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A new dynamic selection rule for pp into two mesons
Volume 157B, number 4
PHYSICS LEq~TERS
18 July 1985
A N E W D Y N A M I C S E L E C T I O N R U L E F O R p~ I N T O T W O M E S O N S J.A. N I S K A N E N t and F. M Y H R E R 2 Research Institute for Theoretical Physics, University of Helsinki, Helsinki, Finland
Received 28 January 1985; revised manuscript received 15 April 1985
We show that quark-antiquark annihilation into gluons gives a suppression of p~ annihilating into 7r±p:~ from p~ 1So and 3p states at threshold.
The observed branching ratios of stopped ~ in hydrogen going to two mesons provide us with a clear preference for a specific annihilation mechanism as we will show below. The annihilation process p~ ~ rr-+p~ comes dominantly from the isospin zero p~ states 3S 1 and 1P1, whereas the isospin one pO states 1S 0 or 3P0,1, 2 seem to be strongly suppressed at least by an order of magnitude [ 1] although these states are not forbidden by any symmetry arguments. Recently several dynamical annihilation models have been proposed. One model is based on the quark rearrangement model of Rubinstein and Stern [2] and is used by Maruyama and Ueda [3], Green and Niskanen [4], Faessler et al. [5] to describe p~ annihilation at low energy into three mesons. Another is the old 3P 0 model where a q~l pair annihilates into vacuum and is used by e.g. Dover and Fishbane [6] and Green et al. [7]. In this model the initial state 33p 1 is allowed [4]. Also topological models have been used to describe annihilations [8]. A third model uses specific quark gluon degrees of freedom and was discussed early by the Seattle group [9] and later by Faessler et al. [10] and recently expanded by Tegen et al. [ 11 ], Weise et al. [12] and Ando [13]. The two latter groups investigated multi-gluon contributions to the annihilation process and we will specifically use this dynamical annihilation model [11,12] to study the suppression of On leave of absence from the Department of Theoretical Physics, University of Helsinki, Helsinki, Finland. 2 Permanent address: Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the p~ ~ p+rr- from the isospin one 1 SO p~ state and we will show that the dynamic process qC:l~ gluons does suppress this annihilation channel and explains the annihilation data. This kind of model has also been applied by Maruyama and Ueda [14] and by Kohno and Weise [ 12] for a study of branching ratios. Indication in ref. [14] seems to be - although the authors do not bring this up explicitly - that the p~ prt is suppressed for the 1S 0 initial state. This one can see from the fact that at rest they obtain rather nearly (within 20%) equal branching ratios for p - n +, p0rr0 and p+Ir-, Since the p0rr0 channel is forbidden from the 1S 0 states the difference in the branching ratios should be a measure of the 1 SO state's contribution to p±Ir; . The qq pair annihilating into one gluon in fig. 1 have to be in an 3S 1 state, so as an example let us choose the q and ~1to have both S z = + ~ (see fig. 1). The gluon from this annihilation vertex couples to the spin (~) of another quark (or antiquark) at the other vertex. Then in order to have an initial N ~ 351 state
y f a
b
c
d
Fig. 1. Annihilation into two mesons by one-gluon exchange. 247
Volume 157B, number 4
PHYSICS LETTERS
we have the combination 1.--
1
1351 , S z = 0) = ~X/~(IN, S z = "~, N , S z = - ~ > + IN, Sz = - ~ ,,
. ~,S
z
=~)), ,
(I)
which means we must add the four diagrams of figs. l a - l d . (The quarks and antiquarks are all in relative S-states in the N and N respectively, so the spatial wave function is symmetric.) To construct the initial NN 1S 0 state we add figs. la and lb and subtract figs. lc and ld 115 o,
Sz
= 0> = ½x/2{ iN,
- IN, S z
=
S z -- ~ ", R , S
z = - ~ *>
- i ,1N. -, S z =~)}.
