Charge independence restrictions on some high energy two-body processes

Nuclear Physics B42 (1972) 153-172. North-Ilolland Publishing Company

CHARGE INDEPENDENCE RESTRICTIONS ON SOME HIGH ENERGY TWO-BODY PROCESSES G . V . D A S S aim J . F R O Y L A N D ~ CERN, Geneva Received 15 December 1971

Abstract: Bounds due to isospin invariance lmve been considered for spin effects and for the unpolarized cross sections in some situations of experimental interest. Dynamical consequences like possible structure in the near-forward direction and some knowledge of the strength of the real part of the forward scattering amplitude follow in a modelindependent way from these isospin bounds for the unpo "larized differential cross sections for np ~ np scattering and K-+n ~ K±n scattering at higli energies. Useful bounds for the differential cross sections for high energy fn ~ ~n scattering and for the highenergy reactions rrN--+ fiN and KN ~ K*(890)N (and also bounds for tim full observed density matrix of the vector meson) have been illustrated. We point out some simple implications of the rotation of the reference axes (about the normal to the production plane) for the resonance density matrix bounds. Some other applications are discussed.

1. I N T R O D U C T I O N The inequalities [ 1 , 2 1 b e t w e e n tile cross sections for three (or m o r e ) processes b e c a u s e o f relations (due to isospin-invariance or a n y o t h e r s y m n l e t r y ) b e t w e e n t h e i r a m p l i t u d e s can be very useful in particular situations (see for e x a m p l e , refs. [ 3 - 5 ] ). This p a p e r is d e v o t e d to isospin b o u n d s in s o m e t w o - b o d y processes at high energies. In sect. 2, we list these b o u n d s s o m e o f w h i c h are applied to data in sect. 3. D y n a m i c a l c o n s e q u e n c e s like a possible change o f slope of the n e a r - f o r w a r d peak for n p -+ n p s c a t t e r i n g , and s o m e k n o w l e d g e o f the relative s t r e n g t h o f t h e real part o f t h e f o r w a r d K~n --* K±n s c a t t e r i n g a m p l i t u d e follow b y c o m p a r i n g (sect. 3) o u r b o u n d s on t h e u n p o l a r i z e d d i f f e r e n t i a l cross s e c t i o n s d u e to isospin i n v a r i a n c e alone, w i t h the existing data. Q u i t e restrictive b o u n d s have b e e n obt a i n e d for the u n p o l a r i z e d d i f f e r e n t i a l cross sections for high energy pn ~ ~n scat* tering a n d for t h e ifigh-energy r e a c t i o n s rrN -+ oN a n d KN -+ K*(8c)0)N. The isospm b o u n d s on t h e o b s e r v e d part o f the r e s o n a n c e d e n s i t y - m a t r i x have b e e n illustrated for tire case o f v e c t o r - m e s o n p r o d u c t i o n . F o r r e s o n a n c e p r o d u c t i o n , s o m e sitnple c o n s e q u e n c e s o f r o t a t i n g t h e r e f e r e n c e f r a m e a r o u n d tile n o r m a l to the p r o d u c t i o n $ On leave of absence from the University of Oslo.

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G. V.Dass and J.Froyland, Charge independence restrictions

plane, and their relevance to isospin bounds, have also been illustrated for the above vector-meson production processes. This rotation provides a simple way for checking the well-known positivity requirements (see, for example, ref. [61 ) on the density-matrix. Some concluding remarks are made in sect. 4.

2. THE BOUNDS When the anaplitudes T i for the processes i = 1,2, ...,N satisfy the linear relation N i= 1

ri--0

(1)

due to some principle like isospin-invariance, the generalization of the usual triangular inequalities [ 1 , 2 ] gives, for the corresponding differential cross sections* oi-~ l Til 2, the relations

N Xfb-j?j~< ~

~

f = 1..... N ,

(2)

which give the bounds

N \°f~M-- ~

t~j,M

N V~t<~x/~j ~< ~

i~]"

V~i,

(3)

where M stands for the process with the largest cross section. For the case of N = 3, one has the well-known triangular inequality

(V~I - ~ 2 ) 2 K 03 ~<( ~ 1 + ~ 2 ) 2-

(4)

By considering tile polarized differential cross sections (1 + P)o, one gets the isospin bounds I2]

(~/(1 +-P1)ol - V ~ T + P2)o2) 2 < (1 -+P3)o3 < (~/(] -+PI)Ol + ~/(1 + P2)o2) 2

(5)

for, for example, the processes (i = 1,2, 3)

* The quantities a i differ from the experimental differential cross sections by Clebsch-Gordan coefficient factors.

