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Coherence counterterms in elastic three-body scattering
Volume 112A, n u m b e r 3,4
PHYSICS LETTERS
21 October 1985
C O H E R E N C E C O U N T E R T E R M S IN ELASTIC THREE-BODY SCATTERING "" S. SERVADIO Dipartimento di Fisica, Universitil degli Studi di Pisa, Piazza Torricelli 2, Pisa 56100, Italy Received 3 July 1985; accepted for publication 27 July 1985
Coherent interference of the q u a n t u m non-relativistic elastic 3 - 3 scattering is shown to yield combinations in the form of two-body time delays. Such terms are relevant counterterms to the unitarity constraint on the "truly three-body" scattering. Their structure is appealing since it announces the possibility of opening up of inelastic channels and thus casts light on the question of coherence of many-particle final states.
There are questions relating to the three-body problem that never seem to find satisfactory answers. Let me consider the competing reactions
a+b+c~
a+b+c (ab)+c
within Schr6dinger's theory. Although a correct framework is available within Faddeev's theory a lot of work is needed to calculate the scattered wavefunction. As a matter of fact "very little is understood about the structure of many-particle final states and about how two-body interaction information is distributed over them" [ 1]. Relevant questions are: (i) how is the rate for the bound pair formation related to the elastic amplitude by the overall unitarity constraint?; (ii) how does the so-called "truly three-body" amplitude enter unitarity?; (iii) what is the contribution from the lowest order multiple scattering elastic terms?. These questions may be compounded into a single one: do we know the coherence and the mutual interferences between the various terms of the wavefunction? " As embarrassing as it may be, this question has been left largely unanswered. Suppose the three particles are of the same species (a = b = e) and that a box full of them is given. The old problem [2] of the third virial coefficient (b3) later turned into a controversy [3-6]. The formula by Dashen, Ma and Bernstein appears to miss "counterterms" that carry one order of differentiation more (with respect to the energy) than the main contribution. This is so even in absence of bound states. Since inclusion of inelastic channels has to be smooth in the coupling strength [7,8] coherence effects in the elastic collision must be non-trivial and signal the possibility of binding. That was my own motivation for investigating the "truly three-body optical theorem" [9]. What I report here is the occurrence of counterterms in the most plausible combination of two.body time delays. To do so I shall limit myself to the lowest order scattering terms, which carry the worst singularities in momentum space. (Incidentally such terms are most important in the low-energy expansion of the S-matrix and in the zero-temperature expansion of b3). Denoting by X the incoming plane wave and labelling pairs by the third particle, I write the wavefunction in I:16
"~ Work supported in part by INFN Sezione di Pisa.
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
111
Volume 112A, number 3,4
PHYSICS LETTERS
21 October 1985
as
Kit = X + xI~l,3 + ....
where 41,3=,1
+,3
+,13
consists of unconnected graphs and a double-scattering graph. The term * 13 can be further analyzed [10,11] into ,13=,13
+,13+U13,
as a sum of respectively a discontinuous "geometrical optics" solution, a discontinuous Fresnel type wave and an everywhere present spherical wave U 13. Let {41, 3, xi,1,3} be the flux of representative points through a sphere of hyperradius P, to be studied as O ~ oo. Analyze it as {xI,1,3, qjI,3} = A
+B+C+D,
where A={X+,I
+,3 +,13,X+,1
C ={X + . 1 + . 3 , q b 1 3 } ,
+,3 +,13),
B=(qbl3,,13}+(,
D={X +*1 +,3+,13,
F13,dpF13}+ {,13, U13) +{U13, U13},
U13}.
