IL NUOVO CIMENTO
VOL. 102 B, N. 1
Luglio 1988
On the Connection between the S-Matrix and the Third Virial Coefficient. S. SERVADIO Dipartimento di Fisica, Universitd di Pisa - Piazza Torricelli 2, 56100 Pisa, Italia I N F N - Sezione di Pisa, 56100 Pisa, Italia
(ricevuto il 24 Marzo 1988)
Summary. - - We prove that the third virial coefficient of the equation of state of a fluid is expressible in terms of on-shell scattering characteristics only. The proof is direct and does not require full specification of the final formula in terms of the two- and three-body S-matrices. The contributions from the regions of nonasymptotic motion are evaluated and found to reduce to first- and second-order derivatives of two-body S-matrices, which are present in Buslaev and Merkuriev's formula as against that by Dashen, Ma and Bernstein. The contributions from the regions of asymptotic motion, which are certainly expressible in terms of S-matrices, are shown to be evaluable through straightforward analysis. PACS 02.90 - Other topics in mathematical methods in physics. PACS 03.65.Nk- Scattering theory: nonrelativistic. PACS 03.80 - General theory of scattering.
1. -
Introduction.
T h e r e exists growing interest in the third virial coefficient b3 of the equation of state of a fluid, both from a theoretical and an experimental point of view (see the review (9 and (2)). Being interested in r a t h e r general properties, we resume the old question of the connection between b3 and the scattering properties of the constituent particles. A direct proof will be given that b3 is expressible in t e r m s of the three-body and two-body (on-shell) S-matrices. This problem was left open when Beth and (') W. G. GIBSON: in Few-Body Methods: Principles and Applications edited by T. K. LIM, C. G. BAO, D. P. Hou and S. HUBER (World Scientific, Singapore, 1986), p. 637. (2) D. BOLLIX:Higher Virial Coefficients in Two and Three Dimensions, Universiteit Leuven Preprint KUP-TF-87/13. 1 - II N u o v o C i m e n t o B.
l
2
s. S E R V A D I O
Unlenbeck's famous formula for b2 was first produced (~). It was recognized as a difficult question to determine if S-matrix information is actually sufficient ('.~) (according to the usual nomenclature, S-matrices, as against t-matrices, are always on-shell). Clearly, a correct understanding of the three-body S-matrix is needed beforehand, which is not trivial. The three-body scattering has many inherent and unavoidable complications giving rise to vicious traps and pitfalls. The problem was recognized (6) as being essentially that of mutual cancellation of divergent terms. Dashen, Ma and Bernstein (7,8) claimed to have proved both the cancellation and the validity of a formula for b3: namely the direct and most obvious generalization of the formula for b2. However, in a series of papers (9-11)Buslaev and Merkuriev showed, by using rigorous methods based on Faddeev's theory in momentum space, that i) the integral for b3 is finite, as it should; ii) Dashen, Ma and Bernstein's formula is incorrect and, taken literally, devoid of meaning; iii) many cancellations of off-shell t-matrices do obtain by repeated use of Hilbert's identity; iv) b3 can be expressed in terms of two- and three-body S-matrices; v) the final formula is very complicated due to the occurrence of counterterms after the cancellations. Unfortunately, other authors (12) have declaredly not been able to reproduce these results. One essential feature of Merkuriev and Buslaev's work is that the proof of iv) is obtained only by working out all the details leading to the final formula. In the present paper we prove property iv) directly, without having to spell all the details that go into b3. From a technical point of view we mimick the xspace calculation leading to the ,,truly three-body optical theorem,, (i~). This approach duplicates the derivation of b2 given in C4), but it is unavoidably much more difficult and complicated. (8) (') (5) (6) (7) (s) (9) (,0) (~9 (,2) ('9 (')
E. F. D. U. R. R. V. V. S. T. S. S.
BETH and G. E. UHLENBECK:Physica, 4, 915 (1937). T. SMITH: Phys. Rev., 131, 2803 (1963). BEDEAUX: Physica, 45, 469 (1970). BAR-GADDA:Physica, 62, 321 (1971). DASHEN, S.-K. MA and H. BERNSTEIN: Phy8. Rev., 187, 345 (1969). DASHEN and S.-K. MA: J. Math. Phys., 12, 689 (1971). S. BUSLAEVand S. P. MERKURIEV: Trudy Mat. Inst. Steklov, 110, 28 (1970). S. BUSLAEVand S. P. MERKURIEV: Teor. Mat. Fiz., 5, 1216 (1970). P. MERKURIEV:Zap. Nauchn. Semin. LOMI, 95 (1976). A. OSBORN and T. Y. TSANG:Ann. Phys., 101, 119 (1976). SERVADIO: Nuovo Cimento B, 100, 565, 587 (1987). SERVADIO: Phys. Rev. A, 4, 1256 (1971).
