Renormalization group study of the four-body problem
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Renormalization group study of the four-body problem
Richard Schmidt ∗ and Sergej Moroz † ∗ Physik Department, Technische Universit¨at M¨unchen,James-Franck-Strasse, D-85748 Garching, Germany † Institut f¨ur Theoretische Physik, Universit¨at Heidelberg,Philosophenweg 16, D-69120 Heidelberg, Germany
We perform a renormalization group analysis of the non-relativistic four-boson problem by meansof a simple model with pointlike three- and four-body interactions. We investigate in particular theregion where the scattering length is infinite and all energies are close to the atom threshold. Wefind that the four-body problem behaves truly universally, independent of any four-body parameter.Our findings confirm the recent conjectures of Platter et al. and von Stecher et al. [1–3] that thefour-body problem is universal, now also from a renormalization group perspective. We calculatethe corresponding relations between the four- and three-body bound states, as well as the full boundstate spectrum and comment on the influence of effective range corrections.
PACS numbers: 03.65.Nk, 34.50.-s,11.10.Hi
I. INTRODUCTION
During the last decade few-body physics experienceda renewed interest due to the advent of experiments withultracold atomic gases. Whereas the study of few-bodyphysics in nuclear systems is hindered by the large com-plexity of the interparticle potentials, the interactions inultracold atomic gases are describable to high accuracywith very simple short-range models. In addition, ul-tracold atomic gases become even more attractive as anideal theoretical and experimental playground since theydo not only offer excellent experimental control but alsothe amazing possibility of tuning the two-body interac-tion strength over a wide range using so-called Feshbachresonances [4].This made it possible, that, about forty years after V.Efimov’s seminal prediction [5] of the existence of univer-sal three-body bound states in systems with large two-body interactions, first evidence in favor of the presenceof these states had been found in the remarkable experi-ment by Kraemer et al. in 2006 [6]. In his work, Efimovpredicted the existence of infinitely many trimer statesfor infinitely large scattering length where the two-bodyinteraction is just on the verge of having a bound state.The energy levels form a geometric spectrum and thethree-body system is found to be universal in the sensethat apart from the s-wave scattering length a only onepiece of information about the three-body system entersin the form of a so-called three-body parameter [7]. Thefindings of Kraemer et al. stimulated extensive activityin the field of three-body physics, both experimentally[8–12] and theoretically; for recent reviews on also thelatter see [7, 13] and references therein. As a result, theEfimov effect in three-body systems is a well-understoodphenomenon today. ∗ Electronic address: [email protected] † Electronic address: [email protected]
The next natural step is to raise the question what thephysics of four interacting particles may be. Early at-tempts towards an understanding of this system weremade in the context of nuclear physics using a varietyof approaches [14–17]. Also the four-body physics of He atoms has been investigated in much detail, for anoverview see, e. g. [18]. The simpler four-body physicsof fermions with two spin states, relevant for the dimer-dimer repulsion, has also been studied [19].In their pioneering work, Platter and Hammer, et al.[1, 2] investigated the four-body problem using effectiveinteraction potentials and made the conjecture that thefour-boson system exhibits universal behavior. They alsofound that no four-body parameter is needed for a self-consistent renormalization of the theory. Calculating theenergy spectrum of the lowest bound states in depen-dence on the scattering length a the existence of twotetramer (four-body bound) states associated with eachtrimer was conjectured.Recently, von Stecher, D’Incao, and Greene [3, 20] in-vestigated the four-body problem in a remarkable quan-tum mechanical calculation. They found that the Efimovtrimer and tetramer states always appear as sets of stateswith two tetramers associated with each of the trimer lev-els and calculated the bound state energy spectrum of thelowest few sets of states. The calculation suggests thatthe energy levels within one set of states are related toeach other by universal ratios, which were obtained fromthe behavior of these lowest sets of states. In accordancewith the results of Platter et al. [1, 2] the absence ofany four-body parameter was also demonstrated. In or-der to find experimental evidence of the tetramer statesextremely precise measurements are required. Remark-ably, Ferlaino et al. were able to observe signatures of thelowest two of the tetramer states in a recent experiment[21].While the calculations by Platter et al. [1, 2] and vonStecher et al. [3, 20] rely on quantum mechanical ap-proaches, in this work we want to shed light onto thefour-body problem from a different perspective. A lot of FIG. 1: (Color online) The generalized Efimov plot for fouridentical bosons. Here, we plot the energy levels of the vari-ous bound states as a function of the inverse s-wave scatteringlength a as numerically calculated in our approximative, ef-fective theory. In order to improve the visibility of the energylevels we rescale both the dimensionless energy E/ Λ and thedimensionless inverse scattering length a − / Λ where Λ de-notes the UV cutoff of our model. Also, we only show thefirst three sets of Efimov levels. The solid black line denotesthe atom-atom-dimer threshold, while the dotted black linegives the dimer-dimer threshold. In the three-body sectorone finds the well known spectrum of infinitely many Efimovtrimer states (green, dashed) which accumulate at the unitar-ity point E ψ = a − = 0, indicated by the orange star. In