- Email: [email protected]
Mandelstam representation for the nonrelativistic many channel problem
ANNALS
OF
PHYSICS:
12:
Mandelstam
68-85
(1961)
Representation Many Channel L. FONDAt Institute
for
for the Nonrelativistic Problem*
AND L. A. RADICATI$
Advanced
Study,
Princeton,
New
Jersey
AND T. REGGE Palmer
Physical
Laboratory,
Princeton
University,
Princeton,
New
Jersey
In this paper we investigate the analytic properties of the transition amplitudes for the collision of two particles each capable of a number of excited states and interacting through a set of generalized Yukawa potentials. Double dispersion relations for each of the amplitudes can be derived only when the reduced mass is the same for all channels and a certain stability condition, which ensures the absence of anomalous thresholds, is satisfied. The Mandelstam representation is then discussed in connection with unitarity and the partial wave expansion is obtained with results similar, to the one-channel case. I. INTRODUCTION
The validity of double dispersion relations in the energy and momentum transfer (1) has been recently proved for the nonrelativistic scattering by a central potential (Z-4). It is the purpose of this paper to extend this result to the more general case of nonrelativistic collisions of two spinless particles A and B each capable of a finite number of excited states. We shall assume t’hat the interaction between the two particles can be described by a set of local potentials (5) and that the Schr6dinger equation which describes the evolution of the syst,em is invariant under Galilean transformations. The last assumption implies that the total mass of the syst,em be the same in all channels.’ In Section III we discuss the analytic properties of the second Born approximation: it will be shown that unless the reduced masses of the particles are assumed to be equal in all channels, it is not even possible to write ordinary * A summary of this paper has been presented at the Tenth High Energy Physics Conference, Rochester, August 1960. t On leave of absence from Istituto di Fisica, Universit& di Trieste, Trieste, Italy. $ On leave of absence from Istituto di Fisica, Universit$ di Pisa, Pisa, Italy. ** On leave of absence from Istituto di Fisica, Universita di Torino, Torino, Italy. 1 By channel we mean the internal state of the pair A + B. We shall label the channels by an index i, j, . . . 68
MANDELSTAM
REPRESENTATION
Ix4
dispersion relations for the second-order transition amplitude 5”$), except for the scattering in the channel of lowest reduced mass. It will also be shown that, t,o prevent the appearance of anomalous thresholds, the excitations energies of the various channels Ej and the ranges of the potent,ials p.7: must satisfy t,he st,ahility condition pL:L> 2m 1Ef - Ei 1,
(1.1 )
m being the common reduced mass in all channels. Equation ( 1.1) is the nonrelativistic analog of the conditions discussed by Karplus et al. (6). We shall t,herefore base our proof of the ordinary and double dispersion relations (Sections IV and V) on the assumptions that: (i) the reduced masses of the two part,icles are the same in all channels though the excitation energies are different, and (ii) that the st,abilit)y condition ( 1.l ) is satisfied. We shall moreover restrict t,he potentials to a superposition of Yukawa potentials. For t,he proof of the Mandelstam representation we shall apply a method discussed by Landau (7’) t’o find the singularities of the 71th Born approximation. In Section VI we derive from unitarity a set of nonlinear integral equations for t,he weight functions pfz which appear in the Mandelstam representation. Finally in Section VII we discuss the dispersion relations for the parGal wave amplitudes llji . The stability condition (1.1) will prove to he necessary to insure that the singularit,ies of Tic be confined to the real axis. II.
PR,ELIMINARY
CONSII>ERATIONS
Let, +(t) be the stat,e vector which describes the syst,em: it is a column matrix whose elements refer t,o the different channels (5). $(t) satisfies the Schrbdinger equation” (2.1) where N is t*he t,otal number of channels, and Hi” is given by
p1 and pj are the momenta of the particles A and H; ml,;, rn?, , and Rj are the masses and the excitation energy in the jth channel; El = 0 by definition. For local and energy independent potentials, Equation (2.1) is not in general invariant, under Galilei transformat’ions unless mlj + ‘rnzj is a constant independent of j, and (Xl
3 We use units
7
x2 ( V,, / x1’? x2’) = kTJk(r)8(r
such that ?L = c = 1.
- r’)6(x,
- xe’)
70
FONDA,
RADICATI,
AND
REGGE
where xg and xe' are the center-of-mass coordinates in the jth and lath channel, respectively, and r = x1 - x2 , r’ = x1’ - x2’. The separation of the center-of-mass motion is straightforward, and the resulting time independent Schrodinger equation can be replaced by
where i specifies the incoming channel and n labels the component of the wave function. &,; is a free wave function which in the coordinate representation is given by 9Li(r)
=
6~
exp[Ik*rl
and the matrix G is the complete Green’s function satisfying outgoing wave boundary conditions. The matrix elements of G between normalized states are analytic in the energy plane cut from E = 0 to + CC,apart from simple poles on the real axis (bound states). In the following we shall restrict the potentials to the sarne class for which the Mandelstam representation has been proved in the one-channel case, namely: vjk (T) = Irn dp ajk(/h) F
&k
>
0,
lrjk
where Ujk(,.&) is in general a distribution. Time reversal invariance requires that vjk(T) = vj*,(r) = vkj(r). From the relative momenta in the initial and final channel ki and kf , we construct the momentum transfer A = k, - ki and a vector q orthogonal to A: q=fr(ki+k,)+aiciat’,ZA, 1 _ 24s
- Ed - 24s
- E,)
t
1
(2.2)
where s = E (total energy) and t = A2 and mj is the reduced mass in channel j. The angle B between ki and k, is related to the momentum transfer and to the total energy by t = 2s(mi + m,) -
2(miEi + m,Ef) -
With these notations
the amplitude
4z/mi mf(s - Ei) (S - Ef) Tf<(syt)
for the transition
i +
cos
0.
f reads
(2.3)
MANDELSTAM
71
REPRESENTATION
where !i”Ti(t) Tfi(s, t; r’,r)
= 1 d3rexp[--iA. = exp
A(r
r] Vfi(r), + r’)
1 II exp
-z
i k,F -
t
kf2
A(r - r’)
exp[iq.(r
III.
THE
ANALYTIC
PROPERTIES
OF
THE
SECOND
BORN
1
(2.5)
- r’)]
APPROXIMATION
In this section we examine the conditions under which T.!l’ satisfies the ordinary and double dispersion relat,ions. For simplicity me shall consider only elastic scattering and pure Yukawa potentials:
with P independent of j and lc. T!?’ 2% (s 7t) is given by
where X= Hi*
d4P2 + t + 2P d/2-
= p2 + 2mi(Ej
- Ei) + 2(s - Ej)(mi
- mj) f id2mj(s
- Ej)(4$
+ t).
To simplify the discussion of (3.1) we shall examine two cases: (A) h’j = 0 for all j’s, but all reduced masses are different. (B) All reduced masses are eyual rni = m, 2m = 1, but all Ej are different. Case (A) : Tj:’ is singular for dHj+ = &zv’H,for any j, i.e., for d/s= and for Hj-
-+ d2m;
f
.~ d2mj ’
(3.2)
= 0, i.e., for (3.3)
Let, us first suppose t,hat for some j, mi > mj . For t > 0 the roots J/-s1
,
72
FONDA,
RADICATI,
AND
4-a of (3.2) and (3.3) are both positive the cuts OIs<
W)
-s2
REGGE
imaginary.
I’,‘:’ has therefore
at least
5 s 2 -s1.
