Dispersion relations and resonance reactions

Nuclear Physics 57 (1964) 134--151; (~) North-Holland Publishing Co., Amsterdam Not to

be reproduced by photoprint or microfilm

without written permission from the

publisher

DISPERSION RELATIONS AND RESONANCE REACTIONS CLASINE VAN WINTER *

Institute for Theoretical Physics, University of Copenhagen Received 15 January 1964 F o r one-channel scattering, a resonance formalism is developed which avoids continuing the scattering function into a region where it has resonance poles. In a non-relativistic terminology, it is assumed that for any particular partial wave the scattering function S is analytic in the energy plane cut from -- ~ to -- 1 and from 0 to ~ . It is further assumed that S satisfies a dispersion relation, and that it takes complex conjugate values at complex conjugate points. U n d e r these assumptions, S can be written as a Blaschke product times an exponential function. The Blaschke product contains the zeros o f S. At energies below the threshold for inelastic processes, it describes the resonances. If in this energy region S has isolated zeros close to the real axis, these give rise to Breit-Wigner peaks in the cross section. The exponential function corresponds to the potential scattering. It refers to a scattering process in which the real part o f the phase shift is connected with the absorption at higher energies through a dispersion relation. The formalism can be extended to energies at which more than one channel is involved. This generalization is, however, not discussed.

Abstract:

1. Introduction In the course of years, various theories have been developed for describing the resonance peaks observed in nuclear-reaction cross sections. In the simple case of isolated resonances, these theories all yield the well-known Breit-Wigner formula. At the same time, they differ widely in their definitions of the resonances, as well as in the mathematical tools used to expand the cross section in terms of the resonance parameters. In the Kapur-Peierls formalism 1), the starting point is an eigenvalue problem with an energy-dependent boundary condition. This has the disadvantage of making the positions of the resonances dependent on the energy of the incident particle. Also, there is a certain arbitrariness in the choice of the surface on which the boundary condition is to be imposed. Corresponding to this, each term in the expansion of the cross section involves a somewhat arbitrary scattering radius, whereas the cross section as such is obviously independent of such an auxiliary quantity. The theory of Wigner and Eisenbud 2) likewise suffers from the complications arising from the introduction of scattering radii. It is an additional difficulty that this formalism is conceived mainly in terms of the R-matrix, which is connected with the cross section in a very indirect way. As a result the parameters of the Wigner-Eisenbud formalism are not at all closely related to the observed resonance energies and widths. On leave o f absence from the University o f Groningen. 134

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135

In a number of more recent papers on the subject, the resonances correspond to poles of the scattering matrix S, the elements of which are considered as analytic functions. Along these lines, very much work has been done by Humblet and Rosenfeld 3.4). In particular, it has been shown that, in the case of scattering by a potential well of finite range, S is a meromorphic function which can be expanded in terms of a Mittag-Leffler series. This approach has also been extended to cases where more than one channel is involved. Adopting the same definition of the resonances, Peierls s) and Le Couteur 6) have discussed the complications arising from S having branch points due to the occurrence of inelastic processes. In expanding the scattering matrix, these authors make use of Cauchy integrals and product representations, respectively. In problems in which there is only one channel and the interaction vanishes beyond a certain distance, infinite products have also been used by Hu 7) and by Van Kampen s). In Hu's work as well as in the papers quoted above, the scattering is investigated within the context of the Schrtidinger equation. Van Kampen has used only conditions of causality and symmetry. It is obvious that, in a formalism in which the resonances are associated with poles of the scattering matrix, the elements of S must be the boundary values of analytic functions which can be continued into a region where they have poles. Let us examine this point under the assumption that, for any particular partial wave, each element S~j(z) is analytic in the energy plane cut along the positive real axis, satisfies s,j(z)

(1.1)

=

and is a continuous function of z for z > 0, the variable z denoting the energy. If there is only one element S (z) and if for z > 0 this satisfies the unitarity relation

IS(z)l = 1 (z > 0),

(1.2)

it follows from the Schwarz reflection principle (Titchmarsh 9) p. 157) that the function 1IS(z) provides the analytic continuation of S(z) onto a second sheet of the energy plane. Clearly there may be poles on the second sheet, corresponding to zeros of S (z) on the original so-called physical one. To discuss an example in which more channels are involved, let us consider a 2 x 2 S-matrix S(z) with elements $11, $12 = $21, $22 such that

ISlx(z)l

= 1 (0 <__ z < cx),

S*(z)S(z)

= 1 (z > c l ) .

(1.3)

Combining eqs. (1.1) and (1.3) shows that the function 1/S~(z) yields the continuation of S~l(z) across the segment 0 < z < cl. Also, since the inverse matrix S-~(z) has elements det

1

S(z----~)$22(z),

det

1

S(z~)S12(z)'

det

iS(z) S~(z),

(1.4)

the functions (1.4) are the continuations of $11 (z), Slz(z), $22(z) across the segment c~ < z < ~ . They will in general have poles where det S(z) vanishes. The continu-