(2)
Now, since the gluon is a vector particle it will couple to the quark like ~ ' ( t X q),
(3)
where t is the polarization of the gluon and a is the quark spin. Since not only the four diagrams in fig. 1 contribute to pff -+ rt+-p~ , but also the four diagrams where the gluon couples to the quarks on the right, we have eight diagrams to add in the one-gluon annihilation mechanism. The latter four diagrams we call a', b', c' and d'. (The gluon can also couple to the other quarks with the same result.) When the annihilation gluon couples to a quark of spin up ( S z = + !2) we have a factor +1 from eq. (3) and if the spin is down we have a contribution of - 1 due to the spin factor in eq. (3), which operates at the non-annihilation vertex. Here we do not include the spin flip. The process of the eight diagrams similar to fig. 1 then give an amplitude A for pp ~ m lm2 from a pp 351 state proportional to A ~ ( - l a + lb) + ( - l c
- ld) + ( , l a ' - lb')
- Id').
(5)
We now calculate the overlaps with the two-meson spin wave function ~02 meson = li'T-sT>IST> 248
/
(4)
and from a p~ 1 SO state an amplitude proportional to
- (lc'
This wave function is the only one contributing with the above spin projection assumptions. Explicitly one gets the desired selection rules: p~r comes only from the p~ 351 and 1P 1 states. In the latter case a + sign is used in eq. (6) since we have an even spatial wave function for the mesons. In this way the parity of the final (and initial) state is taken into account. It should be noted that it is essential to have the quark spin operator a at the non-annihilation q u a r k ~ l u o n vertex in fig. 1 to obtain this result. The same argumentation can be repeated for the case with a spin flip or for S z ( q q pair) = 0 or for Sz(N~ ) ve 0. The result is the same. The overlap for a triplet initial p~ state is nonzero only for an odd meson state, i.e. an e v e n ( 3 S 1 ) p~ state, and for an initial singlet only for an even meson state, i.e. odd (1P1) p~ wave. As noted above the 3P 0 model in the first order for qq annihilation vertices cannot give this experimentally observed selection rule [4], since the 3P 0 model does not have the additional helicity counting vertex of eq. (3). The above arguments concerned only the spin and spatial parity. The result is independent of the flavour because gluons are flavour blind. Inclusion of the flavour complicates the algebra immensely, but the result has been checked in some cases also using the full SU(6) wavefunctions. We have checked that the processes in fig. 2 give the same selection rule for p~ in a relative S-state. Both in diagrams 2a and 2b one has to require that the two annihilation qC:tpairs are in relative 351 states. One of the gluons creates a new qq pair in a relative 351 state and the second gluon then couples with one quark (or antiquark) via the spin operator in eq. (3) to give the selection rule. It will certainly be interesting to use the model for corresponding selectivities in other branch-
- ld) + ( - l a ' - lb')
+ ( l c ' - ld'),
B ~ ( - l a + lb) - ( - l c
18 July 1985
+- I,I,T)ItT - ,I,T>.
(6)
X a
b
Fig. 2. Annihilation into two mesons by two-gluon exchange.
Volume 157B, number 4
PHYSICS LETTERS
ing ratios whenever available information is sufficient. Also the importance of higher order effects (e.g. two pairs annihilated, one created) in both the gluon exchange and 3P 0 models should be investigated in detail before any really definitive conclusion about the annihilation mechanism can be made. We are grateful to Dr. L. Tauscher for pointing out and discussing the experimental results with us during the Nordic conference at Geilo where this work was performed. One of the authors (FM) acknowledges partially support by an NSF-grant PHY82-01910 and a Univ. S.C. Research and Prod. Scholarship grant.
