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Saddle-point approximation of compact integrals
NUCLEAR PHYSICS A Nuclear Physics A585 (1995) 554-564
ELSEVIER
Saddle-point approximation of compact integrals * R.F. Langbein, H. R e i n h a r d t Institut fiir Theoretische Physik, Universitlit Tiibingen, Auf der Morgenstelle 14, D-72076 Tiibingen, Germany Received 10 October 1994
Abstract
We investigate the validity of the saddle-point approximation of integrals over the compact manifold Sn, in which the confining 6-function in the measure has been replaced by its Fourier representation. The approximation is found to deliver good results, particularly when the exact roots of the defining polynomial are used.
1. Introduction
A fundamental technique in non-perturbative quantum field theory is the process of linearizing interaction terms in a given action in terms of an auxiliary field. For example, for some generic field or, the path integral
z= f~a exp[-½(Ka)2-ga4],
(1)
where K is an operator describing kinetic energy and mass and g is a coupling, may be rewritten as
Z=
exp(-½(K
) 2 _ 1 z_i/3a2).
(2)
This expression is now only quadratic in the original field a which may be thus integrated out, leaving an alternative form for (1) in terms of the auxiliary field/3. This technique has been applied in a range of theories, including Yang-Mills field theory [1], ~b4-theory [2], as well as interacting many-body systems [3], and it is the foundation of bosonization approaches [4]. * Supported by the Deutsche Forschungsgemeinschaft under contract DFG-Re 856/1-2. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(94)00533-8
R.F. Langbein, H. Remhardt/ Nuclear PhysicsA585 (1995) 554-564
555
However, if the measure for the field we want to linearize is non-trivial, i.e., non-fiat, as is the case for example in Yang-Mills theory (though of course it is assumed fiat in the perturbative treatment), we cannot apply this technique directly. An often used approach, if the measure contains a 8-function, is to replace the ~-function by its Fourier representation. The linearization technique can then be applied as before and our path integral at the end contains an integration over the Fourier modes of this ~-function. We might question this technique in light of the inevitable approximations that are needed to interpret this final path integral. An approximation scheme will only sum over some of the Fourier modes and we might wonder if this properly reflects the distribution character of the ~-function. We examine here a test case, integration of the exponential of a quadratic form over the n-dimensional sphere S,. This case has a number of features in common with the actual integrals required for example in lattice gauge theory [5]. It is, first of all, as for integrals over a gauge group, a compact integral, and the cases n = 2 and n = 4 are equivalent to integration over the groups U(1) and SU(2) respectively. We apply the chain of techniques mentioned above, i.e., replacing the ~-function in the measure by its Fourier representation, linearizing and then approximating, and see if the final expression makes sense. We do this in the following section for arbitrary n to see what we can deduce in general, whilst in Section 3 we compare the approximation with the exact result in the case n = 2. The final section gives some conclusions.
2. Integration over the n-sphere We examine the compact integral Cn(M )
=
f
...dx
n
~( xTx-
1) e x p ( - - x T M x ) ,
(3)
w h e r e x T = ( x 1. . . . , x , ) , M is an n × n symmetric m a t r i x and the 6 - f u n c t i o n on the
measure impies integration over the n-sphere C,(A) = fdXl...dx
. 6(xTx-
S n.
It is in fact sufficient to study
1) e x p ( - - x T A x ) ,
(4)
where A is the diagonal matrix of eigenvalues of M, i.e., A1 A = RTMR =
,
(5)
A,
where )q . . . . , An are the eigenvalues of M, and R is an orthogonal transformation with det(R) = 1. Eq. (4) is obtained from (3) by making the substitution x -~ Rx.
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R.F. Langbein, H. Reinhardt /Nuclear Physics A585 (1995) 554-564
The non-trivial measure of the integral (4) can be handled by replacing the a-function by its Fourier representation, giving C,(A) = ~
1
f d x x . . . d x n dta exp[--xT(A--ito)x--ito],
(6)
evaluating the gaussian integral over the xi, leaving T~n / 2 - 1
C,(A)
2
f d t o det-1/E(A - ito) e -i°'
,ITn / 2 - 1
#
e -ic°
- ~ / d t o z
(7)
HL-~(~..