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C. VAN WINTER

ations o f Sa,(z) across the segments 0 < z < c, and c 1 < z < oo, respectively, are not onto the same sheet. For if they were, we should have ISla(z)l = 1 also on ca < z < oo, contrary to the assumption. In other words, Sal(Z ) has a branch point at z = ca. Accordingly, Peierls 5) and Le Couteur 6) have considered the elements of a j x j S-matrix as multi-valued functions on a large number of sheets. W o r k along the same lines has been done by Newton lo), Davies and Baranger ,a), and Chart a2). Humblet and Rosenfeld 4) have made use only of the continuation of all matrix elements across the segment cg_ a < z < oo, where cj_ 1 denotes the highest threshold. In the present paper the point of view is adopted that, even if S (z)can be continued across the real axis, its properties are completely determined by its behaviour on the physical sheet. Hence it must be possible to develop a resonance formalism entirely within the framework of the energy plane cut along the positive real axis. The foregoing considerations suggest that in such a formalism the resonances in any interval c~_ ~ < z < c~ between two successive thresholds be associated with the zeros of the determinant of the i x i matrix which refers to the i open channels. This idea is worked out here for the interval 0 < z < c~, in which there is only one open channel. We consider one single function S(z), which is assumed to satisfy a dispersion relation. In its simplest form, this reads

S(z)

_-

~/_ xda(t) _ _

+

~fo ~ 1da(t)

oo t - - z

t--z

+

~

ak

(1.5)

k=l p k - - z

Here, a(t) is of bounded variation, PR satisfies -- 1 < Pk < 0, the coefficients ak are real, K is finite. If, for instance, two subtractions are required, we choose numbers t1 and t2 such that - 1 < t 1 < tz < 0 and tl, 12 :¢- Pk" For this case we assume that S(z)

( z - t l ) ( Z - t2)

q-

S(tl) (z- tl)(t 2 - tl)

+

S(t2) (z - t2)(t I - t2) -1 ___ - + = n f - o o dz(tlt_z

t-z

+

k=l pk--Z

,

(1.6)

and similarly for more complicated cases. It follows f r o m eqs. (1.5) and (1.6) that in the z-plane cut from - oo to - 1 and from 0 to oo, the function S(z) is regular except for a finite number of poles in the gap between the cuts. It satisfies the symmetry relation (1.1). The length of the gap has been chosen to be 1 purely for convenience. It is known ~3-15) that a dispersion relation of the form considered here app'.ies to the scattering by a superposition of Yukawa potentials, the potential 9e-U'/r giving rise to a gap proportional to pz and independent of the coupling constant 9. The same dispersion relation is also used in elementary-particle physics. At present we do not go into the question whether a dispersion relation is actually justified, say in nuclear physics. This point is discussed in a forthcoming paper in

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RESONANCE REACTIONS

which multi-channel scattering is investigated from the point of view of the Schr6dinger equation for n particles. In that paper we also describe how the present formalism can be extended to energies at which inelastic processes take place.

2. Integral Representations and Hardy Classes Let us consider a function F(w) which can be represented by an integral according to V(w) = ~l f - - ~ dx(s s- )

(Im w > 0),

(2.1)

where X(s) is of bounded variation. It follows from eq. (2.1) that F(w) is regular in the open upper half-plane, which is henceforth denoted by o¢. Also, ½X(s-O)+½)~(s+O)-½x(O-O)-½)~(O+O ) = lim

Im F(u+iv)du.

(2.2)

v--*0 d 0

It is obvious that we may write

l

F(w) = C + ~

sw+l+l)dX(s) '

_ ~ (~- ~)(~

if °

C = ;~ _ ~ ~ s+---q dx(s).

(2.3)

To facilitate the use of certain results which are readily available in the literature, we now map o¢ onto the circle I~l < 1, which we denote by c~. The mapping is achieved by the transformation 1+( w = i-1-(

( = '

w-i w+i

(2.4) "

The real axis w = s, with - o o < s < oo, corresponds to the circle ( = ei~', with 0 < q~ < 2rr. Using the notation

G(~) = -iF(w)+iC,

c0(¢p) = 2

oo t 2 + l d3f(t),

(2.5)

we obtain 1 f2~ ei~,+ G(0 = ~ d o e'%-~ dco(q~)

(1¢1 < 1).

(2.6)

Functions which can be represented in this form have been investigated by many authors, notably Riesz 16), Nevanlinna 1~) and Priwalow is). A useful reference is also Hoffman 19). The following discussion is to a large extent based on the book by Priwalow 1a), which is quoted as P. For general theorems in the theory of functions, we refer to the textbook by Titchmarsh 9) as T. Since ~(~o) is of bounded variation, it is the difference of two non-decreasing functions. It follows that G (O is the difference of two regular functions with positive real parts. Such functions are known to belong to the so-called Hardy class ,~>a

138

c. VAN WINTER

(0 < 6 < 1), which consists of all functions G(~) = G(pe l~*) which are regular in c~ and satisfy limf2~lG(pe'~)16dq~ < ~

(2.7)

p---~ 1 4 0

(P pp. 64, 53). Since IGI+G2I ~ ~

261GlIn-+-2alG21a,

(2.8)

the sum of two functions in ~ also belongs to ~6. It is easily checked with H01der's inequality (T p. 382) that, if Gz belongs to ~61 and G2 belongs to ~ 2 , the product G1G2 belongs to ~)~, with 6 = 6162/(61+62). It follows likewise from HOlder's inequality that, if G belongs to ~6, it also belongs to every class ~ , with 0 < 6' < 5. The Hardy classes ~s are all contained in the class 9A formed by the functions G (joe~*) which are regular in ~ and satisfy lira f2"log+ IG(pe~*)l d~p < 0%

(2.9)

p'-~ldO

log+[al being equal to loglal if lal _-__1 and vanishing otherwise (P pp. 53, 54). Since log+lG1G21 <= log+lGll + log+lG2l,