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
3
After this introduction, the development of the paper will be as follows. Section 2. Rewrite b~ as the trace of a surface integral in the relative x-space R6. Section 3. Separate out from the integration domain regions where only Smatrices are known to contribute to the result. Section 4. Prove that the rest of the integration domain contributes, after concellations, a finite result in terms of S-matrices. Section 5. Comment on the low-temperature behaviour of b3. There are three appendices to the paper. Appendix A sets out the full calculation by including the integrals singled out in sect. 3. We do not work out the whole lot of terms involved; we only show that it can be done in a pedestrian manner. The collection of the final formula for b3 in terms of scattering characteristics is left for the future. Appendix B supplies some details of the wave function which had not been worked out in the existing literature. Finally, appendix C discusses what happens when one expands into spherical waves a wavefunction of a different fall-off at infinity. This paper's logical order is different from Merkuriev and Buslaev's i) to v). Section 4 proves property iv) and it obtains contributions expressed in terms of first- and second-order derivatives of two-body S-matrices. Such terms are present in Buslaev and Merluriev's formula, as against that of Dashen, Ma and Bernstein. Note that this can be done without going into the full calculation set out in appendix A.
2. - b3 a s t h e t r a c e o f a s u r f a c e i n t e g r a l .
We assume the three particles i = l, 2, 3 to interact through pair forces of sufficiently short range and with no possibility of binding. With these hypotheses, there are only scattering states which will be labelled by the incoming momenta. We take the masses m~ = 1 and describe the system in the centre of mass. If ri and p~ are positions and momenta, relative coordinates and conjugate momenta can be chosen as X1 =
P1 =
•/•
rl,
Yi =
Pl ,
Q1 ---
1
1
(r2 - rs) ,
(P2 -- P 3 ) ,
or any cyclic permutation thereof. It is seen that Xi describes the distance of
4
S. SERVADIO
particle 1 from the (2, 3) c.m. point, whereas Y~ gives the relative position of the (2, 3) pair. We shall understand that the unlabelled coordinates are X = X1,
Y = Y1.
The configuration space will be the Re of such vectors with volume element denoted by d6X. The Euclidean distance p = ~v/X2+ yz is called the hyperradius. Pairs are often labelled by the third particle, e.g., pair (1) is made up of particles (2, 3). It is well known 05) that the third virial coefficient of a monocomponent fluid of such particles reduces to a three-body problem. It is proportional to (2.1)
b8 ~ f dE exp [ - fiE] Tre As,
where
The outer trace Tr~ is a sum over all the scattering states of energy E, i.e. Tr~ = f d6K ~(/(z _ E). A~ is an excess normalization integral over the whole relative coordinate space. In this section we prove that the volume integral A3 can be converted into a surface integral in the limit as the surface goes to infinity. This will reduce the calculation of bs to integrals in terms of the coordinate space asymptotic expansion of three-body wave functions. For any solution of Shroedinger's equation
( - V2+ V) ~= E~, in the presence of a real potential V one can verify that
I d~X]~]2=~dSa 9Re(-~E , ~ - ~V ~-~E), where V is the 6-gradient, V2 is Laplace' operator, 818E means differentiation with respect to the energy scale of all momenta, and the integration on the righthand side is over the boundary of the integration volume. It is important that the right-hand side does not depend explicitly on V. (Is) K. HUANG:Statistica Mechanics (John Whiley & Sons, New York, N.Y., 1967).
ON THE CONNECTION B E T W E E N THE S-MATRIX AND THE THIRD ETC.
5
Introduce a dimensionless scale parameter d/2 for all momenta and write
3E - E 3s ; then (2.2)
f
X l 12 = l
ds a.
,
where [39*.,
(2.3)
]
Primed momenta will be the incoming ones and we understand that the unlabelled momenta are p , = P~,
Q' = Q~.
The kinetic energy is E = p'2 + Q '2.
We denote by ~, T(~) and F the wave functions which evolve, respectively, freely, according to the interaction within pair i) only and according to the full interaction within pairs; they are understood to arise out of the same incident momenta P ' , Q'. Any such function can be taken as the ~ of eq. (2.2). Accordingly the volume integral A8 is reduced to (2.4) where it is understood that the limit as the surface goes to infinity be taken, to enclose the whole configuration space. By inserting the representation ~(i) = Z + gi(i), in terms of the scattered wave ~(~), and the multiple scattering series (2.5)
• = Z -~- Z ~i + Z ' ~ij jr uhigher, i
/j
6
S. SERVADIO
the unconnected combinations of (2.4) are subtracted out: (2.6)
A 3 = ~ - dSa 9
]
Z+~,~'~ '
i~ + ~ ' [ ~ , ~ ] + ij
§
ij
[
~'r
E
ij
~ § Z ~i § Z ' oij, Uhigher
ij
§
]
+
])
Uhigher, Uhigher
.
Here we have made recourse to the following symbol: (2.7)
[~, ~] = Re
V r - CV
+ Re [ - - ~ - v ~ - ~ v - ~ - j ,
if ~ r ~b; on the other hand, we keep (2.3) in the case ~ = ~b. This is a convenient bookkeeping to shorten the formulae; no confusion should arise from the fact that (2.7) does not go into (2.3) as ~o~b. Equation (2.6) has achieved the reduction of A3 to a surface integral over a surface that we must let grow to enclose the whole space. The subtraction has eliminated the unconnected combinations, but it has certainly not eliminated lower-order scattering terms. To evaluate the surface integrals making up A~ we shall need the coordinate space asymptotic behaviour of the waves. Before doing that it is appropriate to make contact with the analogous calculation of b2. In this case the configurational integral is
where T = Z + U consists of the plane wave Z and the scattered part U; [Z, U] and [U, U] are defined by formulae analogous to (2.7) and (2.3) (but with 3dimensional gradient operators). In(14) it was proved that, in the case in which
U= exp[iEl~](fo+~l/Z) is the appropriate asymptotics of the scattered part, Az reduces to the wellknown time-delay combination of Beth and Uhlenbeck after cancellations induced by the two-body optical theorem on f0. These cancellations do away with terms which would be 0(~). Similar cancellations must occur also in As, or possibly in taking the outer Whereas the trace operation in TrEA2 is quite trivial because of rotational invariance, the situation is different for ba since the pair potentials do not make up a spherically symmetric obstacle in R6. As happens, the
TrEA3.