J
(
(8)
ito ) 1/2
-
- ~r"/2---------~1jd~of exp - i w - -
2
'n
)
~ ln(A,-i~0) ,
2i= 1
(9)
and finally approximating this final expression (9) asymptotically by using the saddle-point method. Several questions now arise. The function in the exponent of (9), 1
~
b.(to) = -ito - ~ ,l•n ( a . . = l - - i t o ) ,
(10)
has in fact n saddle points. Which saddle points does our contour need to pass through? Does an asymptotic approximation place restrictions on the possible values of the ai? And is the approximation any use outside this asymptotic region? We start by considering the general case and assume that the eigenvalues a~ are all real and positive. The saddle points of b.(to) are found by solving b'.(to)= 0, which, after a little algebra, are found to be the roots of the polynomial
zn--(al--ln)z n-l+ [ a 2 - - 1 ( n - 1 ) a l ] z n-e+... +(--1)nan--lan_l=O, (11) where z = ito
(12)
and n
n
a2= ]~AiAj . . . . .
a 1 = ~ Ai, i=1
a. = H A i .
i~j
(13)
i= 1
Using Newton's rule, it can be checked that the real roots of (11) lie between -oo 1 and ~n. If we assume that Ai >> 1 and also I Ai - Aj I >> 1, we find we can approximate the roots of (11) by 1
1 ~i=--Ai'~-2
-
E 4(A/-Ay) j~i
+
1 O( (Ai _ Aj)2 ) .
(14)
R.F. Langbein, H. Reinhardt ~Nuclear Physics A585 (1995) 554-564
557
Fig. 1. Plot of the real part of b4(t~) where to = x +iy, for A1=1, Az =2, A3 =3 and ;t 4 = 4. The logarithmic singularities at each to = A have been truncated for display purposes. Note the saddle points between adjacent singularities plus a further saddle between - i times the smallest eigenvalue and ½n ( = 2 in this case).
We derive this result, and a method of obtaining further higher-order terms in the expansion, in the appendix. W e identify this large, widely separated (i.e., non-degenerate) limit of the eigenvalues with the asymptotic approximation given by the saddle-point mettiod applied to (9). In the asymptotic limit at least, we have n distinct pure-imaginary saddle points (i.e., n real roots of (11)). This is in fact true in general, if we have distinct real eigenvalues, though this seems difficult to prove directly from (11) (it is, however, possible in the cases n = 2, n = 3). We can understand this if we look at the real part of the function b(to) in (10), an example of which is given in Fig. 1 for n = 4. A p a r t from the initial - i t o t e r m we simply have a set of n logarithmic singularities distributed along the imaginary axis. There must therefore be a saddle point between each neighbouring singularity and another between - i times the smallest eigenvalue and ½in (the maximum value for a saddle point) making a total of n saddle points. Furthermore, the path of steepest a s c e n t for each saddle is clearly along the imaginary axis (i.e., traveling from one singularity to the next) which implies that the steepest-descent p a t h for each saddle is perpendicular to this, i.e., parallel to the real axis.
558
R.F. Langbein, H. Reinhardt~Nuclear PhysicsA585 (1995) 554-564
The contour of the integral (6) we are approximating lies along the real axis and we wish to deform it to pass through the saddle points. Since, by Cauchy's theorem, we can only deform it inside the analytic region of bn(¢o), only a single saddle point is thus available due to the presence of the logarithmic singularities. In the case of all positive real eigenvalues this is the largest root on the imaginary axis which is the root corresponding to the smallest eigenvalue. In the asymptotic approximation therefore, the saddle-point approximation to (6) yields
Cs'p'=
'Wn-1 exp[ b"(wmi")] 2 I b'(~Omin) 11/2'
(15)
since the phase through the saddle point is zero [6]. Noting that 1
1
b~(oJ) = - ~ ~. ( A i - / t o ) 2'
(16)
then in the asymptotic approximation, using (14) and ignoring terms in 1/(A i - At), we find
Cs.p.=
exp -/~min q-
H
(at - '~min)-1/2,
(17)
Aj =~Amin
where the product term arises from the logarithmic terms in b~(wmt,). We can now test how good this approximation is by comparing a numerical evaluation of (4) with (17) for certain values of A. In Fig. 2 we plot for n = 4 and with three of the eigenvalues held constant, the exact result and various approximations to it as a function of the fourth eigenvalue. We see that the asymptotic approximation breaks down as we would expect, in the region where two eigenvalues approach each other. Note, however, that this only happens for the smallest eigenvalue, i.e., when A / - Amin << I, the approximation remaining good when any of the larger eigenvalues approach each other, as long as the smallest remains separated. This is because in this case it is integrals over contours passing through the other saddle points which are affected, and not the one we are concerned with. Note the remarkable accuracy of the saddle-point approximation when the exact roots of the polynomial (ii) are used. After some numerical investigation, this appears to remain true even when all eigenvalues are simultaneously very small. It thus appears that the main inaccuracy in the asymptotic expansion is in estimating the roots of (II) rather than the saddle-point method itself. We now look at a special case equivalent to an integration over the group U(1) where we can examine the entire parameter space of the A. 3. Integration over S z
For the case n = 2, the integral (4) is C2(A l, A2)=
½fdO exp(-A1 cos20-Ax sinE0),
(18)
IZF. Langbein, H. Reinhardt / Nuclear PhysicsA585 (1995) 554-564 1.o
559
i! i
i
0.8
0.6
0.4
0.2
m
0.0 0
m
2
m
4
6
m
8
10
Fig. 2. Comparison of various approximations to C4(AI, A2, A3, A4), with A 1 = 3, A 2 = 4 and A 3 = 5 held constant, as a function of A 4. Plotted is the exact result calculated numerically from (4) (solid line), the saddle-point approximation (15) with the roots of b4(oJ) calculated exactly (short dashes), the same with the roots calculated to 0(1) in the asymptotic approximation, i.e., Eq. (17) (long dashes), and with the roots calculated in the next order in the asymptotic approximation, i.e., from (12) and (14) (dotted).