ourpointlike approximation the four-body sector features a singletetramer (solid, red) associated with each trimer state. insight into the three-body problem had been gained fromeffective field theory and renormalization group (RG)methods [7, 13, 22, 23] and it is desirable to apply thesealso to the four-body problem. In this paper we will makea first step towards such a description that is complemen-tary to the previous quantum mechanical approaches.Of special interest is the further investigation of uni-versality in the four-body system. In this context theso-called unitarity point, illustrated by the star in Fig.1, is of particular importance. In this limit not only thescattering length a is infinite but also all binding energiesin the problem accumulate at the atom threshold at zeroenergy. Only at the unitarity point physics becomes trulyuniversal in the sense that for example the ratio betweenthe binding energies of consecutive trimer levels assumesexactly its universal value, E n +1 /E n = exp( − π/s ),with s ≈ . ) three- andfour-body interactions. In the three-body problem uni-versality manifests itself in an RG limit cycle of the three-body coupling. We find that this three-body limit cycleleads in turn to a “self-sustained” limit cycle of the four-body sector leaving no room for any four-body parame-ter.The RG method allows furthermore for computationsaway from the unitarity point. We calculate the boundstate energy spectrum in the pointlike approximation(see Fig. 1) and investigate how the relations betweentetramer and trimer states approach the universal limitas one comes closer to the unitarity point.The paper is structured as follows. In Sec. II weintroduce the functional renormalization group (FRG)method and set up the microscopic model. Secs. IIIand IV are devoted to the FRG analysis of the two- andthree-body sector. In Sec. V we discuss the four-bodysector and present our numerical results. Our findingsare summarized in Sec. VI. II. METHOD AND DEFINITION OF THEMODEL
In this work we are interested in the computation ofthe few-body properties, such as the bound state spec-trum, of four identical bosons. In a quantum field theoryapproach the information about these properties can beextracted from the effective action Γ which is the generat-ing functional of one-particle irreducible vertex functionsΓ ( n ) and which contains all information about a givensystem. The computation of Γ is a very complicated taskas quantum (and, as in the case of nonzero density, statis-tical) fluctuations have to be integrated out on all lengthand therefore momentum scales q . In order to cope withthis task we rely on the functional renormalization group[24], for detailed reviews we refer to [25, 26].The central quantity of the FRG is a scale dependenteffective action functional, the so-called effective flowingaction Γ k . The effective flowing action Γ k , which in-cludes all fluctuations with momenta q & k , interpolatesbetween the classical action S at some ultraviolet (UV)cutoff scale k = Λ and the full quantum effective actionΓ in the limit k →
0. The underlying idea is similarto Wilson’s idea of momentum shell-wise integration offluctuations. The evolution of Γ k is governed by the Wet-terich equation [24], which is an exact, non-perturbativeRG equation. It reads ∂ k Γ k = 12 Tr (cid:16) Γ (2) k + R k (cid:17) − ∂ k R k , (1) Throughout the paper we will use the term momentum inde-pendence to refer to combined spatial momentum and frequencyindependence. where Γ (2) k is the flowing, full inverse propagator andthe trace Tr sums over momentum ~q and Matsubara fre-quency q as well as the internal degrees of freedom suchas species of fields. The dependence on the RG scale k isintroduced by the regulator R k . At the UV scale k = Λthe effective flowing action Γ k equals the classical action S and as we want to consider dilute atomic gases the UVscale Λ is set to be of the order of the inverse Bohr radius a − . For most problems, quantum and statistical fluctu-ations will generate infinitely many terms in Γ k . Dueto this fact it is in practice impossible to solve Eq. (1)exactly. Therefore one has to decide for a truncation ofΓ k , which in turn corresponds to solving the theory onlyapproximately.In this work we investigate the four-boson problem byapproximately solving Eq. (1). Our truncation for theEuclidean flowing action is given by a simple two-channelmodelΓ k = Z x { ψ ∗ ( ∂ τ − ∆ + E ψ ) ψ + φ ∗ (cid:18) A φ ( ∂ τ − ∆2 ) + m φ (cid:19) φ + h φ ∗ ψψ + φψ ∗ ψ ∗ )+ λ AD φ ∗ ψ ∗ φψ + λ φ ( φ ∗ φ ) + β ( φ ∗ φ ∗ φψψ + φφφ ∗ ψ ∗ ψ ∗ ) + γφ ∗ ψ ∗ ψ ∗ φψψ } , (2)where ∆ denotes the Laplace operator and we use thenatural, non-relativistic convention 2 M = ~ = 1 withthe atom mass M . ψ denotes the field of the elementarybosonic atom, while the dimer, the bosonic bound stateconsisting of two elementary atoms, is represented by thefield φ ∼ ψψ . Both the atom and the dimer field are sup-plemented with non-relativistic propagators with energygaps E ψ and m φ , respectively. In our approximation thefundamental four-boson interaction ∼ λ ψ ( ψ ∗ ψ ) is medi-ated by a dimer exchange, which yields λ ψ = − h /m φ inthe limit of pointlike two-body interactions. The dynam-ical dimer field φ allows us to capture essential detailsof the momentum dependence of the two-body interac-tion. We introduce a wave function renormalization fac-tor A φ for the dimer field in order to take into accountan anomalous dimension of the dimer field φ . The onlynonzero interaction, present at the microscopic UV scale k = Λ, is taken to be the Yukawa-type term with the cou-pling h . Together with the microscopic value of A φ theYukawa interaction h at the UV scale can be connectedto the effective range r eff in an effective range expansion.The atom-dimer interaction λ AD as well as the variousfour-body interactions λ φ , β , and γ vanish at the UVscale and are built up via quantum fluctuations duringthe RG flow.At this stage we want to