R-Matrix theory of regge poles

ANSALS OF PHYSICS: 40, 197-220

R-Matrix

(190,6)

Theory

of Regge

Poles*,

f

Wigner ( 1 ) has discussed the analytic*ity of the S matrix as :t fu11c.tion of’ WIPI’~~ in a many channel problem using R-matrix tcchniqucs. Hc also intlic~:ttctl j ~2i that, the same techniques should be applicable t,o :I discussion of thr :Ltl;tly1 i(.ity ot the S matrix in the angular momentum. In this p:1p~, tllis progr:tn~ is c*:\rricbtl out, explicitly to obtain t,he Regge (3) behavior of t,llc S matrix. In the first section, t,he R-mat,rix formnlisnl for single (~h:~nn(~l sca:~ttctrirlg is rederived wit,11 the aim of studying the analyt,icit,y of the $ matrix in t 11~ :111gu1:11. moment,um. The second section contains specific results about t hc locdatiolr oi’ tlrcb Regge poles and about the behavior of thca trajc(~torics. III thr t,hirtl r;octiol\ t li(a results are generalized to a many channel situatiotl. The :~:dogy to the sillgl(k channel case is strikingly close. l~inally, the fourth stcdi ion (1~1s wit,h SOIIW of t lr(b limitat~ions of an R-matrix formulation. The results obtained are not new as far as potential S(XI tc>ring is (*o~l(*~~rn(~(l.’ The point of the paper is to indicat,e how the simple I?-matrix tec*hniclucs wllic*lr have been used to discuss the energy analyticity of t,he S nlat,rix can I,(> t1ircc.t 1~ applied to the study of the angular momeIltunl an:Llyt,ic*it.y, and to SW \vh:~t modifications must be introduced. JIoreover, 11~~ vcrsnt,ilit,y of the form:~lisIn i:: indicat,ed by its almost immediate appliaabili t,y to t hc many c~hannc~l WW. In many ways, the R mat,rix is simpler &an t,ll(l S mat-ris. Its :m:dyt8icL propc~ties are easily formulated and proved. It is real and symmetric for physical v:du(+ of energy and angular momentum. These are linear constraints whrlrt~as unit arit y is a nonlinear const8raint1. The complications in the analytics form of t h(> ,\’ matrix arise from the relat,ively complicat#ed analytic h(xhavior of Hnnkcl I’unc+iolls;. But. * Pa.rt of this paper was contained in a thesis a~~bmit~ted to the iM;tt.hcmatics I)t~p:lrtrnc~lit of Princeton Universit,y in partial fulfillment of the rcqlGrements for a r’h.1). tlcgwc~. i Work supported in part by the U. S. At,omic Energy Commission. 1 The results obtained in potential scattering, together with an castrtlsivc lisl or rc~l’~r ewes will he fonnd in ref. 4. 197

19x

ROSKIES

these are known explicitly. Thus, one of the strengths of the R-matrix formalism is to provide a representation of the S matrix in terms of an analytically simple function and explicitly known functions. It thus provides an excellent starting point for approximations. That is, by approximating R by simple expressions consistent with its general properties, one automatically obtains an S matrix which satisfies unitarity, symmetry, and has the correct analytic properties (5). The formalism is therefore very useful for phenomenological analysis. On the other hand, the R-matrix formalism suffers from certain difficulties. These are also explored at the end. I. R-MATRIX

FORMALISM

The R-matrix formalism was introduced and developed by Wigner and coworkers (6) to study the quantum mechanical theory of scattering and of nuclear reactions. The essential idea is the following. One divides configuration space into two regions, internal and external. The subdivision is chosen so that in the external region one has only stable complexes whose internal states are presumed known, and whose center-of-mass motion can be expressed analytically. For example, in the external region the interactions between complexes might vanish, in which case the centers of mass behave like free particles. (The only other case ever considered is a Coulomb interaction between the centers of mass.) All the complicated interactions occur in the internal region. The achievement of R-matrix t’heory is that one can characterize what occurs in t,he internal region by a matrix-the R matrix-which uniquely determines the S mat’rix. Moreover, the R matrix has certain simple properties which enable one t,o prove various results about t’he S matrix. The formalism is valid even when the interactions are not describable by local pobentials-one requires only that the interactions have a finite range. Thus the R-matrix formalism is more general than the usual formalism based on the Schriidinger equation with local potentials. We shall use the Schrbdinger equation to derive properties of the R matrix. Having found these properties, we will refer only to the R matrix, and not to the original equation, in deriving properties of the S matrix. In this section we introduce and examine some properties of the R matrix in the simplest two-body scattering problem-two structureless particles interacting via a spherically symmet’ric potential. We later assumethat the potential is of finite range, and derive the relation between the R matrix and the S matrix.’ The exposition is patterned on the development in refs. 1 and 2. A. DEFJNITION

OF THE

R

MATRIX

After separating the scattering problem into partial waves, the SchrGdinger 2 For two body scattering

by central forces, the matrix

reduces to a function.

R-AIATRlX

equatiori

hec.omes (setting

(

OF

J'(,),j

190

POLES

j

fi2/2m = i

d" P

REGGE

_ '(I2 + '1 + E +(7")= 0 9 >

i 1)

+(Tj = PP(I-)

izj

\vhere

aucl *I( I’ ) is the true wave function. The depeudence of + Ii:quatiou ( 1 ) cau be rewritteri as

on

E

and

1

is supprw~~~tl.

