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there are several advantages in studying a heavy - light system on a lattice .
our meson is much more simple than in true qcd : one of the quarks is static with the light quark `` orbiting '' it .
this makes it very beneficial for modelling . on the lattice
an abundance of data can be produced , and we know which state we are measuring  the physical states can be a mixture of two or more configurations , but on the lattice this complication is avoided . however , our results on the heavy - light system can still be compared to the @xmath6 meson experimental results .
we have measured both angular and radial excitations of heavy - light mesons , and not just their energies but also some radial distributions . since the heavy quark spin decouples from the game we may label the states as @xmath7 , where l is the angular momentum and @xmath8 is the spin of the light quark .
the measurements were done on a @xmath9 lattice with dynamical clover fermions .
we have two degenerate quark flavours with a mass that is close to the strange quark mass .
the lattice configurations were generated by the ukqcd collaboration .
some details about the different lattices used in this study can be found in table  [ latttable ] .
two different levels of fuzzing ( 2 and 8 iterations of conventional fuzzing ) were used in the spatial directions to permit the extraction of the excited states .
.lattice parameters .
these are ukqcd collaboration s lattices with @xmath10 and @xmath11 . [ cols="<,^,<,<,<,<",options="header " , ]
the 2-point correlation function ( see figure  [ fig : corr ] a ) is defined as @xmath12 where @xmath13 is the heavy ( infinite mass)-quark propagator and @xmath14 the light anti - quark propagator . @xmath15 is a linear combination of products of gauge links at time @xmath16 along paths @xmath17 and @xmath18 defines the spin structure of the operator .
the @xmath19 means the average over the whole lattice .
the energies ( @xmath20 ) and amplitudes ( @xmath21 ) are extracted by fitting the @xmath22 with a sum of exponentials , @xmath23 fuzzing indices have been omitted for clarity .
we introduced two types of smearing in the time direction to allow the stationary quark to move a little , but not too far , from its fixed location .
first we tried ape type smearing , where the original links in the time direction are replaced by a sum over the six staples that extend in the spatial directions ( in fig .
[ fig : smear ] on the left ) .
this smearing is called here `` sum6 '' for short . to smear the static quark even more we then tried hypercubic blocking , again only for the links in the time direction ( in fig .
[ fig : smear ] on the right ) .
now the staples ( the red ones in fig .
[ fig : smear ] ) are not constructed of the original , single links , but from staples ( the blue ones in fig .
[ fig : smear ] ) .
this allows the heavy quark to move within a `` hypercube '' ( the edges of the `` cube '' are 2@xmath0 in spatial directions but only one lattice spacing in the time direction ) .
this smearing is called here `` hyp '' for short .
smearing the heavy quark was expected to improve the measurements , particularly radial distributions - which it did , to some extent .
the energy spectrum obtained is shown in fig .  [
fig : espectr ] . using different smearing for
the heavy quark does not seem to change the energies too much - except for the p@xmath24 state .
it is not understood yet why this state should be more sensitive to changes in the heavy quark than the other states we have considered .
the energy of the d@xmath25 state was expected to be near the spin average of the d@xmath26 and d@xmath24 energies , but it turned out to be a poor estimate of this average .
thus it is not clear whether or not the f@xmath25 energy is near the spin average of the two f - wave states , as was hoped .
our earlier results can be foun in ref .
@xcite .
one interesting point to note here is that the spin - orbit splitting of the p - wave states is small , almost zero .
we extracted the energy difference of the p@xmath24 and p@xmath26 states in two different ways :    1 .   indirectly by simply calculating the difference using the energies given by the fits in equation  [ c2fit ] , when the p@xmath24 and p@xmath26 data are fitted separately .
2 .   directly by combining the p@xmath24 and p@xmath26 data ( taking the ratio ) and fitting everything in one go with @xmath27+b\exp[-m_ct],\ ] ] where @xmath28 , @xmath29 and @xmath30 are fit parameters . @xmath31 and @xmath32 are the energies of the p@xmath26 and p@xmath24 ground states , respectively .
the energy difference , rather than the energies themselves , were varied in the fit .
the second exponential contains the corrections from the excited states .
d - wave spin - orbit splitting was also extracted in a similar manner .
the results are shown in fig .
