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An estimate of Regge cuts from inclusive finite mass sum rules
Volume
40B, number
PHYSICS
5
AN ESTIMATE
LETTERS
7 August
OF REGGE CUTS FROM INCLUSIVE
1912
FINITE MASS SUM RULES
D.P. ROY and R.G. ROBERTS Rutherford
High Energy Laboratory, Received
Chdton, Drdcot, Berkshue,
16 June
England
1972
P
-P-o Inclusrve data on I& Ko,and K +K are used to calculate the pp andAaA2 Regge cuts m forward nucleonnucleon scattering Although the energies for whtch data exist are too low to allow a precise estimate, we can obtam an upper bound for the cut magnitude which IS at the level grven by the erkonal model.
Finite mass sum rules have recently been constructed for inclusive reactions [ 1, 21. They relate the residues of the nonsense wrong signature fixed poles m some particle Reggeon scattering amplitudes to experimentally measurable quantities. For instance, a missing mass integral of the inclusive cross-section for b
a-fc,
(1)
ISrelated to the fixed pole residue m bR scattering, where R IS the leadmg Reggeon m the acchannel. These residues constitute the Gribov vertices m the standard evaluatron of Regge cuts for two body scattering in the framework of Grrbov calcalus [3]. Thus, as remarked by Abarbenel [4], Einhorn, Ellis and Fmkelstein [2], the inclusive data provide us, for the first time, with the possrbrhty of calculatmg such Regge cuts, in a fairly model independent way. In this note we shall use the projectrle fragmentation reactions K+$K”
(2a)
K- SE‘!
(2b)
to estimate the fixed pole residues m pp and A2p amphtudes and thereby evaluate the pp and A,A, Regge cuts m elastic pp scattering. Phenomenologrcally of course, the more srgmfrcant Regge cuts are the ones, where one, or both, of the Regge exchanges are Pomerons. The nature of the Pomeron singularity, however, 1s very unclear, and tt may be more reasonable at this early stage of the game, to work with genume pole exchanges, for whrch the FMSR have been derrved. Moreover the aC channels m (2a, b) are relatively clean; p and A, bemg the only sigmficant Regge poles. The mclusive cross-sectton for (1) IS schemattcally represented in fig. 1 in the missmg mass (M) range M2 < s. Here osi represents the leading Regge poles in the ac channel and the blob represents the Reggeon-particle amphtude A~j. One gets, m this range, the invariant cross-section
where fiLFis the standard Regge residue and < represents the signature factor F(cu(O))( 1 f e-l”(~))/r(cu(C))sinncu(t). For symmetry requirements, 77and v have been defined in terms of s and M2 as [2]
s
Fig. 1 Graph
describmg
the fragmentation
a 4 c cross-section
555
Volume 40B, number 5
PHYSICS LETTERS
7 August 1972
s=Pb.(P,+P,)=(s-mo2-~~)-3(~2t,b2-t), v=p~.(p,-pc>=~~2-c-mb2). The
(4)
FMSR involve the combmations
(5) which are respectively even and odd under crossmg between the two particle-Reggeon scattering channels. 2 t2*, the Reggeon-particle amplitude IQ(V) can be approximated by the leading Regge exIn the region v 9 mb, change in the b& channel. Thus for q/v % 1 and v % rni, t; we can approximate (3) as
(6) where k represents the Regge exchange in the bb channel, and g$ is the triple Regge coupling. The trajectory degeneracy between p and A2 implies that there is no pA2 mterference term in the cross-section (3). Therefore, the line reversed reactions K OP -+ K+ and K- gK” should have equal cross-sections in the range n/v > 1. We can thus form the combinations Z and A of eq. (5) f rom the reactions (2a) and (2b). The odd crossing combinatrons UC and A lead to the FMSR (see ref. [2]) (7)
where IV B rnz, t. Then a Schwarz type FMSR for the even combination
C yields the futed pole residues R,, and
(9)
One has, of course, the familiar triple Regge formula for Z(N), ie. (10)
Finally usmg the strong exchange degeneracy condition crp”ciA 2
-a,
k_
gPP
between p and Ax, we get k -gA2Az
=gk
,
R,,= RAzA2= R
(11)
Thus the fixed pole residue R follows from the finite mass integral of (9), once we know the two triple-Regge terms. The latter can be obtained from any of the three other relations (7), (8) and (lo), using the EXD between g; and g,;: 556
