Volume 106B, number 6 PHYSICS LETTERS 26 November 1981

RADIATIVE DECAYS OF HEAVY VECTOR MESONS INTO PARAQUARKONIA ~r

A. DEVOTO and W.W. REPKO

Department of Physics, Michigan State University, East Lansing, MI48824, USA

Received 16 July 1981 Revised manuscript received 16 September 1981

The width for decays of the type 1 - ~ ~, + 0- is evaluated using the two gluon mechanism of QCD. In particular we compute r(qJ -~ .y + n'), r(~ ~ 7 + r/) and F(T ~ 3' + r~c). The first two are found to be in reasonable agreement with the experimental results, while the latter leads to a branching ratio of 2 × 10 -5.

The study of radiative decays of heavy vector mesons can provide important checks on the dynamical predic- tions of quantum chromodynamics (QCD). Among these is the prediction that inclusive radiative decays should be dominated by the process [1 ] 1 - -+ 7 + 2g, where g denotes a gluon. Given this inclusive process, it is evident that detailed studies of exclusive radiative channels can provide information about how the two gluons materialize into ordinary hadrons. One of the most exciting possibilities would be the observation of a gluonic bound state - a glue ball. This aspect of radiative decays has received considerable attention recently [2,3] in connection with the observation [4] of a KI~Tr enhancement at 1440 MeV in the hadronic final state of the decay ff -+ 7 + X. Whether or not this enhancement is eventually shown to be a glueball, it is important to examine the quantitative accuracy of QCD predictions for decays of the form 1- + 7 + bound state. This is essential because analyses of radiative decays frequently make the attractive assumption that only the two gluon intermediate state is needed to describe the decay amplitude [3].

With this in mind, we have computed the width of the radiative transition 1- -+ 7 + 0 - , where 0 - denotes a 1 SO paraquarkonium state, by evaluating the one loop diagrams illustrated in fig. 1. To assess the validity of re- stricting the calculation to the two gluon intermediate state, we computed the widths for ff -+ 3' + r7 and ~b -+ 3' + ~7". Our results are in good agreement with the observed widths [5] for these transitions. We interpret this agree- ment as direct evidence for the dominance of the two gluon -+ bound state coupling (i.e. the lowest order QCD process) in exclusive radiative amplitudes. This is in accord with previous calculations of ¢ radiative decays [6], which stressed the role of the two gluon process.

To evaluate the amplitudes shown in the figure, we make the usual assumption that the decaying 1 - meson is

Research supported in part by the National Science Foundation.

(a) (b) (c)

Fig. 1. Two gluon diagrams contributing to radiative decays of vector mesons.

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a 3S 1 quark-antiquark ( q - ~ bound state at threshold. The coupling of the O- meson to the virtual gluons is de- scribed by the amplitude

M'uv(q ' q', p ' ) = (6ab/2X/~)(27r)46 (4)(q + q' _ p')g2

X / "d4p' - , 1 , p, [(C~p, (p)"tuSF(~P - - q)Tv) + (la "~" v, q ~ q')] (1)

J(27r)4

where fp , (p') is the Bethe-Salpeter wave function for a q - ~ 1 SO bound state with relative momentum p' and center of mass momentum P', C is the charge conjugation matrix and the angular brackets denote a trace. The factors 6ab[2,v/3 and g2 represent the color dependence and the dependence on the quark-gluon coupling. In prac- tice, it is convenient to replace fp , (p') with fM ' (P'), the 1 SO wave function in rest frame of the 0 - center of mass. These are related by a Lorentz transformation as

C f p , (p') = S(A) C~MI (A- 1 p ' )S - 1 (A), (2)

where Auv is the boost which takes (0, iM') to P~. Using eq. (2) and changing the integration variable to Ap' re- places the trace in eq. (1) by

4 , f d'~p' Auu'Avv' [ (C~-M' (P ' )Tu 'SF(~A-1p ' - p' - A-lq)Tv ,) + (la "~ v ,q oq ' ) ] (3)

- ( 2 ~ ) 4

If we now assume that p' can be neglected in the fermion propagator (i.e. that binding effects are unimportant), the d4p ' integration can be performed yielding f M ' (0), the amplitude of the wave function at the origin. The trace can then be evaluated and eq. (3) reduces to

