The J/psiDD* vertex

Abstract

We employ QCD sum rules to calculate the J/psiDD* form factors and coupling constant by studying the threepoint J/psiD*D correlation function. We find that the momentum dependence of the form factor depends on the off-shell meson. We get a value for the coupling which is in agreement with estimates based on constituent quark model.

The J/yDD^* vertex

R. Rodrigues da Silva; R.D. Matheus; F.S. Navarra; M. Nielsen

Instituto de Física, Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, SP, Brazil

ABSTRACT

We employ QCD sum rules to calculate the J/yDD* form factors and coupling constant by studying the threepoint J/yD*D correlation function. We find that the momentum dependence of the form factor depends on the off-shell meson. We get a value for the coupling which is in agreement with estimates based on constituent quark model.

Hadrons are composites of the underlying quarks whose effective fields describe point-like physics only when all the interacting particles are on mass-shell. When at least one of the particles in a vertex is off-shell, the finite size effects of the hadrons become important. Therefore, the knowledge of the form factors in hadronic vertices is of crucial importance to estimate any hadronic amplitude using hadronic degrees of freedom. This work is devoted to the study of the J/yD*D form factor, which is important, for instance, in the evaluation of the dissociation cross section of J/y by pions and r mesons using effective Lagrangians [1,2,3]. Since a decrease of J/y production in heavy ions collisions might signal the formation of a quark-gluon plasma (QGP) [4], a precise evaluation of the background, i.e., conventional J/y absorption by co-moving pions and r mesons, is of fundamental importance.

The J/yD*D coupling has been studied by some authors using different approaches: vector meson dominance model plus relativistic potential model [1] and constituent quark meson model [5]. Unfortunately, the numerical results from these calculations may differ by almost a factor two. The relevance of this difference can not be underestimated since the cross section is proportional to the square of the coupling constants. In ref. [3] it was shown that the use of different coupling constants and form factors can lead to cross sections that differ by more than one order of magnitude, and that can even have a different behavior as a function of .

In previous works we have used the QCD sum rules (QCDSR) to study the D^*Dp [6,7], DDr [8] and J/yDD [9] form factors, considering two different mesons off mass-shell. In these works the QCDSR results for the form factors were parametrized by analytical forms such that the respective extrapolations to the off-shell meson poles provided consistent values for the corresponding coupling constant. In this work we use the QCDSR approach to evaluate the J/yD*D form factors and use the same procedure described above to estimate the J/yD*D coupling constant.

The three-point function associated with a H₁H₂H₃ vertex (see Fig. 1), where H₁ and H₃ are the incoming and outgoing external mesons respectively and H₂ is the off-shell meson, is given by

where j_i is the interpolating field for H_i. For J/y, D^* and D mesons the interpolating fields are respectively = and with q and c being a light quark and the charm quark fields.

The phenomenological side of the vertex function, G(p,p'), is obtained by the consideration of H₁ and H₃ state contribution to the matrix element in Eq. (1):

where h. r. means higher resonances.

The matrix element of the current j₂ defines the vertex function V_ll'(p,p'):

where q = p' – p. Calling p₁, p₂ and p₃ the four momentum of J/y, D^* and D respectively one has

The vacuum to meson transition amplitudes appearing in Eqs. (2) and (3) are given in terms of the corresponding meson decay constants fH_i by

and

for the vector meson V = J/y or V = D^*. Therefore, using Eqs. (3), (4), (5) and (6) in Eq. (2) we get

where

The contribution of higher resonances and continuum in Eq. (7) will be taken into account as usual in the standard form of ref. [10].