Volume
40B, number
PHYSICS
5
LETTERS
I August
1972
As in any standard Regge expansion the leading Regge terms on the right hand side of (3) and all the subsequent relations describe the corresponding mclusive cross-sectron up to an order v/r&“M2/s) only. This means that one would introduce no extra ambiguity if the n values for the two reactions K+ l?+K” and K -8 I?O differed from each other wrthin a range v or if they varied over the same range during the integration of eqs. (7)-(9). Thus one can use the inclusive data at a fured s for the missing mass integration and the data for the two processes may even have different s-values [2] as long as this difference is less than M2(IV). However, this also means that the triple Regge terms determined from the inclusive data (eg. eqs. (7, 8, 10)) are ambrguous up to order I@(N)/s; and thus ambiguity propagates into the determination of R from eq. (9). In order that the Regge expansion of the Reggeon-particle amplitude is valid, we must take M2(N) at least m the 4-6 GeV2 range. The present inclusive data for reactions 2(a, b) is restricted to rather low energres. We use the ABCLVBC bubble chamber data [6], which corresponds to s - 16 and 20GeV2 for 2(a) and 2(b) respectively. This induces an ambrgurty of about 25% in the estimate of the Regge terms on the right hand srde of (9) of which the Pomeron exchange contribution is the dominant one. Thus last contrrbutron turns out to be roughly as big as the finite mass integral (within 25%), which 1s of course an expected feature for a reasonably large N. Thus the present data are inadequate to provide an estimate for the fixed pole residue, but gives none the less an upper bound on its magnitude at around 25% of the finite mass integral. We shall see below that this bound 1s in the nerghbourhood of the eikonal model result for the Gribov vertex. Our result for the futed pole residue R, as a function of the Regge 4-momentum squared t IS summarrzed in fig. 2. The triple Regge terms on the right hand side of (9) have been evaluated separately by using eqs. (7, 8) and eqs. (7, 10); and the contributions to the FMSR of eq. (9) from resulting estrmates of R are shown as curves 1 and 2 respectively. The indices a and b refer to M(N) = 2 and 2.5 GeV. By taking into account the terms
I 0
02
04 -tlGeV2)
06
08
Fig. 2. Contributions of R (T) to FMSR of eq. (9) usmg 4 methods compared with sum of N and A contribution model estimate). Methods la, b use jvzdu, jZdv and jAdu urlth M(N) = 2, 2.5 Methods 2a, b use jvxdv, jxdv 2, 2.5 GeV.
to ImApp (eikonal and C with M(N) = 557
PHYSICS LETTER C
Volume 408, number 5
7 August 1972
Fig. 3. Convolution of futed pole residues of ap,A2N scattering to @ve pp, Ayl2 cut contrlbutlon
to forward NN scattering.
in eqs (7, 8, 10) one sees that the four estimates of the triple Regge contributions differ v&hin 25%, as m fact they do*. We have tried several other variations, e.g. using higher for C m lieu of eqs. (8) or (10). The resultmgR values were quantitatively very different from bounded roughly by the curves la, b. The sum of nucleon and A pole contrlbutlons to ImA,, typical model estimate for R, IS shown in fig. 2 for comparison. The A contribution is roughly nucleon. The trajectory and residue parameters used in this analysis are N (q/v)2orl-2
a(t)= 0.5+-t ,
L?$(o)//?;D(o) = 4.5 1
$$t)
are expected to moment sum rules each other, but representing a twice as big as the
(12)
= 478 pb GeV
based primarily on two body phenomenology [7]. Finally, the pp and A2A2 cut contribution to forward NN scattermg (corresponding to isospm zero 111the t-channel) is evaluated by the standard convolution sketched m fig. 3. The procedure 1s the same as the eikonal model calculations of ref. [8], except that the blobs - representmg fixed pole residues - have replaced the single baryon intermediate states. We get
FCUt(o) = - LJdt
[(R(~)s~(‘)-~‘~ .$,(t))2 +(R(t)“(f)-“2
.$A2Ct))2]
(13)
n2q where q IS the s-channel CMS momentum.
ucut =*ImFLU’(o)
The corresponding
total cross-section and forward da/dt are
,
(14)
4
docut -&-(0)
=;lPf(o)/2
.
