- Email: [email protected]
Bounds for the correction to the Born term and applications to p-p scattering for a generalized dispersion model
ANNALS
OF
PHYSICS:
36, 435457
Bounds for Applications
Lawrence
(1966)
the Correction to the Born Term and to p-p Scattering for a Generalized Dispersion Model*
Radiation
Laboratory,
Universify
of California,
Berkeley,
California
The unitarizing corrections to the Born term, as given by the Chew-Arndt, MacGregor-Arndt (~‘0 even), and Scotti-Wong models, are shown to be bounded above and below. The bounds follow from the mathematical forms of the models, t,o all of which forms a theorem (proved in this paper) applies. The correction bounds are given for several p-p amplitudes for the range 0 to 300 MeV. The results are shown to explain why the apparently different models mentioned above give similar Born terms after subtraction of the correction from the experimentally determined (p-p) amplit.ude. A generalization of these models for the unitarizing correction is proposed, which has properties leading to partial-wave dispersion relations, and which has a specified relationship between the asymptotic behavior and the fluctuat,ion of the sign of the left-hand discontinuity. Upper and lower bounds are found for the correction term prescribed by this generalized model (which is not restricted, in its applicat,ion, to nucleonnucleon scatteringj. I. INTRODUCTION In t,ions
nucleon-nucleon have
been
meson-exchange exchange terms restrictions old
scattering,
a small
is referred
to
observed
behavior
variable
T.
rect’ion
term
and That has
some
made to fit terms plus
t’he as
these been
correction t,he
Born
asyrnlrtot’ic
restrictions shown
(1-S)
phase
not
Noravcsik
In
(4)
term in
enough
to .l It
dispersion
rela-
with a sum of oneThe sum of one-mcsonthese attempts the only
correction
behavior
are by
ut,ilizing shift’s
term. term.
the form of the
in choosing reasonable
attempt~s
experimental
the lrlace
was
are lab bounds
t,herefore
proper
thresh-
kinetic-energy on not
the
cor-
obvious
* This Tvork was performed under the auspices of t,he United States Atomic Energy Commission. 1 However, Moravcsik’s objection in (4) to the Scotti-Wong model (1) can easily be obviated by considering the unsubtracted dispersion relation to include an additional constant representing the contribution from the contour at infinity, so that t,he asymptotic behavior of the model may be acceptable. Also, his object,ion to the Kantor approximation (S) does not apply if this approximation is considered as a limiting case; because even though the assumptions rule out zero as the contribution of the left-hand integral of the correction term, they do not rule out an arbitrarily small value. 435
436
BINKTOCK
why three apparently dissimilar schemes for the calculation of the unitarizing correction term do in fact give similar results, as was pointed out by Arndt (2). In Section III of this paper it is shown that the three models in question are really special cases of a more general type of model, to which the theorem in Appendix A applies. The generalized model, not restricted to nucleon-nucleon scattering (but restricted to amplitudes connecting states whose quantum numbers exclude bound states and resonances in the direct channel, and also excluding s-waves), is given in Section II. In that section it is shown that the five properties of the generalized model are sufficient to determine bounds on the contribution to the correction from the unphysical cut. These bounds are plot’ted in Appendix B for the p-p amplitudes 3P1, ‘Pp, lDz , Ez , 3F2, 3F3, 3Fl, ‘G, , and Ed for the three different models considered in Section III. As a result, the similarity in the structures of the different p-p corrections (calculated from the three different schemes) and the smallness of the real part of each of the corrections (relative to the real part of the experimental amplitude) are shown t,o be predetermined by the form of the correction scheme. II. A GENERALIZED
SCATTERING
MODEL
First consider t’he case of nucleon-nucleon scattering. For each I-spin state, 0 and 1, we can write Al(T) = Bl(T) + ul(T), w h ere hl( T) represents any one of the five partial-wave amplitudes Jh,i(i = 1, . . . ,4), or “hi(i = 5). Here I equals the orbital angular momentum L when i = 1, . . . , 3; and 1 equals the total angular momentum J when i = 5. By unitarity,
(la> JhL2
=
(lb)
(ezisJJ
-
Jh,3 = $p
[(COS
2cJ)
eXp
(2i6J-l,J)
--I],
(lc)
‘hs4
[(COS
2cJ)
eXp
(2dJ+1.J)
-11,
(Id)
=
gp
z
22p
‘h5 = g sin
2eJ
exp
11,
[i(sJ+l,J
+
8J-l,J)],
where E = >sds, p = [s/4 - M211”, & = total center-of-mass (c.m.) energy in the p-p channel, 4 and d; are the total cm. energies in the two crossed channels, and IM = proton mass. The nuclear-bar convention (5, 6) is used in (lc)-( le) for the phase shifts 6 and the coupling parameters E. Equations (la) and (lb) are singlet and triplet uncoupled amplitudes, respectively, for the T, = J states; (lc) and (Id) are triplet coupled amplitudes for the L = J - 1 and
BOUSIW
FOR
THE
C’ORRECTION
TO
THE
BORN
437
TERM
I; = J + 1 states, respectively; ( le) is the triplet coupling amplitude. The phase shifts 6 are real below the inelastic threshold of T = 290 NIeV, where T = 2p”/M = lab kiuetic energy. Above the inelastic threshold, Im 6 2 0. With this notation’, h l( T j ‘T-0 8( T’ ). If the five independent helicity amplitudes & (defined in ref. 7) each have a nlandelstam representation, and if the pion pole is the only singularity in positive t or u below t’he k-o-pion threshold, then Eqs. ( k23), (1.25)) and (4.26) of ref. 7 give the following results. Each of the amplitudes in ( 1) has, after removal of the one-pion ~)ole contribution, a left cut, in the 7’ I)lane corresponding to the range of 7’ when 1 + [(2m,)“/JIT] runs from 1 to - 1. In other words, this left cut in the T plane rUIIS from - 00 to -L, where -I, = -2r,/,‘/M. The right rut begirls at T = 0. Since BL( T), the Born term, includes the one-pion pole cont’ribution plus a remainder that is generally assigned no left cut above t,hc two-l)ion t)hreshold, it is reasonable for one to assume a uI( T) wit,h a left cut, from - *3 to -L and a right, cut from 0 to T, . (T, is an arbitrary mtoff, and will be taken as 400 ?tIcV in Appendix B.) The rest of the right cut is assumed to be carried by Rl( T). This model may be regarded as a (*rude approximation to a Reggeized Born t’erm for which the imaginary part approaches the imaginary part of the amplitude at high energies. Also, if Bl( T) +T+O B ( T’ ), then 2~~( T) will have similar t’hreshold behavior. Now let us choose a model for the corre(*tion term ul( T) that incorporates the threshold behavior and analytic structure just discussed for nucleon-nucleon scattering. .4s before, we write hl( T) = U,(T) + ul( T ), where hl( T) represents Jhl,i (which 1,LIb e1~ s tl le amplitudes coupling stat,es of the same orbital angular rnonmltunl I,, 1, = 1) or Jh' (which labels amplitudes coupling states of different, L, J = 1). This model is for elastic amplitudes that do not involve s-waves and that do not. have bound states or resonancesin the direct channel. The prol)erties t,hnt#define t’his model are: (i) ~(2’) +T+n B( T’ ) , the st,anclardthreshold behavior; (ii) Y”+’ Re zhi(T) +T+m 0, [the samek that appears in property (v j]; (iii) U!(T) has a left cut from --a to -b (a > b > 0, and aneed not be finite), a right cut from 0 to T, , and is ot,herwise real analytic; (iv) the right-hand discontinuity of ul( T) is 2i Irn hi(T) (Thus Bl(T) is real from 0 to T, and carries t’he full discontinuity above T,) ; (v) the left’-hand discontinuity multiplied by l/2$ denoted by fl(T), has 1 + k sign changes”over the left cut. (Sate that in conjunction with the other 4 properties this property corresJ)ondsto a minimum number of sign changes when hl( T) represents JhL.“;and also when h,( T) represenk Jhi, if the pn’s defined in *The expression h!(T) -+-+o O(F) const,ant. 3 A change of sign for jr(T) occurs definitely negat,ive, or vice versa.
is taken when
to mean its value
that
changes
limT,0 from
hl(2’)/!P definitely
= a finite positive
to
438
BINSTOCK
(7) are all of the same sign and do not sum to zero. This is shown in the discussions following (7) and (Sa) . ) It then follows that UL( T)/T’ has a dispersion relation N(T) Tl
_ _1 -’ fi(T’) P s-a T”(T’
-
Tc Im hL( T’) dT’ + c dT’ T) + i l T’l(T’ - T) ’
by properties (i), (iii), and (iv), where C is a real constant tribution from the contour at infinity. By (2) and (ii),
representing
(2) the con-
Tl+k+’
Tk+’ Reul(T)
= ___ T Tc Im hl( T’) dT’ + rc + ‘1 T’l(T’ - T) in (3) then gives
Expansion of (T’ - T)-’ “f,(T’) dT’ =T’ l-n s --a
s0
Tc Im hl( T’) dT’ > TIl-n
1
+T+W 0.
(3)
(n = 0, *-* ) 1 + AT), (4)
c =o.
i
The result is
by (2) and (4), where &(T)
= I”
,$$)dT;)
.
Now define pn =
s0
Tc Im hl( T’) dT’ 7 T’ l-n
(n = 0, . . . , I + k).
(71
Each set of pLLn’smay be described by one of the following three cases: (a) A 2 0, and En pn Z 0; (b) A 5 0, and cnpn Z 0; (c) either the pCLn’s are not all of the same sign, or elsep,L = 0. By unitarity, we have case (a) for hl( T) = JhLi. For p-p scattering, we have case (b), by calculation (8), if hi(T) = Jh5(J = 2 or 4, T, = 400 MeV). Then by (4), (6), (7), and property (v), the theorem in Appendix A applies. The bounds on I&(T) are as given by Eq. (A.3) and the discussionfollowing it:
(84
BOUNDS
FOR
THE
CORRECTION
TO
THE
BORX
439
TERM
The upper inequalities hold for case (a); the lower ones hold for case (b). Also by the theorem, I + k is the minimum number of sign changes of fl( T) when case (a) or case (b) holds ( and therefore when hl( T) represents any Jh,,i or the p-p amplitude “/l”(J = 2 or -I, T, = 400 i\IcV) ). Assuming a11 upper limit of -I, on the extent of the left cut for t’he term ul( T), where b 2 L > 0, we also have thr less-restrictive equation
from t,he corollary to the theorem. As before, the upper inequalities hold for case (a) ; t#he lower inequalities hold for case ( h 1. For case (c ), we have
(Sb) T 2 0. The constants
c, rl E (a, b) and are chosen from the set (a, b, x1 , . . . , XL), where 1 x1 to xl are solutions of CL=0 1’ p,r-’ = 0. Then if Im hl( T) is assumed to be 0 experimentally known in t’he region 0 5 T 5 T,. it follows that bounds are established on ul( T) when T 2 0. Kow define a correction ul( 7’) to be of type ( UL), with 1~ some specified integer, if ZL~(T) has the properties (i) to (v) for k = 1~. We next show that the ChewArndt model (2) is of type (O), t’he MacGregor-Arndt model (j, even) (8) is of type (r~), where HL 5 - 1, and the Scotti-Wong model (1) with cutoff is of type (-2). Also, the Kantor approximation (3) is shown to be a limiting case of the Chew-Arndt and Scotti-Wong models. III.
