Contact parameters in two dimensions for general three-body systems

We study the two dimensional three-body problem in the general case of three distinguishable particles interacting through zero-range potentials. The Faddeev decomposition is used to write the momentum-space wave function. We show that the large-momentum asymptotic spectator function has the same functional form as derived previously for three identical particles. We derive analytic relations between the three different Faddeev components for three distinguishable particles. We investigate the one-body momentum distributions both analytically and numerically and analyze the tail of the distributions to obtain two- and three-body contact parameters. We specialize from the general cases to examples of two identical, interacting or non-interacting, particles. We find that the two-body contact parameter is not a universal constant in the general case and show that the universality is recovered when a subsystem is composed of two identical non-interacting particles. We also show that the three-body contact parameter is negligible in the case of one non-interacting subsystem compared to the situation where all subsystem are bound. As example, we present results for mixtures of Lithium with two Cesium or two Potassium atoms, which are systems of current experimental interest.


Introduction
The surprising and unexpected predictions from quantum mechanics have been challenging our classical intuition for a century. Since then, efforts in both theoretical and experimental fields have increasingly aimed at a better understanding of quantum systems. In particular, experiments with cold atomic gases [1] are an interesting way of building and probing quantum systems: properties of atomic condensates near absolute zero temperature are governed by pure quantum effects. An interesting example of an unexpected quantum mechanical prediction that was experimentally confirmed in cold atomic gases is the so-called Efimov effect, which predicts that three identical bosons interacting through short range potentials present infinitely many bound states, where the energies between states are geometrically spaced. This effect was predicted by Efimov in 1970 [2] and was experimentally verified in cold atomic gases experiments in 2006 [3]. The Efimov effect happens for three dimensional (3D) systems, while the quantum theory predicts that the same two dimensional (2D) system presents only two bound states and no Efimov effect [4,5,6,7,8]. The theoretical difference arising from changing system dimension could most likely also be verified in cold atomic gases experiments in the near future since experimentalists are already able to change dimensionality of such systems and build experiments in effectively one (1D) or two (2D) spatial dimensions.
Another important theoretical prediction, that was recently reported in [9], is the emergence of a parameter in the study of two-component Fermi gases, the two-body contact parameter, C 2 , which connects universal relations between two-body correlations and many-body properties [9,10,11,12,13,14,15,16,17,18,19]. This parameter can most easily be defined by considering the single-particle momentum distribution of the system, n(q). In the limit where q → ∞, one finds lim q→∞ q 4 n(q) = C 2 . (1) As mentioned above, this parameter appears in a number of universal relations for both few-and many-body properties. These relations also hold for bosons and were confirmed in cold atomic gases experiments for two-component Fermi gases [20,21] and for bosons [22]. One way to determine this parameter is to find the coefficient in the leading order asymptotic behavior of the one-body momentum density of few-body systems. The next order in this expansion defines the three-body contact parameter, C 3 , but since the Pauli principle suppresses the short-range correlations for two-component Fermi gases, the three-body parameter is only important for bosons [14,18]. In a gas of identical bosons in 3D the Efimov effect occurs and one finds a momentum distribution of the form n(q) → C 2 q 4 + C 3 sin(s 0 ln(q) + δ) q 5 for q → ∞, where s 0 is the Efimov parameter [2] and δ is constant [17]. The two-and three-body contact parameters were studied for a 3D system of three identical bosons in [17,16,23] and for mixed-species systems in [24]. These results show that the influence of non-equal masses in three-body systems goes beyond changing the contact parameters values. In 3D the sub-leading term, which defines C 3 , changes to a different functional form when the masses are not equal. In view of that, one may ask what changes could we get when dealing with mixed species systems in 2D. It is known that mixed-species systems have a richer energy spectrum in 2D comparing with symmetric mass systems [25,26,27]. The study of the momentum distribution for three identical bosons in 2D was reported in [28] where the two-body contact parameter is found to be a universal constant, in the sense that C 2 E 3 is the same for both three-body bound states of energy E 3 . The leading order term of the momentum distribution at large momenta has the same inverse quartic form in 1D [29,30], 2D, and 3D. This can be derived on general grounds and is intuitively connected to the behavior of the free propagator for the particles [23,31]. However, the three-body contact parameter and the functional form of the sub-leading term were showed to present a very different behavior in 2D as compared to 3D [28] although no analytical results have been presented to estimate the value of the contact parameter.
