Efimov effect from functional renormalization

S. Moroz,* S. Floerchinger,† R. Schmidt,‡ and C. Wetterich§

Institut für Theoretische Physik, Philosophenweg 16, D-69120 Heidelberg, Germany�Received 15 December 2008; published 3 April 2009�

We apply a field-theoretic functional renormalization-group technique to the few-body �vacuum� physics ofnonrelativistic atoms near a Feshbach resonance. Three systems are considered: one-component bosons with aU�1� symmetry, two-component fermions with a U�1��SU�2� symmetry, and three-component fermions witha U�1��SU�3� symmetry. We focus on the scale-invariant unitarity limit of an infinite scattering length. Theexact solution for the two-body sector is consistent with the unitary fixed-point behavior of the consideredsystems. Nevertheless, the numerical three-body solution in the s-wave sector develops a limit cycle scaling incase of U�1� bosons and SU�3� fermions. The Efimov parameter for the one-component bosons and thethree-component fermions is found to be s�1.006, consistent with the result of Efimov.

DOI: 10.1103/PhysRevA.79.042705 PACS number�s�: 03.65.Nk, 21.45.�v, 34.50.�s

I. INTRODUCTION

The physics of ultracold atoms is a broad area of researchwhich develops rapidly both experimentally and theoretically�for reviews see �1,2��. To a large extent this is due to theexcellent tunability and control of the studied systems. Inparticular the interaction strength of atoms near a Feshbachresonance can be changed in broad ranges by tuning themagnetic field, which makes these systems an ideal play-ground for testing the predictions of theoretical models atstrong coupling. Both few-body and many-body quantumand thermodynamic effects have been extensively studiedwith ultracold gases.

Near a broad Feshbach resonance the atom-atom s-wavescattering length a is large in comparison to the effectiverange reff of the microscopic interaction. The low-energyvacuum physics �for vanishing temperature and density� be-comes universal: some physical observables become insensi-tive to the detailed form of the microscopic interaction anddepend only on the scattering length a �3�. For example, fora�0 the theory admits a stable shallow diatom. For thisatom-atom bound state the universal binding energy is deter-mined simply by dimensional analysis. In the unitarity limitall energy scales drop out of the problem and the theory isscale invariant in the two-body sector. It is a well-establishedresult, derived first by Efimov �4�, that in the three-bodysector of the resonantly interacting particles a spectrum ofshallow three-body bound states develops. At unitarity, thespectrum is geometric, which is a signature of the limit cyclebehavior of the renormalization-group �RG� flow. Even inthe case of a scale symmetry in the two-body sector, therunning of the renormalized three-body couplings indicates aviolation of the dilatation symmetry and may be associatedwith a quantum anomaly �5�.

The low-energy few-body scattering of atoms has beeninvestigated using various computational nonperturbative

techniques ranging from effective field theory �6–9� toquantum mechanics �4,10�. The perturbative � expansionaround critical d=4 and d=2 dimensions has also been ap-plied to this problem �11,12�. A field-theoretical functionalrenormalization-group approach has been used to investigatethe two-body and three-body sectors of two-componentfermions recently �13,14�. As a convenient truncation invacuum, the authors use a vertex expansion and reproducethe Skorniakov–Ter-Martirosian integral equation �15�. Inthis way, the universal ratio of the atom-diatom to the atom-atom scattering length is computed.

In this paper we follow �13� and consider the few-bodyphysics of nonrelativistic atoms near a Feshbach resonancewhich may be described by a simple two-channel modelof particles with short-range interactions. We study three dif-ferent systems: bosons with a U�1� symmetry �system I�,fermions with a U�1��SU�2� symmetry �system II�, and fer-mions with a U�1��SU�3� symmetry �system III�. Bothsystems I and II have been well studied during the last de-cade. The model of SU�3� fermions might be of relevancefor three-component mixtures of 6Li atoms near the broadFeshbach resonances. The many-body properties of thismodel have been studied in �16–20�. Recently, the three-component fermion system has been studied with functionalrenormalization-group methods using an approximation in-cluding a trion field �21� �a trion is a bound state of threeatoms�. The present paper, which is based on a vertex expan-sion, complements and extends the results of �21�. It under-lines the basic finding of the presence of Efimov states forSU�3� fermions and estimates the universal Efimov param-eter s with a higher precision.

The structure of the paper is as follows. In Sec. II wepresent a field-theoretic RG method, which we use to solvethe few-body problem, and the three models we are going toinvestigate in this work. In Sec. III we investigate the effec-tive action in the vacuum state, i.e., for vanishing tempera-ture T=0 and density n=0. The following section, Sec. IV, isdevoted to the exact solution of the two-body sector for posi-tive scattering lengths a�0 �diatom phase�. In Sec. V weturn to the analysis of the three-body sector and derive theRG flow equation for the atom-diatom vertex at unitarity.This RG equation is solved analytically employing a simple

*s.moroz@thphys.uni-heidelberg.de†s.floerchinger@thphys.uni-heidelberg.de‡r.schmidt@thphys.uni-heidelberg.de§c.wetterich@thphys.uni-heidelberg.de

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pointlike approximation in Sec. VI. In Secs. VII and VIII wereproduce the Skorniakov–Ter-Martirosian integral equationand present a numerical solution of the three-body RG flowequation. We draw our conclusions in Sec. IX.

II. METHOD AND DEFINITION OF MODELS

In this work we calculate a scale-dependent effective ac-tion functional �k �22� �for reviews see �23,24��, often calledaverage action, flowing action, or running action. Thisrenormalization-group method is formulated in Euclideanspace-time using the Matsubara formalism. The flowing ac-tion �k includes all fluctuations with momenta q�k. In theinfrared limit k→0 the full effective action �=�k→0 is ob-tained. This dependence on the scale k is introduced by add-ing a regulator Rk to the inverse propagator �k

�2� and theflowing action �k obeys the exact functional flow equation�22�

�k�k = 12STr �kRk��k

�2� + Rk�−1 = 12STr �̃k ln��k

�2� + Rk� .

�1�

This functional differential equation for �k must be supple-mented with the initial condition �k→�=S, where the “clas-sical action” S describes the physics at the microscopicultraviolet �UV� scale, k=�. In Eq. �1� STr denotes a super-trace which sums over momenta, Matsubara frequencies, in-ternal indices, and fields �taking fermions with a minus sign�.The second functional derivative �k

�2� denotes the full inversefield propagator, which is modified by the presence of theinfrared �IR� regulator Rk. As a consequence, the fluctuationswith q2�k2 are suppressed and the effective action dependson the scale k. The choice of the momentum-dependent regu-lator function Rk�q� introduces a scheme dependence whichhas to disappear for the exact solution for k→0. In the sec-ond form of flow equation �1�, �̃k denotes a scale derivative,which acts only on the IR regulator Rk. This form is veryuseful because it can be formulated in terms of one-loopFeynman diagrams. The effective action �k=0 is the generat-ing functional of the 1PI vertices, which can be easily con-nected to the different scattering amplitudes in the case ofvanishing density �n=0� and vanishing temperature �T=0�. Itis also convenient to introduce the RG “time” t� ln�k /��,which flows in the interval t� �− ,0�. In the following wewill use both t and k.

In most cases of interest functional differential equation�1� can be solved only approximately. Usually some type ofexpansion of �k is performed, which is then truncated atfinite order, leading to a finite system of ordinary differentialequations. The expansions do not necessarily involve a smallparameter �such as an interaction coupling constant� and theyare, in general, of nonperturbative nature. As has alreadybeen advocated in Sec. I, we perform a systematic vertexexpansion of �k taking the full momentum dependence of therelevant vertex in the three-body sector into account. Thevertex expansion is an expansion in powers of fields; hencegenerally

�k = �n=0

�k�n� = �k�2� + �k�3� + �k�4� + ¯ , �2�

where the index in parentheses denotes the number of fieldsn in the monomial term �k�n�. In the second equation �k�0�and �k�1� are missing because we are not interested in thefree energy of the vacuum and the term linear in the fields isabsent by construction.

