Two-Photon and Two-gluon Decays of 0^{++} and 2^{++} P-wave Heavy Quarkonium States1footnote 11footnote 1 Talk given at the QCD@Work 2010 International Workshop on QCD: Theory and Experiment, Martina Franca, Valle d’Istria, Italy, 20–23 June 2010

Two-Photon and Two-gluon Decays of
and P-wave Heavy Quarkonium States111 Talk given at the QCD@Work 2010 International Workshop on QCD: Theory and Experiment, Martina Franca, Valle d’Istria, Italy, 20–23 June 2010

T. N. Pham Centre de Physique Théorique, CNRS
Ecole Polytechnique, 91128 Palaiseau, Cedex, France
July 19, 2019
Abstract

By neglecting the relative quark momenta in the propagator term, the two-photon and two-gluon decay amplitude of heavy quarkonia states can be written as a local heavy quark field operator matrix element which could be obtained from other processes or computed with QCD sum rules technique or lattice simulation, as shown in a recent work on two-photon decays. In this talk, I would like to discuss a similar calculation on -wave and two-photon decays. We show that the effective Lagrangian for the two-photon decays of the -wave and is given by the heavy quark energy-momentum tensor local operator and its trace, the scalar density. A simple expression for two-photon and two-gluon decay rate in terms of the decay constant, similar to that of is obtained. From the existing QCD sum rules value for , we get for the two-photon width, somewhat larger than measurement.

pacs:
13.20.Hd,13.25.Gv,11.10.St,12.39Hg

I Introduction

First of all, I would like to dedicate this talk to the memory of Professor Giuseppe Nardulli, who, with great kindness and generosity has initiated the long and fruitful collaboration I have with the members of the Physics Department and INFN at the University of Bari.

In the non-relativisitic bound state calculation Barbieri (); Brambilla (), the two-photon and two-gluon decay rates for -wave quarkonium states depend on the derivative of the spatial wave function at the origin which has to be extracted from potential models, unlike the two-photon decay rate of -wave and quarkonia which can be predicted from the corresponding and leptonic widths using heavy quark spin symmetry(HQSS) Lansberg (), there is no similar prediction for the -wave and states and all the existing theoretical values for the decay rates are based on potential model calculations Barbieri (); Godfrey (); Barnes (); Bodwin (); Gupta (); Munz (); Huang (); Ebert (); Schuler (); Crater (); LWang (); Laverty ().

Since the matrix element of a heavy quark local operator between the vacumm and -wave quarkonium state is also given, in bound state description, by the derivative of the spatial wave function at the origin, one could express the -wave quarkonium two-photon and two-gluon decay amplitudes in terms of the matrix element of a local operator with the appropriate quantum number, like the heavy quark scalar density or axial vector current . We have thus an effective Lagrangian for the two-photon and two-gluon decays of -wave quarkonia in terms of heavy quark field operator instead of the traditional bound state description in terms of the wave function. This effective Lagrangian can be derived in a simple manner by neglecting the relative quark momentum in the heavy quark propagator as in non-relativistic bound state calculation. In this talk, I would like to report on a recent work Lansberg2 () using the effective Lagrangian approach to describe the two-photon and two gluon decays of -wave heavy quarkonia state, similar to that for -wave quarkonia Lansberg () . This was stimulated by the recent new CLEO measurements CLEO (); PDG () of the two-photon decay rates of the charmonium -wave , and states. We obtain an effective Lagrangian for -wave quarkonium decays in terms of the heavy quark energy-momentum tensor and its trace and that the two-photon and two-gluon decay rates of and can be expressed in terms of the decay constants and , similar to that for and , which are given, respectively, by and . Then a calculation of and by sum rules technique Novikov (); Reinders () or lattice simulation Dudek (); Chiu () would give us a prediction of the -wave quarkonia decay rates. In fact, as shown below, obtained in Novikov () implies a value of for the two-photon width, somewhat larger than measurement. In the following I will present only the main results, as more details are given in the published paper Lansberg2 ().

Figure 1: Diagrams for annihilation to two photons.

Ii Effective Lagrangian for and

By neglecting term containing the relative quark momenta in the quark propagatorKuhn () ( being the heavy quark charge), the -wave part of the and amplitudes represented by diagrams in Fig. 1 are

(1)

with the photon part of the amplitude and the heavy quark part given by

(2)
(3)

which can be obtained directly from the following effective Lagrangian for two-photon and two-gluon decay of -wave heavy quarkonia states

(4)

With the matrix element of between the vacuum and or given by ()

(5)

The two-photon decay amplitudes are then easily obtained :

(6)
(7)

with from HQSS and

(8)
(9)

For QCD sum rules calculation or lattice simulation, it is simpler to compute the trace of the energy-momentum tensor given by . We have then

(10)

The problem of computing the two-photon or two-gluon decay amplitude of and states is reduced to computing the decays constants and defined as

(11)

Thus is given directly in terms of without using the derivative of the -wave spatial wave function at the origin.

