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Multiparticle shadow effects on elastic scattering in the dual model
Volume 46B, number 2
PHYSICS LETTERS
MULTIPARTICLE
17 September 1973
SHADOW EFFECTS ON ELASTIC SCATTERING IN THE DUAL MODEL
Chan HONG-MO Rutherford Htgh Energy Laboratory, Berkshtre, England J.E. PATON Department of TheoretscalPhysics, Oxford Umverstty, England Recewed 10 July 1973 Unltarlty effects of multiparticle channels on elastic scattering are studied schematically m the dual model. Explanatmns are offered for (A) the difference m energy dependence between 'Pomeron' and lsospin exchanges, (B) the difference in the slopes of diffraction peaks for exolac and non exotic reactmns, and (C) the 'cross-over' phenomenon. The unitanty condition on the elastic two particle amphtude may be written as Im(f[Tli) = ~
(fiT* In) (nlTll),
(1)
n
where the sum on the right-hand side is taken over all possible intermediate states. Since, however, elastxc scattering is known to contribute no more than about 15 percent of the total cross section at high energy, one may assume the sum to be dominated by inelastic, and especially multiparticle, reactions. Therefore if one has a model tor multiparticle processes, one can, at least in principle, apply it to calculate via (1) the dominant ~maginary part of the elastic amplitude. This is what is meant by the 'multiparticle shadow', on elastic scattering. Now, from experiment, elastic cross sections near the forward direction are known to show a number of distinctive features [e.g 1 ]. For example, (A) the elastic peak shrinks with energy but not as fast as the peak for charge exchange reactions, (B) reactions with non-exotic quantum numbers in the direct channel, such as K-p, have higher cross sections at t = 0 than their exotic counterparts such as K'p, but for t 4: 0, their cross sectmn have a sharper dependence on t, (C) the difference in cross sections between non exotlc (K-p) and exotic (K*p) reactions shows a zero at some small value o f t ~ - 0.2 GeV2; this is the socalled cross-over phenomenon. The explanation of these features is beyond the 228
scope of the usual Regge model. They are generally interpreted as s-channel unitanty effects as indicated m the preceding paragraph. As such, they have been parametrised by various absorption mechanisms but never directly explained in terms of the inelastic channels. It is thus a considerable challenge to any model for multiparticle reactions to reproduce these features from a shadow calculation, particularly since it will involve the phases of production amplitudes which are not available directly from experiment. We show here that within the dual-Regge framework there is a natural qualitative explanation for those phenomena. For this, it is sufficient to consider the multiparticle amplitudes schematically. We shall therefore use the multiperlpheral approximation for simplicity m spite of the fact that we believe the approximation to be a poor one m reahty. We return to this point at the end of the paper In the multiperipheral approximation to dual amplitudes then, the calculation of the multiparticle shadow in (1) reduces to the evaluation of the diagram in fig. 1 where each loop can be either crossed or uncrossed. Notice that in a dual model with internal symmetry such as isospin, loops with only one crossed Reggeon hne are not allowed, as can readily be checked by drawing quark diagrams. The scheme we are considering is thus very similar to that recently studied by Huan Lee who considered eq. (1) for forward scattering [2]. We differ somewhat in philosophy from Huan Lee, however, in that the various components of elastic scattering at high energy are here Identified by the
Volume 46B, number 2
PHYSICS LETTERS
z.
t
17 September 1973
¢
¢
Q
-"
C
Q
f
f
(a)
(b)
Fig. 2
Fig. 1.
quantum numbers exchanged in the t-channel and not by resonance background separation. We consider the separation of an amplitude into a resonance part and a background part as unphysical at high energy since there is no unambiguous way to perform this separa. tion experimentally, even in principle. Consider now one loop in the chain of fig. 1. In terms of dual diagrams, the crossed and uncrossed loops are represented respectively by the diagrams (a) and (b) of fig. 2. The internal quantum numbers carried by the loops can readily be examined by tatong the traces of the appropriate isospin or SU 3 factors [3]. Thus, assuming that we have only mesons carrying isospin, which are exactly exchange degenerate, we can write for the zero loop diagram of fig. 3 the isospin factor: ±2Tr (r a r b r e rd)
(2)
where by convention,
Pauli matrixes, x = 1, 2, 3.
