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Asymptotic bounds on the derivatives of the elastic scattering amplitudes
ANNALS
OF PHYSICS
109,
143-164 (1977)
Asymptotic
Bounds on the Derivatives Elastic Scattering Amplitudes ARVIND
of the
S. VENGURLEKAR
Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India Received October 16, 1976
Using unitarity, analyticity in the Lehmann-Martin ellipse, and asymptotic polynomial boundedness, rigorous high energy upper bounds are established on the derivatives of the absorptive parts A@, t) of elastic scattering amplitudes at i = 0. These are used to obtain a number of useful results including bounds on A@, t) and its derivatives for 0 < I < to, s --, m (t,, = 4m,,a for nn and TN). Lower bounds on the derivatives at t < 0 are derived and used to determine the near-forward physical region in which the derivatives can not vanish. We also obtain upper bounds on the derivatives of elastic differential cross sections for t < tJ4, s ---f co.
1. INTRODUCTION The study of restrictions on elastic scattering amplitudes F(s, t) obtained only from certain general principles such as unitarity and analyticity has been of interest for quite some time. Within this framework, for example, several authors [l-5] have obtained exact results regarding the high energy behavior of F(s, t) and the absorptive parts A(s, t) of F(s, t) near t = 0 in the complex t-plane. These results have many interesting consequences regarding, e.g., “the diffraction peak width,” Regge behavior, and zeros of the amplitudes in the t-plane. Corresponding information with regard to the derivatives of these amplitudes, however, is either incomplete or nonexistent. Previously, Eden [6] obtained high energy upper gounds on d”A(s, t)/dt” ItZO of the form d”A(s, t)/dt” It=,, < const xs(log s)~~+~, s ---f co, 1, 2,.... Both upper and lower bounds on &A(s, t)/& jteO for s --f co, have been recently obtained by Singh [7] in terms of dmA(S, t)/dt” ItXO, m = 0, 1, 2 ,..., n - 1. (For a general n, however, an explicit form for the upper bounds is too hard to get from Singh’s results.) Similar results for t # 0, however, are not to be found, except for recent bounds on din A(s, t)/dt, s --f co, 0 < t < to derived by Chung [S].+ The purpose of the present paper is to establish the best possible upper bounds on the derivatives (d”A(s, t)/dt”) ItEO, n = 1, 2 ,..., s ---f co, given (i) unitarity (ii) analyticity within Leymann-Martin ellipse, and (iii) the Jin-Martin upper bound [9] 4s, t> < wo)*,
s-
a,
(1.1)
+ Recently, asymptotic lower bounds for 0 < t < t, and unitarity upper and lower bounds for t < 0 have been obtained on d In A(s, t)/dt. See Ref. 23.
143 Copyright All rights
Q 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-4916
144
ARVIND S. VENGURLEKAR
where t = 2k2(z - 1) < t, , t, being related to the semimajor axis x of the LehmannMartin ellipse by x = 1 + to/2k2, where k is the c.m. momentum. (For Z-T, Z-N, ~TK,KK, KK sc6ttering we have (to)1/2 = 2r?r,.) These bounds are then used to deduce (i) upper bounds on A@, t) for 0 < t, < to, s + co, (ii) upper bounds on PA@, t)/dt”, n = 1, 2,... for 0 < t < to, s + co, (iii) information on the region near t = 0 in the complex t-plane in which PA@, t)/dt”, n = 1,2,..., s --f co can not vanish, (iv) an infinite series of upper bounds on (~&/a,~), s + co, (v) restrictions on the parameters entering in the usual forms fitting F(s, t) for t < 0. We also obtain, for t < 0, lower bounds on dA(s, t)/dt and on certain combinations of d”A(s, t)/dt”. These again can be used to find the near-forward physical region in t in which the derivatives cannot vanish. Similar results in the case of upper bounds on the derivatives d”(do/dt) dt”, n = 1,2,... are proved. It should be mentioned that our bounds on dnA(s, t)/dt” can accomodate any possible high energy behavior of otot. The information obtained on the zeros of the derivatives of A(s, t) (and of da/dt) may be of some use in enhancing our confidence in the usual extrapolations of da/&) for small [ t 1, in view of the fact that one can not preclude, a priori, the possibility that the amplitude due to strong interaction changes its behavior significantly [lo] in the inaccurately known Coulomb region [Ill. We obtain our basic result, namely, high energy upper bounds on d”A(s, t)/dt” ltzO, in Section 2. The mathematical points which are relevant for this are discussed in detail in appendices. The besic result is used to deduce a number of useful consequences in Section 3. In Section 4, we obtain lower bounds on d”A(s, t)/dt” for t < 0. Upper bounds on d*(du/dt)/dt”, 0 -c t, s -+ co are derived in Section 5. We shall use the notation P(s, t) = d”A(s, t)/dtTL
(1.2)
whenever convenient.
2. UPPER BOUNDS ON P(s,t),
t = 0, s--t
CO
We use the following partial wave expansion of A(s, t): A(s, t) = T
2 (21 + 1) Imf,P,
(1 + &)
(2.1)
1=0
where the partial wave amplitudes
obey the usual unitarity
relations,
Let us define an(s, t = 0) = (2%!)(2k2)” (--&)
= ,$
(21 +
P(s, 0)
1) Imf,NA 4
(2.3) (2.4)
ASYMPTOTIC
145
BOUNDS
where n = 1, 2, 3,... and
n)! m 4 = (I(I _+ n)! = (I + n)(l + n - 1) *** (I - n + 1)
for
I > ir
(2.5)
-0
for
I < n.
(2.6)
We wish to solve the following mathematical (a)
problem. Maximize
Bn(s, 0), given
-$ A@, 0) = %
= go(21+ l>Jmfi,
(b) $4,
(2.7)
1,)= 1 (21+ 1)ImfZ’dx), 2=0
and (c) unitarity, Eq. (2.2). The search for the solution proceeds by the usual variational method. We shall finally prove the upper bound by the “direct subtraction method.” Setting up the auxiliary function 2 = Dn(s, 0) + a k2atot/4T - f (21 + 1) Im fi 0
+ b kA(s, to)/s1i2- f (21 + 1) Imf,P,(x) 0
+ [f
(21 -I- 1) Wmh
1 1
- 1.h121],
(2.9)
0
where a, b, and A, are Lagrange multipliers, an extremum [12]:
we obtain the following condition
R(I, 17)- (a + M’,(x)) + A,(1 - 2 ImfJ
(1)
= 0,
A, Re fl = 0.
(2)
for (2.10)
(2.11)
Here A, > 0 for all 1. Moreover, at least when the extremum is a maximum, we expect This follows because when A(s, 0) is held fixed, G[Max D(s, O)]f 6[A(s, to)] (which in fact is equal to b [12]) is expected on intuitive grounds to be positive. From the above conditions (1) and (2), we should obtain an upper bound on D(s, 0) for the following set of partial wave amplitudes, b to be positive.
Imfi
= 1
if
I E U = (I j R(Z, n) > (a + bP,(x))}
(2.12a)
= 0
if
I E W = {I / R(Z, n) < (a + bP,(x))}
(2.15b)
146
ARVIND S. VENGURLEKAR
and 0 < Imft
< 1
if
I E I/ -= (I 1R(1, n) = a + bP,(x)).
(2.12c)
To determine the sets U, V, and W, we have to study the properties of R(1, n) and P,(X) for I E [0, co], s --f co. This problem is discussedin detail in Appendices A and B, where we find that for an upper bound, U = {I j M > Z > N, K > Z > n, J>ZsO},y,=(ZIE=M,N),andYzO W=(ZjZ>M,N>Z>K,n>Z>JJ) where the integers M, N, K, J are to be obtained from the constraint Eqs. (2.7), (2.Q and (2.12~) and V = V, @ Vz . We now state the solution to the maximization problem in form of a theorem. THEOREM
1.
B&&,
D(s, 0) < D&Js,
0), where, for s + co ,
0) = C (21 + 1) R(z> n) 1ELi + (2~ + 1) E&M,
n) $ (2N t 1) c,R(N, 4.
(2.13)
The nonnegative integers M, N, K, and J and the numbers chl, cN (1 > E&~,Ed > 0) are to be determined by
(2.14)
+ (2N i- 1) ENPN(.~),
-& A(&0)= c (21 + 1)f u
(2M
f 1)EM+ (2N i 1)EN,
B(M, N; I = K, J; n) = 0,
(2.15) (2.16)
where R(M, B(M, N; I; n) =
II)
1 PM(X)
RCZ.n> R(N, 4 1 PAX)
1 . PN(X)
(2.17)
Remarks. (I) Note that we have now four unknowns M, N, K, and J to be obtained from the for equations (2.14), (2.15), and (2.16). Since constraints of A(s, 0) and A(s, f,) (Eqs. (2.7), (2.8)) are nontrivial, we expect to have nontrivial solutions at least for M and N. It is possible that the equation B(M, N; I; n) = 0 has no solution for 1E [M > N > Z > n] and/or ZE [n > Z > 01. In such an event, the unknowns K and/or J and the corresponding equations (2.16) drop out of the problem. (2) If K does not exist, U = {I j M > Z > N, J > Z 3 O}. In addition, if J doesnot exist, U = {I j M > 1 > N} and so on.
ASYMPTOTIC
Proof.
147
BOUNDS
Consider
= C (21 -1 l)(l - Irnft)
R(Z, n) -
zeu
+ (2M + l)(c,t4 - ImfM> Eliminating
n)
R(M, n> + (2N + l)(cN - 1m.L) R(N, n).
(2.18)
(Ed - ImfM) and (Ed - IrnfM) from the constraints
+ l>(EA4- Imfd
(a4
C (21 + 1) ImfiR(Z, lOV,@W
+ W + ~)(CN- Imfid
= - (Fu(21+ 00 - 1m.h) + ls~Bw(2~
-t 1) Imfi
(2.19a)
2 and
(2M + l)(cM - ImfM) = -
P&S-) t (2N + l)tc~ - ImfN)
C (21 + l)(l - Imfi> Pi(x) +
1erJ
P&x)
c (21 + 1) ImfiP2(x) lEVZ@W
(2.19b)
we obtain
[~A&) =
PN(X)l
d
lL (21 + l)(l
- Imfi)
B(M, N; I; ?z) ~
/
C
(2Z+
1) Imf,B(M,
I
(2.20)
20
(from
N; I; tr)
lETz@W
Lemma
5 of Appendix
A).
Q.E.D.
Evaluation of the bound. It is easy to show, by introducing small changes 8~~ m Ed, and SET in Ed (but keeping M, N, and A(s, 0) fixed) that the resultant changes in B&,(s, 0) and A($, t,) are related by ~[&m(s,
o)l = b =
The use of Jin-Martin s--+ 00,
bound, therefore, M(M
+ n)!
(M-n--l)!+ (2M
R(M, n) -
WN, ff) > o
PM(-~)- PN(X)
a4s, [“)I
+ 1) l ,R(M 4 +
is meaningful (N+
l)(N+
(2.21)
.
(see [5]). We now have, for PZ+ l)! +
(N - n)! ON + 1) ENW’C 4,
K(K+
(K-n
n)!
- I)!