In this paper we are interested in the nonrelativistic phys-ics of atoms interacting via a Feshbach resonance, which canbe described by a simple two-channel model. In particular,we consider and compare three different systems:

�a� System I �single bosonic field near a Feshbachresonance�—Our truncation of the scale-dependent flowingaction, written in the Fourier space, is

�k�2� = �Q

��Q��i�q + q2 − ���Q�

+ �Q

��Q�P �Q� �Q� ,

�k�3� =h

2�

Q1,Q2,Q3

� ��Q1��Q2��Q3�

+ �Q1���Q2���Q3����Q1 − Q2 − Q3� ,

�k�4� = − �Q1,. . .,Q4

�3�Q1,Q2,Q3� �Q1��Q2� ��Q3���Q4�

���Q1 + Q2 − Q3 − Q4� , �3�

where Q= �� ,q� and Q=− d�

2�− d3q

�2��3 . The field repre-sents an elementary complex bosonic atom, while �Q� is acomplex bosonic composite diatom which mediates the Fes-hbach interaction. At the initial UV scale we take �3=0. Theaction for becomes Gaussian, and one may integrate out using its field equation . As will be demonstrated inSec. IV, the Yukawa coupling h is simply related to the widthof the Feshbach resonance. For k→0 the coupling�3�Q1 ,Q2 ,Q3� becomes the 1PI vertex which can be con-nected to the atom-diatom scattering amplitude. The systemhas an obvious U�1� symmetry which reflects the conservednumber of atoms.1

�b� System II �fermionic doublet near a Feshbach reso-nance�

We have

�k�2� = �i=1

2 �Q

i��Q��i�q + q2 − ��i�Q�

+ �Q

��Q�P �Q� �Q� ,

1In general, the U�1� symmetry can be spontaneously broken dueto many-body effects and our truncation �3� would be insufficient.In this work, however, we are interested only in the few-body phys-ics �for more details see Sec. III�.

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�k�3� = − h�Q1,Q2,Q3

� ��Q1�1�Q2�2�Q3�

− �Q1�1��Q2�2

��Q3����Q1 − Q2 − Q3� ,

�k�4� = �Q1,. . .,Q4

�3�Q1,Q2,Q3��i=1

2

�Q1�i�Q2� ��Q3�

�i��Q4���Q1 + Q2 − Q3 − Q4� . �4�

Here, the two species of elementary fermionic atoms 1 , 2are described by Grassmann-valued fields, and is a com-posite bosonic diatom. At the UV scale one has 12.This fermionic system has an SU�2��U�1� internal symme-try with �1 ,2� transforming as a doublet and as a singletof the SU�2� flavor subgroup. Two-species fermion systemsnear Feshbach resonances were realized experimentally with6Li and 40K atoms �25�.

�c� System III �fermionic triplet near a Feshbach reso-nance�

We have

�k�2� = �Q�i=1

3

i��Q��i�q + q2 − ��i�Q�

+ �Q�i=1

3

i��Q�P �Q� i�Q� ,

�k�3� =h

2�

Q1,Q2,Q3

�i,j,k=1

3

�ijk� i��Q1� j�Q2�k�Q3�

− i�Q1� j��Q2�k

��Q3����Q1 − Q2 − Q3� ,

�k�4� = �Q1,. . .,Q4

��3a�Q1,Q2,Q3��i=1

3

i�Q1�i�Q2�

��j=1

3

j��Q3� j

��Q4�

+ �3b�Q1,Q2,Q3��i=1

3

i�Q2�i��Q4�

��j=1

3

j�Q1� j��Q3����Q1 + Q2 − Q3 − Q4� . �5�

The three species of the elementary Grassmann-valued fer-mion field can be assembled into a vector = �1 ,2 ,3�.Similarly the three composite Feshbach bosonic diatomsform the vector = � 1 , 2 , 3��23 ,31 ,12�. The ac-tion has an SU�3��U�1� symmetry with transforming as 3and as 3 for the SU�3� flavor subgroup. Two differentcouplings �3a and �3b are allowed by the SU�3� symmetry.This model might be of relevance for three-component mix-tures of 6Li atoms. There are three distinct broad Feshbachresonances for three scattering channels near B�800 G for6Li atoms. As a first approximation we assume that the reso-nances for all channels are degenerate, which leads to the

SU�3� flavor symmetry and to model �5�. A stable three-component mixture of 6Li atoms has been recently created�26,27�. The theoretical investigation of the three-bodylosses in �26,27� has been recently published �28–30�.

To unify our language for the different models consideredin this paper, we refer to the elementary particles as atoms�and denote corresponding quantities with the subscript �,while the composite is called diatom. All considered sys-tems have Galilean space-time symmetry, whose conse-quences we discuss in Appendix A. Our units are �=kB=1.Moreover we choose the energy units such that 2M=1,where M is the mass of the atom.

We should stress that ��2� and ��3� do not have the mostgeneral form. The most general form of the vertex expansionincludes an arbitrary inverse atom propagator P�Q� anda momentum-dependent Yukawa coupling h�Q1 ,Q2 ,Q3�.However, due to special properties of the vacuum state �seeSec. III�, P�Q� and h�Q1 ,Q2 ,Q3� are not renormalized andkeep their microscopic values P�Q�= �i�q+q2−�� andh�Q1 ,Q2 ,Q3�=h during the RG flow. At this point it is alsoimportant to note that our vertex expansion is complete tothe third order in the fields. Possible terms with four fields,which are invariant with respect to the symmetries of ourmodels, can be found in Appendix B. In this appendix wealso present arguments, because of which we do not includethese terms in our truncation. To summarize, the propertiesof the two- and three-body sectors, which are of the maininterest in this work, can be calculated using truncations�3�–�5�.

III. VACUUM LIMIT

The advantage of the method used in this paper is that it isa field-theoretical setting which permits computations for thegeneral case of nonzero temperature �T�0� and density �n�0�. In this paper we are interested only in the scatteringand the bound states of few particles in vacuum. The projec-tion of the effective action �k=0 onto the vacuum state mustbe performed carefully and was developed in �13,31�. Herewe shortly summarize the procedure.

The vacuum projection of �k=0 is performed as follows:

�vac = limkF→0,T→0

�k=0 T�Tc�kF�, �6�

where kF= �3�2n�1/3 is a formal Fermi wave vector �definedfor both bosons and fermions� and n is the atom density ofthe system. Thus we start with the effective action at finitedensity and temperature. The system is then made dilute bytaking limit kF→0. It is crucial, however, to keep the tem-perature T above its critical value in order to avoid many-body effects �e.g., Bose-Einstein condensation�. One mayperform the vacuum limit for a fixed dimensionless T

Tcsuch

that the temperature goes to zero because Tc scales kF2 .

Let us now examine the momentum-independent part ofthe atom inverse propagator P,k=0�Q=0�=−�, as wellas its diatom counterpart m�

2 � P ,k=0�Q=0�, in more

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detail.2 For positive values, i.e., ��0, m 2 �0, they act as

gaps for atoms and diatoms, respectively. There is no Fermisurface in vacuum; hence ��0. The system is above criti-cality in the vacuum limit, i.e., it is in the symmetric phase;hence m

2 �0. These two conditions define a quadrant in them

2-� plane. Moreover, due to the nonrelativistic nature ofthe problem, the zero-energy level can be shifted by an arbi-trary constant. This is a result of the symmetries of our mod-els. The real-time �t=−i�� version of the microscopic actionS=�� in coordinate space �t ,x� is symmetric with respect tothe energy shift symmetry �32�:

→ eiEt, → e2iEt , � → � + E . �7�

Since no anomaly of this symmetry is expected and our cut-off respects this symmetry �see below�, this is a symmetry ofthe flow equations and the effective action �k=0. Hence, bythe appropriate energy shift, we can make one energy stategapless, i.e., put it on the boundary of the quadrant in m

2-�

plane. We end up with three distinct branches �31�:

m 2 � 0, � = 0 atom phase �a−1 � 0� ,

m 2 = 0, � � 0 diatom phase �a−1 � 0� ,

m 2 = 0, � = 0 resonance �a−1 = 0� . �8�

For system II the gapless state is the lowest-energy state. Inthe atom phase �a−1�0� diatoms are gapped and the low-est excitation is an atom . In the diatom phase �a−1�0� thesituation is reversed: is the lowest excitation above thevacuum and has a gap −�, which can be interpreted as ahalf of the binding energy of , �=2�. At resonance �a−1

=0� both and are gapless. For systems I and III, and forsmall values of a −1, one finds a whole spectrum of trions,bound states of three atoms, which have a lower energy thanatoms and diatoms. This effect was first predicted and calcu-lated by Efimov �4� in a quantum-mechanical computation,and modifies the vacuum structure �21�. In the trion phaseboth atoms and diatoms show a gap; i.e., the ground state has��0, m

2 �0. However, for an investigation of the excitedEfimov states we may as well use vacuum fixing condition�8�. At resonance this corresponds to degenerate energy lev-els of the Efimov states and the atoms or diatoms, whichbecomes a good approximation for the high Efimov stateswhich are close to the atom or diatom threshold �21�.

