Three-particle bound states in a finite volume: unequal masses and higher partial waves
aa r X i v : . [ h e p - l a t ] J un Three-particle bound states in a finite volume:Unequal masses and higher partial waves
July 2, 2018Yu Meng a , Chuan Liu a,b , Ulf-G. Meißner c,d and A. Rusetsky ca School of Physics and Center for High Energy Physics,Peking University, Beijing 100871, P.R. China b Collaborative Innovation Center of Quantum Matter,Beijing 100871, P.R. China c Helmholtz–Institut f¨ur Strahlen– und Kernphysik andBethe Center for Theoretical Physics,Universit¨at Bonn, D–53115 Bonn, Germany d Institute for Advanced Simulation (IAS-4), Institut f¨ur Kernphysik (IKP-3),J¨ulich Center for Hadron Physics and JARA-HPCForschungszentrum J¨ulich, D-52425 J¨ulich, Germany
Abstract
An explicit expression for the finite-volume energy shift of shallow three-body boundstates for non-identical particles is obtained in the unitary limit. The inclusion of thehigher partial waves is considered. To this end, the method of Ref. [1] is generalizedfor the case of unequal masses and arbitrary angular momenta. It is shown that in theS-wave and in the equal mass limit, the result from Ref. [1] is reproduced.
Introduction
In the analysis of lattice data, the L¨uscher formalism is used both to evaluate the finite-volume corrections to the stable particle masses [2], as well as to extract the two-bodyscattering lengths and scattering phase shifts from the finite-volume energy spectra of thetwo-particle systems [3, 4]. However, a generalization of the above finite-volume approachfrom two- to three-particle case turned out to be a rather challenging task. Only in the lastfew years, this issue has been addressed extensively in the literature [1, 5–27]. Despite thesignificant effort, the progress has been slow so far. Namely, the finite volume spectrum of thethree-particle system in some simple models has been calculated only very recently [12, 27](see also earlier work [18–21], where exclusively the three-body bound-state sector was ad-dressed). Such calculations are very useful since, at this stage, one does not yet have enoughinsight into the problem and lacks intuition to predict the behavior of the three-particlefinite-volume energy levels. Moreover, these calculations might facilitate the interpretationof a particular behavior of the energy spectrum in terms of various physical phenomena inthe infinite volume.For the reasons given above, it is very interesting to study the few simple three-bodysystems, for which an analytic solution in a finite volume is available. The three-body boundstate is one of these. In Ref. [1], it has been shown that it is possible to obtain an explicitexpression for the leading order finite volume energy shift of the S-wave shallow bound stateof three identical bosons in the unitary limit, i.e., when the two-particle scattering lengthtends to infinity and the effective range (and higher order shape parameters) are zero (theso-called Efimov states, see Ref. [28]). This expression has a remarkably simple form:∆ EE T = c ( κL ) − / | A | exp (cid:18) − κL √ (cid:19) . (1)In this expression, L is the side length of the spatial cubic box, E T and ∆ E denote thebinding energy and the shift, respectively, κ = √ m E T is the bound state momentum ( m denotes the mass of the particle), and c ≃ − .
