A note on the Lipkin model in arbitrary fermion number
aa r X i v : . [ nu c l - t h ] F e b A note on the Lipkin model in arbitrary fermion number
Yasuhiko
Tsue , , Constan¸ca Providˆencia , Jo˜ao da Providˆencia andMasatoshi Yamamura , CFisUC, Departamento de F´ısica, Universidade de Coimbra, 3004-516 Coimbra,Portugal Physics Division, Faculty of Science, Kochi University, Kochi 780-8520, Japan Department of Pure and Applied Physics, Faculty of Engineering Science,Kansai University, Suita 564-8680, Japan
A possible form of the Lipkin model obeying the su (6)-algebra is presented. It is anatural generalization from the idea for the su (4)-algebra recently proposed by the presentauthors. All the relation appearing in the present form can be expressed in terms of thespherical tensors in the su (2)-algebras. For specifying the linearly independent basis com-pletely, twenty parameters are introduced. It is concluded that, in these parameters, theten denote the quantum numbers coming from the eigenvalues of some hermitian operators.The five in these ten determine the minimum weight state. §
1. Introduction
This paper is a continuation of two papers, recently, published by the presentauthors.
Hereafter, these two will be referred to as (I) and (II), respectively. Inthese papers, we treated the Lipkin model with arbitrary single-particle levelsand fermion number. In (I), mainly an idea of how to construct the minimum weightstate, which is the starting point of the algebraic approach, was proposed. In (II), wediscussed how to express the linearly independent basis built on a chosen minimumweight state. Present paper aims mainly at supplementing the results of (II) withthe discussion on the su (6)-algebra, which we promised in (II).First, let us consider the Lipkin model obeying the su ( n )-algebra for the casewith n = 2 m in rather general framework. Here, m denotes integer. Since thetotal fermion number N is a constant of motion, we omit the discussion on N . Ourpresent argument is restricted to the case with even integer n with n = 4 , , · · · ,i.e., n = 2 m with m = 2 , , · · · . If the generators in the su ( n )-algebra are expressedappropriately, we can show that this model includes m su (2)-subalgebras. Thispoint has been shown in (II) and the generators are given in the relation (II.2.3).Therefore, as the total sum, we can define the su (2)-algebra ( e S ± , ) which will playa central role in our approach. Of course, the generators in each su (2)-algebra forma vector. Further, in (II), we showed that the su ( n )-Lipkin model includes one su ( m )-subalgebra, all generators in which are scalar for ( e S ± , ) and the remains form m ( m − / ∼ (II.2.12),respectively. Generally, the minimum weight states in the su ( n )-algebra are specifiedby ( n −
1) quantum numbers. In the present case, we can decompose the number typeset using
PTP
TEX.cls h Ver.0.9 i ( n −
1) into two parts: n − m + ( m − . ( n = 2 m ) (1)As was already mentioned, the case with n = 2 m includes m su (2)-algebras and one su ( m )-algebra. The first and the second term, m and ( m − m su (2)-algebras and of one su ( m )-algebra for the minimum weight state.Next, let us consider the orthogonal set constructed by operating “certain opera-tors” on any minimum weight state. We will call them as the excited state generatingoperators. They should be expressed in terms of n ( n − / (cid:0)(cid:0) n − (cid:1) − ( n − (cid:1) = 12 n ( n −
1) = m (2 m − . ( n = 2 m ) (2)The number n ( n − / su ( n )-generators with the types e S p ( n )( p = 1 , , · · · , n −
