Jastrow functions in double-beta decay

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by  J. Engel

Jastrow functions in double-beta decay
J. Engel
Dept. of Physics and Astronomy, University of North Carolina, Chapel Hill, NC, 27516-3255, USA
J. Carlson
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
R.B. Wiringa
Physics Division, Argonne National Laboratory, Argonne, IL 60439
We use simple analytic considerations and a Monte Carlo calculation of nucleons in a box to
argue that the use of Jastrow functions as short-range correlators in the commonly employed two-
body-cluster approximation causes significant errors in the matrix elements for double-beta decay.
1 The Jastrow approach appears to agree with others, however, if many-body clusters are included.
1 A careful treatment of the charge-changing analog of the nuclear pair density shows, in addition,
0
that differences between Unitary Correlator Operator Method and Bruecknermethods for treating
2
short-range correlations in double-betaare less significant than suggested by previouswork.

n
a
J I. INTRODUCTION overlap preservation and a range of successful applica-

3 tions [7]. Refs. [8] and [9] computed the effects of short-
range correlations within well-defined Brueckner-based
New experiments to measure the rate of neutrinoless
] approximation schemes. All these papers found smaller
h double-beta(0νββ)decaywillprovideinformationabout
effects on matrix elements than the phenomenological
t neutrinomassesifneutrinosareMajoranaparticles. Un-
- Jastrow function yields. Because they all limited their
l fortunately, one must know the value of the nuclear ma-
c analysis to two-body correlations,however, their predic-
trixelementgoverningthedecaytoextractthatinforma-
u
tions for the size of short-rangeeffects do not come with
n tion from an observed rate (or rate limit) [1]. For that
an iron clad guarantee.
[ reason,attempts to better calculate the matrix elements
appear regularly in the literature. Infact,allthesemethodsimplytheexistenceofmany-
1 body effects that are always neglected in applications
The matrix elements involve products of one-body de-
v
to double-beta decay. In Jastrow-based treatments, our
4 cay operators and a sum over intermediate states, but
subject here, the full correlated wave function is written
5 the closureapproximationallowsthem tobe represented
5 to high accuracy [2] by the ground-state-to-ground-state schematically as
0 transition matrix element of a two-body operator
.
1
10 M0ν ≡ Mab. (1) |Ψi= a<Yb<cTabc! aY<bFab!|Ψ0i , (4)
a<b
1 X
: where |Ψ0i is a Slater determinant or a generalization
v The matrix elements Mfi ≡hf|M0ν|ii can therefore be thereof, Fab is a two-body Jastrow correlator depend-
Xi affected by the strong repulsive correlations that alter ing on rab and the spins and isospins of particles a and
pair distribution functions at short distances. b, and T is a similar three-body correlator, which we
r abc
a For many years, theorists were satisfied to simulate will ignore from here on. In recent years, practitioners
these correlations by using a phenomenological Jastrow have developed a range of techniques for moving beyond
functionf(rab)inthetwo-bodyclusterapproximationto the two-body cluster approximation in Eq. (2) and in-
modify the operator M: cluding many-body correlations generated by the prod-
uctofF’s(inadditiontoexplicitthree-bodycorrelations
Mab =⇒f(rab)Mabf(rab), (2) generated by a single T) in Eq. (4). Cluster expansions
andtheFermi-hypernetted-chainmethod(see,e.g.,Refs.
where rab ≡ |ra −rb| is the magnitude of the distance [10, 11] and references therein) include three-and-more-
between the two nucleons. The Jastrow function almost body clusters, andquantum Monte-Carlomethods allow
always had the form prescribed in Ref. [3]: an evaluation of the contributions of all clusters. The
Jastrow approaches now yield accurate observables, in-
f(r)=1−e−1.1r2(1−0.68r2), (3) cluding two-body density distributions with short-range
correlations, in light nuclei [12] and nuclear matter [13].
with r in fm. Recent work has questioned this Here, after analyzing the two-body cluster approxima-
prescription. Refs. [4–6] treated short-range correla- tioninββ-decay,weseewhether aninitialapplicationof
tions through the Unitary-CorrelationOperator Method quantum Monte Carlo with many-body correlations in-
(UCOM), which has the advantages of wave-function cluded supports the phenomenological two-body-cluster
2
Jastrow method used traditionally, or whether it sup- (spin-independent) two-body density:
ports one or moreof the approachesintroducedrecently.
