Three-particle quantization condition in a finite volume: 1. The role of the three-particle force

Using non-relativistic effective Lagrangians in the particle-dimer picture, we rederive the expression for the energy shift of a loosely bound three-particle bound state of identical bosons in the unitary limit. The effective field theory formalism allows us to investigate the role of the three-particle force, which has not been taken into account in the earlier treatment of the problem. Moreover, we are able to relax the requirement of the unitary limit of infinite scattering length and demonstrate a smooth transition from the weakly bound three-particle state to a two-particle bound state of a particle and a deeply bound dimer.


I. INTRODUCTION
For some time now lattice QCD calculations have been addressing hadron physics problems which involve the dynamics of three or more hadrons. As an example, we quote the calculation of the parameters of the Roper resonance [1-10], which decays -at a substantial rate -into a nucleon and two pions. The finite-volume effects in such few-body systems are expected to be rather pronounced. Hence, understanding these effects is a necessary precondition for investigating intriguing puzzles such as the level ordering of the N * (1440) and the N * (1535). Another reason for studying the three-body dynamics in a finite volume now is the recent advent of lattice QCD calculations of light nuclei [11][12][13] and corresponding calculations in nuclear effective field theory on the lattice [14][15][16][17]. In order to fully exploit these advances, a formalism is needed to translate the "raw" lattice results into physical observables like cross sections into various two-and three-body channels, etc. It is also important that the proposed formalism is not overly complicated and can be used even when only a few data points are available.
The quantization of the energy levels of a three-particle system in a finite volume has been considered first in Ref. [18]. In a series of subsequent papers by different groups [19][20][21][22][23][24][25][26][27][28] further important aspects of the problem have been addressed. We would like to stress, however, that despite the substantial progress made, the formalism is still very complicated and, in our opinion, not quite ready to be used straightforwardly by lattice practitioners (in contrast to, e.g., the Lüscher equation for two-particle elastic scattering [29]). In addition, the relation between the different approaches is not obvious and has not been discussed in the literature so far. At the same time, in Refs. [30][31][32][33], the volume dependence of the discrete three-body spectrum has been investigated for bosons as well as nucleons by solving the bound state equations in a finite volume numerically. An effective field theory in the dimer formalism has been used to derive the finite volume bound state equations and relate the bound state properties to scattering parameters in the infinite volume, which greatly simplifies the handling of the three-body problem. These studies also suggest a strategy for formal investigations of three-body dynamics in a finite volume.
The aim of our work is to provide a simple formalism for the analysis of the present and forthcoming lattice data in a straightforward and transparent manner, and to understand the link with the earlier approaches. To this end, we use the particle-dimer approach, which is very convenient and allows one to achieve this goal with a surprising ease. We also found it justified to split the material in two parts. In the first paper, we consider a very simple system -a shallow three-body bound state in a finite volume -which has been already studied in the literature [19,27]. A leading-order analytic expression for the finite-volume energy shift of this system in the unitary limit is available, and we shall see how this result can easily be obtained in the particle-dimer picture. In addition, we address the following issues which were not considered in previous work: (i) The role of the three-body force is studied explicitly. We demonstrate that this shortrange force does not affect the analytic form of the leading-order volume dependence of the shallow bound state energy. However, it is important to know how the three-body force enters in finite-volume observables. Note that omitting the three-body force altogether renders the three-body problem ill-defined in the infinite volume, since the whole renormalization program fails (see, e.g., Refs. [34,35]). 1 Hence, its inclusion is matter of principle and does not simply amount to evaluating corrections to the leading-order result.
(ii) The leading-order result for the finite volume dependence of a shallow three-body bound state is derived for finite scattering length. From the previous derivation in the unitary limit, it is not immediately clear how one can move beyond this approximation, as well as how to build in effective range corrections, mixing with higher partial waves, etc..
(iii) The formalism developed in Refs. [22,23] explicitly excludes deeply-bound dimers. It is not obvious how to proceed if such deeply-bound states exist. We address this issue and show how the leading-order result for a shallow three-body state goes over into the well-known result for the particle-dimer bound state that can be obtained with the use of the Lüscher equation.
In the present paper, we elaborate on the above issues for a system of three identical bosons. The study of this particularly simple model lays the foundation for the treatment of the general three-body quantization condition in the dimer picture, which is considered in our forthcoming publication [37].
1 The case of a covariant formulation was recently investigated in Ref. [36].
The layout of the paper is follows. In Section II, we consider the role of short-range interactions in the finite-volume behavior of the two-body binding energy. This simple example illustrates the pattern, along which the inclusion of the short-range three-particle force is considered. In Section III, we collect all the information about the particle-dimer formalism in the infinite volume, including the analytic solution of the problem in the unitary limit. The leading order formula for the shallow bound-state energy shift in the unitary limit [19] is rederived in Section IV. In addition, we discuss the relation of the asymptotic normalization coefficient in this formula to the short-range three-body force. Section V deals with the calculation of the energy shift beyond the unitary limit and the relation to the particle-dimer bound state picture. Finally, Section VI contains our conclusions.

