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Null-plane model of three bosons with zero-range interaction
Physics Letters B 282 (1992) 409-414 North-Holland
PHYSICS LETTERS B
Null-plane model of three bosons with zero-range interaction T. Frederico Instituto de Estudos Avanqados, Centro TOcnico Aeroespacial, 12.225 Sdo Jos~ dos Campos, S~o Paulo, Brazil Received 15 December 199 l; revised manuscript received 2 March 1992
The relativistic formulation of the three-boson model interacting via a zero-range two-body force in the null plane is given. The Faddeev-like equation for the three-boson bound state is obtained and compared with the non-relativistic formula. The threeboson bound state energy is calculated numerically as a function of the tow-body boson energy.
The wavefunction defined in the null plane allows a relativistic description of bound states that has two simple characteristics: (i) the center of mass coordinate (CM) is easily separated [ 1-3], and (ii) the pair creation process is not relevant [ 3 ]. This feature allows the construction of the bound state wavefunction in terms of only particle degrees of freedom. It has been used extensively to construct relativistic models for the light pseudoscalar mesons [ 4 ] and the nucleon [ 4 ] in terms of bound-state quark systems. It was also applied to nuclear systems in the study of electron-deuteron scattering [3,5 ] for high-momentum transfers. These models are successful in describing the phenomenology. The next step towards understanding the null-plane formulation of relativistic systems is to describe the bound state wavefunction dynamically. In this work, the two- and three-boson systems in the null plane are studied for the zero-range force model and numerical results are presented. We choose this schematic model [ 6 ] because it qualitatively represents the nuclear interaction for the nonrelativistic three-nucleon system. The results are consistent with realistic nuclear force models [ 7 ]. Also, it does not have retardation effects, which simplifies the relativistic dynamical description of the bound state system. The general features of the Faddeev three-body null-plane equation have been discussed some time ago [ 2 ] but a calculation with a specific interaction is still missing. It is thus illustrative to work out the details of the null-plane three-body equation for the given dynamical model. The Weinberg equation [ 8 ] for the two-boson system is solved in the zero-range model of an attractive interaction. It is renormalized by introducing the condition for the existence of the two-boson bound system. Then, the two-body relativistic amplitude thus derived is inserted in the Faddeev-like [ 9 ] three-hoson equation. The important ingredient in both cases is the integration over the null-plane energy in the momentum-loop integral. This integration must be convergent [ 10 ] as it is in this case. The diagram in fig. la shows the amplitude for two-boson scattering which, in the zero-range model, can be solved analytically. The solution in the center-of-mass system is easily obtained as
(a)
(b)
Fig. 1. (a) The diagrammatic representation of the integral equation for the two-boson scattering amplitude. (b) The kernel of the integral equation is given by the bubble diagram ofeq. (2). 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
409
Volume282, number 3,4
PHYSICS LETTERSB
28 May 1992
l
r(M2) -- i 2 - ' - B ( M 2 ) '
( 1)
where 2> 0 is the coupling constant and M2 is the mass of the two-boson system. B(M2) is the kernel of the integral equation for r(M2) and is schematically represented by the bubble diagram of fig. lb, B(M2) = f d4k i i 3 2(n) 4 k2-M2+ie (P-k)Q-M2+ie '
(2)
where Mis the boson mass and p u = (M2, 0, 0, 0). Notations and conventions follow Bjorken and Drell [ 1 1 ]. The momentum loop integral in eq. (2) is integrated in the null-plane energy, k - ( = k ° This procedure gives the solution to the Weinberg equation in this simplified model. In the null-plane variables k - , k + ( = k°+ andk±, eq. (2) is
k3).
1
B(M2)
f dk-dk+d2kx
2(2n) 4 J
k+(P + -k
k3)
1
+ ) [k--(k
2 +M2-ie)/k+]{P--k
- -
[ (P-k) 2
+M2-ie]/(P
+
-k+)}" (3)
The integration over k - in eq. (3) is convergent, so that we can use the Cauchy theorem to perform it. Instead of working with k +, we d e f n e the variable x= k +/p +. There are three regions of integration in x,
x
0 < x < 1 and
x > 1.