1
(2.22)
148
ARVIND
S. VENGURLEKAR
where M, N, K, and Eill , eH are determined by k2atot ___ = (M2 47T
(N i
1)“) t- (K2 - n2 + J”) f (2M + 1) cM $- (2N + 1) cN, (2.23)
(s/&J2,, [PM”(x)-t fLw1
~ [PNW + rv+,(-~-)l
+ [PKyx) + Pi&,(.x)1- P,‘(x) + K&)1 + [Pi(x) + P;-,(x)] +
(mf
+ 1) l ‘dM(X) +
cm + 1) ENPN(ZC),
(2.24)
B(M, N; I = K; n) = 0
if
N > K > n,
(2.25a)
B(M, N; 1 = J; n) = 0
if
n > J > 0.
(2.25b)
and J is given by
Defining,
as in [5], tomot u = 477(ln ~9)~’
s-+00
+!!+
s + co (A > O),
(l>a>O)
(2.26)
and (2.27)
we have A4 ,z &
(2.28)
(1 + 4 ln s + I-.,
0
Nsym -$
(1 + A)(1 - u)l”
In s $- .a.,
(2.29)
0
and K and J are given by p (x) = p (x> R(M 4 - RUG 4 _ p (x) R(N, n) - R(K, n) N K R(M, n) - R(N, n) R(M, n) - R(N, n) ’ M
(2.3Oa)
PM(X) - PN(X) R(K n, R(&f, n> _ R(N, n> .
(2.30b)
PJ(~> = PK(X) -
It can be shown that in the limit s + 00 the values of K and J become inconsequential for the final answer, provided (n/in S) -+ 0, s -+ co. We therefore have, from Eqs. (2.3), (2.22), (2.28)-(2.30), 1 d”A(s, t) A@, 0) dt” which is the basic result of the paper,
(1 + h)2 ln2 s la (1 - (I - u>n+1> (n + l)! cr ’ 4tll
1
(2.31)
ASYMPTOTIC
149
BOUNDS
Remarks. (1) For n = 1, we reproduce an earlier result of Singh and Vengurlekar [5] for the “diffraction peak width,” namely, (2.32) (2) In principle, upper bounds on P(s, 0) are also obtainable from Singh’s work [7] given the bounds on P(s, O), m = 1, 2,..., IZ - 1. The explicit forms of these bounds, however, can be derived only for small n. The bound on Dnz2(s, 0) in terms of Dm(s, 0), m = 0, 1 as given in [7] becomes the same as our bound, Eq. (2.31) for IZ = 2, when we substitute for P=l(s, 0) its upper bound. (3) Our bounds improve over similar earlier results by Eden and Takagi [6]. (4) A weaker form of the bound, Eq. (2.31), is given by
WS, 0) A(s, 0) 5
[
(1 + h)2 ln2 s n 1 %
1 2’
(2.33)
(5) Our bounds are valid for arbitrarily large n, provided n/in s -+ 0, s ---f co. (6) Since the bound is actually achieved for the set: Imf = 1, I E U, Imf = 0, 1 E W and 1 > Imfi 3 0 for 1~ V, as shown by the direct subtraction method, it is the best possible one for the input we have used. (7) Multiplier a, just as b (Eq. (2.21)), can also be found in terms of M and N as
a = R(N, 4 PM(X) - WC 4 f’dx) PM(X) - PNW ’ which is positive unless (1 - ~)~-l < (4to)“/(4Pf1 ln2n-5/2S), s ---f co.
3. SOME CONSEQUENCES OF THE BOUNDS, EQ. (2.31) Let us now briefly consider a few applications
of our basic result.
(i) Upper bounds on A(s, t), to > t > 0, s -+ co. Since A(s, t) is an analytic function oft for I t / < to , we expand A(s, t) into a Taylor series
G f [(t/Ml + hj2 (n2s/4]” 1 - (1 - u)n+l CT (n!)(n + 1) ! 1 1, n=o
(3.1)
150
ARVIND
S. VENGURLEKAR
where we have used Eq. (2.31). Writing u = (t/@~“(l we have 4&
,sm >
F -%!rg-L ( k=l
(1 -
u) f
k”;g)
;
k=l
(3.2)
2 I&4
I
-
+ h) In s and w = (1 - u)lP u,
- 2(1 - u) +,/C7
u
where I,(u) is the modified Bessel function of order 1. This bound may be recognized to be an earlier result obtained by Singh and Vengurlekar [5]. As in [5], the bound, Eq. (3.2), may be weakened to obtain -!Lf!&A < I,((1 + h)(t/to)1~2 In s). A(s, 0) sh= A number of important
(3.3)
results which follow from this bound are discussed in [5].
(ii) Upper bounds on Dn(s, t), t, > t > 0, s -+ co. These can be deduced from Dn(s, 0), s -+ co by using once again the analyticity properties of A(s, t) in 1 t ] < to . Expanding Dn(s, t) into a power series in t, we have
~ f
c
s-tee m=0 m!
. [(I + h)2 In2 s]~+‘~~[ 1 - (1 - r~)%+~+l] (n + m + l)! (4t0)Q+mu
(3.4)
(u2/4)” =- 1 (1 + X)2 In2 s n cm u I m=O m! (m + n + I)! 4to 11 (3.5)
where u = (t/to)l12(l + X) In s and w = (1 - u)li2 u. Using the formula
=mlo m,($yn), 3 (’ydy “1nIO(Y)
[I31 (3.6)
where I,(u) is the modified Bessel function of the zeroth order, we obtain, for t < to, n = 1, 2,..., d”A(s, t) (1 + X)2 ln2 s a A(s, 0) dt” ~2 [ 4to
1
x + [(;
f)”
f&4 - (1 - up+1 (&
6)”
Q”)],
(3.7)
ASYMPTOTIC
BOUNDS
151
where II(u) s 21,(24)/u. A weaker form of the bound, Eq. (3.7), is
D%, 1)
(1 + h)2 ln2 s n 2
A(s, 0) 5
[
I
4t,
k
d n ;lt;)
If t f 0, we may use the asymptotic approximation
(3.8)
IoW
[13]
IO@) “Z &m
(3.9)
to obtain
D%, 0
A(S, 0) 3,%- [
(1 + h)2 In2 s n 2V” zP(2?Tu)l~2 . 4to
1
(3.10)
If we divide this by Dn(s, 0)/&s, 0), we have
JTyst) = D”@, - Q< DYs, 0)
(1 + h)2 In2 s n 2”e* ___ 1 ~--?P(2?72$/2 ; D”(s, 0) I 4to
1
s, u+
co.
(3.11)
Let us now use a theorem due to Bessis [2] which states that if a function f(t) is regular in / t 1 < R, iff(0) = 1 and if /f(t)\ < M(R) on / t j = R, then f(t) has no zeros inside a circle of radius r = R/M(R). Identifying P(.s, t) = f(t), and using the bound, Eq. (3.8), we have from Bessis’ theorem the result that Fn(s, t) cannot vanish in ] t j < t, where t = (2~4~‘~ tou;+3’2g-UR. D”(s, 0) n ! 12 2% ! (1 + X)2 ln2 s ! A(s, O)((l + h)2 In2 s/4to>;iand uR = (R/t0)1/2(1 i- h) In s. Optimizing uR (subject to the condition t
= n
PW2
to@
U, < (1 + @(In s)) we obtain
5/2)n+5’2.
+
(3.12)
2% ! P+5/2(1 + X)2
pa(s)
s--t co,
(Ins)” ’
(3.13)
where p,(s) stands for the curly bracket in Eq. (3.12). Note that when the bound, Eq. (3.Q is saturated, pn = 1. Thus we have the result that if Dn(s, 0)/A@, 0) N (ln2 s)n, s -+ co, the derivatives d”A(s, t)/dt” cannot vanish in / t [ < t, where tn - (In s)--2, s --j co, n = I, 2 ,.... Thus, for example, if the “diffraction-peak width” W-1 = d In A(s, t)/dt ltcO - c ln2 s, s + co, we have the result (dA(s, t)/dt) cannot vanish in I t I < tl where t = 1
3 . 5 7’2 to(?T/2y/2 4toc (
e
1
(1
+
A)”
(Z/z2 s)
’
s+
co.
These results supply some information on sufficient conditions under which the customary smooth extrapolations into the “Coulomb region” do not contradict the general principles such as unitarity and analyticity.
152
ARVIND
S. VENGURLEKAR
(iii) Upper bounds OH &/ue. . If we combine Eq. (2.31) with the lower bounds on (d”A(s, t)/dt” It=,,) obtained by Popov and Mur [15]
Dn(s,0)
1 (212 + 2) ~- &t n A(% 0) 82 n ! (2n + 1) [ (2~ + 1) 167~~~ I ’
(3.14)
where n = 1,2, 3 ,..., we obtain an infinite series of upper bounds on (o~&~oer) given by
( 3lkl)
~ (1 + h)2 lr? s 1 - (1 - a)n+i /*in ;. ( ‘n”211 s-tm u I 4to
)(l+(l’n)),
(3.15)
where n = 1,2, 3 ,.... The best of these results occurs when n -+ co and is given as (3.16) which is the same as obtained by Singh and Roy [18]. (iv) Bounds for experimental fits. It is generally believed that in the high energy limit, Re Fel/lm Fe1 is negligible in the diffraction peak region. Let us further assume that spin effects are negligible. Then we have
Wdt 1- [=I2 ___ (do/d?), s+m A(s, 0)
(3.17)
= G(s, t).
Let us consider a fit of the type (3.18) It is easy to show that dkG(s, t) dt”
/
=k! t=o
c
721.*e,...,nQ
(3.19)
f@&,,k.
j-1
*
where p = CrC1 ini , k = I, 2, 3 . . .. Also from Eq. (3.17), we have dkG(s, t) tzo dt’”
D”(s, 0)
D*-?(s, 0) A(& 0) ’
(3.20)
ASYMPTOTIC
153
BOUNDS
Making use of Eqs. (2.31) (3.14) (3.19), and (3.20), we have
i kc+ Ur
+ 2)/P -I- 111’ . K2r’ + 2)/(2r’ + I)]” r’! (2r’ + 1) r! (2r + 1)
r=o
,sm1$ “CT[l
-
(I
-
r~)~~~][l
-
(1 -
(r + I)! (r’ +
r-0
l)!
h)2k Inzk s (4t,)“-’ ’
(1 +
o)~‘+~]
u2
1
where
r’ = k - r.
As an example of this result, consider the special case: a, = a, a, = b, ai, Equation (3.21) for k = 1 gives (I + Qz (ln2 s)(l - o/2) 3 2tcl
*
l&Q1
s --+ co.
(3.21) = 0.
(3.22)
When utot - (ln2 s), s --+ co, there exists a better lower bound on a [17], namely, a > utot/167r, s + co. In fact for utot = (4n/tJ ln2 s, a = (In2 s)/4to [5, 171. Putting k = 2 ib Eq. (3.21) we obtain, for s + co ( a2 + 2b (, \
(O-683) (+-J2
3(1 + ‘)* ln4 s(i -
u + 5u2/18) . (3 23)
16t02
This, as expected, gives b < 0 for 0 IV 1.
4.
