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NN̄ annihilation into two mesons
Nuclear Physics A486 (1988) 581-603 North-Holland. Amsterdam
NR
ANNIHILATION INTO TWO MESONS Effect of final-state interactions
Received 1 February 1988 (Revised 25 April 1988)
Abstract: An extension
of the usual quark rearrangement model is made in such a NNIS) into two s-wave mesons takes place with the ‘P,, vertex acting or+ at the expense of introducing, into the final state, mesonmeson rescattering. understanding of several S-wave branching ratios but does not explain the
way that the transition otzce. However, this is The result is a partial so-called TP puzzle.
1. Introduction Over the last few years the study of nucleon-antinu~leon (NR) annihilation at Tow energies has received much attention both experimentally and theoretically. On the experimental side the main advance has been the construction of LEAR at CERN and to a lesser extent the antiproton facilities at Rrookhaven and KEK [see ref. ‘) for a recent review]. These have enabled not only elastic and inelastic p and ii cross sections “), but also many branching ratios into specific meson channeis to be measured. The subsequent results have given impetus to the corresponding theoretical studies. However, a newcomer to this field is, unfortunately, confronted with a somewhat confusing situation due to the plethora of alternative models. For example, in the case of NR annihilation into two mesons - the branchings most carefully studied at LEAR - the various models can, broadly speaking, be constructed from the components depicted in fig. 1. Figs. la,b show two possible elementary vertices for the basic quark-antiquark (yq) annihilation. In the first - the ‘P,, or pair creation model - the qq annihilate directly into the vacuum, whereas in the second - the “S, or one-gluon-exchange
model - the qq annihilate
into a gluon
(or effective
gluon). Given the basic qq vertex, the processesin figs. lc,d can be evaluated leading to four different models for NN annihilation into two mesons. In the literature (see e.g. ref. ‘)) three of these have been labelled as: (i) R2 when fig. la is inserted into fig. lc, (ii) S2 when fig. lb is inserted into fig. lc, (iii) A2 when fig. la is inserted into fig. Id. Different groups have different criteria for preferring one or other of the four above models. The most common of these criteria is that the chosen model explains some 0375-9474/M/$03.50 @ Elsevier Science Publishers (North-Holland
Physics
Publishing
Division)
R.V.
582
A.M.
Green, G.Q. Liu / iVfl
annihilation
a) M,
M2
N
b)
d)
Fig. 1. (a) The ‘P,, or pair creation-model for qq annihilation. (b) The ‘S, or one-gluon model for qq annihilation. (c) The yq vertex to first order leading to models R2 and S2 when combined with (a) and (b), respectively. (d) The qq vertex to third order leading to model A2 when combined with (a).
selection of the data with apparent disregard for other data and also the need for some underlying, more microscopic theory. An example of this is the use of the ‘S, vertex of fig. lb. When combined with fig. lc - to give the S2 model - it is able to explain, to some extent, the experimentally observed NN annihilation in relative S-waves into the two s-wave mesons 13). Such a branching is not directly possible for the combination of figs. la and lc - the R2 model. However, it has been shown 4*5) that the basic ‘S, vertex is not capable of explaining the more elementary process of meson decay into two mesons (i.e. M, -+ M,M3), whereas the 3P0 vertex is very successful “). Furthermore, in ref. “) the 3Po vertex has been put on a firmer footing by showing how it can be understood in terms of the flux tube model ‘). In this model - based on the strong coupling limit of QCD - the gluon fields are introduced explicitly as “tubes of gluon flux” connected to the quarks and antiquarks. Meson decay can then be expressed as a breaking of the initial flux tube into two flux tubes both of which give rise to a meson. This indicates that the ‘P,, vertex is to be much preferred over the ‘S, vertex, and that the failure of the former when in fig. fc the R2 model - is a reflection of an inadequate treatment of the more complicated multiquark system entering into NN annihilation. Having dismissed the use of the %, vertex of fig, lb on the grounds that it fails in meson decay and is not based on some underlying microscopic theory, how does the ‘P, vertex explain its most obvious failure to give NN(S) + VP? Two alternatives appear in the literature. In the first, advocated by Dover and his coworkers *“), the 3P0 vertex acting three times is inserted into fig. Id - the A2 model. This readily
A.M. Green,
G.Q. Liu /
NN
annihilation
gives NN( S) + rp. This model gives cu~~urff~le branching
583
ratios in Born approxima-
tion for the annihilations B,=NN(“S,)+TQ, B2= N6J(“So)+
(1.1) xp,
(1.2)
whereas experimentally B, 9 B,. Since @(“‘SC,) + rr -p,, is observed 36), there is no absolute selection rule that B2 = 0. This inhibition of NN + n-p in the isospin one channel has been called the “the mp puzzle” “). A similar, but apparently less extreme, inhibition also occurs in NN P-waves ‘I). However, as shown in refs. ‘,‘), the branching & could, in principle, easily be inhibited with respect to B, since in NN(“S,,) the rrp channel must compete with other strong channels - notably 7r’ff: On the other hand, the second mechanism for explaining NN(S) + 7rp involves only a single ‘P,, vertex and is related to the approach in ref. “). This is based on fig. lc but includes in addition a final-state interaction between the two mesons and can be expressed as the chain NN(S)
:M:M;F
:M:M;(,=O)
:M;M:(I=
1).
(1.3)
Here step 1 is simply a rearrangement of 3 quarks and 3 antiquarks into one s-wave meson MT and two p-wave mesons MP, one of which is the P having the quantum numbers of the vacuum. In step 2 the F is annihilated by the ‘Pt, vertex. In step 3 the resulting s- and p-wave mesons in a relative s-wave (i.e. f = 0) rescafter to give two s-wave mesons in a relative p-wave (i.e. f z=1). More details are given later. However, it should be emphasized that step 1 is a unique procedure and introduces no more parameters than the more usual rearrangement NN(S) --zM:M;M;
.
(1.4)
Expression (1.3) is simply exploiting the fact that the three quarks of the N and the three antiquarks of the N are not necessarily in relative s-waves even though the N and I? centres of mass are in a relative S-wave. Given the quark wavefunctions of the N and N, the above p-wave components M~M~E are simply the next terms in a partial-wave expansion, and so are as unique as, for example, the p-wave component in the partial wave expansion of a plane wave. The fact that p-wave effects can be important, even for the low-energy NN system, should not be a surprise since both experimentally 13) and theoretically it is known that NN annihilation is predominantly in P-waves for p laboratory momenta greater than about 200 MeV/c - see figs. 2 and 1.5 of ref. ‘). This effect is a combination of the strong real distorting potential in the initial state and also the internal momenta of the quarks in the hadrons. The second approach based on fig. lc also has the advantage of partially explaining the rp puzzle without resorting to an inhibition through the competition between the rrp and mf channels as in refs. ‘.“). In terms of quarks the
A.M.
5x4
Green, G.Q. Liu / IV@ ~n~iil~~~t~~n
chain in expression (1.3) can be depicted as in fig. 2a, where V, is the matrix element representing steps 1 and 2, and V, represents the meson rescattering of step 3. Unlike ref. “) the final quark rearrangement in V, is now considered as a model for a final-state interaction between thought of as a mathematical
the two mesons, whereas in ref. I’) it was simply recoupling, which does nof allow NN(S)+ M:Mz.
This latter statement is contrary to the conclusion of ref. “) which contained an error to be discussed later. Topologically, the processes in figs. ld and 2a are less different than appears at first sight. This is because the chain in expression (1.3) employs meson wavefunctions with good spin and isospin. For exampIe, in the transition NN + rp, application of the quark-line-rules (see ref. I”)) demands that fig. 2a must be of the form shown in fig. 2b and not fig. 2c. To see this it is sufficient to only consider the spin and isospin structure of the process and ignore the internal orbital angular momenta. In this case it follows from G-parity that the ‘P,, operator requires an w-like configuration for it to be effective and so in fig. 2c the initial NN state would be expressible as ~pw. Here the n-p simply arise since they are in the final state being considered. This is a clear contradiction, since the final and initial states have opposite G-parity, and shows that the initial quarks and antiquarks must have changed their flavour as in fig. 2b*. Even though, from the point of view of quark lines, fig. 2b is topologi~ally the same as fig. Id their interpretation is very different. In fig. Id the qq annihilate and are recreated only with the vacuum quantum numbers, whereas in fig. 2b the qij annihilate and are recreated with the quantum numbers of the meson under consideration. Furthermore, the flavour transition in the dashed box of fig. 2b only involves Clebsch-Cordan coefficients which introduce factors of J$ and should not to be thought of as some second-order process generated by, for example, two ‘PO vertices as in tig. Id.
b)
al
c)
Fig. 2. (a) Fig. Ic followed by the meson rescattering MYM g- M:M:. (b) A redrawing of fig. 2a where the dashed box represents an interacting meson state with a change of quark flavour. (c) Another redrawing of fig. 2a without a change in quark Ravour. * The authors
thank
Dr. M. Maruyama
for pointing
out this argument
A. h% Green,
As mentioned
earlier,
C.Q.
an attractive
Liu / Nfi
feature
~r~~li~7il~iiun
of the ‘PO model,
585
as the preferred
candidate for qq annihilation, is that it not only gives a better description of meson decays (M, + M,M,) than its rival, the ‘S, model, but can also - to some extent be justified in terms of flux tube breaking “). However, in NN annihilation into two mesons the situation is less clear since neither of the two approaches - figs. lc or Id with la, i.e. R2 or A2 - discussed in the previous paragraph gives obvious disagreement with experiment, provided meson-meson rescattering is included as in the present paper. The question then arises concerning how - and if - they can be justified, to some extent, by some underlying more microscopic description. Unfortunately, at present, nothing as detailed as the flux tube model for meson decays “j has been achieved. However, this framework - based on the strong coupling limit of QCD - suggests that the alternative of fig. lc with la - the R2 model - is more reasonable than the choice of fig. Id with la - the A2 model. For the R2 approach the flux tube model would give the scenario in fig. 3. Namely the N and N are each described in terms of 3 quarks (or antiquarks) with 3 flux tubes meeting at junctions J, and J2 [ref. ‘)I. A qq annihilate by the ‘P,, vertex (Q), in the same way as the flux tube breaking mechanism for meson decay, to give a baryonium state. The two junctions are then annihilated by a plaquette operator (P) to give the final two mesons M,M,. In the strong coupling limit P is of 0(1/g’) and Q of O(1). Therefore, P is in principle only a pe~urbation compared with Q. However, for the validity of the flux tube model g Schulz be ==I, a value such that the corresponding lattice spacing a = (0.1-0.2) fm is smaller than hadronic sizes, but yet large enough so that topological mixing may possibly be treated as a perturbation, see refs. ‘.I’). In this case P and Q could well be comparable. It is seen, therefore, that the R2 approach can be qualitatively explained with the flux tube model. On the other hand, the A2 approach requires the scenario of fig. 4. There, the first qq
Fig. 3. R2 scenario for Nfi annihilation into two mesons via figs. la and lc. (a) Action of the ‘PO operator (0) to give: (h) Baryonjum. (c) In b~ryonium the junctions J,, .f2approach each other to be ~noihilated by the playuette operator (Pi to give: (d) Two mesons M,M2.
586
Fig. 4. A2 scenario for Nti annihilation into operator (Q) to give: (b) Baryonium as in fig. (d) a hybrid meson. (e) In the hybrid meson which
two mesons via figs. la and Id. (a) Action of the ‘P,, 3b. (c) In baryonium Q operates a second time to give: Q operates a third time to give: (f) two mesons one of is still a hybrid.
annihilation with the ‘P, vertex (Q) again results in a baryonium state as in fig. 3. But the application of a second Q results preferentially in a non-exotic hybrid meson - identified by the presence of a flux loop ‘). The hybrid is non-exotic since it must have the same quantum numbers as the original NN state. The third Q then creates another meson, but one of these two mesons is still very likely to be a hybrid. It should be added that this flux-loop criterion for identifying a hybrid is not absolute, since flux loops can arise to some extent in normal mesons through the action of the kinetic-energy operator. However, the latter loops have a size corresponding to that lattice spacing (0.1-0.2 fm) necessary for the validity of the flux-tube model. On the other hand, the flux loop in fig. 4 has dimensions of about 1 fm the typical size of a nucleon. It should be added that this favouring of fig. lc over Id in the flux tube model is in addition to the basic flux tube idea that Q should be treated as a perturbation. The usual argument given for this last statement is the experimental observation
that meson
widths
(proportional
to Q’) are small compared
with their
masses, see ref. “). The scenario of the previous paragraph assumes that the flux tubes of the N and N are separately of the “Mercedes” type - all three meeting at a junction. However, it is possible - but less probable (see appendix B) - for the angle between two sides of the 3-quark triangle to be greater than 120”. In this case the states of minimum energy do not contain an explicit junction - as depicted in fig. 5. The first qq annihilation can still occur with the usual ‘PO vertex (Q) giving a baryonium state but without junctions. But such a state is again less likely than one with junctions (see appendix B). However, the next step for R2 model is less favourable than in fig. 3, since it now needs to bring q, and q2 also together. Earlier the transition to two mesons occurred with a basic plaquette of size 0.1-0.2 fm and left the quarks untouched. For A2 this second step avoids this inhibition.
A.M.