We also point out that apparent differences between the P (r)≡hf| δ(r−r )τ+τ+|ii , (5)
F ab a b
results ofUCOMandBruecknermethods arelargelyfic-
a<b
X
titious.
where F stands for Fermi. If we weight this function
HeavynucleiarestilltoocomplicatedforMonte-Carlo
with H (r), the radial part of the Fermi 0νββ operator
methodsintheircurrentforms,sotoevaluatemany-body F
(given approximately by 1/r), and integrate, we get the
Jastrow effects we look instead at a simplified version of
Fermi piece of the 0νββ matrix element. If we integrate
asymmetric nuclear matter. We make this choice with P (r) without any weighting, we get hf| τ+τ+|ii,
the idea that short-range correlations are nearly univer- F a<b a b
whichmustvanishbecause the isospins of|iiand|fiare
sal in nature, depending little on longer-range structure P
different (in the very good approximationthat isospin is
of the environment in which the correlated nucleons are
conserved), while the operator between them is propor-
embedded, provided that environment has the correct
tional to the square of the isospin-raising operator.
density.
Figure 2 shows P (r) for the shell-model calculation
F
of the ββ-decay of 82Se in Refs. [15] and [16]. The solid
curve contains no Jastrow function and has area of zero
beneath it. The dashed curve is the result of of the
Brueckner-based calculations in Ref. [8]. Its overshoot
II. TWO-BODY CLUSTER APPROXIMATION
at r just greater than one causes the integral to stay
very close to zero despite the suppression at very small
In the S = 0 T = 1 channel that determines the con- r. But the use of the two-body Jastrow function F01
tribution of short distances to the ββ amplitude, realis- `a la Ref. [13] (dotted curve) suppresses contributions at
ticvariationally-determinedcorrelationfunctionsF are small r without an overshoot and thus leads to an in-
ab
notsodifferentfromtheMiller-SpencerJastrowfunction. tegral of 0.006. Substituting the pair distribution func-
Figure1 showsatypicalnuclear-matterexample,follow- tion g01 would only make the problem here worse. The
ing the calculations of Ref. [13], alongside the Miller- Miller-SpencerJastrowfunctionyieldsalittlebitofover-
Spencer function and the effective scaling function, ob- shoot but not nearly enough, and results in an integral
tained from the ratio of calculations with and without of 0.0075.
short-range correlations, that appears in the Brueckner- Itseems,then,thatarealistictreatmentofshort-range
based work of Ref. [9] All the functions go to unity at correlations must yield an overshoot in P (r) if it is to
F
large r, but the Brueckner-based function has a sizeable preserveisospin(TheUCOMproceduredoesthisexactly,
”overshoot” near r = 1 fm. The Miller-Spencer func- by construction). When Jastrow functions are extended
tionhasamuchsmallerovershoot(occurringatlargerr, beyond the two-body cluster approximation, the effec-
which is made less important by the radial falloff of the tive functions that result must therefore look different
0νββ operator) leading to a significantly smaller 0νββ-
matrix element. The variational nuclear-matter resem-
bles the Miller-Spencer function but has essentially no
overshoot,andsoifappliedlikethatfunctionviaEq.(2)
it will produce an even smaller matrix element.
The use of the F from Eq. (4) to multiply a two-body
operator as in Eq. (2) is often called the two-body clus-
terapproximation,becausealltermsarediscardedexcept
thoseinwhichthetransitionoperatorandthecorrelators
actonthesamepairofparticles. Thisapproximationap-
pears to be reasonably good for number-conserving two-
bodydensities. Thedot-dashedlineinFig.1displaysthe
distribution g01(r) in the S = 0,T = 1 channel, follow-
ing Ref. [13], which incorporates the ful product over all
pair correlationsof Eq.(4). This full g01(r) is somewhat
smaller than the corresponding F2 because many-body
tensor correlationspromote a fraction of the spin-singlet
pairs to spin-triplet pairs, so that the number of singlet
FIG.1. (Coloronline) SquaresofJastrowfunctionsFab from
pairs is reduced. The reduction has also been seen in
calculationsfollowingRef.[13](dottedblackline,spin-singlet
light nuclei [14], though the corrections are not large ei-
only),from Miller andSpencer[3](solid red line)andfrom a
ther there or here.
fittotheresultsofamicroscopicBrueckner-basedcalculation
In ββ decay, the picture must be different, however. [9](dashedblueline). Thepurpledot-dashedlinecomesfrom
To see why, consider the charge-changing analog of the three- and more-body corrections tothe dotted line.