The two poles of the integrand in k - of eq. (3) are placed in the lower semi-plane of the k - complex-plane for x > 1, resulting in no contribution to the k - integration. For x < 0 the poles are located in the upper semi-plane, and again they do not contribute to eq. (3). The poles contribute to the integral only if 0
i f dx d2ka 1 (2~)3 x ( 1 - x ) M 2 - ( k 2 + M 2 ) / x ( 1 - x ) "
(4)
Eq. (4) has a logarithmic divergence, which can be absorbed in the redefinition of 2. The physical information introduced in the renormalization of the physical amplitude is the mass of the two bound bosons, M2a. We assume that the two boson amplitude, r(M2), has a pole for M2 =M2~ which implies i2 -1 =B(M2B) . The subtraction imposed by this condition in the denominator of eq. ( 1 ) makes it finite. The appropriate choice of 2 assures the existence of the two-boson bound state. Using eq. (4) and the fact that MzB < 2M, it is possible to show that - i B ( M 2 a ) is positive and thus that 2> 0, meaning the force is attractive. The integrals in eq. (4) can be performed analytically and for M2 < 2M the result is [M ~
1 arctan (2
MM / _~22B
1~-l ~ M 2
I
(2
M~
1) 1] --1
The limit of eq. (5) for M and M2 going to infinity such that (2M-M2 ) / M goes to zero, agrees with the nonrelativistic two-body scattering amplitude of Frederico et al. [ 7 ] where the attractive zero-range two-body potential is used. The non-relativistic model has one two-body bound state. This fact was used for the construction of the integral equation for the three-boson wavefunction [6 ]. We will return to the limit of Mgoing to infinity, when we discuss the non-relativistic limit of the null-plane three-boson integral equation. 410
Volume 282, number 3,4
PHYSICS LETTERS B
28 May 1992
The expression of z(M2) for M2> 2 M is obtained by analytically continuing the inverse tangent function in the physical sheet. For our purposes we do not need this, as will become clear in the following. The Faddeev-like equation for a component of the three-boson vertex is shown in fig. 2. The factor 2 comes from symmetrization of the total vertex. The vertex v, in the center of mass system depends only on the momentum of the spectator particle, q u. The three-boson vertex is the sum of the three components in which each boson is spectator once. The two-boson amplitude, eq. (5), enters in the kernel of the vertex integral equation. Then, fig. 2 gives the following: I v(q u) = 2z(M2)
d4k i (2~r) 4 k 2 - ~ / 2 - i a
i (P3 - q - k ) 2 - M 2 - i e
v(k~') '
(6)
where P~ = (M3, 0, 0, 0) is the three-boson f o u r - m o m e n t u m in the center of mass system and the mass of the two-boson subsystem is given by M22 = (P3 _ q ) 2 . The integration over k - is performed to obtain the integral equation in the null-plane variables. Due to the zero range o f the interaction, the third boson is on the mass shell while the other two are interacting. This is a physically motivated argument. The third boson is propagating freely for any finite distance from the interacting pair due to the zero range of the interaction. As the third boson is free, it is on the mass shell. The interaction of the third boson occurs when the distance between it and one boson of the pair is zero. This configuration belongs to the other Faddeev component of the three-boson system, where the third boson loses its character of spectator. From the discussion above, we conclude that v is a function of q +, q± and q - = (q2 + M 2 ) / q + .
This result should also be obtained if one assumes analytical behavior of the vertex in the k - complex plane. Then, the pole of the propagator of the spectator particle gives the on-mass shell condition after integration over k - . The loop integral in the null-plane variables is _½ f d k - d k + d 2 k ± v ( k - , k + , k ± ) J (270 4 k+(P3-q-k) + 1
× [ k - - (k~ + M a - i e ) / k + l { P y
-q- -k--
[(P3 - q - k ) ~
+ M 2 - i e l / ( P + - k + )}"
(7)
The integration in k - has contributions from O q ÷ > 0. The argument here to obtain the limits in k + using the Cauchy theorem for the integration, follows the same steps as discussed after eq. (3). Before presenting the integral equation for v, let us discuss the limits of the variables qx and y = q +/ M 3 .