Green,
G.Q.
Liu J NN
annihilation
587
Q
w-----+
-
e---+0---+
f)
e) Fig. 5. A? scenario
when the three- and four-quark
configurations
do not contain
flux junctions
The discussion of the previous two paragraphs is of necessity very qualitative. However, it suggests that the scenario of fig. 3 - a two-step process involving one Q and one P - leading to two mesons could be favoured over the scenario of fig. 4 - a three step process involving three Q’s - leading to two mesons one of which is preferentially a hybrid. In the opinion of the author, such considerations involving the gluon degrees of freedom are necessary in order to put the theory of NN annihilation on a firmer footing and so eliminate some of the many competing models involving only the quark degrees of freedom. The main objection to the present line of reasoning is that it depends on an N and N model based on flux tubes - a far from universally accepted notion. However, it does give a definite prescription for discussing NN annihilation in a way that is related to the much simpler problem of meson decay (i.e. M, + M,M,). If it is accepted that NN annihilation into two mesons is dominated by a two-step process involving a qq annihilation with Q and a junction annihilation with P, the next question to be answered concerns the nature of the plaquette operator P. As seen in fig. 3, P only involves the flux tubes in the vicinity of the two junctions, and so the states of the quarks are not directly affected. This suggests that, in terms of quark operators, P acts as a spin and isospin singlet. This is the rationale for saying that the two-step QP process can still be depicted as in fig. lc since this only shows quark lines which are unaffected by P. Of course, P could well generate a complicated radial dependence in terms of r and r' - see fig. lc - but this would only have a secondary effect on calculated branching ratios. It should be added that, for the process of interest in the present paper (i.e. NN(S)+ M:Mz), the rescattering of the two mesons at step 3 in expression (1.3) introduces a further interaction. This is usually thought of as being mediated by a plaquette P as in fig. 6 - see ref. ‘) - or by the flip-flop approximation to this as in ref. I”). Therefore, in all, fig. 2a is proportional to QPP and fig. Id to QQQ. On a
585
A.M.
Green, G.Q. Liu / NR
+
annihilation
e ML
M3 >
Fig. 6. Meson-meson
scattering
[M,MZ+
M,M,]
through
a plaquette
(P)
less microscopic level, meson-meson scattering could possibly be treated in terms of s- or t-channel meson exchange. However, this would require the knowledge of many unknown triple meson vertices. Furthermore, since the total energy is about 2 GeV such s-channel processes as rB+ MR + up are far from being resonant for nodeless s-wave mesons e.g. the w. In sect. 2 the formalism is outlined with details being relegated to appendix A. In sect. 3 the results of this model are presented and compared with those of ref. “). In sect. 4 a summary and conclusion are given.
2. Ni%(S-wave)-+ MFMs In order to generate two s-wave mesons (e.g. VP), when NN annihilate in a relative S-wave by means of the chain in expression (1.3) - depicted as fig. 2, four distinct steps are involved: of the (i) The first stage NN + M~~M~~E~~ is simply a ~ut~e~ff~~cal recoupling angular momentum and isospins of the three quarks of the N and the three antiquarks of the NJ. At this stage the objects Mr,, M& and ~~~ have the angular momentum and isospin of s- and p-wave mesons but nothing is said about their radial wavefunctions. The actual recouplings are
where E, 8, H, 5, etc. have the quantum numbers of the usual p-wave mesons. In this equation, for clarity, the F_3h’~ in each term are not displayed. The amplitudes of the [&MS] configurations have an additional factor of v’?? incorporated to allow . recouplings in ref. “) differ by for the identity of &1?5an d ~36. The corresponding now having an additional factor of fi - in addition to misprints in the [err] and [Sp] channels. The reason for this is purely historical and is explained in ref. I’).
A.M.
Earlier
these particular
ck?ri,
recouplings
were made with the condition
the coefficients (squared) should equal 1296 (=36’). are not all of the same form e.g. MyMliM.7 or MPMIM: a factor of fi
so that comparisons
of like mesons
e.g. M”MzM:
x39
G.Q. Lilf / ,Pm ff?l~~i~~iial~~~~7
that the sum of
However, when the mesons it is convenient to introduce
can be made more directly with the coefficients or MrMeM_!. This factor of JT is the statistical factor incorrectly omitted in ref. “) an d reinstated in ref. “). The sum of the coefficients 48,48 * 11, 48 * 17 and (squared) for the four states ‘I” ‘s-1-’S, are, respectively, 48 * 25 - each of which should now be compared with 3 * 36’. in all it is seen that the four NN states when expressed as M~MFF account for only i of the total probability (4 * 3 * 36’). (ii) The second stage M~,M~,F -+ M:;Mq, simply involves the usual ‘P,, operator for qq annihilation
“‘), namely O(‘P,,) = K,[Y<,X(q,q,,
s = 1)].“““‘-“6(c’),
(2.2)
where c’ = rT - rh in the notation of fig. lc. The value of K, is discussed in ref. “‘), but apart from colour and angular momentum factors it is basically proportional to the corresponding coupling constant extracted from meson As shown in eqs. (A.6) and (A.7) this results in an interaction V,(r,,,
decays M, + MLM3. of the form
Y) =(M:M~(~,,)/O(‘P,,)INN(U)) =1Y3(1-~(ff+p)r’)exp[-:r~(cu+3P)-~~~i,],
(2.3)
where rhll is the vector between the centres of mass of the two mesons. (iii) At this stage the two mesons propagate before rescattering. This is achieved by the propagator (2.4) where dE =2M+E~-h~-JM~+q~-~~+lir(M,)+~ir’(M,) depends on the initial energy of the N and ti i.e. rest mass (2M) plus their energy in the centre of mass (E, h,), and the energies (E, =jMf+ Q’) plus half-widths (~~‘(~~)) of the two mesons. The factor ,fz,) takes into account that the S-, 77 and q’ are not simply qq states with q = u or d. The 9 and 11’ wavefunctions are i( uii + L@T v’%L~ - corresponding to a mixing angle of =- 10” - whereas the z- wavefunction is =J (qCj+ multiquark components) - see ref. ‘I). Therefore, .f‘z,, is taken to be (1)“-’ “I) where are the number of r‘s, T‘S and ~“s in the intermediate state. The factor yc” M,M,/ E, E, is an attempt to simulate the correct meson normalization factor which presumably would appear naturally in a more complete relativistic theory see ref. “) for a recent discussion of such factors. Fortunately, this somewhat controversial factor does not lead to significant changes in the final results. (iv) The fourth stage i.e. MF,Mz,( I = 0) -+ M:,M$( I = 1) involves a second recoupling in which the antiquarks 4 and 5 interchange their quark partners 1 and 2 to
590
A.M.
Green, G.Q. L.iu J hJN un}zi~~iIafio~z
give two s-wave mesons
in a relative p-wave. The actual
recoupling
is
(2.5)
where r” is the vector between the c.m.‘s of the two outgoing s-wave mesons, k the corresponding relative momentum and r14 is the internal radial vector of the p-wave meson Mf, with quark constituents qlq4 - see ref. “) for the more general case involving three mesons. Compared with ref. 12) the recoupling in eq. (2.5) has three differences. Firstly, there is a factor of two less. This arose earlier as [ 1 + ( -l)c’l+Cil] where Cr., were the initial and final G-parities and acted as a check on the programme. Later it will be more convenient to incorporate this effect elsewhere as part of the statistical factor. A second difference compared with the recoupling of ref. “) is that there is now an additional isospin phase (-1)q1’7z multiplying each amplitude for MFM;. This arises due to the difference in definition of the isospin component for the r and p quark wavefunctions. Earlier, for example, the 7~’ wavefunction was written as rrt = ud, whereas now it is more convenient to replace ud by ud - the difference being a sign change for T = I mesons. The final difference compared with ref. “) is that S-coupling (not SL) is used - leading to a phase (_l)S-P.ti-S, iJ,* These last two differences are absolutely necessary, since they correspond to using LS coupling and the same notation for rr, p as was employed in the derivation of eq. (2.1). Their omission in ref. “) was incorrect and their inclusion has a profound effect on the results claimed in ref. “f. There stage 3 of expression (1.3) was considered simply as a mathematical recoupling. As shown in eq. (A.12), including the radial wavefunctions results in an interaction of the form
= K,v”exp
[-$(r”‘+
rz)] Y,(i)
* Y,(P),
(2.6)
where I&(P) is the vector between the centres of mass of M~M~(M~M~) and the factor of r” is characteristic for a p-wave final state and h is the strength of the interaction. This model for meson-meson scattering has the same philosophy as the rearrangement model for NG -+ M,M?M,. The state dependence is assumed to be given completely by the overlap between the initial and Final meson wavefun~tions. In ref. “) it was shown that, in general, the same state dependence emerges if the transition takes place with the same harmonic confining potential that generated the meson wavefunctions. A similar improvement could be incorporated here to give a model for h.
Greerr, G.Q. Liar /
A.M.
Nfl
591
a~l~li~ij~uri~~fl
Combining the four above stages and taking the Fourier transform give the matrix element for NN(S) annihilation into two s-wave mesons of momentum k in a relative p-wave
as
=
,yk
e.
h-./j
q7dq
C
(2.7)
hl:M;
Here the summation is over the 8 p-wave mesons of ref. “) and the 5 s-wave mesons - the n’ included. The two k-dependent factors can be considered as part of the overall P-wave phase-space factor to give [PS]=(2S+l)k”exp(-k’/P),
(2.8)
where S is the spin of the NN state and
N.B. In ref. “) the phase space incorrectly analogous
to .f‘& andJ’;i
kJ instead
contained
in eq. (2.4) must also be introduced.
of k’. Two factors
These are
f” = (]/\I?)“_+‘%, i 3 7111
ii) where II,, n, are the number
of m’s, 71’s and rl”s in the final state.
.f; = JM&LI Et,,,,
(ii)
where E,,,, = 2M -t- EC,,,. . This is the factor advocated in ref. “) and is in place of the more usual &?m. Again this change arises through introducing by hand the relativistic normalization factors (JM,/E,) for the final meson ~vavefunctions. With this notation
the branching
ratios can be expressed
BR(NN-t
M;M;)
as
=[PS]C,j6lf‘;f:,,/‘,
where M is the same as M[ ] in eq. (2.7) but without the two k factors, an overall constant that includes the initial state interaction.
(2.9) and c’rs is
3. Results Given the expression for the branching ratios in eq. (2.9) the ideal approach would be to calculate directly the initial-state interaction (ISI) in CTs using some NN optical potential. However, in the opinion of the authors, this is too ambitious at the present time - for the following reasons: (a) The most complete studies of NN annihilation mechanisms at the quark level are able to achieve only about 70% of the pj3 integrated annihilation cross section, see refs. “*“). It is not known in which channels the models are at fault and so it is not known which of the calculated initial-state interactiotls (ISI) can be reliably
A.M. Green, G.Q. Lilt / NN ~riiii~lilaii(~~
592
estimated. Several authors, e.g. ref. ‘“), have shown that Born approximation can overestimate annihilation cross sections by up to two orders of magnitude and so the C,, are indeed crucial factors for absolute branching ratios. (b) In eq. (2.9) C,, is not the only unknown tion, with a more microscopic
model involving
in eq. (2.2) could well have a strength
factor. As discussed
in the introduc-
gluon effects, the effective 3Po vertex
K, that is diRerent
to the one extracted
from
meson decays, In view of this, here eq. (2.9) is used to predict the values of CT., needed to give the experimentally observed branching ratios. Since, for each value of T and S, there are several possible annihilation channels, the constancy of C~s(~~~~) - for fixed TS - is a measure of the present model’s reliability.
The values of C,,
“SO:
ww
expt (%)“)
1.4*0.6 11.9zt5.1 34.32 28.6 7.3 10.0 13.6 13.2 0.4
(‘,,,,(A) M(1) “1 M(2, wH) ‘) C,,,,(B) C,,,,(C ) G,,( 0 ) C;,,( El BR(ref. ‘))(I%)
“‘S,,:
P”
expt (X) C,,,(A) M(I) M(2, wB) C,,,(B) C,,>(C) c,,,i ut C,,,t E) BR(ref. ‘))(%
“S,: (% ) C,,,(A) M(I) MU, 4 co,(B) c;,,(C) C,,,(D) C,,,(E)
expt
BR(ref. ‘) i( a)
I
211.5”) 4.3 * 3.2 6X.2 65.5 2.9 3.5 4.6 5.1 2.3
w 1.3+0.2 61.9i9.5 14.49 25.5 48.5 62 45.3 46
predicted
by eq. (2.9)
pop” 0.12*0.12 22.2 i- 22.2 7.35 47.4 11.4 21 13.7 5.6 0.4 “I,
CO.2 1) 10.9 9.6
w 5.2 i-0.8 21601330 4.9 I 14.3 1760 1080 1584 2160 4.2
~‘W
4.0 * 0.6 ‘) 25.5
m
TTO
expt (%I 1
<
0.65x0.16 6.43 * 1.S8 31.80 17.1 5.7 6.4 6.4 4.8
1
C,,(A)
C66
M(l) M(2, PC) C,,(R) C,,((‘) C,,(U) C‘,,(E) HR(ref. 31)(?:v)
12.28 5.9 ( 42 r26 <6X ~6% 0.x
<9*3 <178i59 17.6, 14.0 ‘1 15.6, 70. I Cl43 ~185 Cl77 138
C‘,,(A) M(l) M(2, /)F)
(‘,,(U) C’,,(C) C.,,CD) C,,(E) HR(ref. ‘))(‘!:I)
0.48
7rETT
PP
expt (“% i
P77; 0.11 r0.06 0.3210.1x 5X.46 47.73
0.35 * 0.02 8.4-LO.S x 10’ 0.204
0.56 4.5 x 1oJ 1.7X IV 4.6X loJ 8.0 x
I o1 0.IL
“) Seeref. ‘1). “) The value of M( 1)= [ PS]“‘J~.j&J$
used in eq. (2.9). ‘) ‘The largest contribution M( 2, M:MI) from the sum in eq. (7.7). “f An average value of the proposed experimental ratios: 0.7 i 0.3,2.
and 3.9-t 0.6. ‘1 M(pp, s’=O) extract C,,(A). ‘) Predictions
and
M(pp, s’= 21, M’(s’=0)+M’(.s’=2)
assuming
C,,,ig{j)
= C,,,(pw)
being
I + 0.2,
used
to
and CC,,iv’w) = C,,,( 77~).