References [1] M. Foster et al., Nucl. Phys. B6 (1968) 107; S. Ahmad et al., Proc. 7th European Symp. on Antiproton interactions (Durham, July 1984); CERN preprint EP/84-116. [2] H. Rubinstein and H. Stern, Phys. Lett. 21 (1966) 447. [3] M. Maruyama and T. Ueda, Nucl. Phys. A364 (1981) 297; Phys. Lett. 124B (1983) 121.
18 July 1985
[4] A.M. Green and J.A. Niskanen, Nucl. Phys. A412 (1984) 448; A430 (1984) 605; in: International review of nuclear physics, Vol. I - Quarks and nuclei, ed. W. Weise (World Scientific, Singapore, 1985). [5] A. Faessier, S. Furui and S.B. Khadkikar, Nucl. Phys. A424 (1984) 495 ; S. Furui and A. Faessler, Nucl. Phys. A424 (1984) 525. [6] C.B. Dover and P.M. Fishbane, Nucl. Phys. B244 (1984) 349; S. Furui, Orsay preprints (1984). [7] A.M. Green, J.A. Niskanen and S. Wycech, Phys. Lett. 139B (1984) 15; A.M. Green, J.A. Niskanen and V. Kuikka, in preparation. [8] H. Genz, Phys. Rev. D28 (1983) 1094. [9] R.A. Freedman et al., Phys. Rev. C23 (1981) 1103; M. Alberg et al., Phys. Rev. D27 (1983) 536. [10] A. Faessler, G. Lilbeck and K. Shimizu, Phys. Rev. D26 (1982) 3280. [ 11 ] R. Tegen, T. Mizutani and F. Myhrer, University of South Carolina preprint (October 1984); R. Tegen and F. Myhrer, University of South Carolina preprint (January 1985). [12] M. Kohno and W. Weise, Regensburg preprint TPR-8412. [13] F. Ando, in: Proc. Intern. Summer School on Nucleon -nucleon interaction and nucleon many-body problems (Changchun, China, July 1983), eds. S.S. Wu and T.T.S. Kuo (World Scientific, Singapore, 1984). [14] M. Maruyama and T. Ueda, preprint OUAM 84-3-1.
249
PHYSICS LEq~TERS
18 July 1985
A N E W D Y N A M I C S E L E C T I O N R U L E F O R p~ I N T O T W O M E S O N S J.A. N I S K A N E N t and F. M Y H R E R 2 Research Institute for Theoretical Physics, University of Helsinki, Helsinki, Finland
Received 28 January 1985; revised manuscript received 15 April 1985
We show that quark-antiquark annihilation into gluons gives a suppression of p~ annihilating into 7r±p:~ from p~ 1So and 3p states at threshold.
The observed branching ratios of stopped ~ in hydrogen going to two mesons provide us with a clear preference for a specific annihilation mechanism as we will show below. The annihilation process p~ ~ rr-+p~ comes dominantly from the isospin zero p~ states 3S 1 and 1P1, whereas the isospin one pO states 1S 0 or 3P0,1, 2 seem to be strongly suppressed at least by an order of magnitude [ 1] although these states are not forbidden by any symmetry arguments. Recently several dynamical annihilation models have been proposed. One model is based on the quark rearrangement model of Rubinstein and Stern [2] and is used by Maruyama and Ueda [3], Green and Niskanen [4], Faessler et al. [5] to describe p~ annihilation at low energy into three mesons. Another is the old 3P 0 model where a q~l pair annihilates into vacuum and is used by e.g. Dover and Fishbane [6] and Green et al. [7]. In this model the initial state 33p 1 is allowed [4]. Also topological models have been used to describe annihilations [8]. A third model uses specific quark gluon degrees of freedom and was discussed early by the Seattle group [9] and later by Faessler et al. [10] and recently expanded by Tegen et al. [ 11 ], Weise et al. [12] and Ando [13]. The two latter groups investigated multi-gluon contributions to the annihilation process and we will specifically use this dynamical annihilation model [11,12] to study the suppression of On leave of absence from the Department of Theoretical Physics, University of Helsinki, Helsinki, Finland. 2 Permanent address: Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the p~ ~ p+rr- from the isospin one 1 SO p~ state and we will show that the dynamic process qC:l~ gluons does suppress this annihilation channel and explains the annihilation data. This kind of model has also been applied by Maruyama and Ueda [14] and by Kohno and Weise [ 12] for a study of branching ratios. Indication in ref. [14] seems to be - although the authors do not bring this up explicitly - that the p~ prt is suppressed for the 1S 0 initial state. This one can see from the fact that at rest they obtain rather nearly (within 20%) equal branching ratios for p - n +, p0rr0 and p+Ir-, Since the p0rr0 channel is forbidden from the 1S 0 states the difference in the branching ratios should be a measure of the 1 SO state's contribution to p±Ir; . The qq pair annihilating into one gluon in fig. 1 have to be in an 3S 1 state, so as an example let us choose the q and ~1to have both S z = + ~ (see fig. 1). The gluon from this annihilation vertex couples to the spin (~) of another quark (or antiquark) at the other vertex. Then in order to have an initial N ~ 351 state