A.
THE
SPECIAL
CASES
SCOTTI-WONG
In the Scotti-Wong
Now tion
= B,(T)
ul(T)
= (gTJ’-’ this
4 This model with is used, instead
TITE
MOI)EL
GIVEN
IN
SECTION
II
(or S-W) model (I), with rutoff,
h(T)
Fregard
OF
R~ODEL
as
+ ,uz(T),
(9)
c 1”
‘IT”‘;;&
a ’ limiting L -.
case ma
cutoff Z’,. is considered of a cutoff.
in ref.
“~~‘“-‘Irn where .-2. III ref.
hl(T’). (10) _... Y~~~ak-~‘;.~. --,7-u-. (I - 1) disk& poles .-
1 a different
sort
of approxima-
440 Tj(i
BINSTOCK
= 1, ‘..
UL(T) =
all
,I -
1) approach T, , where 7’” < 0. Then Tc dT’nfz:(T’
lim
TZ+To
.i_:;;-
Ti)
T
-
Tj) In1 hL(T’) - T)
T’l(T’
I
*
(10’)
It follows that property (i) is sat’isfied, as is also (ii) for k = -2. The right-hand discontinuity is 2i Im hl( T), and so (iv) holds. The left-hand discontinuity of ul( T) is a sum of (1 - 1) delta functions: linl udT l +o+
+ i~)-dT 2i
Q = &(
-
it>
Z-l
= T
lim all
C a$( T - Ti)
T2-To
Tc dT’ - Ti’ p Q CT’ T; - Ti) so
i=l
Tj) Im h(T’).
(114 (lib)
Thus there are 1 - 1 of the ai, of alternating sign when the T; are arranged in the order T1 < Tt < . . . < Tl-1 . Therefore the number of sign changes of the left-hand discontinuity is I + k(k = -2), satisfying property (v). Property (iii) is also satisfied because the left cut is from TX to Tl-, (which shrinks to a point To), and the right cut is from 0 to T, . The term ul( T) as given in (10’) is real analytic except for these cuts. Hence the S-W model is of type ( -2), when ,?2 2. The shrinking of the left cut corresponds to -a --f To , -b --) To , (To < 0). Thus (7) and (Sa, b) yield
uz(T)
1-Z h T 1 I+ 1To I-r ---cc = ?r T - To n=o ,=o (1 + T/l T,j)n +:l
pn =
Tc Im h(T’) T’l-n s0
Tc Im hl( T’) dT’ T’I(T’T)
dT’
(12)
(13)
This S-W model includes the Kantor approximation (3)) when To --f - to . Equation (12) holds for 2 2 1, where the double sum is zero if 2 = 1, and where hl( T) may represent any partial-wave amplotude, “hLi or Jhi. The requirements To =< -L and L = 21n,‘/M give for p-p scattering the less restrictive bounds
+ :
l
Tc Im ht(T’) dT’ TIL(T’ - T) ’
from (8a’). The upper inequalities hold for case (a), the lower ones for case (b) . Both ( 12) and (12’) hold for I 2 1. The double sum is zero when I = 1.
BOUNDS B.
THE
FOR
THE
MACGREGOR-ARKDT
CORRECTION
TO
BORN
441
TERM
AIODEL
The MacGregor-Arndt)
approximation
(2) is t,he following:
h(T)
= B,(T)
Ul(,T)
czz
jt(T'j
= (a + 2L/T')"'2(2L/1"1)(1
7”
THE
+ tbL(T)
(14)
T') dT' T12(T' - T)
Tc Im hl(
lr
+ d
Z-l
+ 2L/T')'" Jz cQ+~ I
Consider the case when j0 is even. The right-hand discontinuity is 2i Im hl( T) for 0 < T < T, ; t,he leftthand discontinuit~y is 2ifl( T) for - 00 < T < -L. Thus prol,erties (iii) and (iv) are saGstied. Kate that ul( Tj *T-o 0(1") by construct’ion, giving property (i). The second line of Eq. (15) corresponds t,o property (ii) with k 5 - 1. With j” even, fl( T') has at most 1 - 1 sign changes over the region T' = - Q, to T' = - L. This is also the minimum number of sign changes for cases (a) and (b), as was discussed following Eqs. (6) and (7). Thus property (v) is satisfied. The result is t#hat the i\l-A nl~proximation, with j, even, is for cases (a) and (b) a type ( - 1 ) correct,ion with (Sa) applying.
5 -T1 ?" dT' Im hl(T') (,)dT) 7t /0 -T'l(T' - T)
(~)f[cc$@"(;; (16)
TZ +;(&~o&l;;;;)~~
0
The upper inequalities hold if 1~1(7') = JhLi, and the lower ones if (J = 2, 4 and T, = 400 l\IeV, for p-p scattering).
Pn =
T'j dT' > 77,l-n
Tc Im hl( s0
hi(T) = "hj
(n = 0, . . . ) I - 1)
(17)
For case (c) the RI-A approximation, with jo even, is a type (nz) correction, ?)L 5 - 1. When j. is an odd integer, property (v) does not hold and so the theorem does not apply. No bounds are derived in this case. c.
THE
CHEW-ARNDT
In the Chew-Arndt
nlODEL
model @),
h,(T) = B,(T) + w(T) T.2
uk(T) = ~ 1 so
rlT’T’&l(l
(T'~Tj~l(~~/MT')
(1s) +
t/MT)
In1 hz(T'),
(19)
442
BINSTOCK
where Ql( 1 + l/MT) is a Legendre polynomial of the second kind. Now check for properties (i) through (v). The threshold behavior of ul( 3”) goes as T-‘&1( 1 + Z/MT), which corresponds to (i). The term uI( 5”) goes asympt’otically as PQ,( 1 + i/MY’), which gives Us +T-m (3( T-’ Log T). This asymptotic behavior agrees with (ii) for k = 0. The right-hand discontinuity is 2i Im hl( T), giving (iv). The left-hand discontinuity divided by 2i is lim udT e+o+
+ ic)-ul(T 2i
-
Tc T’ Im hl( T’) dT’ so (T’ - T)Ql(l + ?/MT’)
ie) _ P z(1 + Z/MT) 2T
’
(20) for T < -:/2M. This discontinuity has 1 sign changes over the left cut, - 00 < T < - Z/2M, corresponding to (v) for Ic = 0. The right cut runs from 0 to T, , and so (iii) also holds. Now the correction is of type (0) and so (8) applies. -T1 n- s0 I (21)
n /&(t/2M)-r
+T
a T + :‘2w
ni z
0 (1 + 2MT/l)”
t 2 4na,2 I+=
s0
i
TcIm hl (T’) dT’ T’l-r *
(22)
The upper set of inequalities holds when hr( T) = ‘hLi (nucleon-nucleon); the lower set holds when hl( T) = Jh5(J = 2,4, T, = 400 XIeV, for p-p). The largest T for which ul( T) may have a left cut, T = -L = -2am,‘/M, corresponds to a minimum Zof 4mr2. Then by (Sa’), we find that -T1 ” Im hl(T’) dT’ I p s0 Trl(T, _ T) (,)u,o
( z);lTc
l;$,Ty
;’ (21’)
T + ;
&
g
2
0
(1;
;;;l’
with L = 2rn,‘/M and the inequalities subject to the conditions cited following (22). Note that the Kantor approximation (3) is included as the limiting case of t* 00. IV. SUMMARY
The three models for the correction term ut( T) considered in this paper have been shown to be special casesof a more general model. This generalized model
BOUNIW
FOR
THE
CORRECTION
TO
THE
BORN
TERM
443
for ul( T), described in Section II, has bounds t’hat are given by (5), (7), and (Sa, b). Furt#hermore, for all of the p-p amplitudes considered in Appendix B we also have (8a’). For this case the bounded regions for corrections ul( T) (for models of type ( 1~) ) are nest,ed-the bounded region for II! includes the bounded region for HL’, where - 1 5 1~’ 5 ~1. Also, the corrections of tyl)e ( no) all have one of their boutlds in common. This is the lower bound for the amplitudes given by ( la)-( Id), and the upper bound for t’he coupling amplitude given by (1~). The figures in Appendix B show that the bounds are sufficiently limiting to force models of type ( ~1) (,for t’ho p-p amplit,udes considered) to all yield similar results, if 111is small enough. As particular cases when VLis small enough, II) = - 2, - 1, or 0 correspond to the t’hrec models considered--S-W with cutoff, 11-A (j, even), and C-A, respectively. V. CONCLUSION
An important feature of the correction term 1b1(T) is the relat’ionship between it,s asymptot,ic behavior as T + * and t.he number of sign changes (over the left cut) of its discont’inuity. If this and other features are such that properties (i) to (v) of Section II hold, with k = ~1, then one bound on ul( T) (if hl( T) = JhLi, or in general if case (a) or case (b) holds) .1s independent of the value of HZ. Thus some information would be gained without a knowledge of ~1. In other words, knowledge of the precise asymptot’ic behavior of UL( T) and the actual number of sign changes of t,he left-hand discontinuity is not required to obtain one bound. Still postulating that ui(T) has properties (i) to (v) and that case (a) or case (b) holds, we see that, if an upper bound could be placed on the value of 1~ then uL( T) would be sandwiched between two bounds. Each bound on uL(T) corresponds to a bound on Re B1(T) = Re hpP(T) - Re ul(T), over the range for which Re hpP(T) is known. If 1~ could be established to be small enough, then for sufficiently small ~1,‘s the upper and lower bounds on Re BI( 7’) would be close toget’her, and also not, far from Re hpp( T). In this contingency, no further features of UZ( T) would be required to give close bounds on Bl(T) in the region in which Re hpP( T) is l~nown, and the success of t,he one-meson exchange models could be accounted for. It should be emphasized, however, that’ the properties on which this model is based have not been established. APPENDIX THEOREM.
