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Antiprotonic hydrogen. Initial state interactions
Volume 244, number 1
PHYSICS LETTERS B
12 July 1990
Antiprotonic hydrogen. Initial state interactions E. K l e m p t
Institutj~r Physikder Universitdt,D-6500Mainz, FRG Received 25 April 1990
Strong interactions between proton and antiprotons in antiprotonic hydrogen atoms lead to measurable shifts and widths of the S and P levels. In addition, they introduce a nfi component into the f~p wave function. This breakdown of isospin symmetry predicted by models of strong interactions in lbp atoms - is determined from experimental data on f~p annihilation into two mesons. A comparison with the theoretical results shows good agreement.
A n t i p r o t o n i c hydrogen a t o m s have been studied extensively in recent years at the Low-Energy Antiproton Ring ( L E A R ) at C E R N [ 1 ]. The m a i n objective o f these experiments was to d e t e r m i n e the strong interaction shift and width o f the 1S ground state o f the pp a t o m [2 ]. The strong interaction p a r a m e t e r s had been calculated [3] before the data became available by solving the SchriSdinger equation for a C o u l o m b and a nuclear potential. The latter can be o b t a i n e d from the N N potential by G-parity transformation. G o o d agreement holds between experimental results and the theoretical predictions [ 1 ]. There is, however, one aspect o f these calculations which has not yet been tested against e x p e r i m e n t a l data: At annihilation, the distance between proton and a n t i p r o t o n is short, and the nuclear potential energy can be c o m p a r a b l e to the mass difference between the lbp and the fin systems. Therefore the isospin wave function is not necessarily that o f a pure pp system but a nn c o m p o n e n t can be admixed. Hence the isospin wave function o f the 15p a t o m should be written as (1)P a t o m ) 1
-x/~{alI=O, 13=O)+blI=l,I3=O)}.
(1)
In absence o f strong interactions a = b = 1; in any case the relation a 2 + b 2 = 2 holds. In this letter a m e t h o d is p r o p o s e d how this b r e a k d o w n o f isospin invari122
ance can be d e t e r m i n e d from experimental data, and first results are presented. A n n i h i l a t i o n m a y occur via the I = 0 or the I = 1 c o m p o n e n t o f the pp wave function. E.g., the spin triplet, zero angular m o m e n t u m state (3Sl) m a y annihilate into n°o~ via its I = 1 c o m p o n e n t , or into n°p ° via its I = 0 component. The ratio o f branching ratios into n°to and n°p ° from the 3S~ state is given by [4,5 ] R t (/I:°(°) n ¢7+o,o9 Rt(nOpO) . . . . . o,15.
(2a)
R t refers to annihilations from the 3St state, R S from the 'So state. The total hadronic widths o f the 3S~ state, F t, c a n be d i v i d e d into two parts, F t = (7~ + 7~ )Ft; ?~ + 7] = 1. y~ and y] are the fractional contributions o f the two isospin c o m p o n e n t s I = 0 and I = 1 to the total hadronic width. In absence o f strong interactions 7~ = 7 ] = 1. F o r the ~So spin singlet state we define 7~ and 7] in analogy. The ratio ( 2 a ) does not give 7]/Tb, since the hadronic matrix elements for the two a n n i h i l a t i o n modes may be different. It might be argued that the ratio ( 2 a ) is strongly influenced by strong interaction dynamics, and that no conclusions can be drawn on the initial state. But ( 2 a ) can be c o m p a r e d to other annihilation modes. R t ( r l 9 °) --_(0.42+0.05), R'(qo~)
(2b)
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 244, number
1
PHYSICS
Rt(rip °) Rt(ltOpO ) - ( 0 . 2 8 _ + 0 . 0 3 ) = ~ . ( 0 . 4 2 _ + 0 . 0 5 ) ,
(2c)
R_ ' ( _l r % ) Rt(rlo))
(2d)
--
0.15'~ = 3 .
( 0 8 4 +- - 0 . 2 4 )
0.10~
(0.56+- - 0 . 1 6 /
Rt(p°f2) R ' ( of2~ = (0.48 ___0.12),
,
(2e)
The ratios ( 2 a ) - ( 2 e ) refer to annihilations of the 3S~ state of the lop atom. Annihilations from the I = 1 component of the atomic wave function are compared to annihilations from I = 0 . Obviously, the ratio (2a) depends on the isospin of the two mesons in the final state and not on the particular choice of the two mesons. The data are quoted from refs. [4,5]. For lop annihilation into ripo we use the average value, (47 __ 5 ) × 10 -4, of different experiments as given by Weidenauer et al. [ 5 ]; this value differs significantly from the result obtained by Chiba el al. [4]. Likely, the contributions from rip and rlo were not correctly separated in that analysis. Therefore we use the branching ratio for rio from Adiels et al. [ 4 ]. Eq. (2e) is quoted from Armenteros et al. [4]. A correction has been applied to allow for neutral decay modes of the o. For the other reactions this correction had been applied in the original papers. The branching ratio for lop--,n+p- from the 3S~ state can be derived from three data points [4]. May et al. find that (70.9_+4.3)% of ~+Tt-rc ° S wave annihilations proceed via one of the three charge states of the P from the 3S, state. This gives a branching ratio of ( 156_+21 ) × 10 -4. Foster et al. [4] gave amplitudes only, not branching ratios. Following the method of May et al. [4], a branching ratio of ( 180_+ 14) × 10- 4 is calculated. Chiba et al. [ 4 ] give ( 160_+ 10) X 10 -4 for lop__,~opo which is related to lop-,rt+p - from the 3S1 state by charge invariance. The average of the three determinations gives R~)( 7t+p - ) = ( 165 _+8) X 10 -4 . The corresponding mode from the ~So state was also given by May et al. [4]. R] ( ~ + p - ) = (4.6 _+2.0) × 10 -4 . The latter branching ratio will be used below. The factor ~ or 3 in (2c), (2d) is introduced to account for the sg component of the ri which does not couple
LETTERS
B
12 J u l y 1 9 9 0
to lop. Numerically, the factor corresponds to a pseudoscalar mixing angle of - 1 9 . 5 °. The values in brackets on the RHS in (2c) and (2d) give the ratios which one would have measured if the pseudoscalar mesons had ideal mixing. Obviously, the ratio of branching ratios are identical (after applying a correction factor for the 1]). We therefore assume that one can write R i ( M , M2 ) = ( 2 S + 1 ) ' C ( I ; I,, 12)"f( 1, 2)7~ (3)
(i=t,s) .