For t in the interval -4~’ > t > -4p2( mi,‘mj) , the singularities of 2’~~’ are no longer restricted to the imaginary axis of z/s. The location of the singularities is such that it is impossible to write both ordinary and double dispersion relations for T$‘. On the other hand if mi 6 mi for all j’s, the roots of (3.2) and (3.3) lie on the negative imaginary axis of d/s. 7’::’ is therefore analytic in the s plane cut from s = 0 to s = + w and it is possible to derive ordinary and double dispersion relations. Similar features obtain when no restrictions are placed on the Ej and on the (Tkj and also for inelastic processes. Case (B) : The roots of d/Hi+ = +&/Hiand of Hi- = 0 are
4s - Ej zz -ip2;4Ei;tEf EL2 Let t be greater than zero. If for every j, p2 5 Ei - Ei the singularities of 1’~~’ occur for negative imaginary values of l/s - Ej , and are t,herefore unimportant when we map the upper half plane of ds - Ei onto the s plane with the cut Ej < s < + w . Similar conclusions hold for t < 0. It is possible in this case to derive double disperison relations for !i$‘. On the other hand, if p2 < E; - 14~ , T$’ is analytic in the s plane with the cuts
E, _ b* + Ei - Eij2 < s < w. 3 41Jz In this case we have a situation analogous to the relativistic anomalous thresholds (6) and we cannot therefore hope t,o prove any kind of dispersion r&Cons for the full amplitude Tfi .3 The complicated analytic feat,ures of T)f’ which appear when the masses mj are different and/or the excitation energies and the ranges of the potentials do not sat,isfy the stability condition ( 1.l) , show that there is no hope to prove the Mandelstam representation for the full amplitude Tfi unless under very re3 In the general case of excitation singularities defined by the equation the imaginary axis of d\/s if
energies and reduced dHi + = i i z dHi-
&it < 2 (rn.i - rni) (Ej - EC).
masses different, even are no longer restricted
the on
MANDELSTAM
$3
REPRESENTATIOS
stricted assumptions. In the following we shall t,herefore assume that: (i) All the masws mj are equal. For simplicity we shall put in the following 2m = 1. (ii) The stability condition (1 .l ) is satisjied. IV.
ORDINARY
DISPERSION
R.ELATIONS
To derive dispersion relations for positjive momentum t,rsnsfer for l’fi(s,t) one must, first, determine the behavior of t,he Green’s function Gjk(s; r’, r) for large complex s. This can be done by following the same procedure used for t,hc one-channel case (8); it, is sufficient to replace t’he absolute values of t.he quantibies which ent.er t,he proof given in Ref. 8 by the corresponding operator norms. The conclusion is that for large complex s the Born expansion for G converges uniformly to the free Green’s function G’O’. For large c*omplex s we can the]) writ,e Gjk(S; r’, r) = 6ikGi:!‘(s; r’, r) + Rjkis; r’, r),
1s 1---) r~3, (4.1)
where e--I”Q,G
/ r--T’ 1
Rj/s(S; r’, r) = Rjk(s) --F- __-r,,
and Rj,c(s) -+ 0 uniformly as 1s 1--+ QI. Following the procedure suggested by Bogoliubov (9), we next, define a fictit.ious amplitude T,i(.s, t, 5) by putting in (2.4) and (2.5) q2 = 5 - f, ,$ < 0. From (4.1) we get, wit.h the same procedure of Refs. 10 and 11,
where the sum on the right-hand side runs over the bound st,ates. To prove t,hat Im Tri(s’, 1, E) can be analytically cont,inued up to t,he value
WC csonsider
where O,tis t,he solid angle in the direction of /i,, ,
74 and k,(s)
FONDA,
RADICATI,
AND
REGGE
is defined by k&$)
= j &q
1 - Ef ; E;
+ ; 1-
kj2(.g) = s - f + ;
, >
E, - Ei ’ 5 > ’
with similar equations for T,a(k, , ki , t) and ki([). The summation over n in (4.3) is extended only to those values of n such that Ice(s) > 0. Let us choose the z-axis along k, and call Icf , of, , $m and ki , eis , +in , the polar coordinates of kf and ki , respectively. The analytic properties of Im Tji as a function of @, are easily obtainable from those of T,, and !f:i . Tf,b(kf , k, , 6) is analytic wherever the integral (4.4) converges, i.e., inside the ellipse : Im Ofn < 17~= min v/j , i
(4.5)
where
cash qjj
p!! forv=s-~++,,>Ej /= J l + ww , for v < Ej
Pi, Similarly
T,; is analytic
= Pjj + @(Ej
- S) V’Ej -
s
2 p/j *
inside the ellipse: Im
< fi = min vie .
eni
k
(4.6)
If $i(s) < ( v - E, 1, cash q/j < 1 and the integral (3.4) does not converge for any value of cos 8,, . The necessary and sufficient condition for (4.4) to be defined is that the stability condition ( 1 .I ) is satisfied namely /.LF~2 1Ej - E/ 1. Similarly
(4.7)
we get, from Tik : &k
2
1 Ek
Inside the ellipses (4.5) and (4.6) polynomials : Tj, = c
‘3
+
-
Ei
1 .
(4.8)
T,, and Tni can be expanded in Legendre 1)Pt
(cm
ejn)T:nW,
(COS
&&!%f),
L
Tei = c
(21 + l)P2
(4.9)
MANDELSTAM
where for large 1, Z’;,(t) = O(e-l”‘). the angular integrat)ion we obtain:
This expression
converges
75
REPRESENTATION
I nserting
(4.9) int,o (4.3) and carrying
out
inside the ellipse
Im efi < 77/+ 77;,
(4.10)
where cos e,, = a9 z Condition
+ kfw 2h @MU
- t
- *
(4.10) can be expressed solely in t’erms of s, t, ,$as follows: (4.11)
0 < t s b(v, s), where
tdv, s) = 22~- Ei - Ef + 2 z/v-
Ef + &(s)
40 - Ei - &(s) + 2P/i(S)fidS)
is an increasing function of v which is minimum at v = S, i.e., [ = &, . If (4.11) is satisfied for [ = $0 , it is always satisfied for t < & . If to(s, s) > 0 for all s, 0 5 s < + cc, there is a nonvanishing positive interval for t such that (4.11) is satisfied and we can continue analytically Im T,,i up to [ = [o . The requirement, tds, s) > O4 is equivalent to
z/p;j(s) + s - E, + v’&(s)
+ s - E, > 1Hrj(*s) - m(s)1
(4.12)
for all i, f, j, k. A sufficient condition for (4.12) to be satisfied is obtained when we replace the square roots with their minimum value, i.e., for all i, f, j, k: d~;j which
amount,s
+ Ei - E/ + d/.tlk + Ek - Ei > I /J~J - pi/t 19 to
min [pj/,z + z/,~sk - 1Ej -
ik
Finally
I’Fi(t,E)
Ek II > mj”k”bik - .\/I.& - 1Ei - Ek 11.
we examine the residues
= j$l
rT/(t,
m& J d3# e-ik’(‘)‘r’
V/j(/)
5) which
appear in (4.2):
$j(f; b,Z,m) (4.13)
X 1 d3r #k*(r; b, 1, m) Vki(r) eikitE).*, 4 The condition 0 < t S t,,(s, s) is the Iihuri in Ref. 10.
generalization
of condition
t < 4~” discussed
by
76
FONDA,
RADICATI,
AND
REGGE
where b and 1 are the radial and orbital quantum numbers, all the potentials are spherically symmetric, we can write b, 1, m) = Yl”(r)uj(r;
#i(r; By inserting
respectively.
Since
b, 2).
into (4.13) we get rh%,t)
= (21 + l)Pl
(cos ef,) r~‘“(~)(r~‘“(E))*(4?r”),
where
and similarly
for y!“(E).
so that the convergence
uj(r;
b, 1) can be proved to behave asymptotically
like
of -yF” and r%” is insured if Im kf < ~lfj -!- 1/Ej
- sb ,
Im
-
ki
<
Pik
+
.\/Ek
(4.14)
t% .
Condition (4.14) follows from (4.7) and (4.8). We have in this way completed the proof that it is possible to take the limit ,$+ &, on both sides of Eq. (4.2). This leads to the dispersion relations
The sum over b does not include the bound states embedded in the continuum whose possible existence has been proved in Ref. 5. Their contribution is included in the integral on Im TIP . V. THE
MANDELSTAM
REPRESENTATION
To study the analytic properties of T,c(s, t) when both s, 1 are complex, we begin by examining the behavior of each term of the Born expansion. The nth term can be written as
k2 -
m 1 4.h’ %(PL,‘j s - ic + E, so pcL’,2 + (k,-I - k,)2’
where the index a in eraand pee, stands for the pair of summation whereas in E, and k, it stands for era . By definition Ujk(p’)
= 0
for
/I’ <
/ljk
;
(5.1)
indices ( (Y~-~ , a,),
MANDELSTAM
77
REPRESENTATIOK
and
A t,ypical term in the summation nique in the form
(5.1) can be written
with the Feynman
tech-
where 622~
= (ka-1 - k,)’
+ P,? = A-,,,
Qza = k,’ - s + h’, -
+ ~12,
ic,
Z stands for a given sequence of indices:
specifying the intermediate channels. As usual the integrations over the X’S extend from 0 to 1. According to Landau’s analysis (7) the singularities of Icn’ (s, t; Z) may occur only for those values of s, t such that
A,&, = 0
0 $ X, 5 1;
1 5 p 5 2n -
1 (5.3)
and La ‘c’
X,Q,. = 0.