we can then write the crossed (a) and uncrossed (b) loops respectively as. (a) POC,
(b) (P1 +Po) v"
(7)
Notice that the crossed &agram contributes only to isospin zero exchange, but the uncrossed diagram to both isospin states. This is as expected. Next we consider the dynamical factors C and U introduced in (7). They represent essentially convolution integrals of the form: t
O(D)
C, u =fdz I d zl iD--i7 'c,v (Z'l) rc, v (Zl),
(3)
Then for the crossed loop (a) we have the isospin fac tor:
(8)
where t
D = ( 1 - z 2 - z ] - z'12 + 2 z z l z l )
(9)
and z = cos O, which is related to the momentum transfer t of the elastic amplitude by t = - 2q2(1 - c o s 0).
1, x=O
rx =
C
(10)
The symbols T¢, U represent the amphtudes for the reaction a+b~e+f,
(11)
in fig. 2. Assuming now that all particles have equal masses, TC,U are expected to have the forms:
3 e,f=0
~t Tr (ra rf r b re) 7I Tr ( r e rd rf rc)
(4)
TC (Zl) = exp (Atl) (S/So) a(q)
= 4 bac 5 b d '
Tu(Zl) = exp (Atl) (S/So) ~(q )exp [1 rra(q)],
and for the uncrossed loop (b) the isospin factor: 3 e,f~=0 ±2 T r ( r a r b "gf Te) 1 Tr(z e 7"f'rd "rc)
(12)
where (5)
t1 = - 2 q 2 ( 1 - Z l ) .
(13)
= 4 (6ab 6cd -- 6ad 6be) + 4 6ae 6bd" Introduce the isospln projection operators in the t-channel, respectively for I = 0, 1 _1 PO - ~- (Sac 6 b d ,
1 PI - "~ (6 ab (5cd --(5 ad 6be )
(6) Fig. 3.
229
Volume 46B, number 2
PHYSICS LETI'ERS
The convolution integral (8) can immediately be done [e.g. 4] giving, C ' U = 2rr(s/sn)2c~(°)exp( - 4aq 2) ~x / f f sinh x//~ " ..,
(15)
a = A + log (S/So),
(16)
= ,~ 0 for the crossed graph
(17)
in for the uncrossed graph.
Notice that in the forward direction, z = 1, C and U are equal in value For small values of the momentum transfer t, they are approximately exponential in t with slope B gwen by B = ~ 1a (a2 - b2)'
(18)
The uncrossed graph Uhas thus a steeper dependence, on t than the crossed graph C. Although we have worked out in detail only a rather special case, this last statement is seen to be quite general. The t-dependent phases in the U-graph allow for cancellations which are not present in the C-graph when one moves off the forward direction. With these properties of the crossed and uncrossed loops in mind, let us now examine the n-loop diagram, which we write symbolically as: T(n) = [(P1 +P0) U + Po C] n"
(19)
Remembering that P0 and P1 are projection operators, we obtain, T(n) = P1 (U) n + PO (U + C) n.
(20)
Notice that the crossed loop C, being multiplied by P0, acts as an isospin filter, so that I = 1 exchange is absent whenever there is a crossed loop anywhere along the chain One sees immediately then that at high energy, isospin 0 exchange mist dominate over isospin 1 in the elastic amplitude, lndeed, denoting the two components respectively by TO and T1, and assuming as usual that (n) ~ 3 log s. one has T1 ~ r
ro
u
]/31°g s ~
L u+cJ
s -~
log[(U+C)/Ul
(21_)
The fact that U= C at t = 0 but U < C for t < O, then leads to the result (A). One also deduces that the I t = 1 amplitude decreases like a power of the energy 230
X
(o)
P = 8 q4[a2(1 +z) +b2(1 - z ) ] ,
t
X
X
I.