In this paper we study cylindrically symmetric three-body bound states of 2D systems composed of three distinguishable particles with attractive short-range interactions. We derive analytic expressions for C 2 and C 3 and numerically obtain the one-body momentum density to verify our results. Unlike 3D systems, the subleading order in the asymptotic momentum density presents the same functional form for both equal masses and mixed-species systems. We find that C 2 no longer shows universal behavior in the general case but do show that the universality is recovered in at least one special case of two identical non-interacting particles. We also extend our asymptotic formulas to the full range of momenta and use it to give an analytic expression for C 2 for the ground state.
The paper is organized as follows. The formalism and the quantities that appear in the work are properly shown and defined in Sec. 2. The analytic formulas to the asymptotic spectator function are discussed in Sec. 3 and the one-body large momentum behaviors are derived in Sec. 4. The numeric results with an appropriate discussion is presented in Sec. 5. Discussion, conclusions, and an outlook are given in Sec. 6.

Formalism
We consider the two dimensional (2D) problem of three interacting particles of masses m A , m B , m C which are pairwise bound with energies E AB , E AC , E BC . The interaction is assumed to be described as attractive zero-range potentials and the resulting s−wave three-body bound state of energy −E 3 is fully determined by these six parameters: three two-body energies and three masses. We shall only investigate bound states and we therefore let E 3 > 0 denote the absolute value of the binding energy. We use the Faddeev decomposition to write the momentum space wave function as [32] where the integration variable originating from (3) are properly redefined to simplify the arguments of the spectator functions in the integrands. Only n 4 is then left with an angular dependence through the spectator functions. We emphasize that the distributions for the other particles can be obtained by cyclic permutations of (α, β, γ) in these expressions.

Spectator functions
The spectator functions are the key ingredients. They can be characterized by their behavior in small and large momentum limits. We are first of all interested in large momenta, but we shall as well extract the behavior for small momenta. Hopefully these pieces can be put together in a coherent structure.

Large-momentum behavior
For three identical bosons all spectator functions are equal, and the large-momentum behavior was previously found to be [28] lim where the constant Γ 0 depends on which excited state we focus on. Corrections, δf (q), to (10) must vanish faster than ln(q)/q 2 for q → ∞, i.e. δf (q)q 2 / ln(q) → 0. Henceforth, we will refer to (10) as the large-momentum leading order behaviour of the spectator function.
For three distinct particles, we have three different spectator functions. However, their large-momentum asymptotic behavior all remain identical, when all three twobody subsystems are bound, except for individual proportionality factors. To prove this we carry out the angular integrals in (4), which immediately gives The two terms in (11) have the same form, and one can be obtained from the other by interchange of β and γ. It therefore suffices to calculate the first integral in (11). The contribution for large q can in principle be collected from k-values ranging from zero to infinity. To separate small and large k-contributions we divide the integration into two intervals, that is from zero to a large (q-independent) momentum Λ ≫ √ E 3 , and from Λ to infinity. Thus where the dots indicate that the second term in (11) should be added. For q → ∞ the first term, f α,1 , on the right-hand-side of (12) goes to zero as where we used that τ α (q, E 3 ) → [2m βγ ln q] −1 , and that both E 3 and k 2 2m βγ are much smaller than q 2 2mαγ . The integral in (13) is finite and only weakly q−dependent for large q ≫ Λ. The asymptotic spectator function in (10) can be inserted in the second term, f α,2 , on the left-hand-side of (12), because we are in the asymptotic limit where k > Λ. For q → ∞ we then get where we changed integration variable, k = qy, in the last expression. Carrying out the two integrals we get where we used that the integrals in the right-hand-side of (16) and (17) are finite and their contributions can be neglected when q → ∞ in comparison with the terms maintained.
