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Unitarized impulse approximation: KD elastic scattering
ANNALS
OF PHYSICS:
Unitarized
41, 434441
Impulse
(1967)
Approximation: .J. J.
Department
of Physica,
KD Elastic
Scattering*
BREHM
Northwestern
University,
Evanston,
Illinois
An effort is made to incorporate unitarity int,o the dispersive impulse approximation for KD elastic scattering. It is argued that elastic unitarity sufhces since there exist no inelastic two-body states of any kind (involving stable or unstable particles) coupled to the KD system at low energy. The impact parameter representation is used and an approximat,e unitarity relation is obtained. Correct threshold behavior for the partial waves is built in and a unitarized version of the impulse approximation is constructed. The cross sections are calculated at 230 MeV/c. Little data is available for comparison with the computed angular distribution; the total cross section is in excellent agreement with experiment. I. INTRODUCTION
It has been shown that a dispersion theory point of view may be adopted for elastic scattering problems involving composite targets, such as deuterons (1). The most important contribution to the represent’ation is seen to be closely related to the traditional impulse approximation. It should be clear that the result obtained in I is expected, in a sense, to be 110 better than the Born approximation would prove to be in the scattering from particles that are more elementary. Indeed, Brueckner (2) has established that multiple scattering corrections are negligible only if the Born approximation is applicable to the elementary interactions inside the composite system. In a dispersion theory framework, these corrections as well as absorption corrections are to be dealt with in a straightforward way (at least in principle) by incorporating unitarity. The results of I are manifestly in need of this attent’ion in that the amplitude constructed there is not complex. Accordingly the purpose of this paper is to unit’arize the impulse approximation of I in a limited way, in the hope t’hat the corrections which remain unallowed for will not be import,ant. The (*hoice of KD elastic scattering as the means of illustration allows one to argue away correct’ions which would be vital in problems such as ?rD or I?D scattering. II.
A.
ELASTIC
UNITARIZING
THE
AMPLITUDE
UNITARITY
KD scattering is urlique among meson-deuteron problems in that there exist no absorptive channels. Whereas TD is coupled to NN, and l?D to AN and * Research
supported
in part
by the National 434
Science
Foundation.
UNITARIZED
IMPULSE
43.5
APPROXIMATION
zN, no such two-body system of stable particles is coupled to KD. Moreover, there exist no low energy KN resonant states; therefore, all three-body states, coupled to KD, must be treated as such and cannot be approximated by quasitwo-body states (particle plus resonance, such as NN” in the rD case, or Y*N and YN* in the I?‘0 case). Thus the KD problem is vastly less involved from the standpoint of truncating the unit,arity relation. The assumption to be made in this paper will be that only two-body states need be retained. This elastic unitarity approximation amounts to the assumption that rescattering eff e&s are the leading corrections to the impulse approximation. Surely, this argument cannot be stretched to mean that KNN states are unimportant; it is just that feasible methods for including them are not known to this author. A further approximation will be made in this paper. Whereas the full KD elastic amplit,ude is most generally expressed, in the notation of I, as: T = T1 ~(t&~*(rl’)
+ T?
C(d) d’~“(d’) ll!P
.cl
+ T 4(d) .d’E*(d’) .Q + t(d) .&t*(d) 3
.cl + Tq t(d) .Q,t*(d’)
M”
A!12
(1)
.Q ’
it was learned in I that the Tq term does not, occur in the impulse approximation, and that, moreover, the Tz and TX t’erms represent effeck of the order of only 2 ‘X of the T1 term at, low energy. Therefore only t,he T1 term will be unitarized. The elastic unitarit’y relation applied to T = Tlf(d)
.,t*(d’)
(“j
reads : t(d) .(*(d’j
Im Tl(s, t)
k” = [s - (N
+ p)“][S -
(IIf - P)7/4S.
