1. A network-based quark-cluster algorithm Athanasios Petridis Drew Fustin Drake University James Vary Iowa State University April 8, 2003

2. Outline • The quark cluster model • Applications • Clusters as networks • The algorithm • Case studies • Conclusions

3. The quark cluster model • In 1986 Sato, Pirner, Vary, and Coon suggested that when the wavefucntions of nucleons bound in nuclei sufficiently overlap they form larger color singlets (quark clusters). • Using realistic wavefucntions for deuterium they estimated the probability of a 6q cluster in it to be about 4%. • They (and others) predicted an observed deviation in Drell-Yan production due to cluster corrections.

4. Applications • The parton momentum distributions in clusters are softer than for single nucleons. • For proton-like clusters they are (scaling). Isospin relations apply

5. Structure function ratio using up to 12q clusters in Ca (f1 = 0.72) and 6q clusters in d (f1 = 0.96). The triangles are DGLAP-Q2-evolved points. The data are from Amaudriuz et al. (NMC), Z. Phys. C51, 387 (1991)

6. J/Ψ suppression ratio including 6q clusters and final-state absorption (3.5 mb/nucleon). The two lines show the uncertainty of the model. The data are from Alde et al. (E772), Phys. Rev. Lett. 66, 133 (1991).

7. Clusters as networks 2nd order cluster 3rd order cluster 6th order cluster Links connect particles whose distance is smaller than Rc. EACH PARTICLE BELONGS TO ONE AND ONLY ONE CLUSTER

8. The algorithm • For each critical radius first produce a spherically-symmetric distribution of N particles using the Woods-Saxon nuclear density. Care is taken so that the center of mass is at the center. • Define the symmetric distances-array d(i,j), where i, j = 1,…N. • Define the initial network array A0(i,j), where i, j = 1,…N so that A0(i,j) = 1 if d(i,j) < Rc, and 0 otherwise.

9. Reduce A0 to the final network array A(i,j), where i, j = 1,…N. following the steps: • -- compare each column i = 1, …, N – 1 with the column k = i + 1, …, N, counting the rows that contain two 1’s. -- if the count is at least 1 then merge the two columns by replacing A(i,j) with its OR with A(k,j) and then setting A(k,j) to 0. -- count the 1’s in each column (cluster order) and, then, count the clusters. Average over many arrays of particles for each Rc.

10. Case studies • We applied the algorithm to a number of nuclei. • For each Rc there are 10000 iterations (for A = 197 we used 20000 iterations). • The results indicate the expected increase of the cumulative cluster probability with Rc.

11. A = 2

12. A = 4

13. A = 56

14. A = 107

15. A = 197

16. The critical radius may be chosen so that in the case of the deuteron f2 = 0.04. This gives Rc = 0.80 – 0.85 fm. • Then we obtain: A = 4: f1 = 0.96, f2 = 0.02 A = 56: f1 = 0.90, f2 = 0.08 A = 107: f1 = 0.88, f2 = 0.09 A = 197: f1 = 0.85, f2 = 0.11 Computed values approximately agree with the ones used to describe the DIS, D-Y, and J/Ψ data but for large nuclei they are lower.

17. Conclusions • We have developed a network-based algorithm for quark-cluster formation. • The obtained clustering probabilities do not drastically differ from those needed to describe the data. More realistic nuclear density functions including correlations should be introduced.