(2.10)

the product of any two functions in 9~ belongs to 9~. So does the quotient, provided it is regular (P p. 55). Furthermore, by virtue of the relation

log+lG1+G21~ log 2 + log+lGzl + log+lG21,

(2.11)

the sum of two functions in 9Aalso belongs to 9~. The class 9/is identical with the class consisting of all functions which are regular in cg and can be written as quotients of bounded functions (P p. 56). If G (~) is any function in ~, it follows from Fatou's theorem that, if ~ tends to e ~ along a path not tangent to the circle ICI = 1, the function G(ff) tends to a finite boundary value for almost every q~. If this is denoted by G (ei~), we have fo"llog IG(e~)ll d~0 < c~

(2.12)

(P p. 56). 3. The Faetorization Theorem

In the context of the present investigation, the interest of the classes ~ and 9/ derives from the fact that a function which belongs to any of these classes can be expanded in a product with respect to its zeros. In particular, let us assume that G (~) belongs to ~ . If G (() vanishes at the origin, we denote the multiplicity of this zero by r. Otherwise we take r = 0. If there are zeros different from zero, these are

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139

denoted by (, (n = 1, 2 . . . . ), a zero of order r, being counted r, times. With this notation, G (() can be expanded according to

G(¢) = e'~n(£)exp [ 1 I"2~ e ~ + ( (log IG(e'~')l dq~+d/J(q~))l , L2ndo e~*-~

B(0 =

'rI

(3.1)

LQI_c *0j

(P p. 78). Here, m is a real constant, #(r#) is a bounded non-increasing function the derivative of which vanishes almost everywhere. The integral converges owing to eq. (2.12). The function B ( ( ) is called the Blaschke product. It follows from eq. (2.9) that ~, ( 1 -

Iff.I)

<

~

(3.2)

rim1

(P p. 22). Hence, the Blaschke product converges absolutely and uniformly, so that it is regular in 5 . It will be observed that the point ( = 1/(* at which the denominator of B ( 0 vanishes lies outside c~. It might seem that the product expansion suggests that, when continued across the circle I(I = 1, the function G (() has a pole at [ = 1/(*. This, however, need not at all be the case. For the fact that G ( 0 belongs to ~6 does not imply that it can be continued. And among the functions which do have a continuation, let us consider ( - ( 1 , where ](11 < 1. It is clear that this function belongs to every ~96. Hence it can be expanded according to eq. (3.1). But it is obvious that it has no poles, the product representation being valid only in cal. At the same time, it is this peculiar pole-like feature of the Blaschke product which makes it particularly suited to the resonance problem. Since G(() and B ( ( ) are regular in c~ and have the same zeros, the quotient G (O/B(~) is regular and non-vanishing. Hence log[G (~)/B(~)] is also regular. The factorization theorem says that this function can be expressed in terms of a dispersion integral of the form (2.6). When we think of the relation log [G (()/B (~)] = logl G (O/B (()l + i arg [G (~)/B (~)]

(3.3)

combined with the fact that IB(d~)l = 1 for almost every tp, it becomes clear that the exponent in eq. (3.1) is nothing but log[G (()/B(()] expressed in terms of its real part near the boundary I~1 = 1. If G(() is regular not only in c~ but also on the boundary, it follows from the Poisson-Jensen formula that the singular measure #(~0) is constant (T p. 129). In the more general case, choosing #(0) = 0, we may write f~log IG(e'~)l dgo+/l(~k) = lim f~'log [G(pe"e)/B(pe'~')l dgo

(3.4)

p'-*l ~ 0

(P p. 81). For the following it is convenient to replace the function IG(d")l on the left-hand side of this equation by IG (e~')/B (ei~)[. This is allowed, since IB (e~*)[ = 1.

140

c . VAN WINTER

Let us now briefly discuss what is implied by the function p(~b). It is known that, whenever G (~) belongs to f)~, lim f ~ log + IG(pei~*)l d~0 = f~'log+ [G(ei*)[ d~0 p"* l d O

(3.5)

,dO

(P p. 80). This formula applies in particular to G (~)/B(~). For the latter function belongs to ~a by virtue of the converse of the factorization theorem (P p. 78). Hence if we write log [G ([)/B (~)l = log + IG (~)/B ([)1+ log- IG (~)/B (~)l,

(3.6)

it follows that #(~O) = l i m f * l o g - IG(pei'P)/B(pei'p)[ d ~ o - f * l o g p"*ldO

Ia(e"O/n(d~')l

dcp.

(3.7)

dO

This function p(~) is clearly non-increasing, by Fatou's lemma (T p. 346). Let us now assume that

IG(pei'p)/B(pe'q')[ > ~

(0 < ~ < 1)

(3.8)

uniformly in p. Then Ilog-[G(pe'~')/B(pe'q')[[ < [log el.

(3.9)

Hence, in the first term on the right-hand side of eq. (3.7), lim ( dcp = f l i m d~p, p'-* l d

(3.10)

d p--* l

by the theorem of bounded convergence (T p. 345). From this it follows that, if eq. (3.8) holds true, #(~) = 0. Hence if in a particular problem #(~) 4: 0, this has to do with zeros of G (~)/B (~). Since G (~)/B (~) does not vanish in 5 , it must be zeros on the boundary [~[ = 1. We now proceed to investigate these. To this end, we consider the function B(~)/G(~). This is regular in ~ and belongs to 91, being the quotient of two functions in 91. Let us now assume that B(~)/G(~) also belongs to some class ~'~. Then, l i m f * l o g + [B(pei~')/G(pei~)[ dcp = f * l o g + [B(et~')/G(ei~')] d~o, p~ld

0

(3.11)

~ 0

by eq. (3.5). But log + IS (~)/G (~)1 = - log- IG (~)/B (~)l.

