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The continuation in total angular momentum of partial-wave scattering amplitudes for particles with spin
ANNALS
OF PHYSICS:
26,
325-339
The Continuation Wave Scattering
(1963)
in Total Angular Amplitudes for
Momentum of PartialParticles with Spin*
FRANCESCOCALOGERO~ Palmer
Physical
Laboratory,
Princeton
University,
Princeton,
New
Jersey
JOHN M. CHARAP~ Institute
for
Advanced
Study,
Princeton,
New
Jersey
AND EUAN J. SQUIRES Tait
Institute
of Mathematical
Physics,
Edinburgh
8, Scotland
An assumed one-dimensional dispersion relation is used to define a continuation in total angular momentum of the helicity scattering amplitude, for particles with arbitrary spin. The result, which is a generalization of the Froissart-Gribov formula for zero spin, agrees with that discussed previously for potential models. It is shown that cross-channel unitarity imposed certain restrictions on the weight functions. The existence of certain kinematic zeros of the amplitude, corresponding to the decoupling of “sense” and “nonsense” channels, is proved in general, and finally it is shown that the presence of the “nonsense-nonsense” amplitudes does not affect the large z behavior, which is still dominated by Regge poles. I.
INTRODUCTION
There are three main objects of this paper. First, we wish to provide a summary of the essential results of three papers by two of the present authors (1) on the subject of complex angular momentum for nonrelativistic scattering of particles with spin. This summary is contained mainly in Sections II and V of * The study was supported in part by the Air Force Office of Scientific Research Grant AF-AFOSR-42-63. t Harkness Fellow of the Commonwealth Fund, New York, on leave of absence from Istituto Nazionale di Fisica Nucleare, Sezione di Roma, and Istituto di Fisica dell’Universita, Roma, Italy. $ Present address: Istituto di Fisica dell’Universita, Roma, Italy. Present address: CERN, Geneva, Switzerland. 32.5
326
CALOGERO,
CHARAP,
AND
SQUIRES
the present paper. Secondly, in Section III, we use the potential model as an analogy in order to define the appropriate continuation of the partial-wave amplitude in relativistic scattering, i.e., we generalize the Froissart-Gribov definition (2, 3) to apply to arbitrary spins. It is necessary here, of course, to assume that the scattering amplitude satisfies certain analyticity properties in the variables, and is suitably behaved as these variables become large. We further show that these assumptions then lead to certain restrictions on the form of the single spectral functions when particles of high spin are involved. These restrictions, which will be discussed in detail in a later paper by two of the present authors, arise from the requirement of unitarity in the crossed channels and have, therefore, no analogy in the potential model. Finally, we consider the states of positive integral or half-odd integral J, according to whether the total spin is integral or half-odd integral, which are unphysical due to their being less than the maximum value of the helicity. Such states are called “nonsense” states by Gell-Mann (4), and appear to be a source of confusion. In Section IV we prove that in potential theory (and very plausibly also in the relativistic theory) there is no coupling between the “sense” and “nonsense” states. However, contrary to what appears to have been assumed, the “nonsense-nonsense” amplitude is not exactly zero. At first sight this might be expected to vitiate the main result of II, that the scattering amplitude is always dominated, at high energies, by Regge poles. That this is not the case, however, is demonstrated explicitly in Section V, where it is shown that the Regge behavior holds for arbitrary spin. II.
THE
PARTIAL-WAVE
EXPANSION
We are concerned with scattering processes in which both initial and final states contain two particles of arbitrary spins. The formalism can, in fact, also be used where the states contain more than two particles (5, 6) but in this case the relevant matrices have infinite dimensions and the presence of “disconnected graphs” means that Fredholm theory is not immediately applicable. A consequence (5) is that branch points appear in the J-plane. It is possible (and desirable!) that these should cancel in a consistent relativistic theory, but such considerations are beyond the scope of this article. For a transition from an initial state with spins s1 and s2 , helicities Xi and X2 and barycentric momentum k, to a final state in which the corresponding quantities are sl’, a~‘, X,‘, X2’ and x3’,we write the scattering amplitude as (Sl’, s2’; Xl’,
x2’;
k’l f(e, 4 lSlS2; x1x2 ; 12).
Here we have suppressed any other quantum numbers which may be necessary to specify completely the two-particle states. If the masses of the particles are
J CONTINUATION
ml and m2 initially,
FOR
PARTICLES
and ml’ and mzf finally,
WITH
327
SPIN
then
(ml’ + ?c~)~‘~ + (mz” + k2)1’2 = (m:2 + k’y2
+ (,,$ + li2y2
2E.
(2.1)
We introduce x = x1 - x2 x’
=
x1’
-
(2.2)
x2/,
and also A,,, = max (A, A’)
(2.3)
Xmin = min (X, A’). Provided we restrict ourselves to parity conserving interactions of generality in taking
there is no loss
xmax 2 ( Xmin 1. This is because under parity conjugation (SX
x1’x2’; k’l f(O, 4)
+ (&‘;
ISIS2
;
XIX2
(2.4)
we have
; k)
--Al’ - hzl; k’l f( 8, T - 4) ISIS2; --x1 - x2 ; k)
(2.5)
and the two amplitudes are equal apart from a phase (7). If parity is not conserved then it is a simple matter to deal with the other amplitudes in an analogous way to that followed below. The partial-wave expansion of the scattering amplitude can be written in the form (7, I )
xysl
s2’ s’; x1’ - A,‘)
X g C(ZsJ; OX)C(Z’ s’ J’; OX’) (2.6)
328
CALOGERO,
CHARAP,
AND
SQUIRES
where SJ is the conventionally defined S-matrix for definite total angular momentum J. The values of s, s’, 1, and 1’ are integers, or, in the cases of s and .sf, possibly half-odd integers, with ranges restricted by the vanishing of the ClebschGordan coefficients. The elements of the rotation matrix are given by &(-4, and &!(e)
8, +> = exp [i(X -
A’>01C&(O),
(2.7)
can be written in the form*
where we have introduced (2.9)
and (2.10)
x = cos 9.
The quantity P$‘b’( z ) is a Jacobi polynomial of the first kind. Its analytic continuation and asymptotic behavior may be obtained from its connection with the hypergeometric function,
P:‘“‘(x)
=(n~‘)F(-n,n+a+b+l,a+~;~(l--z)).
It is convenient
(2.11)
to rewrite Eq. (2.6) in the condensed form (2.12)
where the partial-wave amplitude AJXIXis defined by comparison of Eq. (2.6) with Eq. (2.12). The important result of II and III that the modified partialwave amplitude XXIX defined by
A;,~ =
1
(J + X’) ! (J - X’) ! 1’2A., (J+x)!(J-A)! x’x
(2.13)
can be continued away from physical J values to give a function of J, which is holomorphic in the whole finite J-plane, apart from isolated poles (Regge poles) which move with energy, and which is suitably bounded as J tends to infinity 1 See Eq. (2.8) of II, and Eq. (10.8) (10) of ref. 8.
J CONTINUATION
POR
PARTICLES
WITH
SPIN
329
to permit the analogy of the Sommerfeld-Watson transformation. Some restrictions on the potential are necessary of course for these results to hold; these are discussed explicitly in ref. 1, and we shall assume them here. The bound at J = m was proved only in the Born approximation; a rigorous proof will be the subject of a later paper by two of the present authors. According to Carlson’s theorem (9) the above properties distinguish the particular continuation uniquely. As discussed in III it follows from these results and from Eq. (2.8) and Eq. (2.11), that the J-summand in Eq. (2.12) is holomorphic apart from Regge poles. III.