C(I; I~, 12) is the squared transition matrix element for lop annihilation into mesons M~ and M2. It depends on the isospin component I of the lop atom via which annihilation proceeds; on the two isospins of M~ and M2, I~ and 12; but - by hypothesis - not on other mesonic quantum numbers. 2 S + l is the usual statistical spin factor; 7~ the fractional isospin contrib u t i o n . f ( 1 , 2 ) contains the dynamics. In some cases, a po meson is replaced by an o meson, and the dynamics can faithfully be assumed not to change much. In other cases, a n ° is replaced by an ri and the assumption of nearby identical dynamical factors needs justification. The phase space for annihilation into two mesons with relative angular m o m e n t u m /5 is proportional to the decay m o m e n t u m p, the amplitude of a point-like particle proportional to I.Ola. The lop system is, of course, not pointlike. Therefore different kinematical assumptions can be used. Hartmann et al. [ 6 ] make the assumption that the amplitudes do not depend on the decay m o m e n t u m . Vandermeulen [7] suggests a preference of lop annihilation into high-mass mesons. This preference for high-mass mesons compensates (to a large extent) the two-body phase space factor. In this letter no kinematical corrections are applied. Inserting (2a) and the right-hand side of (2c) into (3) one finds
C(1;0, 1)7~_ = 0 . 4 4 + 0 0 5 C(0;1, 1)y~ - '
(4a) '
and from (2b), (2d) and (2e) C(1; 0, l)y~ =0.44_+0.05. C(0; 0, 0)y~
(4b)
Eqs. (4a) and (4b) lead to C(0; 0, 0 ) ~ C ( 0 ; 1, 1). From now on we assume that this equality is exact: 123
Volume 244, number 1 C(0;0,0)=C(0;1,
PHYSICS LETTERS B 1).
(5)
12 July 1990
can be derived which will be used below. From (5) we arrive at
In the derivation of (4a) and (4b) partly the same data are used. Therefore we use only (2a), (2b) and (2e) to determine the ratio
C ( l ; 0, 1)y] =0.30_+0.09, C(0; 1, I)7~
C(1; 0, 1)7~
and by dividing by (6) using ( 5 )
c(0;0, 0)%
=0.47_+0.05 .
(6) =0.64_+0.21 .
In the spin singlet ~So state, the corresponding ratio of branching ratio is different: R'(~°f2) R,(~Oa2O ) = 0 . 3 0 + 0 . 0 9 .
(7)
The branching ratio lOp-) ~°f2 from the *So state was determined by Foster et al. and May et al. [4]. Both groups see the 1"2in its x+Tt- decay mode. The measured branching ratios into n°f2 need to be corrected for f2-,n°x ° decays (3) and for other decay modes [8] of the/'2 ( 1 / 0 . 8 6 ( 2 ) ) . Then the two results are (41.9_+ 12.2) × l0 -4 and (38.0-+ 10.3) y 10 -4 with an average value of (39.6_+7.9) × l0 -4. The data on x°a° [ 9-1 l, 5 ] are not consistent: Diaz et al. [ 9 ] find 4 2 3 ( 6 0 ) × l0 -4 for the three charge states of the a2 in the 3x decay mode; Espigat et al. [10] 46( 1 4 ) × l0 -4 for two charge states of the a2 in the rl~ decay mode, Weidenauer et al. [5] 27.3(7.7) X l0 -4 for the latter reaction, Bettini et al. [ l l ] find 1 0 . 9 ( 1 . 8 ) Y l 0 -4 for three charge states of the a2, in the KK decay mode. We normalize the data to one charge state and correct for the known branching ratios of the a2 into prt, rift, and KK [8]. Then we find ( 200 _+30 ) N 10 -4, ( 160 -+ 50) X 10-4, ( 94 _+27 ) × 10- 4, ( 75 _+ 13 ) × 10- 4, respectively. We use the linear average of the four determinations and estimate the error from the variance of the experimental results. This gives ( 132 _+31 ) × 10 -4. All four experiments give also the branching ratio for lOp~rt+a; - from the 3S~ state. These are 118(28)×10-4; 10(5)×10-4; 13.8(5.1)×10-4; 1.5 ( 6 ) × 10-4. Correcting as above these values yield 8 4 ( 2 0 ) × 1 0 - 4 ; 3 4 ( 1 7 ) × 1 0 - 4 ; 4 8 ( 1 8 ) × 1 0 -4 and 15 (6) × 10-4; respectively, With the same method of averaging one finds (45 _+ 15) × 10 4. The ratio of the two contributions does not depend on the absolute normalization. From the four experiments the ratio R'(Tt+a2) =0.34_+0.07 R~(x+a£) 124
(9)
(8)
(I0)
Clearly, 98 and 7~ cannot both be ½. Eq. (10) is incompatible with the absence of initial state interaction at the 2a level. It needs to be mentioned that the result (9) depends completely on one experimental input, Eq. (7). In principle, (9) could also be determined from pp-) of)o, pOpO, coco. The branching ratio into pOco is ( 2 2 6 + 2 3 ) X 1 0 - 4 , pOpO ( 1 2 _ + 1 2 ) X 1 0 - 4 and coco ( 140 _+60) X 10 -4 [ 12,9 ]. The smallness of pOp° in lOp annihilation at rest is not understood in the frame of this analysis. Eq. ( 5 ) requires RS(p°p °) = R s (coco). A special symmetry could be responsible, but it should also be noted that this branching ratio is very difficult to extract from data due to the large number of interfering amplitudes contributing to the rt + 7t- n + x final state, and due to the fact that there are four possible rt+r~- pairs per event. The branching ratio into coco was determined from bubble chamber data with no detection of the two 7t°'s, and the authors consider the result as indicative only. A second relation between 78 and 7~ can be derived from the processes p p - ) x + a ~ - and rt+p -. The branching ratios can be written as
R~(rt+a2)=3C(1; 1, 1)7~f(rta2) ,
(lla)
R ~ ( T t + p - ) = 3 C ( 0 ; 1, 1 ) 7 ~ f ( ~ 9 ) ,
(lib)
Re,( n+a2 ) = C(0; 1, 1 )7~f(rca2) ,
(1 lc)
R~(x+9- )=C(1;1,1);],((~O) .