In total, the spectator functions in (11) are now found by inserting (16) and (17) in (15). Notice that the contribution from (17) has to be multiplied by ln q. With the γ −β interchange we also get the second term in the right-hand-side of (11). The leading order large-momentum behavior of the spectator functions are therefore Replacing f α (q α ) in (18) by its conjectured asymptotic form, (10), we find a system of three linear equations for the three unknown, Γ α = mαγ 2m βγ Γ β + m αβ 2m βγ Γ γ , which can be rewritten as m βγ Γ α = m αγ Γ β = Γ γ m αβ := Γ. The leading order large-momentum asymptotic behavior for the three spectator functions are then This result relates the asymptotic behavior of the three spectator functions for one state. The remaining constant Γ still depends on which excited state we consider, and furthermore also on two-body masses and two-body energies.
The derived large-momentum asymptotic behavior and the coefficients in (19) beautifully agree with the numerical calculation. In figure 1 we plot the difference f α (q)− Γ m βγ ln q q 2 as a function of the momentum q for the two different spectator functions for the 133 Cs-133 Cs- 6 Li system. We also show the same difference for a different system with three identical bosons. First of all, this demonstrates that the large-momentum behavior always is ln q/q 2 for any 2D spectator function. Secondly, for a given state the three cyclic permutations of f α (q) m βγ for large q approach the same constant Γ times ln q/q 2 . This general large-momentum behavior is further demonstrated in figure 2 for a system of three distinct particles. The numerically calculated points are compared to the full lines obtained from (19). This comparison is again consistent with the derived asymptotic behavior, and furthermore exhibit the rate and accuracy of the convergence. The limit is reached within 10% and 1% already for We see that (19) exactly describes the asymptotic spectator function within our accuracy.

Parametrizing from small to large momenta
The asymptotic spectator function in (19) seems to be a good approximation even for moderate values of q, e.g. q ≈ 3 √ E 3 . We also have information about the large- 2.0 Figure 2. Ratios between the three distinct spectator function for a generic case of three distinct particles. Discrete points are the ratios between spectator functions numerically calculated through (4) and full lines are ratios between coefficients in (19).
distance behavior for a given binding energy, that is exp(−κρ), where κ is related to the binding energy and ρ is the hyperradius. Fourier transformation then relates to the small momentum limit with an overall behavior of (D + q 2 ) −1 , where D is a constant related to the energy. This perfectly matches (3) when two Jacobi momenta are present as in the three-body system. We therefore attempt a parametrization combining the expected small momenta with the known large-momentum behavior: where f α (0) is a normalization constant which satisfies d 2 q α n(q α ) = 1.
We should first emphasize that excited states with the same angular structure must have a different number of radial nodes. Therefore we concentrate here only on the ground state. The expression in (20) for the three ground state spectator functions parametrizes the small momentum behavior almost perfectly for the case of three distinguishable particles. This is seen in Fig.3 where we compare numerical and parametrized solution. However, when small momenta are reproduced the large-momentum limit deviates in overall normalization, although with the same qdependence. Surprisingly, the analytic expression is most successful for the spectator function related to the heaviest particle in the three-body system. This large-momentum mismatch is due to the normalization choice in (20), which is chosen to exactly reproduce the q = 0 limit.  Figure 3. Comparison between the analytic spectator function estimated for the ground state, given in (20) and the numeric solution of (4), for a 133 Cs-133 Cs- 6 Li system.

One-body large momentum density
The one-body density functions are observable quantities. The most directly measurable part is the limit of large momenta. We therefore separately consider the largemomentum limit of the four terms in (6) to (9). We employ the method sketched in [28] and used to present numerical results for three identical bosons in 2D. Here we shall give more details and generalize to systems of three distinguishable particles.