The vacuum limit, which we described above, leads tonumerous mathematical simplifications. For example, all dia-grams with loop lines pointing in the same direction vanishin the vacuum limit. This can be demonstrated using theresidue theorem for the frequency loop integration. Indeed,the inverse propagators have non-negative gaps and all con-sidered diagrams have poles in the same half plane of thecomplex loop frequency. Thus we can close the contour suchthat it does not enclose any poles and the frequency integralvanishes. The argument works also for the 1PI vertices pro-vided they have poles in the same half plane as the propaga-

tors. This finding simplifies the RG analysis in vacuum con-siderably. For example, one can show that in vacuum theatom inverse propagator P�Q� is not renormalized �33�. Theonly one-loop diagram, which renormalizes P, has innerlines pointing in the same direction, and therefore vanishes.It is sufficient to analyze only one-loop diagrams becauseRG flow equation �1� has a general one-loop form �23�. An-other very important simplification in vacuum comes from aspecial hierarchy, which is respected by the flow equations.We define the n-body sector as a set of 2n-point 1PI verticeswritten in terms of elementary atoms �in this sense P �Q�belongs to the two-body sector because is composedof two atoms�. The vacuum hierarchy consists in the fact thatthe flow of the n-body sector is not influenced by any higher-body sectors. The flow equations for the n-body sector sim-ply decouple from the flow of the �n+1�-body sector �andhigher�. The observed hierarchy is a consequence of the dia-grammatic simplification in vacuum. At finite density �n�0� or temperature �T�0� the decoupling of the low-n ver-tices from the high-n vertices is not valid anymore.

IV. TWO-BODY SECTOR: EXACT SOLUTIONIN THE DIATOM PHASE FOR A POSITIVE

SCATTERING LENGTH

The two-body sector truncation is defined by

�k = �k�2� + �k�3� �9�

in all three models �3�–�5�. As mentioned in Sec. III the RGflows belonging to the two-body sector decouple fromhigher-body sectors in vacuum. Due to the nonrenormaliza-tion of the atom propagator, it is sufficient to solve the flowequations only for the Yukawa coupling h and the diatominverse propagator P .

It turns out that the Yukawa coupling is not renormalizedin vacuum for all three models:

�th = 0. �10�

Due to the U�1� phase symmetry, there is no one-loop Feyn-man diagram in our truncations �3�–�5�, which renormalizesthe Yukawa coupling h. The only one-loop diagram, whichcould contribute to the flow of h, contains the four-atomvertex ��Q1 ,Q2 ,Q3�. The vertex ��Q1 ,Q2 ,Q3� is notrenormalized in vacuum �see Appendix B� and vanishes onall scales, provided its microscopic value is zero. The argu-ment can be extended to a momentum-dependent Yukawacoupling h�Q1 ,Q2 ,Q3�.

In order to solve the two-body sector, it remains to calcu-late the flow of the diatom inverse propagator P �Q�, whichis schematically shown in Fig. 1 and can be written as fol-lows:

2Flavor indices applicable for systems II and III are suppressed inthis section.

∂t = ∂̃t

FIG. 1. Schematic graphical representation of the flow for theinverse diatom propagator P . Diatoms are denoted by dashed lines,and atoms by solid lines.

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�tP �Q� = −2

3 + p�

L

�̃ t

�h2

�P�L� + R�L���P�Q − L� + R�Q − L��,

�11�

where p= +1 for bosons and p=−1 for fermions. It turns outthat the flow of the inverse diatom propagator in system III isexactly the same as in system II. In the last formula one has

P�Q� = �i�q + q2 − �� �12�

and R�Q� stands for the atom regulator.It is remarkable that using a special choice of the regula-

tor, we can integrate flow �11� exactly. We follow �13,34� bychoosing a regulator, which is frequency and momentum in-dependent and has the form R=k2=�2 exp 2t. This regula-

tor has the advantage that it is Galilean invariant and hencethe Galilean symmetry of the microscopic action is preservedduring the RG evolution. First we perform the frequencyloop integration in Eq. �11� with help of the residue theorem:

�tP �Q� = −2

3 + p� d3l

�2��3 �̃ th2

i�q + l2 + �l − q�2 − 2� + 2R

= −2h2

3 + p�t� d3l

�2��3

1

i�q + l2 + �l − q�2 − 2� + 2R

,

�13�

where the second equality holds due to the nonrenormaliza-tion of the Yukawa coupling h and atom inverse propagatorP, and thus �t→�t. Using the specific values of the regula-tor R�tIR�=0 and R�t=0�=�2 we integrate out the flowequation from the UV scale t=0 to the IR scale tIR=− andobtain

P IR�Q� − P

UV�Q� = −2h2

3 + p� d3l

�2��3� 1

i�q + l2 + �l − q�2 − 2�

−1

i�q + l2 + �l − q�2 − 2� + 2�2�= −

h2

3 + p� dl

2�2� l2

l2 + � i�q

2+

q2

4− �� −

l2

l2 + � i�q

2+

q2

4− � + �2��

= −h2

4��3 + p��� −� i�q

2+

q2

4− �� . �14�

The last identity assumes �� � , q , �q .At this point we must fix the initial condition P

UV�Q� atk=� in order to obtain the physical inverse propagatorP

IR�Q� at k=0. This is done in �13,31,33� and we follow thesame steps here. For broad resonances with h2→ the in-verse diatom propagator at the microscopic scale � is givenby

P UV = ��B� + ��, ��B� = �B�B − B0� . �15�

Here ��B� is the detuning of the magnetic field B whichmeasures the distance to the Feshbach resonance located atB0. The magnetic moment of the diatom is denoted by �B.The counterterm �� depends on the ultraviolet cutoff �. Ne-glecting a possible background scattering length abg, thescattering length a and the detuning ��B� are related by �31�

a = −h2

4��3 + p���B�. �16�

Thus the Yukawa coupling is proportional to the square rootof the width of the Feshbach resonance. For narrow Fesh-bach resonances �h→0� perturbation theory is applicable,while for broad Feshbach resonances �h→�, which are ofmain interest in our work, the problem becomes stronglycoupled. Using Eqs. �15� and �16�, we rewrite Eq. �14� as

P IR�Q� − �� +

h2

4��3 + p�a

= −h2

4��3 + p��� −� i�q

2+

q2

4− �� . �17�

At this point the momentum-independent counterterm �� canbe identified,

�� =h2

4��3 + p�� �18�

and we obtain our final result for the k-dependent inversediatom propagator P ,k�Q�:

P ,k�Q� =h2

4��3 + p��− a−1 +� i�q

2+

q2

4− � + k2� .

�19�

The wave-function renormalization Z ,k can now be defined,

Z�,k � � �P�,k�Q���i�q�

��q=0

=h2

4��3 + p�

Z̃

1

4�k2 − ��

,

�20�

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and the IR inverse diatom propagator P �Q� reads

P �Q� � P ,k=0�Q� =h2

4��3 + p��− a−1 +� i�q

2+

q2

4− �� .

�21�

In vacuum and for positive scattering length �a�0� thevacuum condition �Eq. �8��, P �Q=0�=m

2 =0, must be ful-filled. This leads to

a =1

�− �

. �22�

For positive scattering lengths in vacuum, −� is a positivegap of the atom and can be interpreted as half of thebinding energy of the diatom � . Hence, the binding energycan be expressed as

� = 2� = −2

a2 , � = −1

Ma2 . �23�

The second equation is expressed in conventional units andis the well-known universal relation for the binding energyof the shallow diatom �3�. It should be mentioned here thatthe two-body sector can also be solved exactly using a non-relativistic version of the Litim cutoff �14�, which is opti-mized in the sense of �35,36�. The drawback of this cutoff isthat it breaks Galilean symmetry and one has to put someGalilean noninvariant counterterms into P

UV�Q� to restoreGalilean symmetry in the IR.

It is important to stress the appearance of universality inthe broad-resonance limit �h2→� �31�: the IR physics be-comes insensitive to the initial conditions in the UV. Forexample, one may consider possible momentum-dependentmodifications of the microscopic inverse propagator P ,k=�,which result in deviations from an exactly pointlike form.Their effect on P ,k=0 is suppressed by h−2 with respect to thequantum loop contribution and it therefore becomes irrel-evant in the broad-resonance limit. In vacuum, and for h→, the only physically relevant scale is given by the scat-tering length a.

Physics becomes completely universal if we perform theunitarity limit, h2→ �broad-resonance limit�, and a−1→0�resonance limit� �37�. In vacuum, all scales drop out in thislimit. The atom and diatom inverse propagators take the fol-lowing form:

P�Q� = i�q + q2, P �Q� = Z̃� i�q

2+

q2

4. �24�

An alternative quantum-mechanical derivation of Eq. �24�can be found in Appendix C.

Let us perform a scaling dimension counting in the uni-tary limit.3 We start with the fact that ���=0. In nonrelativ-istic physics, energy scales as two powers of momentum4

and the free field scaling reads

�q� = 1, ��� = 2, �� = 3/2, � � = 3/2, �h� = 1/2.