~
P
(14)
where
b
17 September 1973
(b)
(c)
Fig. 4.
compared with the amplitudes with I t = 0. This last statement is essentially the same as a result of Huan Lee [2], who has shown further that in the more restricted framework of Chew and Pignotti, one can obtain even a relation between the intercepts of the I t = 0, 1 Regge components. Next, consider the pair of elastic reactions: a + b ~ a + b,
(22)
~+b~+b,
(23)
where ~ denotes the antiparticle of a. Further, let the direct channel of ab be exotic, but not so for ~ b. A specific example may be taken as rt*rr÷ for (22) and 7r-lr+ for (23). Now in the dual diagram for the exotic reaction, the two incoming lines a, b are not allowed to adjacent, so that at least one loop along the chain of fig. 1 must be crossed, whereas, for the non-exotic reaction, there is no such restriction. This implies the usual exchange degenerate relation between the isospin 0 and 1 components in the totally uncrossed graph U n, so that at the n-loop level, the two reactions (22) and (23) &ffer by the term: T(~ b) - T(a b) = 2~ U n,
(24)
where # = (a ~ IP0 [b b ) = (~ alPllb b ) --- - (a ~ IPlib b ). Since U = C is positive in the forward direction and Un has a steeper slope in t than any other term in (20), one deduces immediately the result (B). Moreover, since the relative importance of the single term (24) ._ must decrease compared with the total contribution in (20), the amplitude for non-exotic reactions approaches asymptotically that for exotic ones at all values of t near the forward direction. Next, as an example of the 'cross-over' phenomenon, consider specifically K+p -+ K+p,
(25)
K-p ~ K-p.
(26)
It is sufficient to illustrate the point at the one-loop
Volume 46B, number 2
PHYSICS LETTERS
level. Here, one sees that only the crossed diagram (a) of fig. 4 contributes to the exotic reaction (25) but both the diagrams (b) and (c) contribute to the nonexotic reaction (26). Now, the diagrams (a) and (b) both correspond to KN intermediate states. Apart from the charge exchange reaction K-p -* K°n,
(27)
which can only occur in (b), the two diagrams give by line reversal the same contribution to the total cross section. That is, apart from (27), (a) = (b) at t = 0, but because of the difference in phases, (b) has a steeper dependence on t. However, T(K-p) received additional contributions from those intermediate channels in which the initial charge or strangeness of incoming Kis annihilated, namely the reaction (27) and reactions such as: K-p -~ zr°A,
(28)
contained in the diagram (c). These extra contributions raise T(K-p) above T(K+p) at t = 0, and, provided the annihilation cross sections are not as big as the non-annihilation part (a), one obtains a 'cross over'. One sees then that this effect can be quite general, and is expected for pp and ~p elastic scattering as well as for diffractive dissociation reactions such as K÷p ~ Q÷p and K-p ~ Q-p. So we have obtained the result (C). Notice that this 'cross-over' here occurs in the isispin 0 component * which being odd under charge conjugation by definition, has the same quantum numbers as the co-trajectory. In the language of Regge phenomenologists, this means that the co-residue is strongly 'absorbed'. The extension of most of the preceeding discussion to a system with exact SU 3 symmetry is straightforward. All one needs is the replacement of the Pauli matrices r i by the X_ matrices of Gell-Mann [3]. The statements (A) and (B) made above for the I t = 1 and 0 components will then apply to the t-channel octet and signet parts respectively. We do not however believe that such a generalisation is sufficiently realistic to be interesting. Although Regge couplings appear phenomenologicaUy to be SU 3 symmetric to a good approx• The isospin 1 component comes only from the uncrossed diagram (b) and shows no zero at small t. In our picture, therefore, the observed small 'cross-over' I n nip elastic scattering will have to come from exchange degeneracy breaking effects such as pion-exchange.