G. V.Dass and J.Froyland, Charge independence restrictions

1:

pp -+ p p ,

2:

pn--, n p ,

(charge-exchange),

3:

pn -+ pn ,

(elastic),

155

where the polarizations Pi are defined along the direction (Pl X P2), Pl and P2 being the momenta of the initial and final protons; here, elastic (charge-exchange) np scattering corresponds to c.m. scattering angles, between the initial and final protons, less (greater) than 90 °. The relations (5) give the bounds on o3(P3) when ol, 02, P1, P2 and P3 (o3) are known. However, if o 3 is unknown, or poorly known, one may use the bounds on P3, X/o2 [x/~2(P 2 - P I ) - X ] o 1 +o2-Y

~< ( P 3 - P ~ ) <~

,/72 [x/Z2(P2 - e l ) + x] o 1 + 0 2 +Y

(6)

where

X = VOI(I -- ~ ) [ % / ~ + P2)(1 --Pl) + %//~ - P 2 ) ( 1 + PI)] , y = ax/77%1%Ix/(1 + P2) (1 + PI) - x / ( I - P t ) ( l - P 2 ) I , which do not require a knowledge of o 3. Similarly, if P3 is unknown or poorly known, one may use the bounds on o 3

(7)

o I + o 2 - z ~ < o 3~<01 + o 2 + z ,

where

z= °X/~I°~ [x'/[] -P1)( 1 -P2)+~/( 1 +PI)( 1 +P2)] , which do not require a knowledge of P 3. However, if only the cross sections are known, one can get the bounds [3]

½1P3- PI[ ~<

0~2~_1~3

(01 202 Z _03)27j

Icrl + 03--~02

(8a)

for the polarization difference (P1 - P3)" In situations where only P1, °l and o 2 are known, the useful bounds for P~are

X [-X//~I+x/((1-P~I)(1-o~)1

,

(8b)

156

6. V Dass and .I. Froyland. Charge independence restrictio ns

where the upper bounds are applicable only when (1-PI) 2

°l 02

> 1 ,

(8c)

and the lower bound, only when

(1 + PO % 2

> 1.

(8d)

02

The bounds (8b) can be restrictive when o 2 ~ o I . The bounds (5), (6), (7) and (8) are only typical ones: they hold also if the polarization P i s replaced by other spin effects like the D and CNN parameters of nucleon-nucleon scattering, as noted by Phillips [2] ; they also hold for meson-nucleon scattering, also if the spin-parameters R and A are considered [3, 51 instead o f polarization. For resonance production processes, the isospin bounds for the combinations Pll o where Pll is a (non-negative) diagonal element of the density matrix of the produced resonance, are similar to those for the polarized cross sections (1 + P)o of rrN --* rrN scattering. For the off-diagonal elements #lm which can be negative, the situation is different. For the case o f only three charge-modes (i = 1,2, 3), the bounds are X1 +X2

Y12<-X3<-XI +X2+ Y12,

(9)

where Xi= Re(PlmO)i ,

r,2 = [

or hn(PlmO)i ,

(0,,,,)21 ov57 1o .

In fact, the bounds (9) hold for the off-diagonal (l 4: m), as well as for the diagonal (l = m) elements. The diagonal elements are not all independent because of the trace condition EIOIl = 1 which normalizes the density-matrix. For vector-meson production, for example, independently valid bounds oll Ooo can be obtained from the relations (9) for 1 = m = 1 and for I = m = 0. The bounds corresponding to (9) with Y12 replaced by XfOlO2 were already obtained m ref. [3]. If 03 is known, (9) can be treated as bounds on (Plm)Y If (film)3 are unknown, or poorly known, one may use the bounds on 03: o I + o 2 - 2 Z l 2 ~ < o 3~
(10)