y f a
b
c
d
Fig. 1. Annihilation into two mesons by one-gluon exchange. 247
Volume 157B, number 4
PHYSICS LETTERS
we have the combination 1.--
1
1351 , S z = 0) = ~X/~(IN, S z = "~, N , S z = - ~ > + IN, Sz = - ~ ,,
. ~,S
z
=~)), ,
(I)
which means we must add the four diagrams of figs. l a - l d . (The quarks and antiquarks are all in relative S-states in the N and N respectively, so the spatial wave function is symmetric.) To construct the initial NN 1S 0 state we add figs. la and lb and subtract figs. lc and ld 115 o,
Sz
= 0> = ½x/2{ iN,
- IN, S z
=
S z -- ~ ", R , S
z = - ~ *>
- i ,1N. -, S z =~)}.
(2)
Now, since the gluon is a vector particle it will couple to the quark like ~ ' ( t X q),
(3)
where t is the polarization of the gluon and a is the quark spin. Since not only the four diagrams in fig. 1 contribute to pff -+ rt+-p~ , but also the four diagrams where the gluon couples to the quarks on the right, we have eight diagrams to add in the one-gluon annihilation mechanism. The latter four diagrams we call a', b', c' and d'. (The gluon can also couple to the other quarks with the same result.) When the annihilation gluon couples to a quark of spin up ( S z = + !2) we have a factor +1 from eq. (3) and if the spin is down we have a contribution of - 1 due to the spin factor in eq. (3), which operates at the non-annihilation vertex. Here we do not include the spin flip. The process of the eight diagrams similar to fig. 1 then give an amplitude A for pp ~ m lm2 from a pp 351 state proportional to A ~ ( - l a + lb) + ( - l c
- ld) + ( , l a ' - lb')
- Id').
(5)
We now calculate the overlaps with the two-meson spin wave function ~02 meson = li'T-sT>IST> 248
/
(4)
and from a p~ 1 SO state an amplitude proportional to
- (lc'
This wave function is the only one contributing with the above spin projection assumptions. Explicitly one gets the desired selection rules: p~r comes only from the p~ 351 and 1P 1 states. In the latter case a + sign is used in eq. (6) since we have an even spatial wave function for the mesons. In this way the parity of the final (and initial) state is taken into account. It should be noted that it is essential to have the quark spin operator a at the non-annihilation q u a r k ~ l u o n vertex in fig. 1 to obtain this result. The same argumentation can be repeated for the case with a spin flip or for S z ( q q pair) = 0 or for Sz(N~ ) ve 0. The result is the same. The overlap for a triplet initial p~ state is nonzero only for an odd meson state, i.e. an e v e n ( 3 S 1 ) p~ state, and for an initial singlet only for an even meson state, i.e. odd (1P1) p~ wave. As noted above the 3P 0 model in the first order for qq annihilation vertices cannot give this experimentally observed selection rule [4], since the 3P 0 model does not have the additional helicity counting vertex of eq. (3). The above arguments concerned only the spin and spatial parity. The result is independent of the flavour because gluons are flavour blind. Inclusion of the flavour complicates the algebra immensely, but the result has been checked in some cases also using the full SU(6) wavefunctions. We have checked that the processes in fig. 2 give the same selection rule for p~ in a relative S-state. Both in diagrams 2a and 2b one has to require that the two annihilation qC:tpairs are in relative 351 states. One of the gluons creates a new qq pair in a relative 351 state and the second gluon then couples with one quark (or antiquark) via the spin operator in eq. (3) to give the selection rule. It will certainly be interesting to use the model for corresponding selectivities in other branch-
- ld) + ( - l a ' - lb')
+ ( l c ' - ld'),
B ~ ( - l a + lb) - ( - l c
18 July 1985
+- I,I,T)ItT - ,I,T>.