The QCD side, or theoretical side, of the vertex function is evaluated by performing Wilson's operator product expansion (OPE) of the operator in Eq. (1). Writing G_mn in terms of the invariant amplitude:

we can write a double dispersion relation for L, over the virtualities p² and p'² holding Q² = –q² fixed:

where r(s,u,Q²) equals the double discontinuity of the amplitude L(p²,p'²,Q²) on the cuts s_min < s < ¥, < u < ¥, with s_min = in the case of off-shell D^* or D and s_min = in the case of off-shell J/y. We consider diagrams up to dimension three which include the perturbative diagram and the quark condensate. To improve the matching between the two sides of the sum rules, we perform a double Borel transformation in both variables P² = –p² ® M² and P'² = –p'² ® M'² We get one sum rule for each meson considered off-shell. Calling (q²) the yDD^* form factor for the off-shell meson M, we get the following sum rules:

and

with t = q²,

l = (u + s – t)² –4us, l₂ = u + t – s + and

in the case of off-shell D or D^*. In the case of an off-shell J/y we get:

and

In the Eqs. (11), (12) and (13) we have transferred to the QCD side the higher resonances contributions through the introduction of the continuum thresholds s₀ and u₀.

The parameter values used in all calculations are m_c = 1.3 GeV, m_D = 1.87 GeV, m_D* = 2.01 GeV, m_y = 3.1 GeV, f_D = (170 ± 10) MeV, f_D* = (240 ± 20) MeV, f_J/y = (405 ± 15) MeV, áñ = –(0.23)³ GeV³. The continuum thresholds for the sum rules are s₀ = (m₁ + D_s)² and u₀ = (m₃ + D_u)² with D_s = D_u = 0.5 GeV.

We first discuss the J/yDD^* form factor with an off-shell D meson. In Fig. 2 we show the behavior of the form factor (Q²) at Q² = 5.0 GeV² and Q² = –0.21 GeV², as a function of the Borel mass M² using M'² = . We can see that the QCDSR results are rather stable in the interval 7 < M²< 11 GeV². In Fig. 3 we show (Q² = –0.21 GeV²) as a function of M² and M'².

From Fig. 3 we see that the stability is still good even considering the two independent Borel parameters. The same kind of stability is obtained for other values of Q² and for the other two form factors.

Fixing M² = and M'² = we show, in Fig. 4, the momentum dependence of the QCDSR results for the form factors , and through the circles, squares and triangles respectively. Since the present approach cannot be used at Q² << 0, to extract the g_yDD* coupling from the form factors we need to extrapolate the curve to Q² = –: the mass of the off-shell meson.

In order to do this extrapolation we fit the QCDSR results with an analytical expression. We tried to fit our results with a mono-pole form, since this is very often used for form factors, but the fit was only good for . For and we obtained good fits using a Gaussian form. We get:

These fits are also shown in Fig. 4 through the dotted, dashed and solid lines respectively. From Fig. 4 we see that all three form factors lead to compatible values for the coupling constant when the form factors are extrapolated to the off-shell meson mass (shown as open circles in Fig. 4). Considering the uncertainties in the continuum threshold, and the difference in the values of the coupling constants extracted when the D, D^* or J/y mesons are off-shell, our result for the J/yDD^* coupling constant is:

In Table I we show the results obtained for the same coupling constant using different approaches.in refs. [1] and [5].

While our result is compatible with the coupling obtained using constituent quark meson model [5], it is half of the value obtained with the vector meson dominance model plus relativistic potential model [1].

To summarize: we have used the method of QCD sum rules to compute form factors and coupling constant in the J/yDD^* vertex. Our results for the coupling show once more that this method is robust, yielding numbers which are approximately the same regardless of which particle we choose to be off-shell and depending weakly on the choice of the continuum threshold. As for the form factors, we obtain a harder form factor when the off-shell particle is J/y, when compared with the form factors obtained when the off-shell particles are D or D^*.

Acknowledgments

This work was supported by CNPq and FAPESP.

[3] K.L. Haglin and C. Gale, hep-ph/0305174.

[5] A. Deandrea, G. Nardulli and D. Polosa, hep-ph0302273.

Received on 15 August, 2003.

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Publication Dates

• Publication in this collection
  11 May 2004
• Date of issue
  Mar 2004

History

• Accepted
  15 Aug 2003
• Received
  15 Aug 2003