-cl’ - I- f Lw2 + E’v” c$= 0, ( (il.’ >

:; I

L = -Z(Z + I), w = l/l.,

! -1 )

2: = 1. l’hc only relcvaut property of w2 and v2 is that they arc positive defiuite operut,orx. whic*h therefore have positive definite square roots aud positive defiuitck iuvtrrsw. 13’is iutmrodtwcd iuto (3) to make E and L appear iu :I synunet~rical manner. ‘I’hosc~ propcrtics of t’he aolut,ion which one can prow iu the E pla11~ for fiscd I, sho~tl~l I hen also he valid in t,he L plane for fixed k’. I’or the nmueut, we concent,rate on the dependence ou 1:’ for fixed rml 7,. ff ottc dctinw H = t.lwti i :< ) mu he rewrit~teu

$

- T’(T) + Lur’

\ ., 1

= R(vf#‘,.

I ti 1

as (I’-‘Ho-‘)zk$

M:ca uow iutroduce the real functions the self-adjoint operator FIHT1

X,(~j

defined as ckw~teristic

fuuc+ons

ii1

( 2!-‘HU-1) vxx = &A-~ subject. to the boundary

conditions X(0)

for some n > 0. Multiplying

= 0

(6) by XXV, (7) by 4v, aud subtracting

X’xHd, - +HXx

= EX&p

- iT~$?Xx.

ot

gives

200

ROSKIES

Substituting (5) and integrating both sides of (9) from r = 0 to r = a gives, on recalling that o, w are self-adjoint operators on the interval [0, a], that Xh satisfies (8), and that +(T) vanishes at the origin, X~(U)&U)

= (E’x - E) 6’ (UX~>(U~> czr,

(10)

where d(a>

= &dr)

/?.=a.

(11)

The operator v-‘Hv-’ with the boundary condition (8) is presumably selfadjoinP; thus the eigenvalues Elh are real, and the eigenfunctions uXX form a complete set which may be taken orthonormal on the interval [0, a]. In particular, UC#J can be expanded in terms of the vX~ on this interval (u$) (1’) = F ax(vXx) (1’). Mult.iplying gives

both sides of (12) by (uX,) (,v) and integrating

s0 Substituting

a (vXJ(vc$>

(12) from 1‘ = 0 to ‘r = a

dr = up.

(10) in (13), and substituting

the expression

(u+)(r)= cAx;“y;) A

(13) for a, into (12) gives

(vXx)(r),

i.e., 4(r) Substituting

= c

A

x;“py A

XA(l.).

(l-1)

r = a in (14) gives (15)

where we have defined YX = &(a). We define the R matrix

(really

(1’3)

t’he R function)

by (17)

3 That

is, we consider

only

those

potentials

for

which

t.he operat,or

is self-adjoint.

In ( 15 1, the dependence of ye and fdi on t,he parameter I, has licstoring this dependence, one has that for real /I’ and I,,

heen

supprwst~l.

Tlw tleriwtion leading to ( 1.5) is valid provided: ( i ) F’H,l ’ with the boundary condition (8 1 has a pure point spr~trun~, ( ii ) tli:rt ( 14 ) holds pointwisc (not only in the 1 swsc of 2:” LO, a] wltic+ is all th:11 hns hem lcgitirnntely derived); especially, that i 14‘1 holds at 1‘ = a. I’or the tlwivution in tcrnw of E, these conditions xrc ratisfictd for :t wide ~~1:~ oi potcntji:tls.” Hut, in the case of I,, the spectrum of w- ‘I~w-~’ is not tliscwtc~. To tw n1orc praise, one must augment t,he suni in ( 20 ) by an irttc>gr:ll !“li In fact, iii the case T-(V) = 0, one c’an see esplic*itly that tlic wntiuuous t,hr spec*trunl of W-‘Hw? is the interval I?;, = 1. If l’( 1.) is less singul:w

pwt

01

than i’

L The properties of K functions were stlldied by \?‘iglrer (7). They had previl,rlsly IW~II stlldied hy Herglotz. See, e.g., Shohat and Tamarliill (8). ‘I For large energies, the true wave function approaches the wave function iI1 the :tbsr~~~cc~ of illtrract ion, since the potential becomes less significant. Thlls, for 1 = 0 and large X.

so (1.4) converges

pointwise.

202

ROSKIES

at the origin, the integral in (21) wiill still extend over those values of v for which L, lies in the interval [$i, 00 ). From (21), one sees that for fixed real E, R(E, L) can be defined for complex L to be analytic except for poles along the real axis and a cut from L = s/4 to L = 4,. Moreover, R(E, L) maps the upper (lower) half L plane into the upper (lower) half plane. The cut in the L plane is all that distinguishes the function defined in (21) from an ordinary R function. We will therefore call the function in (21) an R’ function. Thus, R(E, L) is an R function of E for real L, L < +i, and an R’ function of L for real E. Several additional properties of R(E, L) follow from the representations (18) and (21). Term-by-ternldifferentiation of (18) yields $ 1 NE,

L) = F

> 0

&$oB)2

for real

E and L.

(22)

Similarly, $ Since Ex(L),

R(E, L) > 0

TX’(L)

for real

E

and

(23)

L

are real for real L, we have R(E*,

L)*

for real

L.

(24)

= R(E, L)”

for real

E.

(25)

L) = R(E,

Similarly, R(E,

L”)

We shall use only those properties of R(E, L) implied by the representations (18) and (21). Since these properties also hold for situations which cannot be described in terms of a Schrodinger equation with a local potential, our discussion is applicable to these more general cases. C.

CONNECTION

BETWEEN

THE

R

&/IATRIX

AND

S MATRIX

Now we assume that the potential occurring in (1) is of finite range, and we suppose that the parameter a is chosen larger than the range. In the region 1‘ > a, the Schrodinger equation (1) reduces to the equation for a free particle. The general solution is a linear combination of ingoing and outgoing waves. We suppose that the waves are normalized to unit flux at infinity. The ingoing wave is denoted by s(k, 1, r), where k2 = E, the outgoing wave is denoted by r(k, 1, r) and we impose (for real k and I); s(k, Asymptotically

1, ?“) = &“(k,

one has (1. -+ 00 )

I, 1’).