Let me first recall the main results from refs. [12,13]. The leading O(01/2) from the matching over the ridge of , 1 3 cancels out B and C. The self-interference B of double-scattering waves reduces to an O(1) integral over intermediate states off the energy shell (this term is in a sense analogous to the ff d ~ l f l 2 of the two-body case). The O(1) contribution from C takes from the ridge and it consists of terms proportional to pair t-matrices on the energy shell. The D sum is easy to deal with since U 13 is a spherical wave (part of the whole truly three-body amplitude); these fluxes are 13(1) and are the counterpart of the Im f(0) term of the usual two-body unitarity. Due to internal cancellations the anomalously large O(0) leading term of A cancels out. I now turn to the new very interesting contributions from A. There are many such terms arising essentially in two ways. By using a better refined wavefunction than derived in ref. [14] and by pushing the stationary phase method to next to leading order in the previous O(0) calculations. In either way the result is expressed as a second order differential operator with respect to the "angles" in R6, to be integrated over last. Take, e.g., the doublescattering *013. The angles in I:16 determine both the deflections of
{x+*l,*13}~Im(t3(Q2))(d--d--1 _ 2-5/2rrd Im
f fda~,(l'dlq(d2)l -
o#
× f 0
112
Re(tl(d2 ))
dr [rational function] ,
3-1/2(2p'3 +p))*
7
)
21 October 1985
PHYSICS LETTERS
Volume 112A, number 3,4
where ff = P~ and E 1 = d 2 = Q~2 must be set after differentiations; and
ffda 2_5/27rQ,3[(iQ,3[t3(Q~)[e,3)2 ( ~ d +d - 2-5/2wrd Im f fdaf (Ydltl(d2)l-3-1/2(P'3 + P))* ~1
( . 3 .r) .w0 ~.13, ,w0 .,.13~ ~
X ? 0
Re(tl(d2))
(Ydltl(d2)l-
3-1/2(2P3 + P ) )
)
d/3 [rational function] ,
where now E 1 = d 2 depends on the direction i into which the scattering (It 3 [)has gone. Furthermore, the quadra. tures over f d/3 are the same. The structure of these terms is exactly the same (apart from the quadratures) as for fluxes out of the so-called "cylinder" C 1 where pair "1" could still be interacting [ 12]. Adding all such contributions together the following result is obtained: {X + qbl + qb3 + qbl3, @13)
f da i
2-43-1rr -3 f
X
(d ~
X K1
Re(tl(d2)) -
PI
o;(e-el )
2-5/2ndlm f
~11 (t'dltl(d2)l-
^' ) Im( t 3(03'2 )) + 2-5/27rQ'3 [(iQ;It3(Q'32)IQ;)[2] t82(i, 03
fdal~
3-1/2(2p3 +P))
)
3-1/2(2p; + P))*
"
It should be noted that the above result cannot be interpreted as an expectation value over the initial state of the operator A+ A+
^
^
S3S1E1(~/dE 1)S1S 3 , where Si is the S-matrix for pair "i" (acting on the three.particle states) and is some kinematical factor. Such an expectation value would be divergent due to unsaturated 6-functions occurring in the diagonal elements. I have thus proved that all counterterms carrying energy derivatives are akin and, quite appealingly, get very large if the pair potential is nearly binding. The occurrence of such counterterms in the "truly three-body optical theorem" was not known on any general ground. Their general structure is convincing and parallels that of the b 3 counterterms (carrying one more order of differentiation than the main contributions). An obvious next endeavour is to apply the present coordinate space techniques to the b 3 problem itself. Preliminary results confirm the occurrence of counterterms with double energy derivatives, exactly as maintained by Merkur'ev and Buslaev.
References [1] [2] [3] [4] [5 ] [6]
R.D. Amado, Phys. Rev. Cll (1975) 719. A. Pais and G.E. Uhtenbeck, Phys. Rev. 116 (1959) 250. R. Dashen, S.-k. Ma and H. Bernstein, Phys. Rev. 187 (1969) 345. V.S. Buslaevand S.P. Merkur'ev, Teor. Mat. Fiz. 5 (1970) 1216. T. Osborn and T.Y. Tsang, Ann. Phys. (NY) 101 (1976) 119. S.P. Merkur'ev, Regularizationformulas for the three-body S-matrix; Zap. Nauci. Semin. Leningr. Otd. Mat. in-ta AN USSR (1976) pp. 95-131 (in Russian).
113
Volume 112A, number 3,4 [7] [8] [9] [10] [11] [12] [13] [14]
114
PHYSICS LETTERS
H.M. Nussenzveig,Acta Phys. Austr. 38 (1973) 130. W. Hoogeveen and J.A. Tjon, Physica 108A (1981) 77. S. Servadio, Acta Phys. Austr. Suppl. 23 (1981) 689. J. NuttaU, J. Math. Phys. 12 (1971) 1896. S.P. MerkuFev, Teor. Mat. Fiz. 8 (1971) 235. S. Servadio, Nuovo Cimento B65 (1981) 57. S. Servadio, Nuovo Cimento B69 (1982) 1. S. Servadio, The geometry of double-scattering waves in 3-3 collisions, IFUP TH/32-84.