emphasize the meaning of theterm pointlike approximation which must not be con-fused with the notion of a zero-range (contact) model.Consider for example the two-body contact interaction ∼ λ ψ ( ψ ∗ ψ ) , which has no momentum dependence onthe UV scale, k = Λ. In order to describe the scatteringof two particles in quantum mechanics one proceeds by solving the two-body Schr¨odinger equation. From thisone obtains the well-known result for the zero-range s-wave scattering amplitude f ( p ) = ( − a − − ip ) − (with p = | ~p | denoting the momentum of the colliding par-ticles). The scattering amplitude becomes momentumdependent. In the RG approach one deals with the effec-tive vertex λ ψ which varies with the RG scale k . On theUV ( k = Λ) scale λ ψ is momentum independent. Whenincluding more and more quantum fluctuations – mean-ing lowering the RG scale k from Λ to eventually k = 0– the effective vertex function λ ψ assumes a momentumdependence which in the IR limit k = 0 is equivalent tothe result for f in the zero-range model. In a pointlikeapproximation one ignores this generated momentum de-pendence. In the simple model Eq. (2) the three- andfour-body sector is treated strictly in the pointlike ap-proximation. However, in the two-body sector the mo-mentum dependence of effective vertex λ ψ is capturedby the exchange of the dynamic (i.e. momentum depen-dent) dimer propagator, such that the two-body sector istreated beyond the pointlike approximation.In the general case of nonzero density and temperatureone works in the Matsubara formalism and the integral inEq. (2) sums over homogenous three-dimensional spaceand over imaginary time R x = R d x R /T dτ . Althoughour method allows us to tackle a full, many-body prob-lem at finite temperature in this way, we are interestedsolely in the few-body (vacuum) physics in this paper, forwhich density n and temperature T vanish. For T = 0, R x reduces to an integral over infinite space and time. Ourtruncation (2) is based on the simple structure of the non-relativistic vacuum and, as demonstrated in [23, 27], nu-merous simplifications occur when solving Eq. (1) com-pared with the general, many-body case. The flowing ac-tion (2) has a global U (1) symmetry which correspondsto particle number conservation. In the vacuum limit itis also invariant under spacetime Galilei transformationswhich restricts the form of the non-relativistic propaga-tors to be functions of ∂ τ − ∆ for the atoms and ∂ τ − ∆ / k we must choose a suitable reg-ulator function R k in order to solve Eq. (1). Based onour recent treatment of the closely related three-fermionproblem [28, 29], we choose optimized regulators, R ψ = ( k − q ) θ ( k − q ) ,R φ = A φ k − q ) θ ( k − q ) , (3)with q = | ~q | . These regulators are optimized in the senseof [26, 30] and allow to obtain analytical results. III. TWO-BODY SECTOR
A remarkable and very useful feature of the vacuumflow equations is comprised by a special hierarchy: theflow equations of the N -body sector do not influence therenormalization group flows of the lower N − N -body sectors canbe solved subsequently. In this spirit we first solve thetwo-body sector, then investigate the three-body sectorin order to finally approach the four-body problem.The solution for the two-body sector can be found ana-lytically in our approximation (for the analogous prob-lem considering fermions, see [28, 31]). The only run-ning couplings in the two-body sector are the dimer gap m φ and its wave function renormalization A φ . The flowequations of the two-body sector are shown in terms ofFeynman diagrams in Fig. 2(a) and read ∂ t m φ = h π k ( k + E ψ ) ,∂ t A φ = − h π k ( k + E ψ ) , (4)where t = ln k Λ . As there are no possible nonzero flowdiagrams for the Yukawa coupling h , it does not flow inthe vacuum limit.The infrared (IR) values of the couplings h and m φ canbe related to the low-energy s-wave scattering length a via a = − h ( k = 0)16 πm φ ( k = 0 , E ψ = 0) . (5)Knowing the analytical solution of the two-body sec-tor, this relation can be used to fix the initial values ofour model. For the UV value of the dimer gap m φ wefind m φ (Λ) = − h π a − + h π Λ + 2 E ψ . (6)The first term fixes the s-wave scattering length ac-cording to Eq. (5), while the second term represents acounterterm taking care of the UV renormalization ofthe two-body sector. Finally, the last term accounts forthe fact that the dimer consists of two elementary atoms.Additionally, we choose A φ (Λ) = 1 which correspondsto the effective range r eff = − πh .The action (2) can also be used for a quite accuratedescription of Feshbach resonances. In this context Eq.(2) is referred to as resonance model. In such a model m φ is proportional to the detuning energy of the molecule in Remarkably, the atom inverse propagator (one-body sector) isnot renormalized in the non-relativistic vacuum. the closed channel with respect to the atom-atom thresh-old [32] and the coupling h is proportional to the width ofthe associated Feshbach resonance being a function of thestrength of the coupling to the closed channel. The choice A φ (Λ) = 1 then corresponds to the so-called characteris-tic length r ∗ = − r eff often used in literature [7, 33].In the limit of large, positive scattering length thereexists a universal, weakly bound dimer state. In or-der to find its binding energy we calculate the pole ofthe dimer propagator, corresponding to the condition m φ ( E ψ , k = 0) = 0, which yields in the limit E ψ / Λ ≪ E D = − E ψ = − h π − s h (64 π ) + h a − π ! = − r − r − r eff a ! . (7)In the limit h → ∞ , corresponding to r eff → E D = − /a . The dimerbound state energy is shown as a function of the inversescattering length in Fig. 1 (black solid line). The de-viation from the universal 1 /a scaling for large inversescattering lengths is due to the finite size of h which istaken to be h / Λ = 10 in Fig. 1. In the regime of smallscattering length a one finds a crossover of the behaviorof the dimer binding energy which then has the limitingbehavior E D = 4 / ( ar eff ). IV. THREE-BODY SECTOR