where we have made a change of variables to polar coordinates, x I r cos 0, x 2 = r sin 0 and carried out the resulting trivial integration over r. This integral is in fact equivalent to (a constant times) an integral representation of a modified Bessel function, i.e., =
C2(A1, A2) =Tr e x p [ - { ( A 1 + A2)]I0(½(A 1 - A 2 ) ) ,
(19)
where Io(z) is the zero-order modified Bessel function of the first kind. Let us compare this with the approximate evaluation of Section 2. From (9) we have
1 C2(•1,•2)
= "~
fdo exp[-b2(00)] ,
(20)
with b2(t0 ) = - i t o - 1[In(A1 - i t o ) + ln(A 2 - ira)].
(21)
The saddle-point equation reveals the two roots of the quadratic equation, from (11), i.e., Wl, 0)2 =
--
li(A 1 +
)k 2 - -
1 +_ v),
v = ~ ( a 1 - / ~ 2 ) 2 Or-1
(22)
as the two saddle points. Expanding the square root for the asymptotic limit (At,
560
R.F. Langbein, H. Reinhardt ~Nuclear Physics A585 (1995) 554-564
A2 >> 1, I ~1 --/~2 I >> 1) gives the expansion (14) for the case n = 2. As discussed in the previous section, we use the saddle point corresponding to the smallest eigenvalue, which yields
C2(/~1, /]1.2)--
exp[ __i~(A 1 + A2 - 1 - v)]
(23)
Without loss of generality we may assume '~2 > '~a, in which case, in the asymptotic limit (14), we find v -- A2 - h a + ~1 so that (23) becomes
C2(Aa, A2) = ~ - -
exp(--A a d- 1)
(24)
aa + Using the appropriate asymptotic form for the modified Bessel function Io(z) = eZ/ ~ we find (19) is the same as (24) ignoring the O(1) compared to A2 - A a in the denominator and apart from the factor (½e) 1/2 = 1.16. It is interesting to note that the contribution from the other saddle point, which yields a purely imaginary, but exponentially suppressed contribution compared to (23) for positive real A,., also appears in the asymptotic form for the modified Bessel function. It needs to be taken into account if the Bessel function's argument is in the negative half-plane. We can now compare the exact result (19) with the approximation (23) with exact roots of the associated quadratic and with the approximation (24) with the 1 asymptotic approximation to the roots. If we use the variables x = ~(/~2 - AI) and Y = 1(/~2 +/~1) then the ratio Cs.p./C2 depends only on x. In Fig. 3, we plot this ratio against x for various approximations Cs~p. We see again the difficulties with the asymptotic evaluation of the roots for x---, 0 whereas the exact roots give a good approximation for all x.
4. Conclusions We have demonstrated that the saddle-point method provides a good approximation to Fourier integrals of &functions as they occur in functional integrals over compact manifolds. For integrals over the sphere Sn, we found in the asymptotic limit (Ai, I Ai - A j [ >> 1 for all i, j ) that the leading order of this approximation depends most strongly on the smallest eigenvalue Atom (see Eq, (17)). The inevitable difficulties with this approximation which occur when Ai - Ami~ << 1 can apparently be rectified by using the exact solutions of (11) instead. These can be found analytically in the eases n = 2 (integration over U(1)), n = 3 and n = 4 (integration over SU(2)), but for larger n they would have to be found numerically. It seems then, that it is the approximate evaluation of the roots of (11) that breaks down in the asymptotic limit rather than a failure of the saddle-point method. A likely reason for the saddle-point method to be working so well, is that there is but a single saddle available to us to deform our contour to pass through,
R.F. Langbein, H. Reinhardt / Nuclear Physics A585 (1995) 554-564
561
2
\ f
I"
f
O0
I 2
I 4
I 6
I 8
10
x
Fig. 3. Comparison of various approximations to C2(,~.1, A2). Plotted is the ratio Cs.p./C 2 against the variable x = ½(A1 - A2) for Cs.p. given by (23) where the roots of bE(tO) have been calculated exactly (solid line), Cs.p. given by (15) in the lowest-order asymptotic approximation (long dashes) Cs.p with the roots calculated to the next order in the asymptotic approximation, i.e., Eq. (17) with roots given by (14) and (12) (dotted).
that corresponding to the largest root or smallest eigenvalue. All other saddles are irrelevant, being protected by logarithmic singularities. So, the process of expressing the g-function as a Fourier integral, linearizing and approximating by the saddle-point method appears to be a relatively safe process, though we must be more careful outside the asymptotic region. Indeed, this would be especially true for an integration over the group SU(2), having investigated this case directly here, and an SU(2) gauge group frequently occurs in field theories. Our only task when we use this approximation is to ensure that we pick out the smallest eigenvalue, in the asymptotic approximation, to use in (15), or the largest root of (11) in the general case, to use in (17).
