Dimer-dimer scattering length for fermions with different masses: analytical study for large mass ratio
DDimer-dimer scattering length for fermions with different masses: analytical study forlarge mass ratio
F. Alzetto ( a ) , R. Combescot ( a ) , ( b ) and X. Leyronas ( a ) (a) Laboratoire de Physique Statistique, Ecole Normale Sup´erieure, UPMC Paris 06,Universit´e Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France. and(b) Institut Universitaire de France, Paris, France (Dated: Received October 30, 2018)We study the dimer-dimer scattering length a for a two-component Fermi mixture in which thedifferent fermions have different masses m ↑ and m ↓ . This is made in the framework of the exactfield theoretical method. In the large mass ratio domain the equations are simplified enough to leadto an analytical solution. In particular we link a to the fermion-dimer scattering length a for thesame fermions, and obtain the very simple relation a = a /
2. The result a (cid:39) a / a with a fairly good precision for any masses. To dominant orders for large mass ratio itagrees with the literature. We show that, in this large mass ratio domain, the dominant processesare the repeated dimer-dimer Born scatterings, considered earlier by Pieri and Strinati. We concludethat their approximation, of retaining only these processes, is a fairly good one whatever the massratio. I. INTRODUCTION
Ultracold atoms are a remarkable playground for a number of other fields of physics, such as condensed matterphysics, nuclear physics and astrophysics. This is due to the simplicity at very low temperature of the effectiveinteraction between atoms and moreover to the experimental ability to choose almost at will the correspondingparameters [1]. The resulting physical systems correspond often to simple limiting situations of high interest inthese other fields. The experimental realization of the BEC-BCS crossover in Fermi gases is a wonderful exampleof the flexibility available in cold gases. As a result of the ability to vary the effective interaction in a very widerange, it has been possible to go from the weakly attractive regime between fermions, where at low temperature aBCS-like condensate arises, to the strongly attractive regime where molecules, or dimers, form and these compositebosons undergo the standard Bose-Einstein condensation at low temperature. This crossover has provided an explicitdemonstration of the deep-seated connection between these two kinds of superfluids, which had been realized longago [2–6]. At the same time it displays, at unitarity where dimers are just appearing, a somewhat new example ofsuperfluidity.When the interaction strength is just beyond the one corresponding to unitarity so that dimers are just forming, thephysical situation is quite complex since the dimer size is very large and they overlap very strongly in the gas leadingto a complicated many-body problem, at higher temperature in the normal state as well as in the superfluid statefound at lower temperature. However when the interaction strength is increased toward the strong coupling regime,the dimer size becomes very small and their overlap becomes negligible. In this case the composite nature of thedimers becomes irrelevant and physically one deals with a simple boson gas. Just as for ultracold Bose gas, the onlyrelevant parameter to describe the low temperature physics is the boson-boson scattering length, in the present casethe dimer-dimer scattering length a . Hence knowing the precise value of this parameter, in terms of the scatteringlength a for fermions making up the dimer, is of utmost importance for the description of this strong coupling limit.This problem was first addressed by Haussmann [7] and by S´a de Melo, Randeria and Engelbrecht [8] by differentmethods, which turn out to be equivalent to the Born approximation for this scattering problem. The correspondingresult is a = 2 a . This result was much improved by Pieri and Strinati [9] who considered repeated dimer-dimerscattering, instead of a single scattering corresponding to the Born approximation. They proceeded to sum up thecorresponding series and obtained numerically a (cid:39) . a . Finally an exact treatment was given by Petrov, Salomonand Shlyapnikov [10, 11] who provided the numerical solution of the corresponding four-body Schr¨odinger equation.This led them a (cid:39) . a . This problem was then taken up by Brodsky, Klaptsov, Kagan, Combescot and Leyronas[12] who gave an exact treatment of the same problem, by making use of field theoretical methods. The numericalsolution of their equations gave naturally the same result a (cid:39) . a as the one obtained by Petrov, Salomon andShlyapnikov.In view of the interest in fermionic mixtures made of different elements, such as Li - K mixtures, Petrov, Salomonand Shlyapnikov extended their treatment [13] to the case where the fermions making up the dimers have differentmasses. The corresponding extension of the exact field theoretical treatment was provided by Iskin and S´a de Melo[14], who provided numerical results for several mixtures of specific interest. This method has been extended recentlyby Levinsen and Petrov [15] to the case of narrow Feshbach resonances, aiming specifically at Li - K mixtures. Here, a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov FIG. 1: Diagrammatic representation of the relation Eq.(1) between the full dimer-dimer scattering vertex T and the vertexΦ. The black strips are for ladder diagrams corresponding to the dimer propagator T given by Eq.(3). The full line is for the ↑ -spin propagator. The dashed line is for the ↓ -spin propagator. in the case of a wide Feshbach resonance, we take over this technique to investigate the large mass ratio analytically.Our aim is the same as in our recent work on the fermion-dimer scattering length [16], namely to gain some insightin the dimer-dimer scattering problem which might be used in dealing with more complicated problems where thisscattering is a building block.In the present case we succeed in obtaining such an insight. We indeed find that the Pieri and Strinati [9] approxi-mation is asymptotically correct for large mass ratio. Taken with the fact that their result for a is also a quite goodapproximation when the two different fermions have equal mass, we come to the conclusion that their approximationis quite good for any mass ratio. This is an important simplification in the dimer-dimer scattering problem since thismeans that we do not have to take into account intermediate states where one of the dimer is broken. Except forthe irreducible process corresponding to the Born contribution, this means that in all the intermediate states in thisscattering process we have to deal with unbroken dimers. II. BASIC EQUATIONS