THE
GENERALIZED
FROISSART-GRIBOV
FORMULA
In order to give an explicit definition of the continued partial-wave amplitude in the relativistic case it is necessary first to write a one-dimensional dispersion relation for the scattering amplitude (cf. refs. 2 and 3). The appropriate dispersion relation can be found by analogy with the potential model.* Explicitly, using Eqs. (2.8) and (2.12), we know that
(!q
(g-b’2jA,A(e)
is an analytic function of z, in the z-plane cut along the positive real axis from 2 = 1 to 2 = m. Note that we keep the energy real and positive here. The convergence of the original partial wave expansion, Eq. (2.6)) within the Lehmann ellipse shows that the cut actually starts at some .zo> 1. Thus, recalling that in the relativistic case there is always an “exchange potential” (corresponding to the existence of two crossed channels), we can write
where, for the moment, we have ignored the question of subtractions which may be necessary to make the integrals in Eq. (3.1) converge. The proof that fhfx(~) has the analyticity required by Eq. (3.1) has not yet been given for the relativistic case in a fully satisfactory way; the problem here, however, is identical to that in the zero spin case. The orthogonality properties of the rotation matrices allow us to insert Eq. (2.12) and obtain A:q = $$ 2 Similar
but
less detailed
considerations
s-1 +l dz.hx(e) are given
&(e>. in ref.
(3.2) IO.
330
CALOGERO,
If we substitute values of J,
CHARAP,
AND
SQUIRES
for ~XIXfrom Eq. (3.1) and use Eq. (2.8) we find, for physical
A:xq = (-1)““in-k
(J + A,,,) ! (J - Xmx) !] “* x ;&G(p) (J + X,i,)! (J - &in)! (3.3)
. (qqb
P.jTf~&)
x I,
dz’ [-
+ &]
.
We now change the order of integration in Eq. (3.3)) which is permissible for suiliciently large J, and perform the integral over x by means of the following generalization of Neumann’s formula3
6?t*“‘
- I)-“(2. + l)-bL;
(/ - $l(l
_ z)“(l + $
(3.4)
*Pcasb) 11 (2’) dx, which holds for n a nonnegative integer and for a and b greater than -1. The Q?Vb’(z’) is a Jacobi function of the second kind; its analytic continuation and asymptotic behavior follow from its relation with the hypergeometric function4
Q;,“‘(x) = f iIn++a,j!Fb’,“:,!! (” ; ‘)-“-’
t+)-b (3.5)
-F(n + 1, n + a -I- 1; 2n + a + b + 2; 2(1 - x)-l). We note that in Eq. (3.3) both a and b are positive, also that, for physical J, J - Xmaxis a nonnegative integer. Thus we can use Eq. (3.5) in Eq. (3.3) and obtain, still for physical J, &,x
(J + X,,,) ! (J - Xmx) ! “’ (J + Amin)! (J - Amin)!
1 md&(x)Q.t~~,,(z) t+y (q) jz. +(qJ--hm,xI m lrs20
= ( -l)xmin-x
where, in obtaining
the last term we have also used the symmetry5
Pp3”‘(z) = ( -l)“PAb*“)( for nonnegative
(3.6)
integral
-x),
(3.7)
n.
8 This follows from Eq. (10.8) (20) Jacobi polynomials. 4 See Eq. (10.8) (18) of ref. 8. 6 See Eq. (10.8) (13) of ref. 8.
of ref.
8, and from
the orthogonality
properties
of the
J
CONTINUATION
FOR
PARTICLES
We shall see below that (at least for potential h(O)
SPIN
331
scattering)
= 0(x*)
where J = ar is the position of the rightmost of energy). It follows that &;a’(~)
WITH
(3.3)
Regge pole (in general a function
= O($--)
(3.9)
so that the integrals in Eq. (3.6) converge provided
J > Re LY.
(3.10)
It is easy to see that the presence of the necessary subtractions in Eq. (3.1) does not alter Eq. (3.6) provided Eq. (3.10) is satisfied, so this formula correctly gives the partial-wave amplitudes for all physical J values in this region. The factor ( - 1) J--xmaxin the second term of Eq. (3.6) makes the expression unsuitable for continuation to nonphysical J, so we define, as in the zero-spin case,6 two amplitudes A$i( J) by the expressions j&(J)
= (-l)Xmi=A
(J + X,,,) ! (J - Amax)! “* (J + &in)! (J - &in)!
1
for all J in the region Re J > Re a. Comparing Eqs. (3.11) and amplitude whenever J is equal if physical values are integral, even (odd) positive integer in tegral. Thus
(3.6) we see that A$h(J) to an even (odd) positive and similarly whenever Re J > Re Q:if physical
(3.12) is equal to the physical integer in Re J > Re 01 J - x is equal to an values are half-odd in-
24:,x = $$[I + GOSTJ + sin aJ]A$h(J) + M[l - cos TJ - sin ?rJ]AGh(J)
(3.13)
for physical values of J in Re J > Re LY.In a potential model we can readily identify A$h( J) with the amplitudes due to the sum or difference of a direct 6 See, for example, E. J. Squires (11).
332
CALOGERO,
CHARAP,
AND
SQUIRES
(Wigner) or exchange (Majorana) potential in a Schrodinger equation (see II). As in II we can write the partial-wave expansion, Eq. (2.12), in the form
fAtA =A=&,, 2 (2.J
+ WG-Q(J) &A(J, 2) + &A(J) &A(J, 211,
(3.14)
where we have used (-l)J+X&~(e) to eliminate
= &&r
- 0)
(3.15)
terms with a bad behavior for large J, and have introduced
d&t (J, x) = c&j f
(cos TX - sin ?TX)d{-~r(?r - /3).
(3.16)
With the amplitude written as in Eq. (3.14) it is possible to make the Sommerfeld-Watson transform as described below. The last statement was proved (in the Born approximation only) in II for a potential model. In general, it can be proved directly from the assumed existence of the integrals of Eq. (3.11). We use the form of Q?~~aX(z) for large J, which follows from Eq. (3.5), Qe;;lx(z)
_ T; (” ; ‘)-”
t
; ‘)-*
.J-“‘exp
(’
; e-‘)-
[-t(J
- La,
(’
;
e-‘~-l’z
(3.17)
+ 1)1[1 + O(J-‘)I,
for 1 J 1 -+ COwith 1arg J 1 < ?r. Here t = cash-’ z
(3.18)
and is real and positive for z real and greater than 1, which for positive energies is true in the region of integration in Eq. (3.11). It is important here that we consider only processes where the masses are such that there are no complex singularities. Otherwise zo in Eq. (3.11) etc. will not be real and the asymptotic form of A$h( J) will apparently not allow the Sommerfeld-Watson transform. This difficulty occurs, for example, when we regard one of the “particles” as a nonbound continuum state of two other particles (5, 6). If we substitute Eq. (3.16) into Eq. (3.11) we obtain ,&(J)
= ( -l)hmin-X &
J-l”
(1 -2 e-cy-1’2 (1 z e-‘>“-“’ (3.19)
nm X exp [--E(J - Lax + 1)1 j
=a
dzL$:(z)
f
&x)ltl + OCJ-‘)I,
which is the required result. From Carlson’s theorem it then follows that the continuation defined by Eq. (3.11) is the same as that previously defined for the potential problem.