17 September 1973
imation, the effects of SU 3 breaking m the trajectory functions a ( t ) which occur usually in the exponent is not negligible, as indicated by the large differences in cross sections between strangeness and non-strangeness exchange reactions [e.g. 5]. Indeed, such symmetry breaking effects are required to explain the cross-over in reactions (25) and (26). In case SU 3 is exact, the diagrams (a) and (c) of fig. 4 are equal, so that T(K-p) will stay above T(K÷p) avoiding thus a 'cross-over' at small t. Taking SU 3 symmetry breaking into account, however, the asymptotic problem for n-loop diagrams becomes much more difficult. We have not been able to find out in what way T(K±p) wdl approach each other, or indeed whether they do so asymptotically within this framework. In spite of its apparent success in explaining some salient features of elastic scattering, the present scheme cannot be regarded as realistic, for the following reasons: (i) the multiperipheral approximation is valid only when all particles are at high energies relative to one anotlxer. In practice, neighbouring particles along the chain have only invariant masses of order 1 GeV. They have thus a strong tendency to form resonances whose effects are grossly neglected by the approximation, (ii) all our preceding arguments relic strongly on exchange degeneracy. However, along the multiperipheral chain of fig. 1, most of the particles emitted will be plons with no exchange degenerate isospin 0 counterparts, (iii) our arguments give the relative slopes between different elastic reactions but not the actual slope of the diffraction peak. As shown already by Michejda [6], however, the large experimental value of the elastic slope poses a serious problem to any multiperipheral calculation. We believe that these difficulties can all be overcome if resonance effects are properly taken into account. Indeed, the schematic results presented in this paper have been abstracted from a model under construction which aims at a realistic calculation of shadow effects in general. Preliminary calculations give reasonable values both for the slopes of elastm peaks and for the cross-over point in K-+p scattering. We hope to present the result of these calculations at a later date. In the course of this investigation we have benefitted greatly from discussions with our colleagues Yung-An 231
Volume 46B, number 2
PHYSICS LETTERS
Chao, Jamle Gough, Paul Hoyer, Hannu Miettinen, Richard Roberts and D.P. Roy.
References [1 ] G. Glacomelh, review presented at the 16th Inter. Conf. on High energy physics, Batavia (1972).
232
17 September 1973
[2] H. Lee, Phys. Rev. Lett. 30 (1973) 719; See also G. Venezlano, Phys. Lett 43B (1973) 413. [3] J.E. Paton and Chart Hong-Mo, Nucl. Phys. B10 (1969) 516. [4] Chan Hong-Mo, K Kajantie and G. Ranft, Nuovo Cim. 49A (1967) 157. [5] P. Dornan et al., Quantum number transfer in K-p interactions, Imperial College prepnnt (1973). [6] L MicheJda, J. Turnau and A. Blalas, Nuovo Clm. A56 (1968) 241.
PHYSICS LETTERS
MULTIPARTICLE
17 September 1973
SHADOW EFFECTS ON ELASTIC SCATTERING IN THE DUAL MODEL
Chan HONG-MO Rutherford Htgh Energy Laboratory, Berkshtre, England J.E. PATON Department of TheoretscalPhysics, Oxford Umverstty, England Recewed 10 July 1973 Unltarlty effects of multiparticle channels on elastic scattering are studied schematically m the dual model. Explanatmns are offered for (A) the difference m energy dependence between 'Pomeron' and lsospin exchanges, (B) the difference in the slopes of diffraction peaks for exolac and non exotic reactmns, and (C) the 'cross-over' phenomenon. The unitanty condition on the elastic two particle amphtude may be written as Im(f[Tli) = ~
(fiT* In) (nlTll),
(1)
n