�25�

For the universal interacting theory the scaling of is modi-fied according to Eq. �24�. The scaling dimension of the dia-tom field and Yukawa coupling h can be computed fromthe Yukawa term and the kinetic term of :

�h� + � � + 2�� = 5 �Yukawa� ,

2�h� + 2� � + 1 = 5 �kinetic� . �26�

This system is degenerate and we get a solution �h�=� and� �=2−�, where � is some real number. The absence of ascaling of h in Eq. �10�, however, fixes �h�=0 and � �=2.Note that the scaling of the diatom field at unitarity is differ-ent from the scaling of the atom field . This is a manifes-tation of the fact that the scaling in the two-body sector isgoverned by a fixed point,5 which is different from theGaussian fixed point �34,38�. Exactly at unitarity no obviousscales are left in the problem and the theory seems to bescale invariant. Even more so, at the two-body sector levelthe theory seems to be an example of a nonrelativistic con-formal field theory �NRCFT�.6 This type of theory is sym-metric with respect to the Schrödinger group, which is anextension of the Galilean symmetry group �for more detailssee Appendix A�. It is known, however, that the Schrödingersymmetry can be broken by a quantum anomaly in higher-body sectors �5�. The fate of the Schrödinger symmetry isdifferent for the different systems considered. For the reso-nantly interacting particles �systems I and III� it was demon-strated by Efimov �4� a long time ago that in the three-bodysector the continuous scaling symmetry, which is a part ofthe Schrödinger symmetry, is broken to the discrete scalingsubgroup Z �3�. This manifests itself in the appearance of ageometric spectrum of bound states in the three-body sector,which is called the Efimov effect. For system II of SU�2�-symmetric fermions it is believed that the Schrödinger sym-metry is not broken in the higher sectors of the theory andthat this is a real example of an NRCFT �39�.

To summarize, in this section we have solved exactly thetwo-body sector in vacuum for a positive scattering length.Solution �21� was obtained for the specific initial conditionshk=��Q1 ,Q2�=h, �,k=�=0, and P ,k=��Q� given by Eq. �15�.This choice corresponds to a pointlike microscopic atom in-teraction. However, the presented calculations can be gener-alized to an arbitrary boson-mediated atom interaction with�,k=�=0, while hk=��Q1 ,Q2� and P ,k=��Q� can be chosenfreely.

V. THREE-BODY SECTOR: FLOW EQUATIONS

The main emphasis of this paper is devoted to the analysisof the three-body sector of the three models �3�–�5� in theunitarity limit. We demonstrate the appearance of the Efimov

3We denote a scaling dimension of some quantity X by �X�.4This is known as the dynamical exponent z=2.

5Called a unitarity fixed point.6Another example of NRCFT in two spatial dimensions is a theory

of anyons �39�.

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effect in systems I and III and its absence in system II fromthe field-theoretical RG perspective. In the present sectionwe formulate a flow equation for the coupling �3�Q1 ,Q2 ,Q3�and make some general simplifications. In Sec. VI we usethe pointlike approximation for �3�Q1 ,Q2 ,Q3� to solve theproblem. Sections VII and VIII are devoted to the solution ofthe general momentum-dependent form of the flow equation.

The closed exact solution for the two-body sector pro-vides a simple strategy for a computation of the coupling �3in the three-body sector. In general, one may introduce sepa-rate cutoffs R and R for the atoms and diatoms . Thepresence of the cutoff R does not affect our computation inthe two-body sector. We may therefore first lower the cutoffR from �2 to zero while keeping R fixed, and subsequentlylower R to zero in a second step �23�. As the result of thefirst step the diatom inverse propagator P is modified ac-cording to Eq. �21�. This step also induces diatom interac-tions, such as, for example, a term � � �2. However, theseinteractions belong to the four-body and higher sectors. Byvirtue of the vacuum hierarchy, they do not influence theflow of �3. For the second step of our computation we cantherefore use a version of the flow equation where only thediatom cutoff R is present. In this flow equation P and P

are fixed according to Eqs. �21� and �12�.For the diatoms we use a sharp cutoff:

R �Q,k� = P �Q�� 1

�� q − k�− 1� . �27�

The special feature of this cutoff is that the regularized dia-tom propagator takes a simple form:

1

P �Q� + R �Q,k�= �� q − k�

1

P �Q�. �28�

Thus the propagator is cut off sharply at the sliding scale k.Our choice of the cutoff is motivated by technical simplicityas well as effective theory �3� and quantum-mechanical �4�approaches to this problem. The advantage of this cutoff isthe property of locality in the momentum space, whichmeans that it chops off momentum shells locally. In thethree-body sector we are interested not only in the IR valueof the atom-diatom vertex �3, but also in the flow at allscales.

Let us now calculate the flow equation of the 1PI atom-diatom vertex �3. For SU�3� fermions there are two atom-diatom vertices, �3a and �3b, and we postpone the analysis ofthis model to the end of the section. In Minkowski space �thereal-time version of our theory� the atom-diatom scatteringamplitude is given by the amputated connected part of theGreen’s function �0 †† 0�, and thus it can be simplycalculated from the knowledge of �3. We first consider thekinematics of the problem. The 1PI atom-diatom vertex�3�Q1 ,Q2 ,Q3� depends generally on three four vectors, i.e.,six independent rotation-invariant variables in the center-of-mass frame. We take the incoming atom and diatom to havemomenta q1 and −q1 and energies E1 and E−E1, while theoutgoing atom and diatom have momenta q2 and −q2 andenergies E2 and E−E2. We denote the vertex in the center-of-mass frame by �3�Q1

,Q2 ,E� �see Fig. 2�. This configura-

tion is in general off-shell, which is necessary since in theflow equations, the vertex also appears inside a loop.

In Minkowski space7 the flow equation for the atom-diatom vertex �3 for systems I and II reads

�t�3�Q1,Q2

;E� = �L

�t

�� l − k�P�L�P �− L + Q�

��C�3�Q1,L;E��3�L,Q2

;E�

+B

2� h2

P�− L + Q1 �

�3�L,Q2;E�

+ �3�Q1,L;E�

h2

P�− L + Q2 ��

7The flow equation of effective action �1� is formulated in Euclid-ean space-time �imaginary-time formalism�. In order to obtain theflow equation in Minkowski space, it is sufficient to take externalfrequencies �ext to be imaginary, i.e., perform a transformation�ext→ i�ext, which is the inverse Wick rotation.

TABLE I. Numerical coefficients A, B, and C in flow equation�29� for the three examined systems. In the case of system III�a� weconsider the scattering of the type i j→i j, with i� j, whilesystem III�b� corresponds to the vertex �3=3�3a+�3b.

Model A B C

System I 1 2 1

System II 1 −2 1

System III�a� 1 −2 1

System III�b� 4 4 1

Qψ2

Qϕ2

Qϕ1

Qψ1

FIG. 2. Kinematics of the vertex �3�Q1 ,Q2

,E� in the center-of-mass frame. The atoms and diatoms have momenta Q1

= �E1 ,q1�,Q1

= �−E1+E ,−q1� and Q2= �E2 ,q2�, Q2

= �−E2+E ,−q2�.

∂t = ∂̃t + ∂̃t + ∂̃t + ∂̃t

� � � � �

FIG. 3. Graphical representation of the flow equation for �3.Full lines denote atoms and dashed lines denote diatoms . Theshaded circle denotes �3.

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+ Ah2

P�− L + Q1 �

h2

P�− L + Q2 �� ,

�29�

where Q=Q1 +Q1

= �E ,0�. The coefficients A, B, and C forsystems I and II can be found in Table I. The graphical rep-resentation of this equation is depicted in Fig. 3. The scalederivative on the RHS acts only on the cutoff and can becomputed easily, �̃t�� l −k�=−k�� l −k�.

Fortunately, the flow equation can be simplified consider-ably. First note that there is only one inverse propagatorP�L� with a loop momentum L of positive sign in Eq. �29�.For this reason the whole integrand in Eq. �29� has a singlefrequency pole in the upper half plane. Thus the frequencyintegration in Eq. �29� can be performed with the help of theresidue theorem by performing the substitution �l→ il2. Thisputs the atom in the loop on-shell, corresponding to P�L� inEq. �29�. We obtain a simpler equation if we also putthe energies of the incoming and outgoing atoms on-shell�Q1

= �iq12 ,q1� ,Q2

= �iq22 ,q2��. The diatoms in the loop in

Eq. �29� are generally off-shell. To solve this “half-off-shell” equation only the values �3�q1 ,q2 ,E���3(Q1

= �iq12 ,q1� ,Q2

= �iq22 ,q2� ,E) are needed �3�.

