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On a class of integrals appearing in the theory of statistical nuclear reactions
ANNALS
OF PHYSICS
176, 140-144 (1987)
On a Class of Integrals Appearing in the Theory of Statistical Nuclear Reactions H. L. HARNEY, E. G. LANZA,* AND P. PEREYRA? Po.s[fuch
Ma.~-Plan~k-ln.Ftitut 103980, 6900 Heidelberg,
,fir Kerphysik, Federal Republic
of Germany
Received November 4. I984
In a recent issue of Annuls qf Physics. Verbaarschot discussed the results of the random matrix theory of statistical nuclear reactions. When the Hauser-Feshbach limit of this theory is considered, a certain class of definite integrals appears. Verbaarschot has given their numerical values. We correct some of his results and describe an analytic and more general way to evaluate the integrals. ( 1987 Academx Press. Inc.
1. INTRODUCTION
In the last years, a unification has been achieved of the random matrix theory of nuclear spectra [l] with the theory of statistical nuclear reactions 121. Recently, Verbaarschot discussed [3] the formal results of the reaction theory, both analytically and numerically. An interesting limit of the reaction theory is the case, where the sum C T,. over the transmission coefficients T, in the open channels c is large compared to unity. It is often referred to as the Hauser-Feshbach or strong absorption limit. In this approximation, the theory of Verbaarschot et al. [Z] may be compared to older theories of compound nuclear reactions. Technically, this approximation consists in an asymptotic expansion of the result of [2] in powers of (x T,,)-‘. The coefficients of this expansion involve a class of definite integrals which are the subject of the present article. In Section 2, we give the asymptotic expansion up to and including the third order of (C T, )- ‘. Some of the interesting integrals are determined by the use of the unitary of the scattering matrix. The same route was taken in [3]. However, part of the results given there has to be corrected. In Section 3, we describe an analytic evaluation of the desired type of integrals and thus generalize the results of [3].
* Now at Istituto di Fisica dell’llniversitl, 95129, Italy. ‘Now at Universidad Autonoma Metropolitana, Azcapotzalco, Pablo 180, 02200 Mexico D.F., Mexico.
140 0003-4916/87
$7.50
CopyrIght ‘x) 1987 by Academic Press. Inc. All rights 01 reproductmn in any form reserved.
Dpto. Ciencias Basicas, Av. San
A CLASS
OF INTEGRALS
2. THE ASYMPTOTIC
141
EXPANSION
Let S,,(E) be an element of the unitary and symmetric nuclear scattering matrix S as defined in [2] and [3]. The argument E is the energy of the scattering system. We define the fluctuating part St,, of the S-matrix through
s$, = S,,h(E) - S<,,(E).
(2.1 I
Here and in the sequel, the overbar denotes the average taken over the ensembleof Gaussian random matrices that generate the bound states in the compound system as explained in [2]. Verbaarschot rt ul. [2] have obtained an expression for the “two-point-function” Si,,( E, ) SFCy( El) which is reproduced in Eqs. (2.6)-( 2.10) of [3]. We do not copy it here and immediately proceed to the expansion of the twopoint function. In order to write it down, some definitions are needed. The transmission coefficients are defined as usual, -7 T, = I - /S,,I-.
(2.3)
and we introduce the sums (2.3) The coefficients of the desired expansion are expressed by T, , S,, and integrals denoted by
Here. j‘ is a rational function of p, and pz. Using the procedure described in Section4 of [3], one finds the expansion of (2.6) of [3]
-=$ ,,,,6,,,4T,,T, S,,,,S,*, S,’
(2.5)
142
HARNEY,
LANZA,
AND
PEREYRA
Here, the functions P, are defined by
PAP* 2Pz)= 1+tc- 1Y+‘(Pf+P$,
(2.6)
and A is
A(P,,P*)=P,-~P*+~PPP*-b.
(2.7)
The expressions (2.5) and (2.6) differ from the corresponding Eqs. (4.8) and (4.5) of [S]. The remaining part of the present article is devoted to the evaluation of the integrals (2.4). The unitarity of the S-matrix requires
which must be satisfied by expression (2.5) up to and including order in S;‘. Whence one finds
the terms of second (2.9a)
Cl>=&
(2.9b)
(P,‘)=$+a(P;2)-~(p,p2PrZ), (P,P;Z)=
-4,
(2.9~)
(P,P;3)=
1,
(2.9d) (2.9e)
(PZP;“>=&
If all this is inserted into Eq. (2.5), one finds complete agreement with Eq. (7.6) of Ref. [4]. The results of Eqs. (2.9) can also be obtained in a more direct way which is described in the next section.
3.
INTEGRATIONS
In applications of the theory of Verbaarschot ef al. [2] one encounters more integrals of the type (2.4) than given in Eqs. (2.9). We therefore briefly indicate a way to obtain them, taking as an example the expression (p,p2P;‘)=;/;
dp,p2
(PI
‘p~~p2’ (1 +p,)-(1 +P2)* (1 +$I
which occurs in the treatment of time-reversal Transform the variables according to p, =p@,
P2)“2
symmetry breaking
pz=peCB
+Pz))”
(3.1)
[S]. (3.2)
A CLASS
143
OF INTEGRALS
with the Jacobian
G,, Pz) J(P,B,=2p
(3.3)
and obtain
p(e” -P“) (I’IP2p,2)=; (,,’CiP s,:4 (I +pe”+pY--“+p~)~
Y’ (I +(1/2)p(e”+r
‘I))?’