(26)

R-MATRIX

OF

REGGE

203

POLES

9( k, 1, Y) -

c2”’

&(k, 1, I’) -

c*eik’

IX)

and t>hephase of c is chosen so that

vanishes at I’ = 0. (This assurest,hat in the absenceof int,cr:lctions, the S m:rt,ris reduces to t,he unit matrix. j The general solution of (1) in t,he external region is”

,~(li~)“‘HI::,,(l~~) + B(lzr)‘/‘Hl~~,*(kl,).

i L’8j

The combination Hi’) + HP’ is regular at the origin. The :rsymptot,i(a behavior of H(l) Y 7 HF’ is given by’

Using (26) and the requirement that &(k, 1, r) -

S(li, 1, r)

vanishes utj the origin, one obtains

The wave function in the external region is qH.7.)= S(k, I, 1,) - S(k, 1jc;tI,:, 1, I’),

I:;1 i

where S(k, I) is the S “matrix” (see footnot,e 2). It, is the coefficient of thn out-going wave if the coefficient of the incoming wave is unity. SCk, I) is chosc~ 50 that the solution (31) can be continued smoothly into the region I’ < a to a solution which vanishes at t’he origin. But a solution in the region 1’ < a is c~har:~ terized by the R matrix-for in t,he derivation of the R-mattrix expansion, W: used explicitly the condition that+(O) = 0. Thus there is :I relation Mwcetl the S matrix and the R matrix, which we now derive. By definition R(E,L)

=$.

! 17)

a

Thus from (31) 6 H”‘, Hi’) are the Hankel 7 scyc ref. 9, p. 198.

functions

of tht? first

xntl

srco~~tl

kitltl

tlcfi~~c~l

ill wf.

B

204

ROSKIES

R(E

1) = g(k I, a) - S(k, oak L a) 7 ’ s’(k, 1, a) - X(k, Z)&‘(k, 1, a)

(32)

1) = g(lc, 1, a) - g’(k 1, a)R(E, &(k, I, a) - &‘(k, 1, u)R(E,

(33)

or fi(k

7

L) L) .

We denote g( k, 1, a) by s(k, g’(k, I(&

1) or simply 9

1, a) by g’(k,

1) or simply g’

I, a) by &(k, 1) or simply

&‘(lc, 1, a) by r’(k,

&

(34)

1) or simply 8’.

Thus, X(k

7

z> = g(k

1) - R(E,

L)s’(k

1) = 9 - Rd

I@, 1) - R(E, L)z'(k, Z)

I-

RG'

(35)

which is the desired relation. Moreover, from (2(i), for real 1~and I g(k, 1) = r*Qc, 1) s’(?c, 1) = &‘“(k, Thus, for real k and I, since R(E,L)

(3(i)

I).

is real

X(k, Z)X”(k,

1) = 1

(37)

which is the unitarity condition. The analytic behavior of S(k, Z) in the lc plane for fixed real 1 is cont~ainedin Eqs. (35) and (18), because 4, g’, E, G’ are explicitly known. Thus (3.5) illustrates one of the simplifications of the R-matrix formalism-that S can be expressed in terms of an analytically simple function and explicitly known (although somewhat complicated) functions. II.

REGGE

POLES

In this section, we use the R-matrix formalism to study some properties of Regge poles. In particular we examine the location of the poles and the behavior of the pole trajectories at threshold. The following contains no new physical results, but it does demonstrate that the R-matrix formalism provides a convenient representation of the X matrix with which one can study analyticity in the angular momentum. The results

R-MATRIX

OF

REGGE

POLES

“05

presented here are not exhaustive, but, merely an indication of the &tn:tt,ris approach. One more point warrants mentioning. Regge’s work (3) was originally illI ended to justify the RIandelstam representation in potential scatJtering. A crucial asp& in this justification was t,he use of t.he Watson-Sonlnterft‘ltl transform, which allows one to relat,e the asymptotic behavior of the 8 mat8ris I’ot 1:trgc tnomettt,unt t>ransfers to the posit,ion of the Regge pole wit,h thtl largcbst, value of Re 1. However, such a relation 110 longer holds for potenlinls vanishing beyond a certain range.’ Thus, for t,he finite range potentials which we cottsitlcr, the original tuotivat,ion is lost,. However, the analytic: nature of the 8 matrix :IS :I, function of 1 remains, and is of interest in studying nnalytkity propcartics ill general. WC restrirt ourselves to pure scattering in which the angular momenfum is :I good quantum nutnber. Then t,he R matrix is diagonal in 2, and t>hc discussion is in terms of functions (one-by-one matrices) rather than matrices. Th(J n~:tn;v channel situat,ion is discussed in Section III. h.

hALYTIClTY

OF

THE

8

RIATRJX

Using (35), which relates the S makix to the ZZ matrix in t~~rtns of esplic*il,lg known funct.ions, and using the properties of R implied by ( al), OIIP cm disc.uss the behavior of S as a function of complex angular n~on~et~t~ut~~for real energy. For real 13, R is meromorphic in I; except for :I c*ut from I, = !$ t,o 1; = z . Recalling (a), one sees that R is meromorphic in 1 in the half plane Kc I > -- 15. Throughout the following, we shall assume that I lies in this half l&me. Since Bessel functions are holomorphic in their order for fixed argumeni,” :md since R is meromorphic in 1, it, follows from (3.5 ) that S is mcromorphic~ itr 1 fat fixed energy. 13.