(12)

Thus by comparing the expression for and we could already have some estimate for the two-photon and two-gluon decay rates. For of , one would expect to be in the range of a few keV.

The decay rates of , states can now be obtained in terms of the decay constant . We have :

(13)
(14)

where and are NLO QCD radiative corrections Barbieri2 (); Kwong (); Mangano ()

This expression is similar to that for  :

(15)

The two-gluon decay rates are :

(16)
(17)

where and are NLO QCD radiative corrections. As with the two-photon decay rates, the expressions for two-gluon decay rates are similar to that for :

(18)

In a bound state calculation, using the relativistic spin projection operator Kuhn (); Guberina () , and are given by

(19)
(20)

which gives the decay amplitudes in agreement with the original calculation Barbieri ().

Comparing with , we have

(22)

which becomes comparable to .

Thus by comparing the expression for and we could already have some estimate for the two-photon and two-gluon decay rates. For of , one would expect to be in the range of a few keV. As shown in Table 1, the predicted two-photon width of from the sum rules value Novikov () is however almost twice the CLEO value, but possibly with large theoretical uncertainties in sum rules calculation as to be expected, while a recent calculation Colangelo () implies a larger decay rates for . The measured ratio is then , somewhat bigger than the predicted value of about as shown in Table 1 together with the CLEO measurement of the decay rates CLEO () which gives and for and respectively.

Reference (keV) (keV)
BarbieriBarbieri ()
GodfreyGodfrey ()
BarnesBarnes ()
GuptaGupta ()
MünzMunz ()
HuangHuang ()
EbertEbert ()
SchulerSchuler ()
CraterCrater ()
WangLWang ()
LavertyLaverty ()
This work
Exp(CLEO)CLEO ()
Exp(Average)CLEO ()
Table 1: Potential model predictions for two-photon widths compared with this work.

The two-photon branching ratios are independent of

(23)
(24)

with . Apart from QCD radiative correction factors, the expressions for branching ratios are very similar to that for and :

(25)

with evaluated at the appropriate scale.

For , to be compared with the measured value of PDG (), but this prediction is rather sensitive to , for example, with , one would get , in better agreement with the measured value of and for , the predicted two-photon branching ratios would be and compared with the measured values of and , for and respcetively. The predicted branching ratio for is rather large and one would need to bring the predicted value closer to experiment.

Recently, the state above threshold found by Belle Belle2 () with mass and width , consistent with , seems to be confirmed by the observation of a similar state by BaBar Babar2 (), with mass and width . Belle Belle2 () gives while Babar Babar2 () gives for this state. If taken to be the excited state and assuming Colangelo2 (); Swanson (); Chao (), one would get . This implies and in the range .

For potential model calculations similar to that for , gives the two-photon width about of that for , which implies , smaller than Cornell potential Quigg () value .

Iii Remark on the two-photon decays

Since the predicted two-photon branching ratios for and for are similar and independent of the decay constants, apart from QCD radiative corrections, as seen in Eq. (23-24) and Eq. (25), one expects a large two-photon decay rates for , it would be relevant here to mention the problem of the decay rate Lansberg (); Pham (). The small value of given previously by CLEO Asner () is obtained by assuming . However, with the recent BaBar measurement of the ratio Babar ()

(26)

and the Belle measurement Belle ()

(27)

BABAR obtains Babar ()

(28)

as quoted by CLEO CLEO2 (). This new BABAR value for is considerably smaller than the corresponding value PDG () for .

Thus with the BaBar result for and the CLEO measurement Asner ()

(29)

one would get CLEO2 ()

(30)

in agreement with the predicted value

(31)

while the assumption of near equality of the branching ratios for and

(32)

and the Belle ratio Choi ()

(33)

would lead to CLEO2 ()

(34)

which is rather small compared with the predicted value given in Eq. (31) above.

Iv Conclusion

In conclusion, we have derived an effective Lagrangian for and two-photon and two-gluon in terms of the decay constants , similar to that for in terms of .

Existing sum rules calculation, however produces a two-photon width about , somewhat bigger than the CLEO measured value. It remains to be seen whether a better determination of from lattice simulation or QCD sum rules calculation could bring the two-photon decay rates closer to experiments or higher order QCD radiative corrections and large relativistic corrections are needed to explain the data.

The problem of two-photon width of would go away if more data could confirm the small BaBar value for compared with .

As relativistic corrections should be small for -wave bottomia states, two-photon and two-gluon decays could provide a test of QCD and a determination of at the the mass scale.

V Acknowledgments

I would like to thank P. Colangelo, F. De Fazio and the organizers for generous support and warm hospitality extended to me at Martina Franca. This work was supported in part by the EU contract No. MRTN-CT-2006-035482, ”FLAVIAnet”.

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