351 is the numerical coefficient. Further, A isthe so-called asymptotic normalization coefficient for the bound state (it is equal to one, if noderivative three-particle forces are present). The formula is valid when κL ≫
1. Later, thesame formula has been obtained in Ref. [11], using the three-particle quantization conditionfrom Ref. [7], and in Ref. [25] by using the finite-volume particle-dimer formalism, formulatedin Refs. [25, 26]. Moreover, in Ref. [25] the role of the three-particle force (encoded in theasymptotic normalization coefficient) has been clarified, and the condition of an infinitelylarge two-body scattering length has been relaxed. By doing this, one can nicely observe acontinuous transition from the bound state of a tightly bound dimer and a spectator to theloosely bound three particle bound state.It should be especially mentioned that the functional L -dependence of the energy shiftdiffers from the one predicted by the two-particle L¨uscher formula [2] (see also Ref. [29] wherethe n -particle bound state is considered), which would be the case, when the three-particlebound state could be represented as a loosely bound state of a tightly bound dimer and aspectator, as well as from the perturbative shift of the three-particle ground state, which2as been derived, e.g., in Refs. [30, 31]. In this sense, the three-body bound state problemrepresents a highly non-trivial testing ground for all theories that describe the spectrum ofthe three-particle system in a finite volume.In the present paper, we generalize the original result of Ref. [1] to the case of non-identical particles and include higher partial waves. This problem is interesting, first andforemost because, to the best of our knowledge, all available explicit results in the three-bodysector so far are limited to the S-wave states only. Carrying out benchmark calculations inhigher partial waves will enable one to carry out more elaborate tests and to understandmuch better the three-particle dynamics in a finite volume that is important for analyzingsimulation data from lattice QCD for the three-particle systems. This is exactly the aim ofthis short, technical article. Eventually, it would be interesting to study the same problemin moving frames and consider the particles with spin. This, however, forms a subject of aseparate investigation and will be addressed in the future. The wave function of three non-identical bosons obeys the Schr¨odinger equation: (cid:26) X i =1 (cid:18) − m i ∇ i + V i ( x i ) (cid:19) + E T (cid:27) ψ ( r , r , r ) = 0 , (2)where ∇ i = ∂/∂ r i . In the following, we always assume that ( ijk ) form an even permutation,and i, j, k can take the values 1 , ,
3. Also, we mainly follow the notations and conventionsof Ref. [32]. The relative coordinates are defined as: x i = µ jk ( r j − r k ) , y i = µ i ( jk ) (cid:18) m j r j + m k r k m j + m k − r i (cid:19) , (3)where µ jk = r m j m k M ( m j + m k ) , µ i ( jk ) = s m i ( m j + m k ) M ( m i + m j + m k ) . (4)Here, M denotes some normalization mass. The observables do not depend on the choice of M . If m = m = m = m , the choice M = m / M = ( m + m + m ) / E T = κ M . (5)There are three different sets of relative coordinates. The relation between them is given by x j = − x i cos γ ij + y i sin γ ij , y j = − x i sin γ ij − y i cos γ ij , (6)3here γ ij = arctan (cid:18)s m k ( m i + m j + m k ) m i m j (cid:19) , − π ≤ γ ij ≤ π . (7)The hyperradius R and the hyperangles α i are defined as: | x i | = R sin α i , | y i | = R cos α i , R = x i + y i . (8)The relation between different hyperangles is given by:sin α j = sin α i cos γ ij + cos α i sin γ ij − α i sin α i cos γ ij sin γ ij cos θ i , (9)where θ i is the angle between the x i and y i .The six-dimensional integration measure is written as d x i d y i = R dR sin α i cos α i dα i d Ω x i d Ω y i , (10)where Ω x i , Ω y i denote the solid angles in the direction of the vectors x i and y i , respectively.The wave function, expressed in terms of the x i , y i , takes the form ψ ( r , r , r ) = ψ i ( x i , y i ) . (11) A straightforward generalization of the energy shift formula of Ref. [1] gives:∆ E = X i =1 X p , q , n , l X k = − ( l + n ) Z d x i d y i × (cid:18) ψ i (cid:0) x i − ( p + q ) µ jk L, y i + µ i ( jk ) Lm j + m k ( p m k − q m j ) (cid:1)(cid:19) ∗ V i ( x i + µ jk k L ) × ψ i (cid:0) x i − ( n + l ) µ jk L, y i + µ i ( jk ) Lm j + m k ( n m k − l m j ) (cid:1) , (12)where p , q , k , l , n ∈ Z . Note that the periodic boundary conditions are assumed.In order to obtain the energy shift at leading order, we use the following procedure. First,we shift the variables x i → x i − µ jk k L , y i → y i − µ i ( jk ) Lm j + m k ( p m k − q m j ) . (13)Next, we take into account the fact that the wave function of the bound state decreases expo-nentially when the hyperradius becomes large. The suppression factor is given by exp( − κR ).The equation (12) contains two wave functions with different arguments – we refer to them4s to the first and the second wave functions in the following. It is immediately seen that inthe sum over p , q , k , l , n the leading contribution is given by those term(s), where the sumof the hyperradii for the first and the second wave functions R + R is minimal as L → ∞ .All other terms will give contributions that are exponentially suppressed with respect to thiscontribution. Writing down explicitly R + R = µ jk L (cid:26) | p + q + k | + (cid:18) ( n + l + k ) + (cid:18) µ i ( jk ) µ jk ( m j + m k ) (cid:19) × (cid:0) ( m j + m k )( − l + q + n − p ) + ( m j − m k )( − l + q − n + p ) (cid:1) (cid:19) / (cid:27) , (14)one can straightforwardly check that the following choices n + l + k = e , p + q + k = , − l + q + n − p = − e , (15)and n + l + k = e , p + q + k = , − l + q + n − p = e , (16)where e is the unit vector with | e | = 1, lead to the minimum of R + R , if all relevantpermutations ( ijk ) = (123) , (231) , (312) are considered . Thus, the energy shift formulasimplifies to∆ E = X e X i =1 Z d x i d y i ( ψ i ( x i , y i )) ∗ V i ( x i ) ψ i (cid:18) x i − µ jk e L, y i − µ i ( jk ) m j e Lm j + m k (cid:19) + X e X i =1 Z d x i d y i ( ψ i ( x i , y i )) ∗ V i ( x i ) ψ i (cid:18) x i − µ jk e L, y i + µ i ( jk ) m k e Lm j + m k (cid:19) , (17)where the sum runs over the six possible orientations of the unit vector e . From Ref. [32] one may read off the explicit form of the wave function of the three-particlebound state in the unitary limit: ψ ( r , r , r ) = X i =1 φ i ( x i , y i ) , (18) Note that the situation here is rather subtle. Namely, if we consider a fixed choice of ( ijk ), for somemass ratios there exist solutions, other than in Eqs. (15,16), which lead to the lower value of R + R . Whatwe claim here, is that this value of R + R is still higher than the value, obtained from Eqs. (15,16) foranother choice of ( ijk ). In other words, we claim that Eq. (17) always contains a leading exponential, alongwith some subleading pieces. On the other hand, one has to retain these subleading pieces as well, if onewants to reproduce the result in the equal mass limit. l and projection m , φ ilm ( x i , y i ) = N l x l y l R − / f ( R ) X l x l y A ( l x l y ) i sin l x α i cos l y α i P + l x , + l y ν ( − cos 2 α i ) × X m x + m y = m c lml x m x ,l y m y Y l x m x (Ω x i ) Y l y m y (Ω y i ) . (19)Here, the P ( a,b ) ν ( x ) denote Jacobi functions, Y lm (Ω) are spherical harmonics, the c lml x m x ,l y m y denote the Clebsch-Gordan coefficients and f ( R ) is the radial function. The wave functions,which in this paper are used in the calculation of the energy shift, obey the Bose-symmetryin case of identical particles, see Refs. [32–34] for more details.The three-particle bound states in the unitary limit exist only if the resonant interactionis in an S-wave, i.e., l x = 0 [32]. Then, l y = l . The coefficients A (0 ,l ) i . = A i obey the linearequations: P Q Q Q P Q Q Q P A A A = 0 , (20)where P = sin(( ν + ) π )sin( π ) ,Q ij = Q ji = Γ( )Γ( ν + + l )Γ( + l )Γ( ν + ) F ( − ν, ν + l + 2 ,