The mass of the two-boson subsystem must be real, so that
Fig. 2. The diagrammatic representation of the integral equation for the Faddeev component of the vertex of the three-boson bound-state. 411
Volume 282, number 3,4
M~=(M3-q
+)
(
M3
PHYSICSLETTERSB qZiq++M2.~ ]-q~
28 May 1992
(8)
>O.
Solving the inequality for q~, we obtain q~ < ( 1 - y ) ( M ~ y - M 2 ) .
(9)
The limits for y are 1>y>M2/M~,
(10)
where the lower bound comes from q~ > O. The integral equation for v results after the integration of eq. ( 7 ) over k - . Introducing eq. (5) and the limits eqs. ( 9 ), (10) in eq. ( 6 ), together with the result of the k - integration, the equation for the Faddeev component of the vertex is
l
1
v(y, q± ) = 2--~x / M E / M 2 a - ~ arctan(2x/ME/M~a - ~ ) - 1 _ x / M E / M E _ ~ a r c t a n ( 2 x / M 2 / M 2 _ ~ )-1 1 --y
×
k T ax
M2/M] X( 1 - - y - - x )
d2k±
M~ - M2o3 '
( 11 )
where M2 is given by eq. (8),
k"2ax=x/ ( l - x ) ( M ~ x - M 2 ) , and the mass of the virtual three-boson state is M23 - k 2 + M 2 + q2 +M___.~2+ ( q + k ) 2 + M 2 x y 1--y--x
(12)
The dependence of v on q - is not specified because the spectator boson is on mass-shell, q + and q+ describe the spectator boson propagation. The limits on the x integration and the condition y > M 2 / M 2 impose a lower bound for the mass of the threeboson system: M3 > x//2M.
(13)
Let us now study the non-relativistic limit of eq. (11 ) (M--,oo). First, the two-boson amplitude, eq. (5), in this limit is (see Frederico et al. [7] )
16z~v/-M
lim z = - i x / ~ z a _ x/~2 ,
M~oo
(14)
where M2B = 2 M - E2B and M2 = 2 M - E2. E2B is the two-boson binding energy. The non-relativistic limit of the integrand ofeq. ( 11 ) involves the change in the variable x to kz: dx x lim M' M~oo dk~ -
( 15 )
and lim x = lim y = ] . Mooo
M~oo
The mass of the virtual three-boson system is substituted in eq. ( 11 ) by 412
(16)
PHYSICS LETTERS B
Volume 282, number 3,4
28 May 1992
3.0
2.5
Ma 2.0
1.5
1.0 ' ~ / / " 0.0 0.4
0.8
1.2
1.6
Me
Mo3
=x/q2+M 2 +
~
2.0
Fig. 3. The three-boson mass as a function of the two-boson mass in units of the mass of the single boson. Shown is the three-boson bound state mass (dashed line) and the threshold for the threeboson mass (solid line).