G. V.Dass and J.Froyland, Charge independence restrictions

157

where Z12 = the relations (10) require a knowledge of the diagonal elements Pll for only the charge modes 1 and 2, but not of (Plm)3" The bounds (10) are better than the bounds (4). If one rotates the frame of axes (with respect to which the density-matrix is defined) about the normal to the production plane, one gets a continuous family of systems in each of which the bounds (9) apply. If in a certain frame of reference axes obtained by rotation through an angle 0, the diagonal element (,Oll)2 vanishes, one gets the simple prediction

(plO/)1 01 = (p/O/)3 0"3

(1 I)

valid in this new frame, indicated by the superscript; the matrix elements pl0m are known functions of the original elements Plm and the angle 0. Similarly, if the off0 vanishes for the charge-mode 1 or/and 2, the bounds (9) for diagonal element P/m (Pfm)3 get simplified. To illustrate the type of information obtained by this rotation, we consider the vector-meson production reactions (e.g., rrN -+ pN and KN -+ K*(890)N). For some of these data, (Re P~0)I and (Re p~0) 2 vanish for the same angle 0 of rotation within errors, in which case the bound (9) becomes

I(ReP°lo)3O31~ [x/iP°oo)2(P°ll)l +x/(P°oo)l (P°ll)2] V/~lO~ .

(12a)

If in a certain frame, (Poo)l 0 = 0, it would mean (Re P{0)l = 0 and, therefore,

(Re 010o.)2 0 -X/~-~Olo2(Poo)2 ~<(UeP~0O)3 ~<(Re P100)2 0 1 0 2. 1 0 + N/fOIO2(PO0) 0

(12b)

0

If, however, (Pll)l = 0, it means (Poo)l = I , (Re Pl0)l o = 0 and (Re PloO)2 0 -X,/Olo2(P{1) 2 < (Re Pl0O)3 0 < (Re p~oO)2 + -V/OlO2(P{ 1)2 .

(12c)

In the relations (12b, c), the normalization Poo + 2Pll = 1

(12d)

has been used. The observed part of the density-matrix, after rotation is given in terms of the

158

G. KDass and J.Froyland, Charge independence restrictions

original matrix elements by 0000 = (Pll - P l - I ) sin20 + 0000 c°s2 0 - v r 2 ( R e Pl0 ) sin20 , --20~ -1 = (0011 -001 - 1 ) C°S20 + 0000 sin20 + X/2-(Re 0010) sin20 -(0011 + P 1 - 1 ) ' - X / 2 Re p~0---~(0011-pl _1) sin20 -~00oo sin 20 - x / 2 ( R e 0010) cos20 ,

(13)

0 0 = Poo + 200011--- 0000 + 2011 constant 0 + P 0l - 1 =0011 + 001 _ 1 = constant • 0011 The relations (9) in the rotated frame can give useful bounds on the density-matrix elements (00lm)3 of the original frame, even in situations where the relation (9) cannot be used in the original frame because of inadequate data. For example, the extraction of the off-diagonal element (Re 0010)1,2 can become difficult because of a poor knowledge of the azimuthal dependence in the angular distribution; one cannot then use the ralations (9), in the original frame, for getting bounds on (Re 0010)3" However, the relations (9), expressed in the rotated frames for suitable values of 0, can give bounds on certain linear combinations of (00/rn)3 - even if only incomplete information on (00lm)2 and (P/m)l is available. For example, the relation (9) for (00ooO)3, tan 2 0 = 2 gives

(N/A1- X/~2) 2 ~ A 3 ~ (~/A~l + ~ 2 ) 2,

(14)

where A i = 3k(1 ¥ 4 Re Pl0 - 201 -1)iOi, (i = l, 2, 3); this can be used to get information on (2 RePl0 + 001-1)3 even if (Poo)t , (Poo)2, (0011)1 and (0011)2 are not known; the normalization (1 2d) has been used in (1 4). Using the expressions (1 3), one can similarly construct other examples where the relations (9) in a suitably rotated frame can provide useful bounds even if this relation in the original frame cannot be used for data reasons. The two Donohue-H6gaasen frames [7] in which Re00~0 = 0 correspond to the angles of rotation [7] tan20 -

- 2X,'~ Re Pl0

(15)

0000 --0011 + JOt -1

In fact, the density-matrix elements 00oo,/911 and 001-1 obtain extremum values in these frames [8]. One gets 2