3
Nowlettheneutronnumberexceedtheprotonnumber
so that the Slater determinants |i0i and |f0i have well-
definedisospinsthatdiffer fromeachother,andconsider
the operatorM =τ+τ+. The matrixelementaboveis
ab a b
then the integral of P (r), i.e. zero. Although the first
F
term in the second line indeed gives zero, the second, as
we’ve seen, does not. The inclusion of all terms first-
orderinhmustrestorethe valuezero,however,because,
as the last line shows, the result can be obtained by act-
ingon|i0iwiththeisospin-preservingtwo-bodyoperator
h before acting with the transition operator. It
a<b ab
is not hard to show that effective four-body term con-
P
tributesthesameamountasthetwo-body-clustercorrec-
tion, andthe effective three-bodyterm contributestwice
that amount with the opposite sign, so that the sum of
termsindeedvanishes. Butthisalsomeansthat,atleast
to first order in h, three- and-four body effective oper-
FIG.2. (Coloronline)Thecharge-changingtwo-bodydensity
PF(r) for the shell-model calculation of 82Se in Refs.[15, 16]. ators are just as important for the quantity PF(r)dr
The solid red line is the result without short-range correla- as is the effective two-body correction generated by the
R
tions, the dashed blue line is that from the Brueckner-based two-body cluster approximation. This perhaps surpris-
calculation of Ref. [8] the dotted black line applies the Jas- ing conclusion leads us to examine the charge-changing
trowfunctionfromtheapproachinRef.[13]inthetwo-body- densityitselfandthe higherordercorrectionsinamodel
clusterapproximation,andthedot-dashedpurplelineapplies amenable to numerical solution.
theMiller-SpencerJastrow function. The inset magnifies the
left-hand part of the figure.
III. MANY NUCLEONS IN A BOX
for charge-changingdensities, whichinvolveonly valence
We consider a cubic box with each side L = 4.85 fm
nucleons,thanforlike-particledensities, towhichallnu-
and periodic boundary conditions. In the box are 2 pro-
cleons contribute coherently.
tonsand16neutrons(sothatthenucleondensityisvery
Itisnothardtogetanideaofhowthishappens. Letus
near nuclear-matter density), which decay to 4 protons
consider spin-and-isospin-independent two-body correla-
and 14 neutrons. The protons in the initial state and
tors F (with no three-body correlatorsT ) in Eq. (4)
ab abc all but the last two neutrons in that state are in filled
and a general charge-changing operator M . Writing
ab fermi levels, and the last two (valence) neutrons are in
F2 ≡1+h , and expanding to first order in h gives
ab ab the spin-zero two-body pairing wave function:
hf|M|ii=hf0| Mab (1+hcd)|i0i (6) |ψvi=N |k,−k;S =0i , (7)
Xa<b cY<d kx,kXy,kz∈K
=hf0| Mab|i0i+hf0| Mabhab|i0i where v stands for valence, N is a normalization con-
a<b a<b stant, and the set K contains vectors in which two k
X X
components are equal to ±2π/L and the third is zero.
+hf0| Mab(hac+hbc)|i0i
In the final state the neutrons and all but the last two
c6=Xa{<ab,b} protonsare in filled fermi levels;the two valence protons
+hf0| Mabhcd|i0i+O(h2) are in the configuration ψv above, but with the set K
containing vectors with one component equal to ±2π/L
c<Xad<6=ba,b and the other two equal to zero.