32 + l, cos γ ij )( − cos γ ij ) l , (21)in terms of Gamma and hypergeometric functions. In order to have a non-trivial solution tothis homogeneous system of linear equations, the determinant of this system must be equalto zero. This determines the discrete values of the parameter ν . One further defines ν = −
12 (2 + l ) + 12 √ λ , λ = − − ξ . (22)The radial wave function is given by the same expression for all l : f ( R ) = R / K iξ ( κR ) , (23)where K µ ( z ) denotes the modified Bessel function. Bound states occur when ξ is real, i.e.,when λ < −
4. In the S-wave, l = 0, this happens for all values of the masses m , m , m .However, if l = 0, one of the masses must be much lighter than other two, in order thatEfimov states can emerge [32] (see also Ref. [35], where the properties of Efimov states inhigher partial waves are discussed). Consequently, the treatment of bound states in higherpartial waves is not possible if only the equal-mass case is considered.6he wave function of a bound state is always normalized to unity. We shall in additionassume that X i =1 A i = 1 . (24)This is equivalent to the assumption that the asymptotic normalization coefficient A = 1or, equivalently, only non-derivative three-particle interactions are present in the system. Inthe following, we shall stick to this assumption. Before considering the case of arbitrary l , we discuss the most interesting cases l = 0 , l = 0 The wave function is given by: ψ i ( x i , y i ) = N √ R − / f ( R ) X i =1 A i sinh( ξ ( π − α i ))sin(2 α i ) . (25)Here, we have introduced an additional factor 2 √ V j ( x j ) ψ j ( x j , y j ) = − δ (3) ( x j ) F ( y j ) , (26)where F ( y j ) = N √ πM A j | y j | K iξ ( κ | y j | ) sinh (cid:18) πξ (cid:19) . (27)The normalization condition gives N = κ c , (28)where c − = 12 π ξ sinh( πξ ) (cid:26)(cid:18) ξ sinh( πξ ) − π (cid:19) X i =1 A i − ξ X i = j A i A j | sin(2 γ ij ) | (( π − | γ ij | ) sinh( ξ | γ ij | ) − | γ ij | sinh( ξ ( π − | γ ij | ))) (cid:27) . (29)7sing the asymptotic behavior for R → ∞ of the radial wave function f ( R ) ∼ r π κ exp( − κR ) , (30)and calculating, as in Ref. [1], the asymptotic form of the second wave function in Eq. (17)as L → ∞ , we arrive at the following expression for the energy shift:∆ E = 6 √ N r π κ L − / sinh (cid:18) ξπ (cid:19) × (cid:26)X i A i exp( − µ i ( jk ) κL )( µ i ( jk ) ) / Z d x i d y i | x i | ( ψ j ( x j , y j )) ∗ V j ( x j ) × exp (cid:18) κµ ki µ i ( jk ) x j e − κµ j ( ki ) m i µ i ( jk ) ( m i + m k ) y j e (cid:19) + (cid:26)X i A i exp( − µ i ( jk ) κL )( µ i ( jk ) ) / Z d x i d y i | x i | ( ψ k ( x k , y k )) ∗ V k ( x k ) × exp (cid:18) κµ ij µ i ( jk ) x k e + κµ j ( ki ) m i µ i ( jk ) ( m i + m j ) y k e (cid:19)(cid:27) . (31)Using Eq. (26) and the normalization condition, we finally arrive at the following expressionfor the energy shift: ∆ EE T = − π r π c sinh (cid:18) πξ (cid:19) ( κL ) − / × X i = j exp( − µ i ( jk ) κL ) A i A j ( µ i ( jk ) ) / I ( | γ ij | ) | sin(2 γ ij ) | , (32)where I ( | γ ij | ) = πξ sinh( πξ ) (cosh( ξ ( π − | γ ij | )) − cosh( ξ | γ ij | )) . (33)It can be checked that, in the equal mass limit, where A = A = A = 1 / √
3, the aboveformulae reduces to the result of Ref. [1] with the asymptotic normalization coefficient A = 1.For illustrative purpose, one may rewrite Eq. (32) as∆ EE T = − ( κL ) − / X i =1 C i exp( − µ i ( jk ) κL ) , (34)where the coefficients C i depend on the masses in the system, but not on L and the bindingenergy. In Fig. 1 we plot the coefficients C and C = C for a particular choice of themasses: m = m and m /m = m /m = z . As can be seen, at z = 1, all C i are equal to96 . . . . / . . . . (cf. with Ref. [1]). 