An integral
A. BOUNDS
ON
ITb(T)
of the four
rTb(T) = I,
-’
fl(T’)
dT’
(T’ _ ,),d3
(A.11
(a > b > 0, a not necessarily jinite, T 2 0), with (1 + 1) fcnite moments c(~ given and with fl( T’) having 1 sign changes’ over the region of integration, is bounded above
444
BINSTOCK
and below. The (1 + 1) moments -b fl(T’) dT’ /&I = - s T’l-n --a
(n = 0, . . . , 1)
(A.2)
need not all have the same sign. The three possible cases are: (a) t.~~2 0, En pn z 0; (b) A, 5 0, En ~Cln# 0; (c) either the t.~~‘sare not all of the same sign, or else t.~~= 0 for all n. If all of the (1 + 1) c(%‘shave the same sign and do not sum to zero,5 which is case (a) or (b), then 1 is the minimum number of sign changes that fi ( T’) may have. For thesetwo cases,the bounds can bewritten
The upper inequalities hold for case (a); the lower set hold for case (6). For case (c), (A.3) holds when a and b are replaced by c and d, ,wherec, d E (a, b) and are chosen from the set (a, b, x1 , . . . , xl}, where x1 to xz are solutions of l--r = 0. All symbols (except XI to XL) stand for real quantities. PJ Proof:&t 2Zfl
fz(T')
=
MJ~G”‘)
r12
CT’
-
(A-4)
Td,
where c is a constant. This form specifiesfl( T’) to be a general real function having at most 1 sign changes in the region of integration. n’ow solve the isoperimetric problem corresponding to the extremizing of ITb( T) . Form HE
from (A.l) gives
(T'
fi(T') - T)T"
+
-$ Lfi(T') n=OTll'"
(AS)
and (A.2). The X, correspond to Lagrangian multipliers. Then (A.4) H = $I~(T’)c 21+1 II ng+2 (T’ - TrJ T
T&
+ n$ AnT’”
1 .
(A.61
Rewrite (A.7) 5 In the applications of the theorem correction bounds for the p-p amplitudes same sign and do not sum to zero.
in this paper (specifically, considered in Appendix
B),
the derivation of the the P~‘S are all of the
BOUNDS
FOR
THE
CORRECTION
TO
THE
BORS
44,5
TERM
where these T,, are t,he (I + 2) rook of the expression. (If Xl were zero there would be fewer root#s in (A.7 ), but this situation is equivalent to that in (A.7)) with some T,, out,side the range of T’ and with some other constant in place of XZ .) Then (A.6 1 and (A.7) ykld H = c+“(T’)X~
n”n”=‘d (T’ T”(T’ - T)
T,)
(AX’)
’
Take (AA) (since this is the Euler-Lagrange equation when H has no explicit dependence on d+( T’),IdT’), and square the result. Then the coefficient, of fl(7”)/7”’ has (21 + 1) roots. So
Kow rewrite
(A. 1) and (A.2))
using (A.4’).
2/+1 ILn
=
-
z
a;
Tin,
(7%
=
0,
* * * ,z>
(A.101
( ai is any real number, and - a < Ti < -b < 0. ) Then the finding of the extrema of I?(T) corresponds to the ext’remizing of IIE( 3”) as given by jA.9), varying the 2(2Z + 1) parameters ai and Ti , subject to the constraint (A.10). However, this prescription can be simplified. Consider the equat’ions correspond ing to variation of a; and Ti : (A.ll) (A.12) for those ai not held fixed at zero; aG -zz c3Ti
-
(Ti !
T)2
+ &
X,nT,“-’
= 0,
for ai ?t 0.
(A.13)
Because of the assumption of 1 sign changes, there must be at least (1 + 1) of the ai which are not zero. Let (i = 1, . . * , I + 1) label these ai . They will be varied. The remaining 1of the ai are not restricted, and they may be either varied or held fixed, wibhout affecting the analysis to follow. Suppose also that one of
446
BINSTOCK
the Ti (associated with a nonidentically-zero ai) is varied. Call this T1 . No restrictions on T; , with i # 1, are made (except that --a < T; < -b); they also may be either varied or held fixed without affecting the following analysis. From (A.12), we obtain xo + XlT, + XzT; + ...
+ XLTI’ = -& 1
Xo+X~Tt+xzTz~+
...
+&Taz
= -p
Xo + XI Q”L+I + XPT;+I + . . . + XI T:+I = Equation (A.14) corresponds to the variation previously. From (A.13), we know that XI + 2XaT1 + 3&T:
+ ...
1 T2 - T 1 T z+1 -
(A.14)
T’
of the first (I + 1) ai , as defined
+ 1x1T:-l
=
(T1 !
T)?’
(A.15)
This corresponds to the variation of T1 . The T1 , Tz , . . , Tl+l in (A.14) may all be regarded as distinct, since they are positions of delta functions. As already pointed out, the assumption that there be I sign changes of fl( T’) requires that there be at least I + 1 distinct Ti , which were chosen to be Ti (i = 1, . . . , I + 1). Now look for solutions of (A.14) plus (A.15) for T1 . Note that in the limit T1 --$ Ti , for i E (2, . . . , 1 + I], the difference between the first and the ith equations of (A.14) yields (A.15). So there are at least 1 roots of T, for the system (A.14) plus (A.15). In fact, these two equations are equivalent to Tl[::f (T1 - Ti) = 0, b ecause there are also no more than 1 roots of T1 in the system. Proof of this statement follows. The last 1 equations of (A.14) give X1 to Xi in terms of X0 , Tz , . . . , Tl+l , in the form of the ratio of two determinants. The finiteness of X1 to Xl is assured because the denominator may be written in the form ( T2 . . . Tl+l) nf:&2,iLz,i
BOCNDS
FOR
THE
CORRECTION
TO
THE
BORN
447
TERM
(AJO), occur when all Ti ---f x, --a < T; < -b. The situation one in which only (1 + 1) of t’he (2l + 1) a;‘s are not identically
is equivalent to zero. So solve (iz.16)
1+1 PI? =
-g
(n
ac TilL,
=
0,
. . .
, Z),
(L4.17)
where all the Ti + .r, --a < T, < -b. Th en vary x over the range of integration of the int,egral, from -a to -b, t’o obtain the bounds of I:“( T). These bounds hold when the number of sign changes of fl( T’) is 1. At this step in the proof, before solving (A.16) and (A.lT), it, is possible to verify the statement) in the theorem that if all of the (I + 1 )h,[‘s have the same sign and do not sum to zero, then 1 is the minimum number of sign changes that fl (T’ ) may have. Suppose the same steps in the entire preceding proof are applied to the case of I$l( T) (with the form given in (A.1 ) ) having (1 + 2) moments T’) is required to have pCLn (n = 0, ... ) 1 + 1) all of t’he same sign. Only ~lowf~+~( 1 sign changes. The result corresponding to (A.17) is 2+1
= -&aiT;“,
l-h
(n=O,
. . ..Z+l).
(A.18)
Consistency requires the augmented determinant, D, to be zero, for nonzero ai . The Ti are distinct, since they are delta-function positions.
D=
PO
a1
a2
...
Pl
alT1
at T2
‘. .
al+1
az+l T~+I I
al
Aa+1
TI
I-Q
1+1
2+1
a2 Tz
. ..
1
1
at+1 5%: . ..
1
. . .
I Tz+ll
...
ITl+l(“+l
(A.19)
2+1
=
( )
-c11
I Tll
17’2
/
pi
( - 1) ‘+‘/~l+l
/ Tl‘/ If1
1T,‘I ‘+I
*(-1)
(l+l)(z+2)/2
.
Since the ai were selected as nonzero, and all the pcliare positive and do not sum t’o zero, it is sufficient to show that the coefficients of pi in the last determinant are each of the same sign and are each not zero. In other words, it is sufficient
448
BINSTOCK
to show that for distinct
Dk =
1
1
I TII
I Tzl
/ Ti / (-a . .. -‘.
( T$’
( T,‘(‘-’
*. .
/ Tl jk+’
I Tz jk+’
. *.
< Ti <
-b < 0), if
1 I TH/ (Tz+i Ik--l ,
(k = 0, ---,
1 + l),
then all D,t have the same sign, and Dk # 0. This is now shown. Since Dk hasall the zeroes of Dl+l , where Dl+l = Tl[f~~l,j~~( 1 Ti 1 Dk can be written
(A.20)
1 Tj I),
(A.21) Dk = Dz+lFz+l--l;( 1 TI 1, * * . , 1 TZ+I j >. Now Fz+I--k(l TI I, . . . , 1 Ti I, . . . , 1 Tj 1, . . . , 1 TI+~ I) is symmetric under the interchange i ++ j (i, j E ( 1, . . . , I + 1) ), because DA/DI+~ has that property. Also, the sum of the exponents of the different 1 Ti I’s in each term of Dk is fixed at (1 + 1) (I + 2)/2 - k. So F~+l--k is a symmetrized sum of terms with (I + 1 - lc) as the total number of different I Ti j’s in each term. No ) Ti /will have an exponent higher than one because the exponent of ( Ti j in any term of Dk is at most one more than the exponent of ( Ti j in Dl+l . It follows that Fz+l--k = ITz+~llTz/
-+. I T~+I I +
symmetry),
(other
terms needed for complete
if I% = 0, . . . , I
= Psi Tz+l 11Tz 1 . . . 1 Tk+l I, = 1, ifk = I+ 1.
(A.22)
if k = 0, e.. ,I
1+1 different ways of k ( > selecting (I + 1 - k) different I Ti I’s from the (1 + 1) available. Since I Ti I > b > 0, it follows that Fl+l-k > 0 for all lc. Also Dz+~ is not equal to 0 by the distinctness of the / Ti 1%. Thus the Dk’S all have the same sign and are nonsero, affirming that D cannot be zero and therefore that fz+l( T’) cannot have 1 sign changes and at the same time satisfy the conditions of the theorem. The same proof holds for f~+~( T’) and 1’ sign changes, when r < 1. This completes the proof that 2 is the minimum number of sign changes of f~ (T’), where fz (T’) is as described in the theorem. The next step is to solve (A.16) and (A.17). Write Here P, is a symmetrizing
operator
that picks out the
BOUSDS
FOR
where I 2 0, --a 5 Ti 5 (A.23) with (A.1) shows (A.23).) This has no right is a sum of delta funct’ions
THE
CORRECTIOS
TO
THE
BORE
449
TERM
-b. (Since Ii”(T) satisfies (A.lj, a comparison of that there exists a function gl(T’) which satisfies cut. The left-hand disrontJinuity multiplied by 1/2i at Ti : (A.24)
The result is simply (A.16) and (A.17) (aft’er applicat’ion of (A.l) and (A.2)). So the limit all Ti -+ x, (--a 5 .C 5 b), should give extrema of IiE( T) for fixed .r. In this limit,
=
cocZT’ gl (T’) so T” (T’ - T)
(z - T’)‘+’ - (5 - T)‘+’ (x-T)l+’
But
(a -
T’ )l+l -
(.c -
T)‘+’
= (T -
T’)go
(z -
T’)p(~
-
1
(A.25)
*
T)z-p.