1) and e S pq ( n ) ( p > q = 1 , , · · · , n − n −
1) + ( n − n − / n ( n − /
2. We will call them as the raising operators and theirhermitian conjugates as the lowering operators. The definition of e S p ( n ) and e S pq ( n )has been given in the relation (I.2.2). We know that there exist m su (2)-subalgebrasand one su ( m )-algebra and, then, in the excited state generating operators, m and m ( m − / m (2 m − − (cid:18) m + 12 m ( m − (cid:19) = 3 · m ( m − . (3)It should be noted that the number m ( m − / e S ± , ), which are presented in (II). With the use of these threetypes of the operators, we can expect to obtain a possible idea for constructing theexcited state generating operators. Further, we notice that the relation (3) can bedecomposed into 3 · m ( m −
1) = 12 m ( m −
1) + 2 · m ( m − . (4)First term corresponds to the number of the raising or the lowering operators in the su ( m )-subalgebra. It is well known that we can construct a tensor operator with twoparameters in terms of a vector, for example, the solid harmonics Y l,l = r l Y ll ( θφ )is constructed in terms of the position vector ( x = r sin θ cos φ, y = r sin θ sin φ, z = r cos φ ). In our case, there exist m ( m − / su (2)-subalgebras, which is specified by two parameters. This argumentleads us to the following: Second term represents the total number of the parame-ters contained in the operators which are built by m ( m − / m ( m − / su ( m )-subalgebra. A merit of our idea may be as follows: All generators of the su ( n )-Lipkin model are expressed in terms of the spherical tensors for ( e S ± , ) and,then, we can apply the technique of the angular momentum coupling rule.As a simple example of the Lipkin model, we will give a brief summary for thecase with n = 4, i.e., m = 2, which has been discussed in (II), but the form in thispaper is a little bit different from that of (II). The generators in the present two su (2)-subalgebras are copied from the relation (II.5.9): e S + (1) = e S , e S − (1) = e S , e S (1) = 12 (cid:16) e S − e S (cid:17) , (4a) e S + (2) = e S , e S − (2) = e S , e S (2) = 12 e S . (4b)The sum is given by e S ± , = e S ± , (1) + e S ± , (2) . (5)The generators in the su ( m = 2)-subalgebra are expressed as e R + = e S + e S , e R − = e S + e S , e R = 12 (cid:16) e S + e S − e S (cid:17) . (6)Since e R ± , are scalars for ( e S ± , ), we have the relation h any of e S ± , , any of e R ± , i = 0 . (7)The vector operators for ( e S ± , ) are given as e R , +1 = − e S , e R , = 1 √ (cid:16) e S − e S (cid:17) , e R , − = e S , (8a) e R , +1 = − e S , e R , = 1 √ (cid:16) e S − e S (cid:17) , e R , − = e S , (8b)The relations (6) ∼ (8) are copied from the relations (II.5.11) ∼ (II.5.14). The vectors e R ± , satisfy h e R + , e R ,ν i = 0 , h e R , e R ,ν i = e R ,ν . ( ν = ± ,
0) (9)The minimum weight state is expressed as | ρ, σ , σ i , where the eigenvaluesof e R , e S (1) and e S (2) are denoted as − ρ , − σ and − σ , respectively. The explicitform of | ρ, σ , σ i is given in (I). Then, the eigenstate of ( e R , e R ), ( e S (1) , e S (1)) and( e S (2) , e S (2)) with the eigenvalues ( ρ ( ρ + 1) , ρ ), ( σ ( σ + 1) , σ ) and ( σ ( σ + 1) , σ )can be expressed as | ρ , σ , σ ; ρ, σ , σ i = f ρρ f σ σ f σ σ e P ρρ ,σ σ ,σ σ | ρ, σ , σ i , (10a) e P ρρ ,σ σ ,σ σ = (cid:16) e R + (cid:17) ρ + ρ (cid:16) e S + (1) (cid:17) σ + σ (cid:16) e S + (2) (cid:17) σ + σ , (10b) f ττ = s ( τ − τ )!(2 τ )!