8 z C Figure 1: The coefficients C (solid line) and C = C (dashed line) as a function of the massratio z = m /m = m /m , see Eq. (34). l = 1 The wave function with l x = 0 and l y = l = 1 is given by ψ i m ( x i , y i ) = X i =1 φ i m ( x i , y i ) , (35) φ i m ( x i , y i ) = N R − / f ( R ) A i φ ( α i ) r π Y m (Ω y i ) , (36)where φ ( α ) = 12 sin(2 α ) cos α (cid:18) sinh (cid:18) ξ (cid:18) π − α (cid:19)(cid:19) sin α − ξ cosh (cid:18) ξ (cid:18) π − α (cid:19)(cid:19) cos α (cid:19) . (37)It can be checked that the wave function obeys the equation V ( x j ) ψ j m ( x j , y j ) = − δ (3) ( x j ) F ( y j ) , (38)where F ( y j ) = − πξA j M cosh (cid:18) ξπ (cid:19) N K iξ ( κ | y j | ) | y j | r π Y m (Ω y j ) . (39)Next, we consider the normalization condition. Here, we have to deal with the angularintegrations of two types. First, there are “diagonal” terms Z d x i d y i H ( R, α i ) Y ∗ m (Ω y i ) Y m ′ (Ω y i ) , (40)9here H ( R, α i ) denotes some function of the arguments R and α i . Using Eq. (10), it isimmediately seen that the angular integrations yield the factor 4 πδ mm ′ . The “non-diagonal”terms have the following structure Z d x i d y i ˜ H ( R, α i , α j ) Y ∗ m (Ω y i ) Y m ′ (Ω y j ) , (41)with some other function ˜ H ( R, α i , α j ). Using Eq. (6), it can be shown that Y m ′ (Ω y j ) = | x i || y j | ( − sin γ ij ) Y m ′ (Ω x i ) + | y i || y j | ( − cos γ ij ) Y m ′ (Ω y i ) . (42)Performing the angular integrations, one should take into account the fact that, owing toEq. (9), the variable α j depends on the orientation of both x i and y i . Using this equation,the integral over d cos θ can be transformed into an integral over α j . The limits on thevariation of α j are given by || γ ij | − α i | ≤ α j ≤ π − (cid:12)(cid:12)(cid:12)(cid:12) π − α i − | γ ij | (cid:12)(cid:12)(cid:12)(cid:12) . (43)Finally, the normalization condition takes the form N = κ c , (44)where c − = πξ πξ ) X i,j =1 A i A j I ij . (45)The diagonal terms can now be written as I ii = π Z π/ dα cos α (cid:18) sinh (cid:18) ξ (cid:18) π − α (cid:19)(cid:19) sin α − ξ cosh (cid:18) ξ (cid:18) π − α (cid:19)(cid:19) cos α (cid:19) , (46)and the non-diagonal terms are given by I ij = − π | sin γ ij | cos γ ij Z π/ dα sin α sin (2 α ) × (cid:18) sinh (cid:18) ξ (cid:18) π − α (cid:19)(cid:19) sin α − ξ cosh (cid:18) ξ (cid:18) π − α (cid:19)(cid:19) cos α (cid:19) J ij ( α ) , (47)where J ij ( α ) = Z α max α min dα ′ cos α ′ (cid:18) sinh (cid:18) ξ (cid:18) π − α ′ (cid:19)(cid:19) sin α ′ − ξ cosh (cid:18) ξ (cid:18) π − α ′ (cid:19)(cid:19) cos α ′ (cid:19) × (cos γ ij + cos α − sin α ′ ) (48)10nd α min = || γ ij | − α | , α max = π − (cid:12)(cid:12)(cid:12)(cid:12) π − α − | γ ij | (cid:12)(cid:12)(cid:12)(cid:12) . (49)Finally, the energy shift, averaged over all values of m , is given by ∆ EE T = − π r π ξ cosh (cid:18) ξπ (cid:19) c ( κL ) − / × X i = j exp( − µ i ( jk ) κL ) A i A j ( µ i ( jk ) ) / γ ji T (cos γ ji ) , (50)where T ( α ) = 1 α Z ∞ K iξ (cid:18) yα (cid:19) ddy (cid:18) sinh yy (cid:19) . (51) l The wave function in case of arbitrary l is given by Eq. (19) with l x = 0 and l y = l (i.e., theresonant interaction is in the S-wave). We can write this expression as φ ilm ( x i , y i ) = N ll R − / f ( R ) A i φ l ( α i ) r π l + 1 Y lm (Ω y i ) , (52)where the Jacobi functions, entering this expression, can be determined from certain recur-rence relations. These relations can be obtained from the definition of the Jacobi functions P a,bν ( x ) = Γ( ν + a + 1)Γ( ν + 1)Γ( a + 1) F (cid:18) − ν, ν + a + b + 1 , a + 1 ,
12 (1 − x ) (cid:19) , (53)as well as the recurrence relations for the hypergeometric functions F , see, e.g., Ref. [37].The recurrence relations for the Jacobi functions take the form (cid:18) ν + a + b (cid:19) (1 − x ) P a +1 ,bν ( x ) = ( ν + a + 1) P a,bν ( x ) − ( ν + 1) P a,bν +1 ( x ) , (cid:18) ν + a + b (cid:19) (1 + x ) P a,b +1 ν ( x ) = ( ν + b + 1) P a,bν ( x ) + ( ν + 1) P a,bν +1 ( x ) , (54)starting from P / , / ν (cos 2 α ) = Γ( ν + 3 / ν + 1)Γ(3 /