(3.12)

Hence in the present case #(~b) also vanishes, by eqs. (3.7) and (3.11). Conversely, if p(¢) vanishes, it follows from the converse of the factorization theorem that B(~)/G (~) belongs to some class 5")a. If G (£) has a simple zero on the boundary, say if in the neighbourhood of ~ = 1, it behaves like 1 - ~, the function B (~)/G (~) has a simple pole. But if this is its most

RESONANCE REACTIONS

141

serious singularity, it belongs to (36 (0 < 6 < 1), since ~"ll-pe'~'l-~dq~ < oo

(0 <

< 1).

(3.13)

Hence a simple zero on the boundary does not contribute to the function/~(~). On the other hand, it is not difficult to construct a function G (() for which #(~O) is not constant. It follows from the foregoing that to do this, we must consider functions in ~ which do not belong to any class ~n. As an example of such a function, Priwalow (p. 61) discusses exp [(1 + ()/(1 - ()]. This would correspond to B (()/G (~). Hence a function G (() with a non-vanishing/l(~k) is provided by exp [((+ 1 ) / ( ( - 1)]. This function is bounded uniformly in the circle I(I =< 1. Hence it belongs to every class ,~6. In particular, since it is in 3)a, it can be represented by a dispersion integral of the form (2.6) (P pp. 67, 68). At the same time, it has an essential singularity at = 1. If ( tends to 1 along a path in Wo,the function G(() tends to 0. But if I(I = 1 and ( 4= 1, we have [G(()I = 1. Hence even information we may have concerning unitarity and a dispersion relation does not prevent G(() from having essential singularities on the boundary. It is in part this consideration which made us find a resonance formalism without appeal to analytic continuation. On the other hand, we do not claim that the singularities encountered here seem to be of major importance in physics. When extracting the Breit-Wigner formula from the product expansion (3.1), we shall in fact assume that/~(~) is constant in the energy region considered. Hence, although the formalism developed here is completely rigorous, we shall make simplifying assumptions when applying it. The foregoing remarks may help in clarifying the nature of these. For future reference we also give the product expansion appropriate to the halfplane J . With the transformation (2.4) this can be obtained easily from eq. (3.1). The transition is straightforward except for the factor involving/~(¢p). For this we have exp F-l-f 2" e'~+( _ d/.z(rp)] L2ndo e' ~ = exp

F L2x(-]-S_¢) [t~(2r0-

= exp

[iaw

#(2~ - 0) + #(0 + 0) - #(0)] + 2--~30 +o eT* ~

apt.cp)J

sW+'dv(s)], -Qo

(3.14)

$~lq

where v(s) is a bounded non-increasing function and a is a real non-negative constant. Writing H(w) = G(O, we thus obtain

H(w) = e"C(w) exp [ iaw- ~f~o sw+l (s2_~_1 log ,n(s)l ds+½dv(s)) 1 -oo

t.,, + ~.J .=~

$--W

Iw.--il(w*-O(w.--w) Iw. + / l ( w . - O(w*-w)"

(3.15)

142

c . VAN WINTER

Since in the sum (3.2) (, # 0, it follows from eq. (3.2) that oo

~, (IC,I - X - 1 ) < oo.

(3.16)

n=l

If we now write w, = d, exp(ig,), eq. (3.16) entails that d2

(3.17)

sin ~, < oo.

n=l

For the following it is more convenient to write w, = - ½ + d ~ exp(i~,). With this interpretation of d~ and 9~, eq. (3.17) also holds true.

4. The Cut Plane We proceed to derive a product expansion for the function S(z), which is analytic in the z-plane cut along the real axis from - o o to - 1 and from 0 to oo, by eqs. (1.5) and (1.6). In the following the open cut z-plane is denoted by J . To discuss S (z), we map J onto the upper half w-plane J . It is convenient to choose the mapping such that the point z = 0 corresponds to w = 0. This is achieved by the transformation w2

z -- - - ,

l+2w

w = z+~/z(z+l),

(4.1)

where we define the square root in such a way that

4z(z+l) =

4- 4 1,

Imx/z > 0,

R e x / z + l > 0.

(4.2)

If w runs from - o o to - 3 , z runs from - o o to - 1 , encircles the point z = - 1 clockwise, and goes back to - o o . If w runs from - ½ to oo, z runs from oo to 0, encircles the point z = 0 clockwise, and goes back to oo. Hence there is a correspondence between the integrals f c - 1 - ) d z + J of(o-) ° dz,

f~o _ ,odw.