The bound state spectrum of the three-body sector ismuch richer than the one of the two-body system. Inhis seminal papers [5] Efimov showed the existence ofan infinite series of three-body bound states for strongtwo-body interactions. These energy levels exhibit auniversal geometric scaling law as one approaches theunitarity point E ψ = a − = 0. Remarkably, thesethree-body bound trimer states exist even for negativescattering lengths a where no two-body bound state ispresent; they become degenerate with the three-atomthreshold for negative scattering length and merge intothe atom-dimer threshold for positive a . An additionalthree-body parameter is needed in order to determinethe actual positions of the degeneracies [22]. In thissection we want to shortly review how Efimov physicscan be treated within our approach. For a more detailedaccount on that matter and an application to thethree-component Li Fermi gas we refer to [23, 28, 29].In our truncation, the three-body sector contains a sin-gle, pointlike φ ∗ ψ ∗ φψ term with a coupling λ AD , whichis assumed to vanish in the UV. It is build up by quan-tum fluctuations during the RG flow and the correspond-ing Feynman diagrams of the flow equation for λ AD isshown in Fig. 2(b). First we investigate the unitaritypoint, E ψ = a − = 0. For this limit we are able to ob-tain an analytical solution for the flow equation of λ AD while away from unitarity we have to rely on a numericalsolution.At the unitarity point all intrinsic length scales dropout of the problem and the system becomes classicallyscale invariant. At unitarity, the Yukawa coupling h is di-mensionless and the only (extrinsic) length scale presentis the inverse ultraviolet cutoff Λ − , which defines thevalidity limit of our effective theory.In our approximation the dimer field φ develops a largeanomalous dimension η = − ∂ t A φ ¯ A φ = 1 at unitarity whichis consistent with the exact solution of the two-body sec-tor [23, 27]. As the atom and dimer propagators havevanishing gaps in the IR the two-body sector respects acontinuous scaling symmetry.In order to find the solution of the three-body sec-tor we switch to the rescaled, dimensionless coupling˜ λ AD ≡ k h λ AD . One finds that the flow equation for˜ λ AD becomes independent of k and h , ∂ t ˜ λ AD = 2425 (cid:16) − η (cid:17)| {z } a ˜ λ AD − (cid:18) − η (cid:19)| {z } b ˜ λ AD + 2625 (cid:16) − η (cid:17)| {z } c . (8)As was demonstrated in [23, 34], the behavior of the so-lution of this type of flow equation is determined by thesign of the discriminant D of the right hand side of Eq.(8) which is D = b − ac <
0. Eq. (8) can be solvedanalytically and one finds˜ λ AD ( t ) = − b + √− D tan (cid:16) √− D ( t + δ ) (cid:17) a , (9)where δ is connected to the three-body parameter anddetermines the initial condition. Most remarkably, thethree-body sector exhibits a quantum anomaly: The RGflow of the renormalized coupling ˜ λ AD exhibits a limitcycle, which, due to its periodicity, breaks the classicallycontinuous scaling symmetry to the discrete subgroup Z .The Efimov parameter can be determined from the pe-riod of the limit cycle [23] and is given in our approxi-mation by s = √− D ≈ . . (10)The exact result is given by s ≈ . k dependent) in-teraction vertex of the three-body sector, the agreementis quite good. In fact, in previous work [23] we haveshown how to obtain the exact value of s using the FRG. FIG. 2: The flow equations in terms of Feynman diagramsfor the (a) two-body, (b) three-body, and (c) four-body sec-tor. All internal lines denote full, regularized propagators.The scale derivative ˜ ∂ t on the right hand side of the flowequations acts only on the regulators. Solid lines representelementary bosons ψ , while dashed lines denote compositedimers φ . The vertices are: Yukawa coupling h (small blackdot), atom-dimer vertex λ AD (open circle), dimer-dimer cou-pling λ φ (black circle), coupling β (two circles), and the atom-atom-dimer vertex γ (black square). Due to the large numberof diagrams for the latter two vertex functions, we only showtwo exemplary diagrams. The presence of N-body bound states leads to diver-gencies in the corresponding N-body vertices. The peri-odic divergencies in the analytical solution of ˜ λ AD in Eq.(9) correspond therefore to the presence of the infinitelymany Efimov trimer states at the unitarity point.We can use the latter correspondence to calculate thebound state spectrum also away from unitarity. Thetrimer binding energies are calculated by determining theatom energies E ψ for which ˜ λ AD exhibits divergencies inthe IR as function of a − . The trimer binding energy isthen given by E T = − E ψ . The result is shown in Fig. 1.In this plot the dashed, green lines indicate the bindingenergies of the Efimov trimer states. For calculationalpurposes we switch to the static trimer approximationwhich is completely equivalent to our two-channel modelin Eq. (2). We describe this procedure in Appendix A.At the unitarity point the trimer binding energies forma geometric spectrum and the ratio between adjacent lev-els is given by E ( n +1) T E ( n ) T = e − πs , (11)which can be understood from the limit cycle flow of λ AD .At each scale k = Λ e t , where λ AD diverges, one hits atrimer state. The RG scale k can in turn be connected tothe atom energy E ψ [23, 28, 29] and as the divergenciesappear periodically in t one easily obtains Eq. (11).There is an additional universal relation obeyed bythe trimer energy levels which we may take as a mea-sure of the quality of our approximation. It is given asthe relation between the trimer binding energy E ∗ for a → ∞ and value of a for which the trimer becomesdegenerate with the atom-dimer ( a ∗ + ) and three-atomthreshold ( a ∗− ), respectively. For comparison we definea wave number κ ∗ by E ∗ = − ~ κ ∗ /M (in our conven-tion, E ∗ = − κ ∗ ) and find a ∗− κ ∗ ≈ − . , a ∗ + κ ∗ ≈ .