In this way a self-consistent procedure is produced. Firstly, the microscopic models are developed until the C,,Y(M:M:) are constant for a given TS. This lends confidence for using these same models to produce effective NR optical potentials. Only then can the C, generated by these potentials be expected to be realistic. In table 1 are given the values of C,, predicted by eq. (2.9). The C,,s are shown using various approximations. Case A is the most realistic model and includes: (a) Meson widths that are energy dependent as in ref. “). (b) The relativistic factor fk of eq. (2.4). (c) Nucleon and quark wavefun~tions given by cy = 1/R’ and p = 2cy/tg with R = 0.6 fm. In case B the _fg are set to unity, case C is the same as case A but with the ,f:” of eqs. (2.4) and (2.9) set to unity. Case D is the same as A but with energy independent meson widths. The meson widths and masses are the same as those in ref. Is) except that ir(e) = 140 MeV and $I’( B) = 75 MeV, which appear to be more in line with the current experimental numbers. These changes, from the earlier values of 200 and 68 MeV, respectively, do not lead to significantly different values
594
A.M. Green, G.Q. Liu / N1\Sannihilation
for the ratios of the Cr,. Case E is the same as A except that the nn’ mixing angle is -19” as opposed to the -10” used elsewhere - for reasons given later. Before discussing the details of table 1, it should be pointed out that if Ph, in eq. (2.4) is treated in the closure approximation (i.e. 3E # 3E(M~M~) andf‘& =J‘;1= 1) then the matrix element M given by eq. (2.7) becomes zero. This complete cancellation between the various terms in the summation over MT and My is a general property of processes that involve the “‘hair-pin” mechanism depicted in the dashed box of fig. 2b”. In ref. “) this rule was the reason for the S =0 matrix elements vanishing. However, the S = 1 matrix elements did not vanish due to the LS++ SL error mentioned earlier. This had the effect of giving the A,-meson contributions with a wrong sign. Since closure gives zero for M in eq. (2.7), it is seen that the present non-zero results are dependent on the fact that the masses and widths of the s-wave and p-wave mesons are not degenerate and that the j$, are not unity. Another limit for P, in eq. (2.4) is when the sum over MTMI is restricted to Mg= F(1.300 MeV) and the ,j’$’ are set at unity. In this case, for a given TS, the branching ratios are the same as those in ref. “) - numerical confirmation of the topological similarity of figs. Id and 2b. In table 1 the values of M(l,2) = rw’~‘l~.f,hd’~,l are given for case A - the M(1) - and also the largest term of the sum in eq. (2.7) - the M(2, MTM!). The comparison between M(1) and M(2) indicates the degree of canceltation - if any - when the sum is carried out. Several comments can be made about table 1. (a) The “S,, channel. Here for case A the C,,,, extracted from the ww and pp branching ratios are consistent with each other. This gives support for the meson rescattering mechanism proposed in this paper. The numbers corresponding to the A2 model of fig. Id and ref. ‘) are also given in table 1. These are not in such good agreement with experiment and correspond to Coo’s that differ by an order of magnitude. However, it should be remarked that - as in many other cases to be discussed later - the experimental branching ratios leave very much to be desired on the question of accuracy e.g. 0.12 Lt0.12 for the p”p” case. (6) The -“S, channel. Here the 7rp branching ratio has not been seen - the “rrp-puzzle”. However, this is not surprising since the present model predicts from eq. (2.9)
BR(w)
M’h)
BR(pw)
M’jpw)
R(np,‘pw)=---=---=
oo26.
’
Admittedly BR(pw) is not well-known, with the experimental numbers ranging from 0.7 * 0.3 to 3.9 ;t 0.6%) but in every case the above R (vp/pw) predicts small BR( rp)‘s ranging from 0.02*0.01 to 0.10*0.02%~. Therefore, with the present model, the “So no-channel is inhibited without introducing the competition from the QT~channel as advocated in refs. ‘.‘). Another interesting feature is that C,,-: C,,,,. In ref. “) an attempt was made to extract Crs from s,,,,(Nl;l(S)jMsM~M3). This resulted in table 2 which shows l
The authors wish to thank
Dr. N. Isgur for pointing
out this sum rule.
A.M. Green, G.Q. Liu / NN artnikilation
595
State dependence of cr.\ ( proportional to C’,\) from ref. “) as a function of h the strength of the NN+M:M;M: rearrangement potential State/A
(MeV
us0 “S I ‘So 3, S:
fm) =
1X
42
90
10.1 2.9
2.7 1.7
2.1
5.3 7.9
51.3 7.x
1.4 22.2 2.4
numbers C, - proportional to CT,s - as a function of /\ the strength of the NN(S)+MTMqN: annihilation mechanism. In that reference, i in the range 1530 MeV fm appeared to be the most favourable - see ref. “) for more details. In this range C,,,- :C,,,,, which suggests that the present picture for NN spin singlet transitions is a consistent one. (c) The ‘-‘S, channel. Here C,,,(nw) -O.O3C,,,( NIL))which shows that the present model cannot be the whole story in this channel. Furthermore, it is the VP-branching that is probably at fault for the following reasons: (i) C,,,(nlw) %5C,,,, - a relationship that is consistent with table 2 in the sense that C,,, can be enhanced considerably over the other Cls’s - an effect attributed to strong tensor effects and to &i components in the initial wavefunctions. This enhancement of T = 0 observables is already seen in NN total integrated cross sections now that lip scattering has made possible a separation of the T =0 and T= 1 cross sections ‘) - a feature already expected from phenomenological NN potentials “). Assuming C,,,(n’w) = C,,,( no) gives the prediction shown in table I _ a surprisingly large number (for possible reservations on this estimate see the later discussion on the pn’ channel). (ii) C,,,(7rp)=2OOC,,,,=5OOC ,() - numbers not consistent with table 2 and also not consistent with the belief that the C,-? usually reduce branching ratios by up to two orders of magnitude compared with Born approximation “‘). Such a large value of C,,, would imply either Born approximation for rp is qualitatively correct or that the S = 0 branchings are anomalously small. Both of the above comments suggest that the present model badly underestimates the np branching in this channel. In fact assuming C,,,(rrf) = C,,,(~W) predicts BR(“S, + np) = 0.15% = O.O3BR(expt). The ideal situation is, therefore the inclusion of some mechanism that enhances the =p branching in the “S, but not “Sg channel i.e. back to the -rrp puzzle with, perhaps, the additional clue that it is in the “S, channel where the problem lies. As will be discussed later in this section and also in the next section all the two-meson branchings that contain n-mesons are in general suspect. (d) The -73SIchannel. Here only C, ,(pn), C, ,(pn’) and C, ,( rr) can be extracted and are seen to differ by orders of magnitude. However, on the positive side it
596
4. M. Green, G.Q. Liu / NN at~}l~~lilatio~
should be noted that C,,(pn)=$Coo again hand, the value of C,,( 7~) c= 104C00 indicates mates very badly indeed the 71n branching.
consistent with table 2. On the other that the present model again underesti-
The result that C, ,(pn’) = O.O5C, ,(pn) is somewhat disturbing so reasonable. Unfortunately, this appears to be unavoidable
since C, I(~~) seems and arises because,
in eq. (2.8), k(pn) = 3.36 fm-’ and k(~n’) = 1.8 fin-’ to give fi( pn)[PS(p~)]‘/‘= 15.9 and .j’L( pn’)[PS(pn’)]“’ = 29.2 fm-“” - all other factors being equal for pn versus pq’. The only parameter in [PS] is p. Increasing this significantly (R going from 0.6 to 0.3 fm) does indeed improve the above 9, n’ problem, but only at the expense of other bratlching ratios “1. However, the above assumes that the n and 77’wavefunctions are of the form given after eq. (2.4), which correspond to a mixing are uncertain in two ways. Firstly, some angle of B(M) = -10”. These wavefunctions radiative decays involving the n indicate O(M) = -19”, which reduces the amplitude of the G-component in the n from -I/& to =I/&, see ref. “). This effect goes in the right direction, but as seen from case E in table 1 this still gives C, ,(pn’) F= O.lOC,,(pn). Another uncertainty in these wavefunctions, that would help this problem, is the amount of “glue-bail” component, which should be larger in the n’. However, at the present time this appears to be consistent with zero, see ref. 15). From table 1 it is seen that the A2 model of ref. ‘f in fig. id does much better for those branchings that contain w mesons. One obvious difference between the present calculation and those of ref. ‘) is that the .& factor after eq. (2.7) has introduced a factor of 4 into the C, ,( rrrr). If this is also put into the calculation of ref. ‘) then they also underestimate the VT branching by a factor 12. But this is still much better than what the present calculation gives. Of course, this pessimism is based on the is essentially the correct value for belief that since C,,(pq) =iCOtjr then C,,(pn) C,, - a point that could be clarified by better limits on the other “S, branchings. Even so, it is hard to see how the X~Tbranching could be significantly increased in the present model, since the exponential term in the phase-space factor of eq. (2.8) becomes so small for the appropriate value of k = 4.7 fm ‘. The comparison of M(2) with M(1) in table 1 shows that in some cases (e.g. pop” and z-r) there can be considerable cancellation when all p-wave mesons are included in the sum of eq. (2.7). However, in spite of this, the present model appears to be quite stable when different approximations are introduced. In particular, as seen by comparing C,,,(B, C, I;? E) with C‘,(A), removing the relativistic factor _fA of eq. (2.4) or settingf:” equal to unity or using energy-dependent meson widths or changing the 7~7’ mixing angle all lead to C,‘s that are qualitatively similar.
4. Summary and conclusions The results in the previous section suggest that the present model of a single ‘P,, vertex followed by meson-meson scattering [as in fig. 2a)] can account reasonably well for the relative branching ratios NN --f M;MQ in the two NN spin singlet states.
A.M. Green, G.Q. Liu / NN nnnihilut~on
597
However, in the spin triplet case, the model has problems - especially with decays involving r-mesons e.g. “S,(np) and 3”S,( rn). The reason for this could be that the pion is, in several ways, qualitatively different from the other mesons and this is not taken
into account
in the above
calculations.
Two of the most important
differences are as follows: (a) All quark models for NN annihilation consider the quarks to contribute only 0.6 fm to the experimental r.m.s. charge radius of 0.8 fm with the difference arising from a pion cloud outside the region where the quarks are confined. However, to the authors’ knowledge, the effect of this cloud is never included in NN annihilation processes. If these pions contribute more than the mechanism of fig. 2a, then in table 1 the values of C, involving pions become smaller. (b) Models for the pion [see refs.h.21.37)] indicate that the radius of the qq component is considerably smaller than in other mesons. Taking this into account has the qualitative property of allowing the meson rescattering mechanism of fig. 2a to give an s-wave meson and p-wave meson in a relative I= 2 state 33) e.g. NN(“S,,) + rf. Such a decay is not possible in the present model where all mesons have the same value of p for their quark wavefunction oscillator parameter. Furthermore, it is found that increasing /3 for the pion by a factor of four, corresponding to a r.m.s. radius of 0.24 fm, increases dramatically matrix elements involving pions. In this way a much more consistent model emerges “). In addition to these problems arising because of the pion structure there is also the problem that, so far, the annihilation mechanism has ignored the explicit effects of gluons. As discussed in the introduction, such gluons in the form of e.g. flux tubes could well modify some of the conclusions reached in their absence. Hopefully, these modifications still leave partially intact the predictions of the ‘quarks-only’ models. For example, if plaquettes play an important role as in fig. 3, then their lack of any spin or isospin dependence would leave unaltered branching ratio predictions. However, any change in the r radial dependence - see fig. Ic - would modify the initial-state interactions and so generate an additional T,S dependence. Also any change in the r” radial dependence in eq. (2.6) would modify the relative branching ratios into different mesons. In this article the main aim was to give, for NN annihilation in S-waves, a partial understanding of the rp puzzle. However, there is a similar problem for P-waves where it is observed that the annihilation into n-p is again inhibited in T = 1 states. In particular, experimentally, R
=
WNN(33~,)+
l
nip.)<
' BR(NN("p,)+ n'pr) Now the basic diagram
shown
in fig. lc contributes
NN(P)+ to give R, = 18 - in gross disagreement
M;M;e
(4.1)
.
directly
through
+ M;“M;
with experiment.
the chain (4.2)
However,
there is now the
59s
chain NN(P)-,MPM~E~M~M,P~M~M,S. This extension
to P-waves
has some additional
earlier S-wave case. For example, eq. (4.3), be continued to include
(4.3)
complications
compared
with the
the chain in eq. (4.2) could, in the philosophy MsMs-+ MfMz.
of
The authors wish to acknowledge very useful conversations with Drs. C. Dover, N. Isgur, M. Maruyama, J. Niskanen, H. Pirner, J.-M. Richard and N. Tornqvist.