II. TWO-BODY BOUND STATE
The energy shift of a shallow two-body bound state of identical bosons of mass m in a finite volume is given by the Lüscher formula [38] where L is the size of the cubic box, κ 2 = √ mE B is the two-body binding momentum and A 2 denotes the two-body asymptotic normalization coefficient (note that we choose a different definition for A 2 as in Ref. [38]). The latter is defined through the behavior of the radial bound-state wave function at large distances: Let us derive the relation of the quantity A 2 to the parameters characterizing the shortrange interactions. We shall do this using the non-relativistic effective Lagrangians with a method that closely resembles the one used in the three-particle case. Dimensional regularization with minimal subtraction is the most convenient choice, albeit the final results do not depend on the regularization used. This is explicitly demonstrated in Appendix A, where we demonstrate that exactly same results are obtained by using cutoff regularization.
The same applies for dimensonal regularization with power divergence subtraction.
In momentum space, the non-relativistic interaction Hamiltonian consists of an infinite tower of operators with increasing mass dimension. Limiting ourselves to O(p 2 ) and to S-waves only, in momentum space we may write where the couplings C 0 and C 2 specifying the effective potential V are related to the scattering length a and the effective range r e through The bound-state wave function obeys the Schrödinger equation where d is the dimension of space.
The wave function Ψ(p) consists of a long-range tail and the short-range part, which can be approximated by a polynomial. Consequently, one may try to solve the Eq. (5) with the following ansatz: Substituting this ansatz into the Schödinger equation and performing the integrals in dimensional regularization with minimal subtraction, it is easy to see that p 2 = p 4 = . . . = 0, while A 2 and p 0 obey the system of linear equations √ Substituting the second equation into the first one, an equation to determine the bound state momentum κ 2 emerges The system of the homogeneous linear equations, Eq. (7) determines only the ratio p 0 A 2 .
In order to determine A 2 , one uses the normalization condition for the wave function Evaluating the integrals and expressing p 0 through A 2 , we finally get where at the final stage we have expressed the non-relativistic couplings through the physical observables.
Below, we briefly summarize the lessons learned: (i) The energy level shift in a finite volume is determined by the asymptotic part of the bound state wave function, which is parameterized by two constants κ 2 and A 2 .
(ii) If the higher-order short-range interactions are absent (r e = 0 together with all higherorder terms), then A 2 = 1. This condition is equivalent to Weinberg's compositeness condition [39], which distinguishes a hadronic molecule from a tightly bound compound.
(iii) The asymptotic normalization coefficient is determined from the normalization of the whole wave function to unity. For example, A 2 ≃ 0 would mean that the short-range component prevails over the long-range one, so that the system is predominately a tight compound. In accordance to this, the energy level has a very little dependence on the volume, and vice versa.
In the next section we shall demonstrate, how the above derivation can be adjusted for the particle-dimer bound state.