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
7
cancellations to prove the main contents of this paper, embodied in sect. 4, occur in taking TrEA~ by virtue of the two-body unitarity, without having to write down the three-body unitarity. However, the full three-body unitarity theorem
~dSa 92 (T* V ~r
T V T * ) = 0,
i.e. zero flux across a closed hypersurface, is needed to prove the finiteness of the contributions dealt with in appendix A. In fact, it is convenient to take the three-body unitarity into account by adding to each [, ] of (2.4) the homonymous {, } multiplied by a relative factor pE v2. To give them explicitly, the {, } are defined by {~, ~} -- Re (i~* ~7~ + i~b*V ~), and {~, ~} = Re (i~* V ~). The algebraic steps from (2.4) to (2.6) can be performed on the {, }'s exactly as on the [, ]'s, to the result that
([ + E [ e , ~] ~
[
"
o
6
]
{~, ~} +
+ ~ - ~ / ~Ui ~h i' -gZhtO e~rY ""
l
"
O
1+
e~, E ' ~'~ + ~,E~ ' e~, E ' ~o + o j l,J
,J
'~-~E1/2
[
~ ~ - ~ 'i~k/-j~' U h i g h e r "
]
"+-
+ [U high"r,Uhigh.~]+ pE v~{U high~ Uhighor}). r
By confronting this with (2.9)
A2 = ~
d ~ 9([z, ~ + ~E''2(z, U} + [U, ~q + ~EI~(U, U}),
one realizes the much longer list of terms that make up A3, due to the much more complex structure of the three-body wave function. If it were true that, in R6, ,~T,, = z + U,
8
S. SERVADIO
with a spherical (2.10)
U = exp
[iE~P(pE~) ~+i~/4] .(~o(~) + ~11(ie)]'
then (2.7) would consist only of (2.11)
,,A3)) = l ~ ds a" ([Z, U] + pEY2(Z, U} + [U, U] + pE~ { U, U}),
and we would have the flux conservation theorem in the form (2.12)
~dSa-({~, U} + {U, U}) = 0.
Identity (2.12) would be
where ,~0(0) is the (,forward~, amplitude, and the third configurational integral would be (2.13)
,,A3>,=
1 ~ . R e ,~0(0) + ImfdSD~ ~-*(fi) ~ 2~
t~o(fi)
This should be considered as the most obvious generalization of Beth and Uhlenbeck's formula. The point is, however, that expansion (2.10) is inadequate and, if one really insisted on it, as if assuming a uniformly valid hyperspherical expansion, the amplitude ,~0(fi) would have divergences that would make TrE ,,As~ infinite. To justify this statement consider a wave of fall-off different from U as .o--) ~, e.g. some ~b of asymptotic behaviour exp [iE 1~~],
with a r 5/2.
Its expansion in terms of spherical waves would yield an outgoing spherical wave
,~o (~)
exp [iE ~2p] p~z
with the amplitude ,~0(fi) bound to be infinite in the directions Vz
=lv l" We are unable to provide a general proof of this phenomenon, which we believe to occur quite universally with respect to dimensionality and to the index
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
9
a. In support of our statement, in appendix C we give two examples in which the alledged divergence can actually be observed. By revealing the geometric character of the phenomenon, these examples point to the correctness of the general statement. Note that wave terms with a = 1 and a = 2 do indeed occur in the three-body wave function; they arise from single- and double-scattering terms and for them one direction
is the forward direction (called/t = 0) with respect to the incoming plane wave. The divergence in this direction occurs at all energies, hence one must expect
-
-
DE
Re ~o(0) = ~ .
The foregoing digression prompts us to extract from ~ its overall spherical part, which we call U, and to rewrite (2.5) as (2.14)
u. i
i]
(Having singled out the lower-order scattering terms of different asymptotic behaviour, the U is a bona-fide spherical wave and it is not affected by the divergence of the preceeding section.) In (2.14) we have used the representation(1~,~7) =
+
+
in terms of the ,,geometrical optics,, part ~g, the ,,diffraction,, ~ and the spherical UiJ; these U ~j have been added to U higher (triple, quadruple, etc.) to make up the overall spherical wave U = E ' U i~+ U higher, ij
which itself can be expressed as (2.10). For a thorough discussion of the asymptotic behaviour of ~ij and U higher w e refer to(1~.17). Strictly speaking this representation of the double-scattering terms is useful only in those angular regions where the motion is fully asymptotic (see appendix A). On the other hand, in sect. 4 we shall make use of representation (4.4) which will be explained in appendix B.