A.l.
DERIVATION
OF V,(r,,r)
In the notation
IN
EQ. (2.3)
of fig. lc a =$(r,+r4),
b=t(r,+r,),
a’ = r, - r, ,
c=t(b+rJ,
b’= rl--rs,
c’=r,-r,,
(A.1)
r and rM are the vectors between the N and N, M, and Mz respectively. R and R’ are the coordinates of the initial and final centres of mass. The matrix element V, in the chain (1.3) can then be expressed symbolically as V,(rM, r) = M,[NN(S)+ =
M:Ml]
da db de da’db’dc’
WY * 0(3Po) * w” e DF, ,
(A.2)
where WN=N~exp[3~uRZ+~(wr2-cu(a2+b;“+c2)-~a(a”+b’2+c”)] are the NN internal
radial
wavefunctions,
W~=N:N~b’Y,(b^‘fexp[-aP(a”+b’“)]Y,(~~) are the two-meson
internal
(A.3)
wavefunctions
(A.4)
and
DF,=G[~(a’+b’+c’)-r]S[(a-b)-r.,]6[~(a+b)-R’]S[f(a+b+c)-R]
(AS)
are the &functions relating the six basic coordinates r, with the “observed” coordinates r, q.,,, R and R’. Here (Y,/3 are the oscillator parameters for the wavefunctions describing the N (or N) and mesons respectively and NO and N; are the normalisations of the nucleon (or N) and the I-wave mesons respectively. The latter have the form
A.M. [see
ref. “1 for more details).
Gwen, G.Q. Liu / NN
599
u~~nihila~ion
@(‘P,,) is the matrix element
of the ‘PC,vertex
ing q&, from eq. (2.2). It is seen that the a, b, c and a’, b’, e’variables dR dR’ -----yexp(iP.R-iP’.R’) (2,).
= S(P-
annihilat-
separate
to give
dadbdcexp[3aR’-cu(a’+h’+c’)]
exp
P’) exp [-P’/24~2](371./2ct)~”
[-_SN&].
(A.61
As always with the ‘P,, model, since the basic interaction O(3P,,) is not galilean invariant, the final matrix element depends on the c.m. momentum P and so the usual prescription P = 0 is imposed. This problem does not enter when the ‘PO model is interpreted in terms of ftux tubes as in ref. ‘1 and justifies this P = 0 condition. On the other hand, a’+b’t-C’ ---r
da’ db’ dc’ 6
N~N~N9exp[-~(a:+P)(a”+h’~)+~ar’]
3
(27r)?‘l
R, exp
=9JZ(n+p)J 2--7;L J _I~$$-
>
1
[lar’]
dk e (k.%ZY,(L) exp [ _,(,z:‘,)]
R,(l -:(cu+/3)r’)
exp[-:r’fcu+3/3)],
(,A.7)
where R, = ~~,N,?,N:N:R,,(NT;J+ and the R,, amplitudes
A.2
DERIVATION
OF
are displayed
V2Cr”, rk,)
IN
EQ.
~7~:s))
in eq. (2.1).
(2.6)
The matrix element V, in the chain (1.3), treated meson wavefunction and of strength A, can likewise VJr“, r&j = liJ,[MeMF+ =: A where
WY is
given
in eq.
as simply the overlap of the be expressed symbolically as
MSM:]
da db da’db’
WY * WY * DF,,
(‘4.8)
(A.4) and
WY = ( NT)’ exp [-$(2(a
- b)‘+~(a’+
b’)‘}],
(A.9) (A.lO)
AM.
600
The spin-isospin
dependence
Green, G.Q. Liu / NN a~tlj~lilati~tt
is given by
([ Y,(k”)[[lJ]“~r~[2;i]~s~“1]S7~]~T~[[~,~Y,(j,J)[14]S~r~]J~“~[2~]”~7~]‘~)+(~r,,~r~~) -RhY,(~).[a’Y,(a^‘)-h’Y,(~‘)]=2R,r”Y,(~)* where
R,, is defined
(A.1 1)
Y,(i”),
by eq. (2.5). At first sight, it might seem that the presence
of
the two terms in the above equation would lead to [a’Y,(a^‘) + b’Y,(bA)]. However, the action of R,, is essentially to reverse the sign of the b’ term w.r.t. the a’ term i.e. effectively [~‘Y,(~‘)+b~Y,(~)]j[~‘Y,(~‘)-b’Y,(~)]=2~”Y,(~).
For convenience factor.
Performing
the factor of 2 in eq. (A.1 1) is incorporated the above
integrals
in the overall statistical
gives
V2(r“, rd)=h(2~/p)3’ZT”exp[-~p(r’~‘+u~)]Y,(R). In eq. 12.7) the quantity
of interest
Y,(?‘)R,.
(A.12)
is
Appendix B THE FtUX
PRORARlLlTY TUBE
FOR
q3 AND q2$ CONFIGURATIONS
TO HAVE A
JUNCTION
In the introduction, it was stated that the nucleon flux-tube configuration contained a junction provided the angles between the sides of the three-quark triangle defining the nucleon were less than 120”. In this appendix, a quantitative estimate is made of this effect. With the notation in fig. 7a b = rL1- r3
a=&,
t = J‘( Y, + r2) - J&,
s=&r,-rJ,
1 Jl
J2 3
2
ai Fig.
7. (a) The limits (it, and h,) on h to ensure the two marked
bl angles are lrss than $r. (b) The
q-‘$‘state.
A.M.
the normalisation
The probability Z(b,, where
integral
Green, G.Q. Liu / NN annihilation
can be written
of a junction
(Y is the oscillator b,
=a
sin $r
as
configuration
b,.x,)=;($)i j,:daa’l:’ constant
is then d.u exp [-:cu(a’+h’+ahx)]
dbb’l,; describing
the nucleon
0,),
h,=
a sin (fn-
b,=O,
b,=x
when
sin ({r-
601
0,)
wavefunction
sin zrr when 0, <
(B.1) and
:rr,
fI,>!,n,
X, = -0.5 . Here 0, is the angle between a and b and xr= -0.5 ensures that the junction does not become coincident with quark 2. The limits b, and b, ensure that the junction does not become coincident with quark 3 and 1 respectively. With I( b,, b,, x,) defined as above, Z(0, cc, -1.0) = 1 is then simply the original normalisation integral. In this case [(O, CC,-0.5) = 0.93(0.86) is the probability that the junction is not at quark 2. The number in parentheses is that for the corresponding calculation in two dimensions. The integral when the junction is not at any of the quark positions is I(b,, b,, -0.5) = 0.78 (0.55). The constraints are seen to follow a regular pattern since I(b,, _ especially that ~80%
b,, -0.5) = [ 1 -3( 1 ~ I(0, cc, -0.5))]
in the three-dimensional case. The conclusion to be drawn of the phase-space integrand for a single nucleon contains
from this is an explicit
junction. The previous paragraph only deals with the three-quark single nucleon problem. However, for NN annihilation - as seen in figs. 3,4 and 5 - four-quark configurations also enter. For these the corresponding calculations - even in two dimensions become very complicated and do not warrant a detailed evaluation in the present article. In the notation of fig. 7b, imposing a single angular constraint to ensure junction J, does not coincide with quark 1 results in the normalisation integral again being reduced to 0.93 (0.86) as in the three-quark case. These numbers are the same since the angular constraint is independent of the positions of q,.? and q,,, and so the integrals separate to again give I(0, co, -0.5). Similarly, to ensure J2 does not coincide with q4 also gives 0.93 (0.86). If now it is assumed that the additional constraints ensuring J, does not coincide with q2 and J, with q, follow the same regular pattern as proven in the three-quark case, then the probability for finding a single junction becomes [ 1 -2( 1 - I(0, m, -0.5))]
= 0.86 (0.72)
A. hf. Green, G.Q. Liu /
602
and for finding
N@urzniililQti#n
two junctions [ 1 - 4( 1 - I(0, a3, -0.5))]
= 0.72 (0.44) .
The above estimates indicate that the probability for configurations involving a junction increases as the number of quarks and antiquarks increases. Therefore, the action of plaquettes is enhanced since single plaquettes with the basic lattice spacing of -0.1-0.2 fm appropriate for the flux tube model can give the transition to two mesons. For configurations without junctions the plaquettes need to operate directly on the quarks, which requires a plaquette or series of plaquettes with nuclear dimensions z-1 fm.
References 1) A.M.
7) 3) 4) 5)
6) 71 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19j 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)
Green and J.A. Niskanen, Prog. Part. Nucl. Phys., Vol. 18, ed. A. Faessler (Pergamon, 1987) p. 9.3 T. Armsstrong er a!., Phys. Rev. D36 (1987) 659 M. Maruyama, S. Furui and A. Faessler, Nucl. Phys. A472 (1987) 643 A.M. Green and J.A. Niskanen, Mod. Phys. Lett. Al (1986) 441 A.M. Green, Proc. of Workshop on the elementary structure of matter, Les Houches, March 1987; ed. J.-M. Richard et al., Proc. in Physics 26 (Springer, 1988) p. 190 E.M. Henley, T. Oka and J.D. Vergados, Nucl. Phys. A476 (1988) 589 R. Kokoski and N. isgur, Phys. Rev. D35 (1987) 907 N. lsgur and J. Paton, Phys. Rev. D31 (19X5) 2910 C. Dover, P. Fishbane and S. Furui, Phys. Rev. Lett. 57 11986) 1538 C. Dover, Proc. 2nd Conf. on the Intersection between particle and nuclear physics, Lake Louise, Canada, 1986, ed. D.F. Geesaman (AIP conference proceedings No. 150) p. 272 G.B. Chadwick et al., Phys. Rev. Lett. IO (1963) 62; M. Foster et al., Nucl. Phys. 66 (1968) 107 S. Ahmad er af., Phys. Lett. Bl57 (198.5) 333 A.M. Green, Mod. Phyz. Lett. A2 t.1987) 617 W. RrSckner el al., Phys. Lett. B166 (1986) 113 H. Gcnz, Phys. Rev. D28 (1983) 1094 J. Merlin and J. Paton, Phys. Rev. D36 (1987) 902 F. Lenz et a/., Ann. of Phys. 170 (1986) 65 A.M. Green and J.A. Niskanen, Phys. Scripta 34 (1986) 550 S. Furui, Z. Phys. A325 (1986) 375; A327 (1987) 241 A.M. Green, V. Kuikka and J.A. Niskanen, Nucl. Phys. A446 (1985) 543 A.M. Green, J.A. Niskanen and S. Wycech, Phys. Lett. 8139 (1984) 15 V. Bernard, R. Brockmann and W. Weise, Nucl. Phys. A412 (1984) 349; A440 (198s) 605 S. Kumano and V.R. Pandharipande, University of Illinois preprint ILL-(NUj-87-50 M. Kohno and W. Weise, Nucl. Phys. A454 (1986) 429 J.A. Niskanen and A.M. Green, in preparation A.M. Green and J.A. Niskanen, Nucl. Phys. A430 (1984) 605 A.M. Green and J.A. Niskancn, Nutl. Phys. A412 (1984) 448 M. Maruyama and T. Ueda, Prog. Theor. Phys. ‘74 (1985) 526 J.A. Niskanen, V. Kuikka and A.M. Green, Nucl. Phys. A443 (1985) 691 A.M. Green, J.A. Niskanen and S. Wycech, Phys. Lett. B172 (1986) 171 B. Moussnllam, Nucl. Phys. A407 (1983) 413; A429 (1984) 429 C. Amsler and G. Smith, Proc. of Workshop on the elementary structure of matter, Les Houches, March 1987; ed J.-M. Richard et al., Proc. in Physics 26 (Springer, 1988) p. 197 and private co~~n~uilication~
4.M.
32) 33) 34) 35) 36) 37)
Grrcx,
G.Q. Litr / NN un~~~~ilufio~
L. Tauscher and M. Chiba et al., Proc. of IV LEAR Workshop, Villars-sur-Ollon, Switzerland, C. Amsler ef al. (Harwood, 1988) p. 397 and 401 C.B. Dover and J.-M. Richard, Phys. Rev. CZS (1982) 1952 A.M. Green and G.Q. Liu, Z. Phys. A, accepted J. Rosner, Proc. 2nd Int. Conf. on hadron spectroscopy, KEK, Japan, April 1987, p. 395 W. Toki, Proc. 2nd Int. Conf. on hadron spectroscopy, KEK. Japan, April 1987, p. 252 R. Bagchi and S. Basu, Mod. Phys. Leti. 3 (198X) 633 D. Bridges cf nl., Phys. Rev. Lett. 56 (1986) 215 G.E. Brown, Nucl. Phys. A358 (1981) 39~
602 ed.