(A.26)
Therefore
k 2
0
=-&p p=or=o [l + (T,,&,‘(T
+ ]xj)
f
=
T”
p=o r=o [l + (T/lx()]pIx(:~(T
where
T 2 0. Note that from (A.l)
and (A.2)
IIE(T)= gogl + something But (A.23) ZL~(T)
ODdT’ ?@(T’)
’
we have
of higher order in l/T.
(A.28)
gives
= go I-
By (A.ZS)
(A.27)
+ jzj)
g
gdT’)
sl
+ something
of higher order in l/T.
(A.29)
and (A.29),
Pi = s 0
mdT’g2 ( T’) TI~-L ’
(i = 0, . . . , I).
(A.30)
450
BINSTOCK
So (A.27)
and (A.30)
give
(A.31) where -a 5 x 5 -b < 0. This is clearly bounded, and so IT”(T) is bounded. In the case when the p, are not all of the same sign, it is necessary to look at the solutions of
as well as at the end points x = -a and x = -b, to determine IIE( T). However, suppose all the pcL,are positive. Then aI,” ~ ax
=
0 + l)lxlL (T-x)+2
5 I (r=o(r)fi}
the bounds on
>‘,
and so the maximum occurs at z = -b. The minimum occurs at x: --+ all the pr’s were negative, the inequalities would be reversed. The result (A.3). COROLLARY. If Gb( T) satisfies the conditions given in the statement theorem, with all of the (I + 1)~~‘s having the same sign, and if a’ 2 a and b 2 then (A.3) holds when a and b are replaced bg a’ and b’ respectively. This trivially from (A.32). APPENDIX
(A.32) -a. If is Eq. of the b’ > 0, follows
B. BOUNDS ON uz(T) AND Bl(T) FOR p-p SCATTERING
The bounds given by (5), (7), and (Sa’) are applied to corrections UI( T) of type (m) (m = -2, -1, 0). The bounds on B,(T) = hi(T) - ul(T) are also shown. Correction C-A (b L 4) M-A (j. even) S-W (To $ - 2m,2/M)
is of
Tw 0 -1 -2
(These corrections are considered in Section III.) In the range 0 to 400 MeV, data hi(T) is taken to be hi(T) Exp, and T, = 400 MeV. The experimental are taken from refs. 2 and 8. The experimental errors are not shown.
BOUNDS
FOR
THE
CORRECTION
TO
THE
h
0 -4
50
FIG.
1. Bounds
180 Re w(T) on ?r (1 f 2M/T)“2
BORN
M correction
451
TERM
MlntypeCm),mZ-I a1sos-w
300
150 TIPb(MeV)
to 61 or et of the Born
term,
compared
to
(a)
3P~ ; (b) Ta.
I IO,
,
I
3
Max
150
300
50
T ,ob (MeV)
FIG.
2. Same
I
Fig.
I
type (ml,mt-2
150 Tlob(MeV)
as for
I
(bl
1. (a) ‘Dz
; (b) Ej
300
452
BINSTOCK
I
0
Min type Cm), m> -3
z aI & d
fi2 E P cl
-5 5
150
300
Tlab (MeV)
FIG.
50
3. Same
150
T,,blMeV)
as for
Fig.
1. (a) 3Fz ; (b)
aFa
300
Slob (Mb’)
0 Min type(m),m>-4
0.2 50
150 T ,ab (MeV)
FIG.
4.
Same
as for
Fig.
1. (a) 3F4 ; (b) 1G4
l!OUNDS
FOR
THE
CORRECTION
50
TO
5. Same
mt
3;0
as for Fig.
1, but
-30
w,f
I 50
for Ed instead
(b)
’
l&f7
-I
z ET P 0 Min type
1
Mm type (ml,mZ-I OISO s-w
\
I 150
/
Tlab
(MeV)
453
TERM
(Mei’)
c Max type h),
BORN
150 T,,,
FIG.
THE
i i
Min type (-II Min type (0)
I
150 T,,,
FIG.
6. Bounds
on
180 Re BI(!!‘) a (1 + 2M/!w where
Re ul(T)
(MeV)
has the
180Re = T bounds
[h:xP(T) (1 +
graphed
-
w(T)]
NN 61 or tl of Born
2M/T)“2
in Figs.
1 to 5. (a) 3P1 ; (b)
3Pz .
term
7
454
RINSTOCK
,
300
150 T,,,
FIG.
7. Same
as for
Fig.
300
Tlab(MeV)
8. Same
150
300
Tlab (MeV)
(MN)
150
FIG.
50
":""-yj
6. (a)
50
IL)1 ; (b) EZ
150 T,,,b(MeV)
as for Fig.
6. (a) 38’2 ; (b) 3Fa
300 ’
BOKVDS
FOR
I
THE
CORRECTION
TO
THE
BORiY
TERM
I Mox type
1.5
(m),m>-4
1.0 5L ,” D 0.5
150
300
50
150
Tlab(MeVl
FIG.
T,o,(MeV)
9. Same
as for
Fig.
6. (a) 3F4 ; (b) ~2~
Max Max
type type
Max type
-
Min
type
(0) (-I) (-21,
Cm),
150 Tlob(MeV)
FIG.
10. Same
as for
Fig.
6, but
for
Eq instead
455
456
BINSTOCK
APPENDIX
C. DISCUSSION
OF
THE
C-A
AND
M-A
MODELS
Since both the C-A and M-A models are taken from an unpublished (a), they are further discussed here.
reference
1. C-A Assuming a Mandelstam representation for the singlet p-p amplitude,
and a strip
approximation
write,
h(T) = Bz(T)+ f j-12 cdc4(T,t’)Qz(1 + &)/W
(C.1)
where h:14(T, t) = i /*‘cZT’[h,t(T’, a- 0
t) + (-l)‘h,,(T’,
and h, t and h,, are the usual double-spectral functions. assumption that the integral in (C.l) can be approximated
00 s
clt?~:‘~(T, t’>Qz (I + A)
t)]/(T’
-
Make as
T),
(C.2)
the additional
= CYL:.~(T, i)Q (1 + A),
(C.3)
4m,2
for some f, where it follows that Im hi(T)
C is independent
= C[h, t( T, f) + (-
of T. Then from real analyticity
l)‘h,,(
T’, i)]/aJJT,
0 < T < T, .
(For this model, Bl( T) is real for 0 < T < T, .) Then (C.1) may be combined to yield h(T)
= &CT)
of hi(T)
through
(C.4) (C.4) (C.Tj)
+ Q(T)
(C.5) and ((3.6) correspond to Eqs. (18) and (19) of Section III. applied by Arndt (2) to all five independent p-p amplitudes.
This model is
2. M-A Again assume both a Mandelstam representation and a strip approximation for the p-p helicity amplitudes, so that only that part of the double-spectral function within the strip 0 < T < T, contributes to the amplitude. Then by real analyticity of hl( T) we may write h(T)
= &CT)
+ w(T)
N(T)
-Lfi(T’) dT’ = ‘, s--oo T, _ T +f6
(C.7) Tc Im hl( T’) clT’ T’-T ’
(C.S)
BOUNDS
FOR
THE
CORRECTION
TO
THE
BORS
457
TERM
where L = 2rn2/M and Im Bl( T) = 0 within the strip. (L corresponds to the 2~ threshold in the crossed channels.) The left-hand discontinuity fl( T) must have the property fz(T)
-T-+--L o( T + L)3’“.
cc.91
This is derived in ref. 2’. Then, requiring normal t’hreshold an esl)ansion of (T’ - T )-’ in ((2.8) yields Q(T)
= ;
-’ fJT’) [S -zc T’l(T’
dT’ Tc Im hi (T’) dT’ - T) + l T’l(T’ - T)
Tc Im hl (T’) dT’ T,n+I ] = 0,
-T” 7r The additional
behavior
of ul( T),
1 (C.S’)
(12 = 0, . . . ) I -
1).
(C.10)
requirement uz(T)
Jvhen combined with for an approximation verge, is
(C.10’)
-+c.m 0,
(C.8’), includes (C.10) as a consequence. A simple choice to Ji ( T), which allows the first integral in (C.10) to COIIZ-l
fl( 7’) = (2 + 3L/‘T)““‘(2L/Tz)(l Then (C.7), (C.S’), of Section III.
(C.lO’j,
+ 2L/‘T)‘“~o
and (‘2.11)
aiT”.
(C.11)
are equivalent8 to Eqs. (14) and (15)
ACKNOWLEDGMENTS I \vould like to thank Professor Geoffrey Chew, who suggested this project, and Dr. Michael .I. Moravcsik, who acted as thesis advisor. I am indebted to them and to Dr. Richard A. Arndt for many helpful discussions. RECEIVED:
*July 19, 1965 REFERENCES
1. A. SCOTTI AND D. Y. WONG, Phys. Rev. Letters 10, 142 (1963); Phys. Rev. 138, B145 (1965). 9. RICHARD A. ARNIIT, “Unitarity Corrections to Peripheral p-p Scattering below 400 MeV,” VCRL 12211-T (Lawrence Radiation Laboratory Report, January 1965, unpublished). 3. P. B. KAXTO~, Phys. Rev. Letters 12, 52 (1964). 4. MICHAEL J. MORAVCSIK, Ann. Phys. (‘V’. ET.) 30, 10 (1964). r. H. P. STAPP, T. J. YPSILANTIS, AND N. METROPOLIS, Phys. Rev. 106, 302 (1958). 1 M. H. MACGREGOR, M. J. MOR.~VCSIIL, END H. P. STAPP, Ann. Rev. Nucl. Sci. 10, 291 (1960). 7. M. L. GOLDBERGER, M. T. GKIS.~RU, S. W. MACDOWELL, AND D. Y. WONG, Phys. Rev.
120, 2250 (19601. 8. RICHARD A. ARTCDT, Lawrence communication, 1965.