(lld)
This allows to calculate the double ratio (10): (9c) ( 9 d ) / ( 9 a ) (9b).
R~(rt+a£)R](rt+p R](~+ay)R~(x+p -) =0.74_+0.44.
)
7~(1-7~) 7~(1-7~) (12)
This ratio is compatible with unity. The small branching ratios R] ( x+p- ) and of R] (~ +aS- ) seem
VoLume244, number 1
PHYSICS LETTERS B
to have their c o m m o n origin in the smallness of the ratio C(1; 1, 1 ) / C ( 0 ; 1, 1 ) = 0 . 1 1 + 0 . 0 2 . A third relation can be derived from the branching ratios lop--,K*I( [11]. There are two dominant contributions:
(13a)
Rg(K*K)=C(O; ½, 1 )f(K*K)?,g = ( 8 . 6 - + 1 . 7 ) × 1 0 -4 ,
In ref. [ 13 ] it is argued that the amplitudes for production of two strange mesons are the same for these two processes, C(0; ~, ½) = C ( 1; ½, ½). Then (14)
yg
y~= (0.58+0.07). The ratio of isospin contributions can also be determined using 7~ + Y'I = 1. ~)~/yD = 1 1'7+0,39
(15a)
y]/~)
( 1
=f~ "/')+0,24 "~. ~z-,-- O, 18
C(1;0,1) =0.43+0.09. (13b)
1 -y~ - =0.95__+0.21 .
y~ = (0.46 + 0 . 0 7 ) ,
. . . . . 0.28,
R~ ( K ' K ) = 3C( 1; ½, ½)f(K*K)7~ =(24.8-+2.4)×10 -4,
12 July 1990
Eqs. (10), (12) and (14) can be used to determine 7~ and 75. Fig. 1 shows the allowed regions for ( 10): region a; for (12): region b; and for (14): region c. The AZ2= 1 contour plot is also shown. We find
c(0; 0, 0)
5b)
(15c)
Eq. 15c follows from (15a), (15b) and (6). Data on 10p annihilation from P states are still scarce. We denote branching ratios from the 3po, L2 states by R °'2'2', and from ~P~ b y / ? ~. The same notation is used for the 7's. Annihilation into nOn°, n°'q, fir1is possible from the states 3po and 3P2. Hence the data represent an unknown mixture of these states. The data of Adiels et al. [4] are used. The data by Chiba et al. [4] disagree with those of Adiels et al., but are statistically less significant. R°'2(n°q) 0.65_+0.14 R 0.2 ( nOn0) -2'2'(0.49_+0.11)
(16a)
10
The factor -~ accounts again for the sg component of the 7, the factor 2 is the usual statistical factor for two identical bosons. Similarly,
0.8
R o,2( nOrl )
1
R°'2(qrl) - (0.61-+0.25) 0.6
=3.2. (0.55 _+0.23).
(16b)
The statistical average is (0.50 -+ 0.10). With ( 3 ) and (15c) this yields
0./,
7°'2 = 1 . 1 6 - + 0 . 3 4 .
yoO,2
(17)
0
.
.
.
.
1.0
v~ Fig. 1. The allowed regions for the fractional isosinglet contribut and 7o. s Region a is defined by tions of the 3S~ and I So states, Yo eq. ( 8 ), region b by (10) and region c by ( 12 ). The AX2= 1 contour is also shown.
Annihilations from P states into n°f2 was studied by May et al. [ 14]. A contribution of 0.2l 1 (7) was determined to the n + n - n o final state. The branching ratio for n ÷ n - n ° is 4.5 ( 6 ) %. Correcting for unseen decay modes of the t"2, a branching ratio of ( 1 8 0 + 25 ) M 10 - 4 for 15p--,n°f2 from the 3P l s t a t e can be determined (100%=all annihilations from P states). Weidenauer et al. [5] find a contribution of 13.1 ( 6 . 9 ) × 10 -4 for lop-,n -+aJ: from P states in the 125
Volume 244, number 1
PHYSICS LETTERS B
tin d e c a y m o d e o f the a2. F o r o n e charge state o f the a2 but for all a2 decays this yields a b r a n c h i n g ratio o f (45 + 24 ) × 10 -4. A n n i h i l a t i o n into ha2 is allowed not only f r o m the 3pl; a n d t h e r e f o r e we can give only an inequality
71/V+>~9+5.