21 October 1985
PHYSICS LETTERS
21 October 1985
C O H E R E N C E C O U N T E R T E R M S IN ELASTIC THREE-BODY SCATTERING "" S. SERVADIO Dipartimento di Fisica, Universitil degli Studi di Pisa, Piazza Torricelli 2, Pisa 56100, Italy Received 3 July 1985; accepted for publication 27 July 1985
Coherent interference of the q u a n t u m non-relativistic elastic 3 - 3 scattering is shown to yield combinations in the form of two-body time delays. Such terms are relevant counterterms to the unitarity constraint on the "truly three-body" scattering. Their structure is appealing since it announces the possibility of opening up of inelastic channels and thus casts light on the question of coherence of many-particle final states.
There are questions relating to the three-body problem that never seem to find satisfactory answers. Let me consider the competing reactions
a+b+c~
a+b+c (ab)+c
within Schr6dinger's theory. Although a correct framework is available within Faddeev's theory a lot of work is needed to calculate the scattered wavefunction. As a matter of fact "very little is understood about the structure of many-particle final states and about how two-body interaction information is distributed over them" [ 1]. Relevant questions are: (i) how is the rate for the bound pair formation related to the elastic amplitude by the overall unitarity constraint?; (ii) how does the so-called "truly three-body" amplitude enter unitarity?; (iii) what is the contribution from the lowest order multiple scattering elastic terms?. These questions may be compounded into a single one: do we know the coherence and the mutual interferences between the various terms of the wavefunction? " As embarrassing as it may be, this question has been left largely unanswered. Suppose the three particles are of the same species (a = b = e) and that a box full of them is given. The old problem [2] of the third virial coefficient (b3) later turned into a controversy [3-6]. The formula by Dashen, Ma and Bernstein appears to miss "counterterms" that carry one order of differentiation more (with respect to the energy) than the main contribution. This is so even in absence of bound states. Since inclusion of inelastic channels has to be smooth in the coupling strength [7,8] coherence effects in the elastic collision must be non-trivial and signal the possibility of binding. That was my own motivation for investigating the "truly three-body optical theorem" [9]. What I report here is the occurrence of counterterms in the most plausible combination of two.body time delays. To do so I shall limit myself to the lowest order scattering terms, which carry the worst singularities in momentum space. (Incidentally such terms are most important in the low-energy expansion of the S-matrix and in the zero-temperature expansion of b3). Denoting by X the incoming plane wave and labelling pairs by the third particle, I write the wavefunction in I:16
"~ Work supported in part by INFN Sezione di Pisa.
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
111
Volume 112A, number 3,4
PHYSICS LETTERS
21 October 1985
as
Kit = X + xI~l,3 + ....
where 41,3=,1
+,3
+,13
consists of unconnected graphs and a double-scattering graph. The term * 13 can be further analyzed [10,11] into ,13=,13
+,13+U13,
as a sum of respectively a discontinuous "geometrical optics" solution, a discontinuous Fresnel type wave and an everywhere present spherical wave U 13. Let {41, 3, xi,1,3} be the flux of representative points through a sphere of hyperradius P, to be studied as O ~ oo. Analyze it as {xI,1,3, qjI,3} = A
+B+C+D,
where A={X+,I
+,3 +,13,X+,1
C ={X + . 1 + . 3 , q b 1 3 } ,
+,3 +,13),
B=(qbl3,,13}+(,
D={X +*1 +,3+,13,
F13,dpF13}+ {,13, U13) +{U13, U13},
U13}.
