 sampled for pairing purposes. The denominator shows a
V]250 10
e N3LO linear behavior both below and above k , with a sharp
M F
χnator, (k) [112050000 01.1 mtzheirenoiF.meWurmmhiilaseutrttfhaheceel,iFnkeeram=ribkseFuh,rafivasicorere.laistTievhxeeplyevcastmleudaellio,nfbaχut(qkun)aosnait--
mi 50 particle-typeapproach,thenon-zerominimumisadirect
Deno 0 (c) (f) 0.01 consequence of the use of a double convolution beyond a
0 0.5 1 1.5 2 0.98 0.99 1 1.01 1.02 quasi-particlepicture. Wenotethat,withoutthemissing
Momentum, k/k Momentum, k/k
F F strength corrections discussed in Appendix A, the near-
Fermi-surface behavior would be erratic. The corrected
FIG. 3. Panels (a)-(c): energy denominator at T = 0 as a
calculations, in contrast, provide a well-defined function
function of momentum for different Fermi momenta, corre-
sponding to the (a) CDBonn, (b) Av18 and (c) N3LO inter- of both momentum and density.
actions. Panels (d)-(f): the same function, around the Fermi In the quasi-particle limit of Eq. (6), the denomina-
surface, plotted in a logarithmic scale. See text for details. tor reflects the momentum and density dependence of
the quasi-particle energy with respect to the chemical
potential. We show the extrapolated zero-temperature
quasi-particle denominator in Fig. 4, in the same condi-
Pairing calculations, particularly in the 3PF channel, tions and for the same NN forces as Fig. 3. The quasi-
2
are very sensitive to the Fermi surface region, and in- particle energies are determined consistently by solving
accuracies on the double folding are amplified in final thecorrespondingimplicitequationfortheSCGFladder
gapsolutions. Inparticular,missingstrengthcorrections, self-energies,
analogoustothosediscussedforn(k)inAppendixA,are
k2
essentialtocomputeacontinuousenergydenominatorin ε (k)= +ReΣN(k,ε (k)), (11)
regions arbitrarily close to k . Panel (b) of Fig. 2 shows qp 2m qp
F
the energy denominators for a CDBonn calculation at
and subtracting the chemical potential.
k = 1.33 fm−1 as a function of momenta for a variety
F Thequasi-particlepictureprovidesanintuitiveunder-
of temperatures. Here, as it was the case with n(k), the
standingforthedensitydependenceoftheenergydenom-
largest modifications due to temperature occur close to
inator. Broadly speaking, the quasi-particle spectrum
k . The low and the high momentum ends of χ(k) are
F is more stretched as density increases. A sign notwith-
less sensitive to temperature, and their details are well
standing,thisgivesrisetoanincreaseofthedenominator
captured by finite temperature calculations. In contrast,
as density rises at low momenta. In turn, when mea-
the region around k = k shows a non-negligible tem-
F sured with respect to the Fermi momentum, the high-
perature dependence even in the lowest temperatures.
momentum quasi-particle energies become more repul-
In the quasi-particle approximation, the Sommerfeld ex-
sive as density increases, which gives rise to the k > k
F
pansionspredictsalineartemperaturedependenceofthe
behavior.
denominatorinregionsclosetok . Panel(b)showsthat
F Two important conclusions can be drawn from a com-
a similar linear temperature dependence is generated in
parisonofFigs.3and4. Ontheonehand,thequalitative
that region.
density and momentum dependence of the quasi-particle
The density dependence of the zero-temperature dou- and the double convolution denominators are similar. In
ble convolution is displayed in panels (a) to (c) of Fig. 3. particular, both are increasing functions of density. As
Each panel represents the results obtained with a differ- functions of momenta, the initial decrease below k is
F
ent NN interaction: (a) CDBonn [58], (b) Av18 [59] and followedbyanincreaseabovetheFermisurface. Further-
(c) the Entem-Machleidt N3LO potential [60]. We note more, a linear behavior is found near the Fermi surface
thatthedifferentNNinteractionsenterthedenominator in both cases, as expected on general grounds.