08 (12)which has to be compared with the exact result a ∗− κ ∗ = − . a ∗ + κ ∗ = 0 . V. THE FOUR-BODY SECTOR
Recently, the solution of the four-body problem in thelow-energy limit has gained a lot of interest. In quantummechanical calculations the existence of two tetramer(four-body bound) states was conjectured for each of theinfinitely many Efimov trimers [1, 2]. By calculating thelowest few sets of bound state levels von Stecher et al.[3, 20] concluded that both ratios of energies between thedifferent tetramers and the trimer state approach univer-sal constants. However, with the quantum mechanicalapproach the calculation directly at the unitarity point ( a − = E ψ = 0), marked explicitly in Fig. 1, turns out tobe difficult, although this point is of great interest whenone wants to gather evidence for universality of the four-body system. In fact, in the three-body sector the infiniteRG limit cycle appears only exactly at the unitarity pointand its universal appearance is directly connected to thebreaking of the continuous scale symmetry. Within ourapproach, the unitarity region is easily accessible.In order to investigate the four-body sector we includeall possible, U(1) symmetric, momentum-independent in-teraction couplings in the effective flowing action Γ k . Ifone assumes all these couplings to be zero at the mi-croscopic UV scale Λ, one can show, by evaluating allpossible Feynman diagrams and using the vacuum hier-archy described in [23], that from all possible four-bodycouplings only the three couplings λ φ , β , and γ are builtup by quantum fluctuations and are therefore includedin Eq. (2). Couplings other than λ φ , β , and γ , such as,for instance, the term ∼ ( ψ ∗ ψ ) are not generated duringthe RG evolution. This consideration leads to our ansatzfor the effective average action (2).For the investigation of the unitarity point we firstswitch to rescaled, dimensionless couplings˜ λ φ = k π h λ φ , ˜ β = k h β, ˜ γ = π k h γ, (13)and obtain the corresponding flow equations by insert-ing the effective flowing action Γ k , Eq. (2), into theWetterich equation (1). By the use of the rescaled cou-plings we find three coupled ordinary differential equa-tions, which are again coupled to the two- and three-body sectors, but become explicitly independent of h and k . We show the diagrammatic representation of the flowequations in Fig. 2(c). Their analytical form at the uni-tarity point is given by ∂ t ˜ λ AD = 128125 − λ AD + 112125 ˜ λ AD , (14) ∂ t ˜ λ φ = 116 + 13 ˜ β −
16 ˜ λ AD + 3˜ λ φ + 12815 ˜ λ φ , (15) ∂ t ˜ β = 188125 ˜ β + 16 ˜ γ + 128125 ˜ λ AD + 224125 ˜ λ AD ˜ β − λ AD + 4384375 ˜ λ φ + 12815 ˜ β ˜ λ φ − λ AD ˜ λ φ , (16) ∂ t ˜ γ = 4592375 + 8768375 ˜ β + 12815 ˜ β + 1125 ˜ γ − λ AD − β ˜ λ AD + 448125 ˜ γ ˜ λ AD + 743681875 ˜ λ AD − λ AD . (17) For illustrative purpose we show the analytical form of the flowequations at the unitarity point only. Away from this limit theirexplicit expressions become much more complex.