+x/(q+k)2+M 2 •
(17)
The result is UNR(q) ~ N / ~
~2
1
f d3k
VNR(k)
X/~2B--X/~2 j M 2 E 3 - q 2 / 2 M - k 2 / 2 M - ( q W k ) 2 / 2 M '
(18)
where E3 is the three-boson binding energy. Eq. (18) is the non-relativistic integral equation obtained in ref. [61. The non-relativistic limit of eq. (11 ) has the qualitative properties of the three-boson bound state when M2B ~ 2M. The Efimov effect [ 12 ] holds true for eq. ( 11 ). However, the short-range behavior is strongly modified and the Thomas collapse [ 13 ] of the system is avoided. The mass of the three-boson state has a lower bound of v/2M. The Faddeev component of the ground state vertex is rotationally symmetric in the x-y plane in eq. ( 11 ). The mass of the boson gives the scale of the system. Here, the solution is presented for M = 1. In fig. 3 the numerical results for the ground-state mass of the three-boson system are shown. The lower bound of Ma is approached by our calculation. The three-boson binding energy increases with the mass of the two-boson system, and saturates at about 0.9M for M2 ~ 1.9M. The binding energy increases with decreasing two-boson binding, because the phase space in the x variable in eq. ( 11 ) also increases. An exception to this behavior occurs near M2 = 2M. The threshold for the three-boson mass is also shown for comparison with the calculations. In summary, we give an example of how null-plane dynamics can be elaborated. We develop a zero-range model of the three-boson bound state in the null plane and solve numerically the dynamical equation for the ground state. The three-boson mass has a lower bound of v/2M and the non-relativistic limit of the equation is consistent with the zero-range theory [ 6 ]. Other situations of interest, such as scattering, will be considered in the future. I am grateful to R. Thaler for calling my attention to this problem, to A. Delfino for discussions, to B. Carlson for reading the manuscript and to H. Fearing for his hospitality at Triumf, University of British Columbia, where part of this work was done. This work was supported by the Conselho Nacional de Desenvolvimento 413
Volume 282, number 3,4
PHYSICS LETTERS B
28 May 1992
Cientifico e Tecnol6gico - CNPq and Fundag~o de A m p a r o / l Pesquisa do Estado de S~o Paulo - FAPESP, Brazil.
References [ 1 ] M.V. Terent'ev, Sov. J. Nucl. Phys. 24 (1976) 106; L.A. Kondratyuk and M.V. Terent'ev, Sov. J. Nucl. Phys. 31 (1980) 561. [ 2 ] B.L.G. Bakker, L.A. Kondratyuk and M.V. Terent'ev, Nucl. Phys. B 158 (1979) 497. [3] L.L. Frankfurt and M.I. Strikman, Nucl. Phys. B 148 (1979) 107; Phys. Rep. 76 ( 1981 ) 215. [4] Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 58 (1987) 2175; Phys. Rev. D 37 (1988) 778; P.L. Chung, F. Coester and W.N. Polizou, Phys. Lett. B 205 (1988) 545; C.R. Ji and S. Cotanch, Phys. Rev. D 21 (1990) 2319; I.G. Aznauryan and K.A. Oganesyan, Sov. J. Nucl. Phys. 47 ( 1988 ) 1097; Phys. Lett. B 249 (1990) 309. [5] I.L. Grach and L.A. Kondratyuk, Sov. J. Nucl. Phys. 39 (1984) 198; P.L. Chung, F. Coester, B.D. Keister and W.N. Polizou, Phys. Rev. 37 ( 1988 ) 2000; T. Frederico, G.A. Miller and E.M. Henley, Nucl. Phys. A 533 ( 1991 ) 617. [6] G.V. Skornyakov and K.A. Ter-Martirosyan, Sov. Phys. JETP 4 (1957) 648. [7] V. Efimov and E.G. Tkachenko, Phys. Lett. B 157 (1985) 108; V.F. Kharchencko, Sov. J. Nucl. Phys. 16 (1973) 173; T. Frederico, I.D. Goldman and A. Delfino, Phys. Rev. C 37 (1988) 497. [ 8 ] S. Weinberg, Phys. Rev. 150 ( 1966 ) 1313. [9] L.D. Faddeev, Sov. Phys. JETP 12 (1960) 1014. [ 10] R. Dashen and M. Gell-Mann, in: Proc. 3rd Coral Gables Conf. on Symmetry principles at high-energy (1966) (Freeman, San Francisco ); S. Fubini, G. Segr6 and D. Walecka, Ann. Phys. 39 (1966) 381; V. de Alfara, S. Fubini, G. Furlan and C. Rossetti, Currents in hadron physics (North-Holland, Amsterdam, 1973 ). [ 11 ] J.D. Bjorken and S.D. Drell, Relativistic quantum fields (McGraw-Hill, New York, 1965 ). [ 12] V. Efimov, Phys. Lett. B 33 (1970) 563; see also R.D. Amado and J.V. Noble, Phys. Rev. D 5 (1972) 1992. [ 13] L.H. Thomas, Phys. Rev. 47 (1935) 903.
