Universal dimer-dimer scattering in lattice effective field theory

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Universal

dimer–dimer

scattering

in

lattice

effective

field

theory

Serdar Elhatisari

a

,

b

,

∗

,

Kris Katterjohn

c

,

Dean Lee

d

,

Ulf-G. Meißner

a

,

e

,

Gautam Rupak

c

a_{Helmholtz-InstitutfürStrahlen- undKernphysik(Theorie)andBetheCenterforTheoreticalPhysics,UniversitätBonn,D-53115Bonn,Germany} b_{DepartmentofPhysics,KaramanogluMehmetbeyUniversity,Karaman70100,Turkey}

c_{DepartmentofPhysics}

_{& Astronomy}

_andHPC2_{CenterforComputationalSciences,MississippiStateUniversity,MississippiState,MS 39762,USA} d_{DepartmentofPhysics,NorthCarolinaStateUniversity,Raleigh,NC 27695,USA}

e_{InstituteforAdvancedSimulation,InstitutfürKernphysik,JülichCenterforHadronPhysicsandJARA–HighPerformanceComputing,ForschungszentrumJülich,} D-52425Jülich,Germany

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received 3 November 2016

Received in revised form 8 March 2017 Accepted 8 March 2017

Available online 14 March 2017 Editor: W. Haxton

We consider two-component fermions with short-range interactions and large scattering length. This systemhas universal propertiesthat are realized inseveraldifferent fieldsofphysics. Inthe limitof largefermion–fermionscatteringlengthaffandzero-rangeinteraction,allpropertiesofthesystemscale proportionallywithaff.Forthecasewithshallowbounddimers,wecalculatethedimer–dimerscattering phaseshiftsusinglatticeeffectivefieldtheory.We extracttheuniversaldimer–dimerscatteringlength add/aff=0.618(30)and effective rangerdd/aff= −0.431(48).Thisresult forthe effectiverangeis the firstcalculation withquantifiedandcontrolledsystematicerrors. Wealsobenchmarkourmethods by computingthefermion–dimerscatteringparametersandtestingsomepredictionsofconformalscaling ofirrelevantoperatorsneartheunitaritylimit.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Two-componentfermionsatlargescatteringlength arean im-portantsystemwithuniversal propertiesandrelevancetoseveral branchesofphysics. Thisuniversalityis dueto theexistence ofa conformalfixedpointcalledtheunitaritylimitwherethefermion– fermion scattering length is infinite and all other length scales are irrelevant at large particle separations or low energies. See forexampleRef.[1]forareview.Innuclearphysics,theneutron– neutronscatteringlength

|

ann

|

∼

19 fm[2]ismuchlargerthanthe inversepionmass 1/Mπ

∼

1.4 fm characterizingthe exponential tailofthenuclearforce[13].Inthephysicsofultracoldatoms,one cantunetheinteractionsarbitrarilyclosetotheunitaritylimit us-inganexternalmagneticfieldnearaFeshbachresonance[3,4].In thisletterwediscussthecasewherethescatteringlengthislarge and positive, and bound dimers composed of two fermions are formed with shallow binding energy. We compute dimer–dimer scatteringand determine the dimer–dimer scattering length and effectiverange.Theseresultscan beused tocompute theenergy densityofadimergasinthedilutelimit

[5–9]

.

*

Corresponding author.

E-mailaddresses:elhatisari@hiskp.uni-bonn.de(S. Elhatisari), kk278@msstate.edu(K. Katterjohn), dean_lee@ncsu.edu(D. Lee),

meissner@hiskp.uni-bonn.de(U.-G. Meißner), grupak@u.washington.edu(G. Rupak).

The elastic scattering phase shift

δ(

p

)

between two non-relativisticfermionswithfinite-rangeinteractionsisparameterized bytheeffectiverangeexpansion(ERE)

[11]

,

p cot

δ

= −

1

aff

+

1 2rffp

2

₊

_O

₍

_p4

_{) ,}

₍₁₎

where p is the relative momentum, aff is the fermion–fermion

scattering length, and rff is the fermion–fermion effective range.