Radiation
Laboratory,
Livermore,
California,
private
OF
PHYSICS:
36, 435457
Bounds for Applications
Lawrence
(1966)
the Correction to the Born Term and to p-p Scattering for a Generalized Dispersion Model*
Radiation
Laboratory,
Universify
of California,
Berkeley,
California
The unitarizing corrections to the Born term, as given by the Chew-Arndt, MacGregor-Arndt (~‘0 even), and Scotti-Wong models, are shown to be bounded above and below. The bounds follow from the mathematical forms of the models, t,o all of which forms a theorem (proved in this paper) applies. The correction bounds are given for several p-p amplitudes for the range 0 to 300 MeV. The results are shown to explain why the apparently different models mentioned above give similar Born terms after subtraction of the correction from the experimentally determined (p-p) amplit.ude. A generalization of these models for the unitarizing correction is proposed, which has properties leading to partial-wave dispersion relations, and which has a specified relationship between the asymptotic behavior and the fluctuat,ion of the sign of the left-hand discontinuity. Upper and lower bounds are found for the correction term prescribed by this generalized model (which is not restricted, in its applicat,ion, to nucleonnucleon scatteringj. I. INTRODUCTION In t,ions
nucleon-nucleon have
been
meson-exchange exchange terms restrictions old
scattering,
a small
is referred
to
observed
behavior
variable
T.
rect’ion
term
and That has
some
made to fit terms plus
t’he as
these been
correction t,he
Born
asyrnlrtot’ic
restrictions shown
(1-S)
phase
not
Noravcsik
In
(4)
term in
enough
to .l It
dispersion
rela-
with a sum of oneThe sum of one-mcsonthese attempts the only
correction
behavior
are by
ut,ilizing shift’s
term. term.
the form of the
in choosing reasonable
attempt~s
experimental
the lrlace
was
are lab bounds
t,herefore
proper
thresh-
kinetic-energy on not
the
cor-
obvious
* This Tvork was performed under the auspices of t,he United States Atomic Energy Commission. 1 However, Moravcsik’s objection in (4) to the Scotti-Wong model (1) can easily be obviated by considering the unsubtracted dispersion relation to include an additional constant representing the contribution from the contour at infinity, so that t,he asymptotic behavior of the model may be acceptable. Also, his object,ion to the Kantor approximation (S) does not apply if this approximation is considered as a limiting case; because even though the assumptions rule out zero as the contribution of the left-hand integral of the correction term, they do not rule out an arbitrarily small value. 435
436
BINKTOCK
why three apparently dissimilar schemes for the calculation of the unitarizing correction term do in fact give similar results, as was pointed out by Arndt (2). In Section III of this paper it is shown that the three models in question are really special cases of a more general type of model, to which the theorem in Appendix A applies. The generalized model, not restricted to nucleon-nucleon scattering (but restricted to amplitudes connecting states whose quantum numbers exclude bound states and resonances in the direct channel, and also excluding s-waves), is given in Section II. In that section it is shown that the five properties of the generalized model are sufficient to determine bounds on the contribution to the correction from the unphysical cut. These bounds are plot’ted in Appendix B for the p-p amplitudes 3P1, ‘Pp, lDz , Ez , 3F2, 3F3, 3Fl, ‘G, , and Ed for the three different models considered in Section III. As a result, the similarity in the structures of the different p-p corrections (calculated from the three different schemes) and the smallness of the real part of each of the corrections (relative to the real part of the experimental amplitude) are shown t,o be predetermined by the form of the correction scheme. II. A GENERALIZED
SCATTERING
MODEL
First consider t’he case of nucleon-nucleon scattering. For each I-spin state, 0 and 1, we can write Al(T) = Bl(T) + ul(T), w h ere hl( T) represents any one of the five partial-wave amplitudes Jh,i(i = 1, . . . ,4), or “hi(i = 5). Here I equals the orbital angular momentum L when i = 1, . . . , 3; and 1 equals the total angular momentum J when i = 5. By unitarity,
(la> JhL2
=
(lb)
(ezisJJ
-
Jh,3 = $p
[(COS
2cJ)
eXp
(2i6J-l,J)
--I],
(lc)
‘hs4
[(COS
2cJ)
eXp
(2dJ+1.J)
-11,
(Id)
=
gp
z
22p
‘h5 = g sin
2eJ
exp
11,
[i(sJ+l,J
+
8J-l,J)],
where E = >sds, p = [s/4 - M211”, & = total center-of-mass (c.m.) energy in the p-p channel, 4 and d; are the total cm. energies in the two crossed channels, and IM = proton mass. The nuclear-bar convention (5, 6) is used in (lc)-( le) for the phase shifts 6 and the coupling parameters E. Equations (la) and (lb) are singlet and triplet uncoupled amplitudes, respectively, for the T, = J states; (lc) and (Id) are triplet coupled amplitudes for the L = J - 1 and
BOUSIW
FOR
THE
C’ORRECTION
TO
THE
BORN
437
TERM
I; = J + 1 states, respectively; ( le) is the triplet coupling amplitude. The phase shifts 6 are real below the inelastic threshold of T = 290 NIeV, where T = 2p”/M = lab kiuetic energy. Above the inelastic threshold, Im 6 2 0. With this notation’, h l( T j ‘T-0 8( T’ ). If the five independent helicity amplitudes & (defined in ref. 7) each have a nlandelstam representation, and if the pion pole is the only singularity in positive t or u below t’he k-o-pion threshold, then Eqs. ( k23), (1.25)) and (4.26) of ref. 7 give the following results. Each of the amplitudes in ( 1) has, after removal of the one-pion ~)ole contribution, a left cut, in the 7’ I)lane corresponding to the range of 7’ when 1 + [(2m,)“/JIT] runs from 1 to - 1. In other words, this left cut in the T plane rUIIS from - 00 to -L, where -I, = -2r,/,‘/M. The right rut begirls at T = 0. Since BL( T), the Born term, includes the one-pion pole cont’ribution plus a remainder that is generally assigned no left cut above t,hc two-l)ion t)hreshold, it is reasonable for one to assume a uI( T) wit,h a left cut, from - *3 to -L and a right, cut from 0 to T, . (T, is an arbitrary mtoff, and will be taken as 400 ?tIcV in Appendix B.) The rest of the right cut is assumed to be carried by Rl( T). This model may be regarded as a (*rude approximation to a Reggeized Born t’erm for which the imaginary part approaches the imaginary part of the amplitude at high energies. Also, if Bl( T) +T+O B ( T’ ), then 2~~( T) will have similar t’hreshold behavior. Now let us choose a model for the corre(*tion term ul( T) that incorporates the threshold behavior and analytic structure just discussed for nucleon-nucleon scattering. .4s before, we write hl( T) = U,(T) + ul( T ), where hl( T) represents Jhl,i (which 1,LIb e1~ s tl le amplitudes coupling stat,es of the same orbital angular rnonmltunl I,, 1, = 1) or Jh' (which labels amplitudes coupling states of different, L, J = 1). This model is for elastic amplitudes that do not involve s-waves and that do not. have bound states or resonancesin the direct channel. The prol)erties t,hnt#define t’his model are: (i) ~(2’) +T+n B( T’ ) , the st,anclardthreshold behavior; (ii) Y”+’ Re zhi(T) +T+m 0, [the samek that appears in property (v j]; (iii) U!(T) has a left cut from --a to -b (a > b > 0, and aneed not be finite), a right cut from 0 to T, , and is ot,herwise real analytic; (iv) the right-hand discontinuity of ul( T) is 2i Irn hi(T) (Thus Bl(T) is real from 0 to T, and carries t’he full discontinuity above T,) ; (v) the left’-hand discontinuity multiplied by l/2$ denoted by fl(T), has 1 + k sign changes”over the left cut. (Sate that in conjunction with the other 4 properties this property corresJ)ondsto a minimum number of sign changes when hl( T) represents JhL.“;and also when h,( T) represenk Jhi, if the pn’s defined in *The expression h!(T) -+-+o O(F) const,ant. 3 A change of sign for jr(T) occurs definitely negat,ive, or vice versa.
is taken when
to mean its value
that
changes
limT,0 from
hl(2’)/!P definitely
= a finite positive
to
438
BINSTOCK
(7) are all of the same sign and do not sum to zero. This is shown in the discussions following (7) and (Sa) . ) It then follows that UL( T)/T’ has a dispersion relation N(T) Tl
_ _1 -’ fi(T’) P s-a T”(T’
-
Tc Im hL( T’) dT’ + c dT’ T) + i l T’l(T’ - T) ’
by properties (i), (iii), and (iv), where C is a real constant tribution from the contour at infinity. By (2) and (ii),
representing
(2) the con-
Tl+k+’
Tk+’ Reul(T)
= ___ T Tc Im hl( T’) dT’ + rc + ‘1 T’l(T’ - T) in (3) then gives
Expansion of (T’ - T)-’ “f,(T’) dT’ =T’ l-n s --a
s0
Tc Im hl( T’) dT’ > TIl-n
1
+T+W 0.
(3)
(n = 0, *-* ) 1 + AT), (4)
c =o.
i
The result is
by (2) and (4), where &(T)
= I”
,$$)dT;)
.
Now define pn =
s0
Tc Im hl( T’) dT’ 7 T’ l-n
(n = 0, . . . , I + k).
(71
Each set of pLLn’smay be described by one of the following three cases: (a) A 2 0, and En pn Z 0; (b) A 5 0, and cnpn Z 0; (c) either the pCLn’s are not all of the same sign, or elsep,L = 0. By unitarity, we have case (a) for hl( T) = JhLi. For p-p scattering, we have case (b), by calculation (8), if hi(T) = Jh5(J = 2 or 4, T, = 400 MeV). Then by (4), (6), (7), and property (v), the theorem in Appendix A applies. The bounds on I&(T) are as given by Eq. (A.3) and the discussionfollowing it:
(84
BOUNDS
FOR
THE
CORRECTION
TO
THE
BORX
439
TERM
The upper inequalities hold for case (a); the lower ones hold for case (b). Also by the theorem, I + k is the minimum number of sign changes of fl( T) when case (a) or case (b) holds ( and therefore when hl( T) represents any Jh,,i or the p-p amplitude “/l”(J = 2 or -I, T, = 400 i\IcV) ). Assuming a11 upper limit of -I, on the extent of the left cut for t’he term ul( T), where b 2 L > 0, we also have thr less-restrictive equation
from t,he corollary to the theorem. As before, the upper inequalities hold for case (a) ; t#he lower inequalities hold for case ( h 1. For case (c ), we have
(Sb) T 2 0. The constants
c, rl E (a, b) and are chosen from the set (a, b, x1 , . . . , XL), where 1 x1 to xl are solutions of CL=0 1’ p,r-’ = 0. Then if Im hl( T) is assumed to be 0 experimentally known in t’he region 0 5 T 5 T,. it follows that bounds are established on ul( T) when T 2 0. Kow define a correction ul( 7’) to be of type ( UL), with 1~ some specified integer, if ZL~(T) has the properties (i) to (v) for k = 1~. We next show that the ChewArndt model (2) is of type (O), t’he MacGregor-Arndt model (j, even) (8) is of type (r~), where HL 5 - 1, and the Scotti-Wong model (1) with cutoff is of type (-2). Also, the Kantor approximation (3) is shown to be a limiting case of the Chew-Arndt and Scotti-Wong models. III.