1

(Poo)DH = ~(Poo + 0011 --001 1) ¥½[(00oo--0011 + 001 - I ) 2 + 8 ( R e P l 0 ) ]~ , (1 6) (001 -1)DH = ~(001t --0000 + 3001-1) ¥ -}[(/900 --0011 + 001 1)2+8(ReP10)2] -~ ,

G. V.Dassand J.Froyland, Charge independence restrictions

159

the upper and lower signs corresponding respectively to the two values of 0 in (15). We shall refer to the frames in which p° ° (and also pl0 _ 1) attain their maximum and minimum values as DH1 and DH2, respectively, and the angles of rotation to these frames as (a -½rr) and a respectively.

3. APPLICATIONS TO SOME HIGH ENERGY TWO-BODY PROCESSES 3.1. Nucleon-nucleon scattering 3.1.1. Differential cross sections l f t h e cross section 02 is much smaller than Ol, the upper and lower bounds (4) on o 3 lie close to each other; this is the case if the indices 1,2 and 3 stand respectively for pp -+ pp, charge-exchange pn -+ np and elastic np ~ np scatterings at high energies. Fig. 1 compares these bounds on the np -+ np elastic differential cross section at 12, 19.2 and 24 GeV/c with the available data [ 9 - 1 1 ] . The input pp data are from ref. [12] ; for the np ~ pn charge exchange cross sections, instead of interpolating* the available data, we have used a phenomenological fit [13] which reproduces very well the data of ref. [14] and assigned generous (30%) errors to it** In this figure and in figs. 2, 3, 4, the optical points have been calculated by using the total cross sections from ref. [15]. Before drawing possible conclusions from the comparison of our bounds of fig. 1 with the data, we make a few simple remarks. The extraction of np -+ np data from pd scattering [9, 16] which usually employs the Glauber theory and some further simplifying assumptions is not a completely unique procedure. On the other hand, the angular distribution of the np -+ np data obtained with neutron beams should be quite reliable; there may be normalization difficulties. In the neutron-beam experiment of ref. [ 10], the normalization is fixed by extrapolating to the optical point and assuming the relative strength of the real part of the forward amplitude to be known. This strength is extracted from pd data [16] and may be ambiguous. This normalization is non-unique also because of the possibility of a change of the diffraction slope for very small Itl values; such a change of slope has been previously reported [17] for pp ~ pp scattering. Fig. 1 shows that the neutron-beam data of ref. [10] lie consistently above the upper bound, and there is an indication of the data having a flatter slope than the * Whenever the data on the different charge modes were not available at the same t (the invariant momentum-transfer squared), we did some simple interpolations; this applies to all our isospin bounds. Also, we have treated the errors as if they were uncorrelated; hence our errors for the density-matrix element calculations of subsect. 3.4 below cannot be taken literally. ** The bounds are not very sensitive to small changes in the charge-exchange cross section and, therefore, our procedure is quite adequate. In fact, the 19 and 24 GeV/c charge-exchange data exist only at ttl < 0.4 GeV2; but we feel that for larger Itl also, this phenomenological fit [which gives a very good representation at those Itl values at lower energies, 8 and 10 GeV/c is good enough for our purposes.

160

G. V.Dass and J.Froyland, Charge independence restrictions 100_

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upper bound. If the normalization of the data is shifted to produce good agreement with our bounds, one sees that for the smallest It[ values, do/dt(np ~ np) < do/dt(pp-~ pp); and also, the data of ref. [10] extrapolate to a point below the

G. V.Dass and J.f'royland, Charge independence restrictions I

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Fig. 2. Isospin b o u n d s on do/dt(mb/GeV 2) for pn -, ~n are shown as a function of t (GeV2). The dotted lines have the same significance as in fig. 1.

optical point*. With the above assumptions, isospin invariance would, therefore, imply a change of slope in np ~ np scattering at small Itl valuest. Moreover, because of the near-equality of the pp and np total cross sections, this change of slope should probably be bigger than the corresponding change seen in pp elastic scattering. It would be interesting to have np --r np experimental results with neutron beams for the small Itl region. We should remark, however, that if one compares * Such a p h e n o m e n o n has recently been observed at Serpukhov energies in rr-p ~ rr'p scattering 118] also. t See the note added in proof.