As it is quite often done, we will call spin-up and spin-down the two different fermions with respective mass m ↑ and m ↓ . This is a convenient convention frequently used in cold gases, although there is in principle no link with thephysical spin of the particles. The dimer is a bound state of one spin-up and one spin-down fermion. Clearly thescattering length a we are looking for is proportional to the spin-up spin-down scattering length a . For dimensionalreasons a depends only on the mass ratio r = m ↓ /m ↑ . Since exchanging the two fermions does not change the dimer,the result is unchanged when r is changed into 1 /r . Hence the large mass limit we are mostly interested in can beseen equally as the r → ∞ limit or the r → T ( p , p ; P ) and Φ( q , q ; p , P ). They are shown in Fig. 1. The first one describes the scatteringof two dimers entering with respective four-momenta P + p and P − p , and outgoing with four-momenta P + p and P − p . In the second one the entering particles are two different fermions with respective four-momenta q and q and a dimer with four-momentum P − q − q , while the outgoing particles are again two dimers with four-momenta P + p and P − p . We will take the convention for Φ that the first variable q corresponds to the spin-up particlewhile the second one q corresponds to the spin-down one, explicitly Φ( q ↑ , q ↓ ; p , P ). In contrast with the equalmass case Φ( q , q ; p , P ) is no longer symmetric in the exchange of q and q .It is clear that the first process in T involves the interaction of a fermion of one dimer with a fermion of the otherdimer. Hence one has first to split open a dimer line into its fermionic components, all the subsequent processes beingdescribed by Φ. This leads to the equality shown in Fig. 1. Algebraically it reads: T ( p , p ; P ) = (cid:88) k G ↑ ( k ) G ↓ ( P + p − k )Φ( k, P + p − k ; p , P ) (1)where (cid:80) k ≡ i (cid:82) d k dω/ (2 π ) is for the summation over momentum and energy and the propagators are given by G ↑ , ↓ ( k ) = [ ω − k / m ↑ , ↓ + i(cid:15) ] − , with (cid:15) → + . In turn we can write an equation for Φ, in the same spirit as aBethe-Salpeter equation. The first process is a break up of the entering dimer, in order to allow the interaction ofone of the free fermions with a fermion from the dimer. Actually other interactions between these same fermionsmay also occur after the first one and the resummation of all these processes leads merely to a dimer propagator T .Afterwards any process may follow, which is again depicted by Φ. However one must have in mind that by definition,in Φ, the first process can not be an interaction between the two free fermions (otherwise one would merely have anentering dimer, instead of two free fermions, which is already accounted for by the entering dimer propagator in T ).But such a process is quite allowed after the first interaction, and since they are not accounted for by Φ we have toadd terms to describe them. The repeated interaction of the free fermions leads to another dimer propagator, so atthis stage we have two dimers. This can be taken as the outgoing state, and the corresponding diagrams correspondmerely to the Born term for the diagrammatic expansion for Φ. However it is also possible to have any other processafter these two dimers have been formed, which is fully described by T . For each of the process we have described,we have actually two possibilities depending on the spin of the involved particles. However this is not true for the last T term we have just described, since one sees easily that exchanging the spins is equivalent to a change in dummyvariables. This leads finally to the diagrammatic equation depicted in Fig. 2, which reads algebraically:Φ( q , q ; p , P ) = − G ↑ ( P − q − p ) G ↓ ( P − q + p ) − G ↓ ( P − q − p ) G ↑ ( P − q + p ) − (cid:88) k G ↑ ( k ) G ↓ (2 P − q − q − k ) T (2 P − q − k )Φ( k, q ; p , P ) − (cid:88) k G ↓ ( k ) G ↑ (2 P − q − q − k ) T (2 P − q − k )Φ( q , k ; p , P ) − (cid:88) Q G ↑ ( Q − q ) G ↓ (2 P − Q − q ) T (2 P − Q ) T ( Q ) T ( P − Q, p ; P ) (2)The dimer propagator is obtained as usual by summing up the ladder diagrams and is given by: T ( P ) = 2 πµ a − − (cid:112) µ ( P / M − E ) ≡ T ( P , E ) (3)for the four-momentum P = { P , E } . Here µ is the reduced mass µ = m ↑ m ↓ /M = m ↑ r/ (1 + r ), while M is the totalmass M = m ↑ + m ↓ = m ↑ (1 + r ).Just as in [12] the scattering length is obtained directly from T , evaluated at the dimer binding energy E b =1 / (2 µa ), the only difference being that we have to use the dimer reduced mass µ and the reduced mass M/ (cid:18) πµ a (cid:19) T (0 , { , − E b } ) = 2 π (2 a ) M/ . (4)Since for the scattering length problem we have to take p = 0 and P = { , − E b } we do not write anymore explicitlythese variables. Hence Eq.(4) becomes: ¯ a ≡ a a = π r ) r T (0)( am ↑ ) (5)In the following we take for convenience a and m ↑ as unit for length and mass, i.e. we set a = 1 and m ↑ = 1. Thislast step breaks apparently the invariance of ¯ a under r → /r , but this symmetry is naturally still satisfied in thefinal results.A very important simplification is that, just as in [12], the calculation of the two Φ terms in the right-hand side ofEq.(2) requires only the knowledge of the ”on the shell” value for Φ with respect to variable k . This allows to considerEq.(2) only for ”on the shell” values for variables q and q . We denote Φ( q , q ) the corresponding ”on the shell”value of Φ (it is not symmetric under the exchange of q and q ). Similarly when Eq.(2) is inserted into Eq.(1), theresulting equation requires only the knowledge of Φ( q , q ). Finally it is again convenient to continue the equationsto purely imaginary values for the frequencies. The corresponding value for T ( Q ) = T ( { Q , i Ω } ) ≡ t ( Q , Ω) is realand the equations contain only real quantities.Let us now give the various terms entering the equations for Φ and t . The first two terms in Eq.(2) correspond tothe Born approximation and give to Φ( q , q ) a contribution:Φ B ( q , q ) = − µ