(1~) S. P. MERKURIEV: Teor. Mat. Fiz., 8, 235 (1971). (17) S. SERVADIO:Nuovo Cimento B, 69, 1 (1981).
l0
S. SERVADIO
We reorganize (2.8) into (2.15)
As = ~
d 5a"
(E
Z+
Z9 +2 ij
ij
i
+ eg), g
+ pE {, ) +
ij
+[U, U] + pEl~{, } ) (where each {, } stands for the flux version of the preceeding [, ]) which, we recall, constitutes the integrand of B2 b8 - J dE exp [ - fiE] TrEA~. A few comments are in order. The terms of leading order as ~-~ ~ from the [, ]'s involving the spherical U wave linearly or bilinearly get subtracted out by the [, ] + ?E 1/2( , ) ,
combinations. These cancellations occur before ~dSa is taken; thus we feel that an adequate use has been made of the three-body unitarity by the bracket arrangement of (2.15). Integrals involving only the plane and the spherical wave are evaluated analytically as (2.11) was, leading to the right-hand side of (2.13). The other integrals must be regarded as very peculiar to the problem of three particles interacting in pairs. Of these, those integrals involving the spherical and some other wave admit a rather straightforward analysis. More involved are those depending only on lower-order scattering terms. Before any evaluation can be done we must analyse the asymptotic fall-off of the waves over the integration domain (the sphere in Re). Then we shall find it convenient to slightly modify the domain so that, though with no change in the net result as p--~ ~, an easier treatment can be given of angular openings where the motion of the particles is not asymptotically free.
3. - C h o i c e o f i n t e g r a t i o n d o m a i n s .
As Merkuriev first showed (16), if the hyperradius goes to infinity in a way that all pairs get infinitely separated, the asymptotic behaviour of the terms making up decomposition (2.14) is described entirely in terms of S-matrices.
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
11
In Merkuriev's words, U is the ~smooth, part of the three-particle scattering; its amplitude is determined by the matrix elements of the three- and two-body Smatrices. The other terms ~ + ~ are known to carry only two-body S-matrices. The S-matrices determine not only the leading terms but the whole asymptotic series of (2.14). This can be viewed as a consequence of the free propagation Helmholtz equation which holds in the angular openings of asymptotic free motion. Of course, the S-matrix dependence of the wave function does not obtain over the angular domains where the relative motion of the pairs is not asymptotic. There are three such nonintersecting domains, the precise definition of which will be given later, called Ci (i = 1, 2, 3) depending on which pair of particles is still close. The rest of the domain of integration, where the motion is asymptotic, will be called S. By splitting
dsa= f dsa+ Z S
~ Ci
we analyse (2.15) as A~ = A~(S) + ~ A~(Ci). i
Let us go back to the problem of showing that only scattering information goes into (3.1) Since A~(S) depends only on asymptotic free motion, TrEA3(S) can certainly be expressed in terms of S-matrices (we shall briefly discuss it in appendix A; one has only to prove that statement for each TrEA~(Ci)). We shall fully calculate (3.2)
TrE A3(C1) ;
the results will show that it is finite and that it does reduce to S-matrix information only. To treat C2 and C~ one would only have to exchange labels. By order of magnitude estimates, no contribution to A3(C1) can come from the spherical wave U, nor from any of Fresnel's terms ~ , which are spherical over C1. Only lower-order graphs of the multiple scattering series contribute to A3(CO and only so if at least in one scattering term pair (1) has scattered last. Let us make the working assumption that scattering between pair (2) is much weaker, which can always be done in a model calculation. This assumption does
12
s. SERVADIO
not at all trivialize the problem; it only saves some bookkeeping. According to the foregoing analysis we need consider TrEA3(C1)--TrE 1 [ dh(r.([Z+O1+~rr~T~13,q~13]+pElP2{,}+[q~l,03]+pEl~{
~,~
}).
We now identify the domain C~. First, we want to isolate an appropriate portion of the hyperspherical surface An + y2 = p2 of global measure 0(~5), where pair (1) is still close. After choosing some R = ,=- c ] with positive C and ~ < 1, we remove the spherical cap for which Y < y~
= ~/p~
-
R 2 "
The untouched spherical surface, call it S, still has measure Y.(S)= 0(~5). Restriction to ~ > - 1 guarantees that
y ~ = 0(~('+~)~)-~ ~ , as p--. ~. The removed spherical cap has measure f d6X r
- V~
+ Y~)o(Y '~~ - Y ) ,
which is 0(p5-(~)(1-~)); this cap is replaced by a flat cylindricaUike part of constant IXI = R and any Y with IYI < Y ' ~ : we call it C,. The original sphere has become flattened and its flat portion has measure
2(CI) = ~dsX ~IX] -
R) O(Y~ " - Y) = 0(p5-(~)(I-~)),
i.e. it is of the same order as the removed cap. As p-~ o: the flattened sphere grows to encompass the whole configuration space; hence the surface integral (2.15) can be taken over the flattened spherical surface and decomposed as the kernel of (3.1). Also, if fl < 1/3, )2(S) - = 0 ( ~ (~2)('-~)) > 0 ( p ) ,
z(Cl)
and the integrals over the main spherical surface can be extended to the whole original spherical surface with relative error less than 0(~). Integrals over the main spherical surface will be shown (see sect. 5) to be not greater than 0(~); their extension to the whole angular domain will be tacitly understood. To s u m m a r i z e : by choosing - 1
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
13
space of the pair. This will yield (3.3)
TrE ~ A3(Ci), i
to be summed to (3.4)
TrE A3(S) ,
from the whole sphere to enter the overall b3. In the next section we deal with (3.3); the separated out contribution from sphere (3.4) will be discussed in appendix A. Some comments are in order to justify the definition of C1. Though we consider short-range interactions we have done in such a way as to make yma~__++; the C1 cylinder is larger than strictly needed to isolate portions where pair (1) is interacting. However, this is a useful choice: having so done, we shah have (see eq. (4.5)) the excess normalization integral
fd Y(kb(1)lz - 1), over the whole R 3 space, i.e. Wigner's time delay combinations (see eq. (4.9)).