A. The Lagrangian
In order to simplify things as much as possible, we shall consider the case of three identical non-relativistic bosons in the CM frame. The inclusion of relativistic kinematics, spin, moving frames, etc., proceeds along the standard path and will be addressed in our future publications.
In the following, we shall mainly follow the Refs. [34,35]. The most general effective Lagrangian that describes the two-and three-particle sectors is given by where ψ denotes the non-relativistic field operator for a boson with a mass m, and ellipses stand for the terms with derivatives. We further introduce a dummy field T (called dimer) with the quantum number of two bosons and consider the Lagrangian Note that the field T is not dynamical -the corresponding Lagrangian does not contain the time derivative. Integrating out this field by using the equations of motion, we arrive at the Expanding this Lagrangian in the power of fields, one sees that it describes exactly the same physics as the Lagrangian from Eq. (11) in the two-and three-particle sectors, if the couplings are fixed in the following manner: The following remarks are in order: (i) As is clear from above, the particle-dimer picture is not an approximation, in fact, being restricted to the two and three particle sectors, it is mathematically equivalent to the original treatment without a dimer field. Hence, the treatment of the finitevolume effects with the use of the particle-dimer approach is as general as the one based on a three-particle Lagrangian without a dimer field.
(ii) Using the dimer formalism does not imply the neglect of the higher partial waves.
wave interactions, can be replaced by a tower of Lagrangians containing dimers with angular momentum 1, 2, etc.. The truncation of the partial-wave expansion is then equivalent to including the dimers with angular momentum below some fixed value (the details of the formalism can be found in Ref. [37]).
One additional remark concerns the inclusion of the kinetic energy term for the dimer. In principle, T is a dummy field, so, instead of Eq. (12), one could consider the Lagrangian with a dynamical dimer field as well where σ = ±1 is sign that depends on the sign of the effective range. The variable T can again be integrated out, leading to an equivalent theory in terms of a field ψ only.
However, as argued, e.g., in Ref. [40], when a shallowly bound two-particle state is present, the convergence radius of the perturbation expansion in the theory with a dynamical dimer should be larger because this theory contains the small scale ∆ explicitly (not hidden in the couplings of the effective theory).
In the following, we shall use the formulation based on the Lagrangian, Eq. (12), neglecting all higher-order terms. The inclusion of the derivative couplings, higher partial waves, etc. will be discussed in our forthcoming paper.

B. The bound-state equation and the normalization condition
As it is well known, the particle-dimer bound state wave function in the theory described by the Lagrangian, Eq. (12), obeys the homogeneous Faddeev equation where Λ denotes an explicit UV cutoff, and Projecting to the S-wave and defining h = 2mg 2 H(Λ) Λ 2 , κ 2 = −mE, we arrive at the equation where Ψ(p) stands for the S-wave wave function (note that τ (q; E) = τ (q; E) depends only on q = q ). It is well known that the particle-dimer coupling constant H(Λ) should be a log-periodic function of the cutoff parameter Λ for the limit Λ → ∞ to exist in this equation [34,35].
Next, we shall derive the normalization condition for the wave function Ψ(p), which has a non-trivial form because the kernel of the integral equation depends on the energy E. The derivation follows the standard pattern (see, e.g., [41]). Namely, we consider the inhomogeneous equation for the scattering amplitude In a compact notation, we have M = Z + Z(8πτ )M. Defining the Green function as G = (8πτ ) + (8πτ )M(8πτ ), we obtain G = (8πτ ) + (8πτ )ZG, and G −1 = (8πτ ) −1 − Z. Further, using the identity GG −1 G = G and the behavior of the Green function in the vicinity of the bound-state pole we arrive at the following normalization condition for the wave function:

C. Minlos-Faddeev solution
Assuming Λ → ∞ and H(Λ) = 0, we obtain the Skornyakov-Ter-Martirosian (STM) equation [42]. Unlike the equation (16), the STM equation is known not to possess a unique solution [43]. Minlos and Faddeev [44] have found an exact solution to the integral equation in the unitary limit a → ∞: where κ = √ −mE is the three-body bound state momentum and s 0 ≃ 1.00624 is a numerical constant, which is a solution of the transcendental equation We hereafter refer to Eq.
where (see Appendix B) and u = ln