Appendix We start with the polynomial (11), i.e., zn-- (al-- ln)zn-l
+ [a2-- l ( n - - 1 ) a l ] z n - 2
+ ... + ( - - 1 ) n a n - 1-~an_1 = 0 ,
(A.1) where the a i are given in (13), and recall that a polynomial Z n - - a l z n-1 + a 2 2 n - 2 +
... +( - 1)"a, =0
(A.2)
R.F. Langbein, H. Reinhardt / Nuclear Physics A585 (1995) 554-564
562
has exactly the roots A1,..., An. Let us construct a polynomial with the roots t
1
~i~'~i
(A.3)
2"
The coefficients of this polynomial will be ' =
al
n 2
al
' --
(A.4)
'
a2 - a 2
2
a; = a~
2
t a, = a,-
a 1+
a~ +
(A.5)
2 '
a(n ) 2
1 ½an_ 1 + ~ a n _ 2 +
+ 0(1),
(A.a)
O(/~n-3),
(A.7)
and we notice that in the asymptotic limit (A i >> 1, I Ai --/~j ] •• 1) these are just the coefficients of (A.1) up to error terms subleading in powers of A. Let us now try to calculate the next-order correction by looking at the polynomial with the roots
(A.8)
h"i = ~ - ~ + ri
instead of (A.3), where the r i are of order 1/A. We thus calculate the coefficients a ", and we want the leading-order ri term in each expression to cancel the leading error term. For example, from a'~ we have Z r i = 0, whilst from a~ we obtain E i ~ , i . j A i r i . Doing this for the n coefficients leads to a set of n linear equations in the n unknowns ri, i.e., 1
1
...
1
r1
Zi~lAi
Ei4~2Ai
...
Ei~nAi
r2
Y~.i~j~IAiAj
Y'.i~j~zAiAj
...
~,,i÷j~nAiAj
r3
1-Iie.lAi
I~i~2Ai
...
I-li~nAi
r~
0
1 n-- 1)a 1
an-2
(A.9)
R.F. Langbein, H. Reinhardt / Nuclear Physics A585 (1995) 554-564
563
Row-reducing this system leads to 1
1
1
...
0 0
0
1
die
d13
...
din
0
d13d23
...
dlnd2n
0
0
...
I-Ii@ndin 0
1
1 4( )al
-
n i~i.1 ) 2A
n -
(A.10)
R where
dij = h i - Aj,
(A.11)
and by an inductive argument, R, the last element on the right-hand side of (A.10) is n-I E 1-I dj, . (A.12) i=1 j#:i Hence, since the system is symmetric in the h i, we find 1
1
r i = - -~ .j~, dj--T .
(A.13)
We can find higher-order corrections by following the same procedure. We add correction terms to the roots which are of the next order in l / A . Forming the polynomial associated with these roots gives rise to a set of equations like (A.9). These equations have the same left-hand side as (A.9), though the right-hand side will differ. Row-reduction thus proceeds in the same way and, since from (A.10) the determinant of the matrix on the left-hand side is
1-I dij i
(A.14)
which is only zero if Ai = Aj for some i, j, we are guaranteed a unique solution unless some of the eigenvalues are degenerate.
References
[1] M.B. Halpern, Phys. Rev. D 16 (1977) 1798; M. Schaden, H. Reinhardt, P. Amundsen and M. Lavelle, Nucl. Phys. B 339 (1990) 595. K. Langfeld and H. Reinhardt, Nucl. Phys. A 579 (1994) 472.
564
I~F. Langbein, H. Reinhardt / Nuclear PhysicsA585 (1995) 554-564
[2] C.M. Bender, F. Cooper and G.S. Guralnik, Ann. of Phys. 109 (1977) 165; K. Langfeld, L.v. Smekai and H. Reinhardt, Phys. Lett. B 308 (1993) 279; B 311 (1993) 207. [3] H. Kleinert Phys. Lett. B 69 (1977) 9; H. Reinhardt, Nucl. Phys. A 298 (1978) 77; A 346 (1980) 1. [4] T. Eguchi, Phys. Rev. D 14 (1976) 947; D. Ebert and H. Reinhardt, Nuci. Phys. B 271 (1986) 188. [5] R.F. Langbein and H. Reinhardt, Nucl. Phys. A 580 (1994) 349. [6] Arfken, Mathematical methods for physicists (Academic, New York, 1966).