LOCATION

OF

THE

POLES

We shall establish that, for physical poles of 8 occur only if a)

k > 0

0)

k =iq

energies (1~ > 0 or Ik = 1’9, 11 > o 1. ~II(‘ and and

Im 1 > 0 Zrenl

The poles of S( k, I) occur either at the poles of the numerator of ( il.3) or :\,t t)he zeros of the denominator. The numerator has poles at, t.he poles of K(E, I, ). But. the location of the poles of R depends on the radius a of the sphere dividittg t,he internal and external regions, whereas the poles of the S nl:tt,rix arc int,rittsitally defined, and are t’hus independent of a. If by accident, the poles of K :u~d ,Y 8 See

Itef.

4, p. 94.

9 see

Ihf.

9, p. 11.

206

ROSKIES

coincide, one can change a slightly, and remove the coincidence.” Thus we need concern ourselves only with the vanishing of the denominator (& - R&I). Since E, considered as a function of r, satisfies the equation

--a2

(39)

dr2 with

& = &(k, I, a)

(40)

6' = g E(k, I, r) (T=a = E'(k, 1, a>

both E and E’ cannot vanish simultaneously, because otherwise I would vanish identically as a function of r. Thus, at a zero of (8 - R&I), 6’ does not vanish, and SO the zeros of (& - R&‘) coincide with the zeros of (&&‘-I - R). We shall now show that (&&l-l - R) can vanish for real E and Re Z > -x only if (38) is valid. We shall establish (38) by showing that if the conditions are violated, the imaginary part of (I&‘-’ - R) does not vanish. If k is real and I is real, then R is real. But 1&‘-l is not, for, Im E&‘-l =

EC’

- g&l

2i

=

&l-y Ed - iJ&‘)P 2i

(41) = --fkl&J-2

< 0

where we have used Wronski’s identity E4’ - g&’ = -2&/a. Now suppose E = k” is real, and Zis not real. I1 The complex conjugate of (39) gives

d2 z*(z*r; l) + k2 &*(k,E,9”)= 0. > ( dr2Alultiplying

(39) by E*(k, I, r), (42) by ~(k, 1, r), and subtracting

& [ E”(k,4r) ; a, 1,r>- UC,I,r) f E”(k,1,r)

gives

1

= [Z(Z + 1) - I*@* + 1>1 ’ G(lc;.;7 r, I2 this would be independent of 10 At zero energy the poles of R amd S might coincide; radius a, since a occurs in the combination ka. Thus if k = 0, changing a will not change value of ka. 11 The following analysis amounts to applying the R matrix results to a solution of p defined in (44) is wave equation between the points T = a and T = b. The quantit,y analogue of the R matrix.

the the the the

R-MATRIX

OF

REGGE

207

POLES

or, integrating

both sides from T = a to 9’ = b(b > a) gives

lh,

I,

1,

b)&*'(k,

b)

-

-

&"(k,

1,

&(lc,

b)&'(k,

I, a)&*‘(k,

1,

b)

I, a) =

+

&*(k,

I, a)&‘(k,

1, a)

[1”(1” + 1) - /CL + l)]

1b w’-‘?

(33) cl,.

Defining p = &(k, I, b)E*‘(k,

1, b) - G(k, I,

I, a),

a)&*'(li,

(13 1 sho\vs tmhat sgn Ill1 p = -sgn !sinw

Re 1 > -!z).

From

In1 1 = sgn 1111I,

t-L-ii)

(44),

&(k, 1, a) = &(k, 1, b)&“‘(h;, I, b) 1, a) 1&‘(k, 6, a) I2

i~F(k,4, ___--- a) j’? ’

E’(k,

i-i(i)

But for large b, the form of &(k, Z, b)&*‘(k, 2, b) is given by (20) and (30). Thus, for real k and large b, from (46), (29), and (30) ia(l*-/)/2 &(l;, I, a) 2&e b4i) ____-- !t:‘~j&qi * E’(k,

1, a) -

-4

G’(k,

I, a) j2

Thus, &eaIm’ Inl &(k, 1, a) N -~~~- ---._ I’@,

1, a)

7r j &‘(k,

Im p

-

I, a) 12

/ G’(k,

1, a) [2

!-4X)

If sgn k = sgn Im L, then by (45), (48) sgn Im

fG, 1,a> &‘(k, 1, a)

=

-SsgIl

Im L.

i4!))

But,, for real ,F:, from (21) sgn Im R(E, L) Thus (&&‘-I

= sgn Im L.

- R) cannot vanish for real k unless sgn k = -sgn

Im I, = sgn Im b

which is condition cr of (38). Throughout the following, let q denote a positive number. If k = in, ( 29) :~nd (30) imply that both &(k, 1, b), &‘(k, 1, h) vanish exponentially as b ;tppro:lcahcs infinity. Thus, from (4.5), (36) sgn Im

G, 1,a) = -sgn Im p = -sgn Xm L, E’(k, 1, a)

( 30 j

208

ROSKIES

and so by the same argument as above, there can be no poles unless Im L = 0. This establishes condition /3) of (38). Thus for physical values of the energy (k > 0 or k = iv), we have restrictions on the possible poles of S(k, 1). For fixed k = ill, the poles of S(k, I) in the 1 plane are simple. We have already seen that such poles occur only for real values of I, when (51) vanishes.

But for real E, 1

except at the poles of R(E, L) which, as we saw before, can be separated the poles of S( k, I). We shall show that

a ~(k, I, d < o z &‘(k, 1,a)

from

(52)

for real I, k = iq. Then

and so the poles are simple as a function of Z for k = ir]. To establish (52), we recall that for k = iv, ei’s’2&(k, I, Y) is real and drops exponentially12 as Y + cc,. We can rewrite (39) as -$ -I- E + $ Differentiation -$

Multiplying

ei1r’2&(k, 1, r) = 0.

with respect to L gives + E + g

>

$

(eiz”%(k,

1, r)) + Jj eiiri2&(k, I, r) = 0.