D. Asymptotic normalization coefficient
As mentioned above, the STM equation does not have unique solutions. First, imposing a cutoff, one arrives at the discrete three-particle spectrum. In order to ensure that the limit Λ → ∞ exists, one has to introduce a short-range interaction parameterized by a constant H(Λ), where the dependence on the cutoff Λ is log-periodic. Furthermore, for a fixed Λ and H(Λ), the low-energy spectrum is discrete, condensing towards zero. For two neighbouring levels whose energy is much smaller than the cutoff Λ, the following relation holds in the unitary limit: Consequently, fixing a single energy level for a given Λ is equivalent to the fixing of the parameter H(Λ).
The Faddeev equation with a finite cutoff and H(Λ) can not be solved analytically.
However, the numerical solution is straightforward. In table I, for illustration, we give several energy eigenvalues for the choice Λ = 10 4 and H(Λ) = 0.
It is clear that, for the momenta much smaller than the cutoff Λ, the wave function will be given by the MF solution Ψ 0 . The difference can arise only at momenta p ≃ Λ. The overall normalization, however, is a subtler issue since the normalization integral includes all momenta. To summarize, the solution of the Faddeev equation with cutoff at a given bound-state momentum κ should be given by where the function A(x) should have a very flat plateau for x ≪ Λ κ. Then, in analogy to the two-body case, we define the particle-dimer asymptotic normalization coefficient as Further, the dimensionless quantity A must be a function of the only dimensionless combination κ Λ that can be composed from the parameters of the theory. If κ Λ → 0, then, obviously, A → 1. Consequently, the asymptotic normalization coefficient is very close to one for the shallow three-particle bound states. The explicit demonstration of the above statements is given in Fig. 1.
The situation changes, when derivative particle-dimer interactions are added. Consider, for example, adding the term H 1 (Λ)(p 2 + q 2 ) to the particle-dimer interaction Hamiltonian (this is an analog of the effective range term in the two-particle case). The ratio will be still flat for p ≪ Λ. However, the statement A → 1, as κ Λ → 0 does not hold any more -in other words, the asymptotic normalization coefficient encodes the effect of the short-range physics, as in the two-particle case. In the particle-dimer case, however, A and H 1 (Λ) can not be related algebraically, and A should be extracted from the numerical solution of the equation with a given value of H 1 (Λ) ≠ 0.

A. Expression of the first order finite-volume energy shift
Our derivation -in the particle-dimer context -will be partly similar to that of Ref. [27].

Consider the Faddeev equation in a finite volume
where q = 2π L n , n ∈ Z 3 and Here, κ 2 L = −mE. Moreover, we have assumed that we are below the particle-dimer breakup threshold, where all denominators are non-singular, and have used Poisson's summation formula. Note that exactly this equation was considered earlier in Refs. [30][31][32][33].
The finite-volume effects in the particle-dimer bound state equation emerge at two different places. First, the integration over q is changed to a sum over discrete values in Eq. (30).
Second, there is an additional term ∆ L (q; E) in Eq. (31). Using again Poisson's summation formula, we may rewrite the Eq. (30) as Excluding now the quantity Z using Eq. (19), we obtain The infinite-volume amplitude M has a pole at the bound-state energy (cf. with Eq, (20)): where Ψ(p) is the infinite-volume wave function. Substituting this ansatz in Eq. (33), we finally obtain the expression for the first-order energy shift of the three-body bound state