NR
ANNIHILATION INTO TWO MESONS Effect of final-state interactions
Received 1 February 1988 (Revised 25 April 1988)
Abstract: An extension
of the usual quark rearrangement model is made in such a NNIS) into two s-wave mesons takes place with the ‘P,, vertex acting or+ at the expense of introducing, into the final state, mesonmeson rescattering. understanding of several S-wave branching ratios but does not explain the
way that the transition otzce. However, this is The result is a partial so-called TP puzzle.
1. Introduction Over the last few years the study of nucleon-antinu~leon (NR) annihilation at Tow energies has received much attention both experimentally and theoretically. On the experimental side the main advance has been the construction of LEAR at CERN and to a lesser extent the antiproton facilities at Rrookhaven and KEK [see ref. ‘) for a recent review]. These have enabled not only elastic and inelastic p and ii cross sections “), but also many branching ratios into specific meson channeis to be measured. The subsequent results have given impetus to the corresponding theoretical studies. However, a newcomer to this field is, unfortunately, confronted with a somewhat confusing situation due to the plethora of alternative models. For example, in the case of NR annihilation into two mesons - the branchings most carefully studied at LEAR - the various models can, broadly speaking, be constructed from the components depicted in fig. 1. Figs. la,b show two possible elementary vertices for the basic quark-antiquark (yq) annihilation. In the first - the ‘P,, or pair creation model - the qq annihilate directly into the vacuum, whereas in the second - the “S, or one-gluon-exchange
model - the qq annihilate
into a gluon
(or effective
gluon). Given the basic qq vertex, the processesin figs. lc,d can be evaluated leading to four different models for NN annihilation into two mesons. In the literature (see e.g. ref. ‘)) three of these have been labelled as: (i) R2 when fig. la is inserted into fig. lc, (ii) S2 when fig. lb is inserted into fig. lc, (iii) A2 when fig. la is inserted into fig. Id. Different groups have different criteria for preferring one or other of the four above models. The most common of these criteria is that the chosen model explains some 0375-9474/M/$03.50 @ Elsevier Science Publishers (North-Holland
Physics
Publishing
Division)
R.V.
582
A.M.
Green, G.Q. Liu / iVfl
annihilation
a) M,
M2
N
b)
d)
Fig. 1. (a) The ‘P,, or pair creation-model for qq annihilation. (b) The ‘S, or one-gluon model for qq annihilation. (c) The yq vertex to first order leading to models R2 and S2 when combined with (a) and (b), respectively. (d) The qq vertex to third order leading to model A2 when combined with (a).
selection of the data with apparent disregard for other data and also the need for some underlying, more microscopic theory. An example of this is the use of the ‘S, vertex of fig. lb. When combined with fig. lc - to give the S2 model - it is able to explain, to some extent, the experimentally observed NN annihilation in relative S-waves into the two s-wave mesons 13). Such a branching is not directly possible for the combination of figs. la and lc - the R2 model. However, it has been shown 4*5) that the basic ‘S, vertex is not capable of explaining the more elementary process of meson decay into two mesons (i.e. M, -+ M,M3), whereas the 3P0 vertex is very successful “). Furthermore, in ref. “) the 3Po vertex has been put on a firmer footing by showing how it can be understood in terms of the flux tube model ‘). In this model - based on the strong coupling limit of QCD - the gluon fields are introduced explicitly as “tubes of gluon flux” connected to the quarks and antiquarks. Meson decay can then be expressed as a breaking of the initial flux tube into two flux tubes both of which give rise to a meson. This indicates that the ‘P,, vertex is to be much preferred over the ‘S, vertex, and that the failure of the former when in fig. fc the R2 model - is a reflection of an inadequate treatment of the more complicated multiquark system entering into NN annihilation. Having dismissed the use of the %, vertex of fig, lb on the grounds that it fails in meson decay and is not based on some underlying microscopic theory, how does the ‘P, vertex explain its most obvious failure to give NN(S) + VP? Two alternatives appear in the literature. In the first, advocated by Dover and his coworkers *“), the 3P0 vertex acting three times is inserted into fig. Id - the A2 model. This readily
A.M. Green,
G.Q. Liu /
NN
annihilation
gives NN( S) + rp. This model gives cu~~urff~le branching
583
ratios in Born approxima-
tion for the annihilations B,=NN(“S,)+TQ, B2= N6J(“So)+
(1.1) xp,
(1.2)
whereas experimentally B, 9 B,. Since @(“‘SC,) + rr -p,, is observed 36), there is no absolute selection rule that B2 = 0. This inhibition of NN + n-p in the isospin one channel has been called the “the mp puzzle” “). A similar, but apparently less extreme, inhibition also occurs in NN P-waves ‘I). However, as shown in refs. ‘,‘), the branching & could, in principle, easily be inhibited with respect to B, since in NN(“S,,) the rrp channel must compete with other strong channels - notably 7r’ff: On the other hand, the second mechanism for explaining NN(S) + 7rp involves only a single ‘P,, vertex and is related to the approach in ref. “). This is based on fig. lc but includes in addition a final-state interaction between the two mesons and can be expressed as the chain NN(S)
:M:M;F
:M:M;(,=O)
:M;M:(I=
1).
(1.3)
Here step 1 is simply a rearrangement of 3 quarks and 3 antiquarks into one s-wave meson MT and two p-wave mesons MP, one of which is the P having the quantum numbers of the vacuum. In step 2 the F is annihilated by the ‘Pt, vertex. In step 3 the resulting s- and p-wave mesons in a relative s-wave (i.e. f = 0) rescafter to give two s-wave mesons in a relative p-wave (i.e. f z=1). More details are given later. However, it should be emphasized that step 1 is a unique procedure and introduces no more parameters than the more usual rearrangement NN(S) --zM:M;M;
.
(1.4)
Expression (1.3) is simply exploiting the fact that the three quarks of the N and the three antiquarks of the N are not necessarily in relative s-waves even though the N and I? centres of mass are in a relative S-wave. Given the quark wavefunctions of the N and N, the above p-wave components M~M~E are simply the next terms in a partial-wave expansion, and so are as unique as, for example, the p-wave component in the partial wave expansion of a plane wave. The fact that p-wave effects can be important, even for the low-energy NN system, should not be a surprise since both experimentally 13) and theoretically it is known that NN annihilation is predominantly in P-waves for p laboratory momenta greater than about 200 MeV/c - see figs. 2 and 1.5 of ref. ‘). This effect is a combination of the strong real distorting potential in the initial state and also the internal momenta of the quarks in the hadrons. The second approach based on fig. lc also has the advantage of partially explaining the rp puzzle without resorting to an inhibition through the competition between the rrp and mf channels as in refs. ‘.“). In terms of quarks the
A.M.
5x4
Green, G.Q. Liu / IV@ ~n~iil~~~t~~n
chain in expression (1.3) can be depicted as in fig. 2a, where V, is the matrix element representing steps 1 and 2, and V, represents the meson rescattering of step 3. Unlike ref. “) the final quark rearrangement in V, is now considered as a model for a final-state interaction between thought of as a mathematical
the two mesons, whereas in ref. I’) it was simply recoupling, which does nof allow NN(S)+ M:Mz.
This latter statement is contrary to the conclusion of ref. “) which contained an error to be discussed later. Topologically, the processes in figs. ld and 2a are less different than appears at first sight. This is because the chain in expression (1.3) employs meson wavefunctions with good spin and isospin. For exampIe, in the transition NN + rp, application of the quark-line-rules (see ref. I”)) demands that fig. 2a must be of the form shown in fig. 2b and not fig. 2c. To see this it is sufficient to only consider the spin and isospin structure of the process and ignore the internal orbital angular momenta. In this case it follows from G-parity that the ‘P,, operator requires an w-like configuration for it to be effective and so in fig. 2c the initial NN state would be expressible as ~pw. Here the n-p simply arise since they are in the final state being considered. This is a clear contradiction, since the final and initial states have opposite G-parity, and shows that the initial quarks and antiquarks must have changed their flavour as in fig. 2b*. Even though, from the point of view of quark lines, fig. 2b is topologi~ally the same as fig. Id their interpretation is very different. In fig. Id the qq annihilate and are recreated only with the vacuum quantum numbers, whereas in fig. 2b the qij annihilate and are recreated with the quantum numbers of the meson under consideration. Furthermore, the flavour transition in the dashed box of fig. 2b only involves Clebsch-Cordan coefficients which introduce factors of J$ and should not to be thought of as some second-order process generated by, for example, two ‘PO vertices as in tig. Id.
b)
al
c)
Fig. 2. (a) Fig. Ic followed by the meson rescattering MYM g- M:M:. (b) A redrawing of fig. 2a where the dashed box represents an interacting meson state with a change of quark flavour. (c) Another redrawing of fig. 2a without a change in quark Ravour. * The authors
thank
Dr. M. Maruyama
for pointing
out this argument
A. h% Green,
As mentioned
earlier,
C.Q.
an attractive
Liu / Nfi
feature
~r~~li~7il~iiun
of the ‘PO model,
585
as the preferred
candidate for qq annihilation, is that it not only gives a better description of meson decays (M, + M,M,) than its rival, the ‘S, model, but can also - to some extent be justified in terms of flux tube breaking “). However, in NN annihilation into two mesons the situation is less clear since neither of the two approaches - figs. lc or Id with la, i.e. R2 or A2 - discussed in the previous paragraph gives obvious disagreement with experiment, provided meson-meson rescattering is included as in the present paper. The question then arises concerning how - and if - they can be justified, to some extent, by some underlying more microscopic description. Unfortunately, at present, nothing as detailed as the flux tube model for meson decays “j has been achieved. However, this framework - based on the strong coupling limit of QCD - suggests that the alternative of fig. lc with la - the R2 model - is more reasonable than the choice of fig. Id with la - the A2 model. For the R2 approach the flux tube model would give the scenario in fig. 3. Namely the N and N are each described in terms of 3 quarks (or antiquarks) with 3 flux tubes meeting at junctions J, and J2 [ref. ‘)I. A qq annihilate by the ‘P,, vertex (Q), in the same way as the flux tube breaking mechanism for meson decay, to give a baryonium state. The two junctions are then annihilated by a plaquette operator (P) to give the final two mesons M,M,. In the strong coupling limit P is of 0(1/g’) and Q of O(1). Therefore, P is in principle only a pe~urbation compared with Q. However, for the validity of the flux tube model g Schulz be ==I, a value such that the corresponding lattice spacing a = (0.1-0.2) fm is smaller than hadronic sizes, but yet large enough so that topological mixing may possibly be treated as a perturbation, see refs. ‘.I’). In this case P and Q could well be comparable. It is seen, therefore, that the R2 approach can be qualitatively explained with the flux tube model. On the other hand, the A2 approach requires the scenario of fig. 4. There, the first qq
Fig. 3. R2 scenario for Nfi annihilation into two mesons via figs. la and lc. (a) Action of the ‘PO operator (0) to give: (h) Baryonjum. (c) In b~ryonium the junctions J,, .f2approach each other to be ~noihilated by the playuette operator (Pi to give: (d) Two mesons M,M2.
586
Fig. 4. A2 scenario for Nti annihilation into operator (Q) to give: (b) Baryonium as in fig. (d) a hybrid meson. (e) In the hybrid meson which
two mesons via figs. la and Id. (a) Action of the ‘P,, 3b. (c) In baryonium Q operates a second time to give: Q operates a third time to give: (f) two mesons one of is still a hybrid.
annihilation with the ‘P, vertex (Q) again results in a baryonium state as in fig. 3. But the application of a second Q results preferentially in a non-exotic hybrid meson - identified by the presence of a flux loop ‘). The hybrid is non-exotic since it must have the same quantum numbers as the original NN state. The third Q then creates another meson, but one of these two mesons is still very likely to be a hybrid. It should be added that this flux-loop criterion for identifying a hybrid is not absolute, since flux loops can arise to some extent in normal mesons through the action of the kinetic-energy operator. However, the latter loops have a size corresponding to that lattice spacing (0.1-0.2 fm) necessary for the validity of the flux-tube model. On the other hand, the flux loop in fig. 4 has dimensions of about 1 fm the typical size of a nucleon. It should be added that this favouring of fig. lc over Id in the flux tube model is in addition to the basic flux tube idea that Q should be treated as a perturbation. The usual argument given for this last statement is the experimental observation
that meson
widths
(proportional
to Q’) are small compared
with their
masses, see ref. “). The scenario of the previous paragraph assumes that the flux tubes of the N and N are separately of the “Mercedes” type - all three meeting at a junction. However, it is possible - but less probable (see appendix B) - for the angle between two sides of the 3-quark triangle to be greater than 120”. In this case the states of minimum energy do not contain an explicit junction - as depicted in fig. 5. The first qq annihilation can still occur with the usual ‘PO vertex (Q) giving a baryonium state but without junctions. But such a state is again less likely than one with junctions (see appendix B). However, the next step for R2 model is less favourable than in fig. 3, since it now needs to bring q, and q2 also together. Earlier the transition to two mesons occurred with a basic plaquette of size 0.1-0.2 fm and left the quarks untouched. For A2 this second step avoids this inhibition.
A.M.
Green,
G.Q.