(18)
F o r the lPl state we h a v e b r a n c h i n g ratios for a n n i h i l a t i o n into p°n° and into 9%1. M a y el al [ 14 ] find a f r a c t i o n o f ( 2 6 . 9 + 4 . 6 ) % o f all P w a v e n + r t - n ° ann i h i l a t i o n s ( = (4.5 + 0.6 ) % ) , but for all three charge states o f the 9. H e n c e /~ l ( p ° n ° ) = ( 4 0 + 9) X 10 - 4 . Weidenauer lop--rip f r o m conservation f r o m the ~Pj ratio
et al. [5] find a b r a n c h i n g ratio for P states o f (9.4 + 5.3) × 10-4; C parity r e q u i r e s that the a n n i h i l a t i o n o c c u r r e d state o f the lop a t o m . H e n c e we find the
T h i s is c o m p a t i b l e with ( 1 5 c ) , h e n c e w i t h no initial state interactions. The ratio of the I = 1 and I = 0
components in the
IP 1 state
gives the range
~3,/~3o = 0 . 8 1 -+ 0 . 5 1 .
(19)
T a b l e 1 collects the results o f this analysis a n d theoretical c a l c u l a t i o n s [ 15 ]. A d m i t t e d l y , there are large u n c e r t a i n t i e s in the num e r i c a l results o f this analysis and, also, c o n c e p t u a l Table 1 7~/7o, theory versus experiment.
15patom
Theory a)
~So 3S1 ~P~ 3p~ 3Po 3P2
0.68 1.22 0.96 9.4 0.03 0.81
126
U s e f u l discussions with P r o f e s s o r J. K 6 r n e r a n d P r o f e s s o r J.M. R i c h a r d are kindly a c k n o w l e d g e d . I w o u l d like to t h a n k P r o f e s s o r L. T a u s c h e r for c o m m u n i c a t i o n o f d a t a p r i o r to p u b l i c a t i o n .
References 1165.
=0.24_+0.14
= ~. ( 0 . 3 5 - + 0 . 2 1 ) .
a) Ref. [151.
difficulties due to the i n c o n s i s t e n c y o f the a p p r o a c h w i t h the e x p e r i m e n t a l results on 10p-, p°p ° a n d ox0. T h e r e are also large e x p e r i m e n t a l d i s c r e p a n c i e s bet w e e n the results o f d i f f e r e n t e x p e r i m e n t s , in particular on n°n °, n°q, rlq, n°o~ and n-+a~ ( 1 3 2 0 ) . But the r a t h e r satisfactory a g r e e m e n t b e t w e e n the result pres e n t e d h e r e a n d the t h e o r e t i c a l calculations o f the initial state i n t e r a c t i o n s gives credibility to both, to o u r u n d e r s t a n d i n g o f the lop w a v e f u n c t i o n and to the ass u m p t i o n s w h i c h were m a d e in this analysis.
[1] See for a review Ch. Batty, Rep. Progr. Phys. 52 (1989)
/~ J ( pOq )
~f( p~xo)
12 July 1990
Experiment 0.68 0.95 0.82 9.7 0.03 0.43
0.8 1.26 0.61 6.5 0.053 0.98
0 •72 +0.24 -o.18 1•17 +0.39 -o.28 0.81 +0.51 >/9 _+5 1.16+_0.34 1.16+0.34
[ 2 ] M. Ziegler et al., Phys. Lett. B 206 ( 1988 ) 15 t; C.A. Baker et al., Nucl. Phys. A 483 ( 1988 ) 631; C.W.E. von Eijk et al., Nucl. Phys. A 486 (1988) 604; R. Bacher et al., Z. Phys. A 334 (1989) 93; [3] J.M. Richard and R. Sainio, Phys. Lett. B 110 (1982) 429; [ 4 ] L. Tauscher (PS 182 ), private communication; L. Adiels et al., Z. Phys. C 42 (1989) 49; R. Armenteros et al., CERN proposal P28 (1980), unpublished; M. Foster et al., Nucl. Phys. B 6 ( 1968 ) 107; B. May et al., Z. Phys. C 46 (1990) 191 ; M. Chiba et al., Phys. Rev. D 38 ( 1988 ) 2021, D 39 ( 1989 ) 329. [ 5 ] P. Weidenauer el al., 11and rI' production in pp annihilation at rest, submitted to Z. Phys. C. [6] U. Hartmann, E. Klempt and J. KOrner, Z. Phys. A 331 (1988) 217. [7] J. Vandermeulen, Z. Phys. C 37 (1988) 563. [8] Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lelt. B 204 (1988) 1. [9] J. Diaz et al., Nucl. Phys. B 16 (1970) 329. [ 10] P. Espigat et al., Nucl. Phys. B 36 (1972) 93. [ 11 ] A. Benini et al., Nuovo Cimento A 63 ( 1969 ) 1199. [ 12] R. Bizzarri et al., Nucl Phys. B 14 (1969) 169; M. Bloch et al., Nucl. Phys. B 23 (1970) 221. [13] E. Klempt, Z. Phys. A 331 (1988) 2tl. [ 14] B. May et al., Z. Phys. C 46 (1990) 203. [ 15] J. Carbonell et al., Z. Phys. A 334 (1989) 329.