In this study we consider the case where aff is large and

posi-tive while all other lengthsscales are negligible.We can express all physical quantities in dimensionless combinations involving powersofaff.Previouscalculationsofthedimer–dimerscattering

lengthhavefoundadd

/

aff

=

0.60

±

0.01[14,15],add

/

aff

=

0.605

±

0.005[16],andadd

/

aff

=

0.60[17].Aperturbativeexpansionabout

fourspatial dimensionsgivesadd

/

aff

≈

0.66[18],andarough

es-timate using the resonating group method in the single-channel approximationgivesadd

/

aff

∼

0.752[19].Ontheotherhand,much

less is known about the higher-order dimer–dimer ERE parame-ters. The effectiverange has been calculatedas rdd

/

aff

≈

0.12 in

Ref.[20],whileaveryroughestimateofrdd

/

aff

∼

2.6 wasgivenin

Ref.[21].

In this work we calculate the low-energy dimer–dimerphase shifts from lattice effective field theory and extract both the dimer–dimerscatteringlengthaddandeffectiverangerdd.Wealso

benchmark our methods by calculating the fermion–dimer

scat-http://dx.doi.org/10.1016/j.physletb.2017.03.011

0370-2693/©2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3_.

tering length afd and effective range rfd. We organize our paper

asfollows.InSec. 2 we introducethe continuum andlattice for-mulations for systems of two-component fermions. In Sec. 3 we discussthemethodsforextractingthescatteringinformationfrom periodic finite volumes. We present our results andanalyses for fermion–dimerand dimer–dimerscatteringin Sec. 4. The results aresummarizedinSec.5.

2. Latticeformalism

FollowingRefs.[22,23],westartbydescribinginteracting two-component fermions in continuous space. Low-energy fermion– fermion scattering is dominated by the s-wave channel, while higher partial wavesbecome more important athigher energies. Inprinciplethetwocomponentscouldhavedifferentmasses, how-everweonlyconsidertheequalmasscaseinthisstudyanddenote thetwocomponentsasupanddownspins.Wewillconsider sys-temsoftwo-componentfermionswithdifferentmassesinafuture publication.

Weworkwithnaturalunitswhereh

_¯

=

1

=

c.Letb_↑,↓(b†_↑,↓)be the annihilation(creation) operators, andlet

ρ

↑,↓ be the density operators,

ρ

_↑

(



r

)

=

b†_↑

(



r

)

b_↑

(



r

),

ρ

_↓

(



r

)

=

b†_↓

(



r

)

b_↓

(



r

) .

(2) ThecontinuumHamiltonianhastheform

H

=



s=↑,↓ 1 2m



d3



r

∇

b†_s

(



r

)

· ∇

b_s

(



r

)

+

C0



d3



r

ρ

_↑

(



r

)

ρ

_↓

(



r

) ,

(3) whereultravioletdivergencesduetothezero-rangeinteractionare regulatedin somemanner. Inourcaseweuse thelatticeto pro-videtheultravioletregularization.

Wedenotethelattice spacingasa.Inourcalculationswe use an

O(

a4

_)-improved

_{lattice action wherethe free lattice}

Hamilto-nian,H0,isdefinedas,

H0

=



s=↑,↓ 1 2m



ˆ l=ˆ1, ˆ2, ˆ3



 n

⎡

⎣



3 k=−3 w_|k|b†s

(

n



)

bs

(

n



+

k

ˆ

l

)

⎤

⎦ ,

(4)

where

ˆ

l

= ˆ

1,2,

ˆ

3 are

ˆ

unit vectors in spatial directions, and the hoppingparameters w0, w1, w2,and w3 are 49/18,

−

3/2, 3/20,

and

−

1/90, respectively. n denotes



the lattice sites on a three-dimensionalL

×

L

×

L periodiccube.

Forthetwo-particle(2N)interactionwe usethesingle-site in-teraction V2N

=

C2N



 n

ρ

_↑

(

n



)

ρ

_↓

(

n



) ,

(5)

where the value of C2N depends on the lattice spacing a. We

tune C2N to produce the desiredvalue of thedimer binding

en-ergyBd.Forconveniencewechooseparameterstypicalfornuclear

physics,withfermionmassm

=

939 MeVanddimerbinding ener-giesrangingfrom1 MeVto10 MeV.Howeverthefinalresultsare completelyindependentofthesedetailswhenexpressedinterms ofthetwo-fermionscatteringlengthaff.