LOWER
BOUNDS
ON
A lower bound on a certain combination obtained easily by using the inequality GIN _ y ?L=O
(1 - 4” d”Pdz) -iZ! dz”
P(s,
t), t
-=c 0, s -+
of Dn(s, t), n = 1, 2,..., t < 0, can be
l _ (1 - 4” (
CL,
N!
dNP,(.4 (
dzN
3 0, Is-l 1)
(4.1)
where -1 < z < 1, N = 1, 2 ,.... This result for N = 1 was proved by Singh [4] and we generalize his proof for arbitrary N as follows. Consider
Cl -
GIN =
c (I --! z)n $ Pl,l(Z) - Pd41 N-1 T&=0
+ (1 - z)” .-. 2N 2NN! N!
(I + N)! (Z-N+ I)!’
(4.2)
154
ARVIND
S. VENGURLEKAR
where we have used dNP,(z)/dzN lzzl = (I + N)!/2NN!(1
- N)!. Now since
1 Pl,l(Z)- Pl(Z> = ;I:-,;) z. w + 1)Pk(4,
(4.3)
and (Z+ N)!
(4.4)
(I - N + l)!
we have
Cl -
GIN =
N-1 --(I c
- z)n
,+,, n! (’ + ‘1
i
kc0
- n --CP-l&(z)
(2k + 1) [(l - 2) 9
+ (I + l)(N ” ;!%(N
dzn-1
- l)!
(1 - z)” i (2k +- 1) = (1-t 1W - l>! lc=o X
(k+ Nl)! _ dN-lPk(z) [ 2N-‘(N - I)! (k - N + 1) ! dzN-l I
> 0,
(4.6)
where we have used (dN-lPk(z)/dzN-l),=, > (dN-lP,(z)/dzN-l) for -1 < z < 1. From inequality (4.6) it follows that GcN is nondecreasing function of 1. Combining this with the fact that GE0 = 0, inequality (4.1) follows. Multiplying GIN by (2Z+ 1) Imfi and summing over Z, we get, for all t f 0, y n=O
C-0” n!
d”-W, 0
> 1 _ (2)”
A(s, 0) dt” ’
to
;Ff$JjN >
/
.
(4.7)
t=o
Weakening this by substituting the bound, Eq. (2.31), for DN(s, t = 0), we have, for t
N!(N+ l)!u l/N (1 _ (1 - ,)N+‘) I (1 + $(lnz
S) ’
s--t co.
We therefore have the result that not all of the derivatives n = 0, l,..., N - l-can vanish in the region --TV < t < 0.
(4.9)
&A(s, t)/dt”,
ASYMPTOTIC
BOUNDS
For N == 1, we of course get back the result due to Singh [4], namely, A@, t> --->(1+$)>0 A@, 0) Consider
(fort
> -TV = --W).
Eq. (4.7) for N = 2. We then get the bound
Alternatively
dA@, t) c--t>A(s, 0) dt It therefore
follows
that dA(s, t)/dt can not vanish in -I,
‘r1)2 =
If -tu/&t(4mel)
(4.12)
l _ 4, t> 1 - Max (H) A(& 0) , d”A(s, t) p 1 Max 2A(s, 0) dt2 t=,, ’ (
< 2.5, we may use the Singh-Roy Max (4#)
and the bound,-Eq.
< t < 0, where
= 1-
(Gz$
bound [IS] (4.14)
+ ...
(2.33) (for IZ = 2) to obtain from Eq. (4.12)
dot MS, t) ___A(s, 0) dt a 36mel
(-t)(l
+ Xl4 ln4 s I .,. 64t02 ’ ’
s+
co.
(4.15)
It may be observed that for t = 0, Eq. (4.15) reduces to the McDowell-Martin lower bound [ 191 on (d In A(s, t)/dt),=, . It is interesting to note from these results that there exists a small but finite, nearforward physical region 0 > t > --I, in which A(s, r) is forbidden by the general principles from having any kinks or oscillations. Remark. The results of Sections 2, 3, and 4 also hold for certain amplitudes in problems of scattering with spin [20]. Thus, for example, the amplitude s1i2(fi + cos t&i) for TN scattering has the samepartial wave expansion which we have utilized in this paper and our results are valid for this amplitude. In general, the helicity nonflip amplitude fA1A2.d1A2 (s, cos 0) in spinful problems will obey properties similar to those proved in absenseof spins.
156
ARVIND S. VENGLJRLEKAR
5. UPPER BOUNDS ON dn(du/dt)/dtn, s-+ co We wish to obtain bounds on d”(do/dt)/dt” for t < to/4, s --t co, II = 0, 1,2,.... Consider first dn 1F 12/dtn for t < 0. We have dnIF12 _____~ dt”
sk2 d” _do rr dt” ( dt )
(5.1)
where r’ = n - r and F’=
f(2Z+
l)if;1$-&
(5.4)
I=0
We now maximize F*, given
2k2 = go (21 +
1) IA. I2
and
where x > I_ Following Singh and Roy [ 161, who have solved this problem for n = 0, we get the result that jn the limit s -+ co, the bound is given by the solution
Ifi ’ = I: If :; i (PC(X) 4 PL(X)) ’
(5.7)
where L = (ks1/2)In s, s + co. The constant a can be known in terms of uet and we get, by putting Max[F”] in Eq. (5.3) d” da G aol GtW/wN2~+2 _dtn ( dt 1t=Os-t-z (4to)*+l
’
(5.8)
where c7z = go (r!)2 (r’!)2 ((2r r;! 1)(2u’ + 1))1/2 and n = 0, 1, 2 ,..., I’ = n - r.
(5.9)
ASYMPTOTIC
157
BOUNDS
(1) As expected, putting n = 0 we reproduce the Singh-Roy 1161 on (da/dt)(t = 0), namely, Remarks.
(
g
)
(t = 0) < cJe* ln2 (pl)
)
s+
bound
co.
0
For It = I, we get (5.11)
which is the same as the Roy-Eden bound [21]. (2) For n 3 2, we obtain new results. For example, if n = 2 we have s+
co.
(5.12)
Consider now the Taylor expansion (5.13) which using analyticity of A(s, t) in 1 t I < to and unitarity, can be shown to converge in ] t ] < t,/4. From Eqs. (5.8) and (5.13) we get the result 0 < t < tJ4,
(5.14)
where C = Max,&&}. Using Eq. (5.14) in combination with the Bessis theorem [Z] (see Section 3), we have the result that dn(dG/dt)/dtn can not vanish in I t / < t, where
If the bound
(5.8) is saturated, it is clear from Eq. (5.15) that t, - (In s)-~,
I?.= 0, 1, 2,...) s -+ cx). Remarks. (1) It is obvious that results (5.9, (5.14), and (5.15) are valid even for an inelastic process ab --f cd with m, = m, , md = mb when ael is replaced by oab-wd
(2) Although our results are proved for the spinless case, spin complications can be handled easily [20]. For helicity nonflip cross sections, we only need to replace dnP,[z)jdzn lZsl by dndjA(z)/dzn lZZ1 and o,i by (T,$,~,~, . For unpolarized cross sections,we replace Ifi I by Ewt I fi1A2,A1rA2,12)112and d”P,(z)/dz” lZC1by Max{,,) [dnd;:,,(z)/dz” I&.
158
ARVIND S. VENGURLEKAR
(3) It may be noticed that the simplifications made to obtain Eq. (5.14) are at the cost of weakening the bounds. Thus, for example, when n = 0, the bound Eq. (5.14) grows faster than any power of s, as s + co.
A
APPENDIX
Here we determine the sets U, V, and W (seeEqs. (2.12)). Consider first the range n > Z > 0. From Eq. (2.6), we have Zg U if a + bPl(x) < 0 and 1E W if a + bPl(x) > 0, where n > I Z 0. Since PE(x) (X > 1) increaseswith Z, there exists an integer J (0 < J < PZ- 1) such that O
if
ZEU
(Ala)
J
if
IEW
@lb)
and
where P.,(x) = -a/b. If (a + b) > 0, J = 0 and if (a + bP,(x)) < 0, J = n - 1. If 0 < J < n - 1, we let JE V. Consider now the range 1 2 n. To obtain the division of {I I n < Z} into sets U, W, and V, it is necessaryto study the zeros of the object R(Z, n) = R(1, n) - (a + bP,(x))
(-42)
in the range I 3 n. Both R(Z, n) and Pi(x) have well-known behavior as functions of 1. Let us note the following properties for Z > n. (1) R(Z, n) defined by Eq. (2.5) for integral values of Z(I > n) can be continued for nonintegral 1. It is a polynomial in Zof degree-2n. Moreover, it is a monotonically increasing, smooth, and strictly positive function of Z for 1 3 IZ. Given I E [n, co], there is a unique value of R(Z, n) E [2n!, co] (and conversely). (2) There are many representations of Pi(x), x > 1 [13]. It is known that Pi(x) is a continuous, monotonically increasing, smooth, and strictly positive function of Zfor I 3 0 (and-hence for I 3 n). Given I E [n, ~1, there is a unique vaIue of Pi(x) E [P,(x), 031 (and conversely). It is clear from these properties that R(Z, n) may be regarded as a continuous, monotonically increasing function of Pi(x) such that given a value of Pi(x) E [P,(x), co], there is a unique value of R(I, n) E [2n!, co] (and conversely). Let us now define y = Pl(x = 1 + 2t,/(s - 4)) and f( JJ) = R(Z, n). Note that the variable y is a function of s. We now state a few lemmas,l proofs of which have been given in Appendix B. These will be useful to obtain information on the zeros of R(Z, n) in the range I > n. 1 Since we eventually use the Jin-Martin bound, Eq. (l.l), to prove these preliminary results only for s --f w .
which is valid for s --f CL),it is sufficient
159
ASYMPTOTIC BOUNDS LEMMA
1.
For s + co, @f(Y) > () dy2 = 0
if y0 ,< y < j, n > 1 if y0 < y < j, n = 1 if j
(A34 W-9 643~)
where y,, = P,(X) and j = P,(x) such that for s-+ CO, n/9/2 constant, whose value depends on the value of n. LEMMA
2.
0 and L/N2 -+
The function g(y) dejined as g(Y) = RL =f(y)
4 -
644)
(a + by>
has at most three zeros in the range y0 < y, s -+ co.
The exact number of zeros of g( y) for y,, ,( y, s -+ 03, depends on the value of IZ and on the multipliers a and b which in turn are in terms of the constraints, Eqs. (2.7) and (2.8). LEMMA 3(a). If g(y) has three zeros in the range y0 < y, s + cq given by Y = tvl , Y, , YJ such fhatyl -=cY, -c y3, then
g(y)<0
for
ybh,
Y2 2 Y z Yl
and
(A5) g(Y) t 0
for
y3
3 Yb
YI 3 Y 3
Y2,
Yo.
LEMMA 3(b). If g(y) has two (simple) zeros in the range y,, < y, s + co, given by y = ( y3 , yZ) such that y, > yZ , then
g(y) G 0
for
Y 3 y3, y2 3 Y 2 y.
g(y) >, 0
for
y3 b Y 3 y2.
and
G46)
Remark. If g(y) has two zeros in y0 ,( y, s + to, one of which is a double zero, then Lemma 3(a) may be modified accordingly. Thus, for example, if yZ = y1 is a double zero of g( y) in y, < y, s + co, we have g(y) < 0 for y > y3 and g(y) 2 0 fory, >Y 3uoLet us also note, for completeness, the following, using Eq. (Al). LEMMA
3(c). g(Y)
< 0
for
Y, > Y > Pk.h$
g(y) > 0
for
Pd4
and where0
(A71 1.