PHYSICS LETTERS B
12 July 1990
Antiprotonic hydrogen. Initial state interactions E. K l e m p t
Institutj~r Physikder Universitdt,D-6500Mainz, FRG Received 25 April 1990
Strong interactions between proton and antiprotons in antiprotonic hydrogen atoms lead to measurable shifts and widths of the S and P levels. In addition, they introduce a nfi component into the f~p wave function. This breakdown of isospin symmetry predicted by models of strong interactions in lbp atoms - is determined from experimental data on f~p annihilation into two mesons. A comparison with the theoretical results shows good agreement.
A n t i p r o t o n i c hydrogen a t o m s have been studied extensively in recent years at the Low-Energy Antiproton Ring ( L E A R ) at C E R N [ 1 ]. The m a i n objective o f these experiments was to d e t e r m i n e the strong interaction shift and width o f the 1S ground state o f the pp a t o m [2 ]. The strong interaction p a r a m e t e r s had been calculated [3] before the data became available by solving the SchriSdinger equation for a C o u l o m b and a nuclear potential. The latter can be o b t a i n e d from the N N potential by G-parity transformation. G o o d agreement holds between experimental results and the theoretical predictions [ 1 ]. There is, however, one aspect o f these calculations which has not yet been tested against e x p e r i m e n t a l data: At annihilation, the distance between proton and a n t i p r o t o n is short, and the nuclear potential energy can be c o m p a r a b l e to the mass difference between the lbp and the fin systems. Therefore the isospin wave function is not necessarily that o f a pure pp system but a nn c o m p o n e n t can be admixed. Hence the isospin wave function o f the 15p a t o m should be written as (1)P a t o m ) 1
-x/~{alI=O, 13=O)+blI=l,I3=O)}.
(1)
In absence o f strong interactions a = b = 1; in any case the relation a 2 + b 2 = 2 holds. In this letter a m e t h o d is p r o p o s e d how this b r e a k d o w n o f isospin invari122
ance can be d e t e r m i n e d from experimental data, and first results are presented. A n n i h i l a t i o n m a y occur via the I = 0 or the I = 1 c o m p o n e n t o f the pp wave function. E.g., the spin triplet, zero angular m o m e n t u m state (3Sl) m a y annihilate into n°o~ via its I = 1 c o m p o n e n t , or into n°p ° via its I = 0 component. The ratio o f branching ratios into n°to and n°p ° from the 3S~ state is given by [4,5 ] R t (/I:°(°) n ¢7+o,o9 Rt(nOpO) . . . . . o,15.
(2a)
R t refers to annihilations from the 3St state, R S from the 'So state. The total hadronic widths o f the 3S~ state, F t, c a n be d i v i d e d into two parts, F t = (7~ + 7~ )Ft; ?~ + 7] = 1. y~ and y] are the fractional contributions o f the two isospin c o m p o n e n t s I = 0 and I = 1 to the total hadronic width. In absence o f strong interactions 7~ = 7 ] = 1. F o r the ~So spin singlet state we define 7~ and 7] in analogy. The ratio ( 2 a ) does not give 7]/Tb, since the hadronic matrix elements for the two a n n i h i l a t i o n modes may be different. It might be argued that the ratio ( 2 a ) is strongly influenced by strong interaction dynamics, and that no conclusions can be drawn on the initial state. But ( 2 a ) can be c o m p a r e d to other annihilation modes. R t ( r l 9 °) --_(0.42+0.05), R'(qo~)
(2b)
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 244, number
1
PHYSICS
Rt(rip °) Rt(ltOpO ) - ( 0 . 2 8 _ + 0 . 0 3 ) = ~ . ( 0 . 4 2 _ + 0 . 0 5 ) ,
(2c)
R_ ' ( _l r % ) Rt(rlo))
(2d)
--
0.15'~ = 3 .
( 0 8 4 +- - 0 . 2 4 )
0.10~
(0.56+- - 0 . 1 6 /
Rt(p°f2) R ' ( of2~ = (0.48 ___0.12),
,
(2e)
The ratios ( 2 a ) - ( 2 e ) refer to annihilations of the 3S~ state of the lop atom. Annihilations from the I = 1 component of the atomic wave function are compared to annihilations from I = 0 . Obviously, the ratio (2a) depends on the isospin of the two mesons in the final state and not on the particular choice of the two mesons. The data are quoted from refs. [4,5]. For lop annihilation into ripo we use the average value, (47 __ 5 ) × 10 -4, of different experiments as given by Weidenauer et al. [ 5 ]; this value differs significantly from the result obtained by Chiba el al. [4]. Likely, the contributions from rip and rlo were not correctly separated in that analysis. Therefore we use the branching ratio for rio from Adiels et al. [ 4 ]. Eq. (2e) is quoted from Armenteros et al. [4]. A correction has been applied to allow for neutral decay modes of the o. For the other reactions this correction had been applied in the original papers. The branching ratio for lop--,n+p- from the 3S~ state can be derived from three data points [4]. May et al. find that (70.9_+4.3)% of ~+Tt-rc ° S wave annihilations proceed via one of the three charge states of the P from the 3S, state. This gives a branching ratio of ( 156_+21 ) × 10 -4. Foster et al. [4] gave amplitudes only, not branching ratios. Following the method of May et al. [4], a branching ratio of ( 180_+ 14) × 10- 4 is calculated. Chiba et al. [ 4 ] give ( 160_+ 10) X 10 -4 for lop__,~opo which is related to lop-,rt+p - from the 3S1 state by charge invariance. The average of the three determinations gives R~)( 7t+p - ) = ( 165 _+8) X 10 -4 . The corresponding mode from the ~So state was also given by May et al. [4]. R] ( ~ + p - ) = (4.6 _+2.0) × 10 -4 . The latter branching ratio will be used below. The factor ~ or 3 in (2c), (2d) is introduced to account for the sg component of the ri which does not couple
LETTERS
B
12 J u l y 1 9 9 0
to lop. Numerically, the factor corresponds to a pseudoscalar mixing angle of - 1 9 . 5 °. The values in brackets on the RHS in (2c) and (2d) give the ratios which one would have measured if the pseudoscalar mesons had ideal mixing. Obviously, the ratio of branching ratios are identical (after applying a correction factor for the 1]). We therefore assume that one can write R i ( M , M2 ) = ( 2 S + 1 ) ' C ( I ; I,, 12)"f( 1, 2)7~ (3)
(i=t,s) .