11 + q
11 + q (6)where now q = | q | and µ = r/ (1 + r ). The two next terms in Eq.(2) give two contributions:Φ Φ1 ( q , q ) = 2(2 π ) (cid:90) d k T ( − /µ − q / − k / r, q + k ) Φ( q , k )2 /µ + q + q /r + k /r + ( q + q + k ) (7) FIG. 2: Diagrammatic representation of the equation Eq.(2) for the vertex Φ. The black strips are for ladder diagramscorresponding to the dimer propagator T given by Eq.(3). Full lines are for ↑ -spin propagators. Dashed lines are for ↓ -spinpropagators. where the integration over the azimuthal angle of k with respect to q can be explicitly performed, since Φ( q , k ) aswell as T depend only on the polar angle between k and q . This leaves a double integration to be performed. Theother contribution is Φ Φ2 ( q , q ) = 2(2 π ) (cid:90) d k T ( − /µ − q / r − k / , q + k ) Φ( k , q )2 /µ + q + q /r + k + ( q + q + k ) /r (8)where the integration over the azimuthal angle of k with respect to q can again be explicitly performed leaving againa double integral. Finally the last term gives, after the change of variable Q → P − Q :Φ T ( q , q ) = 4(2 π ) (cid:90) d Q (cid:90) ∞−∞ d Ω | T ( − / µ − i Ω , Q ) | t ( Q , Ω)[1 /µ − i Ω + q + ( Q − q ) /r ] [1 /µ + 2 i Ω + q /r + ( Q + q ) ] (9)Since t ( Q , Ω) as well as T depend only on the modulus of Q , the angular average over Q can be explicitly performedby making use of: (cid:90) d Ω k π a + k . u b + k . v = 12 A ln ab − k u . v + Aab − k u . v − A (10)where A = (cid:112) ( akv ) + ( bku ) − abk u . v − k [( uv ) − ( u . v ) ] (11)This leaves again a double integral to be performed. To summarize we have explicitly:Φ( q , q ) = Φ B ( q , q ) + Φ Φ1 ( q , q ) + Φ Φ2 ( q , q ) + Φ T ( q , q ) (12)where the four terms in the right-hand side are defined respectively by Eq.(6),Eq.(7),Eq.(8) and Eq.(9).Similarly, substituting Eq.(2) in Eq.(1) we obtain for t : t ( q , ν ) = T B ( q , ν ) + T Φ ( q , ν ) + T T ( q , ν ) (13)After the same change of variable Q → P − Q the last term reads: T T ( q , ν ) = 1(2 π ) (cid:90) d Q (cid:90) ∞−∞ d Ω K ( q , ν ; Q , Ω) | T ( − / µ − i Ω , Q ) | t ( Q , Ω) (14)where the symmetric kernel K ( q , ν ; Q , Ω) = K ( Q , Ω; q , ν ) is given by: K ( q , ν ; Q , Ω) = (cid:88) k G ↑ ( k ) G ↓ ( P + q − k ) G ↑ ( k − q − Q ) G ↓ ( P + Q − k ) (15)with q = { q , iν } . This can be written in a more symmetric way by the change of variable k → k + ( P + q + Q ) / K ( q , ν ; Q , Ω) = (cid:88) k G ↑ ( k + P + q + Q G ↓ ( P + q − Q − k ) G ↑ ( k + P − q − Q G ↓ ( ( P − q + Q − k ) (16)After performing in Eq.(15) the frequency integration over ω and making the change k → k + q , one is left withexpressions which can be written in terms of products of two rational functions, each one being of the form 1 / ( a + k . u ).The angular average can be performed by making use of Eq.(10). One is left with a simple integration over the modulusof k . This means that evaluation of Eq.(14) requires four integrations. However we will not write the lengthy resultingexpressions, which are only necessary to perform the full numerical calculation for the general case of two differentmasses m ↑ and m ↓ .Then the Born contribution T B ( q , ν ) is merely given by: T B ( q , ν ) = − K ( q , ν ; ,
0) (17)Finally the last term T Φ ( q , ν ) is given by: T Φ ( q , ν ) = − π ) (cid:90) d k d k (cid:48) T (cid:0) − /µ − k / − k (cid:48) / r, k + k (cid:48) (cid:1) Φ( k , k (cid:48) )[1 /µ − iν + k + ( k − q ) /r ] [1 /µ + 2 iν + k (cid:48) /r + ( k (cid:48) + q ) ] + ( q → − q ) (18)Since the result depends clearly only on the modulus of q and the dependence on q is explicit, the integrand can beangularly averaged over the direction of q by making use of Eq.(10). This leaves a triple integral over the moduli of k and k (cid:48) , and the angle between them.In order to solve numerically Eq.(12) and Eq.(13) for Φ( q , q ) and t ( q , ν ) and then obtain a from Eq.(4), wehave discretized these equations. This leads to a set of linear equations, the solution corresponding to a matrixinversion. This has been performed making use of the standard LAPACK routines. This works quite well for not toohigh mass ratios, and we estimate the precision of our results to be typically 2%. However for mass ratio typicallyabove 500, the numerics becomes unreliable. This is easily understood from the somewhat singular features whichemerge from our anaytical solution, presented below, for very large mass ratio. We note finally that we never findany zero eigenvalue for the matrix to be inverted. This means that the bound states discussed in [13] do not play anyrole in the calculation of a . III. SUM RULE