---2~3/-2 ~-'(0~) euvc,~qa(P'#/M' ) , (4) (1p, _ q)2 + m,2

where m' is the mass of the 0 - constituent quark, and if'(0) is the scalar wave function evaluated at the origin. With this result, the amplitude for, say, diagram 1 a is expressible as

A - 2 ig4eeq /" d4q_ 3 (2rr)4 ~(0)~'(0) ~ , (~*Sp(- ½P + k)'Y~SF(½P + q - P )q'v(1 + 7 4 ) ~ ) d q2(p _ q)2

euvo~3qo~P'~ X M'[(~P' - q)2 + m,2] ' (5)

where e* is the photon polarization vector, k its four momentum, eq the charge of the decaying quark, ~ the polar- ization vector for the 3S 1 state and ~(0) the heavy quark wave function evaluated at the origin. It is not difficult to show that the amplitude for diagram lc is also given by eq. (5). After evaluation of the remaining trace and a variable change q - - q + P', the expression for A becomes

~ g 4 eeq ~(0) ~'(0) , . A n--'-2 M 4 M' e ~ u v P ~ k ~ u e v ( k ' P ' A 2 +3A4)" (6)

Here, A 2 and A 4 are associated with the dimensionless integral

i 1 d4q qc~qt3 - - - ' ' + ' +P~kc~ ) +A46c~ ~ , (7) k ' P ' f q2(p, q ) 2 ( q 2 _ p . q ) ( q 2 _ p , . q ) AIP'~Pa A2(P"~k# +A3kak~ A ,~ - ~r 2

with the understanding that the heavy quark mass m is unity. The necessary combination of scalar integrals is

_ i 1 ( k . P ' q 2 k ' P ' A 2 + 3 A 4 , 0 ( , . e ' ) 2 f d4q - k ' q P " q ) (8)

q2(p ' _ q)2(q2 _ p . q)(q2 _ p ' . q)"

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The four denominator one loop integral which remains was computed with the aid of the program FORMF devel- oped by Veltman [7]. This program provides a highly accurate numerical evaluation of the real and imaginary parts of any N-point function for most physical values of the external invariants. From eq. (8), it is evident that the desired integrals can be decomposed into sums of three-point functions which FORMF handles very efficiently. As a check, we computed the imaginary part of the eq. (8) directly. The imaginary part of the entire amplitude is given below.

The analysis of diagram lb is similar and we find the expression

B = 16 g4eeq ~k(O) ~ ' (0) , , - 3 - rr 2 M 4 M ' e ~ u v P ' ~ k ~ e v ( - 2 B 4 ) ' (9)

where B 4 is defined in analogy with the scalar integrals in eq. (7). In terms of a five denominator integral, - 2 B 4 is

_ i 1 d4q [(k'p')2q2-21c'P'Ic'qe"q+e'2(k'q)2] (10) - 2 8 4 ~2 (g. e')2 f q2(e' _ q)2(q2 _ e . q)(q2 _ p , . q)[(q _ p,)2 + p . (q _ p,)]

once again, the expression eq. (10) can be reduced to sums of three-point functions. Thus, apart from common kinematic factors, the decay amplitude is determined by the integral

l ( m ' / m ) =- (2k " P ' A 2 + 6A 4 - 2B4) . (11)

In keeping with our assumption that binding can be neglected in loop integrals, we use p2 = _4m 2 and p '2 = ~ m '2 in the evaluation o f 1. The imaginary part of I has the relatively simple form

-Tr K2 In K 2 K = m ' / m . (12) Im(I) = 2(1 -- t~2) 2 (1 + t~ 2) '

For m ' / m = 1/3, a value convenient for both ff and q~ decays * 1,

I (1/3) = 1.004 + 0.4368 i , (13)

which shows that the real parts of the loop integrals are not negligible. With these definitions, the width has the form

F(1 - -+ 7 + 0 - ) = e2aa 4 211(2/3) 3 [1 - (M' /M)2] 3 [1 + ( M ' / M ) 2 ] l I ( m ' / m ) l 2 hb(0)12 1¢'(0)12 (14) M 3 M'2