(15)
The cut terms arising from the four estimates of R(t) are shown in table 1, together with the eikonal model estimate. The upper bound on ucut arising from our fured pole residue estimates 1s a few mb, which is also 111the regon of the eikonal model value for u cut. Such a magnitude may perhaps be too large for ucut for whatever one knows about such cuts from two body phenomenology. In any case there 1s substantial scope for improvmg upon our upper bound on acut, once higher energy data for reactions 2(a, b) are available. At the Serpukhov energies, for instance, one can have M(N)2/s - 5% instead of the present value of 25%. This would make it possible either to give an estimate for acut or else to pin down the upper bound (in case the triple Regge term turns out to be equal to the finite mass mtegral in (9) to within 5%) at about one twenty fifth of the present level. It should be noted in this connection that the eikonal model value for ucti quoted here 1s in fact a lower bound on the general eikonal model result. since all the N* resonances would be additive. * Compare the four estimates to the FMSR at t = -0.1 for mstance, vvlth the finite mass integal
chntributlon
20.5 mb.
Volume
40B, number
Values of pp andAzA*
PHYSICS
5
cut contributions
Comparison -
to NN amphtudes
of 4 methods
LETTERS
I August
Table 1 together with contributrons
is made wrth erkonal
model estrmate
to otot(NN)
and %I,=,
at s = 1OGeV’
usmg N and A contrrbutrons.
~____~
____ __.___
s= 10GeV2
Fpp x m2q
Aqot(NN)
FA2A2 x ln2q
(Gev-‘)
(GeV’)
(mb)
-
6.9 + 11.8 r
Method 1 using fvzdv JZdv and /Adv with cut off 2 GeV
-
05+
1.21
Method 2 usmg JvCdv lCdv and fAdv wrth cut off 2.5 GeV
-
1.5 +
3.4 i
Method 3 usmg JvXdv JCdv and X itself wrth cut off 2 GeV
-
03+
0.41
Method 4 using fvCdv JZZdv and x rtself wrth cut off 2.5 GeV
-11.3
-
6 3i
60.5 - 30.4 r
2.1 -
107-
2.8 -
148
do(NN) dt
1.33
101
2.4 r
0.05
0.19
8.9i
0 23
3.5
1.9 i
0.06
0 21
+631i
3.39
t=O (Crb/GeV2)
.__-
____ Born term contrrbution re. N, A mtermedrate states
1972
689
. It should be remarked that we have relied on a simple iterative approach to Regge cuts, m as much as we describe the Reggeon particle amplitudes by the pole exchanges alone. A theoretically more exact prescription would be to include the cut contribution in the Reggeon particle amphtude as well, and to estrmate it by a boots. trap solutron to the resulting integral equation, i.e. eq. (13). As more extensive data on inclusive cross-sections become available one can extend this analysts to estimate fixed pole residues m bb channel with non zero or even exotic quantum numbers. For instance the combmation of elastic amplitudes
Ap-lr +A/n+ - 2f$Od corresponds
(16)
to isospm 2 in the ‘1771 channel. Thus a finite mass integral
N
s
dv.s d$
[(p “; A++) + (p 2 A++) - 3(p 5 A’)]
(17)
0
would directly give the It = 2 ftxed pole residue in pn and A277scattering. The A and B exchange contrrbutrons can presumably be eliminated by extrapolating the inclusive cross-sections to sufficiently large s. Thus one would be able to estimate the pp and A2A2 cuts 111double charge exchange rrrr scattering. If one assumes the Pomeron to be a factorrsmg Regge pole, then one can estimate the Pomeron-Pomeron cut using the finite mass integral for p 8 p. Similarly a finite mass integral for the difference of the p g p and p & jj 1s expected to give