Inthelow-energylimitofthistheory,three-particleand higher-particleinteractionsareirrelevantoperators.Nevertheless,wefind it useful to include three-particle

(3N)

and four-particle

(4N)

interactions as a diagnostic tool to generate more data for the continuum-limit extrapolations. The three-particle interaction we usefeatures nearest-neighbour andnext-to-nearest-neighbour in-teractions, V3N

=

C(_3N1)



 n



|n−n |2₌₁

ρ

_↑

(

n



)

ρ

_↓

(



n

)

[

ρ

_↑

(



n

)

+

ρ

_↓

(

n



)

]

+

C_3N(2)



 n



|n−n |2₌₂

ρ

_↑

(

n



)

ρ

_↓

(

n



)

[

ρ

_↑

(

n



)

+

ρ

_↓

(



n

)

] .

(6) Similarly, we introducea four-particleinteraction that consistsof nearest-neighbourandnext-to-nearest-neighbourinteractions,

V4N

=

C(_4N1)



 n



|n−n |2₌₁

ρ

_↑

(

n



)

ρ

_↓

(



n

)

ρ

_↑

(

n



)

ρ

_↓

(

n



)

+

C_4N(2)



 n



|n−n |2₌₂

ρ

_↑

(

n



)

ρ

_↓

(

n



)

ρ

_↑

(



n

)

ρ

_↓

(



n

) .

(7)

WekeeptheinteractionstrengthsC_3N(1,2) andC_4N(1,2) atfixedvalues whenmeasuredinlatticeunits.

We can compute the importance oftheseirrelevant operators inthecontinuumlimitneartheconformally-invariantpointwhere the two-fermion scattering length is infinite and the interaction rangeiszero.If

δ

isthescalingdimensionofanoperator O ,then the contributionfromtheinsertion oftheinteraction O†O scales asa2δ−d−2 inthecontinuumlimit,whered isthenumberof spa-tial dimensions

[24]

. The operator-statecorrespondence principle connectsthescalingdimensionofanoperatortothelowestenergy ofthesysteminaharmonictrapwiththecorrespondingnumber of particles andquantum numbers [25]. From numerical calcula-tions of the harmonically-trapped energies, we deduce that the leading behavior ofthe three-particle operators is a3.54544_{, while}

the leading behavior of the four-particle operators is a5.056 _[24].

Inouranalysiswe willcheckexplicitlyifthisdependenceonthe latticespacingcanbeseeninthelatticeresults.

3. Scatteringphaseshift

Lüscher’s finite-volume method relates the two-body energy levels in a cubic periodic box to the elastic scattering phase shifts[26,27].Thetwo-bodyphaseshiftsinaperiodicboxofsize L arerelatedto therelative momentum ofthetwo bodies, p, by therelation p cot

δ(

p

)

=

1

π

L S

(

η

),

η

=



p L 2

π

2

,

(8)

where S

(

η

)

istheregulatedthree-dimensionalzetafunctiongiven by S

(

η

)

=

lim →∞

⎡

⎣



 n

(

2

− 

n2

₎



n2

₋

_η

−

4

π



⎤

⎦ .

(9)

ThesuminEq.(9)isoverthree-dimensionalintegervectorsn.



We use the Lanczos eigenvector method [28] to compute the low-energyspectrumofthelatticeHamiltonianatdifferentvalues ofL.Theseenergieslevelsdeterminethevaluesofp asinputinto Eq.(8),whichthendeterminethetwo-bodyscatteringphaseshifts

δ(

p

).

First we do thesecalculationsfor the three-particlesystem to determine the fermion–dimer scattering parameters. We then do the calculationsforthe four-particle systemto determine the dimer–dimerscatteringparameters.