Liu J NN
annihilation
587
Q
w-----+
-
e---+0---+
f)
e) Fig. 5. A? scenario
when the three- and four-quark
configurations
do not contain
flux junctions
The discussion of the previous two paragraphs is of necessity very qualitative. However, it suggests that the scenario of fig. 3 - a two-step process involving one Q and one P - leading to two mesons could be favoured over the scenario of fig. 4 - a three step process involving three Q’s - leading to two mesons one of which is preferentially a hybrid. In the opinion of the author, such considerations involving the gluon degrees of freedom are necessary in order to put the theory of NN annihilation on a firmer footing and so eliminate some of the many competing models involving only the quark degrees of freedom. The main objection to the present line of reasoning is that it depends on an N and N model based on flux tubes - a far from universally accepted notion. However, it does give a definite prescription for discussing NN annihilation in a way that is related to the much simpler problem of meson decay (i.e. M, + M,M,). If it is accepted that NN annihilation into two mesons is dominated by a two-step process involving a qq annihilation with Q and a junction annihilation with P, the next question to be answered concerns the nature of the plaquette operator P. As seen in fig. 3, P only involves the flux tubes in the vicinity of the two junctions, and so the states of the quarks are not directly affected. This suggests that, in terms of quark operators, P acts as a spin and isospin singlet. This is the rationale for saying that the two-step QP process can still be depicted as in fig. lc since this only shows quark lines which are unaffected by P. Of course, P could well generate a complicated radial dependence in terms of r and r' - see fig. lc - but this would only have a secondary effect on calculated branching ratios. It should be added that, for the process of interest in the present paper (i.e. NN(S)+ M:Mz), the rescattering of the two mesons at step 3 in expression (1.3) introduces a further interaction. This is usually thought of as being mediated by a plaquette P as in fig. 6 - see ref. ‘) - or by the flip-flop approximation to this as in ref. I”). Therefore, in all, fig. 2a is proportional to QPP and fig. Id to QQQ. On a
585
A.M.
Green, G.Q. Liu / NR
+
annihilation
e ML
M3 >
Fig. 6. Meson-meson
scattering
[M,MZ+
M,M,]
through
a plaquette
(P)
less microscopic level, meson-meson scattering could possibly be treated in terms of s- or t-channel meson exchange. However, this would require the knowledge of many unknown triple meson vertices. Furthermore, since the total energy is about 2 GeV such s-channel processes as rB+ MR + up are far from being resonant for nodeless s-wave mesons e.g. the w. In sect. 2 the formalism is outlined with details being relegated to appendix A. In sect. 3 the results of this model are presented and compared with those of ref. “). In sect. 4 a summary and conclusion are given.
2. Ni%(S-wave)-+ MFMs In order to generate two s-wave mesons (e.g. VP), when NN annihilate in a relative S-wave by means of the chain in expression (1.3) - depicted as fig. 2, four distinct steps are involved: of the (i) The first stage NN + M~~M~~E~~ is simply a ~ut~e~ff~~cal recoupling angular momentum and isospins of the three quarks of the N and the three antiquarks of the NJ. At this stage the objects Mr,, M& and ~~~ have the angular momentum and isospin of s- and p-wave mesons but nothing is said about their radial wavefunctions. The actual recouplings are
where E, 8, H, 5, etc. have the quantum numbers of the usual p-wave mesons. In this equation, for clarity, the F_3h’~ in each term are not displayed. The amplitudes of the [&MS] configurations have an additional factor of v’?? incorporated to allow . recouplings in ref. “) differ by for the identity of &1?5an d ~36. The corresponding now having an additional factor of fi - in addition to misprints in the [err] and [Sp] channels. The reason for this is purely historical and is explained in ref. I’).
A.M.
Earlier
these particular
ck?ri,
recouplings
were made with the condition
the coefficients (squared) should equal 1296 (=36’). are not all of the same form e.g. MyMliM.7 or MPMIM: a factor of fi
so that comparisons
of like mesons
e.g. M”MzM:
x39
G.Q. Lilf / ,Pm ff?l~~i~~iial~~~~7
that the sum of
However, when the mesons it is convenient to introduce
can be made more directly with the coefficients or MrMeM_!. This factor of JT is the statistical factor incorrectly omitted in ref. “) an d reinstated in ref. “). The sum of the coefficients 48,48 * 11, 48 * 17 and (squared) for the four states ‘I” ‘s-1-’S, are, respectively, 48 * 25 - each of which should now be compared with 3 * 36’. in all it is seen that the four NN states when expressed as M~MFF account for only i of the total probability (4 * 3 * 36’). (ii) The second stage M~,M~,F -+ M:;Mq, simply involves the usual ‘P,, operator for qq annihilation
“‘), namely O(‘P,,) = K,[Y<,X(q,q,,
s = 1)].“““‘-“6(c’),
(2.2)
where c’ = rT - rh in the notation of fig. lc. The value of K, is discussed in ref. “‘), but apart from colour and angular momentum factors it is basically proportional to the corresponding coupling constant extracted from meson As shown in eqs. (A.6) and (A.7) this results in an interaction V,(r,,,
decays M, + MLM3. of the form
Y) =(M:M~(~,,)/O(‘P,,)INN(U)) =1Y3(1-~(ff+p)r’)exp[-:r~(cu+3P)-~~~i,],
(2.3)
where rhll is the vector between the centres of mass of the two mesons. (iii) At this stage the two mesons propagate before rescattering. This is achieved by the propagator (2.4) where dE =2M+E~-h~-JM~+q~-~~+lir(M,)+~ir’(M,) depends on the initial energy of the N and ti i.e. rest mass (2M) plus their energy in the centre of mass (E, h,), and the energies (E, =jMf+ Q’) plus half-widths (~~‘(~~)) of the two mesons. The factor ,fz,) takes into account that the S-, 77 and q’ are not simply qq states with q = u or d. The 9 and 11’ wavefunctions are i( uii + L@T v’%L~ - corresponding to a mixing angle of =- 10” - whereas the z- wavefunction is =J (qCj+ multiquark components) - see ref. ‘I). Therefore, .f‘z,, is taken to be (1)“-’ “I) where are the number of r‘s, T‘S and ~“s in the intermediate state. The factor yc” M,M,/ E, E, is an attempt to simulate the correct meson normalization factor which presumably would appear naturally in a more complete relativistic theory see ref. “) for a recent discussion of such factors. Fortunately, this somewhat controversial factor does not lead to significant changes in the final results. (iv) The fourth stage i.e. MF,Mz,( I = 0) -+ M:,M$( I = 1) involves a second recoupling in which the antiquarks 4 and 5 interchange their quark partners 1 and 2 to
590
A.M.
Green, G.Q. L.iu J hJN un}zi~~iIafio~z
give two s-wave mesons
in a relative p-wave. The actual
recoupling
is
(2.5)
where r” is the vector between the c.m.‘s of the two outgoing s-wave mesons, k the corresponding relative momentum and r14 is the internal radial vector of the p-wave meson Mf, with quark constituents qlq4 - see ref. “) for the more general case involving three mesons. Compared with ref. 12) the recoupling in eq. (2.5) has three differences. Firstly, there is a factor of two less. This arose earlier as [ 1 + ( -l)c’l+Cil] where Cr., were the initial and final G-parities and acted as a check on the programme. Later it will be more convenient to incorporate this effect elsewhere as part of the statistical factor. A second difference compared with the recoupling of ref. “) is that there is now an additional isospin phase (-1)q1’7z multiplying each amplitude for MFM;. This arises due to the difference in definition of the isospin component for the r and p quark wavefunctions. Earlier, for example, the 7~’ wavefunction was written as rrt = ud, whereas now it is more convenient to replace ud by ud - the difference being a sign change for T = I mesons. The final difference compared with ref. “) is that S-coupling (not SL) is used - leading to a phase (_l)S-P.ti-S, iJ,* These last two differences are absolutely necessary, since they correspond to using LS coupling and the same notation for rr, p as was employed in the derivation of eq. (2.1). Their omission in ref. “) was incorrect and their inclusion has a profound effect on the results claimed in ref. “f. There stage 3 of expression (1.3) was considered simply as a mathematical recoupling. As shown in eq. (A.12), including the radial wavefunctions results in an interaction of the form
= K,v”exp
[-$(r”‘+
rz)] Y,(i)
* Y,(P),
(2.6)
where I&(P) is the vector between the centres of mass of M~M~(M~M~) and the factor of r” is characteristic for a p-wave final state and h is the strength of the interaction. This model for meson-meson scattering has the same philosophy as the rearrangement model for NG -+ M,M?M,. The state dependence is assumed to be given completely by the overlap between the initial and Final meson wavefun~tions. In ref. “) it was shown that, in general, the same state dependence emerges if the transition takes place with the same harmonic confining potential that generated the meson wavefunctions. A similar improvement could be incorporated here to give a model for h.
Greerr, G.Q. Liar /
A.M.
Nfl
591
a~l~li~ij~uri~~fl
Combining the four above stages and taking the Fourier transform give the matrix element for NN(S) annihilation into two s-wave mesons of momentum k in a relative p-wave
as
=
,yk
e.
h-./j
q7dq
C
(2.7)
hl:M;
Here the summation is over the 8 p-wave mesons of ref. “) and the 5 s-wave mesons - the n’ included. The two k-dependent factors can be considered as part of the overall P-wave phase-space factor to give [PS]=(2S+l)k”exp(-k’/P),
(2.8)
where S is the spin of the NN state and
N.B. In ref. “) the phase space incorrectly analogous
to .f‘& andJ’;i
kJ instead
contained
in eq. (2.4) must also be introduced.
of k’. Two factors
These are
f” = (]/\I?)“_+‘%, i 3 7111
ii) where II,, n, are the number
of m’s, 71’s and rl”s in the final state.
.f; = JM&LI Et,,,,
(ii)
where E,,,, = 2M -t- EC,,,. . This is the factor advocated in ref. “) and is in place of the more usual &?m. Again this change arises through introducing by hand the relativistic normalization factors (JM,/E,) for the final meson ~vavefunctions. With this notation
the branching
ratios can be expressed
BR(NN-t
M;M;)
as
=[PS]C,j6lf‘;f:,,/‘,
where M is the same as M[ ] in eq. (2.7) but without the two k factors, an overall constant that includes the initial state interaction.
(2.9) and c’rs is
3. Results Given the expression for the branching ratios in eq. (2.9) the ideal approach would be to calculate directly the initial-state interaction (ISI) in CTs using some NN optical potential. However, in the opinion of the authors, this is too ambitious at the present time - for the following reasons: (a) The most complete studies of NN annihilation mechanisms at the quark level are able to achieve only about 70% of the pj3 integrated annihilation cross section, see refs. “*“). It is not known in which channels the models are at fault and so it is not known which of the calculated initial-state interactiotls (ISI) can be reliably
A.M. Green, G.Q. Lilt / NN ~riiii~lilaii(~~
592
estimated. Several authors, e.g. ref. ‘“), have shown that Born approximation can overestimate annihilation cross sections by up to two orders of magnitude and so the C,, are indeed crucial factors for absolute branching ratios. (b) In eq. (2.9) C,, is not the only unknown tion, with a more microscopic
model involving
in eq. (2.2) could well have a strength
factor. As discussed
in the introduc-
gluon effects, the effective 3Po vertex
K, that is diRerent
to the one extracted
from
meson decays, In view of this, here eq. (2.9) is used to predict the values of CT., needed to give the experimentally observed branching ratios. Since, for each value of T and S, there are several possible annihilation channels, the constancy of C~s(~~~~) - for fixed TS - is a measure of the present model’s reliability.
The values of C,,
“SO:
ww
expt (%)“)
1.4*0.6 11.9zt5.1 34.32 28.6 7.3 10.0 13.6 13.2 0.4
(‘,,,,(A) M(1) “1 M(2, wH) ‘) C,,,,(B) C,,,,(C ) G,,( 0 ) C;,,( El BR(ref. ‘))(I%)
“‘S,,:
P”
expt (X) C,,,(A) M(I) M(2, wB) C,,,(B) C,,>(C) c,,,i ut C,,,t E) BR(ref. ‘))(%
“S,: (% ) C,,,(A) M(I) MU, 4 co,(B) c;,,(C) C,,,(D) C,,,(E)
expt
BR(ref. ‘) i( a)
I
211.5”) 4.3 * 3.2 6X.2 65.5 2.9 3.5 4.6 5.1 2.3
w 1.3+0.2 61.9i9.5 14.49 25.5 48.5 62 45.3 46
predicted
by eq. (2.9)
pop” 0.12*0.12 22.2 i- 22.2 7.35 47.4 11.4 21 13.7 5.6 0.4 “I,
CO.2 1) 10.9 9.6
w 5.2 i-0.8 21601330 4.9 I 14.3 1760 1080 1584 2160 4.2
~‘W
4.0 * 0.6 ‘) 25.5
m
TTO
expt (%I 1
<
0.65x0.16 6.43 * 1.S8 31.80 17.1 5.7 6.4 6.4 4.8
1
C,,(A)
C66
M(l) M(2, PC) C,,(R) C,,((‘) C,,(U) C‘,,(E) HR(ref. 31)(?:v)
12.28 5.9 ( 42 r26 <6X ~6% 0.x
<9*3 <178i59 17.6, 14.0 ‘1 15.6, 70. I Cl43 ~185 Cl77 138
C‘,,(A) M(l) M(2, /)F)
(‘,,(U) C’,,(C) C.,,CD) C,,(E) HR(ref. ‘))(‘!:I)
0.48
7rETT
PP
expt (“% i
P77; 0.11 r0.06 0.3210.1x 5X.46 47.73
0.35 * 0.02 8.4-LO.S x 10’ 0.204
0.56 4.5 x 1oJ 1.7X IV 4.6X loJ 8.0 x
I o1 0.IL
“) Seeref. ‘1). “) The value of M( 1)= [ PS]“‘J~.j&J$
used in eq. (2.9). ‘) ‘The largest contribution M( 2, M:MI) from the sum in eq. (7.7). “f An average value of the proposed experimental ratios: 0.7 i 0.3,2.
and 3.9-t 0.6. ‘1 M(pp, s’=O) extract C,,(A). ‘) Predictions
and
M(pp, s’= 21, M’(s’=0)+M’(.s’=2)
assuming
C,,,ig{j)
= C,,,(pw)
being
I + 0.2,
used
to
and CC,,iv’w) = C,,,( 77~).