G. V.Dass and J.Froyland, Charge independence restrictions

162

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our bounds with another neutron beam experiment [11 ], our conclusion of a possible change of slope becomes weaker; unfortunately, the quoted errors on the data of this latter experiment are larger. 3.1.2. Polarization The relations (5) provide bounds on polarization (P3) in elastic pn -* pn scattering if cross sections for all the three modes and polarizations for the other two

L

10

G. V.Dass and J.Froyland, Charge independence restrictions

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Fig. 4. Isospin bounds (8b) for polarization P in K-n ~ K-n scattering at 10 GeV/c shown as a function of t (GeV)2; the dotted lines have the same significance as in fig. 1. modes are known. Because of the above-mentioned normalization difficulties for the np ~ np elastic cross section (o3) , use of the relations (6) which do not require a knowledge o f e 3 may be better, in order to obtain bounds on P3- In practice, the improvement obtained b y using the bounds (6) or the bounds (8b), instead of the bounds (8a) which do not require as much knowledge o f polarizations, is negligible as compared to the errors on the bounds. At 10 GeV/c, near the forward direction, we find typically, IP1 -P31 ~< about 0.2, where P1 is the polariza'tion in pp elastic scattering. Because P1 ~ 0.I experimentally, this bound is insufficient to determine the sign of P 3. One may give a duality argument that, unlike the case o f TrN scattering [5], the bounds (8a) on I P 3 - P l l are not saturated: the point is that at low energies, P1 ~ P3; and because o f the lack of dibaryon resonances, global duality* expects this near equality to continue to high energies also. Unfortunately, this duality argument can be, at best, qualitative. Comparison of the polarizations P1 and P3 can give useful information on the relative strength of the isoscalar and isovector t channel exchanges; only the isovector contribution changes sign in going from pp to np elastic scattering. At high energies, to a good approximation P1,3 ~ (Re S I r o N ) l , 3 , where S and N are the s-channel single- and zero-flip helicity amplitudes; I m N 1 ~_ IroN3, but R e S 1 and R e S 3 differ b y the amount o f i s o v e c t o r exchange in * The lack of a qualitative change in the experimental P2 (the polarization in charge-exchange pn ~ np scattering) as one goes to higher energies supports this duality argument. Measurements [19,201 ofP 2 exist up to 5.6 GeV/c, wttile P3 measurements [20] exist only at low energies.

164

G. V.Dass and J.Froyland, Charge independence restrictions

them. If isovector exchange dominates the amplitude S, P1 ~ - P3 and the abovementioned bound IP 1 -P31 ~< 0.2 obtained above would be almost saturated, because P1 ~ 0.1. However, if the amplitude S is dominated by isoscalar exchanges, PI'~--P3 and this bound would not be saturated.

3.2. Nucleon-antinucleon scattering Here again, one expects to get good bounds on elastic fin ~ ~n scattering because of the smallness of the charge exchange (~p ~ fin) scattering as compared to elastic (~p -~ ~p) scattering. Our isospin bounds are shown in fig. 2 for 5.7 and 8 GeV/c, to be compared with future data. In fact, our bounds may be a bit too conservative because there is some evidence [21 ] that the charge-exchange data (which we have used as they stand) may be somewhat too highly normalized. The input pp ~ gp data are from ref. [22] and the charge exchange data from ref. [23]. 3.3. Kaon-nucleon scattering As in the previous subsections, since the high-energy cross section for the chargeexchange process K-p -* K°n is much smaller than that for the elastic process K-p -+ K-p, one can get quite good isospin bounds on the cross section for K-n K-n scattering. The resulting bounds are shown in fig. 3a at 7.1,9.5 and 12.3 GeV/e, using input K-p ~ K-p data from refs. [24], and the charge-exchange data from ref. [25]. Similarly, the charge exchange K÷n ~ K°p is much weaker than elastic scattering K+p ~ K÷p and one expects restrictive bounds for K+n ~ K+n scattering. The bounds at 12 GeV/c shown in fig. 3b were obtained by using input K÷p data from ref. [26], and the charge exchange data from ref. [27]. Since the chargeexchange processes K+n ~ K°p and K-p ~ K°n have equal (or nearly equal) cross sections in the near-forward direction at 12 GeV/c, and since also the elastic K±p cross sections are quite similar, the bounds at 12.3 and 12 GeV/c in figs. 3a and 3b are very similar. There is, however, a qualitative difference in the extrapolation of the lower bounds to the forward direction where the known K-+n total cross sections [15] provide the lower bounds, via the optical theorem. Barring a drastic variation with angle, this extrapolation indicates the need of a large real part of the forward amplitude for the K÷n ~ K÷n case; for the K-n -+ K-n case, the extrapolation in t is larger and does not necessarily call for a real part of the forward amplitude. This is consistent with the expectations of an exchange-degenerate dual Regge model [with trajectory intercepts ao(O ) ---aA2(0 ) = ap,(0) = aw(0 ) = 0.5] which makes the contribution of the secondary trajectories to the forward K+n ~ K+n amplitude purely real and the forward K-n -+ K-n amplitude purely imaginary. It is interesting to be able to get such qualitative conclusions from just isospin invariance alone. As an application of the bounds (8b) on the polarization P3 when only PI, °l and o 2 are known, we show in fig. 4 the bounds on polarization in K-n ~ K-n scattering at 10 GeV/e as a function of t. The input K-p -~ K-p data (P1 and Ol) are from ref. [24] and the charge exchange K-p ~ K°n data (o2) are from ref. [25]. Because of the smallness of O2/O1,the bounds of fig. 4 are quite restrictive.