4. - Contributions from asymptotically interacting domains.
In this section we deal with f dE exp [ - fiE] Tr~A~(Ci) and we express them in terms of S-matrices. To save on notations we only treat one such integral, the one for i = 1. The results will be given by (4.11) and (4.12). Consider the flat surface C1 and the integrand (4.1)
A8(Cl) = 1
J da. ([z + +1 _~_•3 .4_ +13, +13] ..~ pE,2 {, } + [+1, +3] .+_ ~E1/2 { , } ) . C1
The surface element d a of CI is directed along the 6-vector (X, 0) and it is of measure Idal = d3X d3Y ~(X - R ) 0(~/~ 2 - R 2 - Y) .
Since the 6-gradient along this direction carries no differentiation with respect to Y , A3(C1) is reduced to density integrals over the coordinates Y of the pair (1). This is what ultimately justifies the recourse to tbe flattening of the sphere.
14
s. SERVADIO The incoming plane wave is
z=(2=)-9/2exp[i{(-1P'+#Q').X+(-#P'-IQ').Y)]. The scattered part if only pair (1) interacts is exactly (4.2)
~1 = 2-9~r-3 Xq~(11) (1) exp [//)1 " '.X],
in terms of the pair scattered wave Z(~) QI" The scattered part if only pair (3) has interacted is asymptotically
"
< Q' 21 t3(Q '2) IQ') -
X
1
§ Q'~ "Y] (Q'XI t3(Q'2)IQ'>)) .
/,~"(exp [ - i l ( v ~ P '
iVQ'X The doubly-scattered ~13 is asymptotically (4.4)
~)13~, 2-4 =-, 3-1~2exp [ _ i [ 1 p ,
[z(,'(Y)
9/
T
,
( q XI t3(Q '2) IQ')
1
L} (z(ql)(Y)
iV~Q,X 2
/
where
q = - l (~/-3p' +Q' f0 . L~ stands for Laplace-Beltrami's angular-momentum operator L~=
1 ~0(0~-~) sin 0 sin
1 a2 sin 20 ~2 .
Equation (4.2) is exact, while eqs. (4.3) and (4.4) give only the first two
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
l~
asymptotic terms of the expansions. The derivation of the last two equations will be sketched in appendix B; it would involve asymptotic evaluation of multidimensional integrals. It must be recognized that the next-to-leading terms given in (4.3) and (4.4) are actually needed to compute A3(C~),and that they are sufficient; that is, no higher-order terms are necessary. This is because A3(C,) turns out to be O(R); to have it to O(1) as ~---~ ~ (and R ~ ~ as well) all estimates must be improved one stop further. By using these formulae A~(C~)can be evaluated in a straightforward manner. It is convenient to split the sum (4.1) into
A3(Ct) = A3[C1]+ A8{C1}, where Aa[C1] includes all the [, ]'s and Aa{, } includes all the (, }'s. The latter is a sum of flux integrals considered in the ,,three-body optical theorem, (la). Many details of that calculation must here be used; not only, the present calculation being pushed to one order higher, those flux estimates must be accordingly refined. To show the many cancellations that occur, we shall now give A~[C1],but not Aa{C~}. Later we shall write the net final result of both. We have found
Aa[C1]~-2-n/2=-63-1(Q')-'E-l f d2D2 9
(4.5)
X) Im (ta) + 2-5/2 =Q '
-
-
{ P--~' . 2 - - ~ Q '
+--
I a~, a
1
V q' 9[82((~,, )~) Im (t3) + 2-5/2 r~Q '
A----~/ 2
"
V Q'
'j
~, a0
"2-
It~l2] f daY(l~l)t 2 -
Q'
20
It3t2] Im f d~Y }F(1)* ~e ~(1) __
, ~Q / ~ Im(t* a0ta) f daY (]~'(1)]2 1) +
1) -
16
s. SERVADIO
1
+ ___=__-
VQ'
3(-~"""~- 2~r Q')2! 3r ~2(1~,,.,,~.)Re(t3)a
$1 30
H e r e ~rf(1)is the full interacting wave function of pair (1) normalized according
to (4.6)
~F(1)(Y)= exp [iq. Y] + =3~z(ql)(Y),
in terms of the incident plane wave of momentum q and its scattered part; ~1 q2 is the energy of this collision. ta is the matrix element {Q'XIts(Q'2)IQ'); by (t3) we denote its forward value, at X = (~'. The unit vector X, to be integrated over, is referred to P ' as its polar axis with polar coordinate 0. The sandwiched differential operators a act, as usual, according to =
A "~B = A ( 3 B ) - (3A) B .