414
PHYSICS LETTERS B
Null-plane model of three bosons with zero-range interaction T. Frederico Instituto de Estudos Avanqados, Centro TOcnico Aeroespacial, 12.225 Sdo Jos~ dos Campos, S~o Paulo, Brazil Received 15 December 199 l; revised manuscript received 2 March 1992
The relativistic formulation of the three-boson model interacting via a zero-range two-body force in the null plane is given. The Faddeev-like equation for the three-boson bound state is obtained and compared with the non-relativistic formula. The threeboson bound state energy is calculated numerically as a function of the tow-body boson energy.
The wavefunction defined in the null plane allows a relativistic description of bound states that has two simple characteristics: (i) the center of mass coordinate (CM) is easily separated [ 1-3], and (ii) the pair creation process is not relevant [ 3 ]. This feature allows the construction of the bound state wavefunction in terms of only particle degrees of freedom. It has been used extensively to construct relativistic models for the light pseudoscalar mesons [ 4 ] and the nucleon [ 4 ] in terms of bound-state quark systems. It was also applied to nuclear systems in the study of electron-deuteron scattering [3,5 ] for high-momentum transfers. These models are successful in describing the phenomenology. The next step towards understanding the null-plane formulation of relativistic systems is to describe the bound state wavefunction dynamically. In this work, the two- and three-boson systems in the null plane are studied for the zero-range force model and numerical results are presented. We choose this schematic model [ 6 ] because it qualitatively represents the nuclear interaction for the nonrelativistic three-nucleon system. The results are consistent with realistic nuclear force models [ 7 ]. Also, it does not have retardation effects, which simplifies the relativistic dynamical description of the bound state system. The general features of the Faddeev three-body null-plane equation have been discussed some time ago [ 2 ] but a calculation with a specific interaction is still missing. It is thus illustrative to work out the details of the null-plane three-body equation for the given dynamical model. The Weinberg equation [ 8 ] for the two-boson system is solved in the zero-range model of an attractive interaction. It is renormalized by introducing the condition for the existence of the two-boson bound system. Then, the two-body relativistic amplitude thus derived is inserted in the Faddeev-like [ 9 ] three-hoson equation. The important ingredient in both cases is the integration over the null-plane energy in the momentum-loop integral. This integration must be convergent [ 10 ] as it is in this case. The diagram in fig. la shows the amplitude for two-boson scattering which, in the zero-range model, can be solved analytically. The solution in the center-of-mass system is easily obtained as
(a)
(b)
Fig. 1. (a) The diagrammatic representation of the integral equation for the two-boson scattering amplitude. (b) The kernel of the integral equation is given by the bubble diagram ofeq. (2). 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
409
Volume282, number 3,4
PHYSICS LETTERSB
28 May 1992
l
r(M2) -- i 2 - ' - B ( M 2 ) '
( 1)
where 2> 0 is the coupling constant and M2 is the mass of the two-boson system. B(M2) is the kernel of the integral equation for r(M2) and is schematically represented by the bubble diagram of fig. lb, B(M2) = f d4k i i 3 2(n) 4 k2-M2+ie (P-k)Q-M2+ie '
(2)
where Mis the boson mass and p u = (M2, 0, 0, 0). Notations and conventions follow Bjorken and Drell [ 1 1 ]. The momentum loop integral in eq. (2) is integrated in the null-plane energy, k - ( = k ° This procedure gives the solution to the Weinberg equation in this simplified model. In the null-plane variables k - , k + ( = k°+ andk±, eq. (2) is
k3).
1
B(M2)
f dk-dk+d2kx
2(2n) 4 J
k+(P + -k
k3)
1
+ ) [k--(k
2 +M2-ie)/k+]{P--k
- -
[ (P-k) 2
+M2-ie]/(P
+
-k+)}" (3)
The integration over k - in eq. (3) is convergent, so that we can use the Cauchy theorem to perform it. Instead of working with k +, we d e f n e the variable x= k +/p +. There are three regions of integration in x,
x
0 < x < 1 and
x > 1.