(5.4)
There are three possibilities: (A) all X, # 0; (B) Xza = 0 for a in a subset A of 1 2 a 5 n - 1, but all &b-I # 0; (C) for someb, &b--1= 0. (A) From (5.3) and (5.4) it follows:
Q, = 0 Xraka
=
X20-A-l.a
+
1 5 p 5 2n h,+A+~,a
1 (5.5) (5.6)
,
where Aab = k, - kb, 0 < a., 6 < n. A;3i = A& = t, Ai,,-l = -PC from (5.5). Equation (5.6) implies that all vectors k, are coplanar. From Eq. (5.6) one can prove by induction : ka = M(a, 6, 6)&a + N(a, 6, c)A,,
,
(5.7)
where b < a < c, ICI > 0, N > 0. Let us multiply both sides of (5.6) with k, : 2Xz,(s - %) = Ll(PLa2 + 2% - Z-1)
+ X2a+l(P:+l + & - Iii,,).
(5.8)
78
FONDA,
RADICATI,
AND
REGGE
Since from (1.1) ~2 > I E, - Eapl 1 and since X, > 0 the left-hand side of Eq. (5.8) is positive. It. follows that s > E, , 1 5 a I n - 1. Furthermore we have Ai, = (&a + Aad2 = &a + A:c + 2(&a + ;
- A,>
= &a + A:,
Ato + ;
A:, - kN
(s - Ect) .
If A;, = -d: , Ai, = -kf , and I.& , I& > 0, it follows that: Ai, 5 pLc)’ since M, N > 0. Moreover A%,,-, = -cl: so that
(5.9)
(pi, +
2
(5.10) After
some elementary
algebra one gets + d(R2,
A:, = A:, + AL - 2(&a&
- Aia) (R:c - A:,)) ,
(5.11)
where &a =
- &a -I- Ea - Ea 2k, a
In deriving Eq. (5.11) we have chosen the positive sign in front of the square root since only this choice is consistent with Eq. (5.10). By repeated applieations of Eq. (5.11) one can obtain t = Ai,, as a function of s, t = ~(s, Z). T is a single valued function of s since at each stage of the application of (5.11) the choice of the sign is unambiguously fixed by (5.10). t = ~(s, Z) defines in the s, t plane a curve lying in the region t s - (c.. P.‘)~, s 2 E> where E> is the largest of the E, , 1 4 a 6 n - 1. The curve is asymptotic to t = - (ca P.‘)~ and to s = E> . From (5.11) it is evident that, if Ai, , At. are nondecreasing functions of s, Ai, is an increasing function of s. Since this is true for AZ,,-1 it is also true for every Ai, and for t. (B) If a E A, we can write X2a-A-1
,a =
X20. +~&.a
tl
.
(5.12)
Aa+ and &,a~~ are therefore pa.rallel and since X2a-1/X2Q+1> 0 they point in the same direction. It follows: &+~a-1
= -(p.’
+ L+d2.
This equation replaces (5.11) for the special values b = (I a E A. If more generally rt, E A for every b < a < c we have A;= = -
(5.13)
1, c = a + 1, (5.14)
Using (5.11) when a does not belong to A, and (5.14) when a E A, we can still express t as a single-valued function of s: t = ~(s, Z, A). This function defines
MANDELSTAM
7!1
REPRESENTATION
a curve in the s, t plane lying in the region s 2 E,(A), t S - ( clz”=l P,‘)‘, where E,(A) is the largest of the E, , a not belonging to A. Just as before, t is an increasing function of s with asymptotes s = E,(A), t = -(CT P.)~. If A’ is the set 1 $ a s n - 1 then T(S, Z, A’) = - (CT pa)‘.5 (C) If &b--1 = 0, Eq. (5.6) reduces to (A2b
Eq. (5.15)
+
must be associated
kzb+&,
=
h2b+lkb+l
(5.15)
.
with
(kb - ka+d2 = -Pi+1 . Since $+I > 1&,+I - &, / this pair of equations has no solution corresponding to positive x26 , h2b+1 . Similar conclusions hold when, together with &b-1 = 0, some other X vanishes. Case (C) does not yield any singularity. From the above analysis it is then clear that I(%)(s, t, Z) is analytic for s > 0 in the t plane with the cut
By integration over all the P’ and summation over all sequences &:ri corresponding to fixed i and f, we deduce that Z’j!‘(s, t) is analytic for s 2 0 in t’he t plane apart from the cut -p&(n)
4 t 2 - w,
(5.16)
where
It follows
that cz=1
Z’jY’ is analytic
in the t plane with the cut
-(InJnp,i(“))2 -
2 t 2 --m.
(5.17)
If pfi > p > 0 we have the inequalit>y IL/~(~) 2 no Let, us now write
and therefore
lim n-tm &i(n)
= ‘30.
Tfi in the form
where
R:S’W)
= /- c T:r’(k,,q)Gjm(S,q,q’) im
5 It will be shown later Case (A) can be formally
(q, _ kf2 +yhdq’1 m22
that there are no singularities of Im I(*) associated to +(s, z, A’). included in Case (B) by considering ‘4 = 0 the empty set.
80
FONDA,
RADICATI,
AND
REGGE
All our conclusions apply to 2’::’ (k, , q) even though 4’ > 0 and q* # s - Ei . If B/j is the angle between k, and q, Tjjn’ (kf , q) is analytic in the region (5.18)
Im 6jj < Im qfj, where COS q/j
=
b2 + 4*+ w(n)* 2bq
*
From this it follows that the analyt.icity domain of R$?’ in the variable cos 8 is at least the ellipse defined by (5.18) and that it can be extended at will by simply increasing n. We can then conclude that Tfi is analytic in an arbitrarily large ellipse with the exclusion of the cut (5.17) and therefore it is analytic in the entire t plane except for the cut (5.17). If in addition to this we suppose t.hat Tfi(.s, t) is bounded by some power tL when t is large, it follows from the above arguments that
and pfi is a suitable distribution, such that pjk = /$j where tfi = (min,& pf;(n))*. By substituting (5.19) into (4.15) we can finally establish the validity of the Mandelstam representation6 :
+ c
(2Z + 1) PLCOS &) YY(YY)* 8 -
b.1
(4?r)2
s,,
(5.20)
6 In Ref. 3 the Mandelstam representation was proved for the one-channel case starting from the partial wave expansion and using a technique involving complex angular momenta. By this method it is possible to prove the inequality (1)
L < -36 + Re VA
+ 56 + iB,
where II Re VAiy)
II 5 A/Y*,
II Im
I/ S
In (I) it has been explicitly assumed plane Re(r) 2 0. This implies that
the
V,(r)
Vjdiy)
Bly2.
possibility of continuing the potential is a superposition of Yukawa potentials.
in the half
MANDELSTAM
81
REPRESENTATION
where -tf2sb-Ei-E, ‘OS e’i = -Z.\/(S~ - Ef)(sb - Ei) ’ VI.
THE
UNITARITY
CONDITION
In analogy to the discussion given in Ref. d for the one-channel case we shall derive from the unitarity condition and the Mandelstam representation a system of coupled integral equations for the weight function pI((s, t) . The unitarity condition reads Im Tfi( s, t) = $2
7’
kj(S)
1 dQj Ti*f(kj 7k,)
Tj;(kj
3ki) ,
(6.1)
where c’ means summation over the intermediate channels j such that s > Ej . For simplicity we shall suppose that in (5.20) there are no bound states and that L = 0. Inserting (5.20) into (6.1) we get aft,er some algebra: P/i(S, t> = T’{
l:,
‘&I ufi (PII) ljI
where E is the sequence f, j, K(s;t,tl,tz,Z)
dp2 uj;(Pz) K(S; t, P?, Pt, S>
i.