In three dimensions (3D), the corresponding problem was solved by inserting the asymptotic spectator function (19) into each of the four terms in (6) to (9), and evaluating the corresponding integrals [17,24]. This procedure is not guaranteed to work in 2D because momenta smaller than the asymptotic values may contribute in the integrands. However, for 3D it was shown that leading order in the integrands is sufficient to provide both leading and next-to-leading order of the one-body momentum distributions. The details of these calculation in 3D can be found in [17] for three identical bosons and in [24] for mass-imbalanced systems.
The large-momentum spectator functions change a lot as dimensionality is changed, going from sin(ln(q))/q 2 in 3D to ln(q)/q 2 in 2D. If we try to naively proceed in 2D as successfully done in 3D, the integrals diverge. We can circumvent this divergence problem by following the procedure used in the derivation of the asymptotic spectator functions. In the following, we work out each of the four momentum components defined in (6) to (9). In addition we must simultaneously consider the next-to-leading order term arising from the dominant n 2 -term.
•n 1 (q α ) : This term is straightforward to calculate. The argument of the spectator function in (6) does not depend on the integration variable. The large-momentum limit is then found by replacing the spectator function by its asymptotic form and taking the large q limit after a simple integration. We get then •n 2 (q α ) : We integrate the two terms in (7) over the angle as allowed by the simple structure where the spectator function is angle independent. The result is then expanded for large q.
where the last equality defines C βγ which we call the two-body contact parameter for the βγ two-body system. The second term on the right-hand-side, n 5 (q α ), gives the next-to-leading term in the expansion of n 2 (q α ). It turns out that this term has the same asymptotic behavior as n 3 (q α ) and n 4 (q α ). We must consequently keep it, but we postpone the derivation. We emphasize that the one-body large-momentum leading order comes only from n 2 (q α ). Here we cannot replace the spectator function by its asymptotic expression, because the main contribution to ∞ 0 dk k |f α (k)| 2 arises from small k. This replacement would therefore lead to a completely wrong result. However, this is not always the case, as we shall see later for n 5 (q α ).
•n 3 (q α ) : The structure of n 3 (q α ) in (8) is similar to n 2 (q α ) in (7). The only difference is that the spectator function under the integration sign now is not squared. This small functional difference leads to a completely different result. As in the previous case, we can still carry out the angular integration, which only involves the denominator. Integrating (8) over the angle we get Here, the difference between n 2 and n 3 becomes important, since ∞ 0 dk k f (k) is divergent and we can not expand (24) as we did for (22). We shall instead proceed as done in obtaining the asymptotic spectator function. We divide the integration in (24) at a large, but finite, momentum, Λ ≫ √ E 3 , and each term on the right-hand-side is split in two others. The two terms only differ by simple factors, and we therefore only give details for the first term. Changing variables to k = q α y, (24) becomes where f β (k) is replaced by its asymptotic form and E 3 is neglected in the second term on the right-hand-side, where √ E 3 ≪ Λ and k > Λ. In the limit q α → ∞, the integral vanishes in the first term which therefore does not contribute to the large-momentum limit. The integrals in the second term are where The function g(y) and its limits ensure that the integrals on the right-hand-side of Eqs. (26) and (27) are finite and their contributions to the momentum distribution can be neglected when q α → ∞. Finally, inserting the results given in (26) and (27) into (25) and replacing the spectator function f α (q α ) by its asymptotic form, the leading order term of the one-body momentum distribution from n 3 (q α ) is given by where the second term in the right-hand-side of (24) is recovered and added by the interchange of m αγ → m αβ in (25) to (27). Although n 2 (q α ) and n 3 (q α ) have rather similar form, their contributions to the one-body large momentum density are quite different. As we shall see later, the nextto-leading order, n 5 (q α ), of n 2 (q α ) is comparable to the n 3 (q α ) leading order, given in (30).