Our aim is the calculation of the atom-diatom scatteringamplitude at low energies and momenta. For low momentathe dominant contribution is given by s-wave scattering. Inprinciple, the right-hand side �RHS� of Eq. �29� has alsocontributions from higher partial waves, which we neglect inour approximation and simplify flow equation �29� by pro-jecting on the s-wave. This is done by averaging Eq. �29�over the cosine of the angle between incoming momentumq1 and outgoing momentum q2. Introducing the averaged 1PIrenormalized vertex, which depends on three scalar vari-ables,

�3�q1,q2,E� �1

2h2�−1

1

d�cos ���3�q1,q2,E� , �30�

we end up with the flow equation

�t�3�q1,q2,E�

= −2�3 + p�

�

k3

�3k2

4−

E

2− i�

�C�3�q1,k,E��3�k,q2,E�

+B

2��3�q1,k,E�G�k,q2� + G�q1,k��3�k,q2,E��

+ AG�q1,k�G�k,q2�� , �31�

where the symmetric function G�q1 ,q2� is defined by8

G�q1,q2� =1

4q1q2ln

q12 + q2

2 + q1q2 −E

2− i�

q12 + q2

2 − q1q2 −E

2− i�

. �32�

The infinitesimally positive i� term arises from the Wickrotation and makes both Eqs. �31� and �32� well defined. It isremarkable that Eq. �31� is completely independent of theYukawa coupling h and thus is a well-defined equation in thelimit of infinite h.

For SU�3� fermions the situation is more complicated be-cause there are two vertices �3a and �3b in our truncation �5�.To extract the flow equation for �3a we consider the scatter-ing channel ii→ j j, with i� j �e.g., 11→ 22�. Afterperforming the same steps as for systems I and II, we end upwith a flow equation:

�t�3a�q1,q2,E�

= −2�3 + p�

�

k3

�3k2

4−

E

2− i�

�3�3a�q1,k,E��3a�k,q2,E�

+ 2�3a�q1,k,E��3b�k,q2,E� + 2��3a�q1,k,E�G�k,q2�

+ G�q1,k��3a�k,q2,E�� + ��3b�q1,k,E�G�k,q2�

+ G�q1,k��3b�k,q2,E�� + G�q1,k�G�k,q2�� , �33�

where p=−1 and G�q1 ,q2� is defined in Eq. �32�. Note thatthe coupling �3b appears in the flow equation for �3a. Theflow equation for �3b can be extracted by consideringthe scattering channel i j→ i j, with i� j �e.g., 21→ 21�:

�t�3b�q1,q2,E�

= −2�3 + p�

�

k3

�3k2

4−

E

2− i�

��3b�q1,k,E��3b�k,q2,E�

+ ��3b�q1,k,E�G�k,q2� + G�q1,k��3b�k,q2,E��

+ G�q1,k�G�k,q2�� . �34�

This equation is completely decoupled from Eq. �33� and hasexactly the same form as Eq. �31� for SU�2� fermions. Thereason for this is simple: RG equation �34� has the graphicalrepresentation depicted in Fig. 3. It turns out that in thischannel only one type of diatom �in our example �2� and twotypes of atoms �1 and 3� appear, which is exactly the sameas in the case of SU�2� fermions. Remarkably, it is possibleto introduce a linear combination �3�3�3a+�3b for SU�3�fermions, which has a simple flow equation of form �31�with coefficients A, B, and C given in Table I �fourth line�.We call this system III�b�. In the SU�3� fermion model thediatom-atom in-state ii can lead to the different diatom-atom out states 11, 22, and 33. If the diatom-atomout- state is not a final but only an intermediate state �e.g.,one is interested in the scattering into a three-atom finalstate�, we must sum the scattering amplitudes for all possibleatom-diatom pairs. It easy to show that �3=3�3a+�3b corre-

8This is in fact an s-wave-projected tree �one-particle-reducible�contribution to the fully connected atom-diatom vertex�3�q1 ,q2 ,E�.

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sponds to the 1PI contribution to the full scattering amplitude ii→anything �e.g., ii→ 11+ 22+ 33�.

To summarize, although at first sight it seems that forSU�3� fermions we must solve a system of two flow equa-tions, it turns out that for the two specific situations it issufficient to solve only one equation �Eq. �31��. This equa-tion is the main result of this section. In Secs. VI–VIII wesolve this final version of the RG flow equation for atom-diatom 1PI vertex for all three systems using various ap-proaches.

VI. THREE-BODY SECTOR: POINTLIKEAPPROXIMATION

In this section flow equation �31� will be solved employ-ing a simple and intuitive pointlike approximation. The 1PIvertex �3�q1 ,q2 ,E� will be replaced by a single momentum-independent coupling �3�E�. In the low-energy limit �E→0� flow equation �31� takes a simple form in the pointlikeapproximation:

�t�3R = −

4�3 + p��3�

�A

4+

B

2�3

R + C��3R�2� + 2�3

R, �35�

where we use G�q→0,k�→ 12k2 from Eq. �32�. The renormal-

ized coupling is defined as �3R=�3k2. This definition is moti-

vated by a simple power counting near the unitary fixedpoint �� ���=7→ ��3�=−2→ ��3

R�=0�. The RHS of Eq.�35� is a quadratic polynomial in �3

R with constant coeffi-cients. This type of equation is discussed in Appendix D. Thebehavior of the solution is governed by the sign of the dis-criminant D of the quadratic polynomial on the RHS of Eq.�35�:

�a� D�0, fixed-point solution.�b� D=0, see Appendix D.�c� D�0, periodic limit cycle solutions with a period

T= 2��−D

.

The discriminant is given by

D = 4�1 −B�3 + p�

�3��2

−16AC�3 + p�2

3�2 . �36�

In the special case of the systems I–III the solution in thepointlike approximation is summarized in Table II. For sys-tems II and III�a� we find the solution with a fixed point withvanishing anomalous dimension �=0 in the IR �see Appen-dix D�. For systems I and III�b� the situation is completely

different. We obtain a periodic limit cycle solution of theform �3

R�t� tan�Tt�. The intuitive interpretation of this solu-tion is that during the RG flow we hit three-body diatom-atom bound states, which manifest themselves as diver-gences of �3

R. In the unitary limit there are infinitely many ofthese bound states, which are equidistant in a logarithmicscale. The continuous scaling symmetry is broken to the dis-crete Z group. This is the well-known Efimov effect �3,4�,which indeed is present for equivalent bosons �system I� andis absent in the case of SU�2� fermions. In the case ofequivalent bosons �system I� the Efimov result is

En+1

En= exp�−

2�

s0� , �37�

with En+1 and En denoting neighboring bound-state energies.The Efimov parameter s0 is given by the solution of a tran-scendental equation and one finds s0�1.006 �4�. By dimen-sional arguments we can connect the artificial sliding scale k2

with the scattering energy E as Ek2 �21�. The proportion-ality factor disappears in the ratio of the energies and hencethe Efimov parameter can be read off from the RG period:

kn+12

kn2 =

En+1

En= exp�− 2T� ⇒ s0 =

�

T. �38�

The values of the Efimov parameter for systems I and III�b�can be found in Table II. We obtain s01.393, which differsfrom the correct result by 40%. In Secs. VII and VIII wedemonstrate that the simple pointlike approximation is toocrude to get the correct quantitative agreement. Neverthelessit provides us with the first hint of how the Efimov effectappears also in the functional renormalization-group frame-work.

VII. THREE-BODY SECTOR: SYSTEMS I AND II

In this section we only discuss systems I and II, leavingthe analysis of system III to Sec. VIII. It turns out that inthese two cases flow equation �31� for E=0 can be formallysolved exactly. For two-component fermions this was shownby Diehl et al. in �13�. To find the exact solution most easilywe perform the redefinition

f t�t1,t2,E� � 4�3 + p�q1q2�3�q1,q2,E� ,

g�t1,t2� � 4�3 + p�q1q2G�q1,q2� , �39�

where, from now on, we prefer to work with logarithms ofmomenta t1=ln�q1 /�� and t2=ln�q2 /��. As before p= +1�p=−1� in the case of bosons �fermions�. The RG scale de-pendence of the reduced atom-diatom vertex f t�t1 , t2 ,E� isdenoted by the subscript t. It is important to stress that we aregenerally interested in the solution of Eq. �31� for the scat-tering of particles of nonzero energy E. Nevertheless, weobserve that the energy E cuts off the RG flow in Eq. �31� ina similar way as regulator �27�. With this relation betweenk2 and E in mind, the coupling for k�0 and E=0 imitatesthe effect of a nonzero energy of the scattering particles, i.e.,k=0 and E�0.

TABLE II. Discriminant D, temporal RG period T �if appli-cable�, and Efimov parameter s0 �if applicable� in the pointlike ap-proximation for systems I, II, III�a�, and III�b�.