In a way completely analogous to what we have found in our study of the atom-dimer scattering length [16], a quiteuseful sum rule can be obtained by analyzing the convergence of the various integrals when the variables go to infinity.It is natural to assume that the solutions Φ( q , q ) and t ( q , ν ) we are looking for have physical ranges correspondingto their variables, and that beyond these ranges these functions go rapidly enough to zero for the various integralsto be convergent. This assumption is confirmed by our numerical calculations. This property allows to study thebehaviour of Φ( q , q ) and t ( q , ν ) for large values of the variables.Let us first consider the t equation Eq.(13) and begin by the Born contribution T B ( q , ν ) given by Eq.(17) andEq.(15). Since we are interested in the case where q and ν are large, P is negligible in this regime and we are leftwith: T B ( q , ν ) ≈ − (cid:88) k G ↑ ( k ) G ↓ ( q − k ) G ↑ ( k − q ) G ↓ ( − k ) (19)It is then convenient to consider more appropriate energy variables. Since energy is homogeneous to momentumsquared, we set ν = q ν and similarly for the integration variable ω = k ω . For large values of q and ν , the naturalvariable to consider is ρ = (cid:112) q + q ν , corresponding to introduce a radial coordinate for | q | and q ν . Similarly we canintroduce r = (cid:112) k + k ω (not to be confused with the mass ratio). Now for example G ↑ ( k ) is homogeneous to r − ,and similarly for the other Green’s functions entering Eq.(19). Hence for homogeneity reasons we have T B ( q , ν ) ∼ ρ − since the summation (cid:80) k introduces a factor homogeneous to r . More precisely, making the change of variable r = ¯ rρ allows to obtain in Eq.(19) a prefactor ρ − , the remaining factor being independent of ρ . One can naturally writemore explicit expressions for T B ( q , ν ), although it is complicated to go to a full analytical result. However we will notneed these expressions and they merely confirm our result that T B ( q , ν ) ∼ ( q + q ν ) − / which comes simply fromour homogeneity analysis.We can now make a similar analysis for the term T T ( q , ν ) in the t equation. Indeed since in Eq.(14) we areinterested in large values of q and ν , while the integration variables Q and Ω are effectively bounded by the factor t ( Q , Ω), the kernel reduces in this limit to: K ( q , ν ; Q , Ω) ≈ K ( q , ν ; ,
0) (20)as it is quite clear from the symmetric form Eq.(16) for K . Hence this kernel factorizes out and we are left with: T T ( q , ν ) = 1(2 π ) K ( q , ν ; , (cid:90) d Q (cid:90) ∞−∞ d Ω | T ( − / µ − i Ω , Q ) | t ( Q , Ω) (21)which has naturally the same behaviour as the Born term, namely T T ( q , ν ) ∼ ( q + q ν ) − / = ρ − .Finally we analyze in the same way the last term T Φ ( q , ν ) in this t equation from its explicit expression Eq.(18).With the k and k (cid:48) variables bounded by the Φ( k , k (cid:48) ) factor, the q and ν dependence comes explicitly from thedenominators and is given by T Φ ( q , ν ) ∼ ( q + q ν ) − = ρ − . Hence it converges toward zero faster than the two otherterms and the overall behaviour of t ( q , ν ) is apparently t ( q , ν ) ∼ ρ − .However when we insert this behaviour in the integral factor found in Eq.(21) for the large ( q + q ν ) behaviour wefind a discrepancy. Indeed from Eq.(3), and for large Q and Ω, we have | T ( − / µ − i Ω , Q ) | ∼ R − , where we haveagain made the change Ω = Q for the energy variable and introduced the radial variable R = (cid:112) Q + Q . This leadsto | T ( − / µ − i Ω , Q ) | t ( Q , Ω) ∼ R − . However we have d Q d Ω ∼ R dR so that the integral in Eq.(21) divergesas dR/R . This is in contradiction with our initial assumption that t ( Q , Ω) insures the convergence of the integral.The only escape is that the contributions from the Born term and from the T T , which have exactly the same powerlaw dependence, cancel out so that the actual decrease of t ( Q , Ω) is faster than R − . This occurs if the correspondingcoefficients cancel exactly, which leads to the sum rule:1(2 π ) (cid:90) d Q (cid:90) ∞−∞ d Ω | T ( − / µ − i Ω , Q ) | t ( Q , Ω) = 2 (22)Upon checking this result numerically, we have found that it is in very good agreement with our calculations.Let us now analyze in the same way the Φ equation Eq.(12). First the Born term is explicit and, for large q and q , it behaves as: Φ B ( q , q ) ≈ − µ q q (23)Then we consider the Φ T ( q , q ) term given by Eq.(9). Again the factor t ( Q , Ω), assumed to go rapidly enough tozero for large Q and Ω, makes these variables Q and Ω effectively bounded. Hence when we consider very large valuesof q and q , we can forget about Q and Ω in the explicit denominators, and the product of these denominators isgiven in this case by q (1 + 1 /r ) q (1 /r + 1) = ( q q /µ ) . This leads for large q and q to:Φ T ( q , q ) ≈ π ) µ q q (cid:90) d Q (cid:90) ∞−∞ d Ω | T ( − / µ − i Ω , Q ) | t ( Q , Ω) (24)We consider finally the Φ Φ1 and Φ Φ2 terms. Our assumption is that Φ( q , q ) goes rapidly enough to zero at large q and q for having the integrals in Φ Φ1 and Φ Φ2 converge for large k due to the Φ factor. This means explicitlythat (cid:82) d k Φ( q , k ) and (cid:82) d k Φ( k , q ) are convergent. However this is not true if we take the behaviour ∼ / ( q q )produced by the Born term and the Φ T term, since this gives an integral (cid:82) d k /k ∼ (cid:82) dk , which is divergent. If thedominant behaviour was coming from the Φ Φ terms themselves, then Φ( q , q ) would go even more slowly to zero atinfinity and the situation would be even worst (one can actually check that this case does not arise). Hence we haveagain a contradiction with our starting hypothesis, but there is also the same way out, namely that the contributionof the Born term and the Φ T cancel precisely, which leads to a decrease of Φ( q , q ) faster than ∼ / ( q q ) for large q and q . We see from Eq.(23) and Eq.(24) that the condition for this to happen is exactly the sum rule alreadyfound Eq.(22). This result is not so surprising when we remember that the t equation is obtained by carrying the Φequation into Eq.(1).We note finally that, in our work on the atom-dimer scattering length [16], we have shown that the sum rule we havefound in this case is merely a direct consequence of the Pauli principle, namely the fact that two identical fermions cannot be found at the same place. In the present case such a physical interpretation is less obvious since the frequencydependence of t ( Q , Ω) enters. Anyway we have not tried to find a specific physical interpretation.