[ fig : sos ] .
for evaluating the radial distributions of the light quark a 3-point correlation function shown in fig .
[ fig : corr ] b is needed .
it is defined as @xmath33 this is rather similar to the 2-point correlation function in eq .
[ 2point ] .
we have now two light quark propagators , @xmath34 and @xmath35 , and a probe @xmath36 at distance @xmath37 from the static quark ( @xmath38 for the vector ( charge ) and @xmath39 for the scalar ( matter ) distribution ) .    knowing the energies @xmath20 and the amplitudes @xmath21 from the earlier @xmath22 fit , the radial distributions , @xmath40 s , are then extracted by fitting the @xmath41 with @xmath42 the results are plotted in figures  [ s11][dmdistr ] .
the error bars in these figures show statistical errors only .
see @xcite for earlier s - wave distribution calculations .
the `` sum6 '' distributions have been published in @xcite , but the `` hyp '' results are still preliminary .
we are currently trying to improve the analysis of the d - wave radial distribution data .
when measuring a radial distribution it is easy to also measure the sumrule by simply summing over the whole lattice .
our results for the charge sumrule are shown in figure  [ dmdistr ] .
we included the vertex correction @xmath43 , so that with our normalisation the result should be one .
this comes out very nicely for the s - wave , but for the other states we get the sumrule to be somewhat smaller .
a simple model based on the dirac equation is used to try to describe the lattice data .
since the mass of the heavy quark is infinite we have essentially a one - body problem .
the potential in the dirac equation has a linearly rising scalar part , @xmath44 , as well as a vector part @xmath45 .
the one gluon exchange potential , @xmath46 , is modified to @xmath47 where @xmath48  mev and the dynamical gluon mass @xmath49  mev ( see  @xcite for details ) .
the potential also has a scalar term @xmath50 , which is needed to increase the energy of higher angular momentum states .
however , this is only a small contribution ( about 30  mev for the f - wave ) .
the solid lines in the radial distribution plots are predictions from the dirac model fit with @xmath51  gev , @xmath52 , @xmath53  gev / fm , @xmath54 and @xmath55 .
these are treated as free parameters with the values obtained by fitting the ground state energies of p- , d- and f - wave states and the energy of the first radially excited s - wave state ( 2s ) .
note that the excited state energies in fig .
[ fig : espectr ] were not fitted .
this fit was done using the energies obtained with ape smearing ( `` sum6 '' ) , and the latest `` hyp '' data was not used .
* there is an abundance of lattice data , energies and radial distributions , available . *
the spin - orbit splitting is small and supports the symmetry @xmath56 as proposed in  @xcite . * the energies and radial distributions of s- ,
p- and d - wave states can be qualitatively understood by using a one - body dirac equation model .
i am grateful to my supervisor a. m. green and to our collaborator , professor c. michael .
i wish to thank the ukqcd collaboration for providing the lattice configurations .
i also wish to thank the center for scientific computing in espoo , finland , for making available resources without which this project could not have been carried out .
the eu grant hprn - ct-2002 - 00311 euridice and financial support from the magnus ehrnrooth foundation are also gratefully acknowledged .
this is a follow - up to our earlier work for the energies and the charge ( vector ) and matter ( scalar ) distributions for s - wave states in a heavy - light meson , where the heavy quark is static and the light quark has a mass about that of the strange quark . 
 we study the radial distributions of higher angular momentum states , namely p- and d - wave states , using a `` fuzzy '' static quark . 
 a new improvement is the use of hypercubic blocking in the time direction , which effectively constrains the heavy quark to move within a 2@xmath0 hypercube ( @xmath0 is the lattice spacing ) .    
 the calculation is carried out with dynamical fermions on a @xmath1 lattice with @xmath2  fm generated using the non - perturbatively improved clover action . 
 the configurations were generated by the ukqcd collaboration using lattice action parameters @xmath3 , @xmath4 and @xmath5 .    in nature 
 the closest equivalent of this heavy - light system is the @xmath6 meson . 
 attempts are now being made to understand these results in terms of the dirac equation .