The two poles of the integrand in k - of eq. (3) are placed in the lower semi-plane of the k - complex-plane for x > 1, resulting in no contribution to the k - integration. For x < 0 the poles are located in the upper semi-plane, and again they do not contribute to eq. (3). The poles contribute to the integral only if 0
i f dx d2ka 1 (2~)3 x ( 1 - x ) M 2 - ( k 2 + M 2 ) / x ( 1 - x ) "
(4)
Eq. (4) has a logarithmic divergence, which can be absorbed in the redefinition of 2. The physical information introduced in the renormalization of the physical amplitude is the mass of the two bound bosons, M2a. We assume that the two boson amplitude, r(M2), has a pole for M2 =M2~ which implies i2 -1 =B(M2B) . The subtraction imposed by this condition in the denominator of eq. ( 1 ) makes it finite. The appropriate choice of 2 assures the existence of the two-boson bound state. Using eq. (4) and the fact that MzB < 2M, it is possible to show that - i B ( M 2 a ) is positive and thus that 2> 0, meaning the force is attractive. The integrals in eq. (4) can be performed analytically and for M2 < 2M the result is [M ~
1 arctan (2
MM / _~22B
1~-l ~ M 2
I
(2
M~
1) 1] --1
The limit of eq. (5) for M and M2 going to infinity such that (2M-M2 ) / M goes to zero, agrees with the nonrelativistic two-body scattering amplitude of Frederico et al. [ 7 ] where the attractive zero-range two-body potential is used. The non-relativistic model has one two-body bound state. This fact was used for the construction of the integral equation for the three-boson wavefunction [6 ]. We will return to the limit of Mgoing to infinity, when we discuss the non-relativistic limit of the null-plane three-boson integral equation. 410
Volume 282, number 3,4
PHYSICS LETTERS B
28 May 1992
The expression of z(M2) for M2> 2 M is obtained by analytically continuing the inverse tangent function in the physical sheet. For our purposes we do not need this, as will become clear in the following. The Faddeev-like equation for a component of the three-boson vertex is shown in fig. 2. The factor 2 comes from symmetrization of the total vertex. The vertex v, in the center of mass system depends only on the momentum of the spectator particle, q u. The three-boson vertex is the sum of the three components in which each boson is spectator once. The two-boson amplitude, eq. (5), enters in the kernel of the vertex integral equation. Then, fig. 2 gives the following: I v(q u) = 2z(M2)
d4k i (2~r) 4 k 2 - ~ / 2 - i a
i (P3 - q - k ) 2 - M 2 - i e
v(k~') '
(6)
where P~ = (M3, 0, 0, 0) is the three-boson f o u r - m o m e n t u m in the center of mass system and the mass of the two-boson subsystem is given by M22 = (P3 _ q ) 2 . The integration over k - is performed to obtain the integral equation in the null-plane variables. Due to the zero range o f the interaction, the third boson is on the mass shell while the other two are interacting. This is a physically motivated argument. The third boson is propagating freely for any finite distance from the interacting pair due to the zero range of the interaction. As the third boson is free, it is on the mass shell. The interaction of the third boson occurs when the distance between it and one boson of the pair is zero. This configuration belongs to the other Faddeev component of the three-boson system, where the third boson loses its character of spectator. From the discussion above, we conclude that v is a function of q +, q± and q - = (q2 + M 2 ) / q + .
This result should also be obtained if one assumes analytical behavior of the vertex in the k - complex plane. Then, the pole of the propagator of the spectator particle gives the on-mass shell condition after integration over k - . The loop integral in the null-plane variables is _½ f d k - d k + d 2 k ± v ( k - , k + , k ± ) J (270 4 k+(P3-q-k) + 1
× [ k - - (k~ + M a - i e ) / k + l { P y
-q- -k--
[(P3 - q - k ) ~
+ M 2 - i e l / ( P + - k + )}"
(7)
The integration in k - has contributions from O
The mass of the two-boson subsystem must be real, so that
Fig. 2. The diagrammatic representation of the integral equation for the Faddeev component of the vertex of the three-boson bound-state. 411
Volume 282, number 3,4
M~=(M3-q
+)
(
M3
PHYSICSLETTERSB qZiq++M2.~ ]-q~
28 May 1992
(8)
>O.
