Volume 61B, number 1 PHYSICS LETTERS 1 March 1976

A NEW S C H E M E O F 77 A N D ~7' M I X I N G A N D V I O L A T I O N S O F T H E O Z I R U L E

T. INAMI, K. KAWARABAYASHI and S. KITAKADO Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan

Received 13 January 1976

A possible mixing scheme for ~ and ~' is proposed based on the picture that 0- mesons form an ideal nonet in the dual planar theory and deviate from it through nonplanar diagrams. Using the new mixing angles (0 H :~ 0rt,), we calcu- late radiative decay widths and production ratio of 77 and ~ which are then compared with experiments. Extension of our scheme to SU(4) leads to rather large violations of the OZI rule in decays of new particles with ~ or ~' in the final states.

The ansatz that SU(3) symmetry breaking is such that vector and tensor mesons form ideal nonets, ex- plains in a very simple way decay properties as well as mass spectra of these mesons (the OZI rule [1 ]) [e.g. 2]. It has been a puzzle, however, that a similar picture is not a sensible approximation for pseudo- scalar mesons. Their mass spectrum and radiative de- cay widths are indeed far from what would be expected from an ideal nonet scheme. Moreover, in the conven- tional picture of ~/-~' mixing, there seems to exist an apparent discrepancy among the mixing angles deter- mined from various types of experiments; namely, (1) mass formula for 0 - mesons gives octet-singlet mixing angle 0 ~ + 10 ° [3], (2) calculation of radiative decay widths o f t leads to 0 ~ - 7 ° and (3) the cross- section ratio of n - p ~ r/n and n - p ~ ~'n scattering gives 0 ~ - 2 0 ° [4].

The recently proposed approach of topological ex- pansion [5] for Reggeon dynamics suggests that the OZI-rule violating interactions of the type of crossed loop diagrams depicted in fig. 1 shift the trajectories o f / = 0 members of nonets which are exchange dege- nerate, equally spaced and approximately linear at the level of planar diagrams [6]. Physical consequences of the application of this idea to natural parity meson trajectories have been discussed by several authors [6-8] .

i 1 *l

Fig. 1. The OZI-rule-violating non-planar interaction.

jP

,.

/(11 iii II

t GeV

Fig. 2. The pattern of bare and shifted unnatural parity tra- jectories.

In this note we apply a similar idea to the pseudo- scalar mesons to study a possible way of deviation of their masses and decay properties from the ideal nonet scheme [9].

Mixing among 71 tra/ectories. When non-planar in- teractions are switched on through the crossed loop diagram (cylinder coupling in the terminology of ref. [5]), I = 0 trajectories, a n and t~ n, deviate from the bare trajectories anN and a~x 0?N and r~ x stand for ideally mixed (p~ + nfi)/x/2 and XX states, respectively), while %r and a K remain unshifted as shown in fig. 2. These trajectories are assumed to recover their nonet structure and approximate linearity for large t in ac- cordance with the ansatz of asymptotic planarity of Chew and Rosenzweig [7].

The shifted trajectories can be obtained by diago-

60

Volume 61B, number 1 PHYSICS LETTERS

Table 1 Ratios of two different decay modes for n and n'.

1 March 1976

Our model Conventional model (o n = - 6 °) (o = - 10 °) (0 n' = ~- 20° )

(o = 10 °) Experiment

P(n --, VV)/r(n -* a'+Tr-3,) 7.4 8.3 2.6 7.60 ± 0.25 r (n ' - , p'r)/r(n'--~'rv) 12 14 19 9.1 - 17

nalizing the following 2 × 2 propagator matrix in J-plane

P + PCP + PCPCP + ... = ( p - I _ C ) - I

= I J - ~ N - 2k - x / ~ k k 1 - 1 (1)

where P is the planar Reggeon propagator and C is the cylinder coupling shown in fig. 1. We have simplified our model by assuming that this coupling C is an SU(3) singlet and is consequently given by a single parameter k. It is a function of both t and J and, in principle, is determined by the dynamics of non-planar loop dia- grams. To have 7/and */' trajectories shifted downward so that they pass through the physical 77 and 77' states,

2 and 2 k has to be negative for J ~ 0 and t ~ m n mn,, which we will assume throughout the paper.

In order to diagonalize the matrix (1) we make a transformation

I*/) = [*/N) cos O - I*/x) sin O = 1./8) cos 0 --I*/1) sin 0 , (2)

17/') = Ir~N)sinO + I*/x) cosO = 1'/8) sin 0 + I*/1) cos 0 ,

where I*/),,Irf), etc. are the basis states corresponding to the trajectories a_, a_,, etc. and 0 = O - 0, , where tan 0 0 = x/2(0 0 ---55 ). Given the bare trajectories

anN and anx (anN= or#, anh= 2 a K - aTr), the resulting mixing angle (9 as well as the shifted trajectories depends on the coupling k*. However, since an(t ) and an,(t ) are zero at t = mn2 and t = ran,2, respectively, it is pos- sible to calculate the unknown coupling k and the mixing angles O n and O n, at these particular t values, with the formulae:

tanOn=--anN(m2n)/x/~anx(m2 ) and _ 2 2 ( 3 )

tan 0 n ' - x/2anN (mn,)/anx(mn,).