In thezero-range limit, the fermion–fermion scatteringlength isrelatedtothedimerbindingenergybytheformula

Bd

=

1

Fig. 1. Theratio of the fermion–fermion scattering length a∗_ff determined using

Lüscher’s finite volume formula and affdetermined from Bd. The results are plotted

versus the lattice spacing a as

a fraction of

aff, and fitted to a polynomial in a/aff. Sincewewilltake thezero-rangelimitinallourcalculations, we candefineaffquitesimplyusingthezero-rangeformulainEq.(10)

andthedimerbindingenergyBd determinedonthelattice.

How-ever,wecanalsodeterminethefermion–fermionscatteringlength more carefully using Lüscher’s finite-volume scattering method. Wecallthisdeterminationofthescatteringlengtha∗_ff.In

Fig. 1

we show the ratio of the scatteringlengths forvarious lattice spac-ings a. In the plot we have fitted a polynomial in a

/

aff to the

results.We seethat the deviationbetween thesetwo definitions ofthescatteringlength vanishes in thecontinuum limit andcan befitwellbyapolynomialina

/

aff.

We will use Lüscher’s finite-volume method to calculate

fermion–dimer scattering and dimer–dimer scattering. In these casesweconsiderthescatteringoftwobodies,whereoneorboth bodiesmay be dimers. Let

μ12

be the reduced massof the two scattering objects. In the infinite volume and continuum limits, therelativemomentum p isrelatedtothetwo-bodyenergylevel E(∞)_as

E(∞)

=

p

2

2

μ

12

−

B1

−

B2

,

(11)

whereB1 andB2 aretheinfinite-volumebindingenergiesforthe

twobodies.Thesewillequal Bdifadimeror0 ifafermion.

Eq. (11) is modified by several effects at finite volume and nonzero lattice spacing. At nonzero lattice spacing, the effective massofthedimerisnotexactlyequaltotwicethefermionmass. Sowenumericallycalculatetheenergyversus momentum disper-sionrelationofthedimertoextract thedimereffectivemass.For thisweusealargeL

=

50a cubicboxinordertominimize finite-volume errors.From the dimereffective mass we can determine thereducedmassofthefermion–dimeranddimer–dimersystems. Wewritethislattice-determinedreducedmassas

μ

∗₁₂.

Atfinitevolume,thereisalsoafinite-volumecorrectiontothe bindingenergies B1 andB2.Thesefinite-volume corrections

van-ishexponentially withthesizeof theboxandso canbe ignored forsufficientlylarge L. Howevercomputational limitsoftenmake very large volume calculationsimpractical, and itso isuseful to remove finite-volume corrections corresponding to binding ener-gieswhenpossible.Itturnsout thatthefinite-volumecorrections tothebindingenergies B1 andB2 aremomentumdependent.We

accountforthesefinite-volumemomentum-dependenteffects us-ingfinite-volume topologicalfactors

τ

(

η

)

dueto thedimer wave functionswrappingaroundtheperiodicbox[29],where

η

was de-finedinEq.(8).Withthesecorrections,Eq.(11)becomes

E(L)

=

p

2

2

μ

∗₁₂

−

B1

−

τ

1

(

η

)

B1(L)

−

B2

−

τ

2

(

η

)

B(L)2

,

(12)

where

B(L)_i isthefinite-volumecorrection

B(L)_i

=

B(L)_i

−

Bi.The topologicalfactorisgivenby[29]

τ

(

η

)

=

⎡

⎣



 k 1

(

k2

₋

_η

₎

2

⎤

⎦

−1



 k

3 i=1cos

(

2

π α

ki

)

3

(

k2

₋

_η

₎

2

,

(13)

where



k runsoverall integervectors, and

α

=

1/2 forthe dimer

bound state. The relative momentum p corresponding to box

length L is computedbysolving Eq.(12)self-consistently forthe givenlatticeenergies E(L)_,B(L)

1,2,andB1,2.

4. Resultsandanalysis 4.1. Fermion–dimerscattering

Before proceeding to the dimer–dimer system, we perform benchmarks of our lattice methods and analysis by computing fermion–dimerscattering.Fermion–dimerscatteringhasbeen cal-culated using semi-analytical methods [12,30–32] in the contin-uum limit. We consider a three-particle system of two spin-up fermionsandonespin-downfermion.OurlatticeHamiltonianhas theform

H

=

H0

+

V2N

+

V3N

,

(14)

wherethefree HamiltonianisdefinedinEq.(4),the two-particle interactionappearsinEq.(5),andthethree-particleinteractionis introducedinEq.(6).Inordertoreducethenumberoffree param-eters in ouranalysis, we define the three-particleparameter c3N

andsetC_3N(1)

=

c3N andC3N(2)

=

c3N

/2.