J CONTINUATION
FOR
PARTICLES
WITH
SPIN
333
Using Eqs. (3.11) and (3.5) we see that A$,(J), defined by an equation analogous to Eq. (2.13), is holomorphic as a function of J in the region given by Eq. (3.12). If there were no coupling to other than two particle systems, as in the potential model of I, II, and III, then the proof that the only singularities of Affh( J) in the J-plane, which move with energy, are Regge poles would follow, exactly as in the work of Gribov (9). In general, however, as in the zero spin case, this must remain an open question. Putting together the various factorial functions in Eqs. (3.5) and (3.11) we see that if the integrals in Eq. (3.11) are defined down to Re J = A,,, - 1, then the summand in Eq. (3.14) contains a pole at J = X,,, - 1 unless the x integrals in Eq. (3.11) become exactly zero at this value of J. Thus, apart from the possibility of this “accidental” cancellation, there is always some singularity in the summand of Eq. (3.14) in the region’ Re J > A,,, - 1. Since, in general, all channels are coupled together and the largest value of x,,, will be .smal, the maximum possible total spin in any one of the coupled channels, this would imply (see next section) fvx(O)
> 0(x”““”
-
1).
(3.20)
Now we know from unitarity in the crossed channel (12) that for negative values of the energy, amplitudes cannot increase with z faster than zl. It follows that the “accidental” cancellation referred to above must take place whenever .smax> 2. Thus, the requirement of cross-channel unitarity enforces certain restrictions upon the single spectral functions, i.e., upon the “potential.” The precise nature of these restrictions will be the subject of a forthcoming paper by Calogero and Charap. The presence of these cancellations would also remove the objections to the Mandelstam representation raised recently by Azimov (15). It should be emphasized here that all our considerations are dependent on the assumption that a single dispersion relation, with a finite number of subtractions, exists. There is, however, at this stage nothing inconsistent in this assumption. Since there is some hope that cross-channel unitarity in fact determines the “potential” uniquely, it need not surprise us that we find some restrictions on the potential so readily! Finally we note that, as in the spin-zero case, we cannot be sure that the amplitudes S$ x( J) actually represent the physical amplitudes at the appropriate physical points in Re J < 1. The assumption that they do, which we shall make in Section V, is usually stated as the assumption that there are no “elementary” particles. This problem of course does not occur in the potential scattering model. 7 Note that the j&d (XII), apart from the added in proof.
poles possible
at J = A,., cancellation,
-
1, A,,, - 2, etc., which are ruled out by unitarity.
would arise from Eq. See, however, note
334
CALOGERO,
IV.
THE
CHARAP,
KINEMATICAL
AND
ZEROS
SQUIRES
OF
THE
AMPLITUDE
Using time reversal invariance* S$x(J>
= A&(J),
(4.1)
and the result that X$X(J) has no fixed singularities on the real J axis, we can immediately see, from Eq. (2.13), that AF7i( J) is finite on the real J axis, except for possible Regge poles, and hence that dffx (J) has certain kinematic zeros. These are obtained by a straightforward analysis of Eq. (2.13) and occur as follows : ALinvAm~x= 0,
for
J = 1 Xmin 1, 1Xmin 1 + 1 . * . X,,, - 1
(4.2)
and Ai!im*x3Xmin= 0,
for
J = -&,,
, -&,,
+ 1, *** -
[ Xmin 1 -
1.
(4.3)
For the A$x( J) these equations imply
A$p.(J)
= 0 for and for
J = I Xmin 1, I Xmin I + 1, *. * X,,, - 1 J = -Ax,,, , -Ax,,,
+ 1, *. . j Xmin 1 - 1.
(4.4)
Note that, in the vicinity of a zero of x at J = Jo , A - (J - Jo), whilst A N (J - Jo)“2. The significance of the vanishing of A at the points given by Eq. (4.4) is that there is decoupling (4) between ‘kense” and “nonsense” channels as mentioned in Section I. We should emphasize that we have only proved this property in potential models; in the relativistic case we depend upon the assumptions made in Section III. Also we note that the “nonsense-nonsense” amplitude, when both X and X’ are unphysical is not necessarily zero (14). It is, however, finite. V.
THE
BEHAVIOR
OF
THE
AMPLITUDE
FOR
LARGE
z
In this section we shall show that, under the assumptions, which are true in potential theory, that x*(J) is meromorphic and equal to the physical amplitude at the appropriate J-values, the large z behavior is dominated by Regge poles. Essentially this section repeats the discussion already given in II, although here we include the effect of the exchange potential and pay special attention to the effect of the nonse amplitudes. We begin with Eq. (3.14) and perform (as in II) the analogue of the Sommerfeld-Watson transform. Using the asymptotic form of A*(J) for large J, to allow us to neglect the contribution of the semicircular contour at infinity,g we obtain 8 If time reversal invariance does not hold, the same results can be obtained by repeating arguments of Section II with X and X’ interchanged. 9 Actually the required bound has been proven only where the integrals in Eq. (3.11) exist, i.e., for Re J > X,,, - 1. We shall assume that it holds for all J provided Re J is bounded below; a rigorous proof for potential scattering will be given in a later paper.
the
J CONTINUATION
fA%(e) =-1s
FOR
PARTICLES
WITH
-1’2+im dJ(2J + 1) MUJ)&(J, sin ?r(J + X) R-~--1/2 (2ai* . dxhr(J, -z>l - T sin 7r (ai* + X) L Xrnin-1 - E2) w 2i
-l/2-im
SPIN
+1>P&A &dJ, -4 +1)&(eL‘Gx , -2)
335
+ &h(J)
(5.1)
where the &hi1 are the residues of A$ x(J) at the Regge poles J = CQ*; the particular Regge pole being labelled by the suffix i. If physical values of J are halfodd integers then the integration in the first term of Eq. (5.1) must be taken just to the right of Re J = - $5. The last sum in Eq. (5.1), which is absent if &in < x, arises from the poles of cosec T(J + h) which occur at J = A,,, - 1, X,,,i,, - 2, . .. O(or x). Only terms with J < Xmin are actually present since (see Appendix) d
(5.2)
for J = Amin, Xmin + 1, . . . , Xl,,, - 1. The asymptotic form of fkrh(e) for large x can immediately be read off from Eq. (5.1): using Eqs. (A.3) and (A.4) of the Appendix. The first term, the socalled background integral term, behaves at most like z-l/2; each Regge pole term behaves like zai*; whilst the terms of the final sum behave like zeJ-‘. The dominance of the Regge poles, provided these lie in Re J > -35, is thus proven. In order to extend this result to apply to poles lying in Re J < - $6 we use the Mandelstam modification of the Sommerfeld-Watson transform (15). First, however, it is convenient to rewrite Eq. (5.1) in terms of the Legendre functions. Th.is is accomplished by means of the Clebsch-Gordan series bmx d:‘z&‘) d;y(e) = c P,+,(x)C(JX,,,J + 7; 7=--hmax (5.4) h, - X)C(JX,n,,J + 7; A’, - A’). For arbitrary J the Clebsch-Gordan coefficients are defined by means of the explicit formula of Wigner or Racah, as explained in I and II. Then Eq. (3.14) can be rewritten as
B:Q(J,
1’) = (2J + l)C(JX,,,J
+ r;
X, - X)C( JX,,,J
+ T; A’, - X’)&(J)
(5.6)
336
CALOGERO,
CHARAP,
AND
SQUIRES
and P*(J,
Whenwe
7, X) = ffJP~+,(x)
perform
f
(cos ~7 - sin 7r~)P~+~( -x)1.
the Sommerfeld-Watson
transform
(5.7)
upon Eq. (5.5) we
obtain
x [BCx(J,T)P+(J,7, -2) + BvxF(J, 7) -z)l
(5.8)
bin--l Pf(J, 7,-2) - c &AU,T)PJ+, (4
-iT
J=O(l/Z)
which is equivalent to Eq. (5.1) and leads to the same conclusions. Here &Jh is the residue of Bxf A(J, r) at J = ai*. In order usefully to move the contour further sum in Eq. (5.5) as
to the left we first write the
(5.9)
J
We then put L = - r, so that the first sum begins at L = 0, L being order of the Legendre functions which occur. The Mandelstam method then be applied to this term exactly as for the spin zero case (15). That S-matrix has the required symmetry on reflection of J about J = -45, values of J such that (J + s) is half-odd integral, was proved in Section of II. There remains the term
the can the for III
The defect in the discussion of Section III of II is that we failed to discuss this term. For potential scattering and plausibly also, in general, it is, however, zero. To show this we first note that one of the Clebsch-Gordan coefficients in Eq. (5.6) is given explicitly bylo
C(J, A, J + T; X, - X)
1
(2J - X + T)! (J + A)! (2X)! (2J + 27. + 1) 1’2 (5.11) (J - X)! (21 + x + T + I)! (A - T)! (x + r)! where we take, for example, X = A,,,, . This is zero for those values of J re=
quired in Eq. (5.10) which lie in 10 See Eq.