= -
22e
(t - 4s,tl, dW(s,
t2,z))
t, 4,
t2 ) E)
(6.3) ’
with W = t2(s - Ej) + tl’(s -
ttl(2s
- ttltz -
- Ei) + tt(s
- Ei - Ei) t(Ej
- Ei)(Ej
-- E/)
- ttz(2S - E/ - Ej) - Ef)
-
tl(E;
- tlt2(2s - El - Ei)
- Ef)(Ei -
- Ej) tz(E, - Ei)(Ef
- E,),
and W(s, T, tl , ta, E) = 0, 7 being the largest positive root of W. In particular 12 12 E) d iscussed in SecGon V, Case (A). 4% Pfi , PLji, S) is the function +s, From the results of Section V it is evident that
82
FONDA,
RADICATI,
AND
REGGE
Eq. (6.2) can be solved with the same iteration procedure used in Ref. 2. The lowest term &’ (s, t) in our expansion is simply the second Born approximation, as given by the first term in the right-hand side of Eq. (6.2). The support of pC2’is the union of the regions t > T(S, P?i 9 Psi
9 Z>f
z = (f,i
s > Ej
i>,
for all
1 5 j 5 N,
j, i fixed
which is contained in the region s > 0, t > $i( 2). The higher terms in the expansion can be classified according to the number n of times the function ajk(p) appears in it. By repeated applications of (6.4) one can easily see, in analogy with the one-channel case, that the support of a term of order n is included in the region s > 0,
t >
(6.5)
/.$i(n>.
Any given point in the s, t plane will belong to at most a finite number of supports. Indeed we can always choose n large enough to violate (6.5). It follows that we can calculate pfi(.s, t) at, any point with a finite number of iterations. This number increases with s and t. We have therefore proved that in absence of bound states and for L = 0 the Mandelstam representation in conjunction with unitarity can replace the Schrtiinger equation in the construction of the scattering matrix. If there are bound states, L must be larger than the highest angular momentum 2 appearing in (5.20). The problem of including bound states into (6.2) has not been completely solved for the one-channel case, we shall not deal with it further. VII.
ANALYTIC
PROPERTIES
OF PARTIAL
WAVES
Let us consider an amplitude !!‘fi(s, t) satisfying the Mandelstam representation (5.20)) where we shall suppose for simplicity that there are no bound states and that L = 0. We define the partial wave transition amplitude Tji(s) by
G(s) = f l; T,i(S, t)lqz) dz, t = 2s
where
(7.1)
- Ei - E, - 2 d(s - Ei)(s - E,).z
From (7.1) and (5.20) we obtain
4?r.P(z)dz
T’i(s) = - f /- uffi(p’) “’ Nfi
/e.m: p2+ 2s -
Ea_ Ef -
2.z,/(s
_ &) (s _ E,)
Pfi(S’,t’)Pl(Z) cl.2 - S - ie)(t’+2~
- Ei- Ef - 22l/(s - E;)(s - Ef))'
(7.2)
MANDELSTAM
The integration
over x can be carried out with 1
l
z s -l dz x Ql(z)
being the Legendre function
-2?r
83
REPRESENTATION
the help of the formula
= f &z(d),
(7.3)
of the second kind. The result is
* s
QZ
udw') dd
Pfl
p2 + 2s - E; - Ef
- Ei)(s - Ef) v'(s - Ei)(s - Ef) 2x4s
t’+ 2s - Ei - Ef QZ 00 dt’ Pfi(S’, 0 ( 2d(s - Ei)(s - E,) 2 J C/i(2) r s’ - s - ie d(s - EJ(s - Ef)
(7.4)
&l(z) is analytic in the z-plane with the cut - 1 5 z 5 1. The values of s on the cut of Tj?’ can be found by solving the equation /.d2 + 2s - Ei - Ef _ 2z/(s - Ei)(s - Ef) - ’ The two roots of (7.5) are s(fv)
s(q) = Ei 7 + Ef
I9 ( < 1.
with
d4 + rl’(Ei - .W*
1
- ii /.A’~+ q&4
- (1 - $)(Ei
Since 1E; - Ef 1 < P’~, s( 7) is a real decreasing function
s(l) = Ei 2 + Ef - g - (Ei - Ed2 ati 4/P
<
s(l,
/.L' =
- E,)2
(7.6)
of 7 with
lim s(q)
= - ~0 .
?p-1
Since s( 1) is a decreasing function of P’ it takes its maximum The Born approximation has therefore the cut --cc
(7.5)
value at ~1’ = @,fi .
/Lfi).
(7.7)
Whenr;i 5 F” < /EfEJ, we already know that it is impossible to prove dispersion relations for Tji . It is however interesting to discuss the analytic properties of the right-hand side of Eq. (7.4) when the stability condition (1.1) is violated. s(s) is real only if v2 > 70 = 1 - [P’~/(E; - E,)2], whereas for rl < 110, s(v) = s*c-v1. The real values of s( 7) lie now on the cut --oo
EC being the smallest of Ei , Ef .
(7.8)
The remaining points of S(V) lie on a closed loop because for j 7 / 6 q. , Im(s( q) ) vanishesat q = qo, q = 0. Wealso haveRe(s(0)) = (Ei + Ef - p”)/2 > EC,
84
FONDA, RADICATI,
AND REGGE
Re( s( 70)) = (Ei + Ef/2) - [(Ei - Ef)“/2#] < E< . The integration on N’ produces a superposition of loops and cuts of the above kind which disappear when p’2 = ]&--&].A na1ogous singularities arise from the second term in (7.4) ; the corresponding cut in s being -cc
5 s $ s(1, /.l = &i(2)).
(7.9)
From the integration over s’ we get the cut o$ss+w.
(7.10)
These cuts do not overlap if p,(2) > fi + @, . Moreover if &(2) < ] Ei - Ef 1, we have, in addition to the cuts, loops similar to the ones encountered in the first Born approximation. Distributions of singularities of this kind have been found with similar techniques in the discussion of specific inelastic processes(1s). If bound states are present they appear as simple poles in addition to the cuts. When the stability condition (1.1) is satisfied we can summarize the analytic properties of Tii(s) in the Cauchy formula:
RECEIVED:
June 20, 1960 ACBNOWLEDGEMENTS
It is a pleasure to thank Dr. M. L. Goldberger, Dr. N. N. Khuri, and Dr. J. Y. Lascoux for many stimulating discussions. Two of the authors (L. F. and L. A. R.) wish to thank Dr. J. R. Oppenheimer for the hospitality extended to them at the Institute for Advanced Study, and the National Science Foundation for a grant-in-aid. The other author (T. R.) acknowledges a grant from the U. S. Air Force.
1.
2. 3.
4. 6.
REFERENCES S. MANDELSTAM, Phys. Rev. 112, 1344 (1958); 116, 1741 (1959); a proof of double dispersion relations in the field theoretic case has been given to any order of the perturbation expansion by R. J. EDEN (unpublished). R. BLANKENBECLER, M. L. GOLDBER~ER, N. N. KHURI, AND S. B. TREIMAN, Annals of Physics 10, 62 (1969). T. REGGE, Nuovo cimento [lo] 14,951 (1959). A. KLEIN, J. Math. Phys. 1, 41 (1969); see also J. BOWCOCK AND A. MARTIN, Nuovo Cimento [lo], 14, 516 (1959). L. FONDA AND R. G. NEWTON, Annals of Physics 10, 499 (1960). Copious references on previous work will be found in this article.
MANDELSTAM
6. R.
KARPLUS,
C. SOMMERFIELD,
REPRESENTATION
AND
E. WICHMANN,
85 Phys. Rev. 111, 1187 (1958); 114,
376 (1959). D. LANDAU, Nuclear Phys. 13, 181 (1959); J. C. POLKINGHORNE AND G. R. SCREATON, Nuovo Cimento [lo] 16, 289 (1960). 8. C. ZEMACH AND A. KLEIN, Nuovo cimento [IO], 10, 1078 (1958); E. P. Wigner (unpublished). 9. N. N. B~GOLIUBOV AND D. V. SHIRKOV, “Introduction to the Theory of Quantized Fields,” Chapter 51. Interscience, New York and London, 1959. 10. PI;. N. KHURI, Phys. Rev. 107, 1148 (1957). 11. A. KLEIN, AND C. ZEMACH, Annals of Physics 7. 440 (1959). 18. S. W. MAC DOWELL, Phys. Rev. 116, 774 (1959). 7. L.