We use quantum Monte Carlo to evaluate P (r) be-
F
=hf0| Mab 1+ hcd |i0i+O(h2) tween states of the form Eq. (4), where now the states
a<b c<d ! |i0i and |f0i are those just described, and with simple
X X
spin-and-isospin two-body correlators F and no three-
ab
where |i0i and |f0i are Slater determinants, and in the body correlatorsTabc, as consideredpreviously. Figure 3
third and fourth lines, hmn ≡hnm if n<m. shows the results with the Miller-Spencer Jastrow func-
The second line in Eq. (6) involves the bare two-body tion. Thetwo-body-clusterapproximationagainhasvery
transition operator and the two-body-cluster correction. little overshoot,but the full results, including clusters of
The third line involves an effective three-body opera- all size, has considerably more, so that the integral van-
tor, and the fourth line an effective four-body operator. ishes as it should. Also shown is the result with the
Terms of higher-order in h generate even higher many- effective Jastrowfunctionfitto the Brueckner-basedcal-
body operators. culation of Ref. [9] (which was done in finite nuclei). It
4
[17]. We use the simplified forms here because they are
sufficient to make our point.
Figure 4 displays C (r) from the Monte Carlo. (In
F
this simple calculation C is just proportional to C
GT F
because the correlator is spin-independent and the two
valence nucleons that participate in the decayare locked
into a spin-zero configuration). The full solution clearly
correctsthe extreme suppressionofthe 0νββ matrix ele-
ment created by the two-body-cluster approximation, in
a way consistent with the analysis of the integral in sec-
tion II. Differences with the Brueckner treatment are
fairly small and due once again at least in part to the
unusualsystemweanalyzehere. Effectsbeyondthetwo-
body cluster approximation are thus both required and
apparently sufficient to describe short-rangecorrelations
in ββ decay.
FIG. 3. (Color online) Monte-Carlo calculation of PF(r) for
2protonsand16neutronsinaboxdecayingto4protonsand
14 neutrons. The red circles are the result with no Jastrow
correlators, the purple diamonds include the Miller-Spencer
Jastrowcorrelatorinthetwo-body-clusterapproximation,the
black upward-pointing triangles are the full result with that
correlator, and the blue downward-pointing triangles apply
the effective Brueckner based two-body-cluster Jastrow from
Ref. [9]. The numerical error associated with the values of
PF(r) are usually smaller than the size of the corresponding
symbols.
is now quite close to the full many-body Miller-Spencer
result,andtheremainingdiscrepancyisprobablymostly
due to the simplicity of our model. Surprisingly, the use
of a Jastrow function with no overshoot at all gives al-
mostthesameresultastheMiller-Spencerfunctionwhen
many-body correlations are taken fully into account. FIG. 4. (Color online) The curves CF(r) corresponding to
Toshowtheeffectsofthesevariousfunctionson0νββ- the distributions PF(r) of Fig. 3, without forbidden currents
decay, we define functions C (r), K = F,GT, for the or nucleon form factors. The symbols indicate the same ap-
K
proximations as in Fig. 3.
Fermi andGamow-Tellerparts ofthe 0νββ operator. (If
wewantedtobeaccuratewewouldalsodefineoneforthe
verysmalltensorterm.) Thesefunctionsaretheproducts
of the densities P (r) and the analogous density
F
IV. UCOM
P (r)≡hf| δ(r−r )σ ·σ τ+τ+|ii (8)
GT ab a b a b
a<b Inthissectionwedigressfromourmainlineofinquiry
X
to take up apparent differences between the results of
withfunctionsH (r)andH (r)thatspecifytheradial
F GT Brueckner methods and UCOM. Fig. 2 of [4], Fig. 9 of
dependence of the 0νββ operators. In other words
Ref.[17]andFig.4of[9]present0νββ distributionfunc-
tions (similar to the function CF(r) presented here in
C (r)=H (r)P (r), (9) 0
K K K
Fig. 4)with the UCOM(and other)treatments ofshort-
with rangecorrelations. Unlikethe Brueckner-basedcurvesin
our Figs. 2 – 4 the UCOM curves show no overshoot.
2R ∞ sinqr
But the reason is that the “contribution from distance
H (r)=H (r)≈ dq .