2) sin(2( ν + 1) α )( ν + 1) sin 2 α . (55) Note that, in higher partial waves, the energy shift depends on m in the two-body bound states as well,see, e.g., Ref. [36]. I ii = 16 π l + 1 Z π/ dα sin α cos α ( φ l ( α )) , (56)whereas the non-diagonal integral (analog of Eq. (47)) is given by I ij = 4 π l + 1 Z d Ω x i d Ω y i dα i sin α i cos α i φ l ( α i ) φ l ( α j ) Y ∗ lm (Ω y i ) Y lm (Ω y j ) . (57)In general, the transformation between the wave functions, depending on different sets ofJacobi coordinates, is given by the Raynal-Revai coefficients [38]. An explicit expression forthese coefficients is known in the literature (see, e.g., Ref. [39] and earlier references therein).However here we do not make use of these rather voluminous formulae. Rather, in order tocalculate the angular integral, in analogy with Eq. (41), we express the quantity Y lm (Ω y j )as a sum of products Y l ′ m ′ (Ω y i ) Y l ′′ m ′′ (Ω x i ) with all possible l ′ + l ′′ ≤ l and m ′ + m ′′ = m . Inorder to do this, is it useful to define the solid harmonics: Y lm ( y j ) = | y j | l Y lm (Ω y j ) . (58)The quantity Y lm ( y j ) is a polynomial of power l in the components of the 3-vector y j .Writing y j = a y i + b x i , one immediately sees that each term in the expression of Y lm ( y j )decomposes into monomials of the components of the vectors y i and x i of power l and l , respectively, with l + l = l . These monomials, in their turn, can be expressed through Y l ′ m ′ ( y i ) and Y l ′′ m ′′ ( x i ), respectively, with l ′ ≤ l and l ′′ ≤ l , leading to the above-mentionedexpansion.Further, one has to calculate integrals of the type I Ω = Z d Ω x i d Ω y i φ l ( α j ) Y ∗ lm (Ω y i ) Y l ′ m ′ (Ω y i ) Y l ′′ m ′′ (Ω x i ) . (59)Let us recall here that α j depends on the scalar product x i y i , so the two angular integra-tions do not immediately decouple. In order to achieve this decoupling, consider first theintegration over d Ω y i , with the direction of the unit vector ˆ x i fixed. Note that it is alwayspossible to find a rotation R x so that R x ˆ x i = e , e = (0 , , . (60)Perform now the variable transformation y i = R − x y ′ i , with d Ω y i = d Ω ′ y i . After this transfor-12ation, we have x i y i = ey ′ i . Further, Y lm (Ω y i ) = l X n = − l D ( l ) mn ( R − x ) Y ln (Ω ′ y i ) ,Y l ′ m ′ (Ω y i ) = l ′ X n ′ = − l ′ D ( l ′ ) m ′ n ′ ( R − x ) Y l ′ n ′ (Ω ′ y i ) ,Y l ′′ m ′′ (Ω x i ) = l ′′ X n ′′ = − l ′′ D ( l ′′ ) m ′′ n ′′ ( R − x ) Y l ′′ n ′′ (Ω e ) , (61)where the D ( l ) denote Wigner D -matrices in the irreducible representation of the rotationgroup, characterized by the angular momentum l . It is now seen that the integration overtwo solid angles decouple: I Ω = X nn ′ n ′′ Z d Ω x i ( D ( l ) mn ( R − x )) ∗ D ( l ′ ) m ′ n ′ ( R − x ) D ( l ′′ ) m ′′ n ′′ ( R − x ) × Z d Ω ′ y i φ l ( α j ) Y ∗ lm (Ω y ′ i ) Y l ′ m ′ (Ω y ′ i ) Y l ′′ m ′′ (Ω n ) . (62)Here, the quantity α i is determined by Eq. (8) with θ i denoting the angle between the unitvectors ˆ y ′ i and e , so that cos θ i = cos θ , d Ω ′ y i = d cos θdϕ and Y lm (Ω ′ y i ) = Y lm ( θ, ϕ ). Theintegral over d Ω x i can be finally performed, yielding a group-theoretical factor, and one isleft only with the integral over the solid angle d Ω ′ y i . It does not make much sense to presentthe (quite voluminous) general result here. If needed, it can be straightforwardly derived ineach particular case along the lines described above.Next, one needs an analog of Eqs. (26), (27) and Eqs. (38), (39) in case of arbitrary l . Tothis end, using the explicit form of φ l ( α ), it suffices to represent the wave function φ ilm ( x i , y i )in Eq. (52) as φ jlm ( x j , y j ) = 14 π | x j | F l ( y j ) + ˜ φ jlm ( x j , y j ) , (63)where the second term on the right-hand side is regular as | x i | →