For application to J/ff radiative decays, the value of if'(0) can be obtained from the leptonic width and is 3.55 × 10 -2 GeV 3. To obtain if '(0) for r/and rl', we use the flavor SU(3) singlet and octet combinations

r /= - s i n0 rT1 + c ° s 0 r ~ 8 , rT'=cos0rT1 + s i n 0 7 / 8 , 0 = - 1 1 ° , (15)

together with a value of the square of the light quark wave function at the origin I ~ 0 ) 12. For the latter, we use the decay 7' -+ 77 whivh gives I ~(O) l 2 = (5.4 + 2.0) × 10-3 GeV 3. With these values for the wave functions and the known particle masses, we find

F(¢ -+ 7 + ~7')/P(~ -+ all) = (1.2 + 0.5) × 10-3 , P(~ -+ ~' + rT)/P(~ -+ all) = (0.2 + 0.1) × 10 -3 , (16)

for a s = 0.21, which is consistent with ff -+ hadrons being given by ~ -+ggg plus ~b -+7* -+ qq [5,8]. Note that the result is very sensitive to the value of a s .

The values for both decay modes are somewhat smaller than the respective average experimental results of (3.7 --- 2.5) × 10 -3 and (1 -+ 0.5) × 10 -3 [4,9]. Given the uncertainties associated with binding corrections and flavor SU(3) breaking, the results are quite reasonable. From this, it appears that the two gluon mechanism does dominate the flavor changing amplitude.

+] For the mass ratio m'/m = 1/5, the corresponding expression is I(1/5) = 0.9801 + 0.2110 i. The difference in Ill 2 for the two cases is less than 20%.

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The result for F ( 1 - ~ 7 + 0 - ) can also be applied to the decay T -~ 7 + ~c, where the various approximations mentioned above should be quite good. Using the leptonic widths of the T and ~ to obtain values for the wave

functions, we find

I ' (T ~ 3' + r/c)/I '(T -+ all) = 2 X 10 -5 (17)

with a s = 0.17 and an T total width of 40 keV [5]. In conclusion, the two gluon mechanism of QCD is capable of providing a quantitative description of radiative

decays involving flavor changes. There are certain aspects of this treatment which deserve additional at tention; in particular, we are examining the question of including binding effects and decays to higher spin bound states in a

systematic way. After completion of this paper, a preprint by Guberina and Ktihn [10] on the decay T ~ 7 + r/c was brought to

our attention. Our calculation differs to some extent from the calculation presented by these authors, but the esti-

mate for T ~ 3' + r/c is quite similar.

We would like to thank M. Chanowitz for a number of valuable conversations.

References

[1] T. Appelquist, A. DeRujula, H.D. Politzer and S.L. Glashow, Phys. Rev. Lett. 34 (1975) 365; M. Chanowitz, Phys. Rev. D12 (1975) 918; L. Okun and M. Voloshin, ITEP-95 (1976).

[2] K. lshikawa, Phys. Rev. Lett. 46 (1981) 978; M. Chanowitz, Phys. Rev. Lett. 46 (1981) 981 ; J. Donohue, K. Johnson and B.A. Li, MITCTP HEP-139 (1980).

[3] C.E. Carlson, J.J. Coyne, P.M. Fishbane, F. Gross and S. Meshkov, Phys. Lett. 98B (1981) 110; 99B (1981) 353. [4] D.L. Scharre, SLAC-PUB-2538 (June 1980). [5] K. Berkelman, in: High energy physics - 1980, eds. L. Durand and L.G. Pondrom (ALP, New York, 1981) p. 1500. [6] H. Fritzsch and J.D. Jackson, Phys. Lett. 66B (1977) 365;

H. Goldberg, Phys. Rev. Lett. 44 (1980) 363. [7] G. 't Hooft and M. Veltman, Nucl. Phys. B153 (1979) 365;

G. Passarino and M. Veltman, Nucl. Phys. B160 (1979) 151. [8] C.W. Kim and W.W. Repko, Phys. Rev. D15 (1977) 913. [9] R. Partridge et al., Phys. Rev. Lett. 44 (1980) 712.

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