•n 4 (q α ) : This is the most complicated of the four additive terms in the one-body momentum density. The angular dependence in both spectator arguments cannot be removed simultaneously by variable change. The formulation in (9) has the advantage that the argument in f γ (|k + q α |) (or in f β (|k + q α |)) is never small in the limit of large q α . This is in contrast to a choice of variables where the numerator in the first term of (9) would be f γ (k)f β (|k − q α |), and the argument in f β would consequently be small as soon as k is comparable to q α . The main contribution to the integrals in (9) arise from small k. For large q α , we can then use the approximation, f γ (|k + q α |) ≈ f γ (q α ) (or f β (|k + q α |) ≈ f β (q α )). The integrals are then identical to the terms of n 3 in (8). By keeping track of the slightly different mass factors we therefore immediately get the asymptotic limit to be •n 5 (q α ) : This is the next-to-leading order contribution from the n 2 (q α ) term. It turns out that this term has the same large-momentum behavior as the leading orders of both n 3 (q α ) and n 4 (q α ). By definition we have which can be rewritten in details as where the dots denote the last term in (22) obtained by interchange of β and γ.
The tempting procedure is now to expand the integrand around q α = ∞ assuming that q α overwhelms all terms in this expression. This immediately leads to integrals corresponding to the cubic moment of the spectator function which however is not converging. On the other hand (33) is perfectly well defined due to the large-k cut-off from the denominator. In fact, the spectator function is multiplied by k 3 and 1/k 3 at small and large k-values, respectively. The integrand therefore has a maximum where the main contribution to n 5 arises. This peak in k moves towards infinity proportional to q. To compute n 5 (q α ) we then divide the integration into two intervals, that is from zero to a finite but very large k-value, Λ s , and from Λ s to infinity. The small momentum interval, k/q α ≪ 1, allows an expansion in k/q α leading to the following contribution n 5,1 (q α ): where η is a constant. Thus the contribution from this small momentum integration vanish with the 6 th power of q α which is faster than the other sub-leading orders we want to keep. We choose Λ s sufficiently large for the spectator function to reach its asymptotic behavior in (19). The large interval integration can now be performed by omitting the small E 3 -terms and change of integration variable to y, k 2 = yq 2 α , i.e. n 5,2 (q α ) = πΓ 2 q 6 α ∞ Λ 2 s /q 2 α dy y 2 ln 2 (y) + ln 2 (q 2 α ) + 2 ln y ln(q 2 α ) where the large y-limit behaves like ln 2 y/y 4 and therefore assuring rapid convergence, whereas the integrand for small y behaves like (ln 2 (y) + ln 2 (q 2 ) + 2 ln y ln(q 2 ))/y. Thus, the integration from an arbitrary minimum value, y L (independent of q α ), of y > Λ 2 s /q 2 α yields a q α -independent value except for the logarithmic factors and q α in the numerator. Thus the large-q α dependence is found from very small values of y close to the lower, and vanishing, limit. In total we get by expansion in small y that the limit for large q α approaches zero as Together with the term from interchange of β and γ in (19) we get in total that

Two-and Three-body contact parameters
We first collect the analytically derived relations, and secondly we compare to numerically calculated values.

Analytic expressions
Two-and three-body contact parameters are defined via the large-momentum one-body density. The two-body contact parameter, C βγ , is the proportionality constant of the leading order q −4 α term, which arises solely from n 2 (q α ) in (23). For three distinguishable particles we have three contact parameters each related to the momentum distribution of one particle. The definition is already given in (23). They are related through and cyclic permutations. For a specific system, where two of the particles are noninteracting in 2D, the corresponding two-body energy vanishes, E βγ = 0 [32,33]. Then, from (4) the spectator function also vanishes, f α (q) = 0, and (38) reduces to In this case, we have this simple relation between the three two-body contact parameters. This relation between different two-body parameters does not depend on system dimension. Although our calculations are in 2D, this relation in (39) applies as well for 3D systems with one non-interacting subsystem. We emphasize that a non-interacting system and a vanishing two-body energy is not the same in 3D where some attraction is necessary to provide a state with zero binding energy. The three-body contact parameter, C βγ,α , is defined as the proportionality constant on the next-to-leading order in the one-body large-momentum density distribution. For distinguishable particles we have again three of these parameters, each related to one of the particle's momentum distributions. The asymptotic behavior, ln 3 (q α )/q 6 α , receives contributions from the three terms specified in (30), (31) and (37). In total we have It is worth emphasizing that only a logarithmic factor distinguishes the behavior of the three-body contact term from the next order, ln 2 (q α )/q 6 α which arises from n 1 as well as from n 2 , n 3 , and n 5 . In practical measurements, it must be a huge challenge to distinguish between terms differing by only one power of ln(q α ).