FIG. 3: (Color online) Renormalization group limit cycle behavior of the three- and four-body sector at the unitarity point E ψ = a − = 0. The real parts of the rescaled, dimensionless couplings ˜ λ AD, , 4˜ λ φ, , ˜ β /
6, and ˜ γ / t = ln( k/ Λ). Not only the three-body coupling ˜ λ AD, (dashed, black) exhibits a limit cycle behavior, but also the four-bodysector couplings ˜ λ φ, (red, dotted), ˜ β (blue, solid), and ˜ γ (green, solid) obey a limit cycle attached to the three-body sectorwith the same period. We pointed out in the last section that the appear-ance of bound states is connected with divergent vertexfunctions Γ ( n ) k and we exploit this behavior to determinethe bound state spectrum of the three- and four-bosonsystem. At this point we must note that these infini-ties are complicated to handle in a numerical solution ofthe theory. In particular, the numerical treatment of un-bounded limit cycles is problematic due to the periodicinfinities during the RG flow. In order to circumvent thisdifficulty we used the method of complex extension, de-veloped in [34]. The basic idea is to extend the domainof the running couplings to the complex plane λ AD → λ AD, + iλ AD, λ φ → λ φ, + iλ φ, β → β + iβ γ → γ + iγ . (18)On the one hand this effectively doubles the number ofreal flow equations and additional initial conditions mustto provided. We choose λ AD, = ǫ = 10 − in our numer-ical calculation and take all other imaginary parts to bezero in the UV. On the other hand this procedure allowsus to perform the numerical integration of the flow equa-tions as it regularizes the periodic infinities in the flowand makes the numerical treatment feasible. Physically,by the complex extension we convert the stable boundstates into metastable resonances and by taking differentvalues of ǫ we are able to vary the decay width of theresonances. One may compare this with the procedureof Braaten and Hammer [35] who introduce a parameter η ∗ in order to model the decay of the trimers to deeplybound states which have not been included in the effec-tive model. In this line we also view our complex exten-sion as a way to include these deeply bound states in theFRG calculation. Specifically, we find that for ǫ ≪ th Efimov trimer Γ ( n )T is given byΓ ( n )T = 4 ǫE ( n )T at unitarity. This is in agreement with theresult in [7] Γ ( n )T ≈ η ∗ s E ( n )T (19) which holds for small η ∗ . Thus, for ǫ ≪
1, the relation tothe parameter η ∗ introduced by Braaten and Hammer isgiven by ǫ = η ∗ s . (20)The result of the numerical calculation of the four-bodysector at unitarity is shown in Fig. 3. Here, we displaythe RG flows of the real parts of all nonzero three- andfour-body sector couplings as a function of the RG scale t = ln( k/ Λ). The three-body coupling ˜ λ AD, (blackdotted line) exhibits the well-known limit cycle behavior,described in Section IV, with the period being connectedto the Efimov parameter s . Remarkably, there is anadditional limit cycle in the flow of the four-body sectorcouplings with a periodic structure of exactly the samefrequency as the three-body sector. This four-bodysector limit cycle exhibits resonances which are shiftedwith respect to the ones of the three-body system. Themagnitude of this shift is given by a new universalnumber, which is inherent to the four-body sector.Our observation is that the four-body sector isintimately connected with the three-body sector atthe unitarity point. It is permanently attached tothe running of the three-body sector from the firstthree-body resonance on. From here on the periodicstructure of the flow remains unchanged as one goes tosmaller values of k . Due to this tight bond between thethree- and four-body sector, there stays no room for anadditional four-body parameter.We also find that the magnitude of the shift beyondthe first resonance is neither dependent on the initialvalues of the four-body sector couplings in the UV noris it influenced by finite range corrections which weare able to check by choosing different values for theYukawa coupling h . Arbitrary choices lead to the samebehavior. Having done this calculation directly at theunitarity point our conclusion is, that, within our simpleapproximation, the four-body sector behaves trulyuniversal and independent of any four-body parameter FIG. 4: (Color online) Calculation of the universal ratios forthe lowest five set of levels. The calculation is done for dif-ferent values of the Yukawa coupling h / Λ which determinesthe effective range in our model. The dotted lines are onlyguide for the eye. (a) Ratios between the values of scatteringlengths a ( n ) Tet and a ( n ) Tri for which the tetramer and correspond-ing trimer become degenerate with the four-atom threshold.(b) Ratios between the values of binding energies E ( n ) Tet and E ( n ) Tri at resonance a → ∞ . confirming the conjecture made by Platter et al. andvon Stecher et al.. We expect that universality will alsohold for an improved truncation.Naively one expects that each resonance in the flowof the vertex functions is connected to the presence of abound state. As one observes there are also additionalresonances in the four-body sector being degenerate withthe three-body sector resonances. However, we arrive atthe conclusion that these resonances are artifacts of ourapproximation. The mathematical structure of the flowequations is of a kind that divergencies in the three-bodysector directly lead to a divergent four-body sector. Weare confident that the resonances at these positions willdisappear as one includes further momentum dependen-cies in the field theoretical model. Therefore we can al-ready infer from the calculation at unitarity that withinour approximation we are only able to resolve a singletetramer state attached to each trimer state also away from unitarity. In contrast, the “exact” quantum me-chanical calculations in [1–3, 20] predict the existence oftwo tetramer states which have recently been observedby Ferlaino et al. [21]. As one includes further momen-tum dependencies, it is well possible that not only thedegenerate resonance disappears but also new, genuineresonances associated with the “missing” tetramer statewill appear at the same time. This effect indeed occursin the three-body problem. There, it is essential to in-clude the momentum dependent two-atom vertex. Onlyunder this condition one arrives at the quadratic equa-tion as in (8) which gives rise to the Efimov effect. Thiscan easily be seen by taking a look at the flow equationof λ AD depicted as Feynman diagrams in Fig. 2(b). Theassumption of a momentum independent two-atom in-teraction corresponds to a momentum (and frequency)independent dimer propagator. In this approximationthe first term on the RHS of Fig. 2(b) vanishes becauseall poles of the loop frequency integration lie on the samecomplex frequency half-plane. This directly leads to theloss of the Efimov effect in this crude level of approxima-tion.We can also use our model to investigate the full boundstate energy spectrum by solving the flow equations forarbitrary