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Effective quark interactions and protonium annihilation into two mesons
i
Physics Letters B 270 ( 1991 ) 11-17 North-Holland
PHYSICS LETTERS B
Effective quark interactions and protonium annihilation into two mesons L. M a n d r u p , A.S. J e n s e n , A. M i r a n d a a n d G.C. O a d e s Institute of Physics, University of Aarhus, DK-8000 Aarhus C, Denmark Received 29 May 1991; revised manuscript received 19 August 1991
Branching ratios for protonium annihilation into two mesons are computed using various effective quark interactions motivated by the 3po annihilation model and by the 3Sinone-gluon exchange model. None of the individual interactions alone are able to reproduce the observed values but several different linear combinations lead to agreement within a factor of about 5 or better.
1. Introduction Protonium annihilation into two mesons has recently been analysed in an almost model independent way in terms of threshold values o f helicity amplitudes [ 1 ] but without any systematic attempt to relate these threshold values to a specific model for annihilation. Detailed experimental data for Nlq annihilation into mesons are available both at rest [ 2 6] and in flight [ 7 ]. The annihilation probabilities for the different final states have been calculated using various simple quark models [8-1 1 ] where the annihilation operators are based on a few lowest order F e y n m a n amplitudes. In an attempt to improve the agreement with experimental data, combinations o f a few of the models have also been tried [ 10,11 ], but without dramatic improvements. Higher order effects, like initial and final state interactions [ 12 ] and relativistic corrections [ 13 ], have then been suggested as the reasons for the difficulties encountered. The complications of such corrections are numerous and implementation of a systematic procedure is extremely difficult. We shall, in this letter, adopt a different philosophy where the selected operator causing the annihilation is considered to be an effective operator accounting for all possible complications. The description is analogous to the formulation of scattering in terms o f the T-matrix and plane waves instead o f the correct interaction and the properly cal-
culated wavefunctions. Our choice o f an effective operator is, to a certain degree, rather arbitrary and alternatives can easily be found. We take linear combinations of operators which have previously been investigated and which are obtained from the basic Feynman amplitude by evaluating to leading order in p / m , where p is the relative quark m o m e n t u m and m is the quark constituent mass. These effective operators are used to compute appropriate threshold helicity amplitudes for the annihilation process into two mesons. Subsequently annihilation from the atomic states are calculated using the general formulation [ l ]. The relative weights o f the different operators in the linear combination are then adjusted to reproduce the observed branching ratios as well as possible.
2. Basic effective interactions The choice o f effective operators is inspired by the two simplest quark-antiquark annihilation and creation mechanisms, i.e. the 3po and 3St interactions. One or two quark-antiquark pairs may annihilate and the remaining pairs rearrange to form the two mesons. The quantum numbers at the vertices correspond to those o f the vacuum and o f a single gluon for the 3Po and 3S I interactions, respectively. The five simplest, different diagrams are shown schematically in fig. 1.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
11
Volume 270, number 1
PHYSICS LETTERS B
7 November 1991
ond row in fig. 1 ) we similarly find a sum of two terms which gives the two effective operators
391~"
V3:0-36" (1191"{-/14) (G2'5' "0"25) ,
Vl +V2
(3)
V4 = i0-36 X [0-11'" (Pl' --Pl ) -- 0-44' " (P4' --P4 ) ]
X (0-2,s, "0-25),
391 ~%~
whereas the second 3Si-annihilation diagram (third row in fig. 1 ) leads to the single effective operator
V 3 +V4
V5[ (0-2'5' "0-25)0-36 "~- (0-2'5'" ~36)0-25 ]" (~2'--/15' ).
391V' ~5 + #A
123654 Fig. 1. Quark model diagrams for plb annihilation into two mesons. The corresponding effective interactions are given in eqs. ( 1 )- (7). The quark numbering applies to all the diagrams but is only given explicitly on the V6 diagram.
Evaluating the 3Sl-rearrangement diagram (first row in fig. 1 ) to leading order in p/m gives a sum of two terms. We take over these two terms and, in the same spirit as many other constituent quark model applications, consider them as independent effective operators of the form VI =0-36" (/11 ~ P 2 + P 4 +/15) ,
( 1)
V2 = i0-36 X [ 0-11" (/iv --/11 ) + 0-22' " (/12' -/1192) --0-55' " (/15' --/15) --0-44' " (/14' --/14) ] ,
(2)
where 0-~kis the Pauli spin operator between the ith and kth quark spinor and/1k is the momentum of the kth quark or antiquark. In eqs. ( 1 ) and (2), as well as in the following effective operators, we have as usual omitted a hopefully obvious number of quark and antiquark creation and annihilation operators. Higher order p/m terms including what might be called recoil terms would, of course, lead to other candidates for effective operators but at this stage we do not want to complicate matters and so we restrict ourselves to the simple forms ofeqs. ( 1 ) and (2). For the first type of 3Sl-annihilation diagram (sec12
(5)
The 3po-rearrangement and annihilation diagrams (last row in fig. 1 ) lead respectively to the operators
1"5" 2"4" 3Po• ~v ~ + V6 +V7
(4)
V6 = 0-36 "P3,
(6)
V7 = (0"36 "/13) (0-25 "P2) (0-2'5' °/12' ) •
(7)
By construction, all but the last of these effective operators show a linear dependence on the quark momenta p while the 1/7 operator exhibits a p3 dependence. Even so, I/7 is retained since it is the first 3Po type operator which is capable of producing strange mesons. In this letter we shall only consider these effective operators and the space spanned by the formation of arbitrary linear combinations of them.