(3.29)
of ref.
16.
J
CONTINUATION
FOR
PARTICLES
WITH
337
SPIN
0 - L) 5 J 6 (x - 1),
(5.12)
so we can replace the expression (5.10) by
Now, for each L in the required range, the terms in J summation cancel in pairs, i. e., the J = J1 and J = - J1 - 1 term add to give zero, for each J1 in the range. To show this let us take the case where physical values of J are integers. Then, from Eq. (5.7) we have P+ (J, r = L -J,z)=P’(-J-l,L+J+l,x)
(5.14)
To obtain the symmetry of B$,( J, T) under the reflection J -+ - (J + 1) we use (2J + 1) = -[2(-J - 1) + l] (5.15) and C(JX,,,J
+ r; X - X) = C( - J -
1, X,,,J
+ r; X - X).
(5.16)
The latter equation can readily be obtained from Eq. (5.11) for X = A,,, , and from the induction formula on the projection quantum numbers, given in Eq. (3.27) of ref. 16, otherwise. The required result then follows from the symmetry &x(J) = A$A(-J - 1) (5.17) for integral values of J in the range required for Eq. (5.13). For potential theory this can be proved from the expressions given in III, by similar methods to those used in Section III of II; we omit the details here. This then completes our derivation that the expression (5.10) is zero when physical values of X are integral. Similar arguments lead to the same result when physical values are half-odd integral. APPENDIX THE
ASYMPTOTIC
I. BEHAVIOR
In this appendix we discuss the asymptotic ~0. Our starting point is
& (0) = ( - l)Xmin-h(J C(J
OF
c&t
69)
behavior of &, (0) as z = cos 0 +
1
+ A,,,) ! (J - Amin) ! “’ + &in)! (J - &ax)!
(A.11 FOmax &in)!
- J, J + Xmax+ 1; hoax - Amin + 1; 3 (1 - x)),
338
CALOGERO,
CHARAP,
AND
SQUIRES
where, without loss of generality (because &x8 (0) = &-x(O)), we suppose x msB2 [ Xmin 1. We also assume x and X’ to be both integer or both half-odd integer. First we note that &, vanishes identically if 1J - X,,,, 1 is an integer and either --X maL 5 J < -Xmin or Xmin 2 J < X,,, , which includes the “sensenonsense” region. In every other case an examination of the asymptotic behavior of the hypergeometric function leads to the result &TX, (0)
=
( _
)x,i,-h(i)x”““-x”i”(2J
+
I)-’
‘(:
‘+
>y,’ m,n
! :J”
1
!
pin’
!I”’
max
!
(2J + l)! ’
i (J
+
Lax)!
(J
-
bin)!
(A.21
(z)p+o(i)] 2
-
(-J-l+~~~~C,:‘~l-h..,i.,!(5)11[1+o(~)]}’
Specifically, unless it vanishes identically, &,(e) - zI ReJ+lP I-112, unless J is real and Xmin instead,
$5 &(e)
-
z-I Jt-112l--1/2.
is a positive Thus if A,,, - 4i-IJ+%l identically, or tends to zero as in Eq. (A.4). RECEIVED:
(A.31
1 J + .$s 1 is a positive integer, in which case,
integer, &f(e)
(A.41 either vanishes
May 6, 1963
Note added in proof. R. G. Newton has pointed out that unitarity does not in general rule out fixed poles in the amplitude when there is spin (see F. Calogero, J. M. Charap, and E. J. Squires, “Complex angular momentum in many-channel problems,” to be published in the Proceedings of the 1963 Sienna Conference, for further discussion of this point). The results of Section IV are therefore dependent on the assumption that there are no fixed poles. It should also be noted that the Froissart unitarity bound has not been proved for general spin values, so the unitarity requirements on the double-spectral-functions, noted below Eq. (3.20), would not necessarily hold if there was some decoupling mechanism which removed the relevant singularities from the amplitude for the scattering of particles with (say) zero spin. In general such a mechanism seems unlikely to exist for singularities which move. The papers by S. Mandelstam, “The Regge formalism for relativistic particles with spin” and “Cuts in the angular momentum plane I and II,” University of Birmingham preprints, 1963, contain further material relevant to the subject of this paper. ACNOWLEDGMENTS
One of us (F. C.) wishes to thank the Commonwealth Physics Department of Princeton University for their Professor J. Robert Oppenheimer for his hospitality
Fund for financial support, hospitality. J. M. C. wishes at the Institute for Advanced
and the to thank Study.
J CONTINUATION
FOR
PARTICLES
WITH
SPIN
339
REFERENCES 1. J. M. CHARAP AND E. J. SQUIRES, Ann. Phys. (N. Y.) 20, 145 (1962); ibid. 21, 8 (1963); also preprint, On complex angular momentum in many channel potential scattering problems III, (1963). These papers will be denoted by I, II, and III respectively. 2. V. N. GRIBOV, Soviet Phys. JETP 16, 873 (1962). 8. M. FROISSART, Phys. Rev. 133, 1053 (1962); and unpublished talk at La Jolla Conference (1961). 4. M. GELL-MANN, 1961 Intern. Conf. High Energy Phys. (CERN), p. 533. 6. I. T. DRUMMOND, preprint, Regge cuts and three particle states in non-relativistic model. Cambridge University, 1962; R. G. NEWTON, preprint, The total angular momentum plane in the non-relativistic three body problem. University of Rome, 1963. 6. T. W. B. KIBBLE, preprint, Feynman rules for Regge particles. London, 1962. 7. M. JACOB AND G. C. WICK, Ann. Phys. (N. I’.) 7, 404 (1959). “Higher Transcendental Functions,” A. Erdelyi, ed. 8. Bateman Manuscript Project, McGraw-Hill, New York, 1953. Section 5.81. Oxford Univ. Press, 1952. 9. E. C. TITCHMARSH, “Theory of Functions,” 10. I. MUZINICH, UCRL 10331 (1962); A. 0. BARUT, I. MUZINICH, AND D. WILLIAMS, UCRL 10463 (1962). 11. E. J. SQUIRES, Nuovo Cimento 26, 242 (1962). 1%‘. M. FROISSART, Phys. Rev. 123, 1053 (1962); A. MARTIN, Phys. Rev. 129, 1432 (1963). 13. YA. I. AZIMOV, Phys. Letters 3, 195 (1963). 14. V. B. BERESTETSKY, Phys. Letters 3, 175 (1963). 16. S. MANDELSTAM, Ann. Phys. (N. I’.) 19, 254 (1962). 16. M. E. ROSE, “Elementary Theory of Angular Momentum.” Wiley, New York, 1957.