OF
PHYSICS:
12:
Mandelstam
68-85
(1961)
Representation Many Channel L. FONDAt Institute
for
for the Nonrelativistic Problem*
AND L. A. RADICATI$
Advanced
Study,
Princeton,
New
Jersey
AND T. REGGE Palmer
Physical
Laboratory,
Princeton
University,
Princeton,
New
Jersey
In this paper we investigate the analytic properties of the transition amplitudes for the collision of two particles each capable of a number of excited states and interacting through a set of generalized Yukawa potentials. Double dispersion relations for each of the amplitudes can be derived only when the reduced mass is the same for all channels and a certain stability condition, which ensures the absence of anomalous thresholds, is satisfied. The Mandelstam representation is then discussed in connection with unitarity and the partial wave expansion is obtained with results similar, to the one-channel case. I. INTRODUCTION
The validity of double dispersion relations in the energy and momentum transfer (1) has been recently proved for the nonrelativistic scattering by a central potential (Z-4). It is the purpose of this paper to extend this result to the more general case of nonrelativistic collisions of two spinless particles A and B each capable of a finite number of excited states. We shall assume t’hat the interaction between the two particles can be described by a set of local potentials (5) and that the Schr6dinger equation which describes the evolution of the syst,em is invariant under Galilean transformations. The last assumption implies that the total mass of the syst,em be the same in all channels.’ In Section III we discuss the analytic properties of the second Born approximation: it will be shown that unless the reduced masses of the particles are assumed to be equal in all channels, it is not even possible to write ordinary * A summary of this paper has been presented at the Tenth High Energy Physics Conference, Rochester, August 1960. t On leave of absence from Istituto di Fisica, Universit& di Trieste, Trieste, Italy. $ On leave of absence from Istituto di Fisica, Universit$ di Pisa, Pisa, Italy. ** On leave of absence from Istituto di Fisica, Universita di Torino, Torino, Italy. 1 By channel we mean the internal state of the pair A + B. We shall label the channels by an index i, j, . . . 68
MANDELSTAM
REPRESENTATION
Ix4
dispersion relations for the second-order transition amplitude 5”$), except for the scattering in the channel of lowest reduced mass. It will also be shown that, t,o prevent the appearance of anomalous thresholds, the excitations energies of the various channels Ej and the ranges of the potent,ials p.7: must satisfy t,he st,ahility condition pL:L> 2m 1Ef - Ei 1,
(1.1 )
m being the common reduced mass in all channels. Equation ( 1.1) is the nonrelativistic analog of the conditions discussed by Karplus et al. (6). We shall t,herefore base our proof of the ordinary and double dispersion relations (Sections IV and V) on the assumptions that: (i) the reduced masses of the two part,icles are the same in all channels though the excitation energies are different, and (ii) that the st,abilit)y condition ( 1.l ) is satisfied. We shall moreover restrict t,he potentials to a superposition of Yukawa potentials. For t,he proof of the Mandelstam representation we shall apply a method discussed by Landau (7’) t’o find the singularities of the 71th Born approximation. In Section VI we derive from unitarity a set of nonlinear integral equations for t,he weight functions pfz which appear in the Mandelstam representation. Finally in Section VII we discuss the dispersion relations for the parGal wave amplitudes llji . The stability condition (1.1) will prove to he necessary to insure that the singularit,ies of Tic be confined to the real axis. II.
PR,ELIMINARY
CONSII>ERATIONS
Let, +(t) be the stat,e vector which describes the syst,em: it is a column matrix whose elements refer t,o the different channels (5). $(t) satisfies the Schrbdinger equation” (2.1) where N is t*he t,otal number of channels, and Hi” is given by
p1 and pj are the momenta of the particles A and H; ml,;, rn?, , and Rj are the masses and the excitation energy in the jth channel; El = 0 by definition. For local and energy independent potentials, Equation (2.1) is not in general invariant, under Galilei transformat’ions unless mlj + ‘rnzj is a constant independent of j, and (Xl
3 We use units
7
x2 ( V,, / x1’? x2’) = kTJk(r)8(r
such that ?L = c = 1.
- r’)6(x,
- xe’)
70
FONDA,
RADICATI,
AND
REGGE
where xg and xe' are the center-of-mass coordinates in the jth and lath channel, respectively, and r = x1 - x2 , r’ = x1’ - x2’. The separation of the center-of-mass motion is straightforward, and the resulting time independent Schrodinger equation can be replaced by
where i specifies the incoming channel and n labels the component of the wave function. &,; is a free wave function which in the coordinate representation is given by 9Li(r)
=
6~
exp[Ik*rl
and the matrix G is the complete Green’s function satisfying outgoing wave boundary conditions. The matrix elements of G between normalized states are analytic in the energy plane cut from E = 0 to + CC,apart from simple poles on the real axis (bound states). In the following we shall restrict the potentials to the sarne class for which the Mandelstam representation has been proved in the one-channel case, namely: vjk (T) = Irn dp ajk(/h) F
&k
>
0,
lrjk
where Ujk(,.&) is in general a distribution. Time reversal invariance requires that vjk(T) = vj*,(r) = vkj(r). From the relative momenta in the initial and final channel ki and kf , we construct the momentum transfer A = k, - ki and a vector q orthogonal to A: q=fr(ki+k,)+aiciat’,ZA, 1 _ 24s
- Ed - 24s
- E,)
t
1
(2.2)
where s = E (total energy) and t = A2 and mj is the reduced mass in channel j. The angle B between ki and k, is related to the momentum transfer and to the total energy by t = 2s(mi + m,) -
2(miEi + m,Ef) -
With these notations
the amplitude
4z/mi mf(s - Ei) (S - Ef) Tf<(syt)
for the transition
i +
cos
0.
f reads
(2.3)
MANDELSTAM
71
REPRESENTATION
where !i”Ti(t) Tfi(s, t; r’,r)
= 1 d3rexp[--iA. = exp
A(r
r] Vfi(r), + r’)
1 II exp
-z
i k,F -
t
kf2
A(r - r’)
exp[iq.(r
III.
THE
ANALYTIC
PROPERTIES
OF
THE
SECOND
BORN
1
(2.5)
- r’)]
APPROXIMATION
In this section we examine the conditions under which T.!l’ satisfies the ordinary and double dispersion relat,ions. For simplicity me shall consider only elastic scattering and pure Yukawa potentials:
with P independent of j and lc. T!?’ 2% (s 7t) is given by
where X= Hi*
d4P2 + t + 2P d/2-
= p2 + 2mi(Ej
- Ei) + 2(s - Ej)(mi
- mj) f id2mj(s
- Ej)(4$
+ t).
To simplify the discussion of (3.1) we shall examine two cases: (A) h’j = 0 for all j’s, but all reduced masses are different. (B) All reduced masses are eyual rni = m, 2m = 1, but all Ej are different. Case (A) : Tj:’ is singular for dHj+ = &zv’H,for any j, i.e., for d/s= and for Hj-
-+ d2m;
f
.~ d2mj ’
(3.2)
= 0, i.e., for (3.3)
Let, us first suppose t,hat for some j, mi > mj . For t > 0 the roots J/-s1
,
72
FONDA,
RADICATI,
AND
4-a of (3.2) and (3.3) are both positive the cuts OIs<
W)
-s2
REGGE
imaginary.
I’,‘:’ has therefore
at least
5 s 2 -s1.