(54) by

a i'-'"&( k, I, r), ze and (55) by eila’z&(k, I, r), subtracting and then integrating from r = a to r = 00, recalling that eiLn/2&(k,I, r) I2 See ref. 9, p. 78.

(55)

R-MATRIX

20!)

OF REGGE POLES

is real and drops exponentially at infinity gives

a i'*'2&(k, I, a) zd eit*'2&(k, 1,a) z" / [ ( But, the second facetor in (3)

I,a)1*< (56) )1[ i'a'2&(k, $ e

0.

is posit’ive, and d/dn commutes with ei’s’2, so that

which is (52 ) . Exactly t,he sametype of proof establishes t,hat for fixed I, the poles of S(b:, I) on the positive imaginary k axis are simple. (I. ~'OLE TRAJECTORIES

We consider 110~ the trajectory of the poles; that is, the locus of the poles of S(k, 1) in the 1 plane considered as a function of k for real k’. The results are tl1a.t for k = iq(E < 0), Z(k) is real and increases as E increases. At threshold (E = 0), the poles leave the real 1 axis and go into t,he upper half plane in :I direction determined by lo = 1(O). The direction of t)he trajectory as :t fun&on of /,I is tabulat,ed in Table I. First we take k = io(E’ < 0)) and consider Z(k) as E increases. By (33/3) , 1 i:: real. The poles of S(k, 1) arise from the vanishing of G(.k, 1), and t.heir locus is given by G(k, Z(kj) = 0.

( 57 )

Different,iation with respect to E’ gives

d(k) __ CL!3 = -& G!k, 1) / ;, But from (S),

G(k, 1).

if 1 is real and k = is

&i G(k, I)

< 0,

i xi)

and an entirely similar proof shows that aGG(k,l)

< 0

so that i .?!I) (since clL,/dl < 0 for Re 1 > -$$).

Equation (591 is t’he familiar result that if

210

ROSKIES

the angular momentum increases, the binding energy of the “corresponding states” decreases. As E increases past threshold (k becomes positive) the poles can only occur if Im I > 0 (38~~). That is, at threshold, the trajectory moves into the upper half I plane. We now investigate the angle at which the trajectory leaves the real axis at threshold. If Z(k) is the pole trajectory, we suppose that as k -+ 0, E(k) -+ 20, where 20 is real and lo > - 35. The equation for the pole is

(57) We expand the right side in powers13 of k and the left side as

R(E,L)

= a(L) + P(L)E + ***

(60)

a(L) = R(O,L)

(61)

where

and so

Similarly,

since for real

& (Y(L) > 0.

(23)

& R(E,L) > o.

(22)

E, L

/3(L) > 0 for real

L.

(62)

A straightforward computation yields the results shown in Table I where all Ci > 0 and the fourth column in the table indicates the angle which the trajectory makes with the positive real 1 axis at threshold. III.

MANY

CHANNEL

FORMULATION

The R-matrix formalism generalizes readily to the many channel case. In fact, this versatility is one of the formalism’s chief assets. We shall show that the many channel S matrix is still meromorphic in I for real E (always assuming Re 2 > -$d), and shall also derive results on the location of the poles. In close analogy to the one channel case, one finds that for the energy below all the thresholds, the angular momentum of the poles is real, whereas if the energy is above all thresholds, all poles lie in the upper half 1 plane. In fact, if Sij is the S I3 See ref.

9,

pp.

40, 74.

R-MATRIX

OF

REGGE

211

POLES

matrix clement between channels ,i and j , t.hen a pole in S,i can occur for real 1 only at energies at which both channels are closed. A. THE

SCHR~DINGER

EQUATION

What is the correct generalization of the partial wave Schriidinger equation in :I many-channel problem? Of necessity, we shall ignore all cahannelswith more than two stable complexes, and shall also assumet#hat each stable complex has no spin.14Even so, the problem is not simple. For the discussion of analyticity in the energy (1 >, one introduces a sphere in configuration space outside which only known, ,nor&eracting complexes exist. However, within t,he sphere, the interactions are t,reated exactly. The approximations come in ignoring the cont.ribut.ions of the many body channels and the closed two-body channels t#othe solution outside the sphere. In the case of angular momentum, the problem is much more difficult, since ill an exact t.reatment of interaction between spinlesspart’icles, the parameter 1does not’ appear in the Schrbdinger equation in a simple manner (11). As a result, most authors (1%))have been forced to use a heuristic Schr6dinger equation

+ 1) --Tjcl2+ 10 7 drr

k” + V d(r)

= 0

cp(r) is a column vector whose ith row is the wave fun&on describing t#hesepnration of complexes in the i channel; k2 is a diagonal mat,rix whose ,ii element is (2~,lF?) (E - E’i) with ~1~and E; respectively the reduced mass and threshold energy in the i channel; r is a diagonal mat,rix whose ii element, is pi , the sepnr:ttion of the complexes in the i channel; V is a real, self-sdjoint operator. We shall also make t’his equation our starting point, but we wish t,o emphasize t#hat it, represents a modification of the SchrBdinger equation. The equation (*an be derived as follows. Suppose there are N channels, each corresponding to a different set of two partichles. Each c*hannelis ort(hogona1 to c&veryot,her--not in the sensethat two wave functions in configuraton space are ort,hogonal, but rather in the sense of elementary parMe physics where, for esample, a ?r+ is orthogonal to a IX+, because the T+ and I? are states in two cl (fwent

spaces.

