Volume 122B, number 3,4 PHYSICS LETTERS l0 March 1983

QCD PREDICTIONS FOR PSEUDOSCALAR PRODUCTION IN RADIATIVE DECAYS OF HEAVY VECTOR MESONS

John F DONOGHUE and Harald GOMM Department of Physws and Astronomy, Umverstty of Massachusetts, Amherst, MA 01003, USA

Received 4 November 1982

We study the reaction tp ~ M7 where M is a pseudoscalar meson, using QCD and a heavy quark mass expansion. We fred that the mamx element is not short dlstance dominated, contrary to conventional assumption, and predict the rates of radiative T decay m terms of those m e decay.

The study of J /~ radmtive decays has recently proven to be a novel way to study meson spectro- scopy, yielding two new states, the t(1440) and 0(1640). In lowest order QCD this process occurs by c~ anmhilanon into a photon and two gluons. The gluons are an a color smglet and dominant ly carry the quantum numbers [ 1 ] jPC = 0++ 0 - + 2++. ~ radla-

tire decay as then an 1deal place to look for glueballs wath these quantum numbers. In particular, the t(1440), wath jPC = 0 - + , appears to be a strong glue- ball candidate [2].

In QCD perturbat ion theory, the process c g ~ GG7 governs all radmtlve transltmns. However per turbatmn theory cannot describe how the gluons can couple to a single pamcle M, producing the reaction ~ -+ MT, as that reqmres knowledge of low energy nonperturbatlve physacs. In the literature [31 it has conventionally been assumed that, due to the large mass of the charm quark, the transition is governed by a matrix element at small &stances, i.e. the wave functmn of M at the origin. In the case of pseudoscalar productmn (~ ~ ~3', rl'~' and tT) the only gauge mvarmnt local matrix ele- ment formed out of two fluons is FuvE "uu (with/~uv = eum~b~¢), so that it has been assumed that the reac- tion as p ropomona l to the transition m a m x element

(M IFu~.ff"~ I 0) .

If true, these assumptmns should follow from a sys- tematic expansmn of the transmon amphtude in pow- ers of 1/rnc. In this paper, we prowde this analysis for

emission of pseudoscalars and find that the conven- tmnal assumptions are not correct. The matr ix ele- ment as not short &stance dominated, but samples a range of distances in the direction of motion of the outgoing meson. The reason for thas is that the charmed quark mass as not the only large scale m the problem, because the final meson's momentum as also O(me). While (x 2) ~ 1/m 2 is small as expected, (k • x) ~ O(1) and all x values m the direction of the momentum k can be important Instead of being short &stance dominated, the matrix element is "light-cone" dominated. As an extra result of this calculation we have an estimate of the radtatwe branching fraction into a given pseudoscalar meson in T decay m terms of that observed an ~ decay, pre&ctmg for the branch- ing ratios (t?)

8 (T ~ M~,) = (m~ Im 6) z~ (~ -~ MT), (I) 2 2 up to corrections of order mM/m e.

In the decay of heavy quark systems there are two parameters which can be used m a perturbative expan- sion. One as the QCD couplang constant, g, when used at vertices anvolving large momentum transfers, so

2 2 2 that a s (me) = g (me) /4n is small. The other small parameter as the ratio of the hght hadronic mass scale, /l, to the heavy par tMe mass scale m e , i.e. p = g2 /m~. , In our problem ~ can be as large as m~ = 1440 MeV, so that for ~b decay p ~ 1/4. This as not small enough to feel completely comfortable with an expansion m powers o f p . Nevertheless such an expansion does

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Volume 122B, number 3,4 PHYSICS LETTERS 10 March 1983

provide a powerful theoretical method for studying the transition. At the least, our results will be very accurate for T decay where p ~ 1/40.

The transmon ¢ ~ My is described by the matrix element

(M(k), T(k ' ) I ae f f I ~ (P)}, (2)

where, to lowest order in the QCD coupling constant,

2e 471~s f d4yd4 z T(~(x)dl a (x) ½X a ~ (x) H e f t - 3 3 !

X ~-(v)jB~y) }Xe~ (y) ~(z)jq,(z)) (3)

2e 47ra s F.4 ,4 - 3 -3i 3 ° y o z T ( ~ ( x ) a 4 ( x ) ½ X A S v ( x - y )

X 4tB(y)½XBSF(y - z)4t (z) ~ (z) + permutat ions) ,

with O/(x) ,AA(x) ,A(x) being the fields for the charmed quark, the gluons, and the photon, respec- twely.

In taking the matrix element of Heff we encounter

Muv(x , y ) = (M(k)]T(A B(x) a B ( y ) ) ] 0 } . (4)

This factor is determined solely by low energy phys- ics, and we cannot calculate it. However Lorentz lnvanance restricts its form to be

(M(k)]T(AB(x)AB (v))]O)

= exp {lk" ½(x +y)} ×uv(r) , (5)

where r = ½(x - y). Furthermore when the meson M is a pseudoscalar, there is a unique form for Xuv whmh has the proper Lorentz and parity transforma- tions

X~u(r) = euva~ kar~x(r2 , k " r) . (6)