In this way a self-consistent procedure is produced. Firstly, the microscopic models are developed until the C,,Y(M:M:) are constant for a given TS. This lends confidence for using these same models to produce effective NR optical potentials. Only then can the C, generated by these potentials be expected to be realistic. In table 1 are given the values of C,, predicted by eq. (2.9). The C,,s are shown using various approximations. Case A is the most realistic model and includes: (a) Meson widths that are energy dependent as in ref. “). (b) The relativistic factor fk of eq. (2.4). (c) Nucleon and quark wavefun~tions given by cy = 1/R’ and p = 2cy/tg with R = 0.6 fm. In case B the _fg are set to unity, case C is the same as case A but with the ,f:” of eqs. (2.4) and (2.9) set to unity. Case D is the same as A but with energy independent meson widths. The meson widths and masses are the same as those in ref. Is) except that ir(e) = 140 MeV and $I’( B) = 75 MeV, which appear to be more in line with the current experimental numbers. These changes, from the earlier values of 200 and 68 MeV, respectively, do not lead to significantly different values
594
A.M. Green, G.Q. Liu / N1\Sannihilation
for the ratios of the Cr,. Case E is the same as A except that the nn’ mixing angle is -19” as opposed to the -10” used elsewhere - for reasons given later. Before discussing the details of table 1, it should be pointed out that if Ph, in eq. (2.4) is treated in the closure approximation (i.e. 3E # 3E(M~M~) andf‘& =J‘;1= 1) then the matrix element M given by eq. (2.7) becomes zero. This complete cancellation between the various terms in the summation over MT and My is a general property of processes that involve the “‘hair-pin” mechanism depicted in the dashed box of fig. 2b”. In ref. “) this rule was the reason for the S =0 matrix elements vanishing. However, the S = 1 matrix elements did not vanish due to the LS++ SL error mentioned earlier. This had the effect of giving the A,-meson contributions with a wrong sign. Since closure gives zero for M in eq. (2.7), it is seen that the present non-zero results are dependent on the fact that the masses and widths of the s-wave and p-wave mesons are not degenerate and that the j$, are not unity. Another limit for P, in eq. (2.4) is when the sum over MTMI is restricted to Mg= F(1.300 MeV) and the ,j’$’ are set at unity. In this case, for a given TS, the branching ratios are the same as those in ref. “) - numerical confirmation of the topological similarity of figs. Id and 2b. In table 1 the values of M(l,2) = rw’~‘l~.f,hd’~,l are given for case A - the M(1) - and also the largest term of the sum in eq. (2.7) - the M(2, MTM!). The comparison between M(1) and M(2) indicates the degree of canceltation - if any - when the sum is carried out. Several comments can be made about table 1. (a) The “S,, channel. Here for case A the C,,,, extracted from the ww and pp branching ratios are consistent with each other. This gives support for the meson rescattering mechanism proposed in this paper. The numbers corresponding to the A2 model of fig. Id and ref. ‘) are also given in table 1. These are not in such good agreement with experiment and correspond to Coo’s that differ by an order of magnitude. However, it should be remarked that - as in many other cases to be discussed later - the experimental branching ratios leave very much to be desired on the question of accuracy e.g. 0.12 Lt0.12 for the p”p” case. (6) The -“S, channel. Here the 7rp branching ratio has not been seen - the “rrp-puzzle”. However, this is not surprising since the present model predicts from eq. (2.9)
BR(w)
M’h)
BR(pw)
M’jpw)
R(np,‘pw)=---=---=
oo26.
’
Admittedly BR(pw) is not well-known, with the experimental numbers ranging from 0.7 * 0.3 to 3.9 ;t 0.6%) but in every case the above R (vp/pw) predicts small BR( rp)‘s ranging from 0.02*0.01 to 0.10*0.02%~. Therefore, with the present model, the “So no-channel is inhibited without introducing the competition from the QT~channel as advocated in refs. ‘.‘). Another interesting feature is that C,,-: C,,,,. In ref. “) an attempt was made to extract Crs from s,,,,(Nl;l(S)jMsM~M3). This resulted in table 2 which shows l
The authors wish to thank
Dr. N. Isgur for pointing
out this sum rule.
A.M. Green, G.Q. Liu / NN artnikilation
595
State dependence of cr.\ ( proportional to C’,\) from ref. “) as a function of h the strength of the NN+M:M;M: rearrangement potential State/A
(MeV
us0 “S I ‘So 3, S:
fm) =
1X
42
90
10.1 2.9
2.7 1.7
2.1
5.3 7.9
51.3 7.x
1.4 22.2 2.4
numbers C, - proportional to CT,s - as a function of /\ the strength of the NN(S)+MTMqN: annihilation mechanism. In that reference, i in the range 1530 MeV fm appeared to be the most favourable - see ref. “) for more details. In this range C,,,- :C,,,,, which suggests that the present picture for NN spin singlet transitions is a consistent one. (c) The ‘-‘S, channel. Here C,,,(nw) -O.O3C,,,( NIL))which shows that the present model cannot be the whole story in this channel. Furthermore, it is the VP-branching that is probably at fault for the following reasons: (i) C,,,(nlw) %5C,,,, - a relationship that is consistent with table 2 in the sense that C,,, can be enhanced considerably over the other Cls’s - an effect attributed to strong tensor effects and to &i components in the initial wavefunctions. This enhancement of T = 0 observables is already seen in NN total integrated cross sections now that lip scattering has made possible a separation of the T =0 and T= 1 cross sections ‘) - a feature already expected from phenomenological NN potentials “). Assuming C,,,(n’w) = C,,,( no) gives the prediction shown in table I _ a surprisingly large number (for possible reservations on this estimate see the later discussion on the pn’ channel). (ii) C,,,(7rp)=2OOC,,,,=5OOC ,() - numbers not consistent with table 2 and also not consistent with the belief that the C,-? usually reduce branching ratios by up to two orders of magnitude compared with Born approximation “‘). Such a large value of C,,, would imply either Born approximation for rp is qualitatively correct or that the S = 0 branchings are anomalously small. Both of the above comments suggest that the present model badly underestimates the np branching in this channel. In fact assuming C,,,(rrf) = C,,,(~W) predicts BR(“S, + np) = 0.15% = O.O3BR(expt). The ideal situation is, therefore the inclusion of some mechanism that enhances the =p branching in the “S, but not “Sg channel i.e. back to the -rrp puzzle with, perhaps, the additional clue that it is in the “S, channel where the problem lies. As will be discussed later in this section and also in the next section all the two-meson branchings that contain n-mesons are in general suspect. (d) The -73SIchannel. Here only C, ,(pn), C, ,(pn’) and C, ,( rr) can be extracted and are seen to differ by orders of magnitude. However, on the positive side it
596
4. M. Green, G.Q. Liu / NN at~}l~~lilatio~
should be noted that C,,(pn)=$Coo again hand, the value of C,,( 7~) c= 104C00 indicates mates very badly indeed the 71n branching.
consistent with table 2. On the other that the present model again underesti-
The result that C, ,(pn’) = O.O5C, ,(pn) is somewhat disturbing so reasonable. Unfortunately, this appears to be unavoidable
since C, I(~~) seems and arises because,
in eq. (2.8), k(pn) = 3.36 fm-’ and k(~n’) = 1.8 fin-’ to give fi( pn)[PS(p~)]‘/‘= 15.9 and .j’L( pn’)[PS(pn’)]“’ = 29.2 fm-“” - all other factors being equal for pn versus pq’. The only parameter in [PS] is p. Increasing this significantly (R going from 0.6 to 0.3 fm) does indeed improve the above 9, n’ problem, but only at the expense of other bratlching ratios “1. However, the above assumes that the n and 77’wavefunctions are of the form given after eq. (2.4), which correspond to a mixing are uncertain in two ways. Firstly, some angle of B(M) = -10”. These wavefunctions radiative decays involving the n indicate O(M) = -19”, which reduces the amplitude of the G-component in the n from -I/& to =I/&, see ref. “). This effect goes in the right direction, but as seen from case E in table 1 this still gives C, ,(pn’) F= O.lOC,,(pn). Another uncertainty in these wavefunctions, that would help this problem, is the amount of “glue-bail” component, which should be larger in the n’. However, at the present time this appears to be consistent with zero, see ref. 15). From table 1 it is seen that the A2 model of ref. ‘f in fig. id does much better for those branchings that contain w mesons. One obvious difference between the present calculation and those of ref. ‘) is that the .& factor after eq. (2.7) has introduced a factor of 4 into the C, ,( rrrr). If this is also put into the calculation of ref. ‘) then they also underestimate the VT branching by a factor 12. But this is still much better than what the present calculation gives. Of course, this pessimism is based on the is essentially the correct value for belief that since C,,(pq) =iCOtjr then C,,(pn) C,, - a point that could be clarified by better limits on the other “S, branchings. Even so, it is hard to see how the X~Tbranching could be significantly increased in the present model, since the exponential term in the phase-space factor of eq. (2.8) becomes so small for the appropriate value of k = 4.7 fm ‘. The comparison of M(2) with M(1) in table 1 shows that in some cases (e.g. pop” and z-r) there can be considerable cancellation when all p-wave mesons are included in the sum of eq. (2.7). However, in spite of this, the present model appears to be quite stable when different approximations are introduced. In particular, as seen by comparing C,,,(B, C, I;? E) with C‘,(A), removing the relativistic factor _fA of eq. (2.4) or settingf:” equal to unity or using energy-dependent meson widths or changing the 7~7’ mixing angle all lead to C,‘s that are qualitatively similar.
4. Summary and conclusions The results in the previous section suggest that the present model of a single ‘P,, vertex followed by meson-meson scattering [as in fig. 2a)] can account reasonably well for the relative branching ratios NN --f M;MQ in the two NN spin singlet states.
A.M. Green, G.Q. Liu / NN nnnihilut~on
597
However, in the spin triplet case, the model has problems - especially with decays involving r-mesons e.g. “S,(np) and 3”S,( rn). The reason for this could be that the pion is, in several ways, qualitatively different from the other mesons and this is not taken
into account
in the above
calculations.
Two of the most important
differences are as follows: (a) All quark models for NN annihilation consider the quarks to contribute only 0.6 fm to the experimental r.m.s. charge radius of 0.8 fm with the difference arising from a pion cloud outside the region where the quarks are confined. However, to the authors’ knowledge, the effect of this cloud is never included in NN annihilation processes. If these pions contribute more than the mechanism of fig. 2a, then in table 1 the values of C, involving pions become smaller. (b) Models for the pion [see refs.h.21.37)] indicate that the radius of the qq component is considerably smaller than in other mesons. Taking this into account has the qualitative property of allowing the meson rescattering mechanism of fig. 2a to give an s-wave meson and p-wave meson in a relative I= 2 state 33) e.g. NN(“S,,) + rf. Such a decay is not possible in the present model where all mesons have the same value of p for their quark wavefunction oscillator parameter. Furthermore, it is found that increasing /3 for the pion by a factor of four, corresponding to a r.m.s. radius of 0.24 fm, increases dramatically matrix elements involving pions. In this way a much more consistent model emerges “). In addition to these problems arising because of the pion structure there is also the problem that, so far, the annihilation mechanism has ignored the explicit effects of gluons. As discussed in the introduction, such gluons in the form of e.g. flux tubes could well modify some of the conclusions reached in their absence. Hopefully, these modifications still leave partially intact the predictions of the ‘quarks-only’ models. For example, if plaquettes play an important role as in fig. 3, then their lack of any spin or isospin dependence would leave unaltered branching ratio predictions. However, any change in the r radial dependence - see fig. Ic - would modify the initial-state interactions and so generate an additional T,S dependence. Also any change in the r” radial dependence in eq. (2.6) would modify the relative branching ratios into different mesons. In this article the main aim was to give, for NN annihilation in S-waves, a partial understanding of the rp puzzle. However, there is a similar problem for P-waves where it is observed that the annihilation into n-p is again inhibited in T = 1 states. In particular, experimentally, R
=
WNN(33~,)+
l
nip.)<
' BR(NN("p,)+ n'pr) Now the basic diagram
shown
in fig. lc contributes
NN(P)+ to give R, = 18 - in gross disagreement
M;M;e
(4.1)
.
directly
through
+ M;“M;
with experiment.
the chain (4.2)
However,
there is now the
59s
chain NN(P)-,MPM~E~M~M,P~M~M,S. This extension
to P-waves
has some additional
earlier S-wave case. For example, eq. (4.3), be continued to include
(4.3)
complications
compared
with the
the chain in eq. (4.2) could, in the philosophy MsMs-+ MfMz.
of
The authors wish to acknowledge very useful conversations with Drs. C. Dover, N. Isgur, M. Maruyama, J. Niskanen, H. Pirner, J.-M. Richard and N. Tornqvist.