212

ROSKIES

Then one can write the over-all wave function in the center of mass frame as

The free Hamiltonian in this frame is the diagonal matrix

Now one introduces a potential operator V insisting that V conserve angular momentum, and that V be a real self-adjoint operator in the large Hilbert space considered. Moreover, in analogy with the single channel case, we assumethat V;j = 0

if

l”i > a or

?'j

> a

for some a > 0. No other assumption about the form of the potential is made.15 One then expands each *Jr,) as a linear combination of states of definite angular momentum, and since V conserves angular momentum, only those components of the qi(ri) with the same angular momentum will couple. One is then led to (63). The condition that the stable complexes have no spin is more reasonable if the different channels correspond to different elementary particles rather than to different configurations of the same elementary particles. In nuclear reactions, where the number of neutrons and prot,ons is a constant, but they are rearranged in different configurations, it is very artificial to assume that the only stable nuclei to consider have no spin. On the other hand, for the elastic and inelastic scattering of the pseudoscalar mesonsin the nonrelativistic limit,, (63) should be a reasonable approximation. B. THE

R MATRIX

Starting from (63)) we define the real column vector r-IX, and the numbers LX to be the eigenvectors and eigenvalues of d2 r ~~ r + rk2r - rVr

(64)

with the boundary conditions

15 One usually of ref. 12.

assumes

a superposition

Xx(0)

= 0

X,‘(a)

= 0

of Yakawa

potent,ials.

See the

first

two

papers

R-MATRIX

where Xx(O) ponents

OF

REGGE

denotes XX(Y; = 0) and Xi’(a)

A derivaCion precisely 1’) gives

2 1:3

POLES

is the column vector

analogous to that given in (9)~(,17)

(with

with

com-

w replacing

R(E:, I,) is a real, symmetric N X N matrix where N is Dhe number of channels. AS in (21), the sum over X should be supplemented by an integral. Moreover, given any vector y z 0, independent of E and L, the function ( 68 1

Y+R(E, L)Y has all the properties of the R function of (21). c.

THE

8

31.4~~1~

The S matrix is given by (1) S = (8 - RI’)-‘@

- R$‘)

( (i!)‘i

where 6, E’, 4, B’ are diagonal matrices whose entries are the elements 6, c’, 9, g’, defined in (30), (34) for the particular chanrlel in question. Since &, G’, g, g’, are holomorphic in I for fixed E while R is meromorphic in 1 ( for Ke 1 > - j/s), S is a meromorphic makix funct.ion of I. For real I and physical energy, S is symmetric. But symmetry is preserved under analytic continuat,ions, and t,herefore S is symmetric for complex 2 as well. D.

I~OC~TION

OF

THE

REGGE

T'OLES

The poles of S for fixed real energy occur for those values of 1at, whkh G = EE’-’ - R

(70)

has zero as an eigenvalue. If one denotes the diagonal elements of EE’-’ by r, , then for Im I, > 0, and physical energy, (49) and ( 50) establish Im ei < 0. IF,T denotes transpose;

t denotes adjoint,.

214

ROSKIES

Thus, for Im

L > 0 Im ytGy = Im( Tgi*

so that G cannot have an eigenvector the poles of S occur only for Im

eiyi - y+Ry) < 0

of eigenvalue

L 2 0, i.e.,

zero for Im

(71)

L > 0. Thus,

Im .Z 2 0.

In a similar manner, one establishes that for energies above all thresholds, there are no poles for real I, whereas for energies below all thresholds, all poles occur for real 1. Just as in (53), one shows that for energies below all thresholds, and for real I,

$

y+Gy < 0

(72)

so that the poles are simple. This does not mean that G cannot have two zero eigenvalues for the same E, I, but rather that if Zi(Zc) describes the location of the poles, then for Z near Zi , the eigenvalues of G go to zero like (1 - Zi) and not like (1 - Zi), for some m > 1. Finally, consider the location of the poles when the energy is above some thresholds and below others (13). Suppose a pole occurs for real angular momentum. For positive ki and real I, (41) shows that Im ei < 0. Thus, we can write

G in block form (73)

where the decomposition corresponds to open channels and closed channels. a, b, c, d are real matrices, b is diagonal with positive elements, and a, c are symmetric.

Suppose 5 = 21 U is an eigenvector 0

of G of eigenvalue zero. Then

(a - zb)u + dv = 0

(74)

dTu + cv = 0.

(75)

Multiplying (74) by ut from the left, and the adjoint of (75) by v from the right and subtracting gives u+(a - zb)u = v+cv. Equating

real and imaginary

parts, using the Hermiticity

u+bu = 0.

(76) of a, b,

C,

gives (77)

215 But b is positive

definite, and so ?A=0

1ix )

ThUS

[= 09.

1791

Also

-$ (+G(L)t

= v+ $

I

v < 0

i SO)

by the same argument that established (72). Thus all poles whicaho(~ur for real I and real E are simple. One can characterize the S matrix near t)he pole by means of the following lemma proved in the appendix: LEMMA: If G(I) is a square matrix whose elements are meromorplric~ill I, anti if G(Zo) has one and only one zero eigenvalue which is simple, then near 1 = i,, , one can write G-1(&j

= - aiPi + terms regular in 1 near I,, 1 - lo

where clliis the eigenvector of G(Zo) belonging to the eigenvalue zero, and /Yj is the eigenvector of the transposed matrix GT( lo), belonging to t,he sanle eigenv:rlllrl. If G is symmetric then one can choose o( = /3. We apply this lemma for the case

assuming that there is only one pole at 1 = 10for the given 8. Because the eigenvector E has components only corresponding to the c*lost~1 chamlels, the singular part of G-’ is

(&-’ - R)-’ E

(41)

But S = E’-‘(E~‘-~ - R)-‘(g&’ = El-1 ( g&l

_ R)-‘(&-’

= s&‘-‘8’ + s&‘-‘(&-’

-

Rjs'

_ R + gg’-’ - &‘)g’

( X6) i j

_ R)-‘($g’--1 _ &‘)g’

and since g, 8’, E, 8’ are diagonal, only the S matrix elements connecting caloseti channels are singular.