3. Numerical results
Measured branching ratios of good quality are available for annihilation from the 1S and 2P states [ 5,6 ] while older and less accurate measurements also exist for annihilation from an unspecified mixture of higher lying S-states [2-4 ]. All these experimental results are statistical averages over the fine and hyperfine structure [ 1 ]. These branching ratios can be calculated as described in ref. [ 1 ], via the threshold helicity amplitudes, the total annihilation widths and protonium atomic wavefunctions. We do not repeat the details here but we emphasize that use of the threshold helicity amplitudes is justified because the binding energy of the atomic protonium bound states is so small on the typical hadronic scale. Even though the CM momentum q of the final state mesons is quite large, the dependence on q is completely specified very close to the pp threshold as an appropriate power of the product (pq). The atomic wavefunctions depend on the optical model potential chosen to describe the
V o l u m e 270, n u m b e r 1
PHYSICS LETTERS B
initial proton-antiproton interaction. The general conclusions are fairly insensitive to the specific choice and we shall here confine ourselves to the D o v e n Richard model 2, see ref. [ 14 ]. The threshold helicity amplitudes are calculated for the various effective interactions using harmonic oscillator quark wavefunctions for the baryons and mesons #~. At this stage we use only the simplest constituent quark model wavefunctions and do not allow flavour symmetry breaking effects although these could be easily included. In the case of the to we assume no s~ component while in the case of the !1 and q' we use a mixing angle of - 20 °. In table 1 measured average branching ratios for annihilation into either two 0 - ground state mesons #1 W e use t h e s a m e h a r m o n i c o s c i l l a t o r b a r y o n a n d m e s o n w a v e f u n c t i o n s as in ref. [ 11 ].
7 N o v e m b e r 1991
or into 0 - and 1 - ground state mesons are compared to computed values for the 1S- and 2P-states for the 7 effective interactions and for the 23 final states for which experimental data are available. The strength of the interaction is, in each case, adjusted so as to minimize the quantity. .
S=
~
L=O,I
2
C [BR(ppL ~l)theo~y'~] i "t _ ,,m',,pvL--,,/exp.~[ln/~] ,
(8)
where the sum over i runs over those of the final states which are allowed for the particular interaction being considered. Strangeness production is only allowed for interactions V3, V4, 115and VT. These interactions generally predict significantly larger branching ratios for strange than for non-strange mesons, the only exception being the 3p0-model V7 in the case of annihilation from the S-state. Another common feature is
Table 1 T h e 1S a n d 2 P b r a n c h i n g r a t i o s c a l c u l a t e d f r o m the s i n g l e - q u a r k m o d e l i n t e r a c t i o n s . T h e first c o l u m n is the t w o - m e s o n c h a n n e l , the s e c o n d c o l u m n is t h e e x p e r i m e n t a l b r a n c h i n g r a t i o s o b t a i n e d f r o m t h e r e f e r e n c e s g i v e n in t h e t h i r d c o l u m n . T h e last s e v e n c o l u m n s c o n t a i n the ratios b e t w e e n c a l c u l a t e d a n d e x p e r i m e n t a l b r a n c h i n g r a t i o s f o r e a c h o f t h e s e v e n q u a r k m o d e l s . E a c h q u a r k m o d e l i n t e r a c t i o n is fitted to r e p r o d u c e the e x p e r i m e n t a l b r a n c h i n g r a t i o s f o r all the t w o - m e s o n c h a n n e l s except t h e 2 P b r a n c h i n g r a t i o s f o r n°l], n ° r f a n d p°rc°. MtM2
IS
n+~t -
Ref.
Model 1
2 1.7
K°I( ° p+np°n° p°rl pOq, con° ~rl K*+K K*°I( °
(3.2+0.3)×10 -3 (9.9-+1.0)×10 -4 (7.8_+0.6)×10 -4 (1.3_+0.3)×10 -2 (1.6_+0.2)×10 -2 (2.5_+0.7)×10 -3 (1.4_+0.3)×10 -3 (5.2_+0.5)×10 -3 (1.0_+0.1)×10 -2 ( 1 . 4 _ + 0 . 2 ) × 10 - 3 (1.2_+0.2)×10 -3
[3] [3] [2] [6] [6] [6] [6] [3] [4] [3] [3]
-
2.6 1.8 0.20 0.32 3.2 0.33
n+~ 7t°Tt° n°rl nOr( firI rlrl' K÷K K°I( ° p+npOlO p°rl p°rf
(4.8+0.5)×10 -3 (2.4+0.3)×10 -3 ( 1 . 3 _ + 0 . 3 ) × 10 - 3 (5.0_+ 1.9) X 10 - 3 (8.1--+3.1)×10 -4 <1.8×10 -3 ( 2 . 9 _ + 0 . 5 ) X 10 - 4 (8.8+2.3)×10 -5 (1.9_+0.8)×10 -3 ( 1 . 0 _ + 0 . 4 ) X 10 - 3 (2.4_+ 1.3) × 10 - 4 ~3×10 -4
[5] [5] [4] [3] [4] [4] [5] [5] [6] [6] [6] [6]
0.43 0.43 3.3)<10 -4 1.3X10 -4 1.7 5.4
-
1.1 0.0063 0.44 1.2
0.90 0.021 1.1 1.8
K+K
2P
B.R. ( e x p )
-
3
4 9 . 8 X 10 - 4 0.34 0.32 0.0068 0.067 0.023 0.049 0.14
0.12 17 17 2 . 9 X 10 - 4 3 . 6 × 10 - 5 36 22 208 738 4.3 0.053 4.1 1.9
5
6
3.4 6.3 8.0 2.1 1.7 0.045 0.015 0.033
0.015 1.1 1.0 0.15 1.8 0.90 1.1
1.8
-
-
0.59 0.51
-
4.1 4.1 7 . 8 X 10 - 3 9.5X10 -4 4.4 0.42 4.8 16 1.4 0.032 1.6 0.13
0.27 0.27 3.7×10 -5 4.7X10 -5 1.6 7.8 1.0 0.0056 0.51 2.2
10 8.8 0.67 0.059 2.2 7.9 0.026 -
7 0.011
-
0.035 0.13 0.096 0.060 0.41 1.86
0.058 0.17 1.9 0.20 11 11 1.0 0.16 11 0.68 37 100 5.6 0.23 15 4.2
13
Volume 270, number I
PHYSICS LETTERSB