(3.4)
Substitute the variable p by q = 1 + pe” + pe ” + /‘?
(3.5)
and find (3.6) which after partial integration (/',/'?P,
Taking
of r/ ’ can be expressed as In
') =;+2
(I +o)? l+2p+Z
/=I‘
(3.7)
the second derivative with respect to Z and substituting x=(1
-p)(l
+/!I)
’
(3.X)
The integrand in this equation is continuous in the interval - 1 d .\: 6 1. All terms but one in Eq. (3.9) can be integrated by help of elementary functions. The remaining term is
see Section (4.231 ) of [6]. One arrives at (p,p2P,
5Y5 ,,h,-I,,
')=&&m352
(3.1 I )
144
HARNEY,
LANZA,
AND
The remaining integrals in Eq. (2.9b)-and the same way. We find
PEREYRA
not only these-can
be worked out in
(P-2)=~--rn2z0.06612 I 12 16
(3.12)
and (3.13) The results (3.11)-( 3.13) fulfill relation (2.9b).
4.
CONCLUSION
Integrals of the type of Eq. (2.4) appear, when the theory of statistical nuclear reactions [2] is considered in the Hauser-Feshbach limit. We have worked out the ones that appear in the (corrected) expansion of [3], and we have described an analytic way to obtain further ones that may be needed in future applications.
REFERENCES 1. T. A. BRODY.
J. FLORES,
J. B. FRENCH.
P. A.
MELLO.
A.
PANDEY,
ANV
Phys. 53 (1981). 385. 2. J. J. M. VERBAARSCHOT. H. A. WEIVENMULLER, AND M. R. ZIKNBAUEK. 3. J. J. M. VERBAARSCHOT, Ann. Phys. (N.Y.) 168 (1986), 368. 4. H. A. WEIDENM~LLER, Ann. Phys. (N. Y. j 158 ( 1984). 120. 5. E. G. LANZA, P. PEREYRA. AND H. L. HARNEY, to be published. 6. I. S. GRADSHTEYN New York, 1980.
AND
1. M.
RYZHIK,
“Table
of Integrals,
Series
and
S. S. M.
WONG.
Rev. Mod.
Phys. Rep. 129 (1985).
Products,”
Academic
367.
Press,
OF PHYSICS
176, 140-144 (1987)
On a Class of Integrals Appearing in the Theory of Statistical Nuclear Reactions H. L. HARNEY, E. G. LANZA,* AND P. PEREYRA? Po.s[fuch
Ma.~-Plan~k-ln.Ftitut 103980, 6900 Heidelberg,
,fir Kerphysik, Federal Republic
of Germany
Received November 4. I984
In a recent issue of Annuls qf Physics. Verbaarschot discussed the results of the random matrix theory of statistical nuclear reactions. When the Hauser-Feshbach limit of this theory is considered, a certain class of definite integrals appears. Verbaarschot has given their numerical values. We correct some of his results and describe an analytic and more general way to evaluate the integrals. ( 1987 Academx Press. Inc.
1. INTRODUCTION
In the last years, a unification has been achieved of the random matrix theory of nuclear spectra [l] with the theory of statistical nuclear reactions 121. Recently, Verbaarschot discussed [3] the formal results of the reaction theory, both analytically and numerically. An interesting limit of the reaction theory is the case, where the sum C T,. over the transmission coefficients T, in the open channels c is large compared to unity. It is often referred to as the Hauser-Feshbach or strong absorption limit. In this approximation, the theory of Verbaarschot et al. [Z] may be compared to older theories of compound nuclear reactions. Technically, this approximation consists in an asymptotic expansion of the result of [2] in powers of (x T,,)-‘. The coefficients of this expansion involve a class of definite integrals which are the subject of the present article. In Section 2, we give the asymptotic expansion up to and including the third order of (C T, )- ‘. Some of the interesting integrals are determined by the use of the unitary of the scattering matrix. The same route was taken in [3]. However, part of the results given there has to be corrected. In Section 3, we describe an analytic evaluation of the desired type of integrals and thus generalize the results of [3].
* Now at Istituto di Fisica dell’llniversitl, 95129, Italy. ‘Now at Universidad Autonoma Metropolitana, Azcapotzalco, Pablo 180, 02200 Mexico D.F., Mexico.
140 0003-4916/87
$7.50
CopyrIght ‘x) 1987 by Academic Press. Inc. All rights 01 reproductmn in any form reserved.
Dpto. Ciencias Basicas, Av. San
A CLASS
OF INTEGRALS
2. THE ASYMPTOTIC
141
EXPANSION
Let S,,(E) be an element of the unitary and symmetric nuclear scattering matrix S as defined in [2] and [3]. The argument E is the energy of the scattering system. We define the fluctuating part St,, of the S-matrix through
s$, = S,,h(E) - S<,,(E).