216

ROSKIES

IV. DRAWBACKS

OF THE

R-MATRIX

APPROACH

We now discuss some drawbacks of the R-matrix method, considering only single channel scattering. Some of these were pointed out by Wigner (1, 2). The first is the usual difficulty that R is not intrinsically defined but dependson the parameter a, which is largely arbitrary. Equation (35) displays a parametrization of the S matrix in terms of explicitly known functions and the R function which is meromorphic in E: for real 2.This by itself already accounts for the multivaluedness st’ructure and some features of the threshold behavior of S(lc, I) as a function of complex energy (1/t). If, in addition, one says that for real I, R(E, L) maps the upper (lower) half R plane into the upper (lower) half plane, and that for real E, it does the same to I, (for Re 1 > -$h), one can derive far-reaching consequemes (14) concerning the position of the poles of S(k, I). Since these results hold for any nonzero choice of a, one might hope that in the limit a + 0, or a + a, the samestructure would be retained. But this is not so, since in these limits the Bessel functions change their analytic structure. SO the only way to overcome the arbitrariness in a is to find some fundamental unit of length (or of energy) in the problem. Another serious drawback in the R-matrix formalism is the difficulty in discussing the behavior of S(& 1) as either E + ~0 or 2 + 00. One knows, from a different approach to the problem (4), that under wide conditions on the potential, the S matrix tends to the unit matrix as E -+ 00. That is, in some sense, R(E, L) approaches the free R matrix (the R matrix in the absence of interactions, denoted by Rf) for large E. But the sensein which this is true is quite complicated since the expression R(E, L) - R,(E, L) has poles for arbitrarily large E’, and so cannot approach zero. We rewrite (35) as

S = (G - R1')-'(s

- Rs')

= (E - RI')-I(9

- Rfd - Rs' + Rfs')

= (8 - RI')-'@

- R,d

= (E - RI')-'[(r

- RE') + R&i

= 1 + (6 - R&')-l(R

- Rd + Rf,'

- R,)(r'

- RI'

- I')

+ RI')

(83)

- R(s' - &')I

- 9')

where we have used the relation 1 = (E - R,&')-'(s

- R@').

(84)

As E + m, the second term in (83) must approach zero. This term appears to have poles at the poles of Rf , but at these points E’ - g’ = 0. At the poles of R, the second term is .d

T

E

E’

-1 - 2i

sin

[ku

-

(1 +

1)rr/2]ei”“ei”+“““.

R-MATRlX

OF

REGGE

POLES

If E:,,f is the position of the nth pole of Rf , we have for large II,, E %f -

.n2.

It also t,urns out that for the true j Is’,~ - l~‘,,~ ) is bounded, and so ( fi

-

R matrix

42;

for rc::~~on:~bl(’

/ --:’ 0.

Sincfio for large l?‘,,f sin [dEnIa it follows

-

(Z + l)ajL)I

- 0,

-

(1 + l)a;i:!]

- 0

t,hat for large E, sin [dza

(G - R&‘)-‘(R

- R,)(s’

- <;I)

contrives to approach zero, because although CR - I?,) h:w poles, 11los;c~thub lo the poles of RI are cancelled by the vanishing of is’ - 8’ 1, thosch tluc to the l)olcs of R are vancelled by (& - R&I)-I, and the term (9’ - f:’ ) :ISS;\~TS th;lf, 111~ rcsiduex get, small. Anot’her serious drawback is the difficultmy in dkussing the :m:~lytic bch:ivior of S(k, 1) as a function of two complex variables. WC have hitherto restric*lc>d our attention to t)he case of real E or of real L. To show that one MI c*ontinuc% I? to a meromorphic function of t’wo complex vnri:lhl(bs tloc*s not s:c(~m easy by llsitlg only “R-matrix techniques.” What one can say is the following. We have shown in S(1c.tiotl I thaw ii’ otI(’ writes the S(~hrijdinger equation as

( where Ph is I positive then

$ + w + XPh$(A,1,)= 0, >

definite operator,

TV is sclkuljoiut

44% w4’0, is an R function

(or R’ function)

(

( !N) i , :ttld x is :I p:lr:mlcbt (lr, I!)1 )

a)

of A. The S&riitlingcr

ctciu:ttiorl

d"

&3

can be rewritten

-

v

+

L/r'

+

4

E

=

()

>

as

Cl2- v + aI + (l-c- &I 1 + &-J i ( dr2 [

I)(b= (1

is)

21s

ROSKIES

where R0 is a real number. Thus, if W=Eo-v X=E-EO

L = c(E - Eo)

c > 0,

then EC%‘, L)

= 4(E, L, 4/&E,

4 a>

(94)