The integration fd3Y over the relative coordinate Y of pair (1) is extended to the domain
IYI <
ymax.
Result (4.5) holds asymptotically as p-* ~, which also implies ymax__>oo. We are then led to consider the volume integrals (4.7)
f d3 Y (IT(')I 2 - 1),
and (4.8)
Im f dSY ~rr ~ , T/i),
over a sphere IYI < ym~ as ym~x__)~. The former is the well-known <,excess normalization integral>>; the latter does not appear to have been considered in the literature. It can be proved that the asymptotic series for both (4.7) and (4.8) are expressible in terms of the pair Smatrix.
17
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
The starting series are (4.9)
f d3 Y(IT(~)I2 - 1) -~ - 23/2=3 . 9Ia~Re (tl) - - ~r~e~/2r - J d2~ Ira(t* '~1 tl) t
(4.10)
I m f d ~ Y ~F(1)*~
+ O((ymax)-l),
T (1) -~ - 2 "2 r? ~i-1/2.
" [ ~ R e ( t l } - - ~7~e]/2 j ( d2t2 Im (t* ~1 t~)] ym~ + 0 (1). Since the t3 matrix elements in (4.5) are all on shell, these facts altogether imply that A3[C1] contains only S-matrix information. The same property can be proved of A3{C1}. Hence the whole A3(CI) contains only S-matrices. This is what we set out to show; it entails that the third virial coefficient b3 can be expressed entirely in terms of on-shell scattering. To make definite sense of this calculation we now compute out the contribution of A3(C~) to b3, that is f dE exp [ - fiE] Tr~A3(C1), where E = p,2 + Q,2 is the energy of the incoming state labelled by the momenta P , Q . Since P
?
TrE =
f
' dSQ ' ~(E -
p,2 _ Q,2),
one must eventually take the angular trace
fd2DQ,A~(C1) = f d2t2q, (A~[C,] + A3{C1}). Consider the trace f d2t2r A3[C1], where As[C~] is given by (4.5). The first three terms of (4.5) depend on (~' only through the bracket [32((~,, )~) Im (ts> + 2-~/2zrQ' ltsl2], which has vanishing angular trace. In fact, the two-body optical theorem, with 2 - Il Nuovo Cimento B.
18
s. SERVADIO
our normalization, is Im (t3} + 2 -5~ r~Q' f d2t)q, I ( Q' 21 ts(Q '2) IQ'>I = o. Then, the first two terms do not contribute at all. The third term also vanishes upon the angular summation over y d2DQ,. The surviving terms can be conveniently rewritten by use of (4.9) and (4.10), and by relabelling the integration variables. A similar analysis can be given of TrEA3 {C1}. Skipping the intermediate details, the result can be cast into the following form: (4.11)
TrEAs(C1) = -- 2-4 r~-~3-1E -1" 9TrE - 2 ( P { - Q ' )
0~3Re (t~} - - - ~ - ~ d2t)(t8 a~3t3) .
. [a~,Re - -=el/2 - ~ f d2t9 (t* '~, tl)]
9Re(t3} O~,Re(tl}---~-jd21?(t*O~ltl)
+ --~(p1.Q')+4q,]lp'x
r~sl/2 r 9Re{ts}0~, O ~ l R e ( t l } - ~ J d 2 t ? ( t
*
9 7) 0~,tl)Jl,
,-,
where 0~ means differentiation with respect to the pair energy ~. This shows that the contribution to b3 from C1 is nonvanishing, finite and it admits the following representation: (4.12)
dE exp [-/~E] TrE
~
i
S~,s, + h Re (S~ - I) 0~1~}sl S~ ~1
+ g Re ($3 - I) ~ where fi g, h are kinematical weights.
i
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
19
More terms of the same order and structure come from f dE exp [ - fiE] TrE A3(S), to be investigated in appendix A. Thus the kinematical weights of (4.12) will have to be summed up with those to have the full expression for bs. We should like to point out that terms carrying double derivatives with respect to the energy of pair S-matrices are actually included in Buslaev and Merkuriev's formula for b3. Thus, our calculation is on line with theirs, confirming that such combinations arise as counterterms after regularization of ortherwise divergent integrals.
5. - L o w - T b e h a v i o u r o f b3 and o f B u s l a e v and M e r k u r i e v ' s c o u n t e r t e r m s .
As Gibson (18)has uncontroversially shown, the low-temperature expansion of the third cluster coefficient (for Boltzmann statistics) is (5.1)
b3 = C 1/~2 ..~ C2/)3 + C3 (log ~)/~4 + 0(~-4),
where = (27~h2fl/m) '/2 is the usual thermal length. The leading behaviour 1/~2 had already been found by Larsen and Mascheroni(TM) by expanding the scattering wave function in hyperspherical waves. This is a dubious procedure since it effectively represents the scattered wave as if it fell off uniformly as p-5/2, which is clearly not so. Also, the power structure (5.1) had been indicated by Adhikari and Amado (20) starting from Dashen, Ma and Bernstein's alledgedly incorrect formula for b3. The question then arises why that incorrect formula is adequate to extract the correct low-T behaviour. To phrase it differently: what is the low-T behaviour of Buslaev and Merkuriev's counterterms of the type of (4.12)? One must expect them to be at most 0(~-4). Take the particles to be hard spheres and consider, following Gibson, the off(is) W. G. GIBSON: Phys. Rev. A, 6, 2469 (1972). (19) S. Y. LARSEN and P. L. MASCHERONI:Phys. Rev. A, 2, 1018 (1970). (~) S. K. ADHIKARIand R. D. AMADO:Phys. Rev. Letters, 27, 485 (1971).