( τ + τ )! . ( τ = ρ, σ , σ ) (11)The eigenstate of ( e S , e S ) with the eigenvalue ( σ ( σ + 1) , σ ) is given by | ρρ , σ , σ , σσ i = X σ σ h σ σ σ σ | σσ i| ρ , σ , σ ; ρ, σ , σ i . (12)We are investigating the case with n = 4, i.e., m = 2. Therefore, as can be seen in therelation (3), further, we must search the operator characterized by three quantumnumbers for obtaining the excited state generating operators. For this task, in (II),the following form is adopted: e R l,l ; l,λ = (cid:16) ⇀ R − (cid:17) l − λ (cid:16) ⇀ S − (cid:17) l − l (cid:16) − e S (cid:17) l . (13)Here, we adopted the notation for e O and e A in the form (cid:16) ⇀ O (cid:17) n e A = h e O, · · · , h e O, h e O | {z } n , e A ii · · · i . (14)Since ⇀ S + (cid:16) − e S (cid:17) l = ⇀ R + (cid:16) − e S (cid:17) l = 0 and ⇀ S (cid:16) − e S (cid:17) l = ⇀ R (cid:16) − e S (cid:17) l = l (cid:16) − e S (cid:17) l , e R l,l ; l,λ can be regarded as spherical tensor operator for the su (2)-algebras ( e S ± , )and ( e R ± , ) with rank l . Operating e R l,l ; l,λ on the state (12) and applying the angularmomentum coupling rule, we have | l, rr , ss , σ ; ρσ σ i = X λ l h lλ ρρ | rr ih ll σσ | ss i e R l,l ; l,λ | ρρ , σ , σ , σσ i . (15)However, the set composed by the states (15) cannot be regarded as orthogonal. Thereasons are as follows: (1) the symbol l is not a quantum number, but a parameter,the value of which is given from the outside and (2) the operator e R l,l ; l,λ contains thedegrees of freedom which do not contain in the states | ρρ , σ , σ , σσ i , for example,such as − e S . Therefore, they form the linearly independent basis and by appropriatemethod, for example, by the Schmidt method, we must construct the orthogonal set.In the sense mentioned above, the excited state generating operator is given by e R l,l ; l,λ · e P ρρ ,σ σ ,σ σ = (cid:16) ⇀ R − (cid:17) l − λ (cid:16) ⇀ S − (cid:17) l − l (cid:16) − e S (cid:17) l × (cid:16) e R + (cid:17) ρ + ρ (cid:16) e S + (1) (cid:17) σ + σ (cid:16) e S + (2) (cid:17) σ + σ . (16)The above is the summary of (II) for the case with n = 4, i.e., m = 2. In thepreparation for the discussion on the case with n = 6, i.e., m = 3, we rewrite therelation (13) in the form slightly different from the original. The form ( ⇀ S − ) l − l ( − e S ) l can be rewritten as (cid:16) ⇀ S − (cid:17) l − l (cid:16) − e S (cid:17) l = (cid:16) ⇀ S − (cid:17) l − l (cid:16) e R , +1 (cid:17) l = e Z l,l . (17)Here, e Z l,l is of the form e Z l,l = s l !(2 l − X λ (cid:18) √ (cid:19) l − λ p ( l + l )!( l − l )! (cid:16) l + l − λ (cid:17) ! λ ! (cid:16) l − l − λ (cid:17) ! × (cid:16) e R , +1 (cid:17) l + l − λ (cid:16) e R , (cid:17) λ (cid:16) e R , − (cid:17) l − l − λ . (18)The sum for λ obeys the following condition : if | l − l | =even or odd, λ cannot beodd or even, respectively. The operator e Z l,l is a tensor with rank l for ( e S ± , ): ⇀ S ± e Z l,l = p ( l ∓ l )( l ± l + 1) e Z l,l ± , ⇀ S e Z l,l = l e Z l,l . (19)In the case that ( e R , ± , e R , ) is a position vector ( r ± = ∓ ( x ± iy ) / √ , r = z ), e Z l,l is reduced to the solid harmonics: e Z l,l −→ s πl !(2 l + 1)!! Y l,l , (cid:16) Y l,l = r l Y ll ( θφ ) (cid:17) . (20)An important property of e Z l,l is as follows: ⇀ R + e Z l,l = 0 , ⇀ R e Z l,l = l e Z l,l . (21)With the use of the relation (9), we are able to obtain the property (21).Under the above preparation, we will investigate the case with n = 6, i.e., m = 3.This case includes three su (2)-subalgebras in the form e S + (1) = e S , e S − (1) = e S , e S (1) = 12 (cid:16) e S − e S (cid:17) , (22a) e S + (2) = e S , e S − (2) = e S , e S (2) = 12 (cid:16) e S − e S (cid:17) , (22b) e S + (3) = e S , e S − (3) = e S , e S (3) = 12 e S . (22c)The total