(4)
Equation (3) is represent’ed by t,he diagram in Fig. 1, in which the other kinematic variables in (3) ar,e defined. The right hand side of (3) contains ,t(d) . t*( d’) and t(d) .&*(d’) .d; again, only the first form is appreciable at low energy, allowing Eq. ( 3) to become:
BREHM
FIG. 1. Elastic unitarity
B. THE
IMPACT
PARAMETER
for KD scattering
REPRESENTATION
In order to construct a unitary amplitude, satisfying, e.g., Eq. (5), the introduction of partial wave amplitudes is often convenient. This is not the casewith a deuteron problem, even at low energy. Here, the angular distributions for elastic scattering are diffractive; the momentum transfer does not have to be very large to suppressevents in which the deuteron is not broken up. Thus many partial waves are important so that an impact parameter expansion would seem to be in order. This approach has long been advocated in the high energy scattering of elementary particles (5), and has been used recently in connection with the peripheral model corrected for absorption (4). The point here is to argue that mesonswith lab momentum as low as 230 MeV/c will exhibit phenomena typical of high energieswhen a deuteron is involved in the collision; the scale indicating what sort of energy is high is established by the small deuteron binding energy. The mathematical device for making the impact parameter expansion is t#he Fourier-Bessel transform. This has been discussed by Predazzi (5), and particularly by Blankenbecler and Goldberger (6), who show how to obtain the unitnrity relation appropriate for this representation. The subscript on T1 is suppressedand the amplitude is written for convenience from here on as a function of s and A where A” = -t
= 4k2 sin’ e/2.
(‘3)
The l’ourier-Bessel transform is introduced by setting: T(s, A) = la b dbJo(bA)H(s, b). If this expansion is introduced into the right hand side of (5)) one eventually gets a unitarity relation for H, approximately valid for each b: * Im H = -apHH (8)
UNITARIZEI~
IMPULSE
437
APPROXIMATION
in which p = 1/[(4?r)21c&].
(9)
That (8) is only valid approximately (unlike the situation with partial waves) follows from the fact that along the way one must assume the dominance of the forward amplitude. At one stage of the derivation an integral over A from 0 to 31~is approximated by letting 2k + w . At high energies this is a good approximation; here it is necessary to assume that the amplitude falls off rapidly away from the forward direction. C.
COKSTRUCTION
0F
H(s,
b)
The toask comes down to one of obtaining a function H( s, b) fulfilling (8) and providing a unitarization of t’he results of I. An additional physical constraint must also be met; the partial waves projected from the final version of T( s, A) must, have the correct threshold behavior for each 1. The procedure turns out to involve making a number of compromises. The requirement of correct threshold behavior for each partial wave is reg:l,rdetl as a principle to be built in at all costs. It implies that H( s, b) must approec~h :I, finite, nonzero constant as s + (M + P)~. This arises as follows. The parGal wzve :~,mplit.udes are given by :
-1 TlC.s.1= 1, I rl cos 0T(s, AjPl(cos -1
0)
and Eq. ( i j holds for T. It follow-s (5, 6) that T1 and H are rel&cl
(IO) by: (11)
In order that Tl vanish :,nt there. The amplitude to be unit,Czed has been constructed in I; it is given there in Eq. (30) and the first of Eqs. ( 31‘). It will be convenient from here on to observe that the funct’ion LT1(in ( 30) and (31) of I) is fit very accurately by the following formula: [‘(s,A)
rzz
-c”Jtu”i.
y2 A?
+
y2
(
12)
in whicah (~’ = -&aoFoZilP/,tr in terms of parameters
explained in I. The rest of (12) is determined
(13) by curve
438 fitting;
BREHM
the parameters
are given in pion mass units: s0
= 150m,’
s1 = 234m,’
(14)
y = 4m,. No physical interpretation can be offered for the constant y. The great virtue of the empirical fit (12) over the exact formula is that the Fourier-Bessel transform of (12) is:
n(s,b) =s
ODA dAJ&A)
U(s, A)
0
(15)
= -Cr2Ko(br)(s Equation
(8) must now be considered.
+ d/so.