that annihilation to the channels p°n°, n°ll and n°ll ' from the P-state are apparently almost always very severely underestimated. The selection rules of V~, V2 and V6 exclude, among other things, strange mesons in the final state but the results for the allowed channels are in general reproduced within an order of magnitude or better, apart from the general exceptions just mentioned. The interaction 113underestimates the S- and overestimates the P-state branching ratios by several orders of magnitude except for the strongly underestimated 2P-state annihilation to the n°ll, n°rf, p°n° channels. The interaction V4 over- or underestimates many of the branching ratios by up to several orders of magnitude. The interaction II5 reproduces the branching ratios to within a factor of 4 except for an underestimate by 1-2 orders of magnitude o f x + x -, p + x - from the S-state and for the usual exceptions of n°ll, n°ll ', p°x° from the P-state. The channel K°K ° is also overestimated by a factor of 16 for the P-state branching ratio. The interaction 116 does not allow annihilation from the S-state while in the case of annihilation from the P-state it reproduces the branching ratios to within a factor of 4 except for an overestimate of the ql l' channel by a factor of 7 and the usual underestimates of the ~°11, n°rf, pOnOchannels. The interaction V7 in general underestimates the S- and overestimates the P-state branching ratios by up to two orders of magnitude. None of the interactions in eqs. ( 1 ) - (7) are individually able to reproduce all the observed branching ratios. This is not so surprising in view of earlier published results and is the basic reason why other workers have tried to include higher order effects such as initial or final state interactions or relativistic corrections. As an alternative possibility we stay with the simple formulation but now try linear combinations of the effective operators with adjustable strength parameters adjusted so as again to minimize the quantity S given by eq. (8) where now the sum is over all of the 23 final states. Several solutions exist which produce comparable minima. Typical results for three different solutions are displayed in fig. 2 and the corresponding relative strengths are given in table 2. Reminimizing to the non-strange meson channels alone or to P-state data alone does not significantly change the results. We also try imposing the additional constraints that the relative normalizations for F~ and 1"2 14
7 November 1991
and for 1/'3and V4 correspond to those given by the original gluon-exchange Feynman amplitude (i.e. the relative normalization in table 2 for V~ is fixed as 2.76 times that for V2 and the relative normalization for V3 is fixed as 1.01 times that for 1/4). In this case the fit becomes marginally worse. The S-state results in the upper half of fig. 2 deviate by at most factor of 7 from the observed values. No systematic behaviour appears to be present. This may be related to the fact that the calculations are for annihilation from the n = 1 state while most of the data refer to annihilation from states of higher principal quantum number. In our experience, this may easily lead to an additional uncertainty of about a factor of 5. The 2P-state results in the lower part of fig. 2 are apparently more systematic. The results of the three different fits differ from each other by at most a factor of three. They reproduce all the measured branching ratios to within a factor of 10, with most of the discrepancies being less than a factor of three. The n + n - and n°n ° channels only receive contributions from isospin 0 and both agree with the measurements. The K + K - and K°I~° are both slightly overestimated. The channel p + n - is systematically overestimated whereas p°n° is slightly underestimated compared to the experimental values which are similar in size. This is due to the isospin 1 amplitude being predicted to be larger than the isospin 0 amplitude which alone the determines the p°n° branching ratio. The channels p°l1 and P%l' (where the experimental value is only an upper limit) with total spin 1 and isospin 1 are apparently systematically overestimated. This is in contrast to the slight underestimate of the channel n°ll and the strong underestimate of the channel ~°11' both of which have total spin 0 and isospin 1. The estimates for the remaining channels rpl and qll' with both spin 0 and isospin 0 are better but perhaps slightly too large. Each of the three different fits are of very similar quality. The individual interactions 'contribute with different relative weights which are ~iven in table 2 in units of the normalization used in table 1. Thus values above unity in table 2 show a preference for the corresponding interaction. The two one-gluon interactions V~ and V2, enter both sets B and C with opposite signs and their relation is therefore quite different from that expected from the original Feyn-
Volume 270, number 1
PHYSICS LETTERS B
10
I
I
I
I
I
I
I
I
7 November 1991 I
I
I
I
S- state
o
o-A []
n. c o []
o
o
o
,
I
rl
I
,
O
O
+ I
t, A
O
O
A
O
~.
O
A D
KoI~0
x+rt-
0.1
I
I
K+K -
I
pOrtO I
p*TC-
I
O
p0-q, I
p0-q
10
I
K"°~,°
to'q (.0/0
I
I
K"÷ K-
I
A
P- stote
o
o o
o
[]
[]
+
I O
A A
r~+rt0.1
I
A
n
pOTtO
KORO I
K+K"
I
I
p+rt-
pO.qJ
I
I
pO-q
I
TtO-q I
n:°Tt °
I
~l'q ?
rt°'rl '
I
I
"qrl'
Fig. 2. The ratios between calculated and experimental branching ratios for various sets of normalization constants (see table 2) as function of the final state two-meson channel. The initial states are the protonium 1S-state (above) and the 2P-state (below) defined by use of the Dover-Richard interaction number 2. The lines represent the experimental error bars.