OF PHYSICS:
26,
325-339
The Continuation Wave Scattering
(1963)
in Total Angular Amplitudes for
Momentum of PartialParticles with Spin*
FRANCESCOCALOGERO~ Palmer
Physical
Laboratory,
Princeton
University,
Princeton,
New
Jersey
JOHN M. CHARAP~ Institute
for
Advanced
Study,
Princeton,
New
Jersey
AND EUAN J. SQUIRES Tait
Institute
of Mathematical
Physics,
Edinburgh
8, Scotland
An assumed one-dimensional dispersion relation is used to define a continuation in total angular momentum of the helicity scattering amplitude, for particles with arbitrary spin. The result, which is a generalization of the Froissart-Gribov formula for zero spin, agrees with that discussed previously for potential models. It is shown that cross-channel unitarity imposed certain restrictions on the weight functions. The existence of certain kinematic zeros of the amplitude, corresponding to the decoupling of “sense” and “nonsense” channels, is proved in general, and finally it is shown that the presence of the “nonsense-nonsense” amplitudes does not affect the large z behavior, which is still dominated by Regge poles. I.
INTRODUCTION
There are three main objects of this paper. First, we wish to provide a summary of the essential results of three papers by two of the present authors (1) on the subject of complex angular momentum for nonrelativistic scattering of particles with spin. This summary is contained mainly in Sections II and V of * The study was supported in part by the Air Force Office of Scientific Research Grant AF-AFOSR-42-63. t Harkness Fellow of the Commonwealth Fund, New York, on leave of absence from Istituto Nazionale di Fisica Nucleare, Sezione di Roma, and Istituto di Fisica dell’Universita, Roma, Italy. $ Present address: Istituto di Fisica dell’Universita, Roma, Italy. Present address: CERN, Geneva, Switzerland. 32.5
326
CALOGERO,
CHARAP,
AND
SQUIRES
the present paper. Secondly, in Section III, we use the potential model as an analogy in order to define the appropriate continuation of the partial-wave amplitude in relativistic scattering, i.e., we generalize the Froissart-Gribov definition (2, 3) to apply to arbitrary spins. It is necessary here, of course, to assume that the scattering amplitude satisfies certain analyticity properties in the variables, and is suitably behaved as these variables become large. We further show that these assumptions then lead to certain restrictions on the form of the single spectral functions when particles of high spin are involved. These restrictions, which will be discussed in detail in a later paper by two of the present authors, arise from the requirement of unitarity in the crossed channels and have, therefore, no analogy in the potential model. Finally, we consider the states of positive integral or half-odd integral J, according to whether the total spin is integral or half-odd integral, which are unphysical due to their being less than the maximum value of the helicity. Such states are called “nonsense” states by Gell-Mann (4), and appear to be a source of confusion. In Section IV we prove that in potential theory (and very plausibly also in the relativistic theory) there is no coupling between the “sense” and “nonsense” states. However, contrary to what appears to have been assumed, the “nonsense-nonsense” amplitude is not exactly zero. At first sight this might be expected to vitiate the main result of II, that the scattering amplitude is always dominated, at high energies, by Regge poles. That this is not the case, however, is demonstrated explicitly in Section V, where it is shown that the Regge behavior holds for arbitrary spin. II.
THE
PARTIAL-WAVE
EXPANSION
We are concerned with scattering processes in which both initial and final states contain two particles of arbitrary spins. The formalism can, in fact, also be used where the states contain more than two particles (5, 6) but in this case the relevant matrices have infinite dimensions and the presence of “disconnected graphs” means that Fredholm theory is not immediately applicable. A consequence (5) is that branch points appear in the J-plane. It is possible (and desirable!) that these should cancel in a consistent relativistic theory, but such considerations are beyond the scope of this article. For a transition from an initial state with spins s1 and s2 , helicities Xi and X2 and barycentric momentum k, to a final state in which the corresponding quantities are sl’, a~‘, X,‘, X2’ and x3’,we write the scattering amplitude as (Sl’, s2’; Xl’,
x2’;
k’l f(e, 4 lSlS2; x1x2 ; 12).
Here we have suppressed any other quantum numbers which may be necessary to specify completely the two-particle states. If the masses of the particles are
J CONTINUATION
ml and m2 initially,
FOR
PARTICLES
and ml’ and mzf finally,
WITH
327
SPIN
then
(ml’ + ?c~)~‘~ + (mz” + k2)1’2 = (m:2 + k’y2
+ (,,$ + li2y2
2E.
(2.1)
We introduce x = x1 - x2 x’
=
x1’
-
(2.2)
x2/,
and also A,,, = max (A, A’)
(2.3)
Xmin = min (X, A’). Provided we restrict ourselves to parity conserving interactions of generality in taking
there is no loss
xmax 2 ( Xmin 1. This is because under parity conjugation (SX
x1’x2’; k’l f(O, 4)
+ (&‘;
ISIS2
;
XIX2
(2.4)
we have
; k)
--Al’ - hzl; k’l f( 8, T - 4) ISIS2; --x1 - x2 ; k)
(2.5)
and the two amplitudes are equal apart from a phase (7). If parity is not conserved then it is a simple matter to deal with the other amplitudes in an analogous way to that followed below. The partial-wave expansion of the scattering amplitude can be written in the form (7, I )
xysl
s2’ s’; x1’ - A,‘)
X g C(ZsJ; OX)C(Z’ s’ J’; OX’) (2.6)
328
CALOGERO,
CHARAP,
AND
SQUIRES
where SJ is the conventionally defined S-matrix for definite total angular momentum J. The values of s, s’, 1, and 1’ are integers, or, in the cases of s and .sf, possibly half-odd integers, with ranges restricted by the vanishing of the ClebschGordan coefficients. The elements of the rotation matrix are given by &(-4, and &!(e)
8, +> = exp [i(X -
A’>01C&(O),
(2.7)
can be written in the form*
where we have introduced (2.9)
and (2.10)
x = cos 9.
The quantity P$‘b’( z ) is a Jacobi polynomial of the first kind. Its analytic continuation and asymptotic behavior may be obtained from its connection with the hypergeometric function,
P:‘“‘(x)
=(n~‘)F(-n,n+a+b+l,a+~;~(l--z)).
It is convenient
(2.11)
to rewrite Eq. (2.6) in the condensed form (2.12)
where the partial-wave amplitude AJXIXis defined by comparison of Eq. (2.6) with Eq. (2.12). The important result of II and III that the modified partialwave amplitude XXIX defined by
A;,~ =
1
(J + X’) ! (J - X’) ! 1’2A., (J+x)!(J-A)! x’x
(2.13)
can be continued away from physical J values to give a function of J, which is holomorphic in the whole finite J-plane, apart from isolated poles (Regge poles) which move with energy, and which is suitably bounded as J tends to infinity 1 See Eq. (2.8) of II, and Eq. (10.8) (10) of ref. 8.
J CONTINUATION
POR
PARTICLES
WITH
SPIN
329
to permit the analogy of the Sommerfeld-Watson transformation. Some restrictions on the potential are necessary of course for these results to hold; these are discussed explicitly in ref. 1, and we shall assume them here. The bound at J = m was proved only in the Born approximation; a rigorous proof will be the subject of a later paper by two of the present authors. According to Carlson’s theorem (9) the above properties distinguish the particular continuation uniquely. As discussed in III it follows from these results and from Eq. (2.8) and Eq. (2.11), that the J-summand in Eq. (2.12) is holomorphic apart from Regge poles. III.