If one defines the function:
N(s, b) = 1 +niTpn
(16)
it is immediately clear that the choice H = iV would satisfy ever, (16) fails to have the correct threshold behavior: N -+ l/i?rp
as
(8) exactly.
k + 0.
How(17)
The final choice calls for a compromise of Eq. (8), which mately valid to begin with. The construction is to let:
was only approxi-
H = N/D
(18)
where ImD
= ?rpReN.
(19)
From (16), (18), and (19) it is clear that H approaches a nonzero, real constant as k -+ 0. The dispersion integral for D, obtained from ( 19), does not converge and point requires a subtraction. The point s = -sl is a convenient subtraction and introduces a subtraction constant D( -SI) :
D(s,b) = D(-SI)
+ (s + SI) [:,--&
P(s’)
Rt,Nt;,b).
(20)
This freedom is helpful in that it allows one last opportunity to improve on the satisfaction of Eq. (8). If the choice D( -a) = 1 is made then (8) is exactly satisfied for the very important value of the impact parameter b = 0. In the b = 0 limit n diverges logarithmically; thus N -+ l/i?rp, D -+ 1 and requirement (8) is fulfilled.
UNITARIZED
IMPULSE
439
APPROXIMATION
The construction of N according to Eqs. (IS), (18)) and (20) accomplishes one of the primary objectives of unitarization. The nonunitarized Fourier-Bessel amplitude 1~of Eq. (15) behaves logarithmically for small b; on the other hand, H is finite at b = 0. Thus the small partial waves are suppressed in the process. If this is to be done accurately it is important that the unitarity relation (8) has been exactly satisfied for b = 0. The final version of the construction is provided by Eqs. ( 16)) ( 18)) and the following formula for D : D(s, b) = 1 + 48, b)lC/(s, b) where
cc ds’ s(M+p)2 sT k2(s’)s’
The desired unitarization (7).
of the results
k(&/s’ + [Cr2Ko(by)(s’
(21)
+ s1)/167130]~’
of I is then obtained
(“)
by means of Eq.
COOS SC,
(a)
FIO.
2. Angular
distribution
at 230 MeV/c
(a) in the
c.m. system,
and
(b)
in the lab
440
BREHM
III.
CALCULATIONS
AND
The s-wave KD scattering length is readily Equation ( 11) in the limit k 4 0 yields : To 4 C[(Jf so that the scattering
RESULTS
obtained
from these formulas.
+ d2 + s11/s0,
(23)
length is:
A = C[(M + Pj2 + sd/[Smo(M + /AL)1 = -0.55
(24)
fermi.
This result is smaller than that obtained by Dass (7)) who claims to need a smaller result for agreement with experiment. The angular distribution at 230 MeV/c may also be calculated using the unitarized amplitude and the cross section formula, Eq. (34) of I. It matters very little whether the Ut and Us terms are included or not; the results for the c.m. system and the lab system are shown in Fig. 2. Again, as in I, a quantitative comparison with experiment (8) is impossible since elastic data have not been extracted from the non-charge-exchange results. If one assumes from Stenger’s data (8) that the inelastic cross section rises gently from a value of the order of 1.5 mb/st in the backward direction, then this, taken in conjunction with the curve of Fig. 2(a), ma.y provide a reasonable fit. The total elastic cross section computed from the elastic angular distribution is gelastic = 7.05 mb.
(25)
This result lies somewhat lower than the figure obtained by Gourdin and Martin (9) at this energy. On the whole the calculated forward cross section appears not to be sharp enough. More will be said about this shortly. Total cross section measurements for all final states arising from KD interactions have been cited in the literature (8, 10-13). As pointed out in I, unitarity TABLE KD
TOTAL
c%
h&kVlc)
R From h From
CROSS
(mb)
I SECTIONS
u;,;p,i (mb)
230
24.9
26.5 + 3.ga - 3.8
377
16.7
23.9 f
3.lb
530
12.5
26.1
3.3’,
refs. 8, 10, and ref. 11.