ELSEVIER
Saddle-point approximation of compact integrals * R.F. Langbein, H. R e i n h a r d t Institut fiir Theoretische Physik, Universitlit Tiibingen, Auf der Morgenstelle 14, D-72076 Tiibingen, Germany Received 10 October 1994
Abstract
We investigate the validity of the saddle-point approximation of integrals over the compact manifold Sn, in which the confining 6-function in the measure has been replaced by its Fourier representation. The approximation is found to deliver good results, particularly when the exact roots of the defining polynomial are used.
1. Introduction
A fundamental technique in non-perturbative quantum field theory is the process of linearizing interaction terms in a given action in terms of an auxiliary field. For example, for some generic field or, the path integral
z= f~a exp[-½(Ka)2-ga4],
(1)
where K is an operator describing kinetic energy and mass and g is a coupling, may be rewritten as
Z=
exp(-½(K
) 2 _ 1 z_i/3a2).
(2)
This expression is now only quadratic in the original field a which may be thus integrated out, leaving an alternative form for (1) in terms of the auxiliary field/3. This technique has been applied in a range of theories, including Yang-Mills field theory [1], ~b4-theory [2], as well as interacting many-body systems [3], and it is the foundation of bosonization approaches [4]. * Supported by the Deutsche Forschungsgemeinschaft under contract DFG-Re 856/1-2. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(94)00533-8
R.F. Langbein, H. Remhardt/ Nuclear PhysicsA585 (1995) 554-564
555
However, if the measure for the field we want to linearize is non-trivial, i.e., non-fiat, as is the case for example in Yang-Mills theory (though of course it is assumed fiat in the perturbative treatment), we cannot apply this technique directly. An often used approach, if the measure contains a 8-function, is to replace the ~-function by its Fourier representation. The linearization technique can then be applied as before and our path integral at the end contains an integration over the Fourier modes of this ~-function. We might question this technique in light of the inevitable approximations that are needed to interpret this final path integral. An approximation scheme will only sum over some of the Fourier modes and we might wonder if this properly reflects the distribution character of the ~-function. We examine here a test case, integration of the exponential of a quadratic form over the n-dimensional sphere S,. This case has a number of features in common with the actual integrals required for example in lattice gauge theory [5]. It is, first of all, as for integrals over a gauge group, a compact integral, and the cases n = 2 and n = 4 are equivalent to integration over the groups U(1) and SU(2) respectively. We apply the chain of techniques mentioned above, i.e., replacing the ~-function in the measure by its Fourier representation, linearizing and then approximating, and see if the final expression makes sense. We do this in the following section for arbitrary n to see what we can deduce in general, whilst in Section 3 we compare the approximation with the exact result in the case n = 2. The final section gives some conclusions.
2. Integration over the n-sphere We examine the compact integral Cn(M )
=
f
...dx
n
~( xTx-
1) e x p ( - - x T M x ) ,
(3)
w h e r e x T = ( x 1. . . . , x , ) , M is an n × n symmetric m a t r i x and the 6 - f u n c t i o n on the
measure impies integration over the n-sphere C,(A) = fdXl...dx
. 6(xTx-
S n.
It is in fact sufficient to study
1) e x p ( - - x T A x ) ,
(4)
where A is the diagonal matrix of eigenvalues of M, i.e., A1 A = RTMR =
,
(5)
A,
where )q . . . . , An are the eigenvalues of M, and R is an orthogonal transformation with det(R) = 1. Eq. (4) is obtained from (3) by making the substitution x -~ Rx.
556
R.F. Langbein, H. Reinhardt /Nuclear Physics A585 (1995) 554-564
The non-trivial measure of the integral (4) can be handled by replacing the a-function by its Fourier representation, giving C,(A) = ~
1
f d x x . . . d x n dta exp[--xT(A--ito)x--ito],
(6)
evaluating the gaussian integral over the xi, leaving T~n / 2 - 1
C,(A)
2
f d t o det-1/E(A - ito) e -i°'
,ITn / 2 - 1
#
e -ic°
- ~ / d t o z
(7)
HL-~(~..