G. V.Dass and J.Froyland, Charge independence restrictions

165

If charge symmetry holds, K+(K-)n scattering is equal to K°(K°)p scattering, and because of their relation 2T(KLP ~ KsP) = T(K°P "+ K°P) - T{KoP -> KoP) to the regeneration amplitude, one can set bounds, using (4), on the regeneration cross section in terms of the K-+n ones. These bounds can be useful, in the absence of actual regeneration measurements, in the study [28] of possible Pomeranchuk theorem violation. The bounds on the regeneration cross section deduced from our K±n bounds at 12 GeV/c are not very useful. Also, of course, if the bounds for K+I~ (K-n) scattering, obtained by using regeneration and K-n (K+n) data (when available) are combined with the independently valid bounds obtained by using K+p K+p (K-p ~ K-p) and K+n ~ K°p (K-p ~ K°n) data, one may make the bounds on K+n (K-n) scattering even more restrictive. 3.4. Vector-meson production

As simple illustrations of the bounds for resonance production, we consider the vector-meson production processes ~N ~ pN (figs. 5 , 6 and 7) and KN ~ K*(890)N (figs. 8, 9 and 10). The bounds (9) for the density matrix elementst (all in the Gottfried-Jackson frame) and (10) for the differential cross sections for 7r-p ~ p % at 4 and 8 GeV/c are compared in fig. 5 with the available data which are found to be generally quite consistent with isospin invariance; the points .(fig. 5f) where the upper bound is lower than the lower bound may be traced to the behaviour of the corresponding differential cross sections. This apparent disagreement with chargeindependence may well be due to our data interpolations; also, we have treated all errors in an uncorrelated way. The 4 GeV/c, ~r-p -+ pOrt data are from ref. [29] and the ~±p ~ p-+p data from ref. [30]. For 8 GeV/c, the pO production data are from ref. [31] and the p± production data from ref. [32]. The S-wave background in the case of 7r-p -> p°n data was accounted for, as in ref. [33]. The effect of rotation of the reference axes on the 15 GeV/e, 7r-p -+ p°n data [34] is illustrated in fig. 6 which shows the value ofc~, the angle of rotation from the s-channel helicity frame to the DH2 frame. It is remarkable to note that for all but the three smallest scattering angles shown, the quantity Poo in the DH2 frame becomes consistent with zero (within one standard deviation). This property of data [(Poo)DH2 ~ 0], typical of the majority of high energy K* and p production data, implies a saturation of the positivity condition on the observed part of the density matrix. Such a saturation could follow in some simple models [33]. The vanishing of p° ° leads to the relation (11) and to the bounds (12b) for the density matrix. For the same reason, rotation around the normal to the production plane provides a simple visual check on the well-known positivity property [6] of the All the density matrix data (except those of fig. 6) used in this paper were given in the Gottfried-Jackson frame.