Inthefollowingwequotethe

valueofc3N inlatticeunits.

We perform lattice calculationsusing the Lanczos eigenvector methodtoobtainthefinite-volumeenergiesofthefermion–dimer systemforvariousinteractioncoefficientsC2N andc3N andlattice

lengthsL.From thefermion–dimerenergies E(L)_fd inthe center-of-massframe,wedeterminetherelativemomentum p using

E(L)_fd

=

p 2

2

μ

∗_fd

−

Bd

−

τ

d

(

η

)

B

(L)

d

,

(15)

andthenuseEq.(8)toextractthefermion–dimerscatteringphase shifts.

The results for the fermion–dimer phase shifts are shown in

Fig. 2.We plot affpcot

δ

versus

(

affp

)

2 forvarious valuesof the

three-particlecouplingc3Nandvariousratiosofthelatticespacing

a tothefermion–fermion scatteringlengthaff.Thevaluesquoted

forc3N are inlattice units.Ineach casewe makea fitusing the

truncatedeffectiverangeexpansion

affp cot

δ

= −

1 afd

/

aff

+

1 2rfd

/

aff

· (

affp

)

2

₊

_O

₍

_p4

_),

₍₁₆₎

where afd and rfd are the fermion–dimer scattering length and

effectiverangerespectively.Asseenin

Fig. 2

,thethree-particle in-teractionshavesome impacton thescatteringphase shiftresults atlarger lattice spacings, whilethe dataat smalla is almost in-dependent of c3N. This is consistent with the conformal scaling

predictionthatthethree-particleinteractions areirrelevantinthe continuumlimit.

Withtheselatticeresultsforafd andrfd,we extrapolatetothe

continuumlimit.There willbe latticecutoffcorrectionsthatscale asintegerpowers ofthelattice spacing. Forthesecorrectionswe

Fig. 2. The

fermion–dimer scattering results for various values of the three-particle coupling

c3Nin lattice units and various ratios of the lattice spacing a to

the fermion–

fermion scattering length aff. We plot affpcotδversus (affp)2in the center-of-mass frame. The points are the lattice data, and the lines are the fits to the effective range

expansion.

fita third-orderpolynomial ina

/

aff withcoefficientsthat are

in-dependentofc3N.Wealsoincludethepredictedleadingorder

cor-rectionfromc3N as

(

a

/

aff

)

3.54544aswellasasubleadingcorrection

atonepowerhigher,

(

a

/

aff

)

4.54544.Wecouldincludeother

correc-tions aswell, however there isa limit to the number ofpowers thatcanbefitreliablyatthesametime.Insummary,weperform thecontinuum-limitextrapolationsforafd andrfd usingthe

func-tionalform

f

(

a

/

aff

)

=

f0

+

f1

(

a

/

aff

)

+

f2

(

a

/

aff

)

2

+

f3

(

a

/

aff

)

3

+

f3.54544

(

a

/

aff

)

3.54544

+

f4.54544

(

a

/

aff

)

4.54544

,

(17)

where f0, f1, f2,and f3areindependentofc3N while f3.54544and

f4.54544 dependonc3N.

The extrapolation fits for the scatteringlength and the effec-tiverangeare shownin

Fig. 3

.Thefinal resultsforthescattering parametersareafd

/

aff

=

1.176(6)andrfd

/

aff

= −

0.029(16),which

areingoodagreement withsemi-analyticcontinuum calculations a_fdcont.

/

aff

=

1.1791(2) andr_fdcont.

/

aff

= −

0.0383(3) [12,30–32]. The

errorbarsincludetheuncertaintyfromtheeffectiverange

expan-sionfits andthe continuumlimit extrapolation.Giventhequality of the extrapolation fits in Fig. 3, we conclude that the three-particle forces make a contribution that is consistent with the conformalscalingpredictionof

(

a

/

aff

)

3.54544.