m a n a m p l i t u d e (see a b o v e ) . This is not the case for set A where the relative n o r m a l i z a t i o n o f Vl to II2 is closer to the F e y n m a n a m p l i t u d e value o f 2.76. II3 a n d 114 are suppressed below their " n a t u r a l " level in all three sets a n d only have a relative n o r m a l i z a t i o n close to the F e y n m a n a m p l i t u d e value o f 1.01 in set C, the relative sign being negative in sets A a n d B. All
the r e m a i n i n g interactions contribute significantly in set A, which indicates that there must be a fair a m o u n t o f cancellation. In both sets B a n d C only two or three interactions are significant, one o f these being the strongly selective 3Po rearrangement interaction. Set A is somewhat similar to the 3Sl situation a d v o c a t e d by K o h n o and Weise [ 10 ], although with significant 15
Volume 270, number 1
PHYSICS LETTERS B
Table 2 Normalization constants in units of those used in table 1 for the seven quark model interactions in the fits of fig. 2. Set
A B C
Model 1
2
3
4
5
6
7
2.7 1.8 5.4
1.2 -0.57 -0.87
0.30 0.26 0.16
-0.49 -0.015 0.11
1.3 0.63 1.2
1.0 1.4 4.5
- 1.2 - 1.0 -0.86
Table 3 Predictions for annihilation to channels which are so far unmeasured. The column labeled B.R. gives the branching ratios together with an uncertainty indicating the spread due to the choice of various equallyacceptablelinear combinations of effectiveoperators, The columns labeled Min. and Max. indicate the lowest and highest branching ratios obtained usingother atomic protonium wavefunctions.All entries are in units of 10-3. Initial state
Final state
B.R.
Min.
Max.
1s 2P 2P 2P 2P 2P
toq' ton° co~ o)rl' K*+KK*°l~°
5 _+4 9 +3 1.4_+0.5 0.8_+0.5 4.7 + 1.0 1.9 + 0.4
0.1 1 0.3 0.2 0.8 0.5
16 22 6 5 6 3
additional contributions, while sets B a n d C are somewhat closer to the 3Po situation advocated by other groups [8,9]. However, in all three cases we also find significant contributions from the remaining interactions so we cannot conclude that one particular model is totally d o m i n a n t . The various solutions can also be used to predict branching ratios which are so far unmeasured. As examples, in table 3 we show predictions for annihilation from the 1S state to o)11'and from the 2P state to ton°, toq, c0~q',K*+K - and K*°K°. We try to give some indication of the uncertainty due to the spread of predictions for the different sets and also due to the spread produced by changing the initial state optical model from DR2 to DR1 [14] or to that of K o h n o a n d Weise [ 10 ].
4. C o n c l u s i o n s
The threshold helicity amplitudes for a n n i h i l a t i o n of N/q into two mesons are calculated using seven 16
7 November 1991
different effective interactions. The branching ratios from the 2P a n d 1S states are then computed using an optical model for the initial state. N o n e of the individual effective operators, based on previously suggested interactions or pieces of such interactions, are alone able to reproduce the measured branching ratios. Either selection rules prohibit observed processes or the results deviate by several orders of magnitude. Properly selected linear c o m b i n a t i o n s significantly improve the comparison and the branching ratios are then reproduced within a factor of about 5 except for P-state a n n i h i l a t i o n to the p0~ a n d n°rl ' final states. Many c o m b i n a t i o n s are almost equally acceptable a n d several interactions are always needed. All reasonable linear combinations seem to need significant contributions from both the 3po a n n i h i l a t i o n model and the 3St one-gluon exchange model, whereas the artificial 3St a n n i h i l a t i o n diagrams F3 + F4 are suppressed although still contributing to several channels. The comparison with measurements is significantly improved by such linear c o m b i n a t i o n s but the agreement is still only semiquantitative indicating that important aspects still ar ,~ missing or too crudely approximated. This could be the need for other effective operators, inadequacy of the optical model (for example the treatment of the tensor interaction) or the need to go beyond the framework of the constituent quark model. In any case, all the above conclusions are based on the extreme low-energy behaviour relevant for annihilation from atomic b o u n d states. As the energy dependence may be significant these results are not directly applicable to the same in-flight a n n i h i l a t i o n processes a n d for this reason comparison with such inflight calculations is not possible.
References
[ 1] L. Mandrup, A.S. Jensen, A. Miranda, G.C. Oades and J. Carbonell, Nucl. Phys. A 512 (1990) 591. [ 2 ] R. Armenteros and B. French, in: High energyphysics,Vol. 4, ed. E.H.S. Burhop (Academic Press, New York, 1969). [ 3 ] S. Mundigl, M. Vieente Vacasand W. Weise,Z. Phys. A 338 (1991) 103, P. Bliim, G. Btiche and H. Koch, Antiproton-proton annihilation at rest, Kernforschungszentrum Karlsruhe preprint (November 1988). [4 ] L. Adiels et al., Z. Phys. C 42 ( 1989) 49.
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[ 5 ] M. Doser et al., Nucl. Phys. A 486 ( 1988 ) 493, Phys. Lett. B 215 (1988) 792. [ 6 ] Asterix Collab., Z. Phys.C 46 (1990) 191,203; C 47 (1990) 353. [ 7 ] U. Gastaldi, R. Klapish, J.-M. Richard and J. Tran Thanh Van, eds., Physics with antiprotons at LEAR in the ACOL era (Editions Fronti~res, Gif-sur-Yvette, 1985); R.S. Dulude et al., Phys. Lett. B 79 (1978) 335. [8] M. Maruyama, S. Furui and A. Faessler, Nucl. Phys. A 472 (1987) 643. [9] S. Furui, G. Strobel, A. Faessler and R. Vinh Mau, Nucl. Phys. A 516 (1990) 643.
7 November 1991
[ 10 ] M. Kohno and W. Weise, Nucl. Phys. A 454 (1986) 429. [ 11 ] M. Maruyama and T. Ueda, Prog. Theor. Phys. 78 (1987) 841. [ 12 ] A.M. Green and G.Q. Liu, Nucl. Phys. A 486 ( 1988 ) 581. [ 13] M. Maruyama, T. Gutsche, A. Faessler and G.L. Strobe1, Relativistic effects in proton-antiproton annihilation into two mesons, Tiibingen preprint (1990). [14] J.M. Richard and M.E. Sainio, Phys. Lett. B 110 (1982) 349.