>
Y
> 1,
160
ARVIND
S. VENGURLEKAR
Let us assume to begin with, that g(y) has three zeros in the range y > y,, , s + CC, given by y = (yl , yZ , y3) such that y3 > ys > y1 . Therefore f(Yi) = a + b’i ,
Yi a Yo ,
i = 1,2,3.
648)
Note that the unknowns in the problem, i.e., a, b, y1 , yz , y, , and Jare to be obtained from the six equations (2.7), (2.8), (Alb), and (A8). Remark. If g(y) has two zeros in y > y. , s -+ 03, the unknown y1 and the corresponding Eq. (A8) for y, drop out of the problem. Eliminating a and b from Eqs. (A8) in favor of yZ and y3, we have g(y)
=
f(JNY2
- Y2) - f(MY - Y2) + f(Y2)(Y (Y3 - YJ
-
YJ
(A%
Identifying (ys , y2, yr) = (P,+,(x), PN(x), P,(x)) (this defines M, N, K) such that it4 > N > K (where K < L because 7 = P=(X) > y1 , L/s1i2 -+ constant, s -+ co), we have, in view of Eqs. (A4) and (A9) the obvious result. 4.
LEMMA
R(Z = [M, N, K], n) = 0.
(AlO)
Using Eq. (A4), we have, since f( y) = R(I, n)
B(M, I;n) RKn,=P&x) -N;PN(X)
(All)
where B(M, N, 1; n) is defined by Eq. (2.17). From Lemma 3, it follows that LEMMA
5.
B(M, N;l;n)
-=c0
for
I > M;
N > I > K;
B(M, N;I;n)
3 0
for
M 3 I > N;
n> I > J
and
@W K 3 12 n;
J > I > 0.
Remark. Although Lemma 5 is valid for continuous values of I, we need it for our purpose only for integral values of 1. It is understood that M, N, K, J in Theorem 1 (Section 2) should be taken to be integers closest, respectively, to the values M, N, K, J appearing in Lemma 5. In view of Lemma 5, we now have determined the sets U, V, and W (defined via Eqs. (2.12)) as follows: U = {I 1M > I > N, K > 13 n,J > 1 > 0}, W={1~l>M,N>I>K,n>l>J}
and
(A.13) V = (111 = M, N, K, J} = V, @ V, , V, E {I 1I = M, N).
ASYMPTOTlC
161
BOUNDS
Note that although B(M, iV, [1 = K, J], n) = 0 (see also IrnfZ = 0 for 1 = K, J (E VJ because, since Imfr’s appear we already know from the theorems in linear programming set V to have (at most) two nonzero partial wave amplitudes. of this and other related points.
APPENDIX
Eq. (2.12c)), we have linearly in the problem, that it is enough for the See [7] for a discussion
B
(i) Proof of Lemma 1. In view of the properties of R(I, n) and Pa(x) as functions of I (I 3 n), we have
@l) and the signature of G is the same as that of H,(s) = Pi’(x) R”(I, n) - P;(x) R’(I, n),
WI
where the dashes denote derivatives with respect to I. Note that Hl depends on s because Pi (x = 1 + 2&,l(s - 4)) d oes. We divide (I I I 3 n] into three regions, (a) I/.s~/~-+ 0, s -+ co; (b) l/s1j2 --L constant, s -+ co; and (c) l/s1f2 + co, s --+ co. (a) Z/,9/2 + 0, s ---f co. Using the Laplace formula
Pi
[13],
= -$-Ln de [x + (X2- 1y2 cos ey
033)
and Eq. (2), we can show in the limit s + co
Hz(s) ,z +
[(21 + 1) R”(1, n) - 2R’(I, n)]
034)
if l/s1J2 --j 0, s + co. Using Eq. (2.6) it is easy to show that
(b) and (c)
Hz(s) = 0,
n = 1,
1/s1J2+ 0,
s+
>o,
n > 1,
I/‘s1/2 -+ 0,
s--+ co.
Z/s’P H 0, s + co. We use the formula
Go,
(B5) VW
[13]
Pi(X) N 1,(2Z(t,/s)*‘2
(B7)
to obtain from Eq. (B2) the expression
Hz(s),ym 2 (+q””
[In’(v) Ryz, n) - VI;(v) -qq,
W)
162
ARVIND
S. VENGURLEKAR
where v = 2Z(t,/s)li2, I,,’ and I;I stand for the derivatives tively. Since l/sllz ft 0 as s + co, we have
dIJdv and d21,/dv2, respec-
R’(I, n) N 2rd2+l, S’S R”(f, n) ,ym 2n(2n -
@9) I) ln7h-2.
@lo)
Using the series expansion
Wl) and Eqs. (B8), (B9), and (BlO), we obtain 0312) If IZ = 1, it is-obvious that H,(s) < 0, l/sl” + 0, s --t 0~). For n > 1, one sees from Eq. (12) that there exists a constant v = v. , such that H,(s) > 0, 0 < l/s112 < L/9/2 = v,/2#2, s + co and H,(s) < 0, llPJ2 > Lls112 = vo/2t,lJ2, s -+ 00 [22]. Hence H,(s)
> 0
for
n < I < L, n > 1
(Bl3a)
= 0
for
n < 1 < L, n = 1
(Bl3b)
for
Z>L,
(Bl3c)
where L/s1i2 = constant, s -+ co, whose (P,(x), Pi(x), PL(x)), Lemma 1 follows.
alln31,
value depends on n. Writing
(y, , y, J) =
(ii) Proof of Lemma 2. By definition, g(y) = f (y) - (a + by). Since d2fldy2 has only one (odd) zero2 in [y. , co] for n > 1, s ---f co, d2/dy2 also has only one (odd) zero in [y. , co] for s + co. Now suppose that g(y) has four zeros in [y. , co], s - co. Since g(y) is a continuous, smooth function of y, by Rolle’s theorem, g’(y) must have at least three zeros and g”(y) must have at least two zeros in [y. , co], s --f co. We, however, know that g”(y) has only one zero in [y. , co], s + co. Hence Lemma 2 follows. (iii) Proof of Lemma 3(a). Since d2f (y)/dy2 changes sign (at most) once (at y = j) in the range y > y. for s + co, and since d2f(y)/dy2 =C 0 for y > J, s - co, f(y) is a concave function of y [14] for y > 7, s --f co. Hence there exists a value y = y3 such that, since b > 0, g(y) = f(u)
- (a + by) < 0
for
y > y3.
(B.14)
* If n = 1, d2f(y)/dyz = 0 for y,, Q y < y and @f(y)/cfy* < 0 for y > 4, s - to. Thus f(y) concave function 1141 of y for y 2 y0 , s -+ co. Therefore g(y) = f(y) - (a + by) shall have at
twozerosiny>y,,s+m.
is a
most
ASYMPTOTIC BOUNDS
163
If g(y) has three zeros in y 3 yO, they all must be simple zeros. (This is so because d2g(y)/dy2 changes sign (at most) once in y > yO). If y3 is so chosen that g(y,) = 0, it follows that
g(Y)G 0 dY>3 0
for Y 3 y3, y23 Y 3 y1, for y3 2 Y Z Y,, Y,2 Y 3 yo,
(B15)
where g(yl) = 0, i = 1, 2, 3, y, > y2 > y1 > y. . Note that in this case y3 > j7 > y1 . Similar considerations prove Lemma 3(b). It should be mentioned that if g(y) has 0nIy two zeros in y > yO, one of them can be a double zero in which case Lemma 3 gets modified accordingly. In view of the solution obtained in Section 2, we know, a posteriori, that we need not discuss separately the possibility that g(y) has only one zeroiny, Cy,s+ co. ACKNOWLEDGMENTS I am grateful to Professor Virendra Singh for help and advice during all stages of this work, which leans heavily upon some of his earlier results. I also thank Dr. A. K. Kapoor for assistance in proving one of the lemmas and Mr. A. K. Raina for useful information.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
For a review see, e.g., S. M. ROY, Phys. Rep. 5 (1972), 125. J. D. BESSIS,Nuovo Cir~~to A 45 (1966), 974. R. J. EDEN AND G. D. KAISER, Phys. Rev. D 3 (1971), 2286. V. SINGH, Phys. Rev. Lett. 26 (1971), 530. V. SINGH AND A. S. VENGURLEKAR, Phys. Rev. D 5 (1972), 2310. R. J. EDEN, J. Math. P&s. 8 (1967), 320. See also F. TAKAGI, Prog. Theor. Phys. 45 (1971), 559. V. SINGH, Ann. Phys. (N.Y.) 92 (1975), 377. B. K. CHUNG, Nucl. Phys. B 105 (1976), 178. See also [23]. Y. S. JIN AND A. MARTIN, Phys. Rev. 135 (1964), 1375. See, e.g., G. GIACOMELLI, Phys. Rep. 23 (1976), 125. This fact was underlined by the results of R. J. EDEN AND G. D. KAISER, Nucl. Pt’zys. B 28 (1971), 253. M. B. EINHORN AND R. BLANKENBECLER Ann. Phys. (N.Y.) 67 (1971), 480. For this and other formulas, see, e.g., G. SZEGO, “Orthogonal Polynomials,” Amer. Math. Sot. Colloquium Publications, N.Y., 1959; I. S. GRADSHTEYN AND I. M. RYZHIK,” Table of Integrals Series and Products,” Academic Press, New York/London, 1965. For properties of convex functions, see, e.g., G. H. HARDY, J. E. LITTLEWOOD, AND G. POLYA, “Inequalities,” 2nd ed., Chap. 3, Cambridge Univ. Press, New York/London, 1952. V. S. POPOV AND V. D. MUR, Sov. J. Nucl. Phys. 3 (1966), 406. V. SINGH AND S. M. ROY, Ann. Phys. (N.Y.) 57 (1970), 461. A. MARTIN, Phys. Rev. 129 (1963), 1432. V. SINGH AND S. M. ROY, Phys. Rev. Lett. 24 (1970), 28; Phys. Rev. D 1 (1970), 2638. S. W. MCDOWELL AND A. MARTIN, Phys. Rev. 135 (1964), 960. See, e.g., A. MARTIN AND F. CHEUNG, “Analyticity Properties and Bounds of the Scattering Amplitudes,” Gordon and Breach, New York 1970, for a review of the spin-complications. R. J. EDEN, Rev. Mod. Phys. 43 (1971), 15; S. M. ROY [l].
164
ARVIND
S. VENGURLEKAR
22. Use of an extension of Descartes’ rule of signs is made. See G. POLYA AND G. SZEGO, “Problems and Theorems in Analysis,” Vol. II, Chap. 1, Part IV, Problem 38, Springer Verlag, English translation, 1976. 23. For recent bounds on the slope parameter for physical and unphysical t, see A. S. VENGURLEKAR, Nucl. Phys. B 119 (1977), 453; B 122 (1977) errata; and A. D. GANGAL AND A. S. VENGURLEKAR, Phyx Reo. D (1977), (in press).