is an R function (or R' function) of (E - E,) for fixed c, i.e., R(E, c(E - Eo)) is an R function (or R' function) of (E - Eo) for fixed c > 0. This allows both L and E to become complex providing L/(E - Eo) is real and positive. For arbitrary real Eo , this allows a definition of R(E, L) in the product of the upper half L and E planes, or in the product of the lower half L and E planes. Since it is an R function of (E - Eo), it follows that R(E, L) maps the product of the upper (lower) half E and L planes into the upper (lower) half plane. But such tricks do not seem suflicient to establish joint meromorphy in E, L without explicit assumptions about the behavior of R(E, L) at infinity. However, the result that R(E, L) can be extended to the product of the upper (lower) half E, L planes, and maps them into the upper (lower) half plane, has consequences for the S matrix. It can be established, by arguments similar to those leading t’o (38~~, /3), that for Im k > 0 and Im L.Im E > 0, the sign of Im &(lc, I, a)&‘-‘( k, I, a) is opposite to the sign of Im R(E, L). If one continues to represent S(k, 1) by (35), it follows that there are no poles on the physical sheet if Im L* Im E > 0. This agrees with the result of Bottino et al. (lb). It is interesting to note that if the potential is a positive definite operator, then one can discuss analyticity in the coupling constant using precisely the same techniques. The results which one obtains with an R-matrix formalism are often weaker than those obt’ained from usual potential scattering. (For example, it has been possible to show that generally the line Re 1 = -x doesnot represent a natural boundary beyond which one cannot discussthe analytic nature of the S matrix (4). ) One obtains stronger results in potential scattering becauseone can use the very powerful techniques of ordinary differential equations, whereas in R matrix theory, one imposesweaker constraints on the potential operator-just sufficient to assure that the Hamiltonian with certain boundary conditions is self-adjoint on a finite interval. Weaker results are the inevitable price of such generality. However, these weaker results can be expected to be valid in caseswhere one does not have a local potential. APPENDIX

Suppose G(Z) is meromorphic in 1. Then the eigenvalues of G(Z) are analytic in 2 except for poles and branch points. If there is only one zero eigenvalue of G( lo)

R-MATRIX

OF

REGGE

‘l!)

POLES

and this zero is simple, then near 1 = lo one ran find an Z-dependent similarity transformation U so that U-‘G(Z)U

zz

f(z) (

G’)

where, near 1 = ZO f(Z) = ~(z - Z,j + O(z - lo)”

a#0

and where G’ is an (n - 1) X (n - 1) matrix invertible at 1 = lo. Then

1

and the only singular terms in G-l near 1 = lo are

with

U%)G(Zo)U(Zo)

=

(”

G’Cl,,,)

so that ZYil(ZOjis the eigenvector of G(Zo) with eigenvalue zero. l’ransposit ioli shows that ij L:-‘)l,(Z,) is the eigenvector of GT(zoj with eigenvalut: zero. CJ.lLlJ. This is the origin of the factorization of the residues of the 8 matrix at a pole (16). Xear a pole, one assumesthat S’ has exactly one zero eigenvaluc. Then the lemma proves t’he factorization and the symmetry of S me;ms that one wn choose(Y; = pi. ACKNOWLEDGMENTS

I would like to thank Professor E. P. Wigner for having suggested this investigat,ion, for many enlightening discussions concerning it. I also wish to thank Dr. I. K&on

:tr~d For

220

ROSKIER

having read the manllscript and suggesting several improvements in presentation. I wish to acknowledge the generous support of the Oak Ridge National Laboratory academic year 19(i4-65 during which most of this work was carried out. RECEIVED:

Finally, for the

April 25, 1966 REFERENCES

1. E. I-‘. WIGNER, Causality, R matrix and the collision matrix. In “Dispersion Relations and Their Connection with Carlsality, Scnola Internazionale di Fisica “Enrico Fermi,” Varenna,” E. P. Wigner, ed. Academic, New York, 1904. 8. E. P. WIGNER, Review of collision theory. 111 “Energy Transfer in Gases, Solvay Institute 12th Chemistry Conference,” It. Stoops, ed., p. 211. Interscience, New York, 1963. 3. T. REGGE, Nuovo Pimento 14,951 (1959); T. REGGE, Xuovo Cimento 18,947 (1960). 4. R. G. NEWTON, “The Complex j-Plane.” Benjamin, New York, 1964. 5. C. B. I)UKE ,INL) E. P. WIGNEH, ZZezl. i\focl. Phys. 36, 584 (1964). 6. L. EISENB~JD AND E. P. WIGNER, Proc. A’atl. =Icad. Sci. U.S. 27, 281 (1941); Phys. Rev. 72, 29 (1947); E. P. WIGNER, Phys. lieu. 70, 15 (1946); ibid. 70, GO6 (1946); T. TEICHMANN, Phys. Rev. 77, 506 (1950); ibid. 83, 141 (1951); T. TEICHM~NN AND E. P. WIGNEH, Phys. hv. 87, 123 (19%). 7. E. P. WIGNEX, flnn. Math.. 63, 36 (1951). 8. J. A. SII~HAT :\NI) J. U. T.\MARKIN, “The Problem of Moments,” p. 2-l. dul. Math. Sot. Publ. (1943). 9. G. N. WATSON, “Theory of Bessel FIII~C~~OUS,” 2nd ed. Cambridge Univ. Press, 1945. 10. J. M. CH,~HAP AND E. J. SQUIXES, =Inn. Phys. (N.Y.) 20, 145 (1962); ibid. 21.8 (1963). il. A. K. BH,\TI.I AND A. TEMKIN, Phys. Rev. 137, A 1335 (1965). 12. A. M. JAPFE 1\~~ Y. S. KIM, Phys. Rev. 127,226l (1962); L. FAVELL.~ AND M. T. REINERI, iVuovo Cimento 23, 616 (1962); R. G. NEWTON, Ann. Phys. (Ar.IY.) 4, 29 (1958); L. FONDA, A’uovo Pimento 13, 956 (1959). IS. I’. KAUS, P. NATH, ;~ND Y. N. SHIVASTAVA, Phys. Rev. 138, B726 (1965). 14. It. ROSKIES, Thesis, Princeton University (1965), unpublished. 15. A. BOTITINO, A. M. LONGONI, AND T. REGGE, A’uovo Cimento 23, 954 (1962). See p. 984. 16. M. GELL-MANN, Phys. Rev. Letters 8, 263 (1962); J. M. CHARAP AND E. J. SQUIRES, Phys. Rev. 127, 1387 (1962).