A.
THE
SPECIAL
CASES
SCOTTI-WONG
In the Scotti-Wong
Now tion
= B,(T)
ul(T)
= (gTJ’-’ this
4 This model with is used, instead
TITE
MOI)EL
GIVEN
IN
SECTION
II
(or S-W) model (I), with rutoff,
h(T)
Fregard
OF
R~ODEL
as
+ ,uz(T),
(9)
c 1”
‘IT”‘;;&
a ’ limiting L -.
case ma
cutoff Z’,. is considered of a cutoff.
in ref.
“~~‘“-‘Irn where .-2. III ref.
hl(T’). (10) _... Y~~~ak-~‘;.~. --,7-u-. (I - 1) disk& poles .-
1 a different
sort
of approxima-
440 Tj(i
BINSTOCK
= 1, ‘..
UL(T) =
all
,I -
1) approach T, , where 7’” < 0. Then Tc dT’nfz:(T’
lim
TZ+To
.i_:;;-
Ti)
T
-
Tj) In1 hL(T’) - T)
T’l(T’
I
*
(10’)
It follows that property (i) is sat’isfied, as is also (ii) for k = -2. The right-hand discontinuity is 2i Im hl( T), and so (iv) holds. The left-hand discontinuity of ul( T) is a sum of (1 - 1) delta functions: linl udT l +o+
+ i~)-dT 2i
Q = &(
-
it>
Z-l
= T
lim all
C a$( T - Ti)
T2-To
Tc dT’ - Ti’ p Q CT’ T; - Ti) so
i=l
Tj) Im h(T’).
(114 (lib)
Thus there are 1 - 1 of the ai, of alternating sign when the T; are arranged in the order T1 < Tt < . . . < Tl-1 . Therefore the number of sign changes of the left-hand discontinuity is I + k(k = -2), satisfying property (v). Property (iii) is also satisfied because the left cut is from TX to Tl-, (which shrinks to a point To), and the right cut is from 0 to T, . The term ul( T) as given in (10’) is real analytic except for these cuts. Hence the S-W model is of type ( -2), when ,?2 2. The shrinking of the left cut corresponds to -a --f To , -b --) To , (To < 0). Thus (7) and (Sa, b) yield
uz(T)
1-Z h T 1 I+ 1To I-r ---cc = ?r T - To n=o ,=o (1 + T/l T,j)n +:l
pn =
Tc Im h(T’) T’l-n s0
Tc Im hl( T’) dT’ T’I(T’T)
dT’
(12)
(13)
This S-W model includes the Kantor approximation (3)) when To --f - to . Equation (12) holds for 2 2 1, where the double sum is zero if 2 = 1, and where hl( T) may represent any partial-wave amplotude, “hLi or Jhi. The requirements To =< -L and L = 21n,‘/M give for p-p scattering the less restrictive bounds
+ :
l
Tc Im ht(T’) dT’ TIL(T’ - T) ’
from (8a’). The upper inequalities hold for case (a), the lower ones for case (b) . Both ( 12) and (12’) hold for I 2 1. The double sum is zero when I = 1.
BOUNDS B.
THE
FOR
THE
MACGREGOR-ARKDT
CORRECTION
TO
BORN
441
TERM
AIODEL
The MacGregor-Arndt)
approximation
(2) is t,he following:
h(T)
= B,(T)
Ul(,T)
czz
jt(T'j
= (a + 2L/T')"'2(2L/1"1)(1
7”
THE
+ tbL(T)
(14)
T') dT' T12(T' - T)
Tc Im hl(
lr
+ d
Z-l
+ 2L/T')'" Jz cQ+~ I
Consider the case when j0 is even. The right-hand discontinuity is 2i Im hl( T) for 0 < T < T, ; t,he leftthand discontinuit~y is 2ifl( T) for - 00 < T < -L. Thus prol,erties (iii) and (iv) are saGstied. Kate that ul( Tj *T-o 0(1") by construct’ion, giving property (i). The second line of Eq. (15) corresponds t,o property (ii) with k 5 - 1. With j” even, fl( T') has at most 1 - 1 sign changes over the region T' = - Q, to T' = - L. This is also the minimum number of sign changes for cases (a) and (b), as was discussed following Eqs. (6) and (7). Thus property (v) is satisfied. The result is t#hat the i\l-A nl~proximation, with j, even, is for cases (a) and (b) a type ( - 1 ) correct,ion with (Sa) applying.
5 -T1 ?" dT' Im hl(T') (,)dT) 7t /0 -T'l(T' - T)
(~)f[cc$@"(;; (16)
TZ +;(&~o&l;;;;)~~
0
The upper inequalities hold if 1~1(7') = JhLi, and the lower ones if (J = 2, 4 and T, = 400 l\IeV, for p-p scattering).
Pn =
T'j dT' > 77,l-n
Tc Im hl( s0
hi(T) = "hj
(n = 0, . . . ) I - 1)
(17)
For case (c) the RI-A approximation, with jo even, is a type (nz) correction, ?)L 5 - 1. When j. is an odd integer, property (v) does not hold and so the theorem does not apply. No bounds are derived in this case. c.
THE
CHEW-ARNDT
In the Chew-Arndt
nlODEL
model @),
h,(T) = B,(T) + w(T) T.2
uk(T) = ~ 1 so
rlT’T’&l(l
(T'~Tj~l(~~/MT')
(1s) +
t/MT)
In1 hz(T'),
(19)
442
BINSTOCK
where Ql( 1 + l/MT) is a Legendre polynomial of the second kind. Now check for properties (i) through (v). The threshold behavior of ul( 3”) goes as T-‘&1( 1 + Z/MT), which corresponds to (i). The term uI( 5”) goes asympt’otically as PQ,( 1 + i/MY’), which gives Us +T-m (3( T-’ Log T). This asymptotic behavior agrees with (ii) for k = 0. The right-hand discontinuity is 2i Im hl( T), giving (iv). The left-hand discontinuity divided by 2i is lim udT e+o+
+ ic)-ul(T 2i
-
Tc T’ Im hl( T’) dT’ so (T’ - T)Ql(l + ?/MT’)
ie) _ P z(1 + Z/MT) 2T
’
(20) for T < -:/2M. This discontinuity has 1 sign changes over the left cut, - 00 < T < - Z/2M, corresponding to (v) for Ic = 0. The right cut runs from 0 to T, , and so (iii) also holds. Now the correction is of type (0) and so (8) applies. -T1 n- s0 I (21)
n /&(t/2M)-r
+T
a T + :‘2w
ni z
0 (1 + 2MT/l)”
t 2 4na,2 I+=
s0
i
TcIm hl (T’) dT’ T’l-r *
(22)
The upper set of inequalities holds when hr( T) = ‘hLi (nucleon-nucleon); the lower set holds when hl( T) = Jh5(J = 2,4, T, = 400 XIeV, for p-p). The largest T for which ul( T) may have a left cut, T = -L = -2am,‘/M, corresponds to a minimum Zof 4mr2. Then by (Sa’), we find that -T1 ” Im hl(T’) dT’ I p s0 Trl(T, _ T) (,)u,o
( z);lTc
l;$,Ty
;’ (21’)
T + ;
&
g
2
0
(1;
;;;l’
with L = 2rn,‘/M and the inequalities subject to the conditions cited following (22). Note that the Kantor approximation (3) is included as the limiting case of t* 00. IV. SUMMARY
The three models for the correction term ut( T) considered in this paper have been shown to be special casesof a more general model. This generalized model
BOUNIW
FOR
THE
CORRECTION
TO
THE
BORN
TERM
443
for ul( T), described in Section II, has bounds t’hat are given by (5), (7), and (Sa, b). Furt#hermore, for all of the p-p amplitudes considered in Appendix B we also have (8a’). For this case the bounded regions for corrections ul( T) (for models of type ( 1~) ) are nest,ed-the bounded region for II! includes the bounded region for HL’, where - 1 5 1~’ 5 ~1. Also, the corrections of tyl)e ( no) all have one of their boutlds in common. This is the lower bound for the amplitudes given by ( la)-( Id), and the upper bound for t’he coupling amplitude given by (1~). The figures in Appendix B show that the bounds are sufficiently limiting to force models of type ( ~1) (,for t’ho p-p amplit,udes considered) to all yield similar results, if 111is small enough. As particular cases when VLis small enough, II) = - 2, - 1, or 0 correspond to the t’hrec models considered--S-W with cutoff, 11-A (j, even), and C-A, respectively. V. CONCLUSION
An important feature of the correction term 1b1(T) is the relat’ionship between it,s asymptot,ic behavior as T + * and t.he number of sign changes (over the left cut) of its discont’inuity. If this and other features are such that properties (i) to (v) of Section II hold, with k = ~1, then one bound on ul( T) (if hl( T) = JhLi, or in general if case (a) or case (b) holds) .1s independent of the value of HZ. Thus some information would be gained without a knowledge of ~1. In other words, knowledge of the precise asymptot’ic behavior of UL( T) and the actual number of sign changes of t,he left-hand discontinuity is not required to obtain one bound. Still postulating that ui(T) has properties (i) to (v) and that case (a) or case (b) holds, we see that, if an upper bound could be placed on the value of 1~ then uL( T) would be sandwiched between two bounds. Each bound on uL(T) corresponds to a bound on Re B1(T) = Re hpP(T) - Re ul(T), over the range for which Re hpP(T) is known. If 1~ could be established to be small enough, then for sufficiently small ~1,‘s the upper and lower bounds on Re BI( 7’) would be close toget’her, and also not, far from Re hpp( T). In this contingency, no further features of UZ( T) would be required to give close bounds on Bl(T) in the region in which Re hpP( T) is l~nown, and the success of t,he one-meson exchange models could be accounted for. It should be emphasized, however, that’ the properties on which this model is based have not been established. APPENDIX THEOREM.