Model D T s0

System I −7.762 2.255 1.393

System II 9.881

System III�a� 9.881

System III�b� −7.762 2.255 1.393

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The flow equation at vanishing energy E=0 now reads

�t f t�t1,t2� = −1

�3��Ag�t1,t�g�t,t2� +

B

2�f t�t1,t�g�t,t2�

+ g�t1,t�f t�t,t2�� + Cft�t1,t�f t�t,t2�� . �40�

We assume that in the UV the reduced atom-diatom 1PI ver-tex is vanishing, i.e., the initial condition is f t=0�t1 , t2�=0.Ingeneral we are dealing with the Riccati differential equa-tion in matrix form, where both matrices g and f t have acontinuous index running in the interval t1 , t2� �− ,0�. TheRHS of Eq. �40� is a complete square, which is a specialfeature of systems I and II �see Table I�. In order to find theformal solution of Eq. �40� we define

f̄ t�t1,t2� = pft�t1,t2� + g�t1,t2� , �41�

which can be recognized as the reduced, fully connectedatom-diatom vertex. The flow equation for the full vertex

f̄ t�t1 , t2� with the initial condition takes the simple form

�t f̄ t�t1,t2� = −p

�3�f̄ t�t1,t� f̄ t�t,t2�, f̄ t=0�t1,t2� = g�t1,t2� .

�42�

It is convenient to rewrite Eq. �42� in matrix notation

� f̄ t�t1 , t2�→ f̄ t�:

�t f̄ t = −p

�3�f̄ t · At · f̄ t, f̄ t=0 = g , �43�

where At has matrix elements At�t1 , t2�=��t− t1���t− t2� andmatrix multiplication denotes t integration. Multiplying both

sides of Eq. �43� from the left and right by f̄ t−1, we obtain

�t f̄ t−1 = − f̄ t

−1 · �t f̄ t · f̄ t−1 =

p�3�

At, f̄ t=0−1 = g−1, �44�

which is formally solved by

f̄ t = �I +p

�3��

0

t

dsg · As�−1

· g �45�

for t� �− ,0�. I denotes the identity matrix.In the IR limit t→−, which corresponds to integration

of all quantum fluctuations, f̄ � f̄ t=− solves the followingmatrix equation:

f̄ = g +p

�3�g · f̄ . �46�

This is the well-known Skorniakov-Ter-Martirosian integralequation for bosons �p= +1� and fermions �p=−1� for thehalf-off-shell, amputated, connected Green’s function9 �3�.

The difference in the signs in Eq. �45� between systems Iand II turns out to be crucial. In order to see that, we solve

Eq. �45� numerically by discretization. A series of cartoonsof the evolution of the reduced 1PI vertex f t�t1 , t2� for bothsystems is shown in Fig. 4. For fermions, first a peak appearsin the UV �t1=0 , t2=0�, which propagates in the diagonaldirection �t1= t2� during the RG evolution. On the other hand,for bosons, a periodic structure �with period Tspatial�6.2 inboth directions� develops gradually. Now it is clear why theapproximation investigated in Sec. VI failed to give thequantitatively correct result. The pointlike approximation,which corresponds to a planar landscape �no t1 and t2 depen-dence; see Sec. VI�, is not valid in the three-body sector �formore details see Appendix C�. The evolution in the RG timet of the UV point f t�t1=0 , t2=0� for both systems is depictedin Fig. 5. While for fermions the evolution is monotonic intime, in the case of bosons we obtain a “temporal” oscillationof period Ttemp�3.1. For different points in the t1-t2 planethe “time” evolution is triggered at the scale tinO�t1 , t2�.

The numerical solution for bosons is consistent with theresults of �8,9�. Spatial and temporal oscillations are corre-lated. As found in �8,9� evolution in the RG time develops alimit cycle behavior. The Efimov parameter s0 can be calcu-lated as s0= �

Ttemp�1.0 �see Sec. VI�, which is in good agree-

ment with the Efimov result s0�1.006 24. The accuracy ofour result is limited by the numerical procedure only.

VIII. THREE-BODY SECTOR: SYSTEM III

As introduced in Sec. V for SU�3� fermions there are twospecific situations �system III�a� and system III�b�� whenthere is a single flow equation instead of the general two.Fortunately, both cases can formally be solved for E=0 in asimilar fashion compared to Sec. VII. In fact, system III�a� iscompletely equivalent to system II �see Table I� such that weobtain a fixed-point solution in this case �see Fig. 5�. Forsystem III�a� we follow similar steps as in Sec. VII: we de-fine a reduced atom-diatom 1PI vertex f t�t1 , t2 ,E� �Eq. �39��and obtain a flow equation for reduced vertex �40� with thecoefficients A=4, B=4, and C=1. These coefficients form acomplete square and thus it is useful to define the fully con-nected atom-diatom vertex:

f̄ t�t1,t2� = f t�t1,t2� + 2g�t1,t2� . �47�

Flow equation �40� now reads

�t f̄ t�t1,t2� = −1

�3�f̄ t�t1,t� f̄ t�t,t2�, f̄ t=0�t1,t2� = 2g�t1,t2� .

�48�

The equation and the initial condition are identical to Eq.�42� for bosons.10 For this reason we expect the appearanceof the Efimov effect for the SU�3� fermionic system III�b�with the Efimov parameter s01.006 24.

9Up to our redefinition �40�.

10The initial condition for SU�3� fermions is f̄ t=0�t1 , t2�=2g�t1 , t2�, while for bosons one has f̄ t=0�t1 , t2�=g�t1 , t2�. However,for bosons g�t1 , t2� is two times larger than for fermions �Eq. �40��.Thus the initial conditions are identical.

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At first sight it seems surprising that both bosons andSU�3� fermions have identical Efimov parameter s0. Asan explanation, we propose a simple possible quantum-mechanical argument: in order to find a bound-state spectrumfor SU�3� fermions, one must solve the three-body

Schrödinger equation. The total wave function must be to-tally antisymmetric for fermions. We can achieve this by tak-ing the total wave function as the product of a totally anti-symmetric flavor part ��ijk i� j� k�� times a totally symmetricorbital part. Hence the orbital part has the same symmetry

(a1) (a2)

t1 t1t2 t2

ft ft

1020

3040

50

10

20

30

40

50

-10

0

1020

3040

1020

3040

50

10

20

30

40

50

-10-505

10

1020

3040

(a3) (a4)

t1 t1t2 t2

ft ft

1020

3040

50

10

20

30

40

50

-100-500

50100

1020

3040

1020

3040

50

10

20

30

40

50

-50-250

2550

1020

3040

(b1) (b2)

t1 t1t2 t2

ft ft

1020

3040

50

10

20

30

40

50

00.20.40.60.8

1020

3040

1020

3040

50

10

20

30

40

50

00.20.40.60.8

1020

3040

(b3) (b4)

t1 t1t2 t2

ft ft

1020

3040

50

10

20

30

40

50

00.20.40.60.8

1020

3040

1020

3040

50

10

20

30

40

50

00.20.40.60.8

1020

3040

FIG. 4. �Color online� The RG evolution of the momentum-dependent modified vertex f t�t1 , t2�=4�3+ p�q1q2�3�q1 ,q2 ,E� for bosons�a1–a4� and SU�2� fermions �b1–b4�. Spatial momenta t1 and t2 and the RG time t are discretized to N=50 intervals with a step �t=0.4. Thecartoons for bosons and fermions correspond to the discretized steps 10, 25, 35, and 50.

EFIMOV EFFECT FROM FUNCTIONAL RENORMALIZATION PHYSICAL REVIEW A 79, 042705 �2009�

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property as the bosonic case. Only the orbital part is neededfor the quantum-mechanical calculation of the bound-stateproblem, which leads to the identical Efimov parameters forbosons and SU�3� fermions.

IX. CONCLUSIONS AND OUTLOOK

This paper applies the method of functional renormaliza-tion to the few-body physics of atoms near a Feshbach reso-nance. We investigate three different systems, namely, iden-tical bosons and two and three species of fermions. The two-body sector is solved exactly. The unitarity limit is governedby a fixed point and all three systems seem to be examplesof the nonrelativistic conformal field theories. In the three-body sector, however, no infrared fixed point exists forbosons and three-component fermions. We solve themomentum-dependent problem of the three-body sector atunitarity. This leads to the Skorniakov–Ter-Martirosian equa-tion, well known from quantum mechanics. A numerical so-lution for U�1� bosons and SU�3� fermions shows the emer-gence of the Efimov effect, the appearance of an infinitegeometric spectrum of triatom states. Hence in these systemsthe continuous scaling symmetry is broken to the discretescaling subgroup Z by a quantum anomaly. Therenormalization-group flow develops a limit cycle behavior�see Fig. 5�. The Efimov parameter s0 for the three-component fermions is found to be identical to the Efimovparameter of the well-studied bosonic case, which agreeswith the quantum-mechanical prediction.