OF PHYSICS
109,
143-164 (1977)
Asymptotic
Bounds on the Derivatives Elastic Scattering Amplitudes ARVIND
of the
S. VENGURLEKAR
Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India Received October 16, 1976
Using unitarity, analyticity in the Lehmann-Martin ellipse, and asymptotic polynomial boundedness, rigorous high energy upper bounds are established on the derivatives of the absorptive parts A@, t) of elastic scattering amplitudes at i = 0. These are used to obtain a number of useful results including bounds on A@, t) and its derivatives for 0 < I < to, s --, m (t,, = 4m,,a for nn and TN). Lower bounds on the derivatives at t < 0 are derived and used to determine the near-forward physical region in which the derivatives can not vanish. We also obtain upper bounds on the derivatives of elastic differential cross sections for t < tJ4, s ---f co.
1. INTRODUCTION The study of restrictions on elastic scattering amplitudes F(s, t) obtained only from certain general principles such as unitarity and analyticity has been of interest for quite some time. Within this framework, for example, several authors [l-5] have obtained exact results regarding the high energy behavior of F(s, t) and the absorptive parts A(s, t) of F(s, t) near t = 0 in the complex t-plane. These results have many interesting consequences regarding, e.g., “the diffraction peak width,” Regge behavior, and zeros of the amplitudes in the t-plane. Corresponding information with regard to the derivatives of these amplitudes, however, is either incomplete or nonexistent. Previously, Eden [6] obtained high energy upper gounds on d”A(s, t)/dt” ItZO of the form d”A(s, t)/dt” It=,, < const xs(log s)~~+~, s ---f co, 1, 2,.... Both upper and lower bounds on &A(s, t)/& jteO for s --f co, have been recently obtained by Singh [7] in terms of dmA(S, t)/dt” ItXO, m = 0, 1, 2 ,..., n - 1. (For a general n, however, an explicit form for the upper bounds is too hard to get from Singh’s results.) Similar results for t # 0, however, are not to be found, except for recent bounds on din A(s, t)/dt, s --f co, 0 < t < to derived by Chung [S].+ The purpose of the present paper is to establish the best possible upper bounds on the derivatives (d”A(s, t)/dt”) ItEO, n = 1, 2 ,..., s ---f co, given (i) unitarity (ii) analyticity within Leymann-Martin ellipse, and (iii) the Jin-Martin upper bound [9] 4s, t> < wo)*,
s-
a,
(1.1)
+ Recently, asymptotic lower bounds for 0 < t < t, and unitarity upper and lower bounds for t < 0 have been obtained on d In A(s, t)/dt. See Ref. 23.
143 Copyright All rights
Q 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-4916
144
ARVIND S. VENGURLEKAR
where t = 2k2(z - 1) < t, , t, being related to the semimajor axis x of the LehmannMartin ellipse by x = 1 + to/2k2, where k is the c.m. momentum. (For Z-T, Z-N, ~TK,KK, KK sc6ttering we have (to)1/2 = 2r?r,.) These bounds are then used to deduce (i) upper bounds on A@, t) for 0 < t, < to, s + co, (ii) upper bounds on PA@, t)/dt”, n = 1, 2,... for 0 < t < to, s + co, (iii) information on the region near t = 0 in the complex t-plane in which PA@, t)/dt”, n = 1,2,..., s --f co can not vanish, (iv) an infinite series of upper bounds on (~&/a,~), s + co, (v) restrictions on the parameters entering in the usual forms fitting F(s, t) for t < 0. We also obtain, for t < 0, lower bounds on dA(s, t)/dt and on certain combinations of d”A(s, t)/dt”. These again can be used to find the near-forward physical region in t in which the derivatives cannot vanish. Similar results in the case of upper bounds on the derivatives d”(do/dt) dt”, n = 1,2,... are proved. It should be mentioned that our bounds on dnA(s, t)/dt” can accomodate any possible high energy behavior of otot. The information obtained on the zeros of the derivatives of A(s, t) (and of da/dt) may be of some use in enhancing our confidence in the usual extrapolations of da/&) for small [ t 1, in view of the fact that one can not preclude, a priori, the possibility that the amplitude due to strong interaction changes its behavior significantly [lo] in the inaccurately known Coulomb region [Ill. We obtain our basic result, namely, high energy upper bounds on d”A(s, t)/dt” ltzO, in Section 2. The mathematical points which are relevant for this are discussed in detail in appendices. The besic result is used to deduce a number of useful consequences in Section 3. In Section 4, we obtain lower bounds on d”A(s, t)/dt” for t < 0. Upper bounds on d*(du/dt)/dt”, 0 -c t, s -+ co are derived in Section 5. We shall use the notation P(s, t) = d”A(s, t)/dtTL
(1.2)
whenever convenient.
2. UPPER BOUNDS ON P(s,t),
t = 0, s--t
CO
We use the following partial wave expansion of A(s, t): A(s, t) = T
2 (21 + 1) Imf,P,
(1 + &)
(2.1)
1=0
where the partial wave amplitudes
obey the usual unitarity
relations,
Let us define an(s, t = 0) = (2%!)(2k2)” (--&)
= ,$
(21 +
P(s, 0)
1) Imf,NA 4
(2.3) (2.4)
ASYMPTOTIC
145
BOUNDS
where n = 1, 2, 3,... and
n)! m 4 = (I(I _+ n)! = (I + n)(l + n - 1) *** (I - n + 1)
for
I > ir
(2.5)
-0
for
I < n.
(2.6)
We wish to solve the following mathematical (a)
problem. Maximize
Bn(s, 0), given
-$ A@, 0) = %
= go(21+ l>Jmfi,
(b) $4,
(2.7)
1,)= 1 (21+ 1)ImfZ’dx), 2=0
and (c) unitarity, Eq. (2.2). The search for the solution proceeds by the usual variational method. We shall finally prove the upper bound by the “direct subtraction method.” Setting up the auxiliary function 2 = Dn(s, 0) + a k2atot/4T - f (21 + 1) Im fi 0
+ b kA(s, to)/s1i2- f (21 + 1) Imf,P,(x) 0
+ [f
(21 -I- 1) Wmh
1 1
- 1.h121],
(2.9)
0
where a, b, and A, are Lagrange multipliers, an extremum [12]:
we obtain the following condition
R(I, 17)- (a + M’,(x)) + A,(1 - 2 ImfJ
(1)
= 0,
A, Re fl = 0.
(2)
for (2.10)
(2.11)
Here A, > 0 for all 1. Moreover, at least when the extremum is a maximum, we expect This follows because when A(s, 0) is held fixed, G[Max D(s, O)]f 6[A(s, to)] (which in fact is equal to b [12]) is expected on intuitive grounds to be positive. From the above conditions (1) and (2), we should obtain an upper bound on D(s, 0) for the following set of partial wave amplitudes, b to be positive.
Imfi
= 1
if
I E U = (I j R(Z, n) > (a + bP,(x))}
(2.12a)
= 0
if
I E W = {I / R(Z, n) < (a + bP,(x))}
(2.15b)
146
ARVIND S. VENGURLEKAR
and 0 < Imft
< 1
if
I E I/ -= (I 1R(1, n) = a + bP,(x)).
(2.12c)
To determine the sets U, V, and W, we have to study the properties of R(1, n) and P,(X) for I E [0, co], s --f co. This problem is discussedin detail in Appendices A and B, where we find that for an upper bound, U = {I j M > Z > N, K > Z > n, J>ZsO},y,=(ZIE=M,N),andYzO W=(ZjZ>M,N>Z>K,n>Z>JJ) where the integers M, N, K, J are to be obtained from the constraint Eqs. (2.7), (2.Q and (2.12~) and V = V, @ Vz . We now state the solution to the maximization problem in form of a theorem. THEOREM
1.
B&&,
D(s, 0) < D&Js,
0), where, for s + co ,
0) = C (21 + 1) R(z> n) 1ELi + (2~ + 1) E&M,
n) $ (2N t 1) c,R(N, 4.
(2.13)
The nonnegative integers M, N, K, and J and the numbers chl, cN (1 > E&~,Ed > 0) are to be determined by
(2.14)
+ (2N i- 1) ENPN(.~),
-& A(&0)= c (21 + 1)f u
(2M
f 1)EM+ (2N i 1)EN,
B(M, N; I = K, J; n) = 0,
(2.15) (2.16)
where R(M, B(M, N; I; n) =
II)
1 PM(X)
RCZ.n> R(N, 4 1 PAX)
1 . PN(X)
(2.17)
Remarks. (I) Note that we have now four unknowns M, N, K, and J to be obtained from the for equations (2.14), (2.15), and (2.16). Since constraints of A(s, 0) and A(s, f,) (Eqs. (2.7), (2.8)) are nontrivial, we expect to have nontrivial solutions at least for M and N. It is possible that the equation B(M, N; I; n) = 0 has no solution for 1E [M > N > Z > n] and/or ZE [n > Z > 01. In such an event, the unknowns K and/or J and the corresponding equations (2.16) drop out of the problem. (2) If K does not exist, U = {I j M > Z > N, J > Z 3 O}. In addition, if J doesnot exist, U = {I j M > 1 > N} and so on.
ASYMPTOTIC
Proof.
147
BOUNDS
Consider
= C (21 -1 l)(l - Irnft)
R(Z, n) -
zeu
+ (2M + l)(c,t4 - ImfM> Eliminating
n)
R(M, n> + (2N + l)(cN - 1m.L) R(N, n).
(2.18)
(Ed - ImfM) and (Ed - IrnfM) from the constraints
+ l>(EA4- Imfd
(a4
C (21 + 1) ImfiR(Z, lOV,@W
+ W + ~)(CN- Imfid
= - (Fu(21+ 00 - 1m.h) + ls~Bw(2~
-t 1) Imfi
(2.19a)
2 and
(2M + l)(cM - ImfM) = -
P&S-) t (2N + l)tc~ - ImfN)
C (21 + l)(l - Imfi> Pi(x) +
1erJ
P&x)
c (21 + 1) ImfiP2(x) lEVZ@W
(2.19b)
we obtain
[~A&) =
PN(X)l
d
lL (21 + l)(l
- Imfi)
B(M, N; I; ?z) ~
/
C
(2Z+
1) Imf,B(M,
I
(2.20)
20
(from
N; I; tr)
lETz@W
Lemma
5 of Appendix
A).
Q.E.D.
Evaluation of the bound. It is easy to show, by introducing small changes 8~~ m Ed, and SET in Ed (but keeping M, N, and A(s, 0) fixed) that the resultant changes in B&,(s, 0) and A($, t,) are related by ~[&m(s,
o)l = b =
The use of Jin-Martin s--+ 00,
bound, therefore, M(M
+ n)!
(M-n--l)!+ (2M
R(M, n) -
WN, ff) > o
PM(-~)- PN(X)
a4s, [“)I
+ 1) l ,R(M 4 +
is meaningful (N+
l)(N+
(2.21)
.
(see [5]). We now have, for PZ+ l)! +
(N - n)! ON + 1) ENW’C 4,
K(K+
(K-n
n)!
- I)!