calculation via the convolution of different extrapolated On the other hand, there are quantitative differences
7
250 1.4 1.4
CDBonn CDBonn
1.3 1.3
200
1.2 1.33 1.2k)
110500 1.11 112...691822 11.1ctor, Z(
a
V] 50 0.9 0.9Z-f
Me (a) 0.8 (a) (d) 0.8
εuasi-particle denominator, |(k)| [qp 2211220550505000000000 NA3vL1O8 (b) ΖDenominator ratio, (k)eff1100011110..34........178912347 NA3vL1O8 (b) (e) 11000111110..........3478912347Z-factor, Z(k)
Q
1.2 1.2k)
150 Z(
100 1.11 11.1ctor,
a
50 (c) 0.9 0.9Z-f
0 0.8 (c) (f) 0.8
0 0.5 1 1.5 2 0.7 0.7
Momentum, k/k 0 0.5 1 1.5 2 0 0.5 1 1.5 2
F Momentum, k/k Momentum, k/k
F F
FIG. 4. Panels (a)-(c): energy denominator at T = 0 in the
FIG.5. Panels(a)-(c): ratioofdenominators,Eq.(12)atT =
quasi-particle limit as a function of momentum for different
0 as a function of momentum for different Fermi momenta.
Fermimomenta,correspondingtothe(a)CDBonn,(b)Av18
The three panels correspond to the (a) CDBonn, (b) Av18
and (c) N3LO interactions.
and (c) N3LO interactions. Panels (d)-(f): the actual Z-
factor as a function of momentum in the same conditions.
See text for details.
between both denominators. The double convolution
takes into account the fragmentation of quasi-particle
states in the normal state. Since strength is removed well above Z (k). It peaks around k = 0 at values of
eff
from the full quasi-particle peak, the denominator be- ≈ 1.3 − 1.5, decreases to a minimum close to k and
F
comeslargerthanthecorrespondingquasi-particlevalue. subsequently raises again to ≈1 at high momenta. Even
Asamatteroffact,thedifferencebetweenthetworesults thoughpairingpropertiesaredominatedbyFermisurface
can be parametrized in terms of an effective Z−factor effects, where both renormalization factors are relatively
[20, 42], similar,thiscomparisonshowsthatarealisticdescription
of the missing strength for pairing purposes can only be
E(k)
Z2 (k)= . (12) achieved approximately by a renormalization-corrected
eff χ(k) BCS-typeapproach[43,53]. Infact,becausetheremoval
ofstrengthisunderestimatedinZ(k)withrespecttoZ ,
The ratio is displayed, for a subset of relevant densities eff
the corresponding gap is larger in a Z−factor corrected
and three NN forces, in panels (a)-(c) of Fig. 5. Z is
eff BCS approach as compared to a fully correlated descrip-
alwaysintherange≈0.8−0.9,andshowsamildmomen-
tion [42]. An additional difficulty is that it is unclear
tumdependence,withaminimumclosetotheFermisur-
howparticlenumberordensitycanbeproperlyobtained
face. Our results suggest that the ratio decreases slowly
from this approach.
with density. This is in accordance to an intuitive pic-
ture, where correlations, measured as a deviation from
one in Z , become more important at higher densities.