The current work can be extended in various ways: onecan go away from unitarity in the three-body sector and de-rive universal properties such as recombination rates and thepositions of diatom-triatom thresholds. Our technique allowsus to investigate equilibrium states with nonzero density andtemperature. This can be achieved by simply changing thechemical potential and introducing the temperature by re-placing the � integrals by the discrete sums of the Matsubaraformalism. In that case the effective approximate descriptionof the models in terms of various composite fields �e.g., tri-ons and density bosons� would be very useful due to a largereduction of the numerical effort. The description of asimple, but efficient effective theory is summarized in Ap-pendix E. The excellent agreement of the vacuum solutionwith high-precision quantum-mechanical computations pro-vides a robust starting point for the investigation of themany-body system of nonzero density and temperature.

ACKNOWLEDGMENTS

S.M. is especially grateful to S. Diehl for enlighteningdiscussions and important suggestions. We acknowledge thediscussions with J. M. Pawlowski, H. Gies, H. C. Krahl, M.Scherer, S. Jochim, T. B. Ottenstein, T. Lompe, M. Kohnen,and A. N. Wenz. We are indebted to J. Hosek and H.-W.Hammer for providing important remarks.

APPENDIX A: GALILEAN AND NONRELATIVISTICCONFORMAL SYMMETRY

All systems we consider in this paper �Eqs. �3�–�5�� havea centrally extended Galilean space-time symmetry.11 Thecentrally extended Galilean algebra consists of 11 generators:particle number N �central charge�, time translation H, threespatial translations Pi, three spatial rotations Mij, and threeGalilean boosts Ki. The nontrivial commutators are �in thereal-time formalism�

�Mij,Mkl� = i��ikMjl − � jkMil + �ilMkj − � jlMki� , �A1�

�Mij,Kk� = i��ikKj − � jkKi�, �Mij,Pk� = i��ikPj − � jkPi� ,

�A2�

�Pi,Kj� = − i�ijN, �H,Kj� = − iPj . �A3�

In the case of a free nonrelativistic field theory the group ofspace-time symmetries is in fact larger than the Galileangroup �40,41� and is called the Schrödinger group.12 For thedynamical exponent z=2 there are two additional generators:the scaling generator D and the special conformal generatorC. The scale symmetry acts on the time and spatial coordi-nates according to

�xi,t� → �xi�,t�� = ��xi,�2t� , �A4�

where � is a scale parameter. A special conformal transfor-mation on time and spatial coordinates is given by �40�

�xi,t� → �xi�,t�� = � xi

1 − ct,

t

1 − ct� , �A5�

where c is a parameter of the special conformal transforma-tion. The additional nontrivial commutators of theSchrödinger algebra are

11This is the nonrelativistic analog of the Poincare group in rela-tivistic QFT.

12This is the nonrelativistic counterpart of the conformal group.

(A)

ft

t (B)

ft

t�8 �6 �4 �2 0

�100

�50

0

50

100

�4 �3 �2 �1 00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7FIG. 5. �Color online� Numeri-

cal evolution in the RG time t off t�t1=0, t2=0� for �A� system Iand �B� system II. For SU�2� fer-mions �panel �B�� the modifiedvertex approaches a fixed-pointsolution. In the case of bosons�panel �A��, a limit cycle behavioris developed with a period Ttemp

�3.1.

MOROZ et al. PHYSICAL REVIEW A 79, 042705 �2009�

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�Pi,D� = − iPi, �Pi,C� = − iKi, �Ki,D� = iKi, �A6�

�D,C� = − 2iC, �D,H� = 2iH, �C,H� = iD . �A7�

It is important to note that besides the free theory there arefew known examples of interacting theories which are sym-metric with respect to the Schrödinger group. These theoriesare called nonrelativistic conformal field theories �NRCFTs�and SU�2� nonrelativistic fermions at unitarity �system II�are believed to constitute one of them.

In analogy to relativistic conformal field theories it is pos-sible to introduce primary operators in an NRCFT �39�. Alocal primary operator O�t ,x� has a well-defined scaling di-mension �O and particle number NO:

�D,O� = i�OO, �N,O� = NOO , �A8�

where O�O�t=0,x=0�. The primary operator O also com-mutes with Ki and C:

�Ki,O� = 0, �C,O� = 0. �A9�

It is possible to show that the operators, constructed by tak-ing spatial and time derivatives of a primary operator O,form an irreducible representation of the Schrödinger group.Similar to the relativistic case the form of the two-bodyGreen’s function of the primary operators is fixed by theconformal symmetry �in the imaginary-time formalism� �39�:

�OO†� �i�̂ +q2

2MNO��

, �A10�

where �=�O−5 /2 for d=3. The simplest examples of pri-mary operators in the theory of SU�2�-symmetric fermionsare the atom operator �N=1,�=3 /2� and the diatomoperator �N =2,� =2�. The form of the inverse propaga-tors at unitarity, which we found to be given by Eq. �24�, isconsistent with Eq. �A10�.

APPENDIX B: COMPLETION OF THE VERTEXEXPANSION TO �k

(4)

In this appendix we complete the vertex expansion tofourth order and argue that our truncations �3�–�5� are suffi-cient to perform exact calculations for the three-body sector.At fourth order in the fields there are only two more vertices,which are compatible with the internal symmetries of theconsidered models:

��4� =1

2�

Q1,. . .,Q4

��Q1,Q2,Q3�†�Q1��Q2�†�Q3��Q4�

���− Q1 + Q2 − Q3 + Q4� ,

��4� =1

2�

Q1,. . .,Q4

� �Q1,Q2,Q3� †�Q1� �Q2� †�Q3� �Q4�

���− Q1 + Q2 − Q3 + Q4� . �B1�

In the two-channel model considered in this paper, wechoose the initial UV value of the vertex � to be zero, �

=0. This means that the interaction between the atoms is

described at the microscopic level by the exchange of diatomstates. With �=0 at the UV scale this coupling is not regen-erated by the flow in vacuum. The one-loop diagrams con-tributing to the flow have inner lines pointing in the samedirection with respect to the loop momentum and thereforevanish in vacuum �see Sec. III�. Thus �=0 is a fixed point.The flow away from this fixed point has been studied for thesystem with two species of fermions in �34�.

The 1PI vertex � belongs to the four-body sector �fordefinition of the n-body sector see Sec. III� and it decouplesfrom the flow equations of the two- and three-body sectorsdue to the vacuum hierarchy �for more details see Sec. III�.Thus our truncations �3�–�5� are sufficient to obtain the exactvacuum physics of the three-body sector.

APPENDIX C: BOUND-STATE APPROXIMATIONAND SEPARABLE POTENTIAL

In this appendix we present an alternative solution of thetwo-body sector using the Lippmann-Schwinger equation ofquantum mechanics, which helps to elucidate the efficiencyof the two-channel model and the limitations of the trionapproximation in �21�.

The one-channel model provides an alternative descrip-tion of ultracold atoms near a broad Feshbach resonance.This model contains the atom field only and the micro-scopic action is given by13

�t=0 = �Q

��Q��i� + q2��Q�

+�

2�

Q1,. . .,Q4

��Q1��Q2���Q3��Q4�

���− Q1 + Q2 − Q3 + Q4� , �C1�

where � is a pointlike four-atom interaction which is relatedto the s-wave scattering length in the ir. Roughly speaking,the quantum-mechanical atom-atom interaction potential ofone-channel model �C1� in Minkowski space is given by14

V�x� =�

2��3��x� . �C2�

Let us now perform a Fourier transformation of this poten-tial:

V�k,k�� =� d3r exp�− i�k� − k�r�V�r� =�

2. �C3�

At this point two important remarks about potential �C3� canbe made:

13For simplicity, we present the one-channel model for U�1�bosons at unitarity only. However, our arguments can be extendedaway from unitarity and are applicable to both SU�2� and SU�3�fermion systems.

14Strictly speaking, the contact interaction is ill defined and mustbe regularized. This can be done by introducing the pseudopotentialV�r��r�=���3��r� �

�r �r�r��. Here we use an alternative regular-ization by introducing a momentum cutoff � directly into theLippmann-Schwinger equation.