B. Evaluation of the first-order energy shift
The energy shift can be written as The evaluation of these integrals in the unitary limit proceeds mainly along the lines described in Ref. [27]. In this limit, one has Λ → ∞, a → ∞ and Ψ(p) is the MF wave function Ψ 0 (p). Using Eq. (22) and performing angular integration, we get where n = n . It is clear that the leading exponential contribution emerges from the term with n = 1. Introducing the variable u defined in Eq. (22), one gets in κL ≫ 1, the integral in the r.h.s. of the above equation has the following asymptotic Using this expansion, one reproduces the result first derived in Ref. [19] and re-derived in Ref. [27] (note that in Ref. [27], an algebraic error contained in the original derivation was corrected): where Taking into account the relation where C −1 0 is defined in Eq. (16) of Ref. [19], it is straightforward to verify that Eq. (41) is identical to the final result of Ref. [19]. 2 However, Eq. (41) contains more information as the original formula from Ref. [19]. It corresponds to the unit asymptotic normalization coefficient A = 1. Now, it is clear, where the non-trivial three-particle force, encoded in the derivative particle-dimer couplings, will reveal itself: the L-dependence in the formula (41) remains the same, only the overall factor will be multiplied by A 2 ≠ 1, where A can be determined from the infinite-volume solution through the procedure described above. The reason for this is that, at small momenta p ≪ Λ, the ratio A defined in Eq. (28) is close to constant and does not affect the large-L behavior of the energy level.
Further, as shown in Ref. [27], the correction ∆E 2 is subleading and behaves as for a large L. The subsequent terms are even more suppressed.
To summarize, we have reproduced the result of Ref. [19] for the leading finite-volume energy shift of the three-body bound state in the unitary limit in the particle-dimer picture.
Moreover, we have shown that in the unitary limit, the asymptotic normalization coefficient emerges from three-particle derivative forces, and this coefficient is equal to one if such forces are absent.
At the next step, we shall investigate the system beyond the unitary limit. 2 Note that the C −1 0 defined in Eq. (16) of Ref. [19] is not related to the C 0 defined in Eq. (3).

A. Energy shift
As seen, the correction ∆E 1 given by Eq. (37), gives the leading contribution to the finite-volume energy level in the unitary limit. We expect that this statement stays valid for finite values of a. Singling out the contribution with n = 1, one may rewrite the leading contribution to ∆E in the following form As we shall demonstrate below, the wave function Ψ(p) is regular near origin (more precisely, the singularities of Ψ(p) are located much farther from the origin than the singularities of the denominator). This means that the singularities of Ψ(p) do not contribute to the large-L behavior of the energy shift at leading order and hence, at this order, Ψ(p) can be replaced by a constant. Performing the angular integration, we arrive at the following result Note that the quantity κ 2 − a −2 is always positive, if a bound state of a particle and a bound dimer is considered (recall that κ 2 = a −1 is the binding momentum of the dimer in the unitary limit). One has to distinguish two limiting cases: A shallow bound state of a particle and a deeply bound dimer In this case, we have κ 2 − a −2 ≪ κ 2 . The singularity at p = ±i 4 3 (κ 2 − a −2 ) is dominant, and the singularity arising from the square root (cut) can be neglected. Performing the Cauchy integration, we get In other words, we reproduce Lüscher's original result for a two particle (particle-dimer) bound state [38]. Note also that this is in a complete agreement with the result of the recent paper [45].
A shallow bound state of three particles This corresponds to the opposite limit κ 2 ≫ a −2 . Then, the first term in Eq. (46) is very small and the energy shift is dominated by the second term. It is straightforward to see that, in this case, In other words, the result of Ref. [19] is reproduced in this limit. To be more precise, for any small but finite value of a −2 the asymptotic behavior of the energy shift is still given by Eq. (47). However, the coefficient of the leading term is very small, whereas the coefficient in front of the subleading term given by Eq. (48) is of order of unity. So, for large (but not asymptotically large) values of L the behavior is given by Eq. (48), whereas Eq. (47) sets in asymptotically.

B. Wave function
The wave function obeys the equation (18). The location of the singularities of Ψ(p) in the complex-p plane is determined, as usual, by the Landau equations. There are two types of singularities (note that the denominator −a −1 + 3q 2 4 + κ 2 does not vanish in the integration region):

Endpoint singularities
The argument of the logarithm is a ± (q) = p 2 ± p ⋅ q + q 2 + κ 2 . At q = 0, the equation a ± (q = 0) = 0 yields p = ±iκ. Examine now this potential singularity in detail. Let us start, for instance, at p = 0 and approach the singular point p → iκ along some path in the complex p-plane (for instance, along the path p = it + 0.05t(1 − t), 0 ≤ t ≤ 1). The four singularities of the logarithm, which are determined by the solutions of the equations a ± (q) = 0, travel along the lines whereas the singularity of the denominator given by the equation −a −1 + 3 4 q 2 + κ 2 = 0 stays fixed in the in the complex q-plane. The trajectories q 1,2,3,4 are shown schematically in Fig. 2, left panel. Two singularities travel from q = ±iκ towards q = 0 and two others return to q = ±iκ. The contour deformation is not needed. Substituting now p = ±iκ into the kernel, we get The logarithm is indeed singular at q = 0, but the integral over q exists, due to the presence of an additional factor q 2 . Consequently, there is no singularity at p = ±iκ.