eff D. Long-range correlations
Itisimportanttostressthattheeffectivedenominator
ratio,Z ,isdifferentfromthestandardrenormalization
eff
factor, The most important effect of LRC on pairing proper-
ties will occur at the level of the effective pairing inter-
(cid:12)
1 (cid:12) action when neutrons near the Fermi surface exchange
Z(k)= (cid:12) . (13)
1−∂ ReΣN(k,ω)(cid:12) possiblycollectivespinanddensitymodes [6,49,50,55,
ω ω=εqp(k) 56, 74]. Following Refs. [51–53], we add to the interac-
We show this quantity, computed in the same conditions tion in the generalized gap equation, VJST, the corre-
LL(cid:48)
as Z , in panels (d)-(f) of Fig. 5. Other than in the sponding contributions accounting for such fluctuations
eff
vicinity of the Fermi surface, the two renormalization in a physically motivated way. We adopt the results of
factors provide very different results. Z(k) is generally Ref. [52], which incorporate an induced interaction that
8
leadstoawell-behavedparticle-holeinteractionthatful- 1.6
F
0
fills appropriate stability criteria, not obeyed by interac- 1.4 G
0
tions that incorporate only the effect of SRC like e.g. G- rs1.2 F0/(1+F0)+G0/(1+G0)
matrices [75]. The coupling to neutrons that are dressed e
et 1
by the full off-shell effect of SRC as described in the pre- m
a
vious section is then governed by the exchange of both a r0.8
a
density fluctuation and a spin mode. The collective fea- p
u 0.6
tures of these modes are controlled by self-consistently a
d0.4
determined Landau parameters [52].Their contribution n
a
to the pairing interaction requires a recoupling from the L0.2
particle-hole channel to the particle-particle channel and 0
isthereforedifferentforspinsingletandspintripletpair-
ing. Wenotethatthisisaphysicallymotivatedapproach 0.5 1 1.5 2 2.5 3
tothetreatmentofLRCthathasonlybeentestedinthe Fermi momentum, k [fm-1]
F
literature in extensions to BCS theory where SRC are
included in terms of renormalization factors [52]. By re-
stricting the effect of LRC to the effective interaction,
FIG.6. LandauparametersF andG extrapolatedtok =
0 0 F
we can test the effect of both SRC and LRC in pairing 3.0fm−1employingtheresultsofRef.[52]. Thecorresponding
properties by turning either correlation effect on or off. value of the forward scattering sum rule is indicated by the
FollowingRef.[52],theinteractionthattreatsLRCfor open circles.
the 1S channel is given by
0
30 0.8
VLSR=C0 = 21Gp0hGp0hΛS=0(q)− 32Gp1hGp1hΛS=1(q), (14) -3m] 20 1S0 VSLR=0C
Vf 0.6
e 10
whereGS representtheverticesthatcoupletothespin-S M
excitatiopnh. Theycanbethoughtofasparticle-holetrans- Val [-100 0.4
formed G-matrix elements averaged around the Fermi on 0.2
energy. As argued in Ref. [52], these vertices are im- Diag-20 (a) (c)
proved by employing the corresponding Landau parame- -30 0
3 0
ters,asintheoriginalworkofBabuandBrownforliquid -3m] 2 3P2 VSLR=1C
3He [65]. The iterated bubble series is then represented Vf 1 -0.2
by: Me 0
V [-1 -0.4
ΛS(q)= Λ0(q) , (15) onal --32 -0.6
1−Λ (q)LS g
0 Dia-4 (b) (d)
-5 -0.8
whereLS correspondstotherelevantLandauparameter. 0 1 2 3 4 5 0 1 2 3 4 5
Momentum, k [fm-1] Momentum, k [fm-1]
The density mode with total spin 0 in the particle-hole
channel is determined by L0 which is attractive at low
density and usually denoted by F . The spin mode with
0 FIG. 7. Panels (a) and (b): diagonal matrix elements of the
total spin 1 is determined by L1, which is repulsive but Av18 interaction in the (a) 1S and (b) 3P channels. Pan-
0 2
has similar magnitude and is often denoted by G0. The els (c) and (d): diagonal matrix elements of the additional
static Lindhard function, Λ0(q) pairing interaction representing the low-energy medium po-
larizationatadensitycorrespondingtok =1.6fm−1 inthe
N(0)1(cid:20) 1(cid:18) q2(cid:19) (cid:12)(cid:12)1−q/2(cid:12)(cid:12)(cid:21) same channels. The scales are different foFr each panel.