0. Then, the analog ofEqs. (26), (26) reads V j ( x j ) ψ j ( x j , y j ) = − δ (3) ( x j ) F l ( y j ) . (64)With these building blocks, the leading contribution to the energy shift expression can bestraightforwardly calculated∆ E m = − X i =1 X e Z d x i d y i δ ( x i )( F l ( y i )) ∗ × (cid:18) φ jlm ( x j , y j + e Lµ j ( ki ) ) + φ klm ( x k , y k − e Lµ k ( ij ) ) (cid:19) . (65)13ere, we take into account the fact that the finite-volume energy shift can explicitly dependon the projection of the angular momentum m .In order to proceed further, we note that, for arbitrary l , the function φ l ( α ) is singularat α = 0: φ l ( α ) = G l α + ˜ φ l ( α ) , (66)where the second term is regular at the origin. The leading contribution in the limit L → ∞ comes from the singular term. Further, in this limit, we havelim L →∞ Y lm (Ω y ′ j ) = Y lm (Ω e ) , lim L →∞ Y lm (Ω y ′′ k ) = ( − l Y lm (Ω e ) , (67)where y ′ j = y j + e Lµ j ( ki ) and y ′′ k = y k − e Lµ k ( ij ) .In the following, we present the averaged shift, defined as∆ E = 12 l + 1 l X m = − l ∆ E m . (68)Defining F l ( y i ) = ¯ F l ( | y | i ) Y lm (Ω y i ), Eq. (65) can be finally transformed into∆ E = − (cid:18) π l + 1 (cid:19) / (cid:18) π κ (cid:19) / X i =1 N ll G l Z − dzP l ( z ) Z ∞ ydy ( ¯ F ( y )) ∗ × (cid:18) A j ( Lµ j ( ki ) ) − / | sin γ ij | exp( − κLµ j ( ki ) ) exp( κ cos γ ij yz )+ ( − l A k ( Lµ k ( ij ) ) − / | sin γ ik | exp( − κLµ k ( ij ) ) exp( κ cos γ ik yz ) (cid:19) . (69)From the above expression, it is clear that the result for general l looks similar to Eqs. (34),(50). Namely, it contains the exponentially vanishing factors together with an overall factor( κL ) − / . Only the numerical coefficients depend on the angular momentum l . i) In this article, we have extended the approach of Ref. [1] and derived explicit expressionsfor the energy shift of the three-particle bound state in the unitary limit with non-equalmass constituents and with the total angular momentum different from zero. All casesof physically relevant angular momenta (i.e., for which the the shallow bound statesexist in the unitary limit) were covered.ii) We show that the behavior of the leading terms in the finite-volume energy shift isuniversal for all l : namely, it contains three exponentially vanishing terms, whose ar-guments are determined by the pertinent reduced masses, i.e., by pure kinematics. In14ddition, there is a common multiplicative factor ( κL ) − / for all l . Only the numericalcoefficients, which stand in front of these universal factors, depend on l , and can becalculated for each l explicitly, using the method described in the paper.iii) On several occasions already, the simple model, considered in Ref. [1], has served as anice testing ground for the different types of the three-particle quantization condition,which are available in the literature (see, e.g., [11, 25]). Moreover, a comparison of theresults has shed more light on the role of a three-particle force in the description of thevolume-dependence of the shallow bound states [25]. A universal formula for arbitrary l and unequal masses, which was derived in this paper, without any doubt, representsa further challenge for the above-mentioned approaches, as well as an opportunity togain a deeper insight in the three-particle dynamics in a finite volume. Acknowledgments
The authors thank H.-W. Hammer for useful discussions. We acknowledge the supportfrom the CRC 110 “Symmetries and the Emergence of Structure in QCD” (DFG grantno. TRR 110 and NSFC grant No. 11621131001). This research is supported in part byVolkswagenstiftung under contract no. 93562, by the Chinese Academy of Sciences (CAS)President’s International Fellowship Initiative (PIFI) (Grant No. 2018DM0034) and by ShotaRustaveli National Science Foundation (SRNSF), grant no. DI-2016-26. It is also supportedin part by the National Science Foundation of China (NSFC) under the project No.11335001and by Ministry of Science and Technology of China (MSTC) under 973 project ”Systematicstudies on light hadron spectroscopy”, No. 2015CB856702.