For t in the interval -4~’ > t > -4p2( mi,‘mj) , the singularities of 2’~~’ are no longer restricted to the imaginary axis of z/s. The location of the singularities is such that it is impossible to write both ordinary and double dispersion relations for T$‘. On the other hand if mi 6 mi for all j’s, the roots of (3.2) and (3.3) lie on the negative imaginary axis of d/s. 7’::’ is therefore analytic in the s plane cut from s = 0 to s = + w and it is possible to derive ordinary and double dispersion relations. Similar features obtain when no restrictions are placed on the Ej and on the (Tkj and also for inelastic processes. Case (B) : The roots of d/Hi+ = +&/Hiand of Hi- = 0 are
4s - Ej zz -ip2;4Ei;tEf EL2 Let t be greater than zero. If for every j, p2 5 Ei - Ei the singularities of 1’~~’ occur for negative imaginary values of l/s - Ej , and are t,herefore unimportant when we map the upper half plane of ds - Ei onto the s plane with the cut Ej < s < + w . Similar conclusions hold for t < 0. It is possible in this case to derive double disperison relations for !i$‘. On the other hand, if p2 < E; - 14~ , T$’ is analytic in the s plane with the cuts
E, _ b* + Ei - Eij2 < s < w. 3 41Jz In this case we have a situation analogous to the relativistic anomalous thresholds (6) and we cannot therefore hope t,o prove any kind of dispersion r&Cons for the full amplitude Tfi .3 The complicated analytic feat,ures of T)f’ which appear when the masses mj are different and/or the excitation energies and the ranges of the potentials do not sat,isfy the stability condition ( 1.l) , show that there is no hope to prove the Mandelstam representation for the full amplitude Tfi unless under very re3 In the general case of excitation singularities defined by the equation the imaginary axis of d\/s if
energies and reduced dHi + = i i z dHi-
&it < 2 (rn.i - rni) (Ej - EC).
masses different, even are no longer restricted
the on
MANDELSTAM
$3
REPRESENTATIOS
stricted assumptions. In the following we shall t,herefore assume that: (i) All the masws mj are equal. For simplicity we shall put in the following 2m = 1. (ii) The stability condition (1 .l ) is satisjied. IV.
ORDINARY
DISPERSION
R.ELATIONS
To derive dispersion relations for positjive momentum t,rsnsfer for l’fi(s,t) one must, first, determine the behavior of t,he Green’s function Gjk(s; r’, r) for large complex s. This can be done by following the same procedure used for t,hc one-channel case (8); it, is sufficient to replace t’he absolute values of t.he quantibies which ent.er t,he proof given in Ref. 8 by the corresponding operator norms. The conclusion is that for large complex s the Born expansion for G converges uniformly to the free Green’s function G’O’. For large c*omplex s we can the]) writ,e Gjk(S; r’, r) = 6ikGi:!‘(s; r’, r) + Rjkis; r’, r),
1s 1---) r~3, (4.1)
where e--I”Q,G
/ r--T’ 1
Rj/s(S; r’, r) = Rjk(s) --F- __-r,,
and Rj,c(s) -+ 0 uniformly as 1s 1--+ QI. Following the procedure suggested by Bogoliubov (9), we next, define a fictit.ious amplitude T,i(.s, t, 5) by putting in (2.4) and (2.5) q2 = 5 - f, ,$ < 0. From (4.1) we get, wit.h the same procedure of Refs. 10 and 11,
where the sum on the right-hand side runs over the bound st,ates. To prove t,hat Im Tri(s’, 1, E) can be analytically cont,inued up to t,he value
WC csonsider
where O,tis t,he solid angle in the direction of /i,, ,
74 and k,(s)
FONDA,
RADICATI,
AND
REGGE
is defined by k&$)
= j &q
1 - Ef ; E;
+ ; 1-
kj2(.g) = s - f + ;
, >
E, - Ei ’ 5 > ’
with similar equations for T,a(k, , ki , t) and ki([). The summation over n in (4.3) is extended only to those values of n such that Ice(s) > 0. Let us choose the z-axis along k, and call Icf , of, , $m and ki , eis , +in , the polar coordinates of kf and ki , respectively. The analytic properties of Im Tji as a function of @, are easily obtainable from those of T,, and !f:i . Tf,b(kf , k, , 6) is analytic wherever the integral (4.4) converges, i.e., inside the ellipse : Im Ofn < 17~= min v/j , i
(4.5)
where
cash qjj
p!! forv=s-~++,,>Ej /= J l + ww , for v < Ej
Pi, Similarly
T,; is analytic
= Pjj + @(Ej
- S) V’Ej -
s
2 p/j *
inside the ellipse: Im
< fi = min vie .
eni
k
(4.6)
If $i(s) < ( v - E, 1, cash q/j < 1 and the integral (3.4) does not converge for any value of cos 8,, . The necessary and sufficient condition for (4.4) to be defined is that the stability condition ( 1 .I ) is satisfied namely /.LF~2 1Ej - E/ 1. Similarly
(4.7)
we get, from Tik : &k
2
1 Ek
Inside the ellipses (4.5) and (4.6) polynomials : Tj, = c
‘3
+
-
Ei
1 .
(4.8)
T,, and Tni can be expanded in Legendre 1)Pt
(cm
ejn)T:nW,
(COS
&&!%f),
L
Tei = c
(21 + l)P2
(4.9)
MANDELSTAM
where for large 1, Z’;,(t) = O(e-l”‘). the angular integrat)ion we obtain:
This expression
converges
75
REPRESENTATION
I nserting
(4.9) int,o (4.3) and carrying
out
inside the ellipse
Im efi < 77/+ 77;,
(4.10)
where cos e,, = a9 z Condition
+ kfw 2h @MU
- t
- *
(4.10) can be expressed solely in t’erms of s, t, ,$as follows: (4.11)
0 < t s b(v, s), where
tdv, s) = 22~- Ei - Ef + 2 z/v-
Ef + &(s)
40 - Ei - &(s) + 2P/i(S)fidS)
is an increasing function of v which is minimum at v = S, i.e., [ = &, . If (4.11) is satisfied for [ = $0 , it is always satisfied for t < & . If to(s, s) > 0 for all s, 0 5 s < + cc, there is a nonvanishing positive interval for t such that (4.11) is satisfied and we can continue analytically Im T,,i up to [ = [o . The requirement, tds, s) > O4 is equivalent to
z/p;j(s) + s - E, + v’&(s)
+ s - E, > 1Hrj(*s) - m(s)1
(4.12)
for all i, f, j, k. A sufficient condition for (4.12) to be satisfied is obtained when we replace the square roots with their minimum value, i.e., for all i, f, j, k: d~;j which
amount,s
+ Ei - E/ + d/.tlk + Ek - Ei > I /J~J - pi/t 19 to
min [pj/,z + z/,~sk - 1Ej -
ik
Finally
I’Fi(t,E)
Ek II > mj”k”bik - .\/I.& - 1Ei - Ek 11.
we examine the residues
= j$l
rT/(t,
m& J d3# e-ik’(‘)‘r’
V/j(/)
5) which
appear in (4.2):
$j(f; b,Z,m) (4.13)
X 1 d3r #k*(r; b, 1, m) Vki(r) eikitE).*, 4 The condition 0 < t S t,,(s, s) is the Iihuri in Ref. 10.
generalization
of condition
t < 4~” discussed
by
76
FONDA,
RADICATI,
AND
REGGE
where b and 1 are the radial and orbital quantum numbers, all the potentials are spherically symmetric, we can write b, 1, m) = Yl”(r)uj(r;
#i(r; By inserting
respectively.
Since
b, 2).
into (4.13) we get rh%,t)
= (21 + l)Pl
(cos ef,) r~‘“(~)(r~‘“(E))*(4?r”),
where
and similarly
for y!“(E).
so that the convergence
uj(r;
b, 1) can be proved to behave asymptotically
like
of -yF” and r%” is insured if Im kf < ~lfj -!- 1/Ej
- sb ,
Im
-
ki
<
Pik
+
.\/Ek
(4.14)
t% .
Condition (4.14) follows from (4.7) and (4.8). We have in this way completed the proof that it is possible to take the limit ,$+ &, on both sides of Eq. (4.2). This leads to the dispersion relations
The sum over b does not include the bound states embedded in the continuum whose possible existence has been proved in Ref. 5. Their contribution is included in the integral on Im TIP . V. THE
MANDELSTAM
REPRESENTATION
To study the analytic properties of T,c(s, t) when both s, 1 are complex, we begin by examining the behavior of each term of the Born expansion. The nth term can be written as
k2 -
m 1 4.h’ %(PL,‘j s - ic + E, so pcL’,2 + (k,-I - k,)2’
where the index a in eraand pee, stands for the pair of summation whereas in E, and k, it stands for era . By definition Ujk(p’)
= 0
for
/I’ <
/ljk
;
(5.1)
indices ( (Y~-~ , a,),
MANDELSTAM
77
REPRESENTATIOK
and
A t,ypical term in the summation nique in the form
(5.1) can be written
with the Feynman
tech-
where 622~
= (ka-1 - k,)’
+ P,? = A-,,,
Qza = k,’ - s + h’, -
+ ~12,
ic,
Z stands for a given sequence of indices:
specifying the intermediate channels. As usual the integrations over the X’S extend from 0 to 1. According to Landau’s analysis (7) the singularities of Icn’ (s, t; Z) may occur only for those values of s, t such that
A,&, = 0
0 $ X, 5 1;
1 5 p 5 2n -
1 (5.3)
and La ‘c’
X,Q,. = 0.