13.
f
UNITARIZEI)
IMPULSE
APP11OXIMATION
441
must be introduced into the calculations in order to obtain this quantity. From t,he optical theorem one needs the imaginary part of (7) in the forward direction: utotal = -1m
T(s, O)/[‘Z&].
( 26)
The results are given in Table I along wit’h existing data. The figures at 230 iUeV/c show excellent agreement. At higher energies there are good reasons to begin to doubt the validity of the calculation. For one thing, elastic unitarity should become less adequate; for another, the assumption of pure s-wave T = 0 KN scattering becomes increasingly less valid. Solutions in the T = 0 state with only an s-wave are ruled out at 377 >IeV/c and above (11) . The use of only s-waves at 230 XeV/c, even in the scattering length approximation, should, however, be valid ( 11). The excellent total cross section result at 230 XeV/c is very encouraging and suggests where the deficiency in t’he calculation might lie in failing to give a desired sharper forward peak in the differential cross section. The implication is that elastic unitarity is quite good for determining the imaginary part at 230 MeV/c. Since the real part is derived from a knowledge of the imaginary part integrated over all energies in the dispersion relation, it seems clear that significant sharpening requires KNN intermediate states to be somehow included. 011 the whole the low momentum results obtained by the methods of this paper are not discouraging. RECEIVED:
June 16, 1866 REFERENCES
1. J. J. BKEHM, Ann. Phys. (iv. Y.), the preceding paper, referred to hereafter as I. d. K. A. BRTJECKNER, Phys. Rev. 89, 534 (1953). 8'. R. GUUBER, “Lectures in Theoretical Physics,” p. 315. Interscience, New York, 1958. References to other earlier work may be found here. 4. J. D. JXKSON, Rev. Mod. Phys. 37, 484 (1965). This review article cites a large number of references on the subject. 5. E. PREDUZI, Ann. Phys. (IV. Y.) 36, 228 (1966). 6. R. BL~NKENBECLER AND M. L. GOLDBERGER, Phys. Rev. 126, 766 (1962). 7. G. V. D.\ss, XZLCZ. Phys. 69, 612 (1965). 8. V. J. STENGER, UCLA Ph.D. thesis (1963), unpublished. 9. M. GOURDIN AND A. MARTIN, ,VUOVO Cimento 11, 670 (1959). 10. W. LEE, UCRL Report-9601 (1961), unpublished. If. T'. J. STENGER, W. E. SL.ITER, D. H. STORK, H. K. TICHO, G. GOLDHABEIL, .IND S. GOLDH.ZBER, Phys. Rev. 134, Bllll (1964). I,?. V. COOK, D. KEEFE, L. T. KERTH, P. Cr. MURPHY, W. A. WENZEI,, AND T. F. ZIPF, Phys. Rev. Letters 7, 182 (1961). 13. W. SLATER, D. H. STORK, H. K. TICHO, W. LEE, W. CHINOIVSKY, C;. GOLDHSBER, S. GOLDHABER, AND T.O'HALLOK.\N, Phys.Kev.Lefters7,378 (1961).
OF PHYSICS:
Unitarized
41, 434441
Impulse
(1967)
Approximation: .J. J.