C(I; I~, 12) is the squared transition matrix element for lop annihilation into mesons M~ and M2. It depends on the isospin component I of the lop atom via which annihilation proceeds; on the two isospins of M~ and M2, I~ and 12; but - by hypothesis - not on other mesonic quantum numbers. 2 S + l is the usual statistical spin factor; 7~ the fractional isospin contrib u t i o n . f ( 1 , 2 ) contains the dynamics. In some cases, a po meson is replaced by an o meson, and the dynamics can faithfully be assumed not to change much. In other cases, a n ° is replaced by an ri and the assumption of nearby identical dynamical factors needs justification. The phase space for annihilation into two mesons with relative angular m o m e n t u m /5 is proportional to the decay m o m e n t u m p, the amplitude of a point-like particle proportional to I.Ola. The lop system is, of course, not pointlike. Therefore different kinematical assumptions can be used. Hartmann et al. [ 6 ] make the assumption that the amplitudes do not depend on the decay m o m e n t u m . Vandermeulen [7] suggests a preference of lop annihilation into high-mass mesons. This preference for high-mass mesons compensates (to a large extent) the two-body phase space factor. In this letter no kinematical corrections are applied. Inserting (2a) and the right-hand side of (2c) into (3) one finds
C(1;0, 1)7~_ = 0 . 4 4 + 0 0 5 C(0;1, 1)y~ - '
(4a) '
and from (2b), (2d) and (2e) C(1; 0, l)y~ =0.44_+0.05. C(0; 0, 0)y~
(4b)
Eqs. (4a) and (4b) lead to C(0; 0, 0 ) ~ C ( 0 ; 1, 1). From now on we assume that this equality is exact: 123
Volume 244, number 1 C(0;0,0)=C(0;1,
PHYSICS LETTERS B 1).
(5)
12 July 1990
can be derived which will be used below. From (5) we arrive at
In the derivation of (4a) and (4b) partly the same data are used. Therefore we use only (2a), (2b) and (2e) to determine the ratio
C ( l ; 0, 1)y] =0.30_+0.09, C(0; 1, I)7~
C(1; 0, 1)7~
and by dividing by (6) using ( 5 )
c(0;0, 0)%
=0.47_+0.05 .
(6) =0.64_+0.21 .
In the spin singlet ~So state, the corresponding ratio of branching ratio is different: R'(~°f2) R,(~Oa2O ) = 0 . 3 0 + 0 . 0 9 .
(7)
The branching ratio lOp-) ~°f2 from the *So state was determined by Foster et al. and May et al. [4]. Both groups see the 1"2in its x+Tt- decay mode. The measured branching ratios into n°f2 need to be corrected for f2-,n°x ° decays (3) and for other decay modes [8] of the/'2 ( 1 / 0 . 8 6 ( 2 ) ) . Then the two results are (41.9_+ 12.2) × l0 -4 and (38.0-+ 10.3) y 10 -4 with an average value of (39.6_+7.9) × l0 -4. The data on x°a° [ 9-1 l, 5 ] are not consistent: Diaz et al. [ 9 ] find 4 2 3 ( 6 0 ) × l0 -4 for the three charge states of the a2 in the 3x decay mode; Espigat et al. [10] 46( 1 4 ) × l0 -4 for two charge states of the a2 in the rl~ decay mode, Weidenauer et al. [5] 27.3(7.7) X l0 -4 for the latter reaction, Bettini et al. [ l l ] find 1 0 . 9 ( 1 . 8 ) Y l 0 -4 for three charge states of the a2, in the KK decay mode. We normalize the data to one charge state and correct for the known branching ratios of the a2 into prt, rift, and KK [8]. Then we find ( 200 _+30 ) N 10 -4, ( 160 -+ 50) X 10-4, ( 94 _+27 ) × 10- 4, ( 75 _+ 13 ) × 10- 4, respectively. We use the linear average of the four determinations and estimate the error from the variance of the experimental results. This gives ( 132 _+31 ) × 10 -4. All four experiments give also the branching ratio for lOp~rt+a; - from the 3S~ state. These are 118(28)×10-4; 10(5)×10-4; 13.8(5.1)×10-4; 1.5 ( 6 ) × 10-4. Correcting as above these values yield 8 4 ( 2 0 ) × 1 0 - 4 ; 3 4 ( 1 7 ) × 1 0 - 4 ; 4 8 ( 1 8 ) × 1 0 -4 and 15 (6) × 10-4; respectively, With the same method of averaging one finds (45 _+ 15) × 10 4. The ratio of the two contributions does not depend on the absolute normalization. From the four experiments the ratio R'(Tt+a2) =0.34_+0.07 R~(x+a£) 124
(9)
(8)
(I0)
Clearly, 98 and 7~ cannot both be ½. Eq. (10) is incompatible with the absence of initial state interaction at the 2a level. It needs to be mentioned that the result (9) depends completely on one experimental input, Eq. (7). In principle, (9) could also be determined from pp-) of)o, pOpO, coco. The branching ratio into pOco is ( 2 2 6 + 2 3 ) X 1 0 - 4 , pOpO ( 1 2 _ + 1 2 ) X 1 0 - 4 and coco ( 140 _+60) X 10 -4 [ 12,9 ]. The smallness of pOp° in lOp annihilation at rest is not understood in the frame of this analysis. Eq. ( 5 ) requires RS(p°p °) = R s (coco). A special symmetry could be responsible, but it should also be noted that this branching ratio is very difficult to extract from data due to the large number of interfering amplitudes contributing to the rt + 7t- n + x final state, and due to the fact that there are four possible rt+r~- pairs per event. The branching ratio into coco was determined from bubble chamber data with no detection of the two 7t°'s, and the authors consider the result as indicative only. A second relation between 78 and 7~ can be derived from the processes p p - ) x + a ~ - and rt+p -. The branching ratios can be written as
R~(rt+a2)=3C(1; 1, 1)7~f(rta2) ,
(lla)
R ~ ( T t + p - ) = 3 C ( 0 ; 1, 1 ) 7 ~ f ( ~ 9 ) ,
(lib)
Re,( n+a2 ) = C(0; 1, 1 )7~f(rca2) ,
(1 lc)
R~(x+9- )=C(1;1,1);],((~O) .
