1. 1 Noncommutative QCDCorrections to the Gluonic Decays of Heavy Quarkonia Stefano Di Chiara A. Devoto, S. Di Chiara, W. W. Repko, Phys. Lett. B 588, 85 (2004). International Workshop on Heavy Quarkonia - 2006

2. 2 Introduction  Noncommutativity Quarkonia decays in NcQCD  Noncommutative QCD  Quarkonium in nonrelativistic approximation  Squared amplitudes for in three gluons  Decay widths of 1 S 0, 3 S 1 Conclusions Outline

3. 3 Space noncommutativity The coordinates do not commute  the point is substituted by a small area Motivations: General Relativity: to measure distance l  energy l -1, if l < l p  black hole  the measure is impossible. String Theory: NcQFT obtained in the background of a strong magnetic field

4. 4 Functions of noncommutative x Dual vector: direction of maximum noncommutativity Function of noncommutative-coordinates Standard product  Moyal-Weyl star-product

5. 5 Noncommutative QCD -product  Moyal *-product  Lagrangian QCD  Lagrangian NcQCD  0i  0  causality, unitarity of action are violated   0i  0 In this case P is satisfied, but C, T, Lorentz invariance are still violated. The infinitesimal variation is given by

6. 6 Feynman rules From the previous formulas we can calculate the Feynman rules in NcQCD for the vertices, while the propagators are the same of QCD since

7. 7 Feynman rules Because of d abc also the abelian theories like QED have three and four boson vertices.

8. 8 Quarkonium Heavy bound state represented by 2 s+1 l j s=0,1  “para-quarkonium”, “ortho-quarkonium” Kinetic energy << M  wave function can be solved from Schroedinger’s eq. Potential needs in part to be fitted to the experiment The amplitude, for Q decaying in any given state, can be expressed in function of a QCD amplitude In the non-relativistic limit Motivations:  Interesting comparation with positronium in NcQED  Much greater masses available to resolve  NC  Great amount of experimental data

9. 9 Decay Therefore the decay width  of quarkonium in three gluons can be expressed as Where the squared matrix elements for the ortho-quarkonium and para-quarkonium are given just by the expressions

10. 10  ggg The diagrams with a single gluon joined to the fermionic line are null, since there is no gluon singlet.

11. 11 Squared amplitudes Helicity squared amplitudes calculated by formal manipulation program FORM (Jos Vermaseren) and simplified by Mathematica. Defining the variables The squared matrix elements summed over the gluon helicities are ORTHO-QUARKONIUM IN THREE GLUONS PARA -QUARKONIUM IN THREE GLUONS

12. 12 Results’ checks QCD:   =0  v=0, w=1  recovered known QCD result. Ward’s identity: P symmetry NcQED: making the substitutions ORTHO-POSITRONIUM IN THREE PHOTONS PARA-POSITRONIUM IN THREE PHOTONS

13. 13 C, P, T, Lorentz invariance C : P : T : For the quarkonium there is only Lorentz invariance violation: A process that violates also C, T, but not P, is for example in NcQED:

14. 14 Ortho-quarkonium decay width QCD result: NcQCD correction: It is not integrable analytically. NcQCD function of the adimensional parameter ORTHO-QUARKONIUM DECAY WIDTH

15. 15 Para-quarkonium decay width Since the pQ can decay in two gluons, it is necessary to introduce a minimum energy detectable: However the NcQCD correction is completely finite for. PARA-QUARKONIUM DECAY WIDTH

16. 16 Experimental limits Lower limit on  NC in particle accelerator (OPAL collaboration): Values of z for the masses of c, b, t : NcQCD relative correction definition: Lorentz violation: Gluons decay preferentially in the (time dependent) plane perpendicular to DECAY WIDTH’S RELATIVE CORRECTIONS DIFFERENTIAL DECAY WIDTH’S R. C.

17. 17 Conclusions The NcQCD corrections to  Q  hadrons are negligible for  NC of order 1 TeV. The infrared contribution of the NcQCD are finite, even for the pQ. The NcQCD correction to  pQ  ggg is negative, nevertheless  pQ  ggg remains positive for large values of z: the theory is not ruled out by the experiment. The decay width differential in the angle  can sensibly improve the experimental limits on  NC.