For the three-body contact parameter, (40) with only one non-interacting two-body system, we get which is obtained by collecting contributions from only the non-vanishing n 4 and n 5 terms (since f α (q) = 0, n 1 and n 3 do not contribute). Cyclic permutations of the indices in (40) and (41) now allow the conclusion that the three different three-body contact parameters are related by the mass factors in (40) and (41). This conclusion holds for all excited states. Universality of independence of excited state is another matter and in fact not found numerically.

Numerical results
The results in the preceding subsection hold for any mass-imbalanced three-body system. Such a system has six independent parameters, which are reduced to four by choosing one mass and one energy as units [33]. This merely implies that all results can be expressed as ratios of masses and energies, and in this way providing very useful scaling relations. Results depending on four independent parameters are still hard to display and digest.
To built up our understanding, we now focus on systems composed of two identical particles, A, and a distinct one, C. Such a system has four independent parameters from the beginning, which are reduced to two after choice of units. From now on we shall use E AC and m A as our energy and mass units, and for simplicity we introduce the mass ratio m = m C m A . In these units energies and momenta appearing in the equations must be multiplied by E AC and √ m A E AC , respectively. For this system the two-body contact parameters in (38) are given by For three identical particles where all masses and interactions are the same, C AA = C AC = C 2 , and the quantity C 2 E 3 is a universal constant in 2D [28,18]. To be explicit, this quantity has the same value for the only two existing bound states, ground and first excited state. Mass-imbalanced systems have a richer energy spectrum with many excited states [33,27]. Maintaining the universal conditions for all excited states is obviously more demanding.
Detailed investigations reveal that when the mass-energy symmetry is broken, the universality of C 2 E 3 does not hold any more. The two two-body contact parameters defined in (42) and (43) divided by the three-body energy are not the same for all possible bound states in the general case. However, in at least one special case of two identical noninteracting particles, E AA = 0, the universality is recovered. This is implied by f C = 0 as seen from the set of coupled homogeneous integrals equations (4). Then the two universal two-body contact parameters are related, that is We illustrate in Fig. 4 how the two-body contact parameters vary with excitation energy for a mass-asymmetric system. We choose 133 Cs− 133 Cs− 6 Li corresponding to A = 133 Cs and C = 6 Li. This system has four excited states at energies depending on the size of E AA , and the large-momentum limit of constants is reached in all cases. For E AA = 0, universality is observed, since all two-body contact values, C AC /E 3 , are equal in units of the three-body energy. This case is rather special because two particles do not interact and the three-body structure is determined by the identical two-body interactions between the other two subsystems. In other words the large-momentum limit of particle A is determined universally by the properties of the A − C subsystem. The other contact parameter, C AA /E 3 , is also universal and following from (44). This picture changes when all particles interact as seen in Fig. 4 for E AA = 1. Now the large-momentum limit, still constants independent of momentum, changes with the excitation energy. The systematics is that both C AA /E 3 and C AC /E 3 as function of excitation energy move towards the corresponding values for E AA = 0, one from below and the other from above. First the non-universality is understandable, since the interaction of the two identical particles now must affect the three-body structure at small distances, and hence at large momenta. However, as the three-body binding energy decreases, the size of the system increases and details of the short-distance structure becomes less important.