the Regge-Pomeron cut. We have tried to estimate the Pomeron-Pomeron cut using the analogue of eq. (9) for p & p. Unfortunately the dominant triple Regge term on the right involves the t-derivative of the trrple Pomeron vertex in the numerator (as the vertex itself vanishes at t = 0) and the Pomeron slope m the denominator. In view of the large ambiguity on these two parameters it seems extremely hard to extract the fixed pole residue. In summary we have tried to estimate the fixed pole residues in pp and A,p amplitudes using the finite mass sum rules for K+ 2: K” and E- e K”. In view of the rather low s values for the present data, we are unable to get a precise estimate for the fKed pole residues or the resulting pp and A2A2 cuts. None the less we are able to get an
s5’)
Volume 40B, number
5
PHYSICS LETTERS
I August 1972
upper bound for these quantities at the level predicted by the standard eikonal model. We have discussed the possible extension of this analysis to Regge-Regge cuts m exotic channels and also to Regge-Pomeron and PomeronPomeron cuts. We are grateful to Dr. R.P. Worden for useful comments and suggestions. We would hke to thank Dr. R. Stroynowslu for providmg low missing mass bins on K+p +K”X and Kp+i?OX We also thank Drs. G. Kane and Chan Hong-MO for critically reading the manuscript. References [l] J. Kwiecmslu, Nuovo Cunento Lett. 3 (1972) 619. [ 21 M.B. Emhorn, J. Elhs and J. Fmkelstem, SLAC-LBL-ITP
preprmt (1972). [3] See e g.: C. Lovelace, Proc. Elementary Particles, Eds. A.G. Tenner and M.J.G. Veltman, Holland) p. 141. [4] H.D.I. Abarbanel, NAL preprmt, THY-28 (1972). [5] J.H. Schwarz, Phys. Rev. 159 (1967) 1269. [6] J V. Beaupre et al., Nucl. Phys B39 (1972) 133, and Private Communication from R. Stroynowsh. [7] V. Barger and R.J N. Philips, Nucl. Phys. B32 (1971) 93. [ 81 F.S. Henyey, G L. Kane and J.J.G. Scamo, Phys. Rev. Lett 27 (1971) 350, C. Qulgg, Nucl. Phys B34 (1971) 77.
Amsterdam
Conf.,
1971 (North-
40B, number
PHYSICS
5
AN ESTIMATE
LETTERS
7 August
OF REGGE CUTS FROM INCLUSIVE
1912
FINITE MASS SUM RULES
D.P. ROY and R.G. ROBERTS Rutherford
High Energy Laboratory, Received
Chdton, Drdcot, Berkshue,
16 June
England
1972
P
-P-o Inclusrve data on I& Ko,and K +K are used to calculate the pp andAaA2 Regge cuts m forward nucleonnucleon scattering Although the energies for whtch data exist are too low to allow a precise estimate, we can obtam an upper bound for the cut magnitude which IS at the level grven by the erkonal model.
Finite mass sum rules have recently been constructed for inclusive reactions [ 1, 21. They relate the residues of the nonsense wrong signature fixed poles m some particle Reggeon scattering amplitudes to experimentally measurable quantities. For instance, a missing mass integral of the inclusive cross-section for b
a-fc,
(1)
ISrelated to the fixed pole residue m bR scattering, where R IS the leadmg Reggeon m the acchannel. These residues constitute the Gribov vertices m the standard evaluatron of Regge cuts for two body scattering in the framework of Grrbov calcalus [3]. Thus, as remarked by Abarbenel [4], Einhorn, Ellis and Fmkelstein [2], the inclusive data provide us, for the first time, with the possrbrhty of calculatmg such Regge cuts, in a fairly model independent way. In this note we shall use the projectrle fragmentation reactions K+$K”
(2a)
K- SE‘!