where the sum on the right-hand side is taken over all possible intermediate states. Since, however, elastxc scattering is known to contribute no more than about 15 percent of the total cross section at high energy, one may assume the sum to be dominated by inelastic, and especially multiparticle, reactions. Therefore if one has a model tor multiparticle processes, one can, at least in principle, apply it to calculate via (1) the dominant ~maginary part of the elastic amplitude. This is what is meant by the 'multiparticle shadow', on elastic scattering. Now, from experiment, elastic cross sections near the forward direction are known to show a number of distinctive features [e.g 1 ]. For example, (A) the elastic peak shrinks with energy but not as fast as the peak for charge exchange reactions, (B) reactions with non-exotic quantum numbers in the direct channel, such as K-p, have higher cross sections at t = 0 than their exotic counterparts such as K'p, but for t 4: 0, their cross sectmn have a sharper dependence on t, (C) the difference in cross sections between non exotlc (K-p) and exotic (K*p) reactions shows a zero at some small value o f t ~ - 0.2 GeV2; this is the socalled cross-over phenomenon. The explanation of these features is beyond the 228
scope of the usual Regge model. They are generally interpreted as s-channel unitanty effects as indicated m the preceding paragraph. As such, they have been parametrised by various absorption mechanisms but never directly explained in terms of the inelastic channels. It is thus a considerable challenge to any model for multiparticle reactions to reproduce these features from a shadow calculation, particularly since it will involve the phases of production amplitudes which are not available directly from experiment. We show here that within the dual-Regge framework there is a natural qualitative explanation for those phenomena. For this, it is sufficient to consider the multiparticle amplitudes schematically. We shall therefore use the multiperlpheral approximation for simplicity m spite of the fact that we believe the approximation to be a poor one m reahty. We return to this point at the end of the paper In the multiperipheral approximation to dual amplitudes then, the calculation of the multiparticle shadow in (1) reduces to the evaluation of the diagram in fig. 1 where each loop can be either crossed or uncrossed. Notice that in a dual model with internal symmetry such as isospin, loops with only one crossed Reggeon hne are not allowed, as can readily be checked by drawing quark diagrams. The scheme we are considering is thus very similar to that recently studied by Huan Lee who considered eq. (1) for forward scattering [2]. We differ somewhat in philosophy from Huan Lee, however, in that the various components of elastic scattering at high energy are here Identified by the
Volume 46B, number 2
PHYSICS LETTERS
z.
t
17 September 1973
¢
¢
Q
-"
C
Q
f
f
(a)
(b)
Fig. 2
Fig. 1.
quantum numbers exchanged in the t-channel and not by resonance background separation. We consider the separation of an amplitude into a resonance part and a background part as unphysical at high energy since there is no unambiguous way to perform this separa. tion experimentally, even in principle. Consider now one loop in the chain of fig. 1. In terms of dual diagrams, the crossed and uncrossed loops are represented respectively by the diagrams (a) and (b) of fig. 2. The internal quantum numbers carried by the loops can readily be examined by tatong the traces of the appropriate isospin or SU 3 factors [3]. Thus, assuming that we have only mesons carrying isospin, which are exactly exchange degenerate, we can write for the zero loop diagram of fig. 3 the isospin factor: ±2Tr (r a r b r e rd)
(2)
where by convention,
Pauli matrixes, x = 1, 2, 3.
we can then write the crossed (a) and uncrossed (b) loops respectively as. (a) POC,
(b) (P1 +Po) v"
(7)
Notice that the crossed &agram contributes only to isospin zero exchange, but the uncrossed diagram to both isospin states. This is as expected. Next we consider the dynamical factors C and U introduced in (7). They represent essentially convolution integrals of the form: t
O(D)
C, u =fdz I d zl iD--i7 'c,v (Z'l) rc, v (Zl),
(3)
Then for the crossed loop (a) we have the isospin fac tor:
(8)
where t
D = ( 1 - z 2 - z ] - z'12 + 2 z z l z l )
(9)
and z = cos O, which is related to the momentum transfer t of the elastic amplitude by t = - 2q2(1 - c o s 0).
1, x=O
rx =
C
(10)
The symbols T¢, U represent the amphtudes for the reaction a+b~e+f,
(11)
in fig. 2. Assuming now that all particles have equal masses, TC,U are expected to have the forms:
3 e,f=0
~t Tr (ra rf r b re) 7I Tr ( r e rd rf rc)
(4)
TC (Zl) = exp (Atl) (S/So) a(q)
= 4 bac 5 b d '
Tu(Zl) = exp (Atl) (S/So) ~(q )exp [1 rra(q)],
and for the uncrossed loop (b) the isospin factor: 3 e,f~=0 ±2 T r ( r a r b "gf Te) 1 Tr(z e 7"f'rd "rc)
(12)
where (5)
t1 = - 2 q 2 ( 1 - Z l ) .