Let me first recall the main results from refs. [12,13]. The leading O(01/2) from the matching over the ridge of , 1 3 cancels out B and C. The self-interference B of double-scattering waves reduces to an O(1) integral over intermediate states off the energy shell (this term is in a sense analogous to the ff d ~ l f l 2 of the two-body case). The O(1) contribution from C takes from the ridge and it consists of terms proportional to pair t-matrices on the energy shell. The D sum is easy to deal with since U 13 is a spherical wave (part of the whole truly three-body amplitude); these fluxes are 13(1) and are the counterpart of the Im f(0) term of the usual two-body unitarity. Due to internal cancellations the anomalously large O(0) leading term of A cancels out. I now turn to the new very interesting contributions from A. There are many such terms arising essentially in two ways. By using a better refined wavefunction than derived in ref. [14] and by pushing the stationary phase method to next to leading order in the previous O(0) calculations. In either way the result is expressed as a second order differential operator with respect to the "angles" in R6, to be integrated over last. Take, e.g., the doublescattering *013. The angles in I:16 determine both the deflections of
{x+*l,*13}~Im(t3(Q2))(d--d--1 _ 2-5/2rrd Im
f fda~,(l'dlq(d2)l -
o#
× f 0
112
Re(tl(d2 ))
dr [rational function] ,
3-1/2(2p'3 +p))*
7
21 October 1985
PHYSICS LETTERS
Volume 112A, number 3,4
where ff = P~ and E 1 = d 2 = Q~2 must be set after differentiations; and
ffda 2_5/27rQ,3[(iQ,3[t3(Q~)[e,3)2 ( ~ d +d - 2-5/2wrd Im f fdaf (Ydltl(d2)l-3-1/2(P'3 + P))* ~1
( . 3 .r) .w0 ~.13, ,w0 .,.13~ ~
X ? 0
Re(tl(d2))
(Ydltl(d2)l-
3-1/2(2P3 + P ) )
)
d/3 [rational function] ,
where now E 1 = d 2 depends on the direction i into which the scattering (It 3 [)has gone. Furthermore, the quadra. tures over f d/3 are the same. The structure of these terms is exactly the same (apart from the quadratures) as for fluxes out of the so-called "cylinder" C 1 where pair "1" could still be interacting [ 12]. Adding all such contributions together the following result is obtained: {X + qbl + qb3 + qbl3, @13)
f da i
2-43-1rr -3 f
X
(d ~
X K1
Re(tl(d2)) -
PI
o;(e-el )
2-5/2ndlm f
~11 (t'dltl(d2)l-
^' ) Im( t 3(03'2 )) + 2-5/27rQ'3 [(iQ;It3(Q'32)IQ;)[2] t82(i, 03
fdal~
3-1/2(2p3 +P))
)
3-1/2(2p; + P))*
"
It should be noted that the above result cannot be interpreted as an expectation value over the initial state of the operator A+ A+
^
^
S3S1E1(~/dE 1)S1S 3 , where Si is the S-matrix for pair "i" (acting on the three.particle states) and is some kinematical factor. Such an expectation value would be divergent due to unsaturated 6-functions occurring in the diagonal elements. I have thus proved that all counterterms carrying energy derivatives are akin and, quite appealingly, get very large if the pair potential is nearly binding. The occurrence of such counterterms in the "truly three-body optical theorem" was not known on any general ground. Their general structure is convincing and parallels that of the b 3 counterterms (carrying one more order of differentiation than the main contributions). An obvious next endeavour is to apply the present coordinate space techniques to the b 3 problem itself. Preliminary results confirm the occurrence of counterterms with double energy derivatives, exactly as maintained by Merkur'ev and Buslaev.
References [1] [2] [3] [4] [5 ] [6]
R.D. Amado, Phys. Rev. Cll (1975) 719. A. Pais and G.E. Uhtenbeck, Phys. Rev. 116 (1959) 250. R. Dashen, S.-k. Ma and H. Bernstein, Phys. Rev. 187 (1969) 345. V.S. Buslaevand S.P. Merkur'ev, Teor. Mat. Fiz. 5 (1970) 1216. T. Osborn and T.Y. Tsang, Ann. Phys. (NY) 101 (1976) 119. S.P. Merkur'ev, Regularizationformulas for the three-body S-matrix; Zap. Nauci. Semin. Leningr. Otd. Mat. in-ta AN USSR (1976) pp. 95-131 (in Russian).
113
Volume 112A, number 3,4 [7] [8] [9] [10] [11] [12] [13] [14]
114
PHYSICS LETTERS
H.M. Nussenzveig,Acta Phys. Austr. 38 (1973) 130. W. Hoogeveen and J.A. Tjon, Physica 108A (1981) 77. S. Servadio, Acta Phys. Austr. Suppl. 23 (1981) 689. J. NuttaU, J. Math. Phys. 12 (1971) 1896. S.P. MerkuFev, Teor. Mat. Fiz. 8 (1971) 235. S. Servadio, Nuovo Cimento B65 (1981) 57. S. Servadio, Nuovo Cimento B69 (1982) 1. S. Servadio, The geometry of double-scattering waves in 3-3 collisions, IFUP TH/32-84.
21 October 1985