References [1] U.-G. Meißner, G. R`ıos and A. Rusetsky, Phys. Rev. Lett. (2015) 091602 Erratum:[Phys. Rev. Lett. (2016) 069902] [arXiv:1412.4969 [hep-lat]].[2] M. L¨uscher, Commun. Math. Phys. (1986) 177.[3] M. L¨uscher, Commun. Math. Phys. (1986) 153.[4] M. L¨uscher, Nucl. Phys. B (1991) 531.[5] K. Polejaeva and A. Rusetsky, Eur. Phys. J. A (2012) 67 [arXiv:1203.1241].[6] R. A. Brice˜no and Z. Davoudi, Phys. Rev. D (2013) 094507 [arXiv:1212.3398].[7] M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2014) 116003 [arXiv:1408.5933].[8] M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2015) 114509 [arXiv:1504.04248[hep-lat]].[9] M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2016) 014506 [arXiv:1509.07929[hep-lat]]. 1510] M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2016) 096006 Erratum: [Phys. Rev.D (2017) 039901] [arXiv:1602.00324 [hep-lat]].[11] M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2017) 034501 [arXiv:1609.04317[hep-lat]].[12] R. A. Brice˜no, M. T. Hansen and S. R. Sharpe, arXiv:1803.04169 [hep-lat].[13] M. Mai and M. D¨oring, Eur. Phys. J. A (2017) 240 [arXiv:1709.08222].[14] R. A. Brice˜no, M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2017) 074510[arXiv:1701.07465 [hep-lat]].[15] P. Guo and V. Gasparian, arXiv:1709.08255 [hep-lat].[16] P. Guo and V. Gasparian, Phys. Lett. B (2017) 441 [arXiv:1701.00438 [hep-lat]].[17] P. Guo, Phys. Rev. D (2017) 054508 [arXiv:1607.03184 [hep-lat]].[18] S. Kreuzer and H.-W. Hammer, Phys. Lett. B (2009) 260 [arXiv:0811.0159].[19] S. Kreuzer and H.-W. Hammer, Eur. Phys. J. A (2010) 229 [arXiv:0910.2191].[20] S. Kreuzer and H.-W. Hammer, Phys. Lett. B (2011) 424 [arXiv:1008.4499].[21] S. Kreuzer and H. W. Grießhammer, Eur. Phys. J. A (2012) 93 [arXiv:1205.0277].[22] M. Jansen, H.-W. Hammer and Y. Jia, Phys. Rev. D (2015) 114031 [arXiv:1505.04099[hep-ph]].[23] S. Bour, H.-W. Hammer, D. Lee and U.-G. Meißner, Phys. Rev. C (2012) 034003[arXiv:1206.1765 [nucl-th]].[24] S. Bour, S. K¨onig, D. Lee, H.-W. Hammer and U.-G. Meißner, Phys. Rev. D (2011)091503 [arXiv:1107.1272 [nucl-th]].[25] H. W. Hammer, J. Y. Pang and A. Rusetsky, JHEP (2017) 109 [arXiv:1706.07700[hep-lat]].[26] H.-W. Hammer, J.-Y. Pang and A. Rusetsky, JHEP (2017) 115 [arXiv:1707.02176[hep-lat]].[27] M. D¨oring, H.-W. Hammer, M. Mai, J.-Y. Pang, A. Rusetsky and J. Wu,arXiv:1802.03362 [hep-lat].[28] V. Efimov, Nucl. Phys. A (1973) 157.[29] S. K¨onig and D. Lee, Phys. Lett. B (2018) 9.[30] S. R. Beane, W. Detmold and M. J. Savage, Phys. Rev. D (2007) 074507[arXiv:0707.1670 [hep-lat]].[31] S. R. Sharpe, Phys. Rev. D (2017) 054515 [arXiv:1707.04279 [hep-lat]].[32] E. Nielsen, D. V. Fedorov, A. S. Jensen and E. Garrido, Phys. Rept. (2001) 371.[33] E. Nielsen, D. V. Fedorov and A. S. Jensen, Phys. Rev. C (1999) 069801.1634] P. Navratil, B. R. Barrett and W. Gloeckle, Phys. Rev. C (1999) 611 [nucl-th/9811074].[35] K. Helfrich and H.-W. Hammer, J. Phys. B (2011) 215301 [arXiv:1107.0869 [cond-mat.quant-gas]].[36] S. K¨onig, D. Lee and H.-W. Hammer, Annals Phys. (2012) 1450 [arXiv:1109.4577[hep-lat]].[37] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products,” SeventhEdition, Elsevier, 2007, Sect 9.137.[38] J. Raynal and J. Revai, Nuovo Cim. A (1970) 612.[39] S. N. Ershov, Phys. Atom. Nucl. (2016) 1010 [Yad. Fiz.79