(2.1 I
Here and in the sequel, the overbar denotes the average taken over the ensembleof Gaussian random matrices that generate the bound states in the compound system as explained in [2]. Verbaarschot rt ul. [2] have obtained an expression for the “two-point-function” Si,,( E, ) SFCy( El) which is reproduced in Eqs. (2.6)-( 2.10) of [3]. We do not copy it here and immediately proceed to the expansion of the twopoint function. In order to write it down, some definitions are needed. The transmission coefficients are defined as usual, -7 T, = I - /S,,I-.
(2.3)
and we introduce the sums (2.3) The coefficients of the desired expansion are expressed by T, , S,, and integrals denoted by
Here. j‘ is a rational function of p, and pz. Using the procedure described in Section4 of [3], one finds the expansion of (2.6) of [3]
-=$ ,,,,6,,,4T,,T, S,,,,S,*, S,’
(2.5)
142
HARNEY,
LANZA,
AND
PEREYRA
Here, the functions P, are defined by
PAP* 2Pz)= 1+tc- 1Y+‘(Pf+P$,
(2.6)
and A is
A(P,,P*)=P,-~P*+~PPP*-b.
(2.7)
The expressions (2.5) and (2.6) differ from the corresponding Eqs. (4.8) and (4.5) of [S]. The remaining part of the present article is devoted to the evaluation of the integrals (2.4). The unitarity of the S-matrix requires
which must be satisfied by expression (2.5) up to and including order in S;‘. Whence one finds
the terms of second (2.9a)
Cl>=&
(2.9b)
(P,‘)=$+a(P;2)-~(p,p2PrZ), (P,P;Z)=
-4,
(2.9~)
(P,P;3)=
1,
(2.9d) (2.9e)
(PZP;“>=&
If all this is inserted into Eq. (2.5), one finds complete agreement with Eq. (7.6) of Ref. [4]. The results of Eqs. (2.9) can also be obtained in a more direct way which is described in the next section.
3.
INTEGRATIONS
In applications of the theory of Verbaarschot ef al. [2] one encounters more integrals of the type (2.4) than given in Eqs. (2.9). We therefore briefly indicate a way to obtain them, taking as an example the expression (p,p2P;‘)=;/;
dp,p2
(PI
‘p~~p2’ (1 +p,)-(1 +P2)* (1 +$I
which occurs in the treatment of time-reversal Transform the variables according to p, =p@,
P2)“2
symmetry breaking
pz=peCB
+Pz))”
(3.1)
[S]. (3.2)
A CLASS
143
OF INTEGRALS
with the Jacobian
G,, Pz) J(P,B,=2p
(3.3)
and obtain
p(e” -P“) (I’IP2p,2)=; (,,’CiP s,:4 (I +pe”+pY--“+p~)~
Y’ (I +(1/2)p(e”+r
‘I))?’
(3.4)
Substitute the variable p by q = 1 + pe” + pe ” + /‘?
(3.5)
and find (3.6) which after partial integration (/',/'?P,
Taking
of r/ ’ can be expressed as In
') =;+2
(I +o)? l+2p+Z
/=I‘
(3.7)
the second derivative with respect to Z and substituting x=(1
-p)(l
+/!I)
’
(3.X)
The integrand in this equation is continuous in the interval - 1 d .\: 6 1. All terms but one in Eq. (3.9) can be integrated by help of elementary functions. The remaining term is
see Section (4.231 ) of [6]. One arrives at (p,p2P,
5Y5 ,,h,-I,,
')=&&m352
(3.1 I )
144
HARNEY,
LANZA,
AND
The remaining integrals in Eq. (2.9b)-and the same way. We find
PEREYRA
not only these-can
be worked out in
(P-2)=~--rn2z0.06612 I 12 16
(3.12)
and (3.13) The results (3.11)-( 3.13) fulfill relation (2.9b).
4.
CONCLUSION
Integrals of the type of Eq. (2.4) appear, when the theory of statistical nuclear reactions [2] is considered in the Hauser-Feshbach limit. We have worked out the ones that appear in the (corrected) expansion of [3], and we have described an analytic way to obtain further ones that may be needed in future applications.
REFERENCES 1. T. A. BRODY.
J. FLORES,
J. B. FRENCH.
P. A.
MELLO.
A.
PANDEY,
ANV
Phys. 53 (1981). 385. 2. J. J. M. VERBAARSCHOT. H. A. WEIVENMULLER, AND M. R. ZIKNBAUEK. 3. J. J. M. VERBAARSCHOT, Ann. Phys. (N.Y.) 168 (1986), 368. 4. H. A. WEIDENM~LLER, Ann. Phys. (N. Y. j 158 ( 1984). 120. 5. E. G. LANZA, P. PEREYRA. AND H. L. HARNEY, to be published. 6. I. S. GRADSHTEYN New York, 1980.
AND
1. M.
RYZHIK,
“Table
of Integrals,
Series
and
S. S. M.
WONG.
Rev. Mod.
Phys. Rep. 129 (1985).
Products,”
Academic
367.
Press,