(4.3)

,#--oo

The interval - 1 < z < 0 is mapped onto the circle w = - 3 + 3

exp(i3), (Tr > 3 > 0).

z*-(4z(z+,l))*

I f z corresponds to w = z + x / z ( z + 1), then z* corresponds to in view of eq. (4.2). To be able to write down a product expansion for S(z), let us start by considering the function

r(z)=

f_ __

1 -1 dot(t) + t-z

lfo0 da(t)

(4.4)

t-z

We want to show that, when J is mapped onto c~, the function T(z) yields a function of ~ which belongs to some class &6. If T(z) yields such a function, we shall say for

143

RESONANCE REACTIONS

simplicity that T(z) belongs to ~ . Now a convenient criterion is that every regular function with a positive imaginary part belongs to the classes (0~ with 0 < 6 < 1 (P p. 64). Also, since a(t) is of bounded variation, it is of the form t r l ( t ) - a 2 ( t ) , where o1(0 and ae(t ) are non-decreasing. However, if z = x + i y , 1

-l

Im ~rf-~o d°'l(t) t-- z

y

-1

dcrl(t) (t-- x) 2 + yZ"

(4.5)

Since this is positive in the upper half of a¢" and negative in the lower half, the mere splitting of tr(t) into two non-decreasing functions is not conclusive. But let us consider the function TLt(Z) =

d i r t ( t ) - 1 w(w+ t- z rr

1)f-_[

d°l(t) ( l + 2 w ) t - w 2"

(4.6)

We know from the beginning of this section that if z is in a¢', the variable w is in J . Hence, if we write w = u + iv, we may restrict ourselves to v > 0. Now it is easily checked that the sign of Im TLI(z) is equal to the sign of (u 2 + v2)(1 + 20 + 2ut + t.

(4.7)

Also, if t < - 1 , (u2+v2)(1 + 2t)+ 2ut + t < - u 2 - v 2 + 2 1 u l

- i

< 0.

(4.8)

Hence TLI(Z ) has a negative imaginary part, so that it belongs to f~ (0 < 6 < 1). Likewise, since for t > 0, ( u 2 + v 2 ) ( l + 2 t ) + 2 u t + t > u2+v2+2t(u2-1ul+¼)+½t > O,

(4.9)

the function TRl(Z) = -rclx / z ( ~

f ° dtrl(t)t_z

(4.10)

belongs to $)6- The same applies to [z(z+ 1)] -½, the imaginary part of this function being negative. We thus see that T(z) can b.~ written as a sum of products of functions in .~3n(0 < 6 < 1), Hence, according to sect. 2, the function T(z) belongs to ~')~,(0 < a' < ½). To handle the poles of S(z), we now introduce the function K

U(z) = S(z)Q(z) = S(z) I-I Pk+X/Pk(Pk+I)--Z--X/Z(Z+I) k =1 Pk -- X/Pk(Pk + 1)-- Z-- X/Z(Z + 1)

(4. ll)

With eqs. (1.5) and (1.6), it is easily checked that this is regular in J . The product Q(z) is also regular itself. In particular, its denominator does not vanish at z = Pk, owing to the fact that at this point th ~. imaginary parts of x/Z(Z + 1) and x/P~pk + l i are both positive. When a¢ is mapp=d onto c~, the function Q(z) is transformed into a rational function of (. With the help of eq. (2.7) this is easily seen to belong to every class g)o. Hence T(z)Q (z) belongs to Ha,.

144

C. VAN WINTER

If S(z) satisfies the dispersion relation (1.5), the difference U(z)-T(z)Q(z) can be transformed into a rational function of ~ which belongs to every class .9~. As a result U(z) belongs to .9~, (0 < 6' < k). If subtractions are required, say if the dispersion relation for S(z) takes the form (1.6), U(z) is a function in the class .~, times a polynomial in z. Now z belongs to .~2~, (0 < 6' < ½). Hence z N belongs to .f-~';N. The conclusion is therefore that in any case there is some 6 such that U(z) belongs to .~n. Since U(z) is in some class .~3~,it allows a product expansion which can be obtained easily from eq. (3.15). In translating eq. (3.15) into an equation for a function of z, we write i

= exp

73

. . . .

[ -i-Y----z-[v(-i+0)-v(-½-0)]2n(2w + 1)

, - .,.

)sw+, dv(s)] s-w

--

= exp [-ibz+ib~/z(z+l)+i~']

x exp1- 2-~(f_ d2+(t)+j d2_(t))[t-lx/t(t+l)l][z+x/z(z+l)]+l] i

-1

-0

t-N/fit + 1)1- z - ~/z(z + 1)

[- 2~(f__-~d2_(t)+fo d 2 + ( t ) ) [ t + I x / t ( t + l ) l ] [ z + x / z ( z + l ) ] + i] i

x

exp

oo

t + Ix/tO+ 1 ) l - z - x / z ( z + 1) = exp [-ibz+ ibx/z(z+ 1)+ ia'l ' (f'-'-'+

r ' ° - ' ) [ t + , ' , ( t + ~)][z+,/z(z + ')]+' d2(t)] .

(4.12)

Here b and :~' are real and b is non-negative. The functions 2+ (t) and 2_ (t) are nonincreasing and non-decreasing, respectively. The sign of ~ / i ( ~ 1) on the various parts of the contour of integration follows from eq. (4.2). In the expansion of U(z), there is a term involving loglU(t)l, where t is on the cut in the z-plane. Now, since - 1 < Pk < 0, the product Q(z) defined by eq. (4.11) has modulus 1 on the cut. Hence from the expansion of U(z) it follows immediately that S(z) can be represented in the form S (z) = e ipR (z)exp [i(a- b)z + i(a + b)~/z(z + 1)] × exp

[ i(f_:'-' --

+

n

t + ~/t(t + 1)- z-~/z(z + 1)

× (\[(t + x/t(-~))' + l]x/t(t + I)log IS(t)Id't + ½d2(')]_ ,~ll .

.

.

.

.