Department
of Physica,
KD Elastic
Scattering*
BREHM
Northwestern
University,
Evanston,
Illinois
An effort is made to incorporate unitarity int,o the dispersive impulse approximation for KD elastic scattering. It is argued that elastic unitarity sufhces since there exist no inelastic two-body states of any kind (involving stable or unstable particles) coupled to the KD system at low energy. The impact parameter representation is used and an approximat,e unitarity relation is obtained. Correct threshold behavior for the partial waves is built in and a unitarized version of the impulse approximation is constructed. The cross sections are calculated at 230 MeV/c. Little data is available for comparison with the computed angular distribution; the total cross section is in excellent agreement with experiment. I. INTRODUCTION
It has been shown that a dispersion theory point of view may be adopted for elastic scattering problems involving composite targets, such as deuterons (1). The most important contribution to the represent’ation is seen to be closely related to the traditional impulse approximation. It should be clear that the result obtained in I is expected, in a sense, to be 110 better than the Born approximation would prove to be in the scattering from particles that are more elementary. Indeed, Brueckner (2) has established that multiple scattering corrections are negligible only if the Born approximation is applicable to the elementary interactions inside the composite system. In a dispersion theory framework, these corrections as well as absorption corrections are to be dealt with in a straightforward way (at least in principle) by incorporating unitarity. The results of I are manifestly in need of this attent’ion in that the amplitude constructed there is not complex. Accordingly the purpose of this paper is to unit’arize the impulse approximation of I in a limited way, in the hope t’hat the corrections which remain unallowed for will not be import,ant. The (*hoice of KD elastic scattering as the means of illustration allows one to argue away correct’ions which would be vital in problems such as ?rD or I?D scattering. II.
A.
ELASTIC
UNITARIZING
THE
AMPLITUDE
UNITARITY
KD scattering is urlique among meson-deuteron problems in that there exist no absorptive channels. Whereas TD is coupled to NN, and l?D to AN and * Research
supported
in part
by the National 434
Science
Foundation.
UNITARIZED
IMPULSE
43.5
APPROXIMATION
zN, no such two-body system of stable particles is coupled to KD. Moreover, there exist no low energy KN resonant states; therefore, all three-body states, coupled to KD, must be treated as such and cannot be approximated by quasitwo-body states (particle plus resonance, such as NN” in the rD case, or Y*N and YN* in the I?‘0 case). Thus the KD problem is vastly less involved from the standpoint of truncating the unit,arity relation. The assumption to be made in this paper will be that only two-body states need be retained. This elastic unitarity approximation amounts to the assumption that rescattering eff e&s are the leading corrections to the impulse approximation. Surely, this argument cannot be stretched to mean that KNN states are unimportant; it is just that feasible methods for including them are not known to this author. A further approximation will be made in this paper. Whereas the full KD elastic amplit,ude is most generally expressed, in the notation of I, as: T = T1 ~(t&~*(rl’)
+ T?
C(d) d’~“(d’) ll!P
.cl
+ T 4(d) .d’E*(d’) .Q + t(d) .&t*(d) 3
.cl + Tq t(d) .Q,t*(d’)
M”
A!12
(1)
.Q ’
it was learned in I that the Tq term does not, occur in the impulse approximation, and that, moreover, the Tz and TX t’erms represent effeck of the order of only 2 ‘X of the T1 term at, low energy. Therefore only t,he T1 term will be unitarized. The elastic unitarit’y relation applied to T = Tlf(d)
.,t*(d’)
(“j
reads : t(d) .(*(d’j
Im Tl(s, t)
k” = [s - (N
+ p)“][S -
(IIf - P)7/4S.
(4)
Equation (3) is represent’ed by t,he diagram in Fig. 1, in which the other kinematic variables in (3) ar,e defined. The right hand side of (3) contains ,t(d) . t*( d’) and t(d) .&*(d’) .d; again, only the first form is appreciable at low energy, allowing Eq. ( 3) to become:
BREHM
FIG. 1. Elastic unitarity
B. THE
IMPACT
PARAMETER
for KD scattering
REPRESENTATION
In order to construct a unitary amplitude, satisfying, e.g., Eq. (5), the introduction of partial wave amplitudes is often convenient. This is not the casewith a deuteron problem, even at low energy. Here, the angular distributions for elastic scattering are diffractive; the momentum transfer does not have to be very large to suppressevents in which the deuteron is not broken up. Thus many partial waves are important so that an impact parameter expansion would seem to be in order. This approach has long been advocated in the high energy scattering of elementary particles (5), and has been used recently in connection with the peripheral model corrected for absorption (4). The point here is to argue that mesonswith lab momentum as low as 230 MeV/c will exhibit phenomena typical of high energieswhen a deuteron is involved in the collision; the scale indicating what sort of energy is high is established by the small deuteron binding energy. The mathematical device for making the impact parameter expansion is t#he Fourier-Bessel transform. This has been discussed by Predazzi (5), and particularly by Blankenbecler and Goldberger (6), who show how to obtain the unitnrity relation appropriate for this representation. The subscript on T1 is suppressedand the amplitude is written for convenience from here on as a function of s and A where A” = -t
= 4k2 sin’ e/2.