4.2. Dimer–dimerscattering

Wenowcompute dimer–dimerscattering.Weconsidera

four-particle system of two spin-up fermions and two spin-down

fermions.OurlatticeHamiltonianhastheform

H

=

H0

+

V2N

+

V3N

+

V4N

,

(18)

wherethefour-particleinteractionisintroducedinEq.(7).Forour analysis we define the parameter c3N,4N and set C_3N(1)

=

2C(_3N2)

=

c3N,4N and C(4N1)

=

2C

(2)

4N

= −

3c3N,4N. In the following we quote

thevalueofc3N,4N inlatticeunits.

As in the fermion–dimer calculations, we use the Lanczos eigenvector method to obtain the finite-volume energies for the dimer–dimer system for various interaction coefficients C2N and

Fig. 3. (Left

panel.) The continuum-limit extrapolation of the fermion–dimer scattering length

afd. (Right panel.) The continuum-limit extrapolation of the fermion–dimer

effective range rfd. The final results are afd/aff=1.176(6)and rfd/aff= −0.029(16). c3N,4N andlattice lengths L.From thedimer–dimerenergies E_dd(L)

inthecenter-of-massframe,wedeterminetherelativemomentum p using E(L)_dd

=

p 2 2

μ

∗_dd

−

2Bd

−

2

τ

d

(

η

)

B (L) d

,

(19)

andthen useEq.(8)toextract thedimer–dimerscatteringphase shifts.

The results for the dimer–dimer phase shifts are shown in

Fig. 4. We plotaffpcot

δ

versus

(

affp

)

2 forvarious values ofthe

multi-particle coupling c3N,4N and various ratios of the lattice

spacing a to the fermion–fermion scattering length aff. The

val-uesquotedforc3N,4N areinlatticeunits.Ineachcasewemakea

fitusingthetruncatedeffectiverangeexpansion

affp cot

δ

= −

1 add

/

aff

+

1 2rdd

/

aff

· (

affp

)

2

₊

_O

₍

_p4

_),

₍₂₀₎

where add and rdd are the dimer–dimer scattering length and

effectiverange respectively. We observein Fig. 4that the multi-particlecouplingc3N,4Nhasastrongerimpactonthedimer–dimer

scattering results than we had seen for c3N in the fermi–dimer

scatteringresults.Thisisduetofinite-volumeeffects.Itisnot pos-sibleat present to go to very large volumes in the four-particle systemcalculations.Theamountofmemoryrequiredscalesas



9_,

where



=

L

/

a is the lattice length measured in lattice units.In practiceitisdifficulttogomuchbeyond



=

12.Thisisincontrast withthethree-particlesystemwherethescalingisas



6,andone canreachvaluesof



severaltimeslarger.

In the absence of the multi-particle coupling c3N,4N, we find

thatthefinite-volumecorrectionsforthefour-particlesystemare significant.ThesefindingsareconsistentwithRef.[34],which dis-cussed a four-particle chain-like excitation wrapping around the lattice boundaries. In addition to the leading

(

a

/

aff

)

3.54544

de-pendence on c3N,4N, we also have corrections proportional to

(

a

/

aff

)

3.54544 times a term proportional to the finite-volume

cor-rectionofthedimerwavefunction

[33]

,

(

a

/

aff

)

3.54544 e−L/aff

L

/

aff

.

(21)

Writtenintermsof

,

thisbecomes

(

a

/

aff

)

3.54544 e−a/aff



a

/

aff

= (

a

/

aff

)

2.54544 e−a/aff



,

(22)

andsoforfixed



weget acorrection thatnaïvelyappears tobe ofalower orderthan theexpected

(

a

/

aff

)

3.54544 scalingandwith

a rather complicated dependence on a

/

aff. Eventhough this

de-pendence ona

/

aff isanartificialcombinationoflatticeandfinite

volume effects,we can still extrapolate thelattice data toobtain thecorrectcontinuumlimit.

Inlightofthecomplicationsfromresidualfinite-volumeeffects, weuseasimplercontinuumextrapolationschemeforaddandrdd.