(4.13)

RESONANCE

145

REACTIONS

the Blaschke product R (z) satisfying

R(-)

= fl Jz~_-F~/~_.{z#+ I)-- il[z* +(w/z.(z.+ l))*-i][zn+x/z.~.+ I)-z-\/z(z + I)] • / ~.(-.+ - - ~ . . . . l)-,][-o ' * + ( , - /. ( "=' I~.+,,"~.(=°+ I)+ ,I[~.+','

z+x/z(z+l)_i

z °+ 1 )) *

- z-,/.-(- + I)]

" ~ pk_,/pk(p,+l)_z--v/Z(--+1)

5. T h e

Symmetry

(4.1.4)

Property

Eqs. (4.13) and (4.14) can be simplified considerably owing to the symmetry property (1.1). This implies that, if S ( z ) vanishes at _- = z,, there is also a zero at -=-*.

At these conjugate zeros, ~ / z ( z + l ) t a k e s

the values \/'~-~iz,,+l)and -* is called - .... say. - ( . , - , ( z , + 1))*, respectively. In the notation used until now, .,, It is convenient, however, to relabel the zeros in such a way that those in the upper half-plane are denoted by z,. those in the lower half-plane by z,, and those on the real axis by qm(-1 < q,,, < 0). Since in eq. (4.14) the product I-I~= * converges absolutely, its factors may be taken in any order. With the new labelling of the zeros, it is particularly convenient to write it in the form

[,_ =+~_z(z+l>I[,-

z+~z(z+,> l

f l [,_ =+./z(z +,> l [,- z +,/z(.. +,> l el,,,,

.'1=1

[,_s=+Wu+,)i"

. xel~

'- --=+i-

•

z

[1-5.

]j

__ q.,_+>(q4q-.,~ ~), (5.1)

+ ~ ~(z-+-,>l' "'fi-'°;'' 1 - - + { Z (: .+--l)--2-;-J

q,,, V'q,,,(q,,+ l)

where exp(iy,), exp(i~), and exp(i?~,) are certain phase factors which can be deduced easily from eq. (4.14). The factors in the expression (5.1) which are raised to the /

.

power r take care of the zero conjugate to the one for which z + \ / z ( z + 1) = i. It is not difficult to show that the products f i e'~'.., n = I

f i e'~"-. m =

(5.2)

l

converge absolutely. If each factor exp(iyn) is expressed in terms of the zero w,, = - ½+ dn exp(ig,) and its conjugate, the convergence of the first product (5.2) follows

146

c. VAN WINTER

directly from eq. (3.17). And similarly for the second product. Hence these phase products may be taken out of the corresponding complicated products in the expression (5.1), and they may then be combined with the phase factor exp(ifl) in eq. (4.13). If this is done, it is no longer necessary to treat the zero for which z+~/z(z+ 1) = i separately. Together with its conjugate, we therefore incorporate it into the product I-I~= ~- We also split off a phase factor from the product 1--I~=1 in eq. (4.14). Summarizing, instead of R (z) we henceforth consider the product

[I-z+x/z(z+{_)_][l-

l

z:

l)rJ l If- z+x/z(z+l) l

= fi

.=1 [I-

z+x/z(z+ l)

z+x/z(z+l)

z +,/z(z +l)

1-

I-

z

+

1)

x fl

qm+X/q,,(q,,+l) ~-I Pk--X/Pk(Pk+I)/_ "=t 1- z+~/z(z+l) k=, 1- z+~gz(z+l) q,,-x/qm(q,,+l) pk +X/~(Pk + 1)

(5.3)

It is easily checked that P (z) satisfies the symmetry relation

P(z) = P*(z*).

(5.4)

From the fact that for S(z) there is a similar relation, it follows that in eq. (4.13) a = b. Furthermore, if S (z) is decomposed according to S(z) = P ( z ) V ( z ) ,

(5.5)

V(z) = V*(z*).

(5.6)

we have Since V(z) does not vanish, we can take square roots in eq. (5.6). This only introduces an over-all ambiguity of sign. Choosing a particular branch of the square roots, we thus obtain V(z) = +_[V(z)]~[V*(z*)]t (5.7) Together with eq. (4.13), this yields

V(z)

= + exp [ 2iagz-~--+1 ) - i ~ / z ( ~ -)

x exp I - ~ x/z(~+l)(f(_:l

(f(_11-)+ f : - ) ) -

+ f:

log [S(t)[

(t-z)#t(t+l)

[t+x/t(t + 1)32 + 1 ))(t-z)[2t+2x/t(t+l)+l]

dt 1

d2(t)]. (5.8)

Owing to the symmetry, [S(t)l takes the same value at conjugate points on the contour of integration. If follows from eqs. (5.6) and (4.13) that there is also a sym-

RESONANCE REACTIONS

147

metry relation between 2 + ( 0 and 2 - ( 0 . For let us choose 2±(0 = ½2±(t-0)+½2±(t+0)

( - o o < t < - 1 , 0 < t < oo),

4 + ( - 1) = 4 _ ( - I) = ½ 4 + ( - 1 - 0 ) + ½ 2 _ ( -

1-0),

(5.9)

4+(0) = 4_(0) = ½ 2 + ( 0 + 0 ) + ½ 2 _ ( 0 + 0 ) .

If s < - 1 , the theory of Cauchy's singular integral (Titehmarsh 20) pp. 28-33) then yields lim f - 1log

IV(x + iy)l dx

y--* 0 d s y>O

=

ffl

log IS(t)l

dt T

f-1 [tT-Ix/t(t+ 1)1]2 + 1 ix/t(t + X)[d2±(t). 2t-T-21~/t(t+1)1 + 1

(5.10)

And similarly for s > 0. Hence in view of eqs. (5.6) and (5.9), V(z)= _exp

2ia _ _ + 1 - - /

____

-

7g

x exp [ - i

x exp

x/z(-~)f-~

[t-]x/~t+l'l]2+l

(t-- z) l~/ fft + l)l d2+(t' 1

Ir (t-z)E2t-21x/t(t+ 1)1 + 1] [ - -i ~/z(z-v,)l ~fo~ [t+l~/t(t+l)[]2+l d2+(t) 1 ~ _ - - n

dtJ

.