17
Physics Letters B 270 ( 1991 ) 11-17 North-Holland
PHYSICS LETTERS B
Effective quark interactions and protonium annihilation into two mesons L. M a n d r u p , A.S. J e n s e n , A. M i r a n d a a n d G.C. O a d e s Institute of Physics, University of Aarhus, DK-8000 Aarhus C, Denmark Received 29 May 1991; revised manuscript received 19 August 1991
Branching ratios for protonium annihilation into two mesons are computed using various effective quark interactions motivated by the 3po annihilation model and by the 3Sinone-gluon exchange model. None of the individual interactions alone are able to reproduce the observed values but several different linear combinations lead to agreement within a factor of about 5 or better.
1. Introduction Protonium annihilation into two mesons has recently been analysed in an almost model independent way in terms of threshold values o f helicity amplitudes [ 1 ] but without any systematic attempt to relate these threshold values to a specific model for annihilation. Detailed experimental data for Nlq annihilation into mesons are available both at rest [ 2 6] and in flight [ 7 ]. The annihilation probabilities for the different final states have been calculated using various simple quark models [8-1 1 ] where the annihilation operators are based on a few lowest order F e y n m a n amplitudes. In an attempt to improve the agreement with experimental data, combinations o f a few of the models have also been tried [ 10,11 ], but without dramatic improvements. Higher order effects, like initial and final state interactions [ 12 ] and relativistic corrections [ 13 ], have then been suggested as the reasons for the difficulties encountered. The complications of such corrections are numerous and implementation of a systematic procedure is extremely difficult. We shall, in this letter, adopt a different philosophy where the selected operator causing the annihilation is considered to be an effective operator accounting for all possible complications. The description is analogous to the formulation of scattering in terms o f the T-matrix and plane waves instead o f the correct interaction and the properly cal-
culated wavefunctions. Our choice o f an effective operator is, to a certain degree, rather arbitrary and alternatives can easily be found. We take linear combinations of operators which have previously been investigated and which are obtained from the basic Feynman amplitude by evaluating to leading order in p / m , where p is the relative quark m o m e n t u m and m is the quark constituent mass. These effective operators are used to compute appropriate threshold helicity amplitudes for the annihilation process into two mesons. Subsequently annihilation from the atomic states are calculated using the general formulation [ l ]. The relative weights o f the different operators in the linear combination are then adjusted to reproduce the observed branching ratios as well as possible.
2. Basic effective interactions The choice o f effective operators is inspired by the two simplest quark-antiquark annihilation and creation mechanisms, i.e. the 3po and 3St interactions. One or two quark-antiquark pairs may annihilate and the remaining pairs rearrange to form the two mesons. The quantum numbers at the vertices correspond to those o f the vacuum and o f a single gluon for the 3Po and 3S I interactions, respectively. The five simplest, different diagrams are shown schematically in fig. 1.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
11
Volume 270, number 1
PHYSICS LETTERS B
7 November 1991
ond row in fig. 1 ) we similarly find a sum of two terms which gives the two effective operators
391~"
V3:0-36" (1191"{-/14) (G2'5' "0"25) ,
Vl +V2
(3)
V4 = i0-36 X [0-11'" (Pl' --Pl ) -- 0-44' " (P4' --P4 ) ]
X (0-2,s, "0-25),
391 ~%~
whereas the second 3Si-annihilation diagram (third row in fig. 1 ) leads to the single effective operator
V 3 +V4
V5[ (0-2'5' "0-25)0-36 "~- (0-2'5'" ~36)0-25 ]" (~2'--/15' ).
391V' ~5 + #A
123654 Fig. 1. Quark model diagrams for plb annihilation into two mesons. The corresponding effective interactions are given in eqs. ( 1 )- (7). The quark numbering applies to all the diagrams but is only given explicitly on the V6 diagram.
Evaluating the 3Sl-rearrangement diagram (first row in fig. 1 ) to leading order in p/m gives a sum of two terms. We take over these two terms and, in the same spirit as many other constituent quark model applications, consider them as independent effective operators of the form VI =0-36" (/11 ~ P 2 + P 4 +/15) ,
( 1)
V2 = i0-36 X [ 0-11" (/iv --/11 ) + 0-22' " (/12' -/1192) --0-55' " (/15' --/15) --0-44' " (/14' --/14) ] ,
(2)
where 0-~kis the Pauli spin operator between the ith and kth quark spinor and/1k is the momentum of the kth quark or antiquark. In eqs. ( 1 ) and (2), as well as in the following effective operators, we have as usual omitted a hopefully obvious number of quark and antiquark creation and annihilation operators. Higher order p/m terms including what might be called recoil terms would, of course, lead to other candidates for effective operators but at this stage we do not want to complicate matters and so we restrict ourselves to the simple forms ofeqs. ( 1 ) and (2). For the first type of 3Sl-annihilation diagram (sec12
(5)
The 3po-rearrangement and annihilation diagrams (last row in fig. 1 ) lead respectively to the operators
1"5" 2"4" 3Po• ~v ~ + V6 +V7
(4)
V6 = 0-36 "P3,
(6)
V7 = (0"36 "/13) (0-25 "P2) (0-2'5' °/12' ) •
(7)
By construction, all but the last of these effective operators show a linear dependence on the quark momenta p while the 1/7 operator exhibits a p3 dependence. Even so, I/7 is retained since it is the first 3Po type operator which is capable of producing strange mesons. In this letter we shall only consider these effective operators and the space spanned by the formation of arbitrary linear combinations of them.