IV. VERY HEAVY MASS EQUATIONS
Let us now consider how the preceding equations simplify when we consider the limiting case where the two masses m ↑ and m ↓ are very different. We can equivalently consider that the mass ratio r = m ↓ /m ↑ goes to zero or to infinity.We will take this last option in the following since it is somewhat more convenient with the asymmetric notations wehave chosen. This implies µ → ∞ B ( q , q ) = − q
11 + q (25)In the Φ Φ1 and Φ Φ2 terms we may replace in Eq.(3) T ( E, P ) by 2 π/ (1 − √− E ) since M → ∞ in this limit. Thisleads to: Φ ∞ Φ1 ( q , q ) = − π (cid:112) q − (cid:90) d k Φ ∞ ( q , k )2 + q + ( q + q + k ) (26)and Φ ∞ Φ2 ( q , q ) = − π (cid:90) d k √ k − ∞ ( k , q )2 + q + k (27)Let us finally consider the Φ T term which gives rise to more problems. If we proceed in the same way as above, wefind: Φ ∞ T ( q , q ) = 4(2 π ) (cid:90) d Q (cid:90) ∞−∞ d Ω 1[1 − i Ω + q ] [1 + 2 i Ω + ( Q + q ) ] t ∞ ( Q , Ω) |√ i Ω − | (28)However we see that the Ω integration diverges as (cid:82) d Ω / Ω for Ω →
0, since |√ i Ω − | (cid:39) in this limit.Indeed the explicit denominators go to finite values and there is no reason to have t ( Q , Ω = 0) = 0. This can anywaybe checked numerically. In particular this does not happen for Q = since, in the large r limit we have from Eq.(5)(with our reduced variables): ¯ a = π r t ( ,
0) (29)The existence of this divergence shows that we have handled the | T | term too rapidly, since it is at the origin ofthe divergence which does not exist naturally in the general equation Eq.(9). We must keep M ≈ r without settingimmediately 1 /M = 0. In this case we have to write: T ( E, P ) = 2 π − (cid:112) P /r − E (30)which leads, instead of Eq.(28), to:Φ ∞ T ( q , q ) = 4(2 π ) (cid:90) d Q (cid:90) ∞−∞ d Ω t ∞ ( Q , Ω)[1 − i Ω + q ] [1 + 2 i Ω + ( Q + q ) ] | (cid:112) i Ω + Q /r + 1 | + Q /r (31)We see that the divergence has disappeared. Instead we find a factor 1 / (4Ω + Q /r ) which, in the limit r → ∞ , isstrongly peaked around Ω = 0 and acts in practice as a δ function:14Ω + Q /r ≈ πr Q δ (Ω) (32)As a result only t ( Q , Ω = 0) appears in the Φ equation so that we may write the equation for t only for Ω = 0,which is a quite convenient simplification. Taking into account that t ( Q , Ω) depends only on the modulus Q = | Q | ,we set: ¯ t ( Q ) = r t ( Q , Ω = 0) (33)which is just the quantity coming in the scattering length we are looking for:¯ a = π t (0) (34)Taking the limit r → ∞ in the numerator of Eq.(31) and performing the angular average over the direction of Q , wefinally end up with: Φ ∞ T ( q , q ) = 2(1 + q ) q (cid:90) ∞ dQ ¯ t ∞ ( Q ) Q ln 1 + ( Q + q ) Q − q ) (35)We can now write the t equation Eq.(13) with the variable ν set to zero, which is the only thing we need as wehave just seen. The Born contribution is given by: T B ( q ,
0) = − K ( q , ,
0) (36)where K ( q , ,
0) is obtained from Eq.(15). Since in the limit m ↓ → ∞ , G ↓ ( k ) reduces to [ ω + i(cid:15) ] − the calculationis fairly simple and leads to: T ∞ B ( q , ≡ t ∞ B ( q ) = 4 π q + 4 (37)The T Φ term is also easily obtained in the limit r → ∞ from Eq.(18), making use of the limiting expression for T and performing the angular average over the direction of q . This gives: T ∞ Φ ( q , ≡ t ∞ Φ ( q ) = 1 π q (cid:90) ∞ dk k k √ k − (cid:90) ∞ dk (cid:48) k (cid:48) ln 1 + ( k (cid:48) + q ) k (cid:48) − q ) (cid:90) d Ω (cid:48) k π Φ ∞ ( k , k (cid:48) ) (38)Finally in the last term of the t equation, coming from Eq.(14), we meet the same troubles as in the last term ofthe Φ equation if we use the simple limiting expression for T . In the same way as what we have done to deriveΦ ∞ T ( q , q ), we have to use Eq.(30) for T . This leads in the same way to a factor proportional to δ (Ω) justifyingthe fact that we write the t equation only for zero frequency. Actually the last two factors in Eq.(14) are the sameas those appearing in Eq.(9). Hence the only difference is that we have now to calculate the kernel K ( q , Q ,
0) for r → ∞ . This proceeds just as for the Born contribution Eq.(37) and leads basically to the same result, provided q isreplaced by q + Q . This gives: K ( q , Q ,
0) = − π q + Q ) + 4 (39)When the angular average over the direction of q is performed, just as in the preceding term, one finds: T ∞ T ( q , ≡ t ∞ T ( q ) = − π q (cid:90) ∞ dQ ¯ t ∞ ( Q ) Q ln 4 + ( Q + q ) Q − q ) (40)To summarize the Φ equation becomes:Φ ∞ ( q , q ) = Φ ∞ B ( q , q ) + Φ ∞ Φ1 ( q , q ) + Φ ∞ Φ2 ( q , q ) + Φ ∞ T ( q , q ) (41)where the four terms in the right-hand side are defined respectively by Eq.(25), Eq.(26), Eq.(27) and Eq.(35), whilethe t equation reads in this limit, with the definition Eq.(33):1 r ¯ t ∞ ( q ) = t ∞ B ( q ) + t ∞ Φ ( q ) + t ∞ T ( q ) (42)where the three terms in the right-hand side are defined respectively by Eq.(37), Eq.(38) and Eq.(40).Although these equations correspond to a very important simplification with respect to the original ones, they arestill too complicated to be solved analytically as such. In the following section we will show that they can be further Solution of Eq. 42Solution of Eqs. 41 − − r ) a / a FIG. 3: (Color online) Dimer-dimer scattering length a as a function of the mass ratio r (logarithmic scale). The red full lineis the exact numerical result obtained from Eq.(12) and Eq.(13). The green dotted line is the numerical result obtained fromEq.(41) and Eq.(42). The blue dashed line is the result obtained from Eq.(42) by setting Φ ∞ ( q , q ) = 0. simplified, leading to an analytical answer. However it is of interest to solve them numerically to obtain the scatteringlength a and to compare the result to the exact numerical solution of the original equations Eq.(12) and Eq.(13).This is done in Fig. 3. It is quite surprising to see that, already for a mass ratio slightly above 10, the result fromthese asymptotic equations coincide with the exact one within numerical precision. This provides naturally a furthervalidation of these asymptotic equations.Finally let us consider the sum rule Eq.(22) in this limit. We have to handle the | T | factor carefully, as we havedone just above. This implies in the same way that only t ( Q , Ω = 0) = 0 appears in the sum rule and we end upwith the very simple relation: (cid:90) ∞ dQ ¯ t ∞ ( Q ) = 1 (43) V. DISCUSSION OF THE VERY HEAVY MASS LIMIT