(lld)
This allows to calculate the double ratio (10): (9c) ( 9 d ) / ( 9 a ) (9b).
R~(rt+a£)R](rt+p R](~+ay)R~(x+p -) =0.74_+0.44.
)
7~(1-7~) 7~(1-7~) (12)
This ratio is compatible with unity. The small branching ratios R] ( x+p- ) and of R] (~ +aS- ) seem
VoLume244, number 1
PHYSICS LETTERS B
to have their c o m m o n origin in the smallness of the ratio C(1; 1, 1 ) / C ( 0 ; 1, 1 ) = 0 . 1 1 + 0 . 0 2 . A third relation can be derived from the branching ratios lop--,K*I( [11]. There are two dominant contributions:
(13a)
Rg(K*K)=C(O; ½, 1 )f(K*K)?,g = ( 8 . 6 - + 1 . 7 ) × 1 0 -4 ,
In ref. [ 13 ] it is argued that the amplitudes for production of two strange mesons are the same for these two processes, C(0; ~, ½) = C ( 1; ½, ½). Then (14)
yg
y~= (0.58+0.07). The ratio of isospin contributions can also be determined using 7~ + Y'I = 1. ~)~/yD = 1 1'7+0,39
(15a)
y]/~)
( 1
=f~ "/')+0,24 "~. ~z-,-- O, 18
C(1;0,1) =0.43+0.09. (13b)
1 -y~ - =0.95__+0.21 .
y~ = (0.46 + 0 . 0 7 ) ,
. . . . . 0.28,
R~ ( K ' K ) = 3C( 1; ½, ½)f(K*K)7~ =(24.8-+2.4)×10 -4,
12 July 1990
Eqs. (10), (12) and (14) can be used to determine 7~ and 75. Fig. 1 shows the allowed regions for ( 10): region a; for (12): region b; and for (14): region c. The AZ2= 1 contour plot is also shown. We find
c(0; 0, 0)
5b)
(15c)
Eq. 15c follows from (15a), (15b) and (6). Data on 10p annihilation from P states are still scarce. We denote branching ratios from the 3po, L2 states by R °'2'2', and from ~P~ b y / ? ~. The same notation is used for the 7's. Annihilation into nOn°, n°'q, fir1is possible from the states 3po and 3P2. Hence the data represent an unknown mixture of these states. The data of Adiels et al. [4] are used. The data by Chiba et al. [4] disagree with those of Adiels et al., but are statistically less significant. R°'2(n°q) 0.65_+0.14 R 0.2 ( nOn0) -2'2'(0.49_+0.11)
(16a)
10
The factor -~ accounts again for the sg component of the 7, the factor 2 is the usual statistical factor for two identical bosons. Similarly,
0.8
R o,2( nOrl )
1
R°'2(qrl) - (0.61-+0.25) 0.6
=3.2. (0.55 _+0.23).
(16b)
The statistical average is (0.50 -+ 0.10). With ( 3 ) and (15c) this yields
0./,
7°'2 = 1 . 1 6 - + 0 . 3 4 .
yoO,2
(17)
0
.
.
.
.
1.0
v~ Fig. 1. The allowed regions for the fractional isosinglet contribut and 7o. s Region a is defined by tions of the 3S~ and I So states, Yo eq. ( 8 ), region b by (10) and region c by ( 12 ). The AX2= 1 contour is also shown.
Annihilations from P states into n°f2 was studied by May et al. [ 14]. A contribution of 0.2l 1 (7) was determined to the n + n - n o final state. The branching ratio for n ÷ n - n ° is 4.5 ( 6 ) %. Correcting for unseen decay modes of the t"2, a branching ratio of ( 1 8 0 + 25 ) M 10 - 4 for 15p--,n°f2 from the 3P l s t a t e can be determined (100%=all annihilations from P states). Weidenauer et al. [5] find a contribution of 13.1 ( 6 . 9 ) × 10 -4 for lop-,n -+aJ: from P states in the 125
Volume 244, number 1
PHYSICS LETTERS B
tin d e c a y m o d e o f the a2. F o r o n e charge state o f the a2 but for all a2 decays this yields a b r a n c h i n g ratio o f (45 + 24 ) × 10 -4. A n n i h i l a t i o n into ha2 is allowed not only f r o m the 3pl; a n d t h e r e f o r e we can give only an inequality
71/V+>~9+5.