3. Numerical results
Measured branching ratios of good quality are available for annihilation from the 1S and 2P states [ 5,6 ] while older and less accurate measurements also exist for annihilation from an unspecified mixture of higher lying S-states [2-4 ]. All these experimental results are statistical averages over the fine and hyperfine structure [ 1 ]. These branching ratios can be calculated as described in ref. [ 1 ], via the threshold helicity amplitudes, the total annihilation widths and protonium atomic wavefunctions. We do not repeat the details here but we emphasize that use of the threshold helicity amplitudes is justified because the binding energy of the atomic protonium bound states is so small on the typical hadronic scale. Even though the CM momentum q of the final state mesons is quite large, the dependence on q is completely specified very close to the pp threshold as an appropriate power of the product (pq). The atomic wavefunctions depend on the optical model potential chosen to describe the
V o l u m e 270, n u m b e r 1
PHYSICS LETTERS B
initial proton-antiproton interaction. The general conclusions are fairly insensitive to the specific choice and we shall here confine ourselves to the D o v e n Richard model 2, see ref. [ 14 ]. The threshold helicity amplitudes are calculated for the various effective interactions using harmonic oscillator quark wavefunctions for the baryons and mesons #~. At this stage we use only the simplest constituent quark model wavefunctions and do not allow flavour symmetry breaking effects although these could be easily included. In the case of the to we assume no s~ component while in the case of the !1 and q' we use a mixing angle of - 20 °. In table 1 measured average branching ratios for annihilation into either two 0 - ground state mesons #1 W e use t h e s a m e h a r m o n i c o s c i l l a t o r b a r y o n a n d m e s o n w a v e f u n c t i o n s as in ref. [ 11 ].
7 N o v e m b e r 1991
or into 0 - and 1 - ground state mesons are compared to computed values for the 1S- and 2P-states for the 7 effective interactions and for the 23 final states for which experimental data are available. The strength of the interaction is, in each case, adjusted so as to minimize the quantity. .
S=
~
L=O,I
2
C [BR(ppL ~l)theo~y'~] i "t _ ,,m',,pvL--,,/exp.~[ln/~] ,
(8)
where the sum over i runs over those of the final states which are allowed for the particular interaction being considered. Strangeness production is only allowed for interactions V3, V4, 115and VT. These interactions generally predict significantly larger branching ratios for strange than for non-strange mesons, the only exception being the 3p0-model V7 in the case of annihilation from the S-state. Another common feature is
Table 1 T h e 1S a n d 2 P b r a n c h i n g r a t i o s c a l c u l a t e d f r o m the s i n g l e - q u a r k m o d e l i n t e r a c t i o n s . T h e first c o l u m n is the t w o - m e s o n c h a n n e l , the s e c o n d c o l u m n is t h e e x p e r i m e n t a l b r a n c h i n g r a t i o s o b t a i n e d f r o m t h e r e f e r e n c e s g i v e n in t h e t h i r d c o l u m n . T h e last s e v e n c o l u m n s c o n t a i n the ratios b e t w e e n c a l c u l a t e d a n d e x p e r i m e n t a l b r a n c h i n g r a t i o s f o r e a c h o f t h e s e v e n q u a r k m o d e l s . E a c h q u a r k m o d e l i n t e r a c t i o n is fitted to r e p r o d u c e the e x p e r i m e n t a l b r a n c h i n g r a t i o s f o r all the t w o - m e s o n c h a n n e l s except t h e 2 P b r a n c h i n g r a t i o s f o r n°l], n ° r f a n d p°rc°. MtM2
IS
n+~t -
Ref.
Model 1
2 1.7
K°I( ° p+np°n° p°rl pOq, con° ~rl K*+K K*°I( °
(3.2+0.3)×10 -3 (9.9-+1.0)×10 -4 (7.8_+0.6)×10 -4 (1.3_+0.3)×10 -2 (1.6_+0.2)×10 -2 (2.5_+0.7)×10 -3 (1.4_+0.3)×10 -3 (5.2_+0.5)×10 -3 (1.0_+0.1)×10 -2 ( 1 . 4 _ + 0 . 2 ) × 10 - 3 (1.2_+0.2)×10 -3
[3] [3] [2] [6] [6] [6] [6] [3] [4] [3] [3]
-
2.6 1.8 0.20 0.32 3.2 0.33
n+~ 7t°Tt° n°rl nOr( firI rlrl' K÷K K°I( ° p+npOlO p°rl p°rf
(4.8+0.5)×10 -3 (2.4+0.3)×10 -3 ( 1 . 3 _ + 0 . 3 ) × 10 - 3 (5.0_+ 1.9) X 10 - 3 (8.1--+3.1)×10 -4 <1.8×10 -3 ( 2 . 9 _ + 0 . 5 ) X 10 - 4 (8.8+2.3)×10 -5 (1.9_+0.8)×10 -3 ( 1 . 0 _ + 0 . 4 ) X 10 - 3 (2.4_+ 1.3) × 10 - 4 ~3×10 -4
[5] [5] [4] [3] [4] [4] [5] [5] [6] [6] [6] [6]
0.43 0.43 3.3)<10 -4 1.3X10 -4 1.7 5.4
-
1.1 0.0063 0.44 1.2
0.90 0.021 1.1 1.8
K+K
2P
B.R. ( e x p )
-
3
4 9 . 8 X 10 - 4 0.34 0.32 0.0068 0.067 0.023 0.049 0.14
0.12 17 17 2 . 9 X 10 - 4 3 . 6 × 10 - 5 36 22 208 738 4.3 0.053 4.1 1.9
5
6
3.4 6.3 8.0 2.1 1.7 0.045 0.015 0.033
0.015 1.1 1.0 0.15 1.8 0.90 1.1
1.8
-
-
0.59 0.51
-
4.1 4.1 7 . 8 X 10 - 3 9.5X10 -4 4.4 0.42 4.8 16 1.4 0.032 1.6 0.13
0.27 0.27 3.7×10 -5 4.7X10 -5 1.6 7.8 1.0 0.0056 0.51 2.2
10 8.8 0.67 0.059 2.2 7.9 0.026 -
7 0.011
-
0.035 0.13 0.096 0.060 0.41 1.86
0.058 0.17 1.9 0.20 11 11 1.0 0.16 11 0.68 37 100 5.6 0.23 15 4.2
13
Volume 270, number I
PHYSICS LETTERSB