Although the equations have been much simplified in this heavy mass limit compared to the general ones, theyare still fairly complicated. We will nevertheless be able to come to a very simple conclusion by showing that aquite natural hypothesis on the behaviour of ¯ t ∞ ( Q ) is fully consistent with the equations, and is in agreement withresults found numerically. However being able to prove that this is the only possible solution looks a very difficultmathematical problem.The natural hypothesis stems from the sum rule Eq.(43) and from the fact that the scattering length a grows whenthe mass ratio increases, as it is known from preceding work [11, 14] and from our own numerical calculations. It is0natural to assume that a grows indefinitely. From Eq.(34) this means that ¯ t ∞ (0) grows indefinitely for large massratio. However the sum rule Eq.(43) puts a constraint. If we assume, as we have already done, that ¯ t ∞ ( Q ) decreasesrapidly when Q is large and has a fairly regular behaviour, the increase of ¯ t ∞ (0), with fixed surface under the curve¯ t ∞ ( Q ) forced by the sum rule, implies that ¯ t ∞ ( Q ) becomes very peaked around the origin for very large mass ratio.In such a case we can further simplify the equations. Let us take first take Φ ∞ T ( q , q ) given by Eq.(35). Since¯ t ∞ ( Q ) is peaked around the origin, Q is actually forced to be small. We can then expand the logarithm intoln[1 + ( Q + q ) ] / [1 + ( Q − q ) ] (cid:39) Qq / (1 + q ). This leads to:Φ ∞ T ( q , q ) (cid:39) q )(1 + q ) (cid:90) ∞ dQ ¯ t ∞ ( Q ) = 8(1 + q )(1 + q ) (44)from the sum rule Eq.(43). Hence Φ ∞ T ( q , q ) cancels exactly the Born contribution Eq.(25). Therefore the onlycontributions left in the right-hand side of the Φ equation are Φ ∞ Φ1 ( q , q ) and Φ ∞ Φ2 ( q , q ). However this means thatthe Φ equation is now an homogeneous linear equation in Φ ∞ ( q , q ), without source term. Barring a singular kernel(which would make impossible in general to solve this Φ equation), the only solution is merely Φ ∞ ( q , q ) = 0.Hence we are only left with the t equation Eq.(42) from which the t ∞ Φ ( q ) term has disappeared. In the equationEq.(40) we can expand, for the same reason as above, the logarithm into ln[4 + ( Q + q ) ] / [4 + ( Q − q ) ] (cid:39) Qq/ (4 + q )which gives: t ∞ T ( q ) (cid:39) − π
14 + q (cid:90) ∞ dQ ¯ t ∞ ( Q ) = − π
14 + q (45)again from the sum rule. And we see again that this term t ∞ T ( q ) cancels exactly the Born term Eq.(37). Hence the t equation Eq.(42) becomes merely ¯ t ∞ ( q ) /r = 0. But this is perfectly consistent with the fact that we deal with r = ∞ limit. In conclusion we find that the equations are perfectly satisfied in this limit by a function ¯ t ∞ ( Q ) verystrongly peaked around the origin and satisfying the sum rule Eq.(43). The only trouble is naturally that we havebeen unable to extract any information. In order to obtain results we have to take more carefully advantage of thefact that ¯ t ∞ ( Q ) is strongly peaked.For this purpose let us go back to the Φ equation Eq.(41). The source term Φ ∞ B ( q , q ) + Φ ∞ T ( q , q ) can befactorized into: Φ ∞ B ( q , q ) + Φ ∞ T ( q , q ) = 2(1 + q ) S ( q ) (46)where: S ( q ) = 1 q (cid:90) ∞ dQ ¯ t ∞ ( Q ) Q ln 1 + ( Q + q ) Q − q ) −
41 + q (47)In the above discussion, valid for an extremely peaked function ¯ t ∞ ( Q ) ∼ δ ( Q ), we had S ( q ) = 0. When r is large,but finite, ¯ t ∞ ( Q ) is peaked around the origin which makes Q effectively bounded. When q is large, we can againexpand the logarithm in Eq.(47) as above and reach again the conclusion that S ( q ) = 0. Hence S ( q ) is also peakedaround the origin.Moreover, even if S ( q ) is not zero, it retains the following exact property: (cid:90) ∞ dk k S ( k ) = 0 (48)Indeed making use of the sum rule Eq.(43) we can write from Eq.(47): (cid:90) ∞ dk k S ( k ) = (cid:90) ∞ dQ ¯ t ∞ ( Q ) (cid:90) ∞ dk (cid:20) kQ ln 1 + ( Q + k ) Q − k ) − k k (cid:21) (49)The integral over k can be calculated analytically, and it is found to be zero whatever the value of Q .These two properties imply that the Φ equation has, to a very good approximation, a factorized solution of theform: Φ ∞ ( q , q ) = F ( q ) S ( q ) (50)Indeed when this factorized expression is substituted in the Φ equation Eq.(41) we see that we can factorize S ( q ) notonly in the source term Eq.(46), but also in Φ ∞ Φ2 ( q , q ) (see Eq.(27)). In the remaining term Φ ∞ Φ1 ( q , q ) given by1Eq.(26), when we substitute Eq.(50), the factor S ( k ) prohibits large values for | k | , as we have just seen. But when | k | issmall, or at most of order unity, it is a very good approximation to neglect it in the denominator 2+ q +( q + q + k ) .Hence we are left in this term with the integral (cid:82) d k S ( k ) = 4 π (cid:82) ∞ dk k S ( k ) = 0 as we have just seen in Eq.