The function X has dimensions of mass, and can In principle be determined without reference to ~ radia- tive decays. For the charmed quarks, we neglect bind- mg corrections. In the normalization of the wave function at the origin such that the total hadronic decay width is [4]

r ( ~ -* 3 gluons) = [160(712 - 9 ) a3/81 rn~] I,I'(0)12 , (7)

we have

<01T(ff~(x) ~e0,))l ~ (e)>

= (2rnq~)l/2 ~(/3) u(p) q~(0)e -~F" x e-~p "Y6a~, (8)

where p(/7) is the momentum of the charmed quark taken to be p = ~ = (m~/2 , 0, 0 0) and a, t3 are color radices. With these definitions the transition matrix element is

H (0) I ~ (P)> = 2e 47ras (2rn~)l/2q I (0) (M(k), ")'(k') eff S 2 - ~

X fd4xd4y d4q d4q' (271)4 (271)4 euml3 k a {O(p)

× % ( ~ - m ) - i % ( 4 ' - m ) - 1 C u ( p )

× exp[i(q +~k - p) x] exp [i(q' - p + k ' )y] ½x ~

X X(¼X 2, -~k- x) + permutat ions}. (9)

If one considers Taylor expanding X(x 2, k • x) around X(0.0) , one can easily see that when the inte- gral over the propagator is done, x 2 becomes O(1/m2c) A Taylor series in x 2 is then equivalent to a Taylor series in la2/m 2. However, upon integration over x, k -x becomes k "p/m 2 ~ O(1). This is to be expected because k • x is dimensionless and an expansion in k • x does not introduce extra factors of/a. We must keep all factors of k • x. We write

X(X 2, k ' x ) = X0(k "X) + x2Xl(k "x) +

and

x0(k- x) = fdz e tzk" Xx0(z ) ,

Xl (k • x) = f dz e lzk "x X1 (z) . (1 O)

To obtain the final answer is now straightforward. We find to lowest order in 1/rn~

(M(k), 7(k ' ) [Heft[ ~ (P)) ( 1 1 )

=(3271ase/3v/3)(2m¢ )l/2 q~(O)M(m3 /mSc )o¢~'75u ,

where

M=fdz/6-8zZ x0(z) 3+4z2 × l ( z ) ] tT7 -47 (12)

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Volume 122B, number 3,4 PHYSICS LETTERS 10 March 1983

The dimensionless number M contains all of the non- perturbative physics associated with the now energy bound state. The reader will note that X0 and Xl enter at the same order in little, contrary to the expecta- tion discussed above. This is due to a cancellation in the leading order effect, proportional to X0, such that the matrix element vanishes when rn~ = 0 This leads to an addmonal suppression by a power of mc 2 over the naive expectations of dimensional counting and puts the effect of X0 and XI at the same order. Higher order terms will be suppressed compared to these

For the decay rate we obtain

P(~b ~ M'y) = ~12rr~2o~lqt(o)i21MI 2

x (m61m8~)(1 -m21,n~) 3 , C131

which implies a branching ratio

[ ' (~ ~ MT)/U(~ ~ hadrons)= [16n15(n 2 - 9 ) l ( c ~ / a 0

x ( . ,6 / , , ,6 )1MI2( 1 _ , , ,2 /m 213. (14)

Because we cannot calculate M we do not have an absolute prediction of the decay rate. However, the matrix element is clearly not short distance domi- nated. There IS a heuristic way to understand this. In the ~ rest frame, the meson M is moving with momen- tum of O(mc) and its wavefunctlon is Lorentz con- tracted In the direction of k. Points that are widely separated along the nine of flight in the rest frame of M are close together in the rest frame of ~. Whine the cg annflaflatlon is dominated by length scales of order lime, this actually samples widely separated points in the M wavefunction due to its Lorentz contraction.

These results suggest that care should be taken in

phenomenologlcal studies of ~ radiative decays. The conventional assumptions are not correct and results based on them are suspect. Presumably, similar results also hold for other spins and parities of the final me- son. Our scaring prediction, eq. (1), relating the branching ratios in ~ and T decays suggests that 77, r?', and t(1440) production should be extremely difficult to observe In T reactions.

Note added in proof" In interpreting our results, it should be kept in nund that we are working to O(a2). In partlculai, there could be corrections to eq. (1) of order aSm41rn4, which would provide less suppres- sion of the T radiative decay modes than is presently implied by eq (1).

References

Il l A Blllmre, R Lacaze, A Morel and H. Navelet, Phys. Lett 80B (1979) 381

[2] J F. Donoghue, K Johnson and B.A. L1, Phys. Lett 99B (1981) 416, M. Chanowitz, Phys Rev. Lett 46 (1981) 981, K lshlkawa, Phys Rev. Lett. 46 (198l) 978, JF. DonoghueandH. Gomm, Phys. Lett l12B (19821 409, J F Donoghue, Proc. XXI Intern. Conf on High energy physms (Pans. 1982) to be pubhshed

[3] tt Goldberg, Phys Rev. Lett. 44 (1980) 363; C. Rosenzwelg, J. Schechter and T. Tudron, Phys. Rev D23 (1981) 1148, K. lshtkawa, Phys Rev. Lett 46 (1981) 978, K. Milton, W Palmer and S. Pmsky, Ohio State preprmt DOF/ER/OI545-321

[41 T. Appelqmst et al, Phys Rev Lett 34 (1975) 365, M Chanowltz, Phys. Rev D12 (1975) 918.

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