G. V.Dass and J.Froyland, Charge independence restrictions

166

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Fig. 5. lsospin bounds of relation (10) on do/dt(mb/GeV 2) and of relation (9) on Poo, Re Pl0 and P1-1 for 7r-p ~ pOn at 4 GeV/c, shown as a function of t (GeV 2) in figs. a, b, c and d respectively, are compared with data; figs. e, f, g and h show the corresponding bounds at 8 GeV/c The dotted lines have the same significance as for fig. 1. original density matrix, as is apparent f r o m the figures below, especially figs. 7 and 9. The behaviour o f P°oo o f eq. (13) is shown as a f u n c t i o n of the angle o f rotation 0, starting f r o m the G o t t f r i e d - J a c k s o n frame, for the 4 G e V / c , nN ~ pN data [29, 30] at t = - 0.12 GeV 2 in fig. 7a. Here again, Poo b e c o m e s consistent w i t h zero in the DH2 frame; the errors have not been shown for clarity.

G. V.Dassand J.Froyland, Chargeindependence restrictions

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Fig. 6. The angle c~of rotation from the s-channel helicity frame to the Donohue-H6gaasen frame DH2, for tile 15 GeV/c rr-p ~ pOn data, shown as a function of t (GeV2). For all but the three lowest Itt values shown by squares, the value of Ooo in the DH2 frame becomes consistent with zero; the dotted line just joins this type of points. Isospin invariance and this vanishing of Poo in the DH2 frame (determined by the angle cQ for one-charge mode (say, ~-p -+ p°n) implies the equality, (11), of the combination [Poo(dO/dt)] for the other two modes 0r-+p ~ p±p), Poo being a known combination Of Poo, P l l , Re Pl0 and Pl 1 of the original frame. This prediction has been compared with relevant data (wherever available for all the modes) and found to be satisfied within the (unfortunately large) errors. For example, this equality due to a vanishing Of Poo at c~ ~_ 113 ° (starting with the Gottfried-Jackson frame) for the 8 GeV/c, 7r+p ~ p+p data [32] at t = 0.25 GeV 2 reads as (0.04 + 0.04) mb/GeV 2 = (0.025 + 0.04) mb/GeV 2 . To illustrate the effect of rotation on the relations (9), the 4 GeV/c, Poo(rr-p ~ p ° n ) b o u n d s , at t = - 0 . 1 2 GeV 2, are compared with the data [29] in fig. 7b as a function of the angle 0 of rotation (starting from the Gottfried-Jackson frame). The input data for fig. 7b are the same as for fig. 5b wherein the 0 = 0 point is shown, As examples of the use of the bounds (9) in a rotated frame, we have evaluated bounds (14) on the quantity (2 Re PI0 -+ Pl - 1); unfortunately, the data are such that the bounds obtained were rather poor, even though the corresponding errors on the P°oobounds are quite reasonable. For KN -+ K*(890)N, we illustrate the effect of the rotation of the reference axes in figs. 8, 9 and 10. The angle ~ of rotation to the DH2 frame (starting from the Gottfried-Jackson frame) is shown in fig. 8 for the 2.64 GeV/c, K-p -+ K*-p data [35] ; here also, the quantity Poo in the DH2 frame does become zero (within one standard deviation) for a number of t values. The prediction (11) for the equali-