An integral
A. BOUNDS
ON
ITb(T)
of the four
rTb(T) = I,
-’
fl(T’)
dT’
(T’ _ ,),d3
(A.11
(a > b > 0, a not necessarily jinite, T 2 0), with (1 + 1) fcnite moments c(~ given and with fl( T’) having 1 sign changes’ over the region of integration, is bounded above
444
BINSTOCK
and below. The (1 + 1) moments -b fl(T’) dT’ /&I = - s T’l-n --a
(n = 0, . . . , 1)
(A.2)
need not all have the same sign. The three possible cases are: (a) t.~~2 0, En pn z 0; (b) A, 5 0, En ~Cln# 0; (c) either the t.~~‘sare not all of the same sign, or else t.~~= 0 for all n. If all of the (1 + 1) c(%‘shave the same sign and do not sum to zero,5 which is case (a) or (b), then 1 is the minimum number of sign changes that fi ( T’) may have. For thesetwo cases,the bounds can bewritten
The upper inequalities hold for case (a); the lower set hold for case (6). For case (c), (A.3) holds when a and b are replaced by c and d, ,wherec, d E (a, b) and are chosen from the set (a, b, x1 , . . . , xl}, where x1 to xz are solutions of l--r = 0. All symbols (except XI to XL) stand for real quantities. PJ Proof:&t 2Zfl
fz(T')
=
MJ~G”‘)
r12
CT’
-
(A-4)
Td,
where c is a constant. This form specifiesfl( T’) to be a general real function having at most 1 sign changes in the region of integration. n’ow solve the isoperimetric problem corresponding to the extremizing of ITb( T) . Form HE
from (A.l) gives
(T'
fi(T') - T)T"
+
-$ Lfi(T') n=OTll'"
(AS)
and (A.2). The X, correspond to Lagrangian multipliers. Then (A.4) H = $I~(T’)c 21+1 II ng+2 (T’ - TrJ T
T&
+ n$ AnT’”
1 .
(A.61
Rewrite (A.7) 5 In the applications of the theorem correction bounds for the p-p amplitudes same sign and do not sum to zero.
in this paper (specifically, considered in Appendix
B),
the derivation of the the P~‘S are all of the
BOUNDS
FOR
THE
CORRECTION
TO
THE
BORS
44,5
TERM
where these T,, are t,he (I + 2) rook of the expression. (If Xl were zero there would be fewer root#s in (A.7 ), but this situation is equivalent to that in (A.7)) with some T,, out,side the range of T’ and with some other constant in place of XZ .) Then (A.6 1 and (A.7) ykld H = c+“(T’)X~
n”n”=‘d (T’ T”(T’ - T)
T,)
(AX’)
’
Take (AA) (since this is the Euler-Lagrange equation when H has no explicit dependence on d+( T’),IdT’), and square the result. Then the coefficient, of fl(7”)/7”’ has (21 + 1) roots. So
Kow rewrite
(A. 1) and (A.2))
using (A.4’).
2/+1 ILn
=
-
z
a;
Tin,
(7%
=
0,
* * * ,z>
(A.101
( ai is any real number, and - a < Ti < -b < 0. ) Then the finding of the extrema of I?(T) corresponds to the ext’remizing of IIE( 3”) as given by jA.9), varying the 2(2Z + 1) parameters ai and Ti , subject to the constraint (A.10). However, this prescription can be simplified. Consider the equat’ions correspond ing to variation of a; and Ti : (A.ll) (A.12) for those ai not held fixed at zero; aG -zz c3Ti
-
(Ti !
T)2
+ &
X,nT,“-’
= 0,
for ai ?t 0.
(A.13)
Because of the assumption of 1 sign changes, there must be at least (1 + 1) of the ai which are not zero. Let (i = 1, . . * , I + 1) label these ai . They will be varied. The remaining 1of the ai are not restricted, and they may be either varied or held fixed, wibhout affecting the analysis to follow. Suppose also that one of
446
BINSTOCK
the Ti (associated with a nonidentically-zero ai) is varied. Call this T1 . No restrictions on T; , with i # 1, are made (except that --a < T; < -b); they also may be either varied or held fixed without affecting the following analysis. From (A.12), we obtain xo + XlT, + XzT; + ...
+ XLTI’ = -& 1
Xo+X~Tt+xzTz~+
...
+&Taz
= -p
Xo + XI Q”L+I + XPT;+I + . . . + XI T:+I = Equation (A.14) corresponds to the variation previously. From (A.13), we know that XI + 2XaT1 + 3&T:
+ ...
1 T2 - T 1 T z+1 -
(A.14)
T’
of the first (I + 1) ai , as defined
+ 1x1T:-l
=
(T1 !
T)?’
(A.15)
This corresponds to the variation of T1 . The T1 , Tz , . . , Tl+l in (A.14) may all be regarded as distinct, since they are positions of delta functions. As already pointed out, the assumption that there be I sign changes of fl( T’) requires that there be at least I + 1 distinct Ti , which were chosen to be Ti (i = 1, . . . , I + 1). Now look for solutions of (A.14) plus (A.15) for T1 . Note that in the limit T1 --$ Ti , for i E (2, . . . , 1 + I], the difference between the first and the ith equations of (A.14) yields (A.15). So there are at least 1 roots of T, for the system (A.14) plus (A.15). In fact, these two equations are equivalent to Tl[::f (T1 - Ti) = 0, b ecause there are also no more than 1 roots of T1 in the system. Proof of this statement follows. The last 1 equations of (A.14) give X1 to Xi in terms of X0 , Tz , . . . , Tl+l , in the form of the ratio of two determinants. The finiteness of X1 to Xl is assured because the denominator may be written in the form ( T2 . . . Tl+l) nf:&2,i
BOCNDS
FOR
THE
CORRECTION
TO
THE
BORN
447
TERM
(AJO), occur when all Ti ---f x, --a < T; < -b. The situation one in which only (1 + 1) of t’he (2l + 1) a;‘s are not identically
is equivalent to zero. So solve (iz.16)
1+1 PI? =
-g
(n
ac TilL,
=
0,
. . .
, Z),
(L4.17)
where all the Ti + .r, --a < T, < -b. Th en vary x over the range of integration of the int,egral, from -a to -b, t’o obtain the bounds of I:“( T). These bounds hold when the number of sign changes of fl( T’) is 1. At this step in the proof, before solving (A.16) and (A.lT), it, is possible to verify the statement) in the theorem that if all of the (I + 1 )h,[‘s have the same sign and do not sum to zero, then 1 is the minimum number of sign changes that fl (T’ ) may have. Suppose the same steps in the entire preceding proof are applied to the case of I$l( T) (with the form given in (A.1 ) ) having (1 + 2) moments T’) is required to have pCLn (n = 0, ... ) 1 + 1) all of t’he same sign. Only ~lowf~+~( 1 sign changes. The result corresponding to (A.17) is 2+1
= -&aiT;“,
l-h
(n=O,
. . ..Z+l).
(A.18)
Consistency requires the augmented determinant, D, to be zero, for nonzero ai . The Ti are distinct, since they are delta-function positions.
D=
PO
a1
a2
...
Pl
alT1
at T2
‘. .
al+1
az+l T~+I I
al
Aa+1
TI
I-Q
1+1
2+1
a2 Tz
. ..
1
1
at+1 5%: . ..
1
. . .
I Tz+ll
...
ITl+l(“+l
(A.19)
2+1
=
( )
-c11
I Tll
17’2
/
pi
( - 1) ‘+‘/~l+l
/ Tl‘/ If1
1T,‘I ‘+I
*(-1)
(l+l)(z+2)/2
.
Since the ai were selected as nonzero, and all the pcliare positive and do not sum t’o zero, it is sufficient to show that the coefficients of pi in the last determinant are each of the same sign and are each not zero. In other words, it is sufficient
448
BINSTOCK
to show that for distinct
Dk =
1
1
I TII
I Tzl
/ Ti / (-a . .. -‘.
( T$’
( T,‘(‘-’
*. .
/ Tl jk+’
I Tz jk+’
. *.
< Ti <
-b < 0), if
1 I TH/ (Tz+i Ik--l ,
(k = 0, ---,
1 + l),
then all D,t have the same sign, and Dk # 0. This is now shown. Since Dk hasall the zeroes of Dl+l , where Dl+l = Tl[f~~l,j~~( 1 Ti 1 Dk can be written
(A.20)
1 Tj I),
(A.21) Dk = Dz+lFz+l--l;( 1 TI 1, * * . , 1 TZ+I j >. Now Fz+I--k(l TI I, . . . , 1 Ti I, . . . , 1 Tj 1, . . . , 1 TI+~ I) is symmetric under the interchange i ++ j (i, j E ( 1, . . . , I + 1) ), because DA/DI+~ has that property. Also, the sum of the exponents of the different 1 Ti I’s in each term of Dk is fixed at (1 + 1) (I + 2)/2 - k. So F~+l--k is a symmetrized sum of terms with (I + 1 - lc) as the total number of different I Ti j’s in each term. No ) Ti /will have an exponent higher than one because the exponent of ( Ti j in any term of Dk is at most one more than the exponent of ( Ti j in Dl+l . It follows that Fz+l--k = ITz+~llTz/
-+. I T~+I I +
symmetry),
(other
terms needed for complete
if I% = 0, . . . , I
= Psi Tz+l 11Tz 1 . . . 1 Tk+l I, = 1, ifk = I+ 1.
(A.22)
if k = 0, e.. ,I
1+1 different ways of k ( > selecting (I + 1 - k) different I Ti I’s from the (1 + 1) available. Since I Ti I > b > 0, it follows that Fl+l-k > 0 for all lc. Also Dz+~ is not equal to 0 by the distinctness of the / Ti 1%. Thus the Dk’S all have the same sign and are nonsero, affirming that D cannot be zero and therefore that fz+l( T’) cannot have 1 sign changes and at the same time satisfy the conditions of the theorem. The same proof holds for f~+~( T’) and 1’ sign changes, when r < 1. This completes the proof that 2 is the minimum number of sign changes of f~ (T’), where fz (T’) is as described in the theorem. The next step is to solve (A.16) and (A.17). Write Here P, is a symmetrizing
operator
that picks out the
BOUSDS
FOR
where I 2 0, --a 5 Ti 5 (A.23) with (A.1) shows (A.23).) This has no right is a sum of delta funct’ions
THE
CORRECTIOS
TO
THE
BORE
449
TERM
-b. (Since Ii”(T) satisfies (A.lj, a comparison of that there exists a function gl(T’) which satisfies cut. The left-hand disrontJinuity multiplied by 1/2i at Ti : (A.24)
The result is simply (A.16) and (A.17) (aft’er applicat’ion of (A.l) and (A.2)). So the limit all Ti -+ x, (--a 5 .C 5 b), should give extrema of IiE( T) for fixed .r. In this limit,
=
cocZT’ gl (T’) so T” (T’ - T)
(z - T’)‘+’ - (5 - T)‘+’ (x-T)l+’
But
(a -
T’ )l+l -
(.c -
T)‘+’
= (T -
T’)go
(z -
T’)p(~
-
1
(A.25)
*
T)z-p.