(2b)
to estimate the fixed pole residues m pp and A2p amphtudes and thereby evaluate the pp and A,A, Regge cuts m elastic pp scattering. Phenomenologrcally of course, the more srgmfrcant Regge cuts are the ones, where one, or both, of the Regge exchanges are Pomerons. The nature of the Pomeron singularity, however, 1s very unclear, and tt may be more reasonable at this early stage of the game, to work with genume pole exchanges, for whrch the FMSR have been derrved. Moreover the aC channels m (2a, b) are relatively clean; p and A, bemg the only sigmficant Regge poles. The mclusive cross-sectton for (1) IS schemattcally represented in fig. 1 in the missmg mass (M) range M2 < s. Here osi represents the leading Regge poles in the ac channel and the blob represents the Reggeon-particle amphtude A~j. One gets, m this range, the invariant cross-section
where fiLFis the standard Regge residue and < represents the signature factor F(cu(O))( 1 f e-l”(~))/r(cu(C))sinncu(t). For symmetry requirements, 77and v have been defined in terms of s and M2 as [2]
s
Fig. 1 Graph
describmg
the fragmentation
a 4 c cross-section
555
Volume 40B, number 5
PHYSICS LETTERS
7 August 1972
s=Pb.(P,+P,)=(s-mo2-~~)-3(~2t,b2-t), v=p~.(p,-pc>=~~2-c-mb2). The
(4)
FMSR involve the combmations
(5) which are respectively even and odd under crossmg between the two particle-Reggeon scattering channels. 2 t2*, the Reggeon-particle amplitude IQ(V) can be approximated by the leading Regge exIn the region v 9 mb, change in the b& channel. Thus for q/v % 1 and v % rni, t; we can approximate (3) as
(6) where k represents the Regge exchange in the bb channel, and g$ is the triple Regge coupling. The trajectory degeneracy between p and A2 implies that there is no pA2 mterference term in the cross-section (3). Therefore, the line reversed reactions K OP -+ K+ and K- gK” should have equal cross-sections in the range n/v > 1. We can thus form the combinations Z and A of eq. (5) f rom the reactions (2a) and (2b). The odd crossing combinatrons UC and A lead to the FMSR (see ref. [2]) (7)
where IV B rnz, t. Then a Schwarz type FMSR for the even combination
C yields the futed pole residues R,, and
(9)
One has, of course, the familiar triple Regge formula for Z(N), ie. (10)
Finally usmg the strong exchange degeneracy condition crp”ciA 2
-a,
k_
gPP
between p and Ax, we get k -gA2Az
=gk
,
R,,= RAzA2= R
(11)
Thus the fixed pole residue R follows from the finite mass integral of (9), once we know the two triple-Regge terms. The latter can be obtained from any of the three other relations (7), (8) and (lo), using the EXD between g; and g,;: 556
Volume
40B, number
PHYSICS
5
LETTERS
I August
1972
As in any standard Regge expansion the leading Regge terms on the right hand side of (3) and all the subsequent relations describe the corresponding mclusive cross-sectron up to an order v/r&“M2/s) only. This means that one would introduce no extra ambiguity if the n values for the two reactions K+ l?+K” and K -8 I?O differed from each other wrthin a range v or if they varied over the same range during the integration of eqs. (7)-(9). Thus one can use the inclusive data at a fured s for the missing mass integration and the data for the two processes may even have different s-values [2] as long as this difference is less than M2(IV). However, this also means that the triple Regge terms determined from the inclusive data (eg. eqs. (7, 8, 10)) are ambrguous up to order I@(N)/s; and thus ambiguity propagates into the determination of R from eq. (9). In order that the Regge expansion of the Reggeon-particle amplitude is valid, we must take M2(N) at least m the 4-6 GeV2 range. The present inclusive data for reactions 2(a, b) is restricted to rather low energres. We use the ABCLVBC bubble chamber data [6], which corresponds to s - 16 and 20GeV2 for 2(a) and 2(b) respectively. This induces an ambrgurty of about 25% in the estimate of the Regge terms on the right hand srde of (9) of which the Pomeron exchange contribution is the dominant one. Thus last contrrbutron turns out to be roughly as big as the finite mass integral (within 25%), which 1s of course an expected feature for a reasonably large N. Thus the present data are inadequate to provide an estimate for the fixed pole residue, but gives none the less an upper bound on its magnitude at around 25% of the finite mass integral. We shall see below that this bound 1s in the nerghbourhood of the eikonal model result for the Gribov vertex. Our result for the futed pole residue R, as a function of the Regge 4-momentum squared t IS summarrzed in fig. 2. The triple Regge terms on the right hand side of (9) have been evaluated separately by using eqs. (7, 8) and eqs. (7, 10); and the contributions to the FMSR of eq. (9) from resulting estrmates of R are shown as curves 1 and 2 respectively. The indices a and b refer to M(N) = 2 and 2.5 GeV. By taking into account the terms
I 0
02
04 -tlGeV2)
06
08
Fig. 2. Contributions of R (T) to FMSR of eq. (9) usmg 4 methods compared with sum of N and A contribution model estimate). Methods la, b use jvzdu, jZdv and jAdu urlth M(N) = 2, 2.5 Methods 2a, b use jvxdv, jxdv 2, 2.5 GeV.