(6)
X a
b
Fig. 2. Annihilation into two mesons by two-gluon exchange.
Volume 157B, number 4
PHYSICS LETTERS
ing ratios whenever available information is sufficient. Also the importance of higher order effects (e.g. two pairs annihilated, one created) in both the gluon exchange and 3P 0 models should be investigated in detail before any really definitive conclusion about the annihilation mechanism can be made. We are grateful to Dr. L. Tauscher for pointing out and discussing the experimental results with us during the Nordic conference at Geilo where this work was performed. One of the authors (FM) acknowledges partially support by an NSF-grant PHY82-01910 and a Univ. S.C. Research and Prod. Scholarship grant.
References [1] M. Foster et al., Nucl. Phys. B6 (1968) 107; S. Ahmad et al., Proc. 7th European Symp. on Antiproton interactions (Durham, July 1984); CERN preprint EP/84-116. [2] H. Rubinstein and H. Stern, Phys. Lett. 21 (1966) 447. [3] M. Maruyama and T. Ueda, Nucl. Phys. A364 (1981) 297; Phys. Lett. 124B (1983) 121.
18 July 1985
[4] A.M. Green and J.A. Niskanen, Nucl. Phys. A412 (1984) 448; A430 (1984) 605; in: International review of nuclear physics, Vol. I - Quarks and nuclei, ed. W. Weise (World Scientific, Singapore, 1985). [5] A. Faessier, S. Furui and S.B. Khadkikar, Nucl. Phys. A424 (1984) 495 ; S. Furui and A. Faessler, Nucl. Phys. A424 (1984) 525. [6] C.B. Dover and P.M. Fishbane, Nucl. Phys. B244 (1984) 349; S. Furui, Orsay preprints (1984). [7] A.M. Green, J.A. Niskanen and S. Wycech, Phys. Lett. 139B (1984) 15; A.M. Green, J.A. Niskanen and V. Kuikka, in preparation. [8] H. Genz, Phys. Rev. D28 (1983) 1094. [9] R.A. Freedman et al., Phys. Rev. C23 (1981) 1103; M. Alberg et al., Phys. Rev. D27 (1983) 536. [10] A. Faessler, G. Lilbeck and K. Shimizu, Phys. Rev. D26 (1982) 3280. [ 11 ] R. Tegen, T. Mizutani and F. Myhrer, University of South Carolina preprint (October 1984); R. Tegen and F. Myhrer, University of South Carolina preprint (January 1985). [12] M. Kohno and W. Weise, Regensburg preprint TPR-8412. [13] F. Ando, in: Proc. Intern. Summer School on Nucleon -nucleon interaction and nucleon many-body problems (Changchun, China, July 1983), eds. S.S. Wu and T.T.S. Kuo (World Scientific, Singapore, 1984). [14] M. Maruyama and T. Ueda, preprint OUAM 84-3-1.
249