J
(
(8)
ito ) 1/2
-
- ~r"/2---------~1jd~of exp - i w - -
2
'n
)
~ ln(A,-i~0) ,
2i= 1
(9)
and finally approximating this final expression (9) asymptotically by using the saddle-point method. Several questions now arise. The function in the exponent of (9), 1
~
b.(to) = -ito - ~ ,l•n ( a . . = l - - i t o ) ,
(10)
has in fact n saddle points. Which saddle points does our contour need to pass through? Does an asymptotic approximation place restrictions on the possible values of the ai? And is the approximation any use outside this asymptotic region? We start by considering the general case and assume that the eigenvalues a~ are all real and positive. The saddle points of b.(to) are found by solving b'.(to)= 0, which, after a little algebra, are found to be the roots of the polynomial
zn--(al--ln)z n-l+ [ a 2 - - 1 ( n - 1 ) a l ] z n-e+... +(--1)nan--lan_l=O, (11) where z = ito
(12)
and n
n
a2= ]~AiAj . . . . .
a 1 = ~ Ai, i=1
a. = H A i .
i~j
(13)
i= 1
Using Newton's rule, it can be checked that the real roots of (11) lie between -oo 1 and ~n. If we assume that Ai >> 1 and also I Ai - Aj I >> 1, we find we can approximate the roots of (11) by 1
1 ~i=--Ai'~-2
-
E 4(A/-Ay) j~i
+
1 O( (Ai _ Aj)2 ) .
(14)
R.F. Langbein, H. Reinhardt ~Nuclear Physics A585 (1995) 554-564
557
Fig. 1. Plot of the real part of b4(t~) where to = x +iy, for A1=1, Az =2, A3 =3 and ;t 4 = 4. The logarithmic singularities at each to = A have been truncated for display purposes. Note the saddle points between adjacent singularities plus a further saddle between - i times the smallest eigenvalue and ½n ( = 2 in this case).
We derive this result, and a method of obtaining further higher-order terms in the expansion, in the appendix. W e identify this large, widely separated (i.e., non-degenerate) limit of the eigenvalues with the asymptotic approximation given by the saddle-point mettiod applied to (9). In the asymptotic limit at least, we have n distinct pure-imaginary saddle points (i.e., n real roots of (11)). This is in fact true in general, if we have distinct real eigenvalues, though this seems difficult to prove directly from (11) (it is, however, possible in the cases n = 2, n = 3). We can understand this if we look at the real part of the function b(to) in (10), an example of which is given in Fig. 1 for n = 4. A p a r t from the initial - i t o t e r m we simply have a set of n logarithmic singularities distributed along the imaginary axis. There must therefore be a saddle point between each neighbouring singularity and another between - i times the smallest eigenvalue and ½in (the maximum value for a saddle point) making a total of n saddle points. Furthermore, the path of steepest a s c e n t for each saddle is clearly along the imaginary axis (i.e., traveling from one singularity to the next) which implies that the steepest-descent p a t h for each saddle is perpendicular to this, i.e., parallel to the real axis.
558
R.F. Langbein, H. Reinhardt~Nuclear PhysicsA585 (1995) 554-564
The contour of the integral (6) we are approximating lies along the real axis and we wish to deform it to pass through the saddle points. Since, by Cauchy's theorem, we can only deform it inside the analytic region of bn(¢o), only a single saddle point is thus available due to the presence of the logarithmic singularities. In the case of all positive real eigenvalues this is the largest root on the imaginary axis which is the root corresponding to the smallest eigenvalue. In the asymptotic approximation therefore, the saddle-point approximation to (6) yields
Cs'p'=
'Wn-1 exp[ b"(wmi")] 2 I b'(~Omin) 11/2'
(15)
since the phase through the saddle point is zero [6]. Noting that 1
1
b~(oJ) = - ~ ~. ( A i - / t o ) 2'
(16)
then in the asymptotic approximation, using (14) and ignoring terms in 1/(A i - At), we find
Cs.p.=
exp -/~min q-
H
(at - '~min)-1/2,
(17)
Aj =~Amin
where the product term arises from the logarithmic terms in b~(wmt,). We can now test how good this approximation is by comparing a numerical evaluation of (4) with (17) for certain values of A. In Fig. 2 we plot for n = 4 and with three of the eigenvalues held constant, the exact result and various approximations to it as a function of the fourth eigenvalue. We see that the asymptotic approximation breaks down as we would expect, in the region where two eigenvalues approach each other. Note, however, that this only happens for the smallest eigenvalue, i.e., when A / - Amin << I, the approximation remaining good when any of the larger eigenvalues approach each other, as long as the smallest remains separated. This is because in this case it is integrals over contours passing through the other saddle points which are affected, and not the one we are concerned with. Note the remarkable accuracy of the saddle-point approximation when the exact roots of the polynomial (ii) are used. After some numerical investigation, this appears to remain true even when all eigenvalues are simultaneously very small. It thus appears that the main inaccuracy in the asymptotic expansion is in estimating the roots of (II) rather than the saddle-point method itself. We now look at a special case equivalent to an integration over the group U(1) where we can examine the entire parameter space of the A. 3. Integration over S z
For the case n = 2, the integral (4) is C2(A l, A2)=
½fdO exp(-A1 cos20-Ax sinE0),
(18)
IZF. Langbein, H. Reinhardt / Nuclear PhysicsA585 (1995) 554-564 1.o
559
i! i
i
0.8
0.6
0.4
0.2
m
0.0 0
m
2
m
4
6
m
8
10
Fig. 2. Comparison of various approximations to C4(AI, A2, A3, A4), with A 1 = 3, A 2 = 4 and A 3 = 5 held constant, as a function of A 4. Plotted is the exact result calculated numerically from (4) (solid line), the saddle-point approximation (15) with the roots of b4(oJ) calculated exactly (short dashes), the same with the roots calculated to 0(1) in the asymptotic approximation, i.e., Eq. (17) (long dashes), and with the roots calculated in the next order in the asymptotic approximation, i.e., from (12) and (14) (dotted).