(A.26)
Therefore
k 2
0
=-&p p=or=o [l + (T,,&,‘(T
+ ]xj)
f
=
T”
p=o r=o [l + (T/lx()]pIx(:~(T
where
T 2 0. Note that from (A.l)
and (A.2)
IIE(T)= gogl + something But (A.23) ZL~(T)
ODdT’ ?@(T’)
’
we have
of higher order in l/T.
(A.28)
gives
= go I-
By (A.ZS)
(A.27)
+ jzj)
g
gdT’)
sl
+ something
of higher order in l/T.
(A.29)
and (A.29),
Pi = s 0
mdT’g2 ( T’) TI~-L ’
(i = 0, . . . , I).
(A.30)
450
BINSTOCK
So (A.27)
and (A.30)
give
(A.31) where -a 5 x 5 -b < 0. This is clearly bounded, and so IT”(T) is bounded. In the case when the p, are not all of the same sign, it is necessary to look at the solutions of
as well as at the end points x = -a and x = -b, to determine IIE( T). However, suppose all the pcL,are positive. Then aI,” ~ ax
=
0 + l)lxlL (T-x)+2
5 I (r=o(r)fi}
the bounds on
>‘,
and so the maximum occurs at z = -b. The minimum occurs at x: --+ all the pr’s were negative, the inequalities would be reversed. The result (A.3). COROLLARY. If Gb( T) satisfies the conditions given in the statement theorem, with all of the (I + 1)~~‘s having the same sign, and if a’ 2 a and b 2 then (A.3) holds when a and b are replaced bg a’ and b’ respectively. This trivially from (A.32). APPENDIX
(A.32) -a. If is Eq. of the b’ > 0, follows
B. BOUNDS ON uz(T) AND Bl(T) FOR p-p SCATTERING
The bounds given by (5), (7), and (Sa’) are applied to corrections UI( T) of type (m) (m = -2, -1, 0). The bounds on B,(T) = hi(T) - ul(T) are also shown. Correction C-A (b L 4) M-A (j. even) S-W (To $ - 2m,2/M)
is of
Tw 0 -1 -2
(These corrections are considered in Section III.) In the range 0 to 400 MeV, data hi(T) is taken to be hi(T) Exp, and T, = 400 MeV. The experimental are taken from refs. 2 and 8. The experimental errors are not shown.
BOUNDS
FOR
THE
CORRECTION
TO
THE
h
0 -4
50
FIG.
1. Bounds
180 Re w(T) on ?r (1 f 2M/T)“2
BORN
M correction
451
TERM
MlntypeCm),mZ-I a1sos-w
300
150 TIPb(MeV)
to 61 or et of the Born
term,
compared
to
(a)
3P~ ; (b) Ta.
I IO,
,
I
3
Max
150
300
50
T ,ob (MeV)
FIG.
2. Same
I
Fig.
I
type (ml,mt-2
150 Tlob(MeV)
as for
I
(bl
1. (a) ‘Dz
; (b) Ej
300
452
BINSTOCK
I
0
Min type Cm), m> -3
z aI & d
fi2 E P cl
-5 5
150
300
Tlab (MeV)
FIG.
50
3. Same
150
T,,blMeV)
as for
Fig.
1. (a) 3Fz ; (b)
aFa
300
Slob (Mb’)
0 Min type(m),m>-4
0.2 50
150 T ,ab (MeV)
FIG.
4.
Same
as for
Fig.
1. (a) 3F4 ; (b) 1G4
l!OUNDS
FOR
THE
CORRECTION
50
TO
5. Same
mt
3;0
as for Fig.
1, but
-30
w,f
I 50
for Ed instead
(b)
’
l&f7
-I
z ET P 0 Min type
1
Mm type (ml,mZ-I OISO s-w
\
I 150
/
Tlab
(MeV)
453
TERM
(Mei’)
c Max type h),
BORN
150 T,,,
FIG.
THE
i i
Min type (-II Min type (0)
I
150 T,,,
FIG.
6. Bounds
on
180 Re BI(!!‘) a (1 + 2M/!w where
Re ul(T)
(MeV)
has the
180Re = T bounds
[h:xP(T) (1 +
graphed
-
w(T)]
NN 61 or tl of Born
2M/T)“2
in Figs.
1 to 5. (a) 3P1 ; (b)
3Pz .
term
7
454
RINSTOCK
,
300
150 T,,,
FIG.
7. Same
as for
Fig.
300
Tlab(MeV)
8. Same
150
300
Tlab (MeV)
(MN)
150
FIG.
50
":""-yj
6. (a)
50
IL)1 ; (b) EZ
150 T,,,b(MeV)
as for Fig.
6. (a) 38’2 ; (b) 3Fa
300 ’
BOKVDS
FOR
I
THE
CORRECTION
TO
THE
BORiY
TERM
I Mox type
1.5
(m),m>-4
1.0 5L ,” D 0.5
150
300
50
150
Tlab(MeVl
FIG.
T,o,(MeV)
9. Same
as for
Fig.
6. (a) 3F4 ; (b) ~2~
Max Max
type type
Max type
-
Min
type
(0) (-I) (-21,
Cm),
150 Tlob(MeV)
FIG.
10. Same
as for
Fig.
6, but
for
Eq instead
455
456
BINSTOCK
APPENDIX
C. DISCUSSION
OF
THE
C-A
AND
M-A
MODELS
Since both the C-A and M-A models are taken from an unpublished (a), they are further discussed here.
reference
1. C-A Assuming a Mandelstam representation for the singlet p-p amplitude,
and a strip
approximation
write,
h(T) = Bz(T)+ f j-12 cdc4(T,t’)Qz(1 + &)/W
(C.1)
where h:14(T, t) = i /*‘cZT’[h,t(T’, a- 0
t) + (-l)‘h,,(T’,
and h, t and h,, are the usual double-spectral functions. assumption that the integral in (C.l) can be approximated
00 s
clt?~:‘~(T, t’>Qz (I + A)
t)]/(T’
-
Make as
T),
(C.2)
the additional
= CYL:.~(T, i)Q (1 + A),
(C.3)
4m,2
for some f, where it follows that Im hi(T)
C is independent
= C[h, t( T, f) + (-
of T. Then from real analyticity
l)‘h,,(
T’, i)]/aJJT,
0 < T < T, .
(For this model, Bl( T) is real for 0 < T < T, .) Then (C.1) may be combined to yield h(T)
= &CT)
of hi(T)
through
(C.4) (C.4) (C.Tj)
+ Q(T)
(C.5) and ((3.6) correspond to Eqs. (18) and (19) of Section III. applied by Arndt (2) to all five independent p-p amplitudes.
This model is
2. M-A Again assume both a Mandelstam representation and a strip approximation for the p-p helicity amplitudes, so that only that part of the double-spectral function within the strip 0 < T < T, contributes to the amplitude. Then by real analyticity of hl( T) we may write h(T)
= &CT)
+ w(T)
N(T)
-Lfi(T’) dT’ = ‘, s--oo T, _ T +f6
(C.7) Tc Im hl( T’) clT’ T’-T ’
(C.S)
BOUNDS
FOR
THE
CORRECTION
TO
THE
BORS
457
TERM
where L = 2rn2/M and Im Bl( T) = 0 within the strip. (L corresponds to the 2~ threshold in the crossed channels.) The left-hand discontinuity fl( T) must have the property fz(T)
-T-+--L o( T + L)3’“.
cc.91
This is derived in ref. 2’. Then, requiring normal t’hreshold an esl)ansion of (T’ - T )-’ in ((2.8) yields Q(T)
= ;
-’ fJT’) [S -zc T’l(T’
dT’ Tc Im hi (T’) dT’ - T) + l T’l(T’ - T)
Tc Im hl (T’) dT’ T,n+I ] = 0,
-T” 7r The additional
behavior
of ul( T),
1 (C.S’)
(12 = 0, . . . ) I -
1).
(C.10)
requirement uz(T)
Jvhen combined with for an approximation verge, is
(C.10’)
-+c.m 0,
(C.8’), includes (C.10) as a consequence. A simple choice to Ji ( T), which allows the first integral in (C.10) to COIIZ-l
fl( 7’) = (2 + 3L/‘T)““‘(2L/Tz)(l Then (C.7), (C.S’), of Section III.
(C.lO’j,
+ 2L/‘T)‘“~o
and (‘2.11)
aiT”.
(C.11)
are equivalent8 to Eqs. (14) and (15)
ACKNOWLEDGMENTS I \vould like to thank Professor Geoffrey Chew, who suggested this project, and Dr. Michael .I. Moravcsik, who acted as thesis advisor. I am indebted to them and to Dr. Richard A. Arndt for many helpful discussions. RECEIVED:
*July 19, 1965 REFERENCES
1. A. SCOTTI AND D. Y. WONG, Phys. Rev. Letters 10, 142 (1963); Phys. Rev. 138, B145 (1965). 9. RICHARD A. ARNIIT, “Unitarity Corrections to Peripheral p-p Scattering below 400 MeV,” VCRL 12211-T (Lawrence Radiation Laboratory Report, January 1965, unpublished). 3. P. B. KAXTO~, Phys. Rev. Letters 12, 52 (1964). 4. MICHAEL J. MORAVCSIK, Ann. Phys. (‘V’. ET.) 30, 10 (1964). r. H. P. STAPP, T. J. YPSILANTIS, AND N. METROPOLIS, Phys. Rev. 106, 302 (1958). 1 M. H. MACGREGOR, M. J. MOR.~VCSIIL, END H. P. STAPP, Ann. Rev. Nucl. Sci. 10, 291 (1960). 7. M. L. GOLDBERGER, M. T. GKIS.~RU, S. W. MACDOWELL, AND D. Y. WONG, Phys. Rev.
120, 2250 (19601. 8. RICHARD A. ARTCDT, Lawrence communication, 1965.
Radiation
Laboratory,
Livermore,
California,
private