(‘3)
The l’ourier-Bessel transform is introduced by setting: T(s, A) = la b dbJo(bA)H(s, b). If this expansion is introduced into the right hand side of (5)) one eventually gets a unitarity relation for H, approximately valid for each b: * Im H = -apHH (8)
UNITARIZEI~
IMPULSE
437
APPROXIMATION
in which p = 1/[(4?r)21c&].
(9)
That (8) is only valid approximately (unlike the situation with partial waves) follows from the fact that along the way one must assume the dominance of the forward amplitude. At one stage of the derivation an integral over A from 0 to 31~is approximated by letting 2k + w . At high energies this is a good approximation; here it is necessary to assume that the amplitude falls off rapidly away from the forward direction. C.
COKSTRUCTION
0F
H(s,
b)
The toask comes down to one of obtaining a function H( s, b) fulfilling (8) and providing a unitarization of t’he results of I. An additional physical constraint must also be met; the partial waves projected from the final version of T( s, A) must, have the correct threshold behavior for each 1. The procedure turns out to involve making a number of compromises. The requirement of correct threshold behavior for each partial wave is reg:l,rdetl as a principle to be built in at all costs. It implies that H( s, b) must approec~h :I, finite, nonzero constant as s + (M + P)~. This arises as follows. The parGal wzve :~,mplit.udes are given by :
-1 TlC.s.1= 1, I rl cos 0T(s, AjPl(cos -1
0)
and Eq. ( i j holds for T. It follow-s (5, 6) that T1 and H are rel&cl
(IO) by: (11)
In order that Tl vanish :
rzz
-c”Jtu”i.
y2 A?
+
y2
(
12)
in whicah (~’ = -&aoFoZilP/,tr in terms of parameters
explained in I. The rest of (12) is determined
(13) by curve
438 fitting;
BREHM
the parameters
are given in pion mass units: s0
= 150m,’
s1 = 234m,’
(14)
y = 4m,. No physical interpretation can be offered for the constant y. The great virtue of the empirical fit (12) over the exact formula is that the Fourier-Bessel transform of (12) is:
n(s,b) =s
ODA dAJ&A)
U(s, A)
0
(15)
= -Cr2Ko(br)(s Equation
(8) must now be considered.
+ d/so.
If one defines the function:
N(s, b) = 1 +niTpn
(16)
it is immediately clear that the choice H = iV would satisfy ever, (16) fails to have the correct threshold behavior: N -+ l/i?rp
as
(8) exactly.
k + 0.
How(17)
The final choice calls for a compromise of Eq. (8), which mately valid to begin with. The construction is to let:
was only approxi-
H = N/D
(18)
where ImD
= ?rpReN.
(19)
From (16), (18), and (19) it is clear that H approaches a nonzero, real constant as k -+ 0. The dispersion integral for D, obtained from ( 19), does not converge and point requires a subtraction. The point s = -sl is a convenient subtraction and introduces a subtraction constant D( -SI) :
D(s,b) = D(-SI)
+ (s + SI) [:,--&
P(s’)
Rt,Nt;,b).
(20)
This freedom is helpful in that it allows one last opportunity to improve on the satisfaction of Eq. (8). If the choice D( -a) = 1 is made then (8) is exactly satisfied for the very important value of the impact parameter b = 0. In the b = 0 limit n diverges logarithmically; thus N -+ l/i?rp, D -+ 1 and requirement (8) is fulfilled.