1
(2.22)
148
ARVIND
S. VENGURLEKAR
where M, N, K, and Eill , eH are determined by k2atot ___ = (M2 47T
(N i
1)“) t- (K2 - n2 + J”) f (2M + 1) cM $- (2N + 1) cN, (2.23)
(s/&J2,, [PM”(x)-t fLw1
~ [PNW + rv+,(-~-)l
+ [PKyx) + Pi&,(.x)1- P,‘(x) + K&)1 + [Pi(x) + P;-,(x)] +
(mf
+ 1) l ‘dM(X) +
cm + 1) ENPN(ZC),
(2.24)
B(M, N; I = K; n) = 0
if
N > K > n,
(2.25a)
B(M, N; 1 = J; n) = 0
if
n > J > 0.
(2.25b)
and J is given by
Defining,
as in [5], tomot u = 477(ln ~9)~’
s-+00
+!!+
s + co (A > O),
(l>a>O)
(2.26)
and (2.27)
we have A4 ,z &
(2.28)
(1 + 4 ln s + I-.,
0
Nsym -$
(1 + A)(1 - u)l”
In s $- .a.,
(2.29)
0
and K and J are given by p (x) = p (x> R(M 4 - RUG 4 _ p (x) R(N, n) - R(K, n) N K R(M, n) - R(N, n) R(M, n) - R(N, n) ’ M
(2.3Oa)
PM(X) - PN(X) R(K n, R(&f, n> _ R(N, n> .
(2.30b)
PJ(~> = PK(X) -
It can be shown that in the limit s + 00 the values of K and J become inconsequential for the final answer, provided (n/in S) -+ 0, s -+ co. We therefore have, from Eqs. (2.3), (2.22), (2.28)-(2.30), 1 d”A(s, t) A@, 0) dt” which is the basic result of the paper,
(1 + h)2 ln2 s la (1 - (I - u>n+1> (n + l)! cr ’ 4tll
1
(2.31)
ASYMPTOTIC
149
BOUNDS
Remarks. (1) For n = 1, we reproduce an earlier result of Singh and Vengurlekar [5] for the “diffraction peak width,” namely, (2.32) (2) In principle, upper bounds on P(s, 0) are also obtainable from Singh’s work [7] given the bounds on P(s, O), m = 1, 2,..., IZ - 1. The explicit forms of these bounds, however, can be derived only for small n. The bound on Dnz2(s, 0) in terms of Dm(s, 0), m = 0, 1 as given in [7] becomes the same as our bound, Eq. (2.31) for IZ = 2, when we substitute for P=l(s, 0) its upper bound. (3) Our bounds improve over similar earlier results by Eden and Takagi [6]. (4) A weaker form of the bound, Eq. (2.31), is given by
WS, 0) A(s, 0) 5
[
(1 + h)2 ln2 s n 1 %
1 2’
(2.33)
(5) Our bounds are valid for arbitrarily large n, provided n/in s -+ 0, s ---f co. (6) Since the bound is actually achieved for the set: Imf = 1, I E U, Imf = 0, 1 E W and 1 > Imfi 3 0 for 1~ V, as shown by the direct subtraction method, it is the best possible one for the input we have used. (7) Multiplier a, just as b (Eq. (2.21)), can also be found in terms of M and N as
a = R(N, 4 PM(X) - WC 4 f’dx) PM(X) - PNW ’ which is positive unless (1 - ~)~-l < (4to)“/(4Pf1 ln2n-5/2S), s ---f co.
3. SOME CONSEQUENCES OF THE BOUNDS, EQ. (2.31) Let us now briefly consider a few applications
of our basic result.
(i) Upper bounds on A(s, t), to > t > 0, s -+ co. Since A(s, t) is an analytic function oft for I t / < to , we expand A(s, t) into a Taylor series
G f [(t/Ml + hj2 (n2s/4]” 1 - (1 - u)n+l CT (n!)(n + 1) ! 1 1, n=o
(3.1)
150
ARVIND
S. VENGURLEKAR
where we have used Eq. (2.31). Writing u = (t/@~“(l we have 4&
,sm >
F -%!rg-L ( k=l
(1 -
u) f
k”;g)
;
k=l
(3.2)
2 I&4
I
-
+ h) In s and w = (1 - u)lP u,
- 2(1 - u) +,/C7
u
where I,(u) is the modified Bessel function of order 1. This bound may be recognized to be an earlier result obtained by Singh and Vengurlekar [5]. As in [5], the bound, Eq. (3.2), may be weakened to obtain -!Lf!&A < I,((1 + h)(t/to)1~2 In s). A(s, 0) sh= A number of important
(3.3)
results which follow from this bound are discussed in [5].
(ii) Upper bounds on Dn(s, t), t, > t > 0, s -+ co. These can be deduced from Dn(s, 0), s -+ co by using once again the analyticity properties of A(s, t) in 1 t ] < to . Expanding Dn(s, t) into a power series in t, we have
~ f
c
s-tee m=0 m!
. [(I + h)2 In2 s]~+‘~~[ 1 - (1 - r~)%+~+l] (n + m + l)! (4t0)Q+mu
(3.4)
(u2/4)” =- 1 (1 + X)2 In2 s n cm u I m=O m! (m + n + I)! 4to 11 (3.5)
where u = (t/to)l12(l + X) In s and w = (1 - u)li2 u. Using the formula
=mlo m,($yn), 3 (’ydy “1nIO(Y)
[I31 (3.6)
where I,(u) is the modified Bessel function of the zeroth order, we obtain, for t < to, n = 1, 2,..., d”A(s, t) (1 + X)2 ln2 s a A(s, 0) dt” ~2 [ 4to
1
x + [(;
f)”
f&4 - (1 - up+1 (&
6)”
Q”)],
(3.7)
ASYMPTOTIC
BOUNDS
151
where II(u) s 21,(24)/u. A weaker form of the bound, Eq. (3.7), is
D%, 1)
(1 + h)2 ln2 s n 2
A(s, 0) 5
[
I
4t,
k
d n ;lt;)
If t f 0, we may use the asymptotic approximation
(3.8)
IoW
[13]
IO@) “Z &m
(3.9)
to obtain
D%, 0
A(S, 0) 3,%- [
(1 + h)2 In2 s n 2V” zP(2?Tu)l~2 . 4to
1
(3.10)
If we divide this by Dn(s, 0)/&s, 0), we have
JTyst) = D”@, - Q< DYs, 0)
(1 + h)2 In2 s n 2”e* ___ 1 ~--?P(2?72$/2 ; D”(s, 0) I 4to
1
s, u+
co.
(3.11)
Let us now use a theorem due to Bessis [2] which states that if a function f(t) is regular in / t 1 < R, iff(0) = 1 and if /f(t)\ < M(R) on / t j = R, then f(t) has no zeros inside a circle of radius r = R/M(R). Identifying P(.s, t) = f(t), and using the bound, Eq. (3.8), we have from Bessis’ theorem the result that Fn(s, t) cannot vanish in ] t j < t, where t = (2~4~‘~ tou;+3’2g-UR. D”(s, 0) n ! 12 2% ! (1 + X)2 ln2 s ! A(s, O)((l + h)2 In2 s/4to>;iand uR = (R/t0)1/2(1 i- h) In s. Optimizing uR (subject to the condition t
= n
PW2
to@
U, < (1 + @(In s)) we obtain
5/2)n+5’2.
+
(3.12)
2% ! P+5/2(1 + X)2
pa(s)
s--t co,
(Ins)” ’
(3.13)
where p,(s) stands for the curly bracket in Eq. (3.12). Note that when the bound, Eq. (3.Q is saturated, pn = 1. Thus we have the result that if Dn(s, 0)/A@, 0) N (ln2 s)n, s -+ co, the derivatives d”A(s, t)/dt” cannot vanish in / t [ < t, where tn - (In s)--2, s --j co, n = I, 2 ,.... Thus, for example, if the “diffraction-peak width” W-1 = d In A(s, t)/dt ltcO - c ln2 s, s + co, we have the result (dA(s, t)/dt) cannot vanish in I t I < tl where t = 1
3 . 5 7’2 to(?T/2y/2 4toc (
e
1
(1
+
A)”
(Z/z2 s)
’
s+
co.
These results supply some information on sufficient conditions under which the customary smooth extrapolations into the “Coulomb region” do not contradict the general principles such as unitarity and analyticity.
152
ARVIND
S. VENGURLEKAR
(iii) Upper bounds OH &/ue. . If we combine Eq. (2.31) with the lower bounds on (d”A(s, t)/dt” It=,,) obtained by Popov and Mur [15]
Dn(s,0)
1 (212 + 2) ~- &t n A(% 0) 82 n ! (2n + 1) [ (2~ + 1) 167~~~ I ’
(3.14)
where n = 1,2, 3 ,..., we obtain an infinite series of upper bounds on (o~&~oer) given by
( 3lkl)
~ (1 + h)2 lr? s 1 - (1 - a)n+i /*in ;. ( ‘n”211 s-tm u I 4to
)(l+(l’n)),
(3.15)
where n = 1,2, 3 ,.... The best of these results occurs when n -+ co and is given as (3.16) which is the same as obtained by Singh and Roy [18]. (iv) Bounds for experimental fits. It is generally believed that in the high energy limit, Re Fel/lm Fe1 is negligible in the diffraction peak region. Let us further assume that spin effects are negligible. Then we have
Wdt 1- [=I2 ___ (do/d?), s+m A(s, 0)
(3.17)
= G(s, t).
Let us consider a fit of the type (3.18) It is easy to show that dkG(s, t) dt”
/
=k! t=o
c
721.*e,...,nQ
(3.19)
f@&,,k.
j-1
*
where p = CrC1 ini , k = I, 2, 3 . . .. Also from Eq. (3.17), we have dkG(s, t) tzo dt’”
D”(s, 0)
D*-?(s, 0) A(& 0) ’
(3.20)
ASYMPTOTIC
153
BOUNDS
Making use of Eqs. (2.31) (3.14) (3.19), and (3.20), we have
i kc+ Ur
+ 2)/P -I- 111’ . K2r’ + 2)/(2r’ + I)]” r’! (2r’ + 1) r! (2r + 1)
r=o
,sm1$ “CT[l
-
(I
-
r~)~~~][l
-
(1 -
(r + I)! (r’ +
r-0
l)!
h)2k Inzk s (4t,)“-’ ’
(1 +
o)~‘+~]
u2
1
where
r’ = k - r.
As an example of this result, consider the special case: a, = a, a, = b, ai, Equation (3.21) for k = 1 gives (I + Qz (ln2 s)(l - o/2) 3 2tcl
*
l&Q1
s --+ co.
(3.21) = 0.
(3.22)
When utot - (ln2 s), s --+ co, there exists a better lower bound on a [17], namely, a > utot/167r, s + co. In fact for utot = (4n/tJ ln2 s, a = (In2 s)/4to [5, 171. Putting k = 2 ib Eq. (3.21) we obtain, for s + co ( a2 + 2b (, \
(O-683) (+-J2
3(1 + ‘)* ln4 s(i -
u + 5u2/18) . (3 23)
16t02
This, as expected, gives b < 0 for 0 IV 1.
4.