Solving the inequality for q~, we obtain q~ < ( 1 - y ) ( M ~ y - M 2 ) .
(9)
The limits for y are 1>y>M2/M~,
(10)
where the lower bound comes from q~ > O. The integral equation for v results after the integration of eq. ( 7 ) over k - . Introducing eq. (5) and the limits eqs. ( 9 ), (10) in eq. ( 6 ), together with the result of the k - integration, the equation for the Faddeev component of the vertex is
l
1
v(y, q± ) = 2--~x / M E / M 2 a - ~ arctan(2x/ME/M~a - ~ ) - 1 _ x / M E / M E _ ~ a r c t a n ( 2 x / M 2 / M 2 _ ~ )-1 1 --y
×
k T ax
M2/M] X( 1 - - y - - x )
d2k±
M~ - M2o3 '
( 11 )
where M2 is given by eq. (8),
k"2ax=x/ ( l - x ) ( M ~ x - M 2 ) , and the mass of the virtual three-boson state is M23 - k 2 + M 2 + q2 +M___.~2+ ( q + k ) 2 + M 2 x y 1--y--x
(12)
The dependence of v on q - is not specified because the spectator boson is on mass-shell, q + and q+ describe the spectator boson propagation. The limits on the x integration and the condition y > M 2 / M 2 impose a lower bound for the mass of the threeboson system: M3 > x//2M.
(13)
Let us now study the non-relativistic limit of eq. (11 ) (M--,oo). First, the two-boson amplitude, eq. (5), in this limit is (see Frederico et al. [7] )
16z~v/-M
lim z = - i x / ~ z a _ x/~2 ,
M~oo
(14)
where M2B = 2 M - E2B and M2 = 2 M - E2. E2B is the two-boson binding energy. The non-relativistic limit of the integrand ofeq. ( 11 ) involves the change in the variable x to kz: dx x lim M' M~oo dk~ -
( 15 )
and lim x = lim y = ] . Mooo
M~oo
The mass of the virtual three-boson system is substituted in eq. ( 11 ) by 412
(16)
PHYSICS LETTERS B
Volume 282, number 3,4
28 May 1992
3.0
2.5
Ma 2.0
1.5
1.0 ' ~ / / " 0.0 0.4
0.8
1.2
1.6
Me
Mo3
=x/q2+M 2 +
~
2.0
Fig. 3. The three-boson mass as a function of the two-boson mass in units of the mass of the single boson. Shown is the three-boson bound state mass (dashed line) and the threshold for the threeboson mass (solid line).