(5.11)

d o ( t - z)[2t + 2Ix/t(t + 1)1 + 1]

Summarizing, the most general product expansion for S(z) compatible with the dispersion relation (1.5) or (1.6) and the symmetry property (1.1) is given by eq. (5.5), with P(z) and V(z) of the form (5.3) and (5.11), respectively. In these expressions, a is a real non-negative constant, 2 + (t) is a bounded non-increasing function the derivative of which vanishes almost everywhere. The above expansion does not apply to the scattering by a potential well of finite range. For the function S(z) relevant to this case does not satisfy a dispersion relation of tbe form (1.5) or (1.6). However, it is known 3,4.15) that, if the potential is sufficiently regular and vanishes at all distances larger than some finite distance A, the function SA(z) = S(z)exp(2iAx/z) is analytic in the z-plane cut from 0 to oo. It is easily shown that in this plane with one cut SA(z) can be represented by a product analogous to the one discussed above. The form of the product can be found from eq. (3.15) by mapping the upper half w-plane onto the cut z-plane according to w = ~/z. The product so obtained can then be simplified owing to the symmetry of SA(z). A further simplification arises from the fact that on the cut IS~(z)l = 1. Also, Sa(z) is analytic not only in the open z-plane, but also on the cut. Hence the singular measure analogous to 2 + ( 0 vanishes. If all this is taken into account, a product is obtained exactly of the form discussed by Van Kampen s) and Newton 15).

148

¢. VAN WINTER

6. Scattering at Low Energies In applying the foregoing results to a scattering problem, we assume that there is an interval 0 < t < c t in which ]S(t)l = 1 and 2 + ( 0 is constant. We further assume that if z tends to 0 along a path not tangent to the positive real axis, S(z) tends to 1. The latter assumption implies that the product P(z) has a finite boundary value P(0). It is clear from eq. (5.3) that if P(0) exists, it must be equal to I. Hence under the present assumptions we have the + sign in eq. (5.11). It is shown below that if P ( z ) has isolated zeros close to the interval 0 _< z < ct, these give rise to resonances in the cross section. In a certain sense P(z) therefore contains the compound features of the scattering. Compared to P(z), the quantity V(-) is a slowly varying function of z. It corresponds more or less to the potentialscattering term one encounters in other resonance formalisms. It describes a scattering process in which the real part of the phase shift is connected with the absorption through a dispersion relation. This is analogous to the situation one has in optics, where the real and the imaginary parts of the index of refraction are related to one another in a similar way. The present case is more complicated, however, owing to the fact that the phase shift also receives contributions from the "unphysical" cut - ~ < z__< - 1 . When Izl is small, V(z) ~ exp ( - 2iR x/z), (6.1) where R follows immediately from eq. (5.11). This formula is reminiscent of the potential-scattering term in the Kapur-Peierls ~) and Wigner-Eisenbud 2) formalisms. There is an important distinction, however. In the well-known theories R is the radius of some sphere inside which the scattering is supposed to take place. By the nature of this definition, it is not unambiguous, neither is the separation into potential and resonance scattering. In the present treatment, the product expansion is unique, and so is the constant R. The interpretation of R is less straightforward. It is clear that it is a scattering length which may have either sign. It must be remarked, however, that unlike the authors mentioned above, we find the same general expansion for each angular momentum. Now, in the case of scattering by a local potential which does not depend on the angular m o m e n t u m l, one m a y expect that for small [z[ the quantity S~(z)-I is proportional to z t+½. The corresponding function V l ( z ) - I does in general not have this behaviour. There does therefore not seem to be much point in trying to obtain the limiting properties of V~(z) from a potential model. More generally, the present approach is not particularly appropriate to describe scattering at extremely low energies. At somewhat higher energies, it might be possible to understand the general features of V(z) in terms of a potential. This point cannot be settled without further investigation. I f there is a potential, it will presumably depend on energy as well as on angular momentum. In the energy region where inelastic processes occur, it must have an imaginary part. In the following discussion of resonance reactions at low energies, we merely

149

RESONANCE REACTIONS

consider V(z) as a slowly varying function. Let us assume for simplicity that we are working in an energy region where to a good approximation V(z) is equal to its value at z = 0, which is 1. The cross section is then determined by P(z). Let us now make the further assumption that the number o f zeros q,. is finite, and that the zeros _-, belong to either of two classes. Firstly the class in which each Iz.I is so large that in the neighbourhood of the interval 0 < z < c 1

Iz+\/~f(z+

1)1 << I z . + x / z . ( z . + 1)l.

(6.2)

And secondly the class with z, = E . + ½ i F , .