values of the scattering length a . The energylevels of the various bound states are then determinedby varying the energy of the fundamental atoms E ψ suchthat one finds a resonant four-body coupling in the IR.The result of this calculation, using the static trimer ap-proximation presented in Appendix A, is shown in Fig.1, where we plot the energy levels of the various boundstates versus the inverse scattering length. We find onetetramer state attached to each of the Efimov trimerstates. These tetramer states become degenerate withthe four-atom threshold for negative scattering lengthand merge into the dimer-dimer threshold for positive a . In the experiment this leads to the measured reso-nance peaks in the four-body loss coefficient. In ordernot to overload the plot we show only the first three setsof levels, although the FRG method allows to calculatean arbitrary number of them. One also observes thatthe shape of the tetramer levels follows the shape of thetrimer levels. In analogy to the three-body sector one cancalculate a universal formula relating a tetramer bindingenergy E ∗ T et = − κ ∗ T at a → ∞ with the correspondingscattering length at which the tetramer becomes degen-erate with the four-atom threshold a ∗ T − and the dimer-dimer threshold a ∗ T + , respectively. We find a ∗ T − κ ∗ T ≈ − . , a ∗ T + κ ∗ T ≈ . . (21)In their recent quantum mechanical calculations vonStecher et al. were able to calculate the lowest few sets ofbound state energy levels [3]. From their behavior it wasinferred that the ratio between the tetramer and trimerbinding energies approaches a universal number withinthese first few sets of levels. Figuratively speaking it istherefore expected that the universal regime in the en-ergy plot in Fig. 1 is reached very fast as one goes tosmaller a − and E ψ .In order to investigate this observation we calculatethe behavior of two ratios as a function of the set of levelfor which they are determined. The first ratio relatesthe negative scattering lengths a ( n ) T et and a ( n ) T ri for whichthe n th tetramer and trimer become degenerate with thefour-atom threshold. The second is the ratio between thebinding energies of the n th tetramer E ( n ) T et and the n th trimer E ( n ) T ri at resonance, a → ∞ . The resulting plotsare shown in Fig. 4. We calculate the ratios for differentvalues of the microscopic couplings in order to test thedegree of universality of the various sets of energy levels.In the plots we show in particular the dependence on thechoice of the Yukawa coupling h determining the effectiverange r eff of the model. As one sees, only the first of theratios depend on the microscopic details. Already fromthe second set of levels on the microscopic details arewashed out and the ratios become independent of thechoice of initial conditions: The regime of universality isreached extremely fast and as a − and E ψ are lowered onewill ultimately find the four-body limit cycle describedabove.For the asymptotic ratios we find a ( n ) T et ≈ . a ( n ) T ri , (22) E ( n ) T et ≈ . E ( n ) T ri . (23)Von Stecher et al. find a ( n ) T et /a ( n ) T ri ≈ .
43 (0 . E ( n ) T et /E ( n ) T ri ≈ .
58 (1 . a ( n ) T et /a ( n ) T ri ≈ .
47 (0 . VI. CONCLUSIONS
In this paper, we investigated the four-body problemwith the help of the functional renormalization group.Employing a simple two-channel model with pointlikethree- and four-body interactions we were able to inves-tigate universal properties at the unitarity point a → ∞ , E ψ = 0 as well as to perform computations away from it.In the RG language the Efimov physics of the three-bodyproblem manifests itself as an infinite RG limit cycle be-havior of the three-body coupling constant at unitarity.We found that also the four-body sector is governed by such a limit cycle which is solely induced by the RG run-ning of the three-body sector, signaling the absence of afour-body parameter.We also computed the energy spectrum away from uni-tarity and were able to obtain the universal relations be-tween four- and three-body observables in our approxi-mation. Our calculation provides an explanation for thefindings of von Stecher et al. [3], who found that theseratios approach universal constants very quickly as theyare computed for higher and higher excited states. Wealso found a dependence of the ratios for the lowest levelon microscopic details such as the effective range. This inturn is of relevance for the experimental observations byFerlaino et al. [21]. In this experiment the lowest stateshave been measured and one can therefore not expect tofind the exact universal relations between them.Considering the simplicity of our model, the agreementwith the previous studies in [1–3] is quite good. Therehad been some disagreement in literature about univer-sality and the absence or existence of a four-body pa-rameter, see e.g. [36–38]. Our RG results support theconclusion that the four-body system is universal and in-dependent of any four-body parameter.An important shortcoming of the pointlike approxima-tion is the absence of the shallower of the two tetramerstates. Obviously the pointlike approximation of thethree- and four-body sectors is not sufficient and in fu-ture work one should include momentum dependent in-teractions. From the energy spectrum in Fig. 1 it be-comes also evident that the excited tetramer states candecay into an energetically lower lying trimer plus atom.The higher excited states in the four-body system aretherefore expected to have an intrinsic finite decay width[2]. Whether this width has a universal character stillremains an opened question as well as in which way thecorresponding imaginary coupling constants will changethe RG analysis.The inclusion of the full momentum dependencies in thethree- and four-body sector seems to be a rather compli-cated task. In the effective field theory study of the three-boson system the introduction of a dynamical dimer field,often called the di-atom trick [7], has been a decisive steptowards the exact solution of the three-body problem.From this perspective we suggest that the inclusion of adynamical trimer field in the effective action might helpto simplify the momentum dependent calculation.The four boson system remains still a subject with manyopen questions. With our RG analysis in the point-like approximation, we made the first step towards arenormalization group description of the four-body prob-lem supplementing the previous quantum mechanical ap-proaches. From this perspective this work provides astarting point for a deeper understanding of universalityin the four-body problem.0 VII. ACKNOWLEDGMENTS
We thank E. Braaten, S. Floerchinger, C. H. Greene,H. W. Hammer, J. Pawlowski, J. von Stecher, C. Wet-terich, and W. Zwerger for stimulating discussions. Weare indebted to F. Ferlaino, R. Grimm, and S. Knoopfor many insightful discussions and for pointing out theproblem to us in the first place. RS thanks the DFG forsupport within the FOR 801 ‘Strong correlations in mul-tiflavor ultracold quantum gases’. SM is grateful to KTFfor support.