+x/(q+k)2+M 2 •
(17)
The result is UNR(q) ~ N / ~
~2
1
f d3k
VNR(k)
X/~2B--X/~2 j M 2 E 3 - q 2 / 2 M - k 2 / 2 M - ( q W k ) 2 / 2 M '
(18)
where E3 is the three-boson binding energy. Eq. (18) is the non-relativistic integral equation obtained in ref. [61. The non-relativistic limit of eq. (11 ) has the qualitative properties of the three-boson bound state when M2B ~ 2M. The Efimov effect [ 12 ] holds true for eq. ( 11 ). However, the short-range behavior is strongly modified and the Thomas collapse [ 13 ] of the system is avoided. The mass of the three-boson state has a lower bound of v/2M. The Faddeev component of the ground state vertex is rotationally symmetric in the x-y plane in eq. ( 11 ). The mass of the boson gives the scale of the system. Here, the solution is presented for M = 1. In fig. 3 the numerical results for the ground-state mass of the three-boson system are shown. The lower bound of Ma is approached by our calculation. The three-boson binding energy increases with the mass of the two-boson system, and saturates at about 0.9M for M2 ~ 1.9M. The binding energy increases with decreasing two-boson binding, because the phase space in the x variable in eq. ( 11 ) also increases. An exception to this behavior occurs near M2 = 2M. The threshold for the three-boson mass is also shown for comparison with the calculations. In summary, we give an example of how null-plane dynamics can be elaborated. We develop a zero-range model of the three-boson bound state in the null plane and solve numerically the dynamical equation for the ground state. The three-boson mass has a lower bound of v/2M and the non-relativistic limit of the equation is consistent with the zero-range theory [ 6 ]. Other situations of interest, such as scattering, will be considered in the future. I am grateful to R. Thaler for calling my attention to this problem, to A. Delfino for discussions, to B. Carlson for reading the manuscript and to H. Fearing for his hospitality at Triumf, University of British Columbia, where part of this work was done. This work was supported by the Conselho Nacional de Desenvolvimento 413
Volume 282, number 3,4
PHYSICS LETTERS B
28 May 1992
Cientifico e Tecnol6gico - CNPq and Fundag~o de A m p a r o / l Pesquisa do Estado de S~o Paulo - FAPESP, Brazil.
References [ 1 ] M.V. Terent'ev, Sov. J. Nucl. Phys. 24 (1976) 106; L.A. Kondratyuk and M.V. Terent'ev, Sov. J. Nucl. Phys. 31 (1980) 561. [ 2 ] B.L.G. Bakker, L.A. Kondratyuk and M.V. Terent'ev, Nucl. Phys. B 158 (1979) 497. [3] L.L. Frankfurt and M.I. Strikman, Nucl. Phys. B 148 (1979) 107; Phys. Rep. 76 ( 1981 ) 215. [4] Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 58 (1987) 2175; Phys. Rev. D 37 (1988) 778; P.L. Chung, F. Coester and W.N. Polizou, Phys. Lett. B 205 (1988) 545; C.R. Ji and S. Cotanch, Phys. Rev. D 21 (1990) 2319; I.G. Aznauryan and K.A. Oganesyan, Sov. J. Nucl. Phys. 47 ( 1988 ) 1097; Phys. Lett. B 249 (1990) 309. [5] I.L. Grach and L.A. Kondratyuk, Sov. J. Nucl. Phys. 39 (1984) 198; P.L. Chung, F. Coester, B.D. Keister and W.N. Polizou, Phys. Rev. 37 ( 1988 ) 2000; T. Frederico, G.A. Miller and E.M. Henley, Nucl. Phys. A 533 ( 1991 ) 617. [6] G.V. Skornyakov and K.A. Ter-Martirosyan, Sov. Phys. JETP 4 (1957) 648. [7] V. Efimov and E.G. Tkachenko, Phys. Lett. B 157 (1985) 108; V.F. Kharchencko, Sov. J. Nucl. Phys. 16 (1973) 173; T. Frederico, I.D. Goldman and A. Delfino, Phys. Rev. C 37 (1988) 497. [ 8 ] S. Weinberg, Phys. Rev. 150 ( 1966 ) 1313. [9] L.D. Faddeev, Sov. Phys. JETP 12 (1960) 1014. [ 10] R. Dashen and M. Gell-Mann, in: Proc. 3rd Coral Gables Conf. on Symmetry principles at high-energy (1966) (Freeman, San Francisco ); S. Fubini, G. Segr6 and D. Walecka, Ann. Phys. 39 (1966) 381; V. de Alfara, S. Fubini, G. Furlan and C. Rossetti, Currents in hadron physics (North-Holland, Amsterdam, 1973 ). [ 11 ] J.D. Bjorken and S.D. Drell, Relativistic quantum fields (McGraw-Hill, New York, 1965 ). [ 12] V. Efimov, Phys. Lett. B 33 (1970) 563; see also R.D. Amado and J.V. Noble, Phys. Rev. D 5 (1972) 1992. [ 13] L.H. Thomas, Phys. Rev. 47 (1935) 903.
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