F GT πr Z0 q+E¯ −(Ei+Ef)/2 r” hasbeentreateddifferently whenUCOMcorrelations
(10) are considered than when other methods are used. The
UCOMprescriptionrequiresthatthe operatorr be re-
ab
The quantities E¯, Ei, and Ef are energies to which the placed by a shifted version R+(rab) (where the function
HK arenotverysensitive. Equation(10)holdsonlyifwe R+ is usually determined variationally) in any opera-
neglect nucleon form factors and forbidden currents; in tor that doesn’t depend on momentum. Thus, to eval-
the complete, more complicated expressions H 6= H uate the 0νββ matrix element, one replaces H (r ) by
F GT K ab
5
HK(R+(rab)). V. DISCUSSION
The shifting implies that UCOM produces a function
C (which we use as an example because it is simpler
F
than CGT) given by The main point of this paper, to which we now re-
turn, is that the use of Jastrow functions in the two-
CFU(r)= hf|HF(R+(rab))δ(r−R+(rab))τa+τb+|ii body-cluster approximation suppresses short-range con-
Xa<b tributions too much, and that the problem is fixed by
=HF(r)hf| δ(r−R+(rab))τa+τb+|ii including many-body correlations. This discovery raises
thequestionofwhetherexistingtreatmentsareadequate.
a<b
X
≡H (r)PU(r), (11) Theyincludelong-rangemany-bodycorrelationsinshell-
F F modelorQRPAwavefunctionsbutallowonlytwoparti-
(where U stands for UCOM). In prior work on UCOM clestobecorrelatedatshortdistances. Isthatsufficient?
ββ-decay, however, the “(Fermi) contribution from a
givenr”wasdefinedinsteadbysimplyreplacingr with It is hard to answer the question definitively because
ab
R+(rab) in H, viz: the approach taken here is so different from the others.
Wecansaythatthevery-short-rangecorrelationsareun-
CFU′(r)=hf| HF(R+(rab))δ(r−rab)τa+τb+|ii likely to be altered; our figs. 3 and 4 show that correc-
a<b tions to the two-body-cluster approximation are barely
X
=HF(R+(r))PF(r). (12) noticeable below about r = 0.7 fm. But corrections are
large at 1 fm or so. It is far from obvious that the
This definition, which leaves rab unshifted in the delta marriage of UCOM or Brueckner treatments of short-
function, gives the correct result for the Fermi matrix rangecorrelationstoshell-modelorQRPAtreatmentsof
element when r is integrated over, but does not define longer-range correlations incorporates all important ef-
an observable and is not what is calculated in other ap- fects at the intermediate range r ≈ 1 fm. The UCOM
proaches. The correctexpression,Eq. (11), is a bit more procedure generates three- and higher-body correlations
complicatedtoevaluate,buthasapronouncedovershoot. that have been neglected in almost all applications to
Figure5comparesthedistributionCGT(r)fromtheshell- date, and the Brueckner-based double-beta work has so
model-Brueckner treatment of 82Se −→ 82Kr in Ref. [8] far not included contributions from, e.g., three-particle
(with no forbidden currents or nucleon form factors) to ladders. As for the shell model and QRPA, they leave
theproperlydefinedUCOMdistributionforthesamede- untreated a significant range of single-particle energies
cay. The two curves are extremely close to one another, betweenthosecontainedinthecalculationandthoserep-
and quite different from CGUT′ (r), which is also shown. resentedas short-rangeeffects. Whether these omissions
The UCOM and Brueckner pictures are therefore more are significant is still an open question.
similar than previously thought.
In the meantime, however,it appears that the UCOM
and Brueckner methods give reasonable short-range cor-
relations. Like the full Jastrow calculations here, they
supply the overshoot required to preserve isospin sym-
metry. Higher-body corrections in these schemes appear
unlikely to be as large as they are in the Jastrow ap-
proach,whichviolatesisospinsymmetryinthetwo-body
cluster approximation. Short-range effects in ββ-decay
thus seem to be mostly under control.
ACKNOWLEDGMENTS
FIG. 5. (Color online) The function CGT(r), defined in Eqs.
(8)and((9)solid redcurve),thecorrespondingUCOMfunc- We gratefully acknowledge the support for this
tion CGUT(r), defined as in Eq. (11) (dashed blue curve), the work of the U.S. Department of Energy through the
previously-used UCOM function CGUT′ (r), defined as in Eq. LANL/LDRD Program and through Contract Nos.
(12)(dot-dashedpurplecurve),andtheBrueckner-basedver- DE-AC52-06NA25396, DE-FG02-97ER41019, and DE-
sion (dotted black curve). The new UCOM and Brueckner AC02-06CH11357. Computer time was made available
curvesare very similar. by Los Alamos Open Supercomputing.
6
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