(5.4)
There are three possibilities: (A) all X, # 0; (B) Xza = 0 for a in a subset A of 1 2 a 5 n - 1, but all &b-I # 0; (C) for someb, &b--1= 0. (A) From (5.3) and (5.4) it follows:
Q, = 0 Xraka
=
X20-A-l.a
+
1 5 p 5 2n h,+A+~,a
1 (5.5) (5.6)
,
where Aab = k, - kb, 0 < a., 6 < n. A;3i = A& = t, Ai,,-l = -PC from (5.5). Equation (5.6) implies that all vectors k, are coplanar. From Eq. (5.6) one can prove by induction : ka = M(a, 6, 6)&a + N(a, 6, c)A,,
,
(5.7)
where b < a < c, ICI > 0, N > 0. Let us multiply both sides of (5.6) with k, : 2Xz,(s - %) = Ll(PLa2 + 2% - Z-1)
+ X2a+l(P:+l + & - Iii,,).
(5.8)
78
FONDA,
RADICATI,
AND
REGGE
Since from (1.1) ~2 > I E, - Eapl 1 and since X, > 0 the left-hand side of Eq. (5.8) is positive. It. follows that s > E, , 1 5 a I n - 1. Furthermore we have Ai, = (&a + Aad2 = &a + A:c + 2(&a + ;
- A,>
= &a + A:,
Ato + ;
A:, - kN
(s - Ect) .
If A;, = -d: , Ai, = -kf , and I.& , I& > 0, it follows that: Ai, 5 pLc)’ since M, N > 0. Moreover A%,,-, = -cl: so that
(5.9)
(pi, +
2
(5.10) After
some elementary
algebra one gets + d(R2,
A:, = A:, + AL - 2(&a&
- Aia) (R:c - A:,)) ,
(5.11)
where &a =
- &a -I- Ea - Ea 2k, a
In deriving Eq. (5.11) we have chosen the positive sign in front of the square root since only this choice is consistent with Eq. (5.10). By repeated applieations of Eq. (5.11) one can obtain t = Ai,, as a function of s, t = ~(s, Z). T is a single valued function of s since at each stage of the application of (5.11) the choice of the sign is unambiguously fixed by (5.10). t = ~(s, Z) defines in the s, t plane a curve lying in the region t s - (c.. P.‘)~, s 2 E> where E> is the largest of the E, , 1 4 a 6 n - 1. The curve is asymptotic to t = - (ca P.‘)~ and to s = E> . From (5.11) it is evident that, if Ai, , At. are nondecreasing functions of s, Ai, is an increasing function of s. Since this is true for AZ,,-1 it is also true for every Ai, and for t. (B) If a E A, we can write X2a-A-1
,a =
X20. +~&.a
tl
.
(5.12)
Aa+ and &,a~~ are therefore pa.rallel and since X2a-1/X2Q+1> 0 they point in the same direction. It follows: &+~a-1
= -(p.’
+ L+d2.
This equation replaces (5.11) for the special values b = (I a E A. If more generally rt, E A for every b < a < c we have A;= = -
(5.13)
1, c = a + 1, (5.14)
Using (5.11) when a does not belong to A, and (5.14) when a E A, we can still express t as a single-valued function of s: t = ~(s, Z, A). This function defines
MANDELSTAM
7!1
REPRESENTATION
a curve in the s, t plane lying in the region s 2 E,(A), t S - ( clz”=l P,‘)‘, where E,(A) is the largest of the E, , a not belonging to A. Just as before, t is an increasing function of s with asymptotes s = E,(A), t = -(CT P.)~. If A’ is the set 1 $ a s n - 1 then T(S, Z, A’) = - (CT pa)‘.5 (C) If &b--1 = 0, Eq. (5.6) reduces to (A2b
Eq. (5.15)
+
must be associated
kzb+&,
=
h2b+lkb+l
(5.15)
.
with
(kb - ka+d2 = -Pi+1 . Since $+I > 1&,+I - &, / this pair of equations has no solution corresponding to positive x26 , h2b+1 . Similar conclusions hold when, together with &b-1 = 0, some other X vanishes. Case (C) does not yield any singularity. From the above analysis it is then clear that I(%)(s, t, Z) is analytic for s > 0 in the t plane with the cut
By integration over all the P’ and summation over all sequences &:ri corresponding to fixed i and f, we deduce that Z’j!‘(s, t) is analytic for s 2 0 in t’he t plane apart from the cut -p&(n)
4 t 2 - w,
(5.16)
where
It follows
that cz=1
Z’jY’ is analytic
in the t plane with the cut
-(InJnp,i(“))2 -
2 t 2 --m.
(5.17)
If pfi > p > 0 we have the inequalit>y IL/~(~) 2 no Let, us now write
and therefore
lim n-tm &i(n)
= ‘30.
Tfi in the form
where
R:S’W)
= /- c T:r’(k,,q)Gjm(S,q,q’) im
5 It will be shown later Case (A) can be formally
(q, _ kf2 +yhdq’1 m22
that there are no singularities of Im I(*) associated to +(s, z, A’). included in Case (B) by considering ‘4 = 0 the empty set.
80
FONDA,
RADICATI,
AND
REGGE
All our conclusions apply to 2’::’ (k, , q) even though 4’ > 0 and q* # s - Ei . If B/j is the angle between k, and q, Tjjn’ (kf , q) is analytic in the region (5.18)
Im 6jj < Im qfj, where COS q/j
=
b2 + 4*+ w(n)* 2bq
*
From this it follows that the analyt.icity domain of R$?’ in the variable cos 8 is at least the ellipse defined by (5.18) and that it can be extended at will by simply increasing n. We can then conclude that Tfi is analytic in an arbitrarily large ellipse with the exclusion of the cut (5.17) and therefore it is analytic in the entire t plane except for the cut (5.17). If in addition to this we suppose t.hat Tfi(.s, t) is bounded by some power tL when t is large, it follows from the above arguments that
and pfi is a suitable distribution, such that pjk = /$j where tfi = (min,& pf;(n))*. By substituting (5.19) into (4.15) we can finally establish the validity of the Mandelstam representation6 :
+ c
(2Z + 1) PLCOS &) YY(YY)* 8 -
b.1
(4?r)2
s,,
(5.20)
6 In Ref. 3 the Mandelstam representation was proved for the one-channel case starting from the partial wave expansion and using a technique involving complex angular momenta. By this method it is possible to prove the inequality (1)
L < -36 + Re VA
+ 56 + iB,
where II Re VAiy)
II 5 A/Y*,
II Im
I/ S
In (I) it has been explicitly assumed plane Re(r) 2 0. This implies that
the
V,(r)
Vjdiy)
Bly2.
possibility of continuing the potential is a superposition of Yukawa potentials.
in the half
MANDELSTAM
81
REPRESENTATION
where -tf2sb-Ei-E, ‘OS e’i = -Z.\/(S~ - Ef)(sb - Ei) ’ VI.
THE
UNITARITY
CONDITION
In analogy to the discussion given in Ref. d for the one-channel case we shall derive from the unitarity condition and the Mandelstam representation a system of coupled integral equations for the weight function pI((s, t) . The unitarity condition reads Im Tfi( s, t) = $2
7’
kj(S)
1 dQj Ti*f(kj 7k,)
Tj;(kj
3ki) ,
(6.1)
where c’ means summation over the intermediate channels j such that s > Ej . For simplicity we shall suppose that in (5.20) there are no bound states and that L = 0. Inserting (5.20) into (6.1) we get aft,er some algebra: P/i(S, t> = T’{
l:,
‘&I ufi (PII) ljI
where E is the sequence f, j, K(s;t,tl,tz,Z)
dp2 uj;(Pz) K(S; t, P?, Pt, S>
i.