20
S. S E R V A D I O
shell t-matrix expansion in powers of the sphere diameter
( k'l t(z) [k } = (2~)-l la + a2(- z)l/2-1a3(2z + k'2 + k2) + a3 k t
.k +
O(a4)].
It is simple to estimate the order of the counterterms (4.12). The singlederivative counterterms like, e.g.,
f dE exp [- flE] TrE g Re [ (S3 - I) S~~ielS1] are O((a/2) 4) ;
the double-derivative counterterms like,
e.g.,
f dE exp [- #El TrEf S3t'~e3$3 S1 ~e181 are
O((a/2)~). We conclude that both types are too small to be revealed in a low-T expansion. However, this does not dispense us from investigating the full correct expression which does include the counterterms.
The author is pleased to thank Prof. C. N. Yang for first acquainting him with the problem of the virial series, Prof. F. Calogero for a useful discussion and Prof. S. P. Merkuriev for discussions and encouragement.
APPENDIX A Contributions from asymptotically free domains. We investigate f dE exp [ - fiE] TrE A3(S). We would like to prove that it is finite and we must express it in terms of Smatrices, which is certainly possible, as established in sect. 3.
ON THE CONNECTION B E T W E E N THE S-MATRIX AND THE THIRD ETC.
21
The inner kernel is (A.1)
A3(S)=-~
dSa 9
Z+
, ~J
+ ~+' [~i, q~] + pE1/2 ~ {, } + I ~~j' j ( ~ + ~ ) , ~~j' ~ ] + ~E1/2{, } + 2'+~,~
~j
~0|+pE1/2{,}+
~
Z+~+~+~
(+~++~),U +pEl/2{,}+
j
+[U, U]+pE~{, }/" /
This is to be evaluated to O(1) as ~ ~ by substituting for the wave functions in the intergrands the free asymptotic expressions. The integration operator is
fd D fi, where fi is the unit radial vector; the gradients V in [, ] and {, } are effectively
The last two lines of (A1) involving the spherical U wave can be dealt with in a similar manner as (2.10). The [, ] integrals are 0(~) but the p E~/2{, } exactly cancel such ,,divergences,. The result is 0(1) as p o ~. It resembles (2.12), but 1) it includes additional terms arising from interferences of
Ei ~ + Eij ' ( ~ + ~), with U; 2) the integral bilinear in .~0 is to be regularized. This regularization is brought about by the nineth and tenth integrals of (A.1), in much the same manner as the the analogous regularization of flux integrals in deriving the unitarity relation C). The first eight integrals of (A.1) require longer calculations. The reason is that the lower-order scattering waves are ,,larger~ than U and they are such as to give an unfavourable asymptotic series for each [, ] of the type f da.[,]-~a2p2+ai~ + a0. s
Calculation of the coefficients is straightforward by use of standard asymptotic analysis (21,~), but the identification of higher ones is rather long. (21) V. P. MASLOV:Operational Methods (Mir Publ., Moscow, 1976), English translation. (~) YU. V. SIDOROV,M. V. FEDORYUKand M. I. SHABUNIN:Lectures on the Theory of Functions of a Complex Variable (Mir. Publ., Moscow, 1985), English translation.
22
s. SERVADIO
The series (A.1) is to be confronted with
~ da. {, } ~- b2~ + b, + bop -1, S
which can be gathered from the investigation of the unitarity relation. Thus
A3(S) ~- C2~2 + C, ~ + Co with some overall coefficients C2, C, and Co. It is mandatory, since b3 must be finite, to prove that (A.2)
TrE C., = 0,
(A.3)
TrE C1 = 0.
There is no difficulty in proving (A.2). Also, we have almost completed the proof of (A.3). In fact, we can show that all the terms in C1 carrying some differentiation of S-matrices cancel upon taking TrE. To complete the proof of (A.3) there remains to show that also the undifferentiated terms do cancel: this is under investigation. Then, one should proceed to evaluate TrsCo to have, by complementing it with the results of sect. 4, the final expression of b3 in terms of S-matrices. Some technical considerations are now appropriate. Most cancellations intervening in the proof of (A.2) and (A.3) can be observed ,,by inspection,, after application of the stationary phase method. The labor gets rapidly heavier as the evaluation is pushed to higher order. Very significant advantages are obtained by writing the stationary-phase formula for a multidimensional integral over a compact domain (as S dO~ is) in terms of Laplace-Beltrami's differential operators in a manner that resembles the recurrence relations (4.3) and (4.4) for the wave functions. It might be that a more general understanding in geometrical terms of the stationary-phase method would bring about decisive simplifications in the whole analysis.
APPENDIX B Derivation of some asymptotic formulae.