(18)
F o r the lPl state we h a v e b r a n c h i n g ratios for a n n i h i l a t i o n into p°n° and into 9%1. M a y el al [ 14 ] find a f r a c t i o n o f ( 2 6 . 9 + 4 . 6 ) % o f all P w a v e n + r t - n ° ann i h i l a t i o n s ( = (4.5 + 0.6 ) % ) , but for all three charge states o f the 9. H e n c e /~ l ( p ° n ° ) = ( 4 0 + 9) X 10 - 4 . Weidenauer lop--rip f r o m conservation f r o m the ~Pj ratio
et al. [5] find a b r a n c h i n g ratio for P states o f (9.4 + 5.3) × 10-4; C parity r e q u i r e s that the a n n i h i l a t i o n o c c u r r e d state o f the lop a t o m . H e n c e we find the
T h i s is c o m p a t i b l e with ( 1 5 c ) , h e n c e w i t h no initial state interactions. The ratio of the I = 1 and I = 0
components in the
IP 1 state
gives the range
~3,/~3o = 0 . 8 1 -+ 0 . 5 1 .
(19)
T a b l e 1 collects the results o f this analysis a n d theoretical c a l c u l a t i o n s [ 15 ]. A d m i t t e d l y , there are large u n c e r t a i n t i e s in the num e r i c a l results o f this analysis and, also, c o n c e p t u a l Table 1 7~/7o, theory versus experiment.
15patom
Theory a)
~So 3S1 ~P~ 3p~ 3Po 3P2
0.68 1.22 0.96 9.4 0.03 0.81
126
U s e f u l discussions with P r o f e s s o r J. K 6 r n e r a n d P r o f e s s o r J.M. R i c h a r d are kindly a c k n o w l e d g e d . I w o u l d like to t h a n k P r o f e s s o r L. T a u s c h e r for c o m m u n i c a t i o n o f d a t a p r i o r to p u b l i c a t i o n .
References 1165.
=0.24_+0.14
= ~. ( 0 . 3 5 - + 0 . 2 1 ) .
a) Ref. [151.
difficulties due to the i n c o n s i s t e n c y o f the a p p r o a c h w i t h the e x p e r i m e n t a l results on 10p-, p°p ° a n d ox0. T h e r e are also large e x p e r i m e n t a l d i s c r e p a n c i e s bet w e e n the results o f d i f f e r e n t e x p e r i m e n t s , in particular on n°n °, n°q, rlq, n°o~ and n-+a~ ( 1 3 2 0 ) . But the r a t h e r satisfactory a g r e e m e n t b e t w e e n the result pres e n t e d h e r e a n d the t h e o r e t i c a l calculations o f the initial state i n t e r a c t i o n s gives credibility to both, to o u r u n d e r s t a n d i n g o f the lop w a v e f u n c t i o n and to the ass u m p t i o n s w h i c h were m a d e in this analysis.
[1] See for a review Ch. Batty, Rep. Progr. Phys. 52 (1989)
/~ J ( pOq )
~f( p~xo)
12 July 1990
Experiment 0.68 0.95 0.82 9.7 0.03 0.43
0.8 1.26 0.61 6.5 0.053 0.98
0 •72 +0.24 -o.18 1•17 +0.39 -o.28 0.81 +0.51 >/9 _+5 1.16+_0.34 1.16+0.34
[ 2 ] M. Ziegler et al., Phys. Lett. B 206 ( 1988 ) 15 t; C.A. Baker et al., Nucl. Phys. A 483 ( 1988 ) 631; C.W.E. von Eijk et al., Nucl. Phys. A 486 (1988) 604; R. Bacher et al., Z. Phys. A 334 (1989) 93; [3] J.M. Richard and R. Sainio, Phys. Lett. B 110 (1982) 429; [ 4 ] L. Tauscher (PS 182 ), private communication; L. Adiels et al., Z. Phys. C 42 (1989) 49; R. Armenteros et al., CERN proposal P28 (1980), unpublished; M. Foster et al., Nucl. Phys. B 6 ( 1968 ) 107; B. May et al., Z. Phys. C 46 (1990) 191 ; M. Chiba et al., Phys. Rev. D 38 ( 1988 ) 2021, D 39 ( 1989 ) 329. [ 5 ] P. Weidenauer el al., 11and rI' production in pp annihilation at rest, submitted to Z. Phys. C. [6] U. Hartmann, E. Klempt and J. KOrner, Z. Phys. A 331 (1988) 217. [7] J. Vandermeulen, Z. Phys. C 37 (1988) 563. [8] Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lelt. B 204 (1988) 1. [9] J. Diaz et al., Nucl. Phys. B 16 (1970) 329. [ 10] P. Espigat et al., Nucl. Phys. B 36 (1972) 93. [ 11 ] A. Benini et al., Nuovo Cimento A 63 ( 1969 ) 1199. [ 12] R. Bizzarri et al., Nucl Phys. B 14 (1969) 169; M. Bloch et al., Nucl. Phys. B 23 (1970) 221. [13] E. Klempt, Z. Phys. A 331 (1988) 2tl. [ 14] B. May et al., Z. Phys. C 46 (1990) 203. [ 15] J. Carbonell et al., Z. Phys. A 334 (1989) 329.