* Possible shifts o f /= 0 1+-trajectories will occur by the same mechanism considered in this paper. However, they seem to be quite sensitive to the specific form of J-dependence of k and will not be studied here.

We assume the bare r/N and */x trajectories to be linear and of the same slope, in which case the mixing angles are independent of a particular value for the slope. We then obtain

O n = 49 ° (O n = - 6 ° ) , O n, = 35 ° (O n, = - 2 0 ° ) . (4)

To have a rough idea on the uncertainty of the values of O n and On, due to that of the trajectories in question, we will allow an uncertainty of ~ 10% to the right side of eq. (3). This gives AO n ~ AO n, ~ 3 °. Contrary to the usual treatment of*/-*/' mixing, */and */' mesons are not related to each other by a common mixing angle. One may recall here that a similar phenomenon occured in the current mixing model for 6o and ~b mesons [10].

Experimental consequences. With the mixing angles given by eq. (4), the radiative decay widths of* /and */' can be calculated in a usual manner by assuming exact nonet coupling for Reggeon vertices modified only through the mixing effect.

However, one also has to take into account devia- tions of the shifted */and r/' trajectories from the universal slope. In fact, it is not difficult to show that partial decay rates of r/(*/') are proportional to a factor

t l . I I P t . .

of aO/an(mn)(C~O/an,(mn,2)), where a 0 is the universal slope ofr the bare trajectories. In terms of k(t) , the slope of 77 evaluated at t = m 2 is written as

2 2 , 2 _ or' + [ anN(mn) + 2an~(rnn) '~

an(mn) - 0 ~CtnNt-~:~--~,--~-~-~.~-7,~2)]v,,nj_,,nxv,,n, ' " w " n " k'(m2) . (5)

The corresponding slope for 7/' is obtained from eq. (5) by substituting the subscript */for */'. Numerically, we estimate a'n(m2 ) ~ a~) + 0.02 k'(rn 2) and a'n,(m2,) ~ a~) + 2.6k ' (m2,) , so that for reasonable values of k ' , we can safely neglect this effect for r~ decays but not for r/' decays.

The ratios F(* /~ 73')/P(*/-~ *r+n-') ') and P(*/ '-+77)/ P(*/'-~O07) are, of course, independent of such un- known factors (see table 1). In order to determine the

61

Volume 61B, number 1 PHYSICS LETTERS 1 March 197

Table 2 Partial decay widths of r t and 7' into 3"7.

Our model Conventional model Experi- (07=-6 ° ) ( 0 = - 1 0 ° ) (O=10 °) ment (0 n' = ~20° )

F(7--'77) (eV) 296 387 43 324+-46 r (7'-*3,7) (keY) 4.3 6.4 8.3 <20

slope of r/' trajectory at t = m2,, we compare r/and r/' production ratio in reactions such as n - p ~ X0n at small t, where X 0 denotes r/or r/'. At high energies, this reaction is governed by the t channel exchange I = 1 mesons. Consequently, only the nucleon quark component r/N will contribute to the scattering. Hence we have *,

d a ( n - p -+ r/n)/dt : d o 0 r - p - + r /n) /d t

= cos2Ona,7,(m2,)/sin2On,czn(m2),. (6)

We have calculated the cross-section ratio from the data in n - p scattering for PL > 10 GeV/c [4, 11 ]. Taking the branching ratio P ( r / ' - 77) /P(r / -+ all)

' 2 , 2 0.019 [3], we estimate ~n,(mn,)/an(mn) ~ 1.2+_0.2. Using eq. (4) and the slope parameters estimated above, it is then straightforward to calculate the ra- diative decays of *7 and r/'. The results of our calcula- tions are then summarized in tables 1 and 2, where a comparison has also been made with standard pre- dictions, using 0 = +_10 °. Thus we conclude that pre- dictions of our mixing scheme are in good agreement with existing data.