sum of the above is denoted as e S ± , = e S ± , (1) + e S ± , (2) + e S ± , (3) . (23)The eight generators in the su ( m = 3)-subalgebra, which are scalars for ( e S ± , ), arewritten down as follows: e R + = e S + e S , e R − = e S + e S , e R = 12 (cid:16) e S + e S − e S − e S (cid:17) , (24a) e R , = e S + e S , e R , = e S + e S , e R , − = e S + e S , e R , − = e S + e S , (24b) e R = 12 (cid:16) e S + e S + e S + e S (cid:17) − e S . (24c)The set (24a) forms the su (2)-algebra and ( e R , , e R , − ) and ( − e R , − , e R , ) arespinors for ( e R ± , ). The present case includes three vectors for ( e S ± , ): e R , +1 (1) = − e S , e R , (1) = 1 √ (cid:16) e S − e S (cid:17) , e R , − (1) = e S , (25a) e R , +1 (2) = − e S , e R , (2) = 1 √ (cid:16) e S − e S (cid:17) , e R , − (2) = e S , (25b) e R , +1 (3) = − e S , e R , (3) = 1 √ (cid:16) e S − e S (cid:17) , e R , − (3) = e S . (25c)The hermitian conjugates of the vectors are omitted to give. The expressions (22),(24) and (25) are obtained by putting m = 3 in the relations (II.2.3) and (II.2.9) ∼ (II.2.12).For ν = ± ,
0, the vector ( e R ,ν (1)) satisfies the relation ⇀ R + e R ,ν (1) = ⇀ R , e R ,ν (1) = ⇀ R , − e R ,ν (1) = 0 , (26a) ⇀ R e R ,ν (1) = 12 e R ,ν (1) , ⇀ R e R ,ν (1) = 32 e R ,ν (1) . (26b)For ν = ± , k = 2 ,
3, two vectors ( e R ,ν ( k )) obey the relation ⇀ R + e R ,ν ( k ) = δ k, e R ,ν ( k ) , ⇀ R , e R ,ν ( k ) = 0 , ⇀ R , − e R ,ν ( k ) = − δ k, e R ,ν ( k ) , (27a) ⇀ R e R ,ν ( k ) = (cid:18) δ k, − δ k, · (cid:19) e R ,ν ( k ) , ⇀ R e R ,ν ( k ) = δ k, · e R ,ν ( k ) . (27b)The above are the relations for our present case ( n = 6). However, e R ,ν (2) and e R ,ν ′ (3) do not commute mutually, but, they commute with e R ,ν ′′ (1). We will treatthe case with n = 6 as a natural generalization from the case with n = 4.Now, in parallel with the su (4)-Lipkin model, we are possible to give our schemefor obtaining the linearly independent basis for the su (6)-Lipkin model. In orderto avoid unnecessary complication, we will not apply the angular momentum cou-pling rule, together with the associating numerical factors, for example, such as theform (11). The minimum weight state is expressed in the form | ρ, ρ , σ , σ , σ i ,where ρ , ρ , σ , σ and σ denote the eigenvalues of − e R , − e R , − e S (1), − e S (2) and − e S (3) defined in the relations (24) and (22), respectively. As a possible extensionof e P ρρ ,σ σ ,σ σ shown in the relation (10b), we introduce the following operator: e P µµ ,ρ ρ ,σ σ ,σ σ ,σ σ = (cid:16) e R , (cid:17) µ + µ (cid:16) e R , − (cid:17) µ − µ (cid:16) e R + (cid:17) ρ + ρ (cid:16) e S + (1) (cid:17) σ + σ (cid:16) e S + (2) (cid:17) σ + σ (cid:16) e S + (3) (cid:17) σ + σ . (28)With the use of the operator (28), we define the state | µµ , ρ , σ , σ , σ ; ρ, ρ , σ , σ , σ i = e P µµ ,ρ ρ ,σ σ ,σ σ ,σ σ | ρ, ρ , σ , σ , σ i . (29)The state (29) corresponds to the state (10a) and is expressed in terms of elevenparameters, some of which play the role of quantum numbers.The above is first step in our approach to the case with n = 6. The linearlyindependent basis of our present case should be expressed totally in terms of twentyparameters. Therefore, we must, further, investigate how to consider nine parameterswhich may be related with three vectors. For this task, we extend the idea adoptedin the case with n = 4. This case starts in the relation (9). The relation (9) forthe vector (17) with (18) for the definition of e Z l,l leads us to the relation (21) forthe raising operator e R + and the hermitian operator e R . These two correspond to( e R + , e R , , e R , − ) and ( e R , e R ), respectively for the present case. As can be seen inthe relation (27a), the operation of ⇀ R + and ⇀ R , ± on the vectors labeled k = 2 , R ,ν (1) = e R ,ν (1) , ( ν = ± ,