Λ0(q)= g 2 −1+ q 1− 4 ln(cid:12)(cid:12)1+q/2(cid:12)(cid:12) (16)
is employed in Eq. (15), with the appropriate density of (k < 1.5 fm−1). The parameters are adopted from
F
statesN(0)= 8mkF anddegeneracyfactorg=2. Weas- Ref. [52] and are plotted in Fig. 6. The first term in
(cid:126)2
sume as in Ref. [52] that for neutron matter the effective Eq. (14) is attractive, whereas the second term is repul-
massinthedensityofstatescanbeapproximatedbythe sive. In the density domain relevant for singlet pairing
bare mass. The static Lindhard function is iterated to this repulsion dominates on account of the spin factor
all orders according to Eq. (15) and generates negative leadingtoaninevitableadditionalsuppressionofthegap
particle-hole propagators, ΛS. Projecting Eq. (14) onto in this channel. Figure 7 illustrates that the additional
L=0,theresultinginteractioncanthenbeincludedinto term, VS [panels (c) and (d)], is relatively small com-
LRC
the gap equation for 1S pairing for a given density and pared to the bare interaction [panels (a) and (b)]. For
0
appropriate values of the Landau parameters. the 1S channel, LRC reduce the attraction of the bare
0
Both Landau parameters exhibit a modest density Av18 interaction.
dependence in the domain relevant for singlet pairing TheprocedureproposedinRef.[52]isgeneralizedhere
9
to the case of the 3PF coupled channel. For the 3PF
2 2 3.0
channel which involves spin-1 pairs, the sampling over
CDBonn (a)
density and spin modes becomes: V]2.5 1S
e 0
M
VS=1 = 1G0 G0 ΛS=0(q)+ 1G1 G1 ΛS=1(q), (17) ∆ [2.0
LRC 2 ph ph 2 ph ph p, 1.5
a
g
w1Sit0hchbaontnhelt,ertmhiss cyoienltdriinbgutaiotntrawcitliloanl.waCysonletaradrytotaonttihse- airing 1.0 BBCCSS+LRC
creening of the gap, as it represents an attractive inter- P0.5 SRC
SRC+LRC
action. This point is illustrated in panel (d) of Fig. 7,
0.0
which shows the relatively small but nevertheless attrac- 3.0
tive contribution of the LRC interaction in the S = 1 Av18 (b)
channel. This is to be compared to the bare interaction V]2.5 1S
e 0
in the 3P channel, shown in panel (b). M
The La2ndau parameters F and G from Ref. [52] ∆ [2.0
are extrapolated to higher den0sities in0a smooth way as ap, 1.5
g
shown in Fig. 6. In Fig. 6, we also include the contri- ng 1.0
bution to the forward scattering sum rule of the Landau airi
parameters F and G indicated by the open circles (see P0.5
0 0
e.g. Refs. [75–77]). While the extrapolated Landau pa-
0.0
rameters are both positive at higher density, the forward 3.0
scatteringsumruleisneverthelessapproximatelyfulfilled N3LO (c)
when one allows for a negative contribution of the Lan- V]2.5 1S
e 0
dthaeuspuamramruelete,rgiFv1en(fboyr eFxa/m(1pl+e Fof/a3b)o.utTh−e0.e5xt[r7a7p])oltao- ∆ [M2.0
tionintroducessomeuncer1taintyin1theeffectofLRCfor ap, 1.5
g
tripletpairingathigherdensity,butitshouldbeempha- ng 1.0
sized that Eq. (17) leads to antiscreening whatever the airi
numerical values or sign of the Landau parameters F P0.5
0
and G . Moreover, this small correction is motivated by
0 0.0
well-explored many-body theory principles. 0.0 0.5 1.0 1.5
Fermi momentum, k [fm-1]
In future work, we intend to generate the Landau pa- F
rameters from a consistent evaluation starting from the
ladder-summed effective interaction. A proper inclusion
FIG. 8. Pairing gaps at the Fermi surface as a function of
oftheinducedinteractionwiththisstartingpointishow-
Fermi momentum in the 1S channel. The three panels cor-
ever considerably beyond the scope of the present work. 0
respond to the (a) CDBonn, (b) Av18 and (c) N3LO inter-
Further, the possibility that the presence of the pion-
actions. Results for different approximations are presented:
exchange tensor interaction strongly influences the spin
BCS (solid lines), beyond BCS with short-range correlations
mode[78]shouldalsobeinvestigated(seealsoRef.[79]). (empty circles), beyond BCS with long-range correlations
A proper treatment of retardation implied by the possi- (emptysquares)andbeyondBCSwithbothshort-andlong-
bility of exchanging low-lying density and spin modes in range correlations included (solid circles). The dashed lines
principle generates a complex solution of the gap equa- represent the fits provided in Table I.