0 < F~ << [ E , - E , . [

(6.3)

(n # n').

where E, and F, are real and it is assumed that the zeros of the second class are all simple. If the zeros are distributed in this way, the product P(z) tends to a well-behaved boundary value P(t) as z tends to a point t in the interval 0 < t < c t. To a good approximation, the zeros z. of the first class only contribute a factor 1 to P(t). If we restrict ourselves to t-values for which

t = E.+At,

x/t(t+

1)

__>0,

Izttl << I E . - E . , [

(n 4: n'),

(6.4)

the zeros z. of the second class approximately yield a total contribution P.(t) = [ l -

E-"-+At+[(-E-"+-AO(-E-~---I--At+I)-,]~---1 E. + kit. + [(E. + ½ir.)(E. + ½iF. + 1)]*l x [ 1-

E.+A_!+_[(E.+At)(E.+At+I__)]~ ]-' e . - ½Jr. +([(e. + ½ir.)(e. + ½ir. + 1)]~)*_i

(6.5)

We now assume that this is in fact a good approximation to P(t), i.e. we assume that ~o and l'-lk--1 in eq. (5.3) are apin the energy region considered the products l-I.,=t proximately equal to 1. Summarizing, it is assumed that in the neighbourhood of t = E, the behaviour of P(t) is entirely dominated by the zero z.. Expanding the square roots in eq. (6.5) and taking into account the fact that At and F. are small quantities, we obtain

2~/E.(E. + 1)

P (t) "~ n

Il+ 2x/E.(E. + 1)31 Il +

2x/E,(E. + 1~

t-E.-½iFn

1 ]

t - E. + ½ir~

(t ~ F~.). (6.6)

150

C. VAN WINTER

If this function P.(t) refers to the angular m o m e n t u m / , the cross section the approximate form

a,(t) ~

( 2 1 + 1 ) ~ [1-P.(t)[ 2 ~ (21+1)~

t

-~

2

t ( t - e . ) +Ir.

a~(t) takes

(t ,~ E.),

(6.7)

which is nothing but the Breit-Wigner formula• Hence under suitable circumstances the zeros of S(z) correspond to resonance peaks in the cross section. Special resonance shapes may occur if in the neighbourhood of t = 0 the behaviour of P(t) is dominated by a zero q., or by a pole Pk on the real axis. For instance, if there is an isolated zero q., with - 1 << q,, < 0, it is useful to write q,, = - E.,, and to consider

e.(t) = [ a -

t+x/t(t+l)

~I1_

t+~/t(t+l)

]-t

,,, i~/E.-~/t ix/Em+x/t If l = O, this function

P,~(t) gives

(6.8)

rise to a cross section of the form

4re %(0 g - -

t + E,~

(t ~ 0),

(6.9)

which takes its maximum at t = 0. For a discussion of this and related cases, the reader is referred to the paper by Humblet and Rosenfeld 4). If the potential scattering cannot be neglected, the result (6.7) may be improved by replacing P.(t) by an expression such as P.(t)exp(-2iR~/t), or by some other approximation to P.(t)V(t). Overlapping resonances can be treated with the help of products of the form P.(t)P.+ 1(0, say. It is expected that with these refinements a fairly good description can be obtained of resonances at low energies. As the energy approaches the threshold for inelastic processes, special effects may occur owing to the contribution V(z) receives from log IS(t)], cf. eq. (5.11). Above the threshold, the function V(t) may fluctuate appreciably. It is then no longer true that fluctuations in the cross section correspond to fluctuations in the product P(t). In other words, the resonances are no longer determined by the zeros of the function S(z). This is not surprising. For it is obvious that, in the energy region where more channels are involved, we want a whole scattering matrix rather than one single function. A discussion of this more complicated case will be given in a separate paper. It is a pleasure to thank Professor L. Rosenfeld for stimulating discussions, as well as for his active interest in the work which led to this paper. The investigation described here is part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie" (F.O•M.), which is financially supported bythe"Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek" (Z.W.O.).

RESONANCE

REACTIONS

151

The author is indebted to F.O.M. for making her stay in Copenhagen possible. The hospitality shown by the members of the Institute for Theoretical Physics is gratefully acknowledged. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

P. L. Kapur and R. Peierls, Proc. Roy. Soc. A166 (1938) 277 E. P. Wigner and L. Eisenbud, Phys. Rev. 72 (1947) 29 J. Humblet, M6m. Soc. Roy. Sc. Liege (8°) 12, no. 4 (1952) J. Humblet and L. Rosenfeld, Nuclear Physics 26 (1961) 529 R. E. Peierls, Proc. Roy. Soc. A253 (1959) 16 K. J. Le Couteur, Proc. Roy. Soc. A2.56 (1960) 115 Ning Hu, Phys. Rev. 74 (1948) 131 N. G. van Kampen, Phys. Rev. 89 (1953) 1072, 91 (1953) 1267 E. C. Titchmarsh, The theory of functions (Oxford University Press, Oxford, 1952) R. G. Newton, J. Math. Phys. 2 (1961) 188 K. T. R. Davies and M. Baranger, Ann. of Phys. 19 (1962) 383 Chan Hong-Mo, J. Math. Phys. 4 (1963) 1042 A. Martin, Nuovo Cim. 14 (1959) 403, 15 (1960) 99 D. I. Fivel and A. Klein, J. Math. Phys. 1 (1960) 274 R. G. Newton, J. Math. Phys. 1 (1960) 319 F. Riesz, Math. Z. 18 (1922) 87 R. Nevanlinna, Eindeutige analytische Funktionen (Julius Springer, Berlin, 1953) I. I. Priwalow, Randeigenschaften analytischer Funktionen (VEB Deutscber Verlag der Wissenschaften, Berlin, 1956) 19) K. Hoffman, Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, N. J. ,1962) 20) E. C. Titchmarsh, Introduction to the theory of Fourier integrals (Oxford University Press, Oxford, 1948)