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�a� V�k ,k�� is a separable potential because it can be writ-ten in the form

�

2 U�k�U�k��.�b� V�k ,k�� is k and k� independent, i.e., U�k�=U�k��

=1.We investigate the atom-atom scattering in the center-of-

mass frame. The Lippmann-Schwinger integral equation forthe K matrix is �42�

K�k,k�,E� = V�k,k�� + P�� d3q

�2��3

V�k,q�K�q,k�,E�E − 2q2 ,

�C4�

where P denotes the Cauchy principle value and � is a mo-mentum cutoff, which regularizes the contact interaction.The K matrix is similar to the T matrix but uses a standing-wave boundary condition which leads to the principal-valueprescription in Eq. �C4�. The kinematics of K�k ,k� ,E� issimilar to the kinematics depicted in Fig. 2. Integral equation�C4� can be easily solved in the special case of a separablepotential. The solution factorizes

K�k,k�,E� = −U�k�U�k��

D�E�, �C5�

where D�E� is given by

D�E� = −2

�

+ P�� d3q

�2��3

U2�q�E − 2q2 . �C6�

In the special case of the contact interaction, Eq. �C5� de-pends only on E. This means that the exact atom-atom scat-tering amplitude in the center-of-mass frame can be rewrittenin terms of the exchange of the composite diatom with in-verse propagator P��E ,p=0�D�E�. For the contact inter-action D�E� is given by

D�E� = −2

�

+ P�� dq

2�2

q2

E − 2q2 = −2

�

−�

4�2

+E

8�2P�� dq

q2 − E/2. �C7�

The microscopic � can be adjusted such that

2

�

+�

4�2 a−1. �C8�

At unitarity the first two terms in the second equation in Eq.�C7� cancel. The last integral is convergent; hence we take�→. By dimensional analysis we obtain

D�E� �E . �C9�

To summarize, the atom-atom scattering amplitude is mo-mentum independent in the center-of-mass frame. Hence thetwo-body sector can be solved exactly by introducing a dia-tom exchange in the s-channel. By Galilean symmetry thisresult can be extended to a general reference frame:

D�E,k� �E −k2

4M

�C10�

The functional form of the inverse diatom propagator is con-sistent with our findings �Eq. �24�� in Sec. IV.

In the three-body channel the atom-diatom interacting po-tential is momentum dependent in the center-of-mass frame.The momentum dependence is generated by the box diagram�see first diagram on RHS of Fig. 3�. For this reason the trionapproximation, which we used in �21�, and in particular thepointlike approximation �Sec. VI� do not fully capture thismomentum dependence and is not as efficient as the “diatomtrick.” It leads to the quantitative inaccuracy of the Efimovparameter s0.

APPENDIX D: ANALYSIS OF ddt f(t)=�f(t)2+�f(t)+�

In this appendix we perform an analysis of the differentialequation, which we encountered in the calculation of thethree-body sector in the pointlike approximation:

d

dtf�t� = �f�t�2 + �f�t� + �, f�t0� = f , �D1�

with � , � , ��R. The form of the solution is determinedby the sign of the discriminant of the � function D��2

−4��. There are three different cases �without loss of gen-erality we take ��0, which is the case for systems I–III�:

�1� D�0: In this case the � function has two fixed pointsf1 �IR stable� and f2 �IR unstable� with f1� f2 �see Fig. 6�.For the initial condition f � f1 the solution is attracted to thefixed point f1 in the IR. If the initial condition is f � f2, thesolution is repelled from the fixed point f2 and is driventoward a Landau pole. Formally, we can extend the solutionbeyond the Landau pole. For an initial condition f1� f � f2the solution is attracted by the IR fixed point f1 and has theform

f�t� =

− � + �D tanh��D

2�t + ���

2�, �D2�

with � fixed by the initial condition.�2� D=0: In this case there is one fixed point and we have

four distinct subcases:�a� �=�=�=0⇒ A trivial solution f�t�= f is obtained.�b� �=�=0 and ��0⇒ We have a linear solution f�t�

= f +��t− t0�.

FIG. 6. � function of RG equation �D1� for �A� D�0 and �B�D�0. The arrows show the direction of the RG flow toward the IR.

MOROZ et al. PHYSICAL REVIEW A 79, 042705 �2009�

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�c� ��0 and �=�=0⇒ We obtain the Landau pole solu-tion

f�t� =f

1 + f��t − t0�. �D3�

This solution can be formally extended beyond the Landaupole.

�d� ��0, ��0, and ��0⇒ This case can be put in theform of case �c� by a transformation f → f +� �

� .�3� D�0: In this case there are no fixed points. The for-

mal solution can be written as

f�t� =

− � + �− Dtan��− D

2�t + ���

2�, �D4�

where � is fixed by the initial condition. This solution isperiodic with a period T= 2�

�−D.

It is important to mention that the discriminant D isinvariant under the multiplicative reparametrization f�t�→�f�t� with some constant �. This means that both theclassification and the period T are not sensitive to the multi-plicative reparametrization of �3

R in Sec. VI.

APPENDIX E: BOSONIZATION AND FERMIONIZATION

It was shown in Secs. VII and VIII that for a quantitativeprecise description of the three-body physics the momentumdependence of the vertex �3 is quite important. Although itwas eventually possible to take this momentum dependenceinto account, we had to pay a price for it. Solving the flowequation numerically while taking the full s-wave-projectedmomentum dependence of the vertices into account needs arelatively large numerical effort. While this is still manage-able in the vacuum where both the density and the tempera-ture vanish, the numerical cost would be significantly largerin the more general case of nonzero density or temperature.It is therefore reasonable to look for an effective approximatedescription that is numerically less expensive but neverthe-less leads to good numerical precision.

For this task it is crucial to find a simple way to take atleast the qualitative features of the momentum dependenceinto account. For simplicity we concentrate this discussionon system III only. Let us start with the discussion of �3b.From Eq. �5� we can read off that it corresponds to a channelwhere the fermionic and bosonic spins are contracted sepa-rately,

− �3bi�i j

� j . �E1�

One might describe this vertex by the exchange of a realtwo-component boson field �= �� ,� � which couples to thecomposite operators † and † with some Yukawa-typeinteractions. More explicitly, we could use the following ac-tion:

�� = �Q

��Q�P��Q�� �− Q� − �Q1,Q2

h��Q1,Q2�

�†�− Q1��Q2���Q1 + Q2� − �Q1,Q2

h� �Q1,Q2�

� †�− Q1� �Q2�� �Q1 + Q2� . �E2�

Since � is a real boson, its propagator fulfills P��Q�= P��−Q�. Together with Galilean invariance this implies thatP� does not have any frequency dependence P��Q�= P��q��43�. The exchange of a � boson corresponds to an instanta-neous interaction. Note, however, that Galilean invariance isbroken spontaneously by a condensate or a Fermi surface atnonzero density. In that case the � boson becomes dynamicaland corresponds to a propagating phonon. An expectationvalue of � corresponds to a shift in the effective chemicalpotential �31�.

In Eq. �E2� the field � appears quadratic and we caneliminate it by solving its field equation. This results in

�� = − �Q1,. . .,Q4

h��Q1,Q2�h� �Q3,Q4�P��Q1 + Q2�

†�− Q1��Q2�

� †�− Q3� �Q4���Q1 + Q2 + Q3 + Q4� .

We observe that we get a tree-level contribution that has thespin structure of the term �3b. Assuming Yukawa couplingsthat are independent of the momenta and using the conven-tions of Sec. V, the contribution to �3b reads

�3b,�-exchange�p1,p2;E� =h�h�

P��p1 − p2�. �E3�

For an inverse propagator of the form P��q�=m�2 +q2 we find

after the s-wave projection

�3b,�-exchange�p1,p2;E� =h�h�

4p1p2ln� p1

2 + p22 + 2p1p2 + m�

2

p12 + p2

2 − 2p1p2 + m�2 � .

�E4�

The parameters m�2 and h� , h� can be chosen such that the

form of �3b is resembled closely. One can compare this to thetree-level contribution to �3b by the exchange of a fermion .It is obtained from Eq. �5� by solving the field equation for and while using the fact that the propagator for and theYukawa coupling h are not renormalized:

�3b,-exchange�p1,p2;E� =2h2

p12 + p2

2 + �p1 + p2�2 − E − �

.

�E5�After s-wave projection, this reads

�3b,-exchange�p1,p2;E�

=h2

2p1p2ln� p1

2 + p22 + p1p2 − �� + E�/2

p12 + p2

2 − p1p2 − �� + E�/2� . �E6�

One can see that the functional form of the two tree-levelcontributions after s-wave projection is quite similar. This isalso the form of the momentum dependence found in the

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numerical solution of the flow equation for �3b without the �boson �Secs. VII and VIII�. We therefore expect that thedescription of �3b as the exchange of a � boson withmomentum-independent Yukawa couplings h� and h�

leadsto results that are comparable to the inclusion of the full�s-wave-projected� momentum dependence. However, thisdescription would be much more efficient with respect to thenumerical effort. For the translation between the descriptionused in the main part of this paper, where �3b is included asits own vertex and the description of �3b in terms of theexchange of a � boson, one might use the method of re-bosonization �44�.

The vertex �3a in Eq. �5� can also be described by theexchange of some particle, which corresponds in this case toa bound state of three atoms, the trimer or trion �21�. Al-though the vertex �3b and the momentum dependence of theYukawa-type couplings were neglected in �21�, the behaviorfound there was already qualitatively correct. Why this tria-tom approximation is not sufficient to describe the completemomentum dependence of �3 is discussed in Appendix C.

We conclude that an effective description of the three-body physics with only a few couplings seems possible andwould facilitate the study of systems at nonzero density andtemperature.

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