Foraddweuseasimplefunctionalform

f

(

a

/

aff

)

=

f0

+

f1

(

a

/

aff

)

+

f2

(

a

/

aff

)

2

+

f3

(

a

/

aff

)

3 (23)

with f0 independent of c3N,4N, but allowing f1, f2, and f3 to

vary with c3N,4N. In Fig. 5 we show the continuum-limit

ex-trapolation for the dimer–dimer scattering length add. The final

result we obtain is add

/

aff

=

0.618(30), which is in good

agree-ment with the most accurate determinations in the literature, add

/

aff

=

0.60

±

0.01 [14,15], add

/

aff

=

0.605

±

0.005 [16], and

add

/

aff

=

0.60 [17]. The error bar includes the uncertainty from

theeffectiverangeexpansionfitsandthecontinuumlimit extrap-olation.

As we increase c3N,4N, the three-particle interaction becomes

more repulsive. This repulsive interaction impedes the formation ofthechain-likeexcitationwrappingaroundtheperiodicboundary whichwas observedinRef. [34].So thefinite-volumecorrections are smaller forlarge positive values of c3N,4N, andwe expect to

recovertheusual

(

a

/

aff

)

3.54544 dependenceonc3N,4N.Thisis

con-sistent withtheresultsin

Fig. 5

. Forthelargestvaluesofc3N,4N,

thecoefficientsofthefirstandsecondpowersofa

/

aff are

approx-imatelyindependentofc3N,4N.

Forthe continuum extrapolation ofrdd, we useonly the data

with c3N,4N

≥

0.40, where the three-particle interaction is quite

repulsiveandfinite-volumeeffectsaresmall.Weusethefunctional form

f

(

a

/

aff

)

=

f0

+

f1

(

a

/

aff

)

+

f2

(

a

/

aff

)

2 (24)

with f0 independent of c3N,4N, but allowing f1 and f2 to vary

withc3N,4N.In Fig. 6weshow thecontinuum-limitextrapolation

forthedimer–dimereffectiverangerdd.Wenotethatasc3N,4N

in-creases,thecoefficientsofthefirstandsecondpowersofa

/

affare

approximately independentof c3N,4N.This isconsistent with the

expected

(

a

/

aff

)

3.54544 dependenceon c3N,4N. Thefinal resultwe

obtain isrdd

/

aff

= −

0.431(48). The errorbar includesthe

uncer-tainty fromthe effectiverangeexpansion fits andthecontinuum limitextrapolation.Thisvalueisdifferentfrompreviousestimates

Fig. 4. The

dimer–dimer scattering results for various values of the multi-particle coupling

c3N,4Nin lattice units and various ratios of the lattice spacing a to

the fermion–

fermion scattering length aff. We plot affpcotδversus (affp)2in the center-of-mass frame. The points are the lattice data, and the lines are the fits to the effective range

expansion.

intheliterature,rdd

/

aff

≈

0.12[20]andrdd

/

aff

∼

2.6[21].However

ineachofthepreviousestimates,thesizeofthesystematicerrors has notbeenquantified.InparticulartheanalysisinRef.[21]was plaguedbythesamelargefinite-volumeeffectswehavediscussed here.Inour analysiswe usedtherepulsivethree-particle interac-tiontoreducethesizeofthefinite-volumecorrections.

Tofurtheranalyzethesignoftheeffectiverange,wehave con-sidereda simple modelof thedimer–dimer systemconsistingof two fundamental particles. Due to the Pauli repulsion between

identicalparticles,itisveryplausiblethat thedimer–dimer inter-actionhasthecharacteristics ofa repulsiveYukawainteractionat longdistances,sincethisisthefunctionalformofthedimerwave function[35].Therefore,wehaveconsideredadimer–dimer inter-actionoftheform

V

(

r

)

=

VG

(

r

) θ (

Rg

−

r

)

+

VS

(

r

) θ (

r

−

Rg

) θ (

Ry

−

r

)

Fig. 5. The continuum-limit extrapolation of the dimer–dimer scattering length add. The final result is add/aff=0.618(30).