THE
GENERALIZED
FROISSART-GRIBOV
FORMULA
In order to give an explicit definition of the continued partial-wave amplitude in the relativistic case it is necessary first to write a one-dimensional dispersion relation for the scattering amplitude (cf. refs. 2 and 3). The appropriate dispersion relation can be found by analogy with the potential model.* Explicitly, using Eqs. (2.8) and (2.12), we know that
(!q
(g-b’2jA,A(e)
is an analytic function of z, in the z-plane cut along the positive real axis from 2 = 1 to 2 = m. Note that we keep the energy real and positive here. The convergence of the original partial wave expansion, Eq. (2.6)) within the Lehmann ellipse shows that the cut actually starts at some .zo> 1. Thus, recalling that in the relativistic case there is always an “exchange potential” (corresponding to the existence of two crossed channels), we can write
where, for the moment, we have ignored the question of subtractions which may be necessary to make the integrals in Eq. (3.1) converge. The proof that fhfx(~) has the analyticity required by Eq. (3.1) has not yet been given for the relativistic case in a fully satisfactory way; the problem here, however, is identical to that in the zero spin case. The orthogonality properties of the rotation matrices allow us to insert Eq. (2.12) and obtain A:q = $$ 2 Similar
but
less detailed
considerations
s-1 +l dz.hx(e) are given
&(e>. in ref.
(3.2) IO.
330
CALOGERO,
If we substitute values of J,
CHARAP,
AND
SQUIRES
for ~XIXfrom Eq. (3.1) and use Eq. (2.8) we find, for physical
A:xq = (-1)““in-k
(J + A,,,) ! (J - Xmx) !] “* x ;&G(p) (J + X,i,)! (J - &in)! (3.3)
. (qqb
P.jTf~&)
x I,
dz’ [-
+ &]
.
We now change the order of integration in Eq. (3.3)) which is permissible for suiliciently large J, and perform the integral over x by means of the following generalization of Neumann’s formula3
6?t*“‘
- I)-“(2. + l)-bL;
(/ - $l(l
_ z)“(l + $
(3.4)
*Pcasb) 11 (2’) dx, which holds for n a nonnegative integer and for a and b greater than -1. The Q?Vb’(z’) is a Jacobi function of the second kind; its analytic continuation and asymptotic behavior follow from its relation with the hypergeometric function4
Q;,“‘(x) = f iIn++a,j!Fb’,“:,!! (” ; ‘)-“-’
t+)-b (3.5)
-F(n + 1, n + a -I- 1; 2n + a + b + 2; 2(1 - x)-l). We note that in Eq. (3.3) both a and b are positive, also that, for physical J, J - Xmaxis a nonnegative integer. Thus we can use Eq. (3.5) in Eq. (3.3) and obtain, still for physical J, &,x
(J + X,,,) ! (J - Xmx) ! “’ (J + Amin)! (J - Amin)!
1 md&(x)Q.t~~,,(z) t+y (q) jz. +(qJ--hm,xI m lrs20
= ( -l)xmin-x
where, in obtaining
the last term we have also used the symmetry5
Pp3”‘(z) = ( -l)“PAb*“)( for nonnegative
(3.6)
integral
-x),
(3.7)
n.
8 This follows from Eq. (10.8) (20) Jacobi polynomials. 4 See Eq. (10.8) (18) of ref. 8. 6 See Eq. (10.8) (13) of ref. 8.
of ref.
8, and from
the orthogonality
properties
of the
J
CONTINUATION
FOR
PARTICLES
We shall see below that (at least for potential h(O)
SPIN
331
scattering)
= 0(x*)
where J = ar is the position of the rightmost of energy). It follows that &;a’(~)
WITH
(3.3)
Regge pole (in general a function
= O($--)
(3.9)
so that the integrals in Eq. (3.6) converge provided
J > Re LY.
(3.10)
It is easy to see that the presence of the necessary subtractions in Eq. (3.1) does not alter Eq. (3.6) provided Eq. (3.10) is satisfied, so this formula correctly gives the partial-wave amplitudes for all physical J values in this region. The factor ( - 1) J--xmaxin the second term of Eq. (3.6) makes the expression unsuitable for continuation to nonphysical J, so we define, as in the zero-spin case,6 two amplitudes A$i( J) by the expressions j&(J)
= (-l)Xmi=A
(J + X,,,) ! (J - Amax)! “* (J + &in)! (J - &in)!
1
for all J in the region Re J > Re a. Comparing Eqs. (3.11) and amplitude whenever J is equal if physical values are integral, even (odd) positive integer in tegral. Thus
(3.6) we see that A$h(J) to an even (odd) positive and similarly whenever Re J > Re Q:if physical
(3.12) is equal to the physical integer in Re J > Re 01 J - x is equal to an values are half-odd in-
24:,x = $$[I + GOSTJ + sin aJ]A$h(J) + M[l - cos TJ - sin ?rJ]AGh(J)
(3.13)
for physical values of J in Re J > Re LY.In a potential model we can readily identify A$h( J) with the amplitudes due to the sum or difference of a direct 6 See, for example, E. J. Squires (11).
332
CALOGERO,
CHARAP,
AND
SQUIRES
(Wigner) or exchange (Majorana) potential in a Schrodinger equation (see II). As in II we can write the partial-wave expansion, Eq. (2.12), in the form
fAtA =A=&,, 2 (2.J
+ WG-Q(J) &A(J, 2) + &A(J) &A(J, 211,
(3.14)
where we have used (-l)J+X&~(e) to eliminate
= &&r
- 0)
(3.15)
terms with a bad behavior for large J, and have introduced
d&t (J, x) = c&j f
(cos TX - sin ?TX)d{-~r(?r - /3).
(3.16)
With the amplitude written as in Eq. (3.14) it is possible to make the Sommerfeld-Watson transform as described below. The last statement was proved (in the Born approximation only) in II for a potential model. In general, it can be proved directly from the assumed existence of the integrals of Eq. (3.11). We use the form of Q?~~aX(z) for large J, which follows from Eq. (3.5), Qe;;lx(z)
_ T; (” ; ‘)-”
t
; ‘)-*
.J-“‘exp
(’
; e-‘)-
[-t(J
- La,
(’
;
e-‘~-l’z
(3.17)
+ 1)1[1 + O(J-‘)I,
for 1 J 1 -+ COwith 1arg J 1 < ?r. Here t = cash-’ z
(3.18)
and is real and positive for z real and greater than 1, which for positive energies is true in the region of integration in Eq. (3.11). It is important here that we consider only processes where the masses are such that there are no complex singularities. Otherwise zo in Eq. (3.11) etc. will not be real and the asymptotic form of A$h( J) will apparently not allow the Sommerfeld-Watson transform. This difficulty occurs, for example, when we regard one of the “particles” as a nonbound continuum state of two other particles (5, 6). If we substitute Eq. (3.16) into Eq. (3.11) we obtain ,&(J)
= ( -l)hmin-X &
J-l”
(1 -2 e-cy-1’2 (1 z e-‘>“-“’ (3.19)
nm X exp [--E(J - Lax + 1)1 j
=a
dzL$:(z)
f
&x)ltl + OCJ-‘)I,
which is the required result. From Carlson’s theorem it then follows that the continuation defined by Eq. (3.11) is the same as that previously defined for the potential problem.