0) (30a)˘ R , ( k ) = 1 √ (cid:16) e R , (1) e R , ( k ) − e R , ( k ) e R , (1) (cid:17) , ˘ R , ( k ) = 1 √ (cid:16) e R , (1) e R , − ( k ) − e R , ( k ) e R , − (1) (cid:17) , ˘ R , − ( k ) = 1 √ (cid:16) e R , (1) e R , − ( k ) − e R , ( k ) e R , − (1) (cid:17) . (30b)The vector (30b) is derived by the formula Z ν = P λ,µ h λ µ | ν i X λ Y µ for two vectors X and Y . The vectors (30) satisfy the following relation for ν = ± , k =1 , , ⇀ R + ˘ R ,ν ( k ) = ⇀ R , ± ˘ R ,ν ( k ) = 0 , (31a) ⇀ R ˘ R ,ν ( k ) = 12 (1 + δ k, − δ k, ) ˘ R ,ν ( k ) , ⇀ R ˘ R ,ν ( k ) = 32 (1 + δ k, ) ˘ R ,ν ( k ) . (31b)With the use of the vectors (31), we can define the operator˘ Z l l ,l l ,l l = e Z l l (1) e Z l l (2) e Z l l (3) . (32)Here, e Z l k l k ( k ) ( k = 1 , ,
3) is obtained by replacing ll and e R ,ν in the relation(18) with l k l k and ˘ R ,ν ( k ). As was already mentioned, some pairs of ( e R ,ν ( k )) donot commute and, therefore, the ordering of e Z l k l k ( k ) in the definition (32) should befixed beforehand, for example, as is shown in the definition (32). The operator (32)satisfies ⇀ R + ˘ Z l l ,l l ,l l = ⇀ R , ± ˘ Z l l ,l l ,l l = 0 , (33a) ⇀ R ˘ Z l l ,l l ,l l = 12 (cid:0) l + 3 l (cid:1) ˘ Z l l ,l l ,l l , ⇀ R ˘ Z l l ,l l ,l l = 32 (cid:0) l + l + 2 l (cid:1) ˘ Z l l ,l l ,l l . (33b)We can see that the above is a natural extension from the relation (21).The operator (32) is expressed in terms of six parameters ( l , l ), ( l , l ) and( l , l ). Then, in order to accomplish our task, we must search, further, three pa-rameters. As can be seen in the relation (13), the case with n = 4 is completed bytaking into account the lowering operator e R − in the form ⇀ R − . The relation (33a)tells us that the operation of the raising operators e R + and e R , ± makes the resultsvanish. It may be enough to consider three lowering operators ⇀ R − and ∓ ⇀ R , ± on the operator (32). First, we note that the operator (32) is nothing but tensorspecified by l = l = 12 ( l + 3 l ) . (34)Then, tensor operator specified by l and l ( l = − l, − l + 1 , · · · , l − , l ) can begiven in the form ˘ Z ll ( l l ,l l ,l l ) = (cid:16) ⇀ R − (cid:17) l − l ˘ Z l l ,l l ,l l . (35)Next, we consider the operators ∓ e R , ∓ , which form the spinor for ( e R ± , ). As iswell known, tensor operator specified by λ and λ ( λ = − λ, − λ + 1 , · · · , λ − , λ )is constructed in the form e Y λλ = (cid:16) − ˜ R , − (cid:17) λ + λ (cid:16) e R , (cid:17) λ − λ . (36)In the case of applying the angular momentum couping rule, it may be convenientto attach the numerical factor g λλ : g λλ = s (2 λ )!( λ + λ )!( λ − λ )! . (37)Then, we introduce the operator ⇀ Y λλ = (cid:16) − ⇀ R , − (cid:17) λ + λ (cid:16) ⇀ R , (cid:17) λ − λ . (38)Product of the operators (35) and (38) gives us the operator with nine parameters: e R λλ ,ll ( l l ,l l ,l l ) = ⇀ Y λλ ˘ Z ll ( l l ,l l ,l l ) = (cid:16) − ⇀ R − (cid:17) λ + λ (cid:16) ⇀ R , (cid:17) λ − λ (cid:16) ⇀ R − (cid:17) l − l ˘ Z l l ,l l ,l l . (39)Thus, multiplying the operator (40) by (28), we obtain the state generating operatorwith fifteen parameters.Until the present, we have not contacted with the spherical tensor representationof our model with respect to the su (2)-algebras ( e R ± , ) and ( e S ± , ). Its explicitexpression is omitted to show, but, we will discuss the basic idea