Appendix A: The trimer approximation andrebosonization
In this appendix we will apply the rebosonizationmethod developed in [40] to our model (2). We intro-duce an additional trimer field χ , representing the boundstate of three bosons, which then mediates the atom-dimer interaction. A similar procedure had already beenused long time ago by Fonseca and Shanley [17] in thecontext of nuclear physics and was recently employed byus in [28, 29] for the treatment of the three-componentFermi gas. There are several reasons for employing thisprocedure. First, it is useful to reduce the number of res-onances one has to integrate through in the RG flow. In-stead of calculating the divergent coupling λ AD one onlyhas to calculate zero-crossings of the trimer energy gapwhich is numerically much easier to handle. Secondly, bythe introduction of a dynamical trimer field one may beable to mimic some of the complicated momentum struc-ture of the atom-dimer interaction in a simple way whichcould probably be sufficient to find the missing tetramerstate in our calculation. The third point is of a more tech-nical nature and concerns the method of rebosonization,which we will employ here in a quite extensive manner.In the three-body sector the real atom-dimer coupling λ AD exhibits divergencies when the energy gap of thefundamental atoms E ψ is tuned such that one hits thetrimer bound state in the IR. In the static trimer approx-imation, the coupling λ AD is mediated by the exchangeof a trimer field χ with the non-dynamical, inverse prop-agator P χ = m χ , which can be depicted asThe trimer field χ ∼ ψ is introduced on the microscopicscale by a Hubbard-Stratonovich transformation and ouransatz for the effective average action, motivated by the resulting classical action, readsΓ k = Z x { ψ ∗ ( ∂ τ − ∆ + E ψ ) ψ + φ ∗ (cid:18) A φ ( ∂ τ − ∆2 ) + m φ (cid:19) φ + χ ∗ m χ χ + h φ ∗ ψψ + φψ ∗ ψ ∗ ) + λ AD φ ∗ ψ ∗ φψ + g ( χ ∗ φψ + χφ ∗ ψ ∗ )+ λ φ ( φ ∗ φ ) + β ( φ ∗ φ ∗ φψψ + φφφ ∗ ψ ∗ ψ ∗ )+ γφ ∗ ψ ∗ ψ ∗ φψψ + δ χ ∗ ψ ∗ χψ + δ ( χ ∗ ψ ∗ φφ + χψφ ∗ φ ∗ )+ δ ( χ ∗ ψ ∗ φψψ + χψφ ∗ ψ ∗ ψ ∗ ) } . (A1)The Yukawa interaction g couples the trimer field to thedimer and atom field. The δ i are the additional U(1)symmetric four-body couplings which are generated byquantum fluctuations. All other possible couplings canbe shown to stay zero during the RG evolution providedthey are zero at the UV scale. Also the coupling λ AD is regenerated through a box diagram in the RG flow.However, it is possible to absorb all these emerging cou-plings by the use of the rebosonization procedure.For this matter we promote the trimer field χ to be ex-plicitly scale dependent, χ → χ k , χ ∗ → χ ∗ k and the Wet-terich equation generalizes to ∂ k Γ k [Φ k ] = 12 Tr (cid:16) Γ (2) k [Φ k ] + R k (cid:17) − ∂ k R k + (cid:18) δδ Φ k Γ k [Φ k ] (cid:19) ∂ k Φ k , (A2)where Φ k now includes all fields including the trimerfields ( χ, χ ∗ ). The additional term in the generalized flowequation (A2) allows for the absorption of the reemergingcouplings since one has the freedom to choose the scaledependence of the trimer fields as a function of fields. Inorder to continuously eliminate the couplings λ AD and δ i we choose ∂ k χ k = φψζ a,k + ψ ∗ χ k ψζ b,k + ψ ∗ φφζ c,k + ψ ∗ φψψζ d,k ,∂ k χ ∗ k = φ ∗ ψ ∗ ζ a,k + ψχ ∗ k ψ ∗ ζ b,k + ψφ ∗ φ ∗ ζ c,k + ψφ ∗ ψ ∗ ψ ∗ ζ d,k . (A3)Upon inserting Eq. (A3) into the generalized Wet-terich equation (A2) the condition that the flows of λ AD and δ i vanish leads to ζ a = − ∂ k λ AD g , ζ b = − ∂ k δ m χ ζ c = − ∂ k δ m χ , ζ d = − ∂ k δ + gζ b m χ . (A4)When one calculates now the flow equations of the re-maining flowing couplings by projecting Eq. (A2) ontothem, one obtains new contributions due to the presence1of additional terms arising from Eq. (A3).In our static trimer approximation the trimer field has nodynamical propagator and the model given by Eq. (A1)is completely equivalent to the two-channel model in Eq.(2). 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