We also note that the sign of 0 is left undetermined in the conventional treatment based on the mass mixing formula. In our approach, however, this sign is uniquely predicted to be negative. This is because, by introducing the cylinder coupling, *7 trajectory, which was pure (p~ + nn)/x/~ and degenerate with pion trajectory C%r , is shifted downward so that it passes through physical r/state with 2 smaller than m 2= (am 2 m2)13, mn

Extension to SU(4}. Conseciuences of the present approach will become most dramatic when we apply our scheme to r/c, the hypothetical 0 - c~ state. In this

c - ~ c - -

a b

Fig. 3. Two possible diagrams contributing to the processes q -~ n(n')3'.

case, we consider, instead of eqs. (1) and (2), mixings among (p~ + n~)/x/~, kX and c~ states,

I~> = In> - 15x Inc>,

I~'> = In') - 6qblnc), (7)

Inc> = 8 x In> + 8q~ In'> + Ir/c>,

where 5× and 5q~ represent small r/'r/c and r/"r/c mixing angles, respectively. The states Ir/> and Ir/'> have been defined by eq. (2), and the state Ir/c> corresponds to the bare trajectory anc" Similarly to the case of SU(3), we can estimate the angles 5~ and 5X at t = m 2 and t = m~,, respectively, by using the formulae: 7

6× n : - x / ~ k(m 2 ) sin O (m2 )/anc(m2) ,

= 2 2 ( 8 ) 60 n, -x/-3k(m2,) cos 0 (m 7, )/O~7c(m 7,) .

We tentatively identify the newly discovered 2.8 GeV state [13] with ~c meson. Assuming a linear bare *7 c trajectory with the same slope as that of r/N and 77 x trajectories and m 2 ~ m 2 nc ~c in accord with asymptotic planarity, we obtain

6X n ~ 0.02 - 0.03 , 6~ n, ~ 0.05 . (9)

This means that r/and r/' mesons contain the c~ compo- nent of the order of 0 .04-0 .1% and 0.3%, respectively*

Let us apply our mixing scheme to decays of new particles in the SU(4) framework. We consider radia- tive decays of ff/J particle into r /and r/' mesons. There are possibly two types of diagrams contributing to these processes (a and b o f fig. 3) [15]. The amplitudes cor- responding to the diagram a are proportional to 6Xn and 6q~ n, defined at 7/and r/' mass shells respectively,

* In our picture, the ratio of singlet/octet n couplings is fixed to the ideal nonet value x/2. For different attempt to this problem see ref. [12].

* These values are significantly small compared with those speculated by Harari [14]. Note, however, that they are much larger than the corresponding mixing (< 10 -4) for 4.

62

Volume 61B, number 1 PHYSICS LETTERS 1 March 1976

while those of the diagram b are proportional to mixing angles similarly defined between the 1 - c~ and q~ states (q represents old quarks, p, n and X) at ~/J mass shell. The latter angles are likely to be much smaller than the formers because of the following two reasons: (i) assymptotic planarity [7], (ii) empi- rically, the OZI rule for 1 - mesons is much more accurate than that for 0 - mesons. If this is the caset, the contribution from the diagram b can be ignored and we have

r ( ~ / J -+ #~/ ) l r ( ~ lJ --, ~ / ) (10)

(phase space factor) ' ' 2 = (%/%,) (~i,:I,~,/,~ × n ) "

Using the mixing angles (9), we then obtain

r ( ~ / J ~ n '~/ ) / r ( f f /J ~ n'r) ~ 2 - 4 , (11)

which is well within the latest experimental upper limit ( ~ 5 ) [13].

It is important to note at this place that in order to obtain our numerical values for 6Xn and 6qbn,, it was crucial to take into account the t-dependence of the cylinder coupling k. In fact, any model in which the coupling k is treated as constant in t, would have predicted a very large branching ratio for (1 l) ( ~ 34 for 0 = - 1 0 °) [15]. This would clearly be in contra- diction with the above mentioned upper limit, unless there were a fair amount of contribution from the diagram b. As noted before, the relative importance of the contributions from the diagrams a and b could be estimated from the ratio of ff/J -~ ~r0~ , to ~/J-~r/3'. Thus, further experimental determ!nation of the radia- tive decay widths of ff/J into 7r 0 , r /and r/' is highly desirable to discriminate our scheme from various other models for new particles.

On the other hand, the state r/c is expected to con- tain much smaller amount of old quarks than that of c~ component contained in ~7 and 77'. This is because the coupling k and consequently the mixing angles 5X and 5(I) presumably decrease as t increases (asymptotic planarity). How rapidly they decrease depends on the structure of the cylinder coupling k and therefore is

t There is an indirect evidence for this assumption. The decay of ~V/J ~ trO.r has the contribution only from the diagram b. Its width estimated from the observed branching ratio of ~/J ~ p~r with p dominance for the photon is of the order of 1 eV [14, 15] ,'which is indeed very small. ,

difficult to predict. We want to emphasize, however, that the difficulty with respect to the narrow decay widths for rTc ~ pp(~6o) and ~ and large mixtures of c~ component in r7 and r/' [14] can naturally be re- solved in our scheme by a suppression of the order of i6x(t = m2c)/6×(t., =m2)l ~ 0.1. This is indeed an ex-

pec ted order of magnitude if we note that Ik(m2,)/ k(m2) I ~ 0.25.