for this problem.The linearly independent basis obtained in this paper is expressed in terms of twentyparameters, which, at least, five are related with the quantum numbers for theminimum weight state, i.e., ρ , ρ , σ , σ and σ . Then, it may be important toinvestigate which parameters play a role of the quantum numbers or not. It is ourfinal problem of this paper.Application of the angular momentum coupling rule to the state (29) leads usto the following: µµ , ρ ρ −→ µρ ; ηη , (40a) σ σ , σ σ , σ σ −→ σ σ ( σ ) σ ; σ σ , (40b) | µ − ρ | ≤ η ≤ µ + ρ , (41a) | σ − σ | ≤ σ ≤ σ + σ , | σ − σ | ≤ σ ≤ σ + σ . (41b)The inequalities (41a) and (41b) are related with the coupling rule for ( e R ± , ) and( e S ± , ), respectively. If we note that ρ , σ , σ and σ denote the quantum numberscoming from the Casimir operator, µ , η , σ and σ play a role of the parametersobeying the inequality (41). Next, we consider the operator (38). In this case, thecoupling rule gives us λλ , ll −→ λl ; ξξ , (42a) l l , l l , l l −→ l l ( l ) l ; l l , (42b) | λ − l | ≤ ξ ≤ λ + l, (43a) | l − l | ≤ l ≤ l + l , | l − l | ≤ l ≤ l + l . (43b)Under the relation (34), the seven, λ , ξ , l , l , l , l and l have to be regarded asparameters obeying the inequality (43). Finally, we consider e R , e R , e S and e S and,further, e R . Let the eigenvalues of e R , e R , e S and e S denote r ( r + 1), r , s ( s + 1)and s , respectively. For ( e R , e R ) and ( e S , e S ), we have the relations ξξ , ηη −→ ξη ; rr , (44a) l l , σ σ −→ l σ ; ss , (44b) | ξ − η | ≤ r ≤ ξ + η , (45a) | l − σ | ≤ s ≤ l + σ . (45b)At the present stage, we can summarize our discussion as follows: The eight symbols ρ , σ , σ , σ , ( r, r ) and ( s, s ) denote the quantum numbers and the eleven λ , µ , ξ , η , σ , σ , l , l , l , l and l must be regarded as the parameters, totallythe nineteen. However, in the above analysis, there are one point missing. We musttake account of ρ and R (the eigenvalue of e R ). After straightforward calculation, we0can show that our linearly independent basis is a set of the eigenstates of e R and R can be expressed in the form R = 3( µ − λ ) + 32 ( l + l + 2 l ) − ρ . (46)Then, we have to add R and ρ to the quantum numbers. Therefore, one of threeparameters µ , λ and ( l + l + 2 l ) /
2, for example, µ , depends on the others if R and ρ are given.Thus, we have a summary: In our linearly independent basis, ρ , ρ , σ , σ , σ , r , r , s , s and R denote the quantum numbers and σ , σ , l , l , l , l , l , ξ and η play a role of the parameters. Then, with the use of an appropriatemethod, for example, the Schmidt method, we can construct the orthogonal set forthe su (6)-Lipkin model with arbitrary fermion numbers.Including (I) and (II), this note has been devoted to the discussion on the Lipkinmodel, which belongs to the classical model in nuclear many-body theories. Theauthors tried to stress that there exist still some problems which remain unsolveduntil the present. Acknowledgment
Two of the authors (Y.T. and M.Y.) would like to express their thanks to Profes-sor J. da Providˆencia and Professor C. Providˆencia, two of co-authors of this paper,for their warm hospitality during their visit to Coimbra in spring of 2015. The au-thor, M.Y., would like to express his sincere thanks to Mrs K. Yoda-Ono for hercordial encouragement. The authors, Y.T., is partially supported by the Grants-in-Aid of the Scientific Research (No.26400277) from the Ministry of Education,Culture, Sports, Science and Technology in Japan.
References
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