A.l.
DERIVATION
OF V,(r,,r)
In the notation
IN
EQ. (2.3)
of fig. lc a =$(r,+r4),
b=t(r,+r,),
a’ = r, - r, ,
c=t(b+rJ,
b’= rl--rs,
c’=r,-r,,
(A.1)
r and rM are the vectors between the N and N, M, and Mz respectively. R and R’ are the coordinates of the initial and final centres of mass. The matrix element V, in the chain (1.3) can then be expressed symbolically as V,(rM, r) = M,[NN(S)+ =
M:Ml]
da db de da’db’dc’
WY * 0(3Po) * w” e DF, ,
(A.2)
where WN=N~exp[3~uRZ+~(wr2-cu(a2+b;“+c2)-~a(a”+b’2+c”)] are the NN internal
radial
wavefunctions,
W~=N:N~b’Y,(b^‘fexp[-aP(a”+b’“)]Y,(~~) are the two-meson
internal
(A.3)
wavefunctions
(A.4)
and
DF,=G[~(a’+b’+c’)-r]S[(a-b)-r.,]6[~(a+b)-R’]S[f(a+b+c)-R]
(AS)
are the &functions relating the six basic coordinates r, with the “observed” coordinates r, q.,,, R and R’. Here (Y,/3 are the oscillator parameters for the wavefunctions describing the N (or N) and mesons respectively and NO and N; are the normalisations of the nucleon (or N) and the I-wave mesons respectively. The latter have the form
A.M. [see
ref. “1 for more details).
Gwen, G.Q. Liu / NN
599
u~~nihila~ion
@(‘P,,) is the matrix element
of the ‘PC,vertex
ing q&, from eq. (2.2). It is seen that the a, b, c and a’, b’, e’variables dR dR’ -----yexp(iP.R-iP’.R’) (2,).
= S(P-
annihilat-
separate
to give
dadbdcexp[3aR’-cu(a’+h’+c’)]
exp
P’) exp [-P’/24~2](371./2ct)~”
[-_SN&].
(A.61
As always with the ‘P,, model, since the basic interaction O(3P,,) is not galilean invariant, the final matrix element depends on the c.m. momentum P and so the usual prescription P = 0 is imposed. This problem does not enter when the ‘PO model is interpreted in terms of ftux tubes as in ref. ‘1 and justifies this P = 0 condition. On the other hand, a’+b’t-C’ ---r
da’ db’ dc’ 6
N~N~N9exp[-~(a:+P)(a”+h’~)+~ar’]
3
(27r)?‘l
R, exp
=9JZ(n+p)J 2--7;L J _I~$$-
>
1
[lar’]
dk e (k.%ZY,(L) exp [ _,(,z:‘,)]
R,(l -:(cu+/3)r’)
exp[-:r’fcu+3/3)],
(,A.7)
where R, = ~~,N,?,N:N:R,,(NT;J+ and the R,, amplitudes
A.2
DERIVATION
OF
are displayed
V2Cr”, rk,)
IN
EQ.
~7~:s))
in eq. (2.1).
(2.6)
The matrix element V, in the chain (1.3), treated meson wavefunction and of strength A, can likewise VJr“, r&j = liJ,[MeMF+ =: A where
WY is
given
in eq.
as simply the overlap of the be expressed symbolically as
MSM:]
da db da’db’
WY * WY * DF,,
(‘4.8)
(A.4) and
WY = ( NT)’ exp [-$(2(a
- b)‘+~(a’+
b’)‘}],
(A.9) (A.lO)
AM.
600
The spin-isospin
dependence
Green, G.Q. Liu / NN a~tlj~lilati~tt
is given by
([ Y,(k”)[[lJ]“~r~[2;i]~s~“1]S7~]~T~[[~,~Y,(j,J)[14]S~r~]J~“~[2~]”~7~]‘~)+(~r,,~r~~) -RhY,(~).[a’Y,(a^‘)-h’Y,(~‘)]=2R,r”Y,(~)* where
R,, is defined
(A.1 1)
Y,(i”),
by eq. (2.5). At first sight, it might seem that the presence
of
the two terms in the above equation would lead to [a’Y,(a^‘) + b’Y,(bA)]. However, the action of R,, is essentially to reverse the sign of the b’ term w.r.t. the a’ term i.e. effectively [~‘Y,(~‘)+b~Y,(~)]j[~‘Y,(~‘)-b’Y,(~)]=2~”Y,(~).
For convenience factor.
Performing
the factor of 2 in eq. (A.1 1) is incorporated the above
integrals
in the overall statistical
gives
V2(r“, rd)=h(2~/p)3’ZT”exp[-~p(r’~‘+u~)]Y,(R). In eq. 12.7) the quantity
of interest
Y,(?‘)R,.
(A.12)
is
Appendix B THE FtUX
PRORARlLlTY TUBE
FOR
q3 AND q2$ CONFIGURATIONS
TO HAVE A
JUNCTION
In the introduction, it was stated that the nucleon flux-tube configuration contained a junction provided the angles between the sides of the three-quark triangle defining the nucleon were less than 120”. In this appendix, a quantitative estimate is made of this effect. With the notation in fig. 7a b = rL1- r3
a=&,
t = J‘( Y, + r2) - J&,
s=&r,-rJ,
1 Jl
J2 3
2
ai Fig.
7. (a) The limits (it, and h,) on h to ensure the two marked
bl angles are lrss than $r. (b) The
q-‘$‘state.
A.M.
the normalisation
The probability Z(b,, where
integral
Green, G.Q. Liu / NN annihilation
can be written
of a junction
(Y is the oscillator b,
=a
sin $r
as
configuration
b,.x,)=;($)i j,:daa’l:’ constant
is then d.u exp [-:cu(a’+h’+ahx)]
dbb’l,; describing
the nucleon
0,),
h,=
a sin (fn-
b,=O,
b,=x
when
sin ({r-
601
0,)
wavefunction
sin zrr when 0, <
(B.1) and
:rr,
fI,>!,n,
X, = -0.5 . Here 0, is the angle between a and b and xr= -0.5 ensures that the junction does not become coincident with quark 2. The limits b, and b, ensure that the junction does not become coincident with quark 3 and 1 respectively. With I( b,, b,, x,) defined as above, Z(0, cc, -1.0) = 1 is then simply the original normalisation integral. In this case [(O, CC,-0.5) = 0.93(0.86) is the probability that the junction is not at quark 2. The number in parentheses is that for the corresponding calculation in two dimensions. The integral when the junction is not at any of the quark positions is I(b,, b,, -0.5) = 0.78 (0.55). The constraints are seen to follow a regular pattern since I(b,, _ especially that ~80%
b,, -0.5) = [ 1 -3( 1 ~ I(0, cc, -0.5))]
in the three-dimensional case. The conclusion to be drawn of the phase-space integrand for a single nucleon contains
from this is an explicit
junction. The previous paragraph only deals with the three-quark single nucleon problem. However, for NN annihilation - as seen in figs. 3,4 and 5 - four-quark configurations also enter. For these the corresponding calculations - even in two dimensions become very complicated and do not warrant a detailed evaluation in the present article. In the notation of fig. 7b, imposing a single angular constraint to ensure junction J, does not coincide with quark 1 results in the normalisation integral again being reduced to 0.93 (0.86) as in the three-quark case. These numbers are the same since the angular constraint is independent of the positions of q,.? and q,,, and so the integrals separate to again give I(0, co, -0.5). Similarly, to ensure J2 does not coincide with q4 also gives 0.93 (0.86). If now it is assumed that the additional constraints ensuring J, does not coincide with q2 and J, with q, follow the same regular pattern as proven in the three-quark case, then the probability for finding a single junction becomes [ 1 -2( 1 - I(0, m, -0.5))]
= 0.86 (0.72)
A. hf. Green, G.Q. Liu /
602
and for finding
N@urzniililQti#n
two junctions [ 1 - 4( 1 - I(0, a3, -0.5))]
= 0.72 (0.44) .
The above estimates indicate that the probability for configurations involving a junction increases as the number of quarks and antiquarks increases. Therefore, the action of plaquettes is enhanced since single plaquettes with the basic lattice spacing of -0.1-0.2 fm appropriate for the flux tube model can give the transition to two mesons. For configurations without junctions the plaquettes need to operate directly on the quarks, which requires a plaquette or series of plaquettes with nuclear dimensions z-1 fm.
References 1) A.M.
7) 3) 4) 5)
6) 71 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19j 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)
Green and J.A. Niskanen, Prog. Part. Nucl. Phys., Vol. 18, ed. A. Faessler (Pergamon, 1987) p. 9.3 T. Armsstrong er a!., Phys. Rev. D36 (1987) 659 M. Maruyama, S. Furui and A. Faessler, Nucl. Phys. A472 (1987) 643 A.M. Green and J.A. Niskanen, Mod. Phys. Lett. Al (1986) 441 A.M. Green, Proc. of Workshop on the elementary structure of matter, Les Houches, March 1987; ed. J.-M. Richard et al., Proc. in Physics 26 (Springer, 1988) p. 190 E.M. Henley, T. Oka and J.D. Vergados, Nucl. Phys. A476 (1988) 589 R. Kokoski and N. isgur, Phys. Rev. D35 (1987) 907 N. lsgur and J. Paton, Phys. Rev. D31 (19X5) 2910 C. Dover, P. Fishbane and S. Furui, Phys. Rev. Lett. 57 11986) 1538 C. Dover, Proc. 2nd Conf. on the Intersection between particle and nuclear physics, Lake Louise, Canada, 1986, ed. D.F. Geesaman (AIP conference proceedings No. 150) p. 272 G.B. Chadwick et al., Phys. Rev. Lett. IO (1963) 62; M. Foster et al., Nucl. Phys. 66 (1968) 107 S. Ahmad er af., Phys. Lett. Bl57 (198.5) 333 A.M. Green, Mod. Phyz. Lett. A2 t.1987) 617 W. RrSckner el al., Phys. Lett. B166 (1986) 113 H. Gcnz, Phys. Rev. D28 (1983) 1094 J. Merlin and J. Paton, Phys. Rev. D36 (1987) 902 F. Lenz et a/., Ann. of Phys. 170 (1986) 65 A.M. Green and J.A. Niskanen, Phys. Scripta 34 (1986) 550 S. Furui, Z. Phys. A325 (1986) 375; A327 (1987) 241 A.M. Green, V. Kuikka and J.A. Niskanen, Nucl. Phys. A446 (1985) 543 A.M. Green, J.A. Niskanen and S. Wycech, Phys. Lett. 8139 (1984) 15 V. Bernard, R. Brockmann and W. Weise, Nucl. Phys. A412 (1984) 349; A440 (198s) 605 S. Kumano and V.R. Pandharipande, University of Illinois preprint ILL-(NUj-87-50 M. Kohno and W. Weise, Nucl. Phys. A454 (1986) 429 J.A. Niskanen and A.M. Green, in preparation A.M. Green and J.A. Niskanen, Nucl. Phys. A430 (1984) 605 A.M. Green and J.A. Niskancn, Nutl. Phys. A412 (1984) 448 M. Maruyama and T. Ueda, Prog. Theor. Phys. ‘74 (1985) 526 J.A. Niskanen, V. Kuikka and A.M. Green, Nucl. Phys. A443 (1985) 691 A.M. Green, J.A. Niskanen and S. Wycech, Phys. Lett. B172 (1986) 171 B. Moussnllam, Nucl. Phys. A407 (1983) 413; A429 (1984) 429 C. Amsler and G. Smith, Proc. of Workshop on the elementary structure of matter, Les Houches, March 1987; ed J.-M. Richard et al., Proc. in Physics 26 (Springer, 1988) p. 197 and private co~~n~uilication~
4.M.
32) 33) 34) 35) 36) 37)
Grrcx,
G.Q. Litr / NN un~~~~ilufio~
L. Tauscher and M. Chiba et al., Proc. of IV LEAR Workshop, Villars-sur-Ollon, Switzerland, C. Amsler ef al. (Harwood, 1988) p. 397 and 401 C.B. Dover and J.-M. Richard, Phys. Rev. CZS (1982) 1952 A.M. Green and G.Q. Liu, Z. Phys. A, accepted J. Rosner, Proc. 2nd Int. Conf. on hadron spectroscopy, KEK, Japan, April 1987, p. 395 W. Toki, Proc. 2nd Int. Conf. on hadron spectroscopy, KEK. Japan, April 1987, p. 252 R. Bagchi and S. Basu, Mod. Phys. Leti. 3 (198X) 633 D. Bridges cf nl., Phys. Rev. Lett. 56 (1986) 215 G.E. Brown, Nucl. Phys. A358 (1981) 39~
602 ed.