where we have made a change of variables to polar coordinates, x I r cos 0, x 2 = r sin 0 and carried out the resulting trivial integration over r. This integral is in fact equivalent to (a constant times) an integral representation of a modified Bessel function, i.e., =
C2(A1, A2) =Tr e x p [ - { ( A 1 + A2)]I0(½(A 1 - A 2 ) ) ,
(19)
where Io(z) is the zero-order modified Bessel function of the first kind. Let us compare this with the approximate evaluation of Section 2. From (9) we have
1 C2(•1,•2)
= "~
fdo exp[-b2(00)] ,
(20)
with b2(t0 ) = - i t o - 1[In(A1 - i t o ) + ln(A 2 - ira)].
(21)
The saddle-point equation reveals the two roots of the quadratic equation, from (11), i.e., Wl, 0)2 =
--
li(A 1 +
)k 2 - -
1 +_ v),
v = ~ ( a 1 - / ~ 2 ) 2 Or-1
(22)
as the two saddle points. Expanding the square root for the asymptotic limit (At,
560
R.F. Langbein, H. Reinhardt ~Nuclear Physics A585 (1995) 554-564
A2 >> 1, I ~1 --/~2 I >> 1) gives the expansion (14) for the case n = 2. As discussed in the previous section, we use the saddle point corresponding to the smallest eigenvalue, which yields
C2(/~1, /]1.2)--
exp[ __i~(A 1 + A2 - 1 - v)]
(23)
Without loss of generality we may assume '~2 > '~a, in which case, in the asymptotic limit (14), we find v -- A2 - h a + ~1 so that (23) becomes
C2(Aa, A2) = ~ - -
exp(--A a d- 1)
(24)
aa + Using the appropriate asymptotic form for the modified Bessel function Io(z) = eZ/ ~ we find (19) is the same as (24) ignoring the O(1) compared to A2 - A a in the denominator and apart from the factor (½e) 1/2 = 1.16. It is interesting to note that the contribution from the other saddle point, which yields a purely imaginary, but exponentially suppressed contribution compared to (23) for positive real A,., also appears in the asymptotic form for the modified Bessel function. It needs to be taken into account if the Bessel function's argument is in the negative half-plane. We can now compare the exact result (19) with the approximation (23) with exact roots of the associated quadratic and with the approximation (24) with the 1 asymptotic approximation to the roots. If we use the variables x = ~(/~2 - AI) and Y = 1(/~2 +/~1) then the ratio Cs.p./C2 depends only on x. In Fig. 3, we plot this ratio against x for various approximations Cs~p. We see again the difficulties with the asymptotic evaluation of the roots for x---, 0 whereas the exact roots give a good approximation for all x.
4. Conclusions We have demonstrated that the saddle-point method provides a good approximation to Fourier integrals of &functions as they occur in functional integrals over compact manifolds. For integrals over the sphere Sn, we found in the asymptotic limit (Ai, I Ai - A j [ >> 1 for all i, j ) that the leading order of this approximation depends most strongly on the smallest eigenvalue Atom (see Eq, (17)). The inevitable difficulties with this approximation which occur when Ai - Ami~ << 1 can apparently be rectified by using the exact solutions of (11) instead. These can be found analytically in the eases n = 2 (integration over U(1)), n = 3 and n = 4 (integration over SU(2)), but for larger n they would have to be found numerically. It seems then, that it is the approximate evaluation of the roots of (11) that breaks down in the asymptotic limit rather than a failure of the saddle-point method. A likely reason for the saddle-point method to be working so well, is that there is but a single saddle available to us to deform our contour to pass through,
R.F. Langbein, H. Reinhardt / Nuclear Physics A585 (1995) 554-564
561
2
\ f
I"
f
O0
I 2
I 4
I 6
I 8
10
x
Fig. 3. Comparison of various approximations to C2(,~.1, A2). Plotted is the ratio Cs.p./C 2 against the variable x = ½(A1 - A2) for Cs.p. given by (23) where the roots of bE(tO) have been calculated exactly (solid line), Cs.p. given by (15) in the lowest-order asymptotic approximation (long dashes) Cs.p with the roots calculated to the next order in the asymptotic approximation, i.e., Eq. (17) with roots given by (14) and (12) (dotted).
that corresponding to the largest root or smallest eigenvalue. All other saddles are irrelevant, being protected by logarithmic singularities. So, the process of expressing the g-function as a Fourier integral, linearizing and approximating by the saddle-point method appears to be a relatively safe process, though we must be more careful outside the asymptotic region. Indeed, this would be especially true for an integration over the group SU(2), having investigated this case directly here, and an SU(2) gauge group frequently occurs in field theories. Our only task when we use this approximation is to ensure that we pick out the smallest eigenvalue, in the asymptotic approximation, to use in (15), or the largest root of (11) in the general case, to use in (17).