= -
22e
(t - 4s,tl, dW(s,
t2,z))
t, 4,
t2 ) E)
(6.3) ’
with W = t2(s - Ej) + tl’(s -
ttl(2s
- ttltz -
- Ei) + tt(s
- Ei - Ei) t(Ej
- Ei)(Ej
-- E/)
- ttz(2S - E/ - Ej) - Ef)
-
tl(E;
- tlt2(2s - El - Ei)
- Ef)(Ei -
- Ej) tz(E, - Ei)(Ef
- E,),
and W(s, T, tl , ta, E) = 0, 7 being the largest positive root of W. In particular 12 12 E) d iscussed in SecGon V, Case (A). 4% Pfi , PLji, S) is the function +s, From the results of Section V it is evident that
82
FONDA,
RADICATI,
AND
REGGE
Eq. (6.2) can be solved with the same iteration procedure used in Ref. 2. The lowest term &’ (s, t) in our expansion is simply the second Born approximation, as given by the first term in the right-hand side of Eq. (6.2). The support of pC2’is the union of the regions t > T(S, P?i 9 Psi
9 Z>f
z = (f,i
s > Ej
i>,
for all
1 5 j 5 N,
j, i fixed
which is contained in the region s > 0, t > $i( 2). The higher terms in the expansion can be classified according to the number n of times the function ajk(p) appears in it. By repeated applications of (6.4) one can easily see, in analogy with the one-channel case, that the support of a term of order n is included in the region s > 0,
t >
(6.5)
/.$i(n>.
Any given point in the s, t plane will belong to at most a finite number of supports. Indeed we can always choose n large enough to violate (6.5). It follows that we can calculate pfi(.s, t) at, any point with a finite number of iterations. This number increases with s and t. We have therefore proved that in absence of bound states and for L = 0 the Mandelstam representation in conjunction with unitarity can replace the Schrtiinger equation in the construction of the scattering matrix. If there are bound states, L must be larger than the highest angular momentum 2 appearing in (5.20). The problem of including bound states into (6.2) has not been completely solved for the one-channel case, we shall not deal with it further. VII.
ANALYTIC
PROPERTIES
OF PARTIAL
WAVES
Let us consider an amplitude !!‘fi(s, t) satisfying the Mandelstam representation (5.20)) where we shall suppose for simplicity that there are no bound states and that L = 0. We define the partial wave transition amplitude Tji(s) by
G(s) = f l; T,i(S, t)lqz) dz, t = 2s
where
(7.1)
- Ei - E, - 2 d(s - Ei)(s - E,).z
From (7.1) and (5.20) we obtain
4?r.P(z)dz
T’i(s) = - f /- uffi(p’) “’ Nfi
/e.m: p2+ 2s -
Ea_ Ef -
2.z,/(s
_ &) (s _ E,)
Pfi(S’,t’)Pl(Z) cl.2 - S - ie)(t’+2~
- Ei- Ef - 22l/(s - E;)(s - Ef))'
(7.2)
MANDELSTAM
The integration
over x can be carried out with 1
l
z s -l dz x Ql(z)
being the Legendre function
-2?r
83
REPRESENTATION
the help of the formula
= f &z(d),
(7.3)
of the second kind. The result is
* s
QZ
udw') dd
Pfl
p2 + 2s - E; - Ef
- Ei)(s - Ef) v'(s - Ei)(s - Ef) 2x4s
t’+ 2s - Ei - Ef QZ 00 dt’ Pfi(S’, 0 ( 2d(s - Ei)(s - E,) 2 J C/i(2) r s’ - s - ie d(s - EJ(s - Ef)
(7.4)
&l(z) is analytic in the z-plane with the cut - 1 5 z 5 1. The values of s on the cut of Tj?’ can be found by solving the equation /.d2 + 2s - Ei - Ef _ 2z/(s - Ei)(s - Ef) - ’ The two roots of (7.5) are s(fv)
s(q) = Ei 7 + Ef
I9 ( < 1.
with
d4 + rl’(Ei - .W*
1
- ii /.A’~+ q&4
- (1 - $)(Ei
Since 1E; - Ef 1 < P’~, s( 7) is a real decreasing function
s(l) = Ei 2 + Ef - g - (Ei - Ed2 ati 4/P
<
s(l,
/.L' =
- E,)2
(7.6)
of 7 with
lim s(q)
= - ~0 .
?p-1
Since s( 1) is a decreasing function of P’ it takes its maximum The Born approximation has therefore the cut --cc
(7.5)
value at ~1’ = @,fi .
/Lfi).
(7.7)
Whenr;i 5 F” < /EfEJ, we already know that it is impossible to prove dispersion relations for Tji . It is however interesting to discuss the analytic properties of the right-hand side of Eq. (7.4) when the stability condition (1.1) is violated. s(s) is real only if v2 > 70 = 1 - [P’~/(E; - E,)2], whereas for rl < 110, s(v) = s*c-v1. The real values of s( 7) lie now on the cut --oo
EC being the smallest of Ei , Ef .
(7.8)
The remaining points of S(V) lie on a closed loop because for j 7 / 6 q. , Im(s( q) ) vanishesat q = qo, q = 0. Wealso haveRe(s(0)) = (Ei + Ef - p”)/2 > EC,
84
FONDA, RADICATI,
AND REGGE
Re( s( 70)) = (Ei + Ef/2) - [(Ei - Ef)“/2#] < E< . The integration on N’ produces a superposition of loops and cuts of the above kind which disappear when p’2 = ]&--&].A na1ogous singularities arise from the second term in (7.4) ; the corresponding cut in s being -cc
5 s $ s(1, /.l = &i(2)).
(7.9)
From the integration over s’ we get the cut o$ss+w.
(7.10)
These cuts do not overlap if p,(2) > fi + @, . Moreover if &(2) < ] Ei - Ef 1, we have, in addition to the cuts, loops similar to the ones encountered in the first Born approximation. Distributions of singularities of this kind have been found with similar techniques in the discussion of specific inelastic processes(1s). If bound states are present they appear as simple poles in addition to the cuts. When the stability condition (1.1) is satisfied we can summarize the analytic properties of Tii(s) in the Cauchy formula:
RECEIVED:
June 20, 1960 ACBNOWLEDGEMENTS
It is a pleasure to thank Dr. M. L. Goldberger, Dr. N. N. Khuri, and Dr. J. Y. Lascoux for many stimulating discussions. Two of the authors (L. F. and L. A. R.) wish to thank Dr. J. R. Oppenheimer for the hospitality extended to them at the Institute for Advanced Study, and the National Science Foundation for a grant-in-aid. The other author (T. R.) acknowledges a grant from the U. S. Air Force.
1.
2. 3.
4. 6.
REFERENCES S. MANDELSTAM, Phys. Rev. 112, 1344 (1958); 116, 1741 (1959); a proof of double dispersion relations in the field theoretic case has been given to any order of the perturbation expansion by R. J. EDEN (unpublished). R. BLANKENBECLER, M. L. GOLDBER~ER, N. N. KHURI, AND S. B. TREIMAN, Annals of Physics 10, 62 (1969). T. REGGE, Nuovo cimento [lo] 14,951 (1959). A. KLEIN, J. Math. Phys. 1, 41 (1969); see also J. BOWCOCK AND A. MARTIN, Nuovo Cimento [lo], 14, 516 (1959). L. FONDA AND R. G. NEWTON, Annals of Physics 10, 499 (1960). Copious references on previous work will be found in this article.
MANDELSTAM
6. R.
KARPLUS,
C. SOMMERFIELD,
REPRESENTATION
AND
E. WICHMANN,
85 Phys. Rev. 111, 1187 (1958); 114,
376 (1959). D. LANDAU, Nuclear Phys. 13, 181 (1959); J. C. POLKINGHORNE AND G. R. SCREATON, Nuovo Cimento [lo] 16, 289 (1960). 8. C. ZEMACH AND A. KLEIN, Nuovo cimento [IO], 10, 1078 (1958); E. P. Wigner (unpublished). 9. N. N. B~GOLIUBOV AND D. V. SHIRKOV, “Introduction to the Theory of Quantized Fields,” Chapter 51. Interscience, New York and London, 1959. 10. PI;. N. KHURI, Phys. Rev. 107, 1148 (1957). 11. A. KLEIN, AND C. ZEMACH, Annals of Physics 7. 440 (1959). 18. S. W. MAC DOWELL, Phys. Rev. 116, 774 (1959). 7. L.