tion which should also be explored further (see e.g. the
work of Ref. [80] for a calculation with a dynamic pion-
exchange interaction). byaddingthedressedeffectiveinteractionsofEq.(14)to
thebareNNforcesinthegapequation,Eq.(3). SRCre-
sults (empty circles) are computed with bare NN forces,
III. RESULTS but double convolution denominators in the gap equa-
tion. Finally, the full circles are obtained from the full
A. Singlet pairing denominators and LRC-corrected effective interactions.
The BCS results (solid line) are very similar for all
Figure 8 provides the pairing gaps in the 1S channel forces, which confirms that phase-shift equivalence is
0
computedatthecorrespondingFermisurfacesinneutron enough to fix the value of the gap in this channel [1].
matter for four different approximations. Results for the The BCS gap peaks at about 3 MeV around k = 0.7-
F
CDBonn, Av18 and N3LO NN forces are displayed in 0.8 fm−1, and closes at k ≈ 1.5 fm−1. As mentioned
F
panels(a)-(c),respectively. Solidlinesrepresentthestan- above, LRC in this channel screen part of the attrac-
dard BCS solution, computed using free sp spectra. The tion of the NN forces. Consequently, BCS+LRC gaps
BCS+LRC results (empty squares) have been obtained (squares) are generally smaller than BCS results. While
10
the overall Fermi momentum dependence is similar, in-
TABLE I. Parameters generated by a fit to the calculated
cluding a similar closure density, the maximum of ∆ de-
gaps for the CDBonn, Av18, and N3LO interactions in the
creasestoabout≈2.5MeV.Itisimportanttostressthat
1S channel. For each interaction, the first line contains the
0
the screening is the same for all interactions.
resultsfortheinclusionofSRConly,andthesecondtheeffect
Including SRC within the Green’s function formalism
of both SRC and LRC.
outlined above (empty circles), we find that the over-
Singlet ∆ k k k k
all gap is reduced, with a maximum that now sits just 0 0 1 2 3
[MeV] [fm−1] [fm−2] [fm−1] [fm−2]
above 2 MeV. This result is expected: by removing
strength from the Fermi surface, the pairing phase space CDBonn SRC 26.59 0.05 1.79 1.46 0.76
is quenched and the corresponding pairing gap decreases CDBonn SRC+P 18.18 0.05 1.39 1.45 0.81
by about 30%. The mild density dependence of Z also Av18 SRC 32.22 0.04 3.46 1.40 0.43
eff
explains why the BCS and the SRC results have simi- Av18 SRC+P 14.07 0.04 1.00 1.44 0.78
lar density dependences. There is a tendency to have a
N3LO SRC 7.77 0.00 0.56 1.49 0.38
slightly lower closure density for the SRC results, a fea-
N3LO SRC+P 5.85 0.00 0.46 1.48 0.42
tureweshalldiscussfurtherwhenweintroducenumerical
parametrizations below.