(48).Therefore the contribution of the Φ ∞ Φ1 term is zero to a very good approximation, which shows that the solution of theΦ equation is indeed of the form Eq.(50). We have checked that this decoupling is properly satisfied asymptoticallyby the results of our numerical solution.The resulting equation for F ( q ) is: F ( q ) = 21 + q − π (cid:90) ∞ dk k √ k − F ( k )2 + k + q (51)This equation is easily solved and gives a well behaved solution. This is most easily understood if one notes thatEq.(51) has an interesting physical interpretation. Indeed, as we have seen, Φ( q , q ) describes the scattering of twofree fermions on a dimer. In the large mass ratio limit, it is reasonable to assume that the free heavy fermion doesnot play any role and that only the scattering of the light fermion on the dimer is relevant. In this case we are backto a problem first investigated by Skorniakov and Ter-Martirosian [17] for the case of equal masses, and for which wehave recently found an analytical solution in the case of very different masses [16]. However there is a slight differencebetween our case and the fermion-dimer scattering length problem. In this last one, since the kinetic energy is zero,the total energy is just the dimer binding energy − E b . In our case, since by definition of Φ the final state is made oftwo dimers with zero kinetic energy, the total energy is − E b , that is twice the dimer binding energy. The two terms2 present in the integral in the right-hand side of Eq.(51) can be tracked back to this total energy, taking into accountthat we have used reduced units. If we really had a fermion-dimer problem, these two 2 should be replaced by two 1.In this case, making this substitution, we would rather have the equation: F (cid:48) ( q ) = 21 + q − π (cid:90) ∞ dk k √ k − F (cid:48) ( k )1 + k + q (52)Making the change F (cid:48) ( q ) = 2 a ( q ) / ( (cid:112) q + 1) we end up with: a ( q ) (cid:112) q + 1 = 11 + q − π (cid:90) ∞ dk a ( k )1 + k + q (53)which is exactly the equation we had obtained [16] when one fermion in the dimer is very heavy while the two otherones are very light. We have found the analytical solution a ( q ) = 1 / (1 + q ). This makes it easy to understand thatthe solution of Eq.(51) is very similar and indeed we have found the analytical solution: F ( q ) = 4(1 + q )( (cid:112) q + 1) (54)This analysis makes it also possible to understand physically the factorization Eq.(50). Indeed since the evolution ofthe heavy and of the light fermions decouples, it is quite natural that in the vertex Φ, which describes their evolutionin the presence of a dimer, their contributions factorize as it is the case for the wavefunction of two independentsystems.Having seen that the solution Eq.(50) for Φ is perfectly acceptable, we can substitute it in the equation for t Eq.(42). However in the term t ∞ Φ ( q ), given by Eq.(38), we can make use of the fact that we are only interestedin small values of the variable q since we know that ¯ t ∞ ( q ) is peaked around the origin. In this case writing againln[1 + ( k (cid:48) + q ) ] / [1 + ( k (cid:48) − q ) ] (cid:39) k (cid:48) q/ (1 + q ) and substituting Eq.(50) leads to: t ∞ Φ ( q ) = 4 π
11 + q (cid:90) ∞ dk k k F ( k ) √ k − (cid:90) ∞ dk (cid:48) k (cid:48) S ( k (cid:48) ) = 0 (55)where the last equality comes again from the property Eq.(48). Hence Φ disappears entirely from the equation for t and we are left explicitly with:1 r ¯ t ∞ ( q ) = 4 π q + 4 − π q (cid:90) ∞ dQ ¯ t ∞ ( Q ) Q ln 4 + ( Q + q ) Q − q ) (56)We have checked numerically that taking Φ ∞ ( q , q ) = 0 in Eq.(41) and Eq.(42) leads to the correct solution. Theexact numerical solution of Eq.(42) with Φ ∞ ( q , q ) = 0 leads for a to the results displayed in Fig. 3. We see that,2for a mass ratio above 10, it gives a fairly good result for a , and that when increasing further the mass ratio theresult converges toward the exact one. We note that the disappearance of Φ from the equation for t means thatthe approximation made by Pieri and Strinati [9] in their work on the dimer-dimer scattering length is fully valid inthe limit of large mass ratio. Indeed they made a ladder approximation, neglecting the possible breaking of a dimerdescribed by Φ and retaining only repeated scattering between the two dimers. We see that, since for the equal masscase, this approximation gives a (cid:39) .
78 instead of the exact [10, 12] a (cid:39) .
60 and since it becomes exact in the limitof very different masses, it turns out to be quite a good approximation for any value of the mass ratio.If we perform as a final step the change of function ˜ t ∞ ( q ) = π ¯ t ∞ ( q ), Eq.(56) becomes:1 r ˜ t ∞ ( q ) = 1 q + 1 − π q (cid:90) ∞ dQ ˜ t ∞ ( Q ) Q ln 1 + ( Q + q ) Q − q ) (57)This equation is identical to the one we had in the fermion-dimer problem [16] when we considered the large massratio domain. The only apparent difference is that, in the left-hand side of the equation, the coefficient in [16] is m ↓ /m ↑ , while in the present case it is 1 /r = m ↑ /m ↓ . However since in [16], the considered limit is m ↓ /m ↑ → m ↓ /m ↑ → ∞ , and since the scattering length a is unchanged under the exchange m ↑ ↔ m ↓ , theequations are indeed identical.Hence we have ˜ t ∞ (0) = ¯ a = a /a in terms of the result obtained in [16]. Using Eq.(34) and making the exchange m ↑ ↔ m ↓ , we obtain our final result for the value of the dimer-dimer scattering length in this large mass ratio limit m ↑ /m ↓ → ∞ : a = a a (cid:20) ln( m ↑ /m ↓ ) − ln(ln( m ↑ /m ↓ )) + 2 C (cid:21) (58)where C = 0 . ... is the Euler constant. On the other hand for m ↑ = m ↓ we have a (cid:39) .