(13)
= 4 (6ab 6cd -- 6ad 6be) + 4 6ae 6bd" Introduce the isospln projection operators in the t-channel, respectively for I = 0, 1 _1 PO - ~- (Sac 6 b d ,
1 PI - "~ (6 ab (5cd --(5 ad 6be )
(6) Fig. 3.
229
Volume 46B, number 2
PHYSICS LETI'ERS
The convolution integral (8) can immediately be done [e.g. 4] giving, C ' U = 2rr(s/sn)2c~(°)exp( - 4aq 2) ~x / f f sinh x//~ " ..,
(15)
a = A + log (S/So),
(16)
= ,~ 0 for the crossed graph
(17)
in for the uncrossed graph.
Notice that in the forward direction, z = 1, C and U are equal in value For small values of the momentum transfer t, they are approximately exponential in t with slope B gwen by B = ~ 1a (a2 - b2)'
(18)
The uncrossed graph Uhas thus a steeper dependence, on t than the crossed graph C. Although we have worked out in detail only a rather special case, this last statement is seen to be quite general. The t-dependent phases in the U-graph allow for cancellations which are not present in the C-graph when one moves off the forward direction. With these properties of the crossed and uncrossed loops in mind, let us now examine the n-loop diagram, which we write symbolically as: T(n) = [(P1 +P0) U + Po C] n"
(19)
Remembering that P0 and P1 are projection operators, we obtain, T(n) = P1 (U) n + PO (U + C) n.
(20)
Notice that the crossed loop C, being multiplied by P0, acts as an isospin filter, so that I = 1 exchange is absent whenever there is a crossed loop anywhere along the chain One sees immediately then that at high energy, isospin 0 exchange mist dominate over isospin 1 in the elastic amplitude, lndeed, denoting the two components respectively by TO and T1, and assuming as usual that (n) ~ 3 log s. one has T1 ~ r
ro
u
]/31°g s ~
L u+cJ
s -~
log[(U+C)/Ul
(21_)
The fact that U= C at t = 0 but U < C for t < O, then leads to the result (A). One also deduces that the I t = 1 amplitude decreases like a power of the energy 230
X
(o)
P = 8 q4[a2(1 +z) +b2(1 - z ) ] ,
t
X
X
I.
~
P
(14)
where
b
17 September 1973
(b)
(c)
Fig. 4.
compared with the amplitudes with I t = 0. This last statement is essentially the same as a result of Huan Lee [2], who has shown further that in the more restricted framework of Chew and Pignotti, one can obtain even a relation between the intercepts of the I t = 0, 1 Regge components. Next, consider the pair of elastic reactions: a + b ~ a + b,
(22)
~+b~+b,
(23)
where ~ denotes the antiparticle of a. Further, let the direct channel of ab be exotic, but not so for ~ b. A specific example may be taken as rt*rr÷ for (22) and 7r-lr+ for (23). Now in the dual diagram for the exotic reaction, the two incoming lines a, b are not allowed to adjacent, so that at least one loop along the chain of fig. 1 must be crossed, whereas, for the non-exotic reaction, there is no such restriction. This implies the usual exchange degenerate relation between the isospin 0 and 1 components in the totally uncrossed graph U n, so that at the n-loop level, the two reactions (22) and (23) &ffer by the term: T(~ b) - T(a b) = 2~ U n,
(24)
where # = (a ~ IP0 [b b ) = (~ alPllb b ) --- - (a ~ IPlib b ). Since U = C is positive in the forward direction and Un has a steeper slope in t than any other term in (20), one deduces immediately the result (B). Moreover, since the relative importance of the single term (24) ._ must decrease compared with the total contribution in (20), the amplitude for non-exotic reactions approaches asymptotically that for exotic ones at all values of t near the forward direction. Next, as an example of the 'cross-over' phenomenon, consider specifically K+p -+ K+p,
(25)
K-p ~ K-p.