J CONTINUATION
FOR
PARTICLES
WITH
SPIN
333
Using Eqs. (3.11) and (3.5) we see that A$,(J), defined by an equation analogous to Eq. (2.13), is holomorphic as a function of J in the region given by Eq. (3.12). If there were no coupling to other than two particle systems, as in the potential model of I, II, and III, then the proof that the only singularities of Affh( J) in the J-plane, which move with energy, are Regge poles would follow, exactly as in the work of Gribov (9). In general, however, as in the zero spin case, this must remain an open question. Putting together the various factorial functions in Eqs. (3.5) and (3.11) we see that if the integrals in Eq. (3.11) are defined down to Re J = A,,, - 1, then the summand in Eq. (3.14) contains a pole at J = X,,, - 1 unless the x integrals in Eq. (3.11) become exactly zero at this value of J. Thus, apart from the possibility of this “accidental” cancellation, there is always some singularity in the summand of Eq. (3.14) in the region’ Re J > A,,, - 1. Since, in general, all channels are coupled together and the largest value of x,,, will be .smal, the maximum possible total spin in any one of the coupled channels, this would imply (see next section) fvx(O)
> 0(x”““”
-
1).
(3.20)
Now we know from unitarity in the crossed channel (12) that for negative values of the energy, amplitudes cannot increase with z faster than zl. It follows that the “accidental” cancellation referred to above must take place whenever .smax> 2. Thus, the requirement of cross-channel unitarity enforces certain restrictions upon the single spectral functions, i.e., upon the “potential.” The precise nature of these restrictions will be the subject of a forthcoming paper by Calogero and Charap. The presence of these cancellations would also remove the objections to the Mandelstam representation raised recently by Azimov (15). It should be emphasized here that all our considerations are dependent on the assumption that a single dispersion relation, with a finite number of subtractions, exists. There is, however, at this stage nothing inconsistent in this assumption. Since there is some hope that cross-channel unitarity in fact determines the “potential” uniquely, it need not surprise us that we find some restrictions on the potential so readily! Finally we note that, as in the spin-zero case, we cannot be sure that the amplitudes S$ x( J) actually represent the physical amplitudes at the appropriate physical points in Re J < 1. The assumption that they do, which we shall make in Section V, is usually stated as the assumption that there are no “elementary” particles. This problem of course does not occur in the potential scattering model. 7 Note that the j&d (XII), apart from the added in proof.
poles possible
at J = A,., cancellation,
-
1, A,,, - 2, etc., which are ruled out by unitarity.
would arise from Eq. See, however, note
334
CALOGERO,
IV.
THE
CHARAP,
KINEMATICAL
AND
ZEROS
SQUIRES
OF
THE
AMPLITUDE
Using time reversal invariance* S$x(J>
= A&(J),
(4.1)
and the result that X$X(J) has no fixed singularities on the real J axis, we can immediately see, from Eq. (2.13), that AF7i( J) is finite on the real J axis, except for possible Regge poles, and hence that dffx (J) has certain kinematic zeros. These are obtained by a straightforward analysis of Eq. (2.13) and occur as follows : ALinvAm~x= 0,
for
J = 1 Xmin 1, 1Xmin 1 + 1 . * . X,,, - 1
(4.2)
and Ai!im*x3Xmin= 0,
for
J = -&,,
, -&,,
+ 1, *** -
[ Xmin 1 -
1.
(4.3)
For the A$x( J) these equations imply
A$p.(J)
= 0 for and for
J = I Xmin 1, I Xmin I + 1, *. * X,,, - 1 J = -Ax,,, , -Ax,,,
+ 1, *. . j Xmin 1 - 1.
(4.4)
Note that, in the vicinity of a zero of x at J = Jo , A - (J - Jo), whilst A N (J - Jo)“2. The significance of the vanishing of A at the points given by Eq. (4.4) is that there is decoupling (4) between ‘kense” and “nonsense” channels as mentioned in Section I. We should emphasize that we have only proved this property in potential models; in the relativistic case we depend upon the assumptions made in Section III. Also we note that the “nonsense-nonsense” amplitude, when both X and X’ are unphysical is not necessarily zero (14). It is, however, finite. V.
THE
BEHAVIOR
OF
THE
AMPLITUDE
FOR
LARGE
z
In this section we shall show that, under the assumptions, which are true in potential theory, that x*(J) is meromorphic and equal to the physical amplitude at the appropriate J-values, the large z behavior is dominated by Regge poles. Essentially this section repeats the discussion already given in II, although here we include the effect of the exchange potential and pay special attention to the effect of the nonse amplitudes. We begin with Eq. (3.14) and perform (as in II) the analogue of the Sommerfeld-Watson transform. Using the asymptotic form of A*(J) for large J, to allow us to neglect the contribution of the semicircular contour at infinity,g we obtain 8 If time reversal invariance does not hold, the same results can be obtained by repeating arguments of Section II with X and X’ interchanged. 9 Actually the required bound has been proven only where the integrals in Eq. (3.11) exist, i.e., for Re J > X,,, - 1. We shall assume that it holds for all J provided Re J is bounded below; a rigorous proof for potential scattering will be given in a later paper.
the
J CONTINUATION
fA%(e) =-1s
FOR
PARTICLES
WITH
-1’2+im dJ(2J + 1) MUJ)&(J, sin ?r(J + X) R-~--1/2 (2ai* . dxhr(J, -z>l - T sin 7r (ai* + X) L Xrnin-1 - E2) w 2i
-l/2-im
SPIN
+1>P&A &dJ, -4 +1)&(eL‘Gx , -2)
335
+ &h(J)
(5.1)
where the &hi1 are the residues of A$ x(J) at the Regge poles J = CQ*; the particular Regge pole being labelled by the suffix i. If physical values of J are halfodd integers then the integration in the first term of Eq. (5.1) must be taken just to the right of Re J = - $5. The last sum in Eq. (5.1), which is absent if &in < x, arises from the poles of cosec T(J + h) which occur at J = A,,, - 1, X,,,i,, - 2, . .. O(or x). Only terms with J < Xmin are actually present since (see Appendix) d
(5.2)
for J = Amin, Xmin + 1, . . . , Xl,,, - 1. The asymptotic form of fkrh(e) for large x can immediately be read off from Eq. (5.1): using Eqs. (A.3) and (A.4) of the Appendix. The first term, the socalled background integral term, behaves at most like z-l/2; each Regge pole term behaves like zai*; whilst the terms of the final sum behave like zeJ-‘. The dominance of the Regge poles, provided these lie in Re J > -35, is thus proven. In order to extend this result to apply to poles lying in Re J < - $6 we use the Mandelstam modification of the Sommerfeld-Watson transform (15). First, however, it is convenient to rewrite Eq. (5.1) in terms of the Legendre functions. Th.is is accomplished by means of the Clebsch-Gordan series bmx d:‘z&‘) d;y(e) = c P,+,(x)C(JX,,,J + 7; 7=--hmax (5.4) h, - X)C(JX,n,,J + 7; A’, - A’). For arbitrary J the Clebsch-Gordan coefficients are defined by means of the explicit formula of Wigner or Racah, as explained in I and II. Then Eq. (3.14) can be rewritten as
B:Q(J,
1’) = (2J + l)C(JX,,,J
+ r;
X, - X)C( JX,,,J
+ T; A’, - X’)&(J)
(5.6)
336
CALOGERO,
CHARAP,
AND
SQUIRES
and P*(J,
Whenwe
7, X) = ffJP~+,(x)
perform
f
(cos ~7 - sin 7r~)P~+~( -x)1.
the Sommerfeld-Watson
transform
(5.7)
upon Eq. (5.5) we
obtain
x [BCx(J,T)P+(J,7, -2) + BvxF(J, 7) -z)l
(5.8)
bin--l Pf(J, 7,-2) - c &AU,T)PJ+, (4
-iT
J=O(l/Z)
which is equivalent to Eq. (5.1) and leads to the same conclusions. Here &Jh is the residue of Bxf A(J, r) at J = ai*. In order usefully to move the contour further sum in Eq. (5.5) as
to the left we first write the
(5.9)
J
We then put L = - r, so that the first sum begins at L = 0, L being order of the Legendre functions which occur. The Mandelstam method then be applied to this term exactly as for the spin zero case (15). That S-matrix has the required symmetry on reflection of J about J = -45, values of J such that (J + s) is half-odd integral, was proved in Section of II. There remains the term
the can the for III
The defect in the discussion of Section III of II is that we failed to discuss this term. For potential scattering and plausibly also, in general, it is, however, zero. To show this we first note that one of the Clebsch-Gordan coefficients in Eq. (5.6) is given explicitly bylo
C(J, A, J + T; X, - X)
1
(2J - X + T)! (J + A)! (2X)! (2J + 27. + 1) 1’2 (5.11) (J - X)! (21 + x + T + I)! (A - T)! (x + r)! where we take, for example, X = A,,,, . This is zero for those values of J re=
quired in Eq. (5.10) which lie in 10 See Eq.