60 while a (cid:39) . a (cid:39) a /
2. Hence it can be guessed that the equality a = a / a and a / ∼ .
1, obtained for ln( m ↑ /m ↓ ) ∼
2. This difference is naturally expected since a isinvariant under m ↑ ↔ m ↓ while this is not the case for a .The last expression in our result Eq.(58), which is displayed in Fig. 4, is in agreement with the work of Marcelis,Kokkelmans, Shlyapnikov and Petrov (MKSP) [18] for a in this large m ↑ /m ↓ regime. They addressed this problemwith the 4-body Schr¨odinger equation, which they solved in this regime by a Born-Oppenheimer approximation. Theyfound the approximate relation 2¯ a = 2 C +ln( m ↑ / m ↓ ¯ a ). To dominant order (i.e. omitting the 2¯ a in the right-handside) it gives 2¯ a = 2 C + ln( m ↑ /m ↓ ) in agreement with two terms of our Eq.(58). Our third term is recovered byinserting this expression in the right-hand side of their relation, corresponding to the next step in a recursive solutionof this equation, and keeping only the dominant contribution.In this spirit one can find an analytical formula, slightly different from Eq.(58) but equivalent for large m ↑ /m ↓ ,which is much closer to the exact numerical result than our Eq.(58) or also than the exact numerical solution (alsodisplayed in Fig. 4) of MKSP approximate relation. In Eq.(58) the trouble comes at low m ↑ /m ↓ from the divergenceof ln(ln( m ↑ /m ↓ )) when m ↑ /m ↓ →
1. This is cured by introducing a constant in the logarithm. This constant couldbe adjusted for fine-tuning. But taking it equal to 1 is both simple and in very good agreement with numerics. Thisleads to: a = a (cid:20) ln( m ↑ /m ↓ ) + 2 C − ln[ln( m ↑ /m ↓ ) + 1] (cid:21) (59)As it can seen from Fig. 4, it is fairly close to the exact numerical result for a (and even extremely close to a / m ↑ ↔ m ↓ (which leads to a zero derivative with respect to ln( m ↑ /m ↓ ) for m ↑ /m ↓ = 1), while an approximateanalytical expression will not have this property. However we see that Eq.(59) has precisely a zero derivative withrespect to ln( m ↑ /m ↓ )) for m ↑ /m ↓ = 1. VI. CONCLUSION
In this paper we have studied the dimer-dimer scattering length a for a two-component Fermi mixture with differentfermion masses m ↑ and m ↓ respectively. For this purpose we have made use of the exact field theoretical methodalready present in the literature [12, 14, 15]. The corresponding equations have been solved numerically for any valueof the mass ratio m ↑ /m ↓ . However our main aim has been to study the large mass ratio domain. In this range we have3 Eq.58Eq.59MKSP solution a / −
13 ln( r ) a / a FIG. 4: (Color online) Dimer-dimer scattering length a as a function of the mass ratio r (logarithmic scale). The red full lineis the exact numerical result obtained from Eq.(12) and Eq.(13). The purple dotted-dashed line is the exact numerical resultfor a /
2. The black dashed line is Eq.(58) The blue dotted line is Eq.(59). The light blue double-dotted-dashed line is thenumerical solution of the equation 2¯ a = 2 C + ln( m ↑ / m ↓ ¯ a ) found by MKSP [18]. been able to simplify the equations enough to obtain an analytical solution. More specifically we have shown thatour final equation is essentially the same as the one obtained in the fermion-dimer scattering problem with scatteringlength a . In this way we have shown that, for large mass ratio, we have the very simple result a = a /
2. Since thisrelation is also correct with a very good precision for m ↑ = m ↓ , a (cid:39) a / m ↑ = m ↓ . In this case there is no general justification forthis approximation. Nevertheless it gives for the scattering length a result a (cid:39) . a which is not so far from theexact one a (cid:39) . a (mostly if one keeps in mind the simple Born result a = 2 a ). As a consequence we come to theimportant conclusion that the Pieri and Strinati approximation is a fairly good one whatever the mass ratio. This isquite interesting since the processes they retain are much simpler than the ones which have to be considered in full4generality. [1] For a review, see S. Giorgini, L. P. Pitaevskii and S. Stringari, Rev.Mod.Phys. , 1215 (2008).[2] V. N. Popov, Zh. Eksp. Teor. Phys. , 1550 (1966), [Sov. Phys. JETP , 1034 (1966)].[3] L. V. Keldysh and A. N. Kozlov, Zh. Eksp. Teor. Phys. , 978 (1968) [Sov. Phys. JETP , 521 (1968)][4] D. M. Eagles, Phys. Rev. , 456 (1969); D.M. Eagles, R.J. Tainsh, C. Andrikidis, Physica C , 48 (1989).[5] A. J. Leggett, J. Phys. (Paris), Colloq. , C7-19 (1980); in Modern Trends in the Theory of Condensed Matter , edited byA. Pekalski and J. Przystawa (Springer, Berlin)[6] P. Nozi`eres and S. Schmitt-Rink, J. LowTemp. Phys. , 195 (1985).[7] R. Haussmann, Z. Phys. B: Condens. Matter , 291 (1993).[8] S.A.R. S´a de Melo, M. Randeria and J.R. Engelbrecht, Phys.Rev.Lett. , 3202 (1993).[9] P. Pieri and G. C. Strinati, Phys. Rev. B , 15370 (2000).[10] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. Lett. , 090404 (2004).[11] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. A , 012708 (2005).[12] I.V. Brodsky, A.V. Klaptsov, M.Yu. Kagan, R. Combescot and X. Leyronas, J.E.T.P. Letters , 273 (2005) and Phys.Rev. A , 032724 (2006).[13] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, J. Phys. B: At. Mol. Opt. Phys. S645 (2005); Proceedings ofthe International School of Physics Enrico Fermi Course CLXIV, Edited by M. Inguscio, W. Ketterle and C. Salomon :Ultracold Fermi Gases (Varenna, June 2006, IOS Press, Amsterdam 2008), p.385[14] M. Iskin and C. A. R. S´a de Melo, Phys. Rev. A , 013625 (2008).[15] J. Levinsen and D. S. Petrov, Eur. Phys. J. D , 67 (2011).[16] F. Alzetto, R. Combescot and X. Leyronas, Phys. Rev. A , 062706 (2010).[17] G. V. Skorniakov and K. A. Ter-Martirosian, Zh. Eksp. Teor. Fiz. , 775 (1956) [Sov. Phys. JETP , 648 (1957)].[18] B. Marcelis, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and D. S. Petrov, Phys. Rev. A77