Finally, we wish to make a few more remarks. (i) In calculating radiative decays of r7 and rl', we

have assumed an exact nonet coupling at the planar level. The physical couplings between cylinder shifted Regge poles are then given in terms of the mixing angles and the renormalization of Regge slopes. Considering the approximations which are presumably reliable to the accuracy of 10% level, it is quite remark- able that such a simple mixing scheme can offer an overall consistent picture for r /and r/'. Since our model gives predictions for 77 and r/' decays substantially dif- ferent from the conventional approach, it will be desir- able to have more detailed information about these decays.

(ii) From our point of view, the so called r~ problem of current algebra [16] may be understood by assum- ing that a n and a n , are shifted downwards by the coupling k which is strongly t dependent and therefore usual assumption of mass mixing with constant k would not be valid here. This appears somewhat ad hoc, but as we have shown in this paper, the t-dependence we need to explain rr-r~-~' mass splittings ( tk(m2,) /k(m2)l ~0.25) is consistent with the ansatz of asymptotic planarity and leads to predictions for decays and pro- ductions of ~ and r?' in good agreement with experiments

(iii) By extending our arguments to SU(4), we have estimated small contaminations of c~ component in r7 and t/'. These values were obtained by assuming the same slope for rl, r/' and r/c trajectories. If the slope of r/c trajectory turns out to be different (smaller) from that of r/and r~' trajectories, we would predict larger amounts t t o f cg component in ~ and r/'. As empha- sized before, it is likely that our estimation will be tested by the more accurate determination of the de- cay rates of ~ / J ~ n'07, ~ / J ~ r /" /and ~ / J ~ rl'3 t in the near future. It remains, of course, to be seen whether

t t For instance, if the slope o fnc trajectory is one half of that of n and n', cOntaminations of c~- component in n and n' would be of the order of 0.2-0.4% and 1%, respectively.

63

Volume 61B, number 1 PHYSICS LETTERS 1 March 1975

our picture will give correct predictions for other de-

cay 15rocesses of new particles with r/or r/' in the final state, like the decay of f f ' ~ ~br~. These problems will

be discussed in a later communication.

We are most thankful to Professor M. Kato for

reading the manuscript and for valuable comments and Professor Y. Fujii, Mr. S. Wada and Dr. N. Sakai

for useful discussions. After our paper was submitted for publication, we

were informed of a similar analysis made by C. Rosen- zweig (preprint, PITT-156 (1975)). We thank him for sending us the preprint.

References

[1] S. Okubo, Phys. Letters 4 (1963) 14; G. Zweig, CERN Report No. 8419/Th 412, (1964) un- published; J. Iizuka, K. Okada and O. Shito, Prog. Theor. Phys. 35 (1966) 1061.

[2] R.P. Feynman, Photon hadron interactions (Benjamin, New York, 1972).

[3] Particle Data Group, Phys. Letters 50B (1974) 1.

[4] V.N. Bolotov et al., Phys. Letters 48B (1974) 280; W.D. Apel et al., Serpuchov preprint IHEP 74-118.

[5] G. Veneziano, Phys. Letters 52B (1974) 220; Nucl. Phys B74 (1974) 365.

[6] S. Yazaki, Proc. Discussion Meeting on the New Reso- nances (Tokyo) UT-247, 1975.

[7] C. Rosenzweig and G.F. Chew, Phys. Letters 58B (1975) 93.

[8] C. Schmid and C. Sorensen, Nucl. Phys. 96B (1975) 209. [9] For related but somewhat different approaches on this

subject, see, P.G.O. Freund and Y. Nambu, Phys. Letters 34 (1975) 1645; K. Akama and S. Wada, in preparation.

[10] N.M. KroU, T.D. Lee and B. Zumino, Phys. Rev. 157 (1967) 1376; V.S. Mathur, S. Okubo and J. Subba Rao, Phys. Rev. D1 (1970) 2058.

[11] W.D. Apel et al., Intern. Conf..on High energy physics, London, 1974, paper 537; V.N. Bolotov et al., Serpukhov preprint IHEP 73-57.

[12] H. Nagai and A. Nakamura, Prog. Theor. Phys. 53 (19751 523.

[13] B. Wiik, preprint DESY 75/37 (1975). [14] H. Harari, WlS-75/39h, 1975. [15] T.F. Walsh, DESY 75/21 (1975);

R.N. Cahn and M.S. Chanowitz, Phys. Letters 59B (1975 277.

[16] S. Weinberg, Proc. XVII Intern. Conf. on High energy physics, London, 1974.

64