to ImApp (eikonal and C with M(N) = 557
PHYSICS LETTER C
Volume 408, number 5
7 August 1972
Fig. 3. Convolution of futed pole residues of ap,A2N scattering to @ve pp, Ayl2 cut contrlbutlon
to forward NN scattering.
in eqs (7, 8, 10) one sees that the four estimates of the triple Regge contributions differ v&hin 25%, as m fact they do*. We have tried several other variations, e.g. using higher for C m lieu of eqs. (8) or (10). The resultmgR values were quantitatively very different from bounded roughly by the curves la, b. The sum of nucleon and A pole contrlbutlons to ImA,, typical model estimate for R, IS shown in fig. 2 for comparison. The A contribution is roughly nucleon. The trajectory and residue parameters used in this analysis are N (q/v)2orl-2
a(t)= 0.5+-t ,
L?$(o)//?;D(o) = 4.5 1
$$t)
are expected to moment sum rules each other, but representing a twice as big as the
(12)
= 478 pb GeV
based primarily on two body phenomenology [7]. Finally, the pp and A2A2 cut contribution to forward NN scattermg (corresponding to isospm zero 111the t-channel) is evaluated by the standard convolution sketched m fig. 3. The procedure 1s the same as the eikonal model calculations of ref. [8], except that the blobs - representmg fixed pole residues - have replaced the single baryon intermediate states. We get
FCUt(o) = - LJdt
[(R(~)s~(‘)-~‘~ .$,(t))2 +(R(t)“(f)-“2
.$A2Ct))2]
(13)
n2q where q IS the s-channel CMS momentum.
ucut =*ImFLU’(o)
The corresponding
total cross-section and forward da/dt are
,
(14)
4
docut -&-(0)
=;lPf(o)/2
.
(15)
The cut terms arising from the four estimates of R(t) are shown in table 1, together with the eikonal model estimate. The upper bound on ucut arising from our fured pole residue estimates 1s a few mb, which is also 111the regon of the eikonal model value for u cut. Such a magnitude may perhaps be too large for ucut for whatever one knows about such cuts from two body phenomenology. In any case there 1s substantial scope for improvmg upon our upper bound on acut, once higher energy data for reactions 2(a, b) are available. At the Serpukhov energies, for instance, one can have M(N)2/s - 5% instead of the present value of 25%. This would make it possible either to give an estimate for acut or else to pin down the upper bound (in case the triple Regge term turns out to be equal to the finite mass mtegral in (9) to within 5%) at about one twenty fifth of the present level. It should be noted in this connection that the eikonal model value for ucti quoted here 1s in fact a lower bound on the general eikonal model result. since all the N* resonances would be additive. * Compare the four estimates to the FMSR at t = -0.1 for mstance, vvlth the finite mass integal
chntributlon
20.5 mb.
Volume
40B, number
Values of pp andAzA*
PHYSICS
5
cut contributions
Comparison -
to NN amphtudes
of 4 methods
LETTERS
I August
Table 1 together with contributrons
is made wrth erkonal
model estrmate
to otot(NN)
and %I,=,
at s = 1OGeV’
usmg N and A contrrbutrons.