that annihilation to the channels p°n°, n°ll and n°ll ' from the P-state are apparently almost always very severely underestimated. The selection rules of V~, V2 and V6 exclude, among other things, strange mesons in the final state but the results for the allowed channels are in general reproduced within an order of magnitude or better, apart from the general exceptions just mentioned. The interaction 113underestimates the S- and overestimates the P-state branching ratios by several orders of magnitude except for the strongly underestimated 2P-state annihilation to the n°ll, n°rf, p°n° channels. The interaction V4 over- or underestimates many of the branching ratios by up to several orders of magnitude. The interaction II5 reproduces the branching ratios to within a factor of 4 except for an underestimate by 1-2 orders of magnitude o f x + x -, p + x - from the S-state and for the usual exceptions of n°ll, n°ll ', p°x° from the P-state. The channel K°K ° is also overestimated by a factor of 16 for the P-state branching ratio. The interaction 116 does not allow annihilation from the S-state while in the case of annihilation from the P-state it reproduces the branching ratios to within a factor of 4 except for an overestimate of the ql l' channel by a factor of 7 and the usual underestimates of the ~°11, n°rf, pOnOchannels. The interaction V7 in general underestimates the S- and overestimates the P-state branching ratios by up to two orders of magnitude. None of the interactions in eqs. ( 1 ) - (7) are individually able to reproduce all the observed branching ratios. This is not so surprising in view of earlier published results and is the basic reason why other workers have tried to include higher order effects such as initial or final state interactions or relativistic corrections. As an alternative possibility we stay with the simple formulation but now try linear combinations of the effective operators with adjustable strength parameters adjusted so as again to minimize the quantity S given by eq. (8) where now the sum is over all of the 23 final states. Several solutions exist which produce comparable minima. Typical results for three different solutions are displayed in fig. 2 and the corresponding relative strengths are given in table 2. Reminimizing to the non-strange meson channels alone or to P-state data alone does not significantly change the results. We also try imposing the additional constraints that the relative normalizations for F~ and 1"2 14
7 November 1991
and for 1/'3and V4 correspond to those given by the original gluon-exchange Feynman amplitude (i.e. the relative normalization in table 2 for V~ is fixed as 2.76 times that for V2 and the relative normalization for V3 is fixed as 1.01 times that for 1/4). In this case the fit becomes marginally worse. The S-state results in the upper half of fig. 2 deviate by at most factor of 7 from the observed values. No systematic behaviour appears to be present. This may be related to the fact that the calculations are for annihilation from the n = 1 state while most of the data refer to annihilation from states of higher principal quantum number. In our experience, this may easily lead to an additional uncertainty of about a factor of 5. The 2P-state results in the lower part of fig. 2 are apparently more systematic. The results of the three different fits differ from each other by at most a factor of three. They reproduce all the measured branching ratios to within a factor of 10, with most of the discrepancies being less than a factor of three. The n + n - and n°n ° channels only receive contributions from isospin 0 and both agree with the measurements. The K + K - and K°I~° are both slightly overestimated. The channel p + n - is systematically overestimated whereas p°n° is slightly underestimated compared to the experimental values which are similar in size. This is due to the isospin 1 amplitude being predicted to be larger than the isospin 0 amplitude which alone the determines the p°n° branching ratio. The channels p°l1 and P%l' (where the experimental value is only an upper limit) with total spin 1 and isospin 1 are apparently systematically overestimated. This is in contrast to the slight underestimate of the channel n°ll and the strong underestimate of the channel ~°11' both of which have total spin 0 and isospin 1. The estimates for the remaining channels rpl and qll' with both spin 0 and isospin 0 are better but perhaps slightly too large. Each of the three different fits are of very similar quality. The individual interactions 'contribute with different relative weights which are ~iven in table 2 in units of the normalization used in table 1. Thus values above unity in table 2 show a preference for the corresponding interaction. The two one-gluon interactions V~ and V2, enter both sets B and C with opposite signs and their relation is therefore quite different from that expected from the original Feyn-
Volume 270, number 1
PHYSICS LETTERS B
10
I
I
I
I
I
I
I
I
7 November 1991 I
I
I
I
S- state
o
o-A []
n. c o []
o
o
o
,
I
rl
I
,
O
O
+ I
t, A
O
O
A
O
~.
O
A D
KoI~0
x+rt-
0.1
I
I
K+K -
I
pOrtO I
p*TC-
I
O
p0-q, I
p0-q
10
I
K"°~,°
to'q (.0/0
I
I
K"÷ K-
I
A
P- stote
o
o o
o
[]
[]
+
I O
A A
r~+rt0.1
I
A
n
pOTtO
KORO I
K+K"
I
I
p+rt-
pO.qJ
I
I
pO-q
I
TtO-q I
n:°Tt °
I
~l'q ?
rt°'rl '
I
I
"qrl'
Fig. 2. The ratios between calculated and experimental branching ratios for various sets of normalization constants (see table 2) as function of the final state two-meson channel. The initial states are the protonium 1S-state (above) and the 2P-state (below) defined by use of the Dover-Richard interaction number 2. The lines represent the experimental error bars.