The quantities C AA E 3 and 2π E 3 ∞ 0 dk k |f C (k)| 2 are defined by the limiting large-q behavior of n 2 in (23). Plotting the corresponding pieces of n 2 (q)q 4 as function of q lead to figures much similar to Fig. 4, where different excitation dependent lines emerge for E AA = 1, while they all coincide for E AA = 0. The constant values for 0 dk k |f C (k)| 2 in the limit q → ∞, are shown in Table 1, for two different interactions and two different systems represented by C = 6 Li and A = 133 Cs or A = 39 K. These results of numerical calculations confirm the systematics described above in complete agreement with (43) and (44).
In general, for two identical particles the two-body contact parameters divided by the three-body energy depend on the mass ratio m. The dependence change from universal for E AA = 0 to non-universal for E AA = 1. The mass dependence for ground states is shown in Fig. 5, where we see that the ratio C AA C AC = 2 in (44) holds for E AA = 0 in the entire mass interval investigated. We also see how the second term on the righthand-side of (43) affects the relation between the two two-body contact parameters. Fig. 5 shows that the values rapidly increase from small m up to 1 and become almost constant above m ≈ 5. This behavior is similar to mass-imbalanced system in 3D [24].
We can estimate the two-body contact parameter dependence on excitation energy by use of the approximation to the ground state in (20). Inserted in (42) we find an   Fig. 4.
. A comparison between this approximation and the numerical results is shown in Fig.  6. We see that (45) provides a fairly good estimate, which is accurate within 5% for small m, around 10% for m > 1, and within about 20% deviation in the worst case of m = 1. The divergence in (45) for E 3 → 1 means that the two-body contact parameters diverge when the three-body system approaches this threshold of binding. This does not reveal the full energy dependence since the normalization factor, f 2 A (0), also is state and energy dependent.
The non-universality of the two-body contact parameters does not encourage universality investigations of the three-body contact parameter, which is related to a sub-leading order. However, at least the system with two non-interacting identical particles turned out to be universal and may lead to an interesting large-momentum three-body structure. As before, by inserting E AA = 0 in the set of coupled integral equations (4) we find f C (q C ) = 0. Then (6) to (9) show directly that n 1 (q c ) and n 3 (q C ) vanish when f C (q C ) = 0, leaving only possible contributions from n 4 (q C ) and n 5 (q C ).
We show in Fig. 7 the sub-leading order of the large-momentum distribution multiplied by q 6 C / ln 3 (q C ), that is C AA,C , as functions of q C for the four bound states for a system where A = 133 Cs and C = 6 Li and for both E AA = 1 and E AA = 0. We only show one of these three-body contact parameters defined in (40) and (41) since the other one, C AC,A , is related state-by-state through the mass factors in (40) and (41). The momentum dependence approach the predicted constancy at large q C but by increasing or decreasing for interacting or non-interacting identical particles, respectively. We divided by the three-body energy to see if a simple energy scaling could explain the differences. Not surprisingly, a more complicated and non-universal behavior is present. However, it is striking to see that this sub-leading order in the large-momentum limit is negligibly small for non-interacting compared to interacting identical particles. This implies that a negligible three-body contact parameter combined with a universal two-body contact parameter can be taken as a signature of a two-body non-interacting subsystem within a three-body system in 2D. for each bound state labeled as n in a system composed with two identical ( A = 133 Cs ) particles and a distinct one ( C = 6 Li ) as a function of the momentum q for both E AA = 1 and E AA = 0.

Discussion and Outlook
In this work, we have considered three-body systems with attractive zero range interactions for general masses and interaction strengths in two dimensions using the Faddeev decomposition to write the momentum-space wave function, through which the one-body momentum density is obtained. The momentum density tail gives the two-and three-body contact parameters, namely C 2 and C 3 , respectively. We derived analytic expressions for the asymptotic spectator functions and for both C 2 and C 3 for three distinguishable bosons.