In this appendix we sketch the derivation of eqs. (4.3) and (4.4) giving to nextto-leading order the single- and the double-scattering wave functions ~3 and q~l~. The leading term of r is ~3 _ _ 2-5 ~-5~ exp [/P' .X~]
exp [iQ' Y3]
t3(Q'~) IQ'}
in which pair (3) of relative coordinate ]13 has interacted, whereas particle (3) of coordinate X3 has gone through in a plane wave. The next term is determined by
ON THE CONNECTION BETWEEN THE S-MATRIX AND THE THIRD ETC.
~-3
the pair (3) scattering, as if it were a two-body problem. Then, as well known, (B.1)
Y3 1 y L,23(It~l} ~z ) ~--2-5~-5/2exp[iP'.X3] exp[iQ'Y3](
where/~'~3 is Laplace-Beltrami's angular momentum operator with respect to the vector 173. It is of advantage to write (B.1) by means offlifferential operators with respect to X~, which we denote by the unlabelled X. This is a straightforward exercise by use of the relations
Y3:I(~x-Y), (the unlabelled Y stands for Y1). The result is exactly eq. (4.3). We now prove (4.4) whose leading term was known to be(~)
913~-2-47c-13-1/2expl-i{~.P'.X-~Q'Z}]~(q'x 't3(Q'2)IQ>' with
First, write 9X -
A~ +
...)
with Ao = - 2 .4 =-' 3 -'/2 Z~)(Y)
(V~c+V~-V~(Y)+E)exp
-i
P'.X-~Q'Y
+-~+... =0
and the interacting pair (1) function Z(q1) is solution of the two-body equation ( V ~ - V2~(Y) + r
(za)
S. SERVADIO:J.
Phys. A, 19, 725 (1986).
= 0.
24
s. SERVADIO
As usual in deriving Luneburg's type recurrence relations, we analyse (B.2) asymptotically as X--, co for fixed Y. To O(X -1) we have the eikonal equation -
V~
P'.X-
--+(V~-V~(Y)+E)~-=0
q'x
which is identically satisfied. To O(X -2) we obtain the transport equation
9~2~1..,
.x
.X_~23Q,X).V:r
.
+ - Vy: P ' . X -
Q'X
+(V~y-V23(Y)+E)
=0.
which must be solved for A1. By looking back at (4.3) one formulates the ansatz A1 =
-
-1
iV Q'
~2 LxAo,
which can be verified through a straightforward calculation. This is the content of eq. (4.4).
APPENDIX C
Expansions in spherical waves. We want to investigate what happens when one expands in spherical waves a wave whose as)-mptotic behaviour is not akin to a spherical wave. We assert that the amplitude of the outgoing spherical-wave component is infinite in the ,,forward,~ direction (for whose definition see below). We maintain that this is a general fact, though we do not have a general proof. We can however supply some examples in which the divergence can be observed and intuitively we feel that the general statement must likewise be correct. The examples are in R3, but they serve the purpose as well. Take the plane wave ~"= exp [ikz], of phase profile z = z and wave normal
N= W = ~ .
iwl
ON T H E C O N N E C T I O N B E T W E E N
T H E S - M A T R I X A N D T H E T H I R D ETC.
25
,,forward,,
This is its direction. Everyone is familiar with its expansion in terms of spherical waves. Asymptotically, as the radius r--) 2, it is
exp[ikz]~-l ~_oit(21+l)P~(cosO)sin(kr-l l=). The outgoing spherical part is proportional to 9~o(0) exp
[ikz] r
with ,~o(0) = ~ (2l + 1)Pt(cos0). I o
In the ,,forward, direction ,~o(0)= 2. As a second example consider the cylindrical wave ?" = J o ( k ~ ) ,
where p is the distance from the z-axis and Jo is the Bessel function
Its wave normals 5f are the unit vectors in the (x, y)-plane, they serve as the set of the ,,forwards, directions. By expanding in spherical waves we have found
Jo(kp) J~
~o~
(2l)! P2z(cos O)sin ~-~,,.-2
~ ' ( ~ .~)
(kr - l=).
In the ,,forward, directions (0 = r,/2, and any ~ in the (x, y)-plane), the spherical amplitude is proportional to (2l)! (-)~ z o 22l(/!)-----------~ P2t(0). Since (_)l p2t(O) -
2l/!! '
this is f (2/)! ~2 ~ ~22l(/!)2] " The series diverges. In fact, by use of Stirling's formula (1 + 1)/*1 l! -~ exp [ - (l + 1)] - V/+I
26
S. SERVAD~
t h e g e n e r a l t e r m i s found to b e h a v e like 4e 2 2/+1 and the reference series 1
~2/ + 1 diverges logarithmically.
9
RIASSUNT0
Si dimostra che il terzo coefficiente del viriale dell'equazione di stato di un fluido esprimibile in termini dei soli dati di scattering. La dimostrazione ~ diretta e non necessita della completa specifica della formula finale in termini delle matrici S dei due e dei tre corpi. I contributi dalle regioni di moto non asintotico sono calcolati e si esprimono per mezzo di derivate del primo e del secondo ordine delle matrici S dei due corpi. Simili termini sono presenti nella formula di Buslaev e Merkuriev, ma non in quella di Dashen, Ma e Bernstein. I contributi dalle regioni di moto asintotico, che sono certamente esprimibili in termini delle matrici S, sono calcolabili mediante calcolo analitico diretto.
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