Fig. 6. The continuum-limit extrapolation of the dimer–dimer effective range rdd. The final result is rdd/aff= −0.431(48). whereVG

(

r

)

isaGaussianpotentialuptoradialdistanceRg,VY

(

r

)

isalong-rangerepulsiveYukawapotentialstartingfromradial dis-tanceof Ry, VS

(

r

)

isa cubicsplinefunction,and

θ

isa unitstep function. In all cases where the scatteringlength is positive, we findthat the effective rangeis negative. We find that the repul-sive Yukawapotential plays an importantrole making the effec-tive rangenegative. Thesefindings supportour latticeresult ofa negativeeffectiverangeforthe dimer–dimersystemandare also consistentwiththenegativevalueforthefermion–dimereffective range.

5. Summaryandconclusions

We have used lattice effective field theory to compute the scattering length and effective range of dimer–dimer scattering inthe universal limit of largefermion–fermion scatteringlength. Tobenchmark our numerical lattice methods, we first calculated fermion–dimerscattering. The scattering phase shifts were com-putedbycalculatingfinite-volumescatteringenergiesandapplying Lüscher’sfinite-volumemethod.Inourcalculationsweincludeda three-particle interaction in order to generate additional data to beusedinthecontinuum-limitextrapolations.Thedependenceon

the three-particle interaction coefficient c3N was consistent with

the

(

a

/

aff

)

3.54544dependencepredictedbyconformalscalinginthe

unitaritylimit. Extrapolatingtothecontinuum limit,we obtained thevaluesafd

/

aff

=

1.176(6)andrfd

/

aff

= −

0.029(16),inexcellent

agreementwithpreviouscalculationsoffermion–dimerscattering lengthandeffectiverange.

Wethenusedthesamemethodstocalculatedimer–dimer scat-tering and extracted the dimer–dimer scattering length add and

effectiverange rdd.Inthis casewe useda multi-particle

interac-tioncoefficientc3N,4N.Wefoundthatthefinite-volumecorrections

could be reduced by making the three-particle interaction suffi-cientlyrepulsive.UsingLüscher’sfinite-volumemethod,we deter-minedthedimer–dimerscatteringphaseshifts.Wethenextracted thevaluesadd andrdd andperformedcontinuum-limit

extrapola-tions. For the scattering length we obtained add

/

aff

=

0.618(30),

ingoodagreement withpublished results.Fortheeffectiverange we found rdd

/

aff

= −

0.431(48), which is different fromprevious

estimates.Howeverthisnewresultrepresentsthefirstcalculation withquantifiedandcontrolledsystematicerrors.

Finally,weconsideredasimplemodelofthedimer–dimer sys-tem as two fundamental particles interacting via a short-range Gaussian interaction and a repulsive Yukawa potential to mimic

thePaulirepulsionbetweenidenticalparticles.Wefoundthatthe effective range is negative for cases where the scattering length is positive. This may explain why both the dimer–dimer and fermion–dimereffectiverangesarenegative.

Ourresultsshould haveimmediateapplicationstothe univer-salphysicsofshallowdimers.Oneparticularlyusefulapplicationis inthedeterminationoftheground-stateenergydensityofadilute gasofshallowdimers.Thedimers behaveasrepulsiveBose parti-cles,andtheenergydensityhasbeendetermineduptoorder

ρ

a3

dd [6–9] aswell asthe firstcorrection proportional tordd

/

add [10].

Thisnewvalue forthedimer–dimereffectiverangesuggeststhat higher-order correctionsto the energy density ofa dilute gas of shallowdimerscouldbelargerthanpreviouslythought.

Acknowledgments

The authors are grateful for discussions with Yusuke Nishida and acknowledge partial support from the U.S. National Science Foundation grant No. PHY-1307453, the U.S. Department of En-ergy (DE-FG02-03ER41260), the DFG (SFB/TR 110, “Symmetries and the Emergence of Structure in QCD”) and the BMBF (con-tract No. 05P2015 – NUSTAR R&D). The work of UGM was also supported in part by The Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) grant No. 2017VMA0025.ComputingresourceswereprovidedbytheHigher Performance Computing centers at Mississippi State University, NorthCarolinaStateUniversityandRWTHAachen.

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