(26)
It is sufficient to illustrate the point at the one-loop
Volume 46B, number 2
PHYSICS LETTERS
level. Here, one sees that only the crossed diagram (a) of fig. 4 contributes to the exotic reaction (25) but both the diagrams (b) and (c) contribute to the nonexotic reaction (26). Now, the diagrams (a) and (b) both correspond to KN intermediate states. Apart from the charge exchange reaction K-p -* K°n,
(27)
which can only occur in (b), the two diagrams give by line reversal the same contribution to the total cross section. That is, apart from (27), (a) = (b) at t = 0, but because of the difference in phases, (b) has a steeper dependence on t. However, T(K-p) received additional contributions from those intermediate channels in which the initial charge or strangeness of incoming Kis annihilated, namely the reaction (27) and reactions such as: K-p -~ zr°A,
(28)
contained in the diagram (c). These extra contributions raise T(K-p) above T(K+p) at t = 0, and, provided the annihilation cross sections are not as big as the non-annihilation part (a), one obtains a 'cross over'. One sees then that this effect can be quite general, and is expected for pp and ~p elastic scattering as well as for diffractive dissociation reactions such as K÷p ~ Q÷p and K-p ~ Q-p. So we have obtained the result (C). Notice that this 'cross-over' here occurs in the isispin 0 component * which being odd under charge conjugation by definition, has the same quantum numbers as the co-trajectory. In the language of Regge phenomenologists, this means that the co-residue is strongly 'absorbed'. The extension of most of the preceeding discussion to a system with exact SU 3 symmetry is straightforward. All one needs is the replacement of the Pauli matrices r i by the X_ matrices of Gell-Mann [3]. The statements (A) and (B) made above for the I t = 1 and 0 components will then apply to the t-channel octet and signet parts respectively. We do not however believe that such a generalisation is sufficiently realistic to be interesting. Although Regge couplings appear phenomenologicaUy to be SU 3 symmetric to a good approx• The isospin 1 component comes only from the uncrossed diagram (b) and shows no zero at small t. In our picture, therefore, the observed small 'cross-over' I n nip elastic scattering will have to come from exchange degeneracy breaking effects such as pion-exchange.
17 September 1973
imation, the effects of SU 3 breaking m the trajectory functions a ( t ) which occur usually in the exponent is not negligible, as indicated by the large differences in cross sections between strangeness and non-strangeness exchange reactions [e.g. 5]. Indeed, such symmetry breaking effects are required to explain the cross-over in reactions (25) and (26). In case SU 3 is exact, the diagrams (a) and (c) of fig. 4 are equal, so that T(K-p) will stay above T(K÷p) avoiding thus a 'cross-over' at small t. Taking SU 3 symmetry breaking into account, however, the asymptotic problem for n-loop diagrams becomes much more difficult. We have not been able to find out in what way T(K±p) wdl approach each other, or indeed whether they do so asymptotically within this framework. In spite of its apparent success in explaining some salient features of elastic scattering, the present scheme cannot be regarded as realistic, for the following reasons: (i) the multiperipheral approximation is valid only when all particles are at high energies relative to one anotlxer. In practice, neighbouring particles along the chain have only invariant masses of order 1 GeV. They have thus a strong tendency to form resonances whose effects are grossly neglected by the approximation, (ii) all our preceding arguments relic strongly on exchange degeneracy. However, along the multiperipheral chain of fig. 1, most of the particles emitted will be plons with no exchange degenerate isospin 0 counterparts, (iii) our arguments give the relative slopes between different elastic reactions but not the actual slope of the diffraction peak. As shown already by Michejda [6], however, the large experimental value of the elastic slope poses a serious problem to any multiperipheral calculation. We believe that these difficulties can all be overcome if resonance effects are properly taken into account. Indeed, the schematic results presented in this paper have been abstracted from a model under construction which aims at a realistic calculation of shadow effects in general. Preliminary calculations give reasonable values both for the slopes of elastm peaks and for the cross-over point in K-+p scattering. We hope to present the result of these calculations at a later date. In the course of this investigation we have benefitted greatly from discussions with our colleagues Yung-An 231
Volume 46B, number 2
PHYSICS LETTERS
Chao, Jamle Gough, Paul Hoyer, Hannu Miettinen, Richard Roberts and D.P. Roy.
References [1 ] G. Glacomelh, review presented at the 16th Inter. Conf. on High energy physics, Batavia (1972).
232
17 September 1973
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