UNITARIZED
IMPULSE
439
APPROXIMATION
The construction of N according to Eqs. (IS), (18)) and (20) accomplishes one of the primary objectives of unitarization. The nonunitarized Fourier-Bessel amplitude 1~of Eq. (15) behaves logarithmically for small b; on the other hand, H is finite at b = 0. Thus the small partial waves are suppressed in the process. If this is to be done accurately it is important that the unitarity relation (8) has been exactly satisfied for b = 0. The final version of the construction is provided by Eqs. ( 16)) ( 18)) and the following formula for D : D(s, b) = 1 + 48, b)lC/(s, b) where
cc ds’ s(M+p)2 sT k2(s’)s’
The desired unitarization (7).
of the results
k(&/s’ + [Cr2Ko(by)(s’
(21)
+ s1)/167130]~’
of I is then obtained
(“)
by means of Eq.
COOS SC,
(a)
FIO.
2. Angular
distribution
at 230 MeV/c
(a) in the
c.m. system,
and
(b)
in the lab
440
BREHM
III.
CALCULATIONS
AND
The s-wave KD scattering length is readily Equation ( 11) in the limit k 4 0 yields : To 4 C[(Jf so that the scattering
RESULTS
obtained
from these formulas.
+ d2 + s11/s0,
(23)
length is:
A = C[(M + Pj2 + sd/[Smo(M + /AL)1 = -0.55
(24)
fermi.
This result is smaller than that obtained by Dass (7)) who claims to need a smaller result for agreement with experiment. The angular distribution at 230 MeV/c may also be calculated using the unitarized amplitude and the cross section formula, Eq. (34) of I. It matters very little whether the Ut and Us terms are included or not; the results for the c.m. system and the lab system are shown in Fig. 2. Again, as in I, a quantitative comparison with experiment (8) is impossible since elastic data have not been extracted from the non-charge-exchange results. If one assumes from Stenger’s data (8) that the inelastic cross section rises gently from a value of the order of 1.5 mb/st in the backward direction, then this, taken in conjunction with the curve of Fig. 2(a), ma.y provide a reasonable fit. The total elastic cross section computed from the elastic angular distribution is gelastic = 7.05 mb.
(25)
This result lies somewhat lower than the figure obtained by Gourdin and Martin (9) at this energy. On the whole the calculated forward cross section appears not to be sharp enough. More will be said about this shortly. Total cross section measurements for all final states arising from KD interactions have been cited in the literature (8, 10-13). As pointed out in I, unitarity TABLE KD
TOTAL
c%
h&kVlc)
R From h From
CROSS
(mb)
I SECTIONS
u;,;p,i (mb)
230
24.9
26.5 + 3.ga - 3.8
377
16.7
23.9 f
3.lb
530
12.5
26.1
3.3’,
refs. 8, 10, and ref. 11.
13.
f
UNITARIZEI)
IMPULSE
APP11OXIMATION
441
must be introduced into the calculations in order to obtain this quantity. From t,he optical theorem one needs the imaginary part of (7) in the forward direction: utotal = -1m
T(s, O)/[‘Z&].
( 26)
The results are given in Table I along wit’h existing data. The figures at 230 iUeV/c show excellent agreement. At higher energies there are good reasons to begin to doubt the validity of the calculation. For one thing, elastic unitarity should become less adequate; for another, the assumption of pure s-wave T = 0 KN scattering becomes increasingly less valid. Solutions in the T = 0 state with only an s-wave are ruled out at 377 >IeV/c and above (11) . The use of only s-waves at 230 XeV/c, even in the scattering length approximation, should, however, be valid ( 11). The excellent total cross section result at 230 XeV/c is very encouraging and suggests where the deficiency in t’he calculation might lie in failing to give a desired sharper forward peak in the differential cross section. The implication is that elastic unitarity is quite good for determining the imaginary part at 230 MeV/c. Since the real part is derived from a knowledge of the imaginary part integrated over all energies in the dispersion relation, it seems clear that significant sharpening requires KNN intermediate states to be somehow included. 011 the whole the low momentum results obtained by the methods of this paper are not discouraging. RECEIVED:
June 16, 1866 REFERENCES
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