LOWER
BOUNDS
ON
A lower bound on a certain combination obtained easily by using the inequality GIN _ y ?L=O
(1 - 4” d”Pdz) -iZ! dz”
P(s,
t), t
-=c 0, s -+
of Dn(s, t), n = 1, 2,..., t < 0, can be
l _ (1 - 4” (
CL,
N!
dNP,(.4 (
dzN
3 0, Is-l 1)
(4.1)
where -1 < z < 1, N = 1, 2 ,.... This result for N = 1 was proved by Singh [4] and we generalize his proof for arbitrary N as follows. Consider
Cl -
GIN =
c (I --! z)n $ Pl,l(Z) - Pd41 N-1 T&=0
+ (1 - z)” .-. 2N 2NN! N!
(I + N)! (Z-N+ I)!’
(4.2)
154
ARVIND
S. VENGURLEKAR
where we have used dNP,(z)/dzN lzzl = (I + N)!/2NN!(1
- N)!. Now since
1 Pl,l(Z)- Pl(Z> = ;I:-,;) z. w + 1)Pk(4,
(4.3)
and (Z+ N)!
(4.4)
(I - N + l)!
we have
Cl -
GIN =
N-1 --(I c
- z)n
,+,, n! (’ + ‘1
i
kc0
- n --CP-l&(z)
(2k + 1) [(l - 2) 9
+ (I + l)(N ” ;!%(N
dzn-1
- l)!
(1 - z)” i (2k +- 1) = (1-t 1W - l>! lc=o X
(k+ Nl)! _ dN-lPk(z) [ 2N-‘(N - I)! (k - N + 1) ! dzN-l I
> 0,
(4.6)
where we have used (dN-lPk(z)/dzN-l),=, > (dN-lP,(z)/dzN-l) for -1 < z < 1. From inequality (4.6) it follows that GcN is nondecreasing function of 1. Combining this with the fact that GE0 = 0, inequality (4.1) follows. Multiplying GIN by (2Z+ 1) Imfi and summing over Z, we get, for all t f 0, y n=O
C-0” n!
d”-W, 0
> 1 _ (2)”
A(s, 0) dt” ’
to
;Ff$JjN >
/
.
(4.7)
t=o
Weakening this by substituting the bound, Eq. (2.31), for DN(s, t = 0), we have, for t
N!(N+ l)!u l/N (1 _ (1 - ,)N+‘) I (1 + $(lnz
S) ’
s--t co.
We therefore have the result that not all of the derivatives n = 0, l,..., N - l-can vanish in the region --TV < t < 0.
(4.9)
&A(s, t)/dt”,
ASYMPTOTIC
BOUNDS
For N == 1, we of course get back the result due to Singh [4], namely, A@, t> --->(1+$)>0 A@, 0) Consider
(fort
> -TV = --W).
Eq. (4.7) for N = 2. We then get the bound
Alternatively
dA@, t) c--t>A(s, 0) dt It therefore
follows
that dA(s, t)/dt can not vanish in -I,
‘r1)2 =
If -tu/&t(4mel)
(4.12)
l _ 4, t> 1 - Max (H) A(& 0) , d”A(s, t) p 1 Max 2A(s, 0) dt2 t=,, ’ (
< 2.5, we may use the Singh-Roy Max (4#)
and the bound,-Eq.
< t < 0, where
= 1-
(Gz$
bound [IS] (4.14)
+ ...
(2.33) (for IZ = 2) to obtain from Eq. (4.12)
dot MS, t) ___A(s, 0) dt a 36mel
(-t)(l
+ Xl4 ln4 s I .,. 64t02 ’ ’
s+
co.
(4.15)
It may be observed that for t = 0, Eq. (4.15) reduces to the McDowell-Martin lower bound [ 191 on (d In A(s, t)/dt),=, . It is interesting to note from these results that there exists a small but finite, nearforward physical region 0 > t > --I, in which A(s, r) is forbidden by the general principles from having any kinks or oscillations. Remark. The results of Sections 2, 3, and 4 also hold for certain amplitudes in problems of scattering with spin [20]. Thus, for example, the amplitude s1i2(fi + cos t&i) for TN scattering has the samepartial wave expansion which we have utilized in this paper and our results are valid for this amplitude. In general, the helicity nonflip amplitude fA1A2.d1A2 (s, cos 0) in spinful problems will obey properties similar to those proved in absenseof spins.
156
ARVIND S. VENGLJRLEKAR
5. UPPER BOUNDS ON dn(du/dt)/dtn, s-+ co We wish to obtain bounds on d”(do/dt)/dt” for t < to/4, s --t co, II = 0, 1,2,.... Consider first dn 1F 12/dtn for t < 0. We have dnIF12 _____~ dt”
sk2 d” _do rr dt” ( dt )
(5.1)
where r’ = n - r and F’=
f(2Z+
l)if;1$-&
(5.4)
I=0
We now maximize F*, given
2k2 = go (21 +
1) IA. I2
and
where x > I_ Following Singh and Roy [ 161, who have solved this problem for n = 0, we get the result that jn the limit s -+ co, the bound is given by the solution
Ifi ’ = I: If :; i (PC(X) 4 PL(X)) ’
(5.7)
where L = (ks1/2)In s, s + co. The constant a can be known in terms of uet and we get, by putting Max[F”] in Eq. (5.3) d” da G aol GtW/wN2~+2 _dtn ( dt 1t=Os-t-z (4to)*+l
’
(5.8)
where c7z = go (r!)2 (r’!)2 ((2r r;! 1)(2u’ + 1))1/2 and n = 0, 1, 2 ,..., I’ = n - r.
(5.9)
ASYMPTOTIC
157
BOUNDS
(1) As expected, putting n = 0 we reproduce the Singh-Roy 1161 on (da/dt)(t = 0), namely, Remarks.
(
g
)
(t = 0) < cJe* ln2 (pl)
)
s+
bound
co.
0
For It = I, we get (5.11)
which is the same as the Roy-Eden bound [21]. (2) For n 3 2, we obtain new results. For example, if n = 2 we have s+
co.
(5.12)
Consider now the Taylor expansion (5.13) which using analyticity of A(s, t) in 1 t I < to and unitarity, can be shown to converge in ] t ] < t,/4. From Eqs. (5.8) and (5.13) we get the result 0 < t < tJ4,
(5.14)
where C = Max,&&}. Using Eq. (5.14) in combination with the Bessis theorem [Z] (see Section 3), we have the result that dn(dG/dt)/dtn can not vanish in I t / < t, where
If the bound
(5.8) is saturated, it is clear from Eq. (5.15) that t, - (In s)-~,
I?.= 0, 1, 2,...) s -+ cx). Remarks. (1) It is obvious that results (5.9, (5.14), and (5.15) are valid even for an inelastic process ab --f cd with m, = m, , md = mb when ael is replaced by oab-wd
(2) Although our results are proved for the spinless case, spin complications can be handled easily [20]. For helicity nonflip cross sections, we only need to replace dnP,[z)jdzn lZsl by dndjA(z)/dzn lZZ1 and o,i by (T,$,~,~, . For unpolarized cross sections,we replace Ifi I by Ewt I fi1A2,A1rA2,12)112and d”P,(z)/dz” lZC1by Max{,,) [dnd;:,,(z)/dz” I&.
158
ARVIND S. VENGURLEKAR
(3) It may be noticed that the simplifications made to obtain Eq. (5.14) are at the cost of weakening the bounds. Thus, for example, when n = 0, the bound Eq. (5.14) grows faster than any power of s, as s + co.
A
APPENDIX
Here we determine the sets U, V, and W (seeEqs. (2.12)). Consider first the range n > Z > 0. From Eq. (2.6), we have Zg U if a + bPl(x) < 0 and 1E W if a + bPl(x) > 0, where n > I Z 0. Since PE(x) (X > 1) increaseswith Z, there exists an integer J (0 < J < PZ- 1) such that O
if
ZEU
(Ala)
J
if
IEW
@lb)
and
where P.,(x) = -a/b. If (a + b) > 0, J = 0 and if (a + bP,(x)) < 0, J = n - 1. If 0 < J < n - 1, we let JE V. Consider now the range 1 2 n. To obtain the division of {I I n < Z} into sets U, W, and V, it is necessaryto study the zeros of the object R(Z, n) = R(1, n) - (a + bP,(x))
(-42)
in the range I 3 n. Both R(Z, n) and Pi(x) have well-known behavior as functions of 1. Let us note the following properties for Z > n. (1) R(Z, n) defined by Eq. (2.5) for integral values of Z(I > n) can be continued for nonintegral 1. It is a polynomial in Zof degree-2n. Moreover, it is a monotonically increasing, smooth, and strictly positive function of Z for 1 3 IZ. Given I E [n, co], there is a unique value of R(Z, n) E [2n!, co] (and conversely). (2) There are many representations of Pi(x), x > 1 [13]. It is known that Pi(x) is a continuous, monotonically increasing, smooth, and strictly positive function of Zfor I 3 0 (and-hence for I 3 n). Given I E [n, ~1, there is a unique vaIue of Pi(x) E [P,(x), 031 (and conversely). It is clear from these properties that R(Z, n) may be regarded as a continuous, monotonically increasing function of Pi(x) such that given a value of Pi(x) E [P,(x), co], there is a unique value of R(I, n) E [2n!, co] (and conversely). Let us now define y = Pl(x = 1 + 2t,/(s - 4)) and f( JJ) = R(Z, n). Note that the variable y is a function of s. We now state a few lemmas,l proofs of which have been given in Appendix B. These will be useful to obtain information on the zeros of R(Z, n) in the range I > n. 1 Since we eventually use the Jin-Martin bound, Eq. (l.l), to prove these preliminary results only for s --f w .
which is valid for s --f CL),it is sufficient
159
ASYMPTOTIC BOUNDS LEMMA
1.
For s + co, @f(Y) > () dy2 = 0
if y0 ,< y < j, n > 1 if y0 < y < j, n = 1 if j
(A34 W-9 643~)
where y,, = P,(X) and j = P,(x) such that for s-+ CO, n/9/2 constant, whose value depends on the value of n. LEMMA
2.
0 and L/N2 -+
The function g(y) dejined as g(Y) = RL =f(y)
4 -
644)
(a + by>
has at most three zeros in the range y0 < y, s -+ co.
The exact number of zeros of g( y) for y,, ,( y, s -+ 03, depends on the value of IZ and on the multipliers a and b which in turn are in terms of the constraints, Eqs. (2.7) and (2.8). LEMMA 3(a). If g(y) has three zeros in the range y0 < y, s + cq given by Y = tvl , Y, , YJ such fhatyl -=cY, -c y3, then
g(y)<0
for
ybh,
Y2 2 Y z Yl
and
(A5) g(Y) t 0
for
y3
3 Yb
YI 3 Y 3
Y2,
Yo.
LEMMA 3(b). If g(y) has two (simple) zeros in the range y,, < y, s + co, given by y = ( y3 , yZ) such that y, > yZ , then
g(y) G 0
for
Y 3 y3, y2 3 Y 2 y.
g(y) >, 0
for
y3 b Y 3 y2.
and
G46)
Remark. If g(y) has two zeros in y0 ,( y, s + to, one of which is a double zero, then Lemma 3(a) may be modified accordingly. Thus, for example, if yZ = y1 is a double zero of g( y) in y, < y, s + co, we have g(y) < 0 for y > y3 and g(y) 2 0 fory, >Y 3uoLet us also note, for completeness, the following, using Eq. (Al). LEMMA
3(c). g(Y)
< 0
for
Y, > Y > Pk.h$
g(y) > 0
for
Pd4
and where0
(A71 1.