~____~
____ __.___
s= 10GeV2
Fpp x m2q
Aqot(NN)
FA2A2 x ln2q
(Gev-‘)
(GeV’)
(mb)
-
6.9 + 11.8 r
Method 1 using fvzdv JZdv and /Adv with cut off 2 GeV
-
05+
1.21
Method 2 usmg JvCdv lCdv and fAdv wrth cut off 2.5 GeV
-
1.5 +
3.4 i
Method 3 usmg JvXdv JCdv and X itself wrth cut off 2 GeV
-
03+
0.41
Method 4 using fvCdv JZZdv and x rtself wrth cut off 2.5 GeV
-11.3
-
6 3i
60.5 - 30.4 r
2.1 -
107-
2.8 -
148
do(NN) dt
1.33
101
2.4 r
0.05
0.19
8.9i
0 23
3.5
1.9 i
0.06
0 21
+631i
3.39
t=O (Crb/GeV2)
.__-
____ Born term contrrbution re. N, A mtermedrate states
1972
689
. It should be remarked that we have relied on a simple iterative approach to Regge cuts, m as much as we describe the Reggeon particle amplitudes by the pole exchanges alone. A theoretically more exact prescription would be to include the cut contribution in the Reggeon particle amphtude as well, and to estrmate it by a boots. trap solutron to the resulting integral equation, i.e. eq. (13). As more extensive data on inclusive cross-sections become available one can extend this analysts to estimate fixed pole residues m bb channel with non zero or even exotic quantum numbers. For instance the combmation of elastic amplitudes
Ap-lr +A/n+ - 2f$Od corresponds
(16)
to isospm 2 in the ‘1771 channel. Thus a finite mass integral
N
s
dv.s d$
[(p “; A++) + (p 2 A++) - 3(p 5 A’)]
(17)
0
would directly give the It = 2 ftxed pole residue in pn and A277scattering. The A and B exchange contrrbutrons can presumably be eliminated by extrapolating the inclusive cross-sections to sufficiently large s. Thus one would be able to estimate the pp and A2A2 cuts 111double charge exchange rrrr scattering. If one assumes the Pomeron to be a factorrsmg Regge pole, then one can estimate the Pomeron-Pomeron cut using the finite mass integral for p 8 p. Similarly a finite mass integral for the difference of the p g p and p & jj 1s expected to give the Regge-Pomeron cut. We have tried to estimate the Pomeron-Pomeron cut using the analogue of eq. (9) for p & p. Unfortunately the dominant triple Regge term on the right involves the t-derivative of the trrple Pomeron vertex in the numerator (as the vertex itself vanishes at t = 0) and the Pomeron slope m the denominator. In view of the large ambiguity on these two parameters it seems extremely hard to extract the fixed pole residue. In summary we have tried to estimate the fixed pole residues in pp and A,p amplitudes using the finite mass sum rules for K+ 2: K” and E- e K”. In view of the rather low s values for the present data, we are unable to get a precise estimate for the fKed pole residues or the resulting pp and A2A2 cuts. None the less we are able to get an
s5’)
Volume 40B, number
5
PHYSICS LETTERS
I August 1972
upper bound for these quantities at the level predicted by the standard eikonal model. We have discussed the possible extension of this analysis to Regge-Regge cuts m exotic channels and also to Regge-Pomeron and PomeronPomeron cuts. We are grateful to Dr. R.P. Worden for useful comments and suggestions. We would hke to thank Dr. R. Stroynowslu for providmg low missing mass bins on K+p +K”X and Kp+i?OX We also thank Drs. G. Kane and Chan Hong-MO for critically reading the manuscript. References [l] J. Kwiecmslu, Nuovo Cunento Lett. 3 (1972) 619. [ 21 M.B. Emhorn, J. Elhs and J. Fmkelstem, SLAC-LBL-ITP
preprmt (1972). [3] See e g.: C. Lovelace, Proc. Elementary Particles, Eds. A.G. Tenner and M.J.G. Veltman, Holland) p. 141. [4] H.D.I. Abarbanel, NAL preprmt, THY-28 (1972). [5] J.H. Schwarz, Phys. Rev. 159 (1967) 1269. [6] J V. Beaupre et al., Nucl. Phys B39 (1972) 133, and Private Communication from R. Stroynowsh. [7] V. Barger and R.J N. Philips, Nucl. Phys. B32 (1971) 93. [ 81 F.S. Henyey, G L. Kane and J.J.G. Scamo, Phys. Rev. Lett 27 (1971) 350, C. Qulgg, Nucl. Phys B34 (1971) 77.
Amsterdam
Conf.,
1971 (North-