(3.29)
of ref.
16.
J
CONTINUATION
FOR
PARTICLES
WITH
337
SPIN
0 - L) 5 J 6 (x - 1),
(5.12)
so we can replace the expression (5.10) by
Now, for each L in the required range, the terms in J summation cancel in pairs, i. e., the J = J1 and J = - J1 - 1 term add to give zero, for each J1 in the range. To show this let us take the case where physical values of J are integers. Then, from Eq. (5.7) we have P+ (J, r = L -J,z)=P’(-J-l,L+J+l,x)
(5.14)
To obtain the symmetry of B$,( J, T) under the reflection J -+ - (J + 1) we use (2J + 1) = -[2(-J - 1) + l] (5.15) and C(JX,,,J
+ r; X - X) = C( - J -
1, X,,,J
+ r; X - X).
(5.16)
The latter equation can readily be obtained from Eq. (5.11) for X = A,,, , and from the induction formula on the projection quantum numbers, given in Eq. (3.27) of ref. 16, otherwise. The required result then follows from the symmetry &x(J) = A$A(-J - 1) (5.17) for integral values of J in the range required for Eq. (5.13). For potential theory this can be proved from the expressions given in III, by similar methods to those used in Section III of II; we omit the details here. This then completes our derivation that the expression (5.10) is zero when physical values of X are integral. Similar arguments lead to the same result when physical values are half-odd integral. APPENDIX THE
ASYMPTOTIC
I. BEHAVIOR
In this appendix we discuss the asymptotic ~0. Our starting point is
& (0) = ( - l)Xmin-h(J C(J
OF
c&t
69)
behavior of &, (0) as z = cos 0 +
1
+ A,,,) ! (J - Amin) ! “’ + &in)! (J - &ax)!
(A.11 FOmax &in)!
- J, J + Xmax+ 1; hoax - Amin + 1; 3 (1 - x)),
338
CALOGERO,
CHARAP,
AND
SQUIRES
where, without loss of generality (because &x8 (0) = &-x(O)), we suppose x msB2 [ Xmin 1. We also assume x and X’ to be both integer or both half-odd integer. First we note that &, vanishes identically if 1J - X,,,, 1 is an integer and either --X maL 5 J < -Xmin or Xmin 2 J < X,,, , which includes the “sensenonsense” region. In every other case an examination of the asymptotic behavior of the hypergeometric function leads to the result &TX, (0)
=
( _
)x,i,-h(i)x”““-x”i”(2J
+
I)-’
‘(:
‘+
>y,’ m,n
! :J”
1
!
pin’
!I”’
max
!
(2J + l)! ’
i (J
+
Lax)!
(J
-
bin)!
(A.21
(z)p+o(i)] 2
-
(-J-l+~~~~C,:‘~l-h..,i.,!(5)11[1+o(~)]}’
Specifically, unless it vanishes identically, &,(e) - zI ReJ+lP I-112, unless J is real and Xmin instead,
$5 &(e)
-
z-I Jt-112l--1/2.
is a positive Thus if A,,, - 4i-IJ+%l identically, or tends to zero as in Eq. (A.4). RECEIVED:
(A.31
1 J + .$s 1 is a positive integer, in which case,
integer, &f(e)
(A.41 either vanishes
May 6, 1963
Note added in proof. R. G. Newton has pointed out that unitarity does not in general rule out fixed poles in the amplitude when there is spin (see F. Calogero, J. M. Charap, and E. J. Squires, “Complex angular momentum in many-channel problems,” to be published in the Proceedings of the 1963 Sienna Conference, for further discussion of this point). The results of Section IV are therefore dependent on the assumption that there are no fixed poles. It should also be noted that the Froissart unitarity bound has not been proved for general spin values, so the unitarity requirements on the double-spectral-functions, noted below Eq. (3.20), would not necessarily hold if there was some decoupling mechanism which removed the relevant singularities from the amplitude for the scattering of particles with (say) zero spin. In general such a mechanism seems unlikely to exist for singularities which move. The papers by S. Mandelstam, “The Regge formalism for relativistic particles with spin” and “Cuts in the angular momentum plane I and II,” University of Birmingham preprints, 1963, contain further material relevant to the subject of this paper. ACNOWLEDGMENTS
One of us (F. C.) wishes to thank the Commonwealth Physics Department of Princeton University for their Professor J. Robert Oppenheimer for his hospitality
Fund for financial support, hospitality. J. M. C. wishes at the Institute for Advanced
and the to thank Study.
J CONTINUATION
FOR
PARTICLES
WITH
SPIN
339
REFERENCES 1. J. M. CHARAP AND E. J. SQUIRES, Ann. Phys. (N. Y.) 20, 145 (1962); ibid. 21, 8 (1963); also preprint, On complex angular momentum in many channel potential scattering problems III, (1963). These papers will be denoted by I, II, and III respectively. 2. V. N. GRIBOV, Soviet Phys. JETP 16, 873 (1962). 8. M. FROISSART, Phys. Rev. 133, 1053 (1962); and unpublished talk at La Jolla Conference (1961). 4. M. GELL-MANN, 1961 Intern. Conf. High Energy Phys. (CERN), p. 533. 6. I. T. DRUMMOND, preprint, Regge cuts and three particle states in non-relativistic model. Cambridge University, 1962; R. G. NEWTON, preprint, The total angular momentum plane in the non-relativistic three body problem. University of Rome, 1963. 6. T. W. B. KIBBLE, preprint, Feynman rules for Regge particles. London, 1962. 7. M. JACOB AND G. C. WICK, Ann. Phys. (N. I’.) 7, 404 (1959). “Higher Transcendental Functions,” A. Erdelyi, ed. 8. Bateman Manuscript Project, McGraw-Hill, New York, 1953. Section 5.81. Oxford Univ. Press, 1952. 9. E. C. TITCHMARSH, “Theory of Functions,” 10. I. MUZINICH, UCRL 10331 (1962); A. 0. BARUT, I. MUZINICH, AND D. WILLIAMS, UCRL 10463 (1962). 11. E. J. SQUIRES, Nuovo Cimento 26, 242 (1962). 1%‘. M. FROISSART, Phys. Rev. 123, 1053 (1962); A. MARTIN, Phys. Rev. 129, 1432 (1963). 13. YA. I. AZIMOV, Phys. Letters 3, 195 (1963). 14. V. B. BERESTETSKY, Phys. Letters 3, 175 (1963). 16. S. MANDELSTAM, Ann. Phys. (N. I’.) 19, 254 (1962). 16. M. E. ROSE, “Elementary Theory of Angular Momentum.” Wiley, New York, 1957.