>
Y
> 1,
160
ARVIND
S. VENGURLEKAR
Let us assume to begin with, that g(y) has three zeros in the range y > y,, , s + CC, given by y = (yl , yZ , y3) such that y3 > ys > y1 . Therefore f(Yi) = a + b’i ,
Yi a Yo ,
i = 1,2,3.
648)
Note that the unknowns in the problem, i.e., a, b, y1 , yz , y, , and Jare to be obtained from the six equations (2.7), (2.8), (Alb), and (A8). Remark. If g(y) has two zeros in y > y. , s -+ 03, the unknown y1 and the corresponding Eq. (A8) for y, drop out of the problem. Eliminating a and b from Eqs. (A8) in favor of yZ and y3, we have g(y)
=
f(JNY2
- Y2) - f(MY - Y2) + f(Y2)(Y (Y3 - YJ
-
YJ
(A%
Identifying (ys , y2, yr) = (P,+,(x), PN(x), P,(x)) (this defines M, N, K) such that it4 > N > K (where K < L because 7 = P=(X) > y1 , L/s1i2 -+ constant, s -+ co), we have, in view of Eqs. (A4) and (A9) the obvious result. 4.
LEMMA
R(Z = [M, N, K], n) = 0.
(AlO)
Using Eq. (A4), we have, since f( y) = R(I, n)
B(M, I;n) RKn,=P&x) -N;PN(X)
(All)
where B(M, N, 1; n) is defined by Eq. (2.17). From Lemma 3, it follows that LEMMA
5.
B(M, N;l;n)
-=c0
for
I > M;
N > I > K;
B(M, N;I;n)
3 0
for
M 3 I > N;
n> I > J
and
@W K 3 12 n;
J > I > 0.
Remark. Although Lemma 5 is valid for continuous values of I, we need it for our purpose only for integral values of 1. It is understood that M, N, K, J in Theorem 1 (Section 2) should be taken to be integers closest, respectively, to the values M, N, K, J appearing in Lemma 5. In view of Lemma 5, we now have determined the sets U, V, and W (defined via Eqs. (2.12)) as follows: U = {I 1M > I > N, K > 13 n,J > 1 > 0}, W={1~l>M,N>I>K,n>l>J}
and
(A.13) V = (111 = M, N, K, J} = V, @ V, , V, E {I 1I = M, N).
ASYMPTOTlC
161
BOUNDS
Note that although B(M, iV, [1 = K, J], n) = 0 (see also IrnfZ = 0 for 1 = K, J (E VJ because, since Imfr’s appear we already know from the theorems in linear programming set V to have (at most) two nonzero partial wave amplitudes. of this and other related points.
APPENDIX
Eq. (2.12c)), we have linearly in the problem, that it is enough for the See [7] for a discussion
B
(i) Proof of Lemma 1. In view of the properties of R(I, n) and Pa(x) as functions of I (I 3 n), we have
@l) and the signature of G is the same as that of H,(s) = Pi’(x) R”(I, n) - P;(x) R’(I, n),
WI
where the dashes denote derivatives with respect to I. Note that Hl depends on s because Pi (x = 1 + 2&,l(s - 4)) d oes. We divide (I I I 3 n] into three regions, (a) I/.s~/~-+ 0, s -+ co; (b) l/s1j2 --L constant, s -+ co; and (c) l/s1f2 + co, s --+ co. (a) Z/,9/2 + 0, s ---f co. Using the Laplace formula
Pi
[13],
= -$-Ln de [x + (X2- 1y2 cos ey
033)
and Eq. (2), we can show in the limit s + co
Hz(s) ,z +
[(21 + 1) R”(1, n) - 2R’(I, n)]
034)
if l/s1J2 --j 0, s + co. Using Eq. (2.6) it is easy to show that
(b) and (c)
Hz(s) = 0,
n = 1,
1/s1J2+ 0,
s+
>o,
n > 1,
I/‘s1/2 -+ 0,
s--+ co.
Z/s’P H 0, s + co. We use the formula
Go,
(B5) VW
[13]
Pi(X) N 1,(2Z(t,/s)*‘2
(B7)
to obtain from Eq. (B2) the expression
Hz(s),ym 2 (+q””
[In’(v) Ryz, n) - VI;(v) -qq,
W)
162
ARVIND
S. VENGURLEKAR
where v = 2Z(t,/s)li2, I,,’ and I;I stand for the derivatives tively. Since l/sllz ft 0 as s + co, we have
dIJdv and d21,/dv2, respec-
R’(I, n) N 2rd2+l, S’S R”(f, n) ,ym 2n(2n -
@9) I) ln7h-2.
@lo)
Using the series expansion
Wl) and Eqs. (B8), (B9), and (BlO), we obtain 0312) If IZ = 1, it is-obvious that H,(s) < 0, l/sl” + 0, s --t 0~). For n > 1, one sees from Eq. (12) that there exists a constant v = v. , such that H,(s) > 0, 0 < l/s112 < L/9/2 = v,/2#2, s + co and H,(s) < 0, llPJ2 > Lls112 = vo/2t,lJ2, s -+ 00 [22]. Hence H,(s)
> 0
for
n < I < L, n > 1
(Bl3a)
= 0
for
n < 1 < L, n = 1
(Bl3b)
for
Z>L,
(Bl3c)
where L/s1i2 = constant, s -+ co, whose (P,(x), Pi(x), PL(x)), Lemma 1 follows.
alln31,
value depends on n. Writing
(y, , y, J) =
(ii) Proof of Lemma 2. By definition, g(y) = f (y) - (a + by). Since d2fldy2 has only one (odd) zero2 in [y. , co] for n > 1, s ---f co, d2/dy2 also has only one (odd) zero in [y. , co] for s + co. Now suppose that g(y) has four zeros in [y. , co], s - co. Since g(y) is a continuous, smooth function of y, by Rolle’s theorem, g’(y) must have at least three zeros and g”(y) must have at least two zeros in [y. , co], s --f co. We, however, know that g”(y) has only one zero in [y. , co], s + co. Hence Lemma 2 follows. (iii) Proof of Lemma 3(a). Since d2f (y)/dy2 changes sign (at most) once (at y = j) in the range y > y. for s + co, and since d2f(y)/dy2 =C 0 for y > J, s - co, f(y) is a concave function of y [14] for y > 7, s --f co. Hence there exists a value y = y3 such that, since b > 0, g(y) = f(u)
- (a + by) < 0
for
y > y3.
(B.14)
* If n = 1, d2f(y)/dyz = 0 for y,, Q y < y and @f(y)/cfy* < 0 for y > 4, s - to. Thus f(y) concave function 1141 of y for y 2 y0 , s -+ co. Therefore g(y) = f(y) - (a + by) shall have at
twozerosiny>y,,s+m.
is a
most
ASYMPTOTIC BOUNDS
163
If g(y) has three zeros in y 3 yO, they all must be simple zeros. (This is so because d2g(y)/dy2 changes sign (at most) once in y > yO). If y3 is so chosen that g(y,) = 0, it follows that
g(Y)G 0 dY>3 0
for Y 3 y3, y23 Y 3 y1, for y3 2 Y Z Y,, Y,2 Y 3 yo,
(B15)
where g(yl) = 0, i = 1, 2, 3, y, > y2 > y1 > y. . Note that in this case y3 > j7 > y1 . Similar considerations prove Lemma 3(b). It should be mentioned that if g(y) has 0nIy two zeros in y > yO, one of them can be a double zero in which case Lemma 3 gets modified accordingly. In view of the solution obtained in Section 2, we know, a posteriori, that we need not discuss separately the possibility that g(y) has only one zeroiny, Cy,s+ co. ACKNOWLEDGMENTS I am grateful to Professor Virendra Singh for help and advice during all stages of this work, which leans heavily upon some of his earlier results. I also thank Dr. A. K. Kapoor for assistance in proving one of the lemmas and Mr. A. K. Raina for useful information.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
For a review see, e.g., S. M. ROY, Phys. Rep. 5 (1972), 125. J. D. BESSIS,Nuovo Cir~~to A 45 (1966), 974. R. J. EDEN AND G. D. KAISER, Phys. Rev. D 3 (1971), 2286. V. SINGH, Phys. Rev. Lett. 26 (1971), 530. V. SINGH AND A. S. VENGURLEKAR, Phys. Rev. D 5 (1972), 2310. R. J. EDEN, J. Math. P&s. 8 (1967), 320. See also F. TAKAGI, Prog. Theor. Phys. 45 (1971), 559. V. SINGH, Ann. Phys. (N.Y.) 92 (1975), 377. B. K. CHUNG, Nucl. Phys. B 105 (1976), 178. See also [23]. Y. S. JIN AND A. MARTIN, Phys. Rev. 135 (1964), 1375. See, e.g., G. GIACOMELLI, Phys. Rep. 23 (1976), 125. This fact was underlined by the results of R. J. EDEN AND G. D. KAISER, Nucl. Pt’zys. B 28 (1971), 253. M. B. EINHORN AND R. BLANKENBECLER Ann. Phys. (N.Y.) 67 (1971), 480. For this and other formulas, see, e.g., G. SZEGO, “Orthogonal Polynomials,” Amer. Math. Sot. Colloquium Publications, N.Y., 1959; I. S. GRADSHTEYN AND I. M. RYZHIK,” Table of Integrals Series and Products,” Academic Press, New York/London, 1965. For properties of convex functions, see, e.g., G. H. HARDY, J. E. LITTLEWOOD, AND G. POLYA, “Inequalities,” 2nd ed., Chap. 3, Cambridge Univ. Press, New York/London, 1952. V. S. POPOV AND V. D. MUR, Sov. J. Nucl. Phys. 3 (1966), 406. V. SINGH AND S. M. ROY, Ann. Phys. (N.Y.) 57 (1970), 461. A. MARTIN, Phys. Rev. 129 (1963), 1432. V. SINGH AND S. M. ROY, Phys. Rev. Lett. 24 (1970), 28; Phys. Rev. D 1 (1970), 2638. S. W. MCDOWELL AND A. MARTIN, Phys. Rev. 135 (1964), 960. See, e.g., A. MARTIN AND F. CHEUNG, “Analyticity Properties and Bounds of the Scattering Amplitudes,” Gordon and Breach, New York 1970, for a review of the spin-complications. R. J. EDEN, Rev. Mod. Phys. 43 (1971), 15; S. M. ROY [l].
164
ARVIND
S. VENGURLEKAR
22. Use of an extension of Descartes’ rule of signs is made. See G. POLYA AND G. SZEGO, “Problems and Theorems in Analysis,” Vol. II, Chap. 1, Part IV, Problem 38, Springer Verlag, English translation, 1976. 23. For recent bounds on the slope parameter for physical and unphysical t, see A. S. VENGURLEKAR, Nucl. Phys. B 119 (1977), 453; B 122 (1977) errata; and A. D. GANGAL AND A. S. VENGURLEKAR, Phyx Reo. D (1977), (in press).