Finally, the complete results including both SRC and
of Ref. [9] shows that our results are close to the Cao-
LRC are shown in full circles in Fig. 8. Screening effects
Lombardo-Schuck (CLS) [52] and Margueron-Sagawa-
in this channel are repulsive, and as a consequence the
Hagino (MSH) [81] singlet gaps. MSH is fit to the CLS
corresponding LRC+SRC gaps decrease in size by about
results, so the agreement between the two is not surpris-
25%withrespecttotheSRConlydata. Thesegapspeak
ing. Our results include LRC in a way that is similar
at values of around 1.8 MeV, for Fermi momenta close
to CLS, but we note that the SRC physics is considered
to 0.75 fm−1. While the overall density dependence is
only at the Z−factor level in a Brueckner–Hartree–Fock
comparable to the previous results, we note a tendency
calculation, and hence misses a complete description of
to find a lower gap closure density.
hole-hole correlation effects.
For a given channel, a convenient parametrization of
All in all, the picture that arises for the singlet gaps
the density dependence of the gap function is given by:
provided by the three interactions is remarkably robust.
(k −k )2 (k −k )2 The small variation of the result among NN forces pro-
∆JST(k )=∆ F 0 F 2 , (18)
L F 0(k −k )2+k (k −k )2+k videsaninsightfulconstraintonthemodeldependenceof
F 0 1 F 2 3
the gap properties. In spite of their different short-range
with ∆0, k0, k1, k2 and k3 numerical parameters [9]. In and(lessrelevantforneutronmatter)tensorcomponents,
particular, k0 and k2 represent the Fermi momenta at the 3 interactions considered here predict singlet gaps
which the gap opens and closes, respectively. Details on which are very close to each other. More importantly,
thenumericalfittothisfunctionaregiveninAppendixB. the many-body effects are very similar in all cases. SRC
We note that this parametrization is particularly sharp deplete the gap by about 25%. When LRC are included
aroundtheclosurepoints,andthatinthesingletchannel on top of SRC, the gap that remains is around 60% of
we supplement the fit with a zero value at zero density. the original BCS result for all forces. The effect of the
WeshowinTableIthevaluesoftheparametersobtained correlation-inducedgapquenchinpairingproperties,like
for these fits. Further, we note that the dashed lines the Cooper pair coherence length [82], or neutron star
shown in Fig. 8 correspond to the fit functions. properties [9], go beyond the scope of the present paper
The fit does reproduce the qualitative shape of the and will be studied elsewhere.
pairinggap. Wetakek ,displayedincolumn5ofTableI,
2 Therobustnessofthesingletgapresultswithandwith-
as a measure of the gap closure density. The confidence
out correlation effects is one of the major conclusions of
interval associated to the fit is within 0.12 fm−1 (for the
this work. We note, however, that this result is not nec-
worst fit) from the central value. We note that there
essarily easily anticipated. The gap itself is a function of
is a robust agreement and for all forces and many-body
both Fermi momentum, k , and sp momentum, k. So
F
approachesthegapclosuresitsbetween1.4and1.5fm−1.
far, we have focused on the values at the Fermi surface,
This parametrization also allows a simple quantitative ∆JST(k =k ), but the momentum dependence provides
L F
estimate of the gap maxima, and their location. For in-
usefulinformation,too. Inparticular,asweareaboutto
stance, the SRC maximum gap lies between k = 0.78
F show, very different momentum dependences can lead to
(N3LO), 0.81 (CDBonn) and 0.84 fm−1 (Av18), at a
similar gaps at the Fermi surface.
value between ∆1S0 = 2.1 MeV (Av18) and 2.3 MeV The pairing gap is shown as a function of the momen-
max
(CDBonn and N3LO). Similarly, the SRC+LRC results tum,k,foravarietyofFermimomentainthefenceplots
peak between kF = 0.75 (N3LO), 0.76 (Av18) and of Fig. 9. Panels (a), (b) and (c) correspond to the three
0.79 fm−1 (CDBonn) to maximum gaps of the order of NN forces, CDBonn, Av18 and N3LO, respectively. We
∆1S0 =1.8 MeV for all three NN interactions. show results for both the BCS (dotted lines) and SRC-
max
We note that similar gaps have already been obtained only (solid lines) approximations. In all cases, as ex-
in the literature. A comparison with the compilation pected in the singlet case, the gap peaks at zero mo-