Appendix We start with the polynomial (11), i.e., zn-- (al-- ln)zn-l
+ [a2-- l ( n - - 1 ) a l ] z n - 2
+ ... + ( - - 1 ) n a n - 1-~an_1 = 0 ,
(A.1) where the a i are given in (13), and recall that a polynomial Z n - - a l z n-1 + a 2 2 n - 2 +
... +( - 1)"a, =0
(A.2)
R.F. Langbein, H. Reinhardt / Nuclear Physics A585 (1995) 554-564
562
has exactly the roots A1,..., An. Let us construct a polynomial with the roots t
1
~i~'~i
(A.3)
2"
The coefficients of this polynomial will be ' =
al
n 2
al
' --
(A.4)
'
a2 - a 2
2
a; = a~
2
t a, = a,-
a 1+
a~ +
(A.5)
2 '
a(n ) 2
1 ½an_ 1 + ~ a n _ 2 +
+ 0(1),
(A.a)
O(/~n-3),
(A.7)
and we notice that in the asymptotic limit (A i >> 1, I Ai --/~j ] •• 1) these are just the coefficients of (A.1) up to error terms subleading in powers of A. Let us now try to calculate the next-order correction by looking at the polynomial with the roots
(A.8)
h"i = ~ - ~ + ri
instead of (A.3), where the r i are of order 1/A. We thus calculate the coefficients a ", and we want the leading-order ri term in each expression to cancel the leading error term. For example, from a'~ we have Z r i = 0, whilst from a~ we obtain E i ~ , i . j A i r i . Doing this for the n coefficients leads to a set of n linear equations in the n unknowns ri, i.e., 1
1
...
1
r1
Zi~lAi
Ei4~2Ai
...
Ei~nAi
r2
Y~.i~j~IAiAj
Y'.i~j~zAiAj
...
~,,i÷j~nAiAj
r3
1-Iie.lAi
I~i~2Ai
...
I-li~nAi
r~
0
1 n-- 1)a 1
an-2
(A.9)
R.F. Langbein, H. Reinhardt / Nuclear Physics A585 (1995) 554-564
563
Row-reducing this system leads to 1
1
1
...
0 0
0
1
die
d13
...
din
0
d13d23
...
dlnd2n
0
0
...
I-Ii@ndin 0
1
1 4( )al
-
n i~i.1 ) 2A
n -
(A.10)
R where
dij = h i - Aj,
(A.11)
and by an inductive argument, R, the last element on the right-hand side of (A.10) is n-I E 1-I dj, . (A.12) i=1 j#:i Hence, since the system is symmetric in the h i, we find 1
1
r i = - -~ .j~, dj--T .
(A.13)
We can find higher-order corrections by following the same procedure. We add correction terms to the roots which are of the next order in l / A . Forming the polynomial associated with these roots gives rise to a set of equations like (A.9). These equations have the same left-hand side as (A.9), though the right-hand side will differ. Row-reduction thus proceeds in the same way and, since from (A.10) the determinant of the matrix on the left-hand side is
1-I dij i
(A.14)
which is only zero if Ai = Aj for some i, j, we are guaranteed a unique solution unless some of the eigenvalues are degenerate.
References
[1] M.B. Halpern, Phys. Rev. D 16 (1977) 1798; M. Schaden, H. Reinhardt, P. Amundsen and M. Lavelle, Nucl. Phys. B 339 (1990) 595. K. Langfeld and H. Reinhardt, Nucl. Phys. A 579 (1994) 472.
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I~F. Langbein, H. Reinhardt / Nuclear PhysicsA585 (1995) 554-564
[2] C.M. Bender, F. Cooper and G.S. Guralnik, Ann. of Phys. 109 (1977) 165; K. Langfeld, L.v. Smekai and H. Reinhardt, Phys. Lett. B 308 (1993) 279; B 311 (1993) 207. [3] H. Kleinert Phys. Lett. B 69 (1977) 9; H. Reinhardt, Nucl. Phys. A 298 (1978) 77; A 346 (1980) 1. [4] T. Eguchi, Phys. Rev. D 14 (1976) 947; D. Ebert and H. Reinhardt, Nuci. Phys. B 271 (1986) 188. [5] R.F. Langbein and H. Reinhardt, Nucl. Phys. A 580 (1994) 349. [6] Arfken, Mathematical methods for physicists (Academic, New York, 1966).