168

G. V.Dass and J.Froyland, Charge independence restrictions I0

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Fig. 7a. The density matrix element p0oo is shown as a function of the angle 0 of rotation (starting from the Gottfried-Jackson frame) for the 4 GeV/c data at t = - 0.12 GeV 2 for the three charge modes n±p ~ p±p and n-p --, pOn. 7b. Isospin bounds (9) on pOo shown as a function of the rotation angle 0 (starting from the Gottfried-Jackson frame) for the mode ~r-p ~ pOn at 4 GeV/c, t = - 0 . 1 2 GeV 2 are compared with data. This figure shows the 0 dependence of the t = - 0.12 GeV 2 point of fig. 5b. The shaded area shows the upper (lower) part of the error on the upper (lower) bound. Some typical error-bars on the data are also shown. ty of the remaining two modes w h e n Poo for the first m o d e vanishes is shown as a f u n c t i o n o f t in fig. 9 for the 4.5 G e V / c , K - p -> K * - p and K - n -~ K * - n data [36] and for the 4.57 G e V / c , K - p -~ K*°n data [37]. In fact, as for a lot of the nN -~ p N data (see fig. 7a), the behaviour o f P°oo as a f u n c t i o n o f the angle 0 o f r o t a t i o n (starting f r o m the G o t t f r i e d - J a c k s o n frame) is qualitatively the same for all the three charge modes: the angle a o f r o t a t i o n to the DH2 frame is the same (within errors), and Poo ~ 0 for all these modes. F o r clarity, Poo is shown in fig. 9 for the three charge m o d e s at only some " a v e r a g e " value (104 °) o f this angle of rotation; the equality (11) is fulfilled, b o t h sides being consistent with zero. Fig. 10 shows 0 the isospin b o u n d s (9) on Poo do/dt(K - p ~ -K- * O n) at 4.5 GeV/c, t = - 0.15 GeV 2 as a f u n c t i o n of the angle 0 o f rotation, starting from the G o t t f r i e d - J a c k s o n frame.

G. V.Dass and J.Froyland, Charge independence restrictions

169

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G. V.Dass and J.Froyland, Charge independence restrictions

170

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Fig. 10. lsospin bounds (9) on Ooo do/dt(mb/GeV2) for the K-p --, K*On mode at 4.5 GeV/c, t = -0.15 GeV2 are shown as a function of the rotation angle 0 (starting from the GottfriedJackson frame). The line with points(e)and typical error-bars shows the data. The shaded area shows the errors on the bounds, as for fig. 7b. The "data" for comparison come from the density matrix elements of the 4.57 GeV/c, K-p ~ K*°n experiment [37] and the t = - 0 . 1 5 GeV 2 differential cross section interpolated between the 4.1 and 5.5 GeV/c experiments [38]. The input 4.5 GeV/c, K-p, n ~ K*-p, n data are from ref. [36]. The corresponding P l l bounds are not restrictive.

4. CONCLUDING REMARKS Strong interactions are believed to be isospin invariant to a very good approximation. Isospin invariant models would fail to fit data on the different charge modes if the data do not fulfil the simple triangular inequalities. On the other hand, isospin invariant models with certain specific dynamical assumptions may misfit a particular piece of data rather strongly even though they fit the remaining data reasonably well, and all the relevant data satisfy charge independence requirements quite well; for an example, see ref. [5]. Our np ~ np and K±n ~ K-+nbounds seem to point out another use of the isospin inequalities: that charge independence alone can teach one something about dynamics in certain situations. For resonance production, rotation around the normal to the production plane can provide a simple way of checking the well-known positivity requirements [6] on the density matrix of the produced resonance; this is illustrated for high energy 7rN ~ oN and KN ~ K*(890)N data for a majority of which the density matrix element 0oo in one of the Donohue-H~Sgaasen frames becomes consistent with zero. This rotation allows one to calculate isospin bounds on certain density matrix ele-merits even in situations where only a partial knowledge of the density matrix for

G. V.Dass and J.Froyland, Charge independence restrictions

171

the other two modes is available. Also, isospin invariance predicts relations [like (11) and (12)] between the density matrices in the rotated frame for the other two modes, if the density matrix data on the first mode satisfy certain conditions. The apparent inconsistencies with positivity and with isospin invariance that came up at some points cannot be taken literally because of our inadequate treatment of errors, especially for the density matrix. Since data on all the three charge modes are available in 7rN elastic scattering, it would be interesting to get dynamical information from a study of the relative positioning (especially its energy dependence) of the data within the isospin bounds, similar to that in fig. 1. We intend to study this, along with a study of the bounds on A 1 and A 2 production (also the density matrix bounds). We are thankful to K.Runge and A.Wetherell for providing us with their data, and to B.H.Kellet for sending us his listing of the 7rN -+ pN data. One of us (G.V.D,) is thankful to P.K. Kabir for enlightening discussions and to C.D. Froggatt for useful comments.

NOTE ADDED IN PROOF A recent measurement by F.E. Ringia et al. (Phys. Rev. Letters 28 (1972) 185) of np elastic scattering at 4.8 GeV/c for 0.002 GeV 2 < Itl < 0.05 GeV 2 seems to support our conclusion of a change o f slope at small rtl values.

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