m a n a m p l i t u d e (see a b o v e ) . This is not the case for set A where the relative n o r m a l i z a t i o n o f Vl to II2 is closer to the F e y n m a n a m p l i t u d e value o f 2.76. II3 a n d 114 are suppressed below their " n a t u r a l " level in all three sets a n d only have a relative n o r m a l i z a t i o n close to the F e y n m a n a m p l i t u d e value o f 1.01 in set C, the relative sign being negative in sets A a n d B. All
the r e m a i n i n g interactions contribute significantly in set A, which indicates that there must be a fair a m o u n t o f cancellation. In both sets B a n d C only two or three interactions are significant, one o f these being the strongly selective 3Po rearrangement interaction. Set A is somewhat similar to the 3Sl situation a d v o c a t e d by K o h n o and Weise [ 10 ], although with significant 15
Volume 270, number 1
PHYSICS LETTERS B
Table 2 Normalization constants in units of those used in table 1 for the seven quark model interactions in the fits of fig. 2. Set
A B C
Model 1
2
3
4
5
6
7
2.7 1.8 5.4
1.2 -0.57 -0.87
0.30 0.26 0.16
-0.49 -0.015 0.11
1.3 0.63 1.2
1.0 1.4 4.5
- 1.2 - 1.0 -0.86
Table 3 Predictions for annihilation to channels which are so far unmeasured. The column labeled B.R. gives the branching ratios together with an uncertainty indicating the spread due to the choice of various equallyacceptablelinear combinations of effectiveoperators, The columns labeled Min. and Max. indicate the lowest and highest branching ratios obtained usingother atomic protonium wavefunctions.All entries are in units of 10-3. Initial state
Final state
B.R.
Min.
Max.
1s 2P 2P 2P 2P 2P
toq' ton° co~ o)rl' K*+KK*°l~°
5 _+4 9 +3 1.4_+0.5 0.8_+0.5 4.7 + 1.0 1.9 + 0.4
0.1 1 0.3 0.2 0.8 0.5
16 22 6 5 6 3
additional contributions, while sets B a n d C are somewhat closer to the 3Po situation advocated by other groups [8,9]. However, in all three cases we also find significant contributions from the remaining interactions so we cannot conclude that one particular model is totally d o m i n a n t . The various solutions can also be used to predict branching ratios which are so far unmeasured. As examples, in table 3 we show predictions for annihilation from the 1S state to o)11'and from the 2P state to ton°, toq, c0~q',K*+K - and K*°K°. We try to give some indication of the uncertainty due to the spread of predictions for the different sets and also due to the spread produced by changing the initial state optical model from DR2 to DR1 [14] or to that of K o h n o a n d Weise [ 10 ].
4. C o n c l u s i o n s
The threshold helicity amplitudes for a n n i h i l a t i o n of N/q into two mesons are calculated using seven 16
7 November 1991
different effective interactions. The branching ratios from the 2P a n d 1S states are then computed using an optical model for the initial state. N o n e of the individual effective operators, based on previously suggested interactions or pieces of such interactions, are alone able to reproduce the measured branching ratios. Either selection rules prohibit observed processes or the results deviate by several orders of magnitude. Properly selected linear c o m b i n a t i o n s significantly improve the comparison and the branching ratios are then reproduced within a factor of about 5 except for P-state a n n i h i l a t i o n to the p0~ a n d n°rl ' final states. Many c o m b i n a t i o n s are almost equally acceptable a n d several interactions are always needed. All reasonable linear combinations seem to need significant contributions from both the 3po a n n i h i l a t i o n model and the 3St one-gluon exchange model, whereas the artificial 3St a n n i h i l a t i o n diagrams F3 + F4 are suppressed although still contributing to several channels. The comparison with measurements is significantly improved by such linear c o m b i n a t i o n s but the agreement is still only semiquantitative indicating that important aspects still ar ,~ missing or too crudely approximated. This could be the need for other effective operators, inadequacy of the optical model (for example the treatment of the tensor interaction) or the need to go beyond the framework of the constituent quark model. In any case, all the above conclusions are based on the extreme low-energy behaviour relevant for annihilation from atomic b o u n d states. As the energy dependence may be significant these results are not directly applicable to the same in-flight a n n i h i l a t i o n processes a n d for this reason comparison with such inflight calculations is not possible.
References
[ 1] L. Mandrup, A.S. Jensen, A. Miranda, G.C. Oades and J. Carbonell, Nucl. Phys. A 512 (1990) 591. [ 2 ] R. Armenteros and B. French, in: High energyphysics,Vol. 4, ed. E.H.S. Burhop (Academic Press, New York, 1969). [ 3 ] S. Mundigl, M. Vieente Vacasand W. Weise,Z. Phys. A 338 (1991) 103, P. Bliim, G. Btiche and H. Koch, Antiproton-proton annihilation at rest, Kernforschungszentrum Karlsruhe preprint (November 1988). [4 ] L. Adiels et al., Z. Phys. C 42 ( 1989) 49.
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[ 5 ] M. Doser et al., Nucl. Phys. A 486 ( 1988 ) 493, Phys. Lett. B 215 (1988) 792. [ 6 ] Asterix Collab., Z. Phys.C 46 (1990) 191,203; C 47 (1990) 353. [ 7 ] U. Gastaldi, R. Klapish, J.-M. Richard and J. Tran Thanh Van, eds., Physics with antiprotons at LEAR in the ACOL era (Editions Fronti~res, Gif-sur-Yvette, 1985); R.S. Dulude et al., Phys. Lett. B 79 (1978) 335. [8] M. Maruyama, S. Furui and A. Faessler, Nucl. Phys. A 472 (1987) 643. [9] S. Furui, G. Strobel, A. Faessler and R. Vinh Mau, Nucl. Phys. A 516 (1990) 643.
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[ 10 ] M. Kohno and W. Weise, Nucl. Phys. A 454 (1986) 429. [ 11 ] M. Maruyama and T. Ueda, Prog. Theor. Phys. 78 (1987) 841. [ 12 ] A.M. Green and G.Q. Liu, Nucl. Phys. A 486 ( 1988 ) 581. [ 13] M. Maruyama, T. Gutsche, A. Faessler and G.L. Strobe1, Relativistic effects in proton-antiproton annihilation into two mesons, Tiibingen preprint (1990). [14] J.M. Richard and M.E. Sainio, Phys. Lett. B 110 (1982) 349.
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