Pinch singularities
In order to find the location of the pinch singularities, we have to solve the equation this gives p = ±2q. Substituting this back to the argument of the logarithm, we get a ± (q) = q ± p 2 2 + 3 4 p 2 + κ 2 , i.e., the argument vanishes at p = ± 2i √ 3 κ.
Consider again the trajectories of the singularities of the logarithm in the complex q-plane, when p varies, according to, for instance, along the path p = 2 √ 3(it+0.05t(1 −t)), 0 ≤ t ≤ 1.
These trajectories are shown in Fig. 2, right panel. In this case, it is necessary to deform the integration contour, in order to avoid the singularities of the logarithm. At p = ± 2i √ 3 κ the contour gets pinched between two singularities. It is, however, straightforward to check that the singularity of the integrand along the new integration contour is of an integrable type: logarithmic for a −1 ≠ 0 and of square-root type for a −1 = 0. Consequently, the function Ψ(p), defined by this integral, is non-singular there (albeit the derivatives become, in general, singular).

VI. CONCLUSIONS
Our conclusions are as follows: (i) In this paper, we have rederived the well-known result [19] for the leading-order finite volume energy shift of a shallow three-particle bound state in the unitary limit using the dimer formalism. While this result was not unexpected, since the particle-dimer picture is algebraically equivalent to the three-particle description, it provides a useful check on the particle-dimer formalism in a finite volume.
(ii) Our treatment goes beyond Refs. [19,27]. Namely, we concentrate on the role of the three-particle force, which is necessary to carry out the renormalization program in the infinite volume. We have shown that the algebraic form of the leading-order formula does not change in the presence of the three-particle force, and only the numerical value of the particle-dimer asymptotic normalization constant is altered.
This constant is equal to one for the STM equation and differs from unity in the presence of the derivative three-particle interactions -similar to the two-body case.
(iii) Finally, we go beyond the unitary limit and derive the leading-order formula in this case. This formula smoothly interpolates between two extremes: the well-known threeparticle bound state in the unitary limit and the bound state of a particle and a deeply bound dimer, for which the usual Lüscher formula applies. The study of these limits enables us to explore the region of applicability of the energy shift formula from Ref. [19].
(iv) A host of additional effects awaits to be included, namely, the effective range expansion in the two particle sector, higher partial waves and partial wave mixing, non-rest frames, relativistic kinematics, etc. Moreover, a general and tractable quantization condition, which could be used by lattice practitioners to analyze the data in the three-particle sector, remains to be worked out. The particle-dimer language allow one to achieve most of the above goals with an impressive ease. However, we relegate the proof of this statement to our forthcoming publication [37].

(A1)
Here, µ denotes the renormalization scale. The coupling constants C 0 (µ) and C 2 (µ) can be determined from matching to the effective range expansion for the two-body scattering amplitude. We obtain where a ≫ r e was assumed.
Using again the ansatz from Eq. (6) with p 2 = . . . = 0, we get The equation for the bound-state momentum κ 2 takes the form Inserting Eqs. (A2), we can rewrite this expression as The normalization condition, yields the following expression for the asymptotic normalization coefficient: Using the equation aκ 2 = 1 + O(κ 3 2 ), one may rewrite the above equation as A −2 2 = 1 − κ 2 r e + O(κ 3 2 ) .
As we see, the final result for A 2 does not depend on the regularization used.
momentum cutoff Λ but we refrain from showing explicit expressions here. In this case, divergences up to fifth order in the cutoff Λ appear. Moreover, lower-order couplings are renormalized by higher orders, which leads to more complicated equations.