We found that the asymptotic spectator function for each of the three distinguishable particles has the same functional form as calculated for three identical particles in [28]. Moreover, we showed that the three distinct spectator functions relate to each other through a constant, Γ, properly weighted by reduced masses. These analytic results are supported by accurate numerical calculation, which confirmed both the asymptotic behavior and the relation between the asymptotic expressions for different spectator functions in a generic case of three distinguishable particles.
The spectator functions and their asymptotic behavior define both two-and threebody contact parameters, C 2 and C 3 . The parameters C 2 arise from integration of the spectator functions over all momenta, and both small and large momenta contribute.
In the case of the ground state, we are able to use our knowledge of the asymptotics of the spectator function to infer the behavior for all momenta. We found that the three two-body parameters for a system of three distinguishable particles are related by simple mass scaling. However, these two-body contact parameters are in general not universal in the sense of being independent of the state when more than one excited state is present. In contrast, we find universality for three-body systems with one distinguishable and two identical, non-interacting particles. In that case the third particle apparently does not disturb the short-distance structure arising from the two interacting particles. Hence, the two-body contact parameter turn out to be universal. This is similar to the 3D case and three identical bosons where C 2 is universal in the scaling or Efimov limit where the binding energy is negligible [17].
In 3D systems, the two-body contact has been observed in experiments using time-of-flight and mapping to momentum space [20], Bragg spectroscopy [20,21,22,34,35], or momentum-resolved photoemission spectroscopy (similar to angle-resolved photoemission spectroscopy) [36]. Measuring the sub-leading term and thus accessing C 3 requires more precision which has so far only produced upper limits for the particular case of 87 Rb [22]. In 2D systems the functional form of the sub-leading term is different from the 3D case so it is difficult to compare the cases. However, given that the precision improves continuously it should be possible to also probe the 2D case when tightly squeezing a 3D sample. As we have shown here, the mass ratio can change the values of the contact parameters significantly. We thus expect that mixtures of different atoms is the most promising direction to make a measurement of a 2D contact parameter.
We have analyzed in details two different systems of the heavy-heavy-light type that is relevant for current experiments with cold gas mixtures. In both cases the light particle is 6 Li while the two heavy particles are either both 133 Cs or both 39 K. We find that, unlike the equal mass scenario, the two-body contact parameters are not universal constants when all subsystems are interacting. Here universal means that C 2 divided by the three-body binding energy is independent of which excited state is considered. However, if the two identical particles are not interacting, the heavy-heavy and the heavy-light two-body contact parameters become universal and are related to each other by a factor of two.
The methods presented here are in principle also applicable to 1D setups and it would be interesting to investigate the question of universality of the contact parameters for three-body states there as well. In some respects 1D is easier to handle since zero-range interactions do not require regularization and one can in fact map the 3D scattering length to a 1D equivalent [37] which provides access to confinementinduced resonances that allow the study of the infinite 1D coupling strength limit. This was recently demonstrated for trapped few-fermion systems in 1D [40,41]. In that case the two-body contact can be determined fully analytically using the methods described in [38,39]. It would be very interesting to consider the bosonic case where three-body bound states are possible with or without an in-line trap. In the case of quasi-1D setups where the transverse trapping energy is a relevant scale compared to the binding energies, one needs to also take into account the transverse (typically harmonic) degrees of freedom [42,23]. Our formalism can be adapted to this case as well. Another interesting pursuit would be to long-range interactions using either heteronuclear molecules or atoms with large magnetic dipole moments [43], where the momentum distribution has in fact already been probed in experiments [44]. Bound state formation has been predicted in both single- [45], bi- [46,47,48], and multi-layer systems [49,50,51], as well as in one or several quasi-1D tubes [52,53,54,55,56]. The current formalism should be adaptable to dipolar particles and the contact parameters could subsequently be studied. In particular, in the limit of small binding energy one may in some cases use effective short-range interaction terms to mimic dipolar interactions [56] which makes the implementation through the Faddeev equations considerably simpler. This is of course also the limit in which the contact parameters are most interesting.