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in a recent paper@xcite , the bgr - collaboration presented results on light hadron ( down to @xmath4 ) quenched spectroscopy with two different actions : the parametrized fixed point ( fp ) action@xcite and the ci action which is a combination of the lscher - weisz gauge action@xcite with a chirally improved dirac operator@xcite . in this paper
we extend the scaling analysis in ref.@xcite on the fp action by adding new results on an @xmath5 lattice with lattice unit @xmath6fm .
this extension is necessary , since our scaling test in @xcite referred to a small @xmath7fm spatial size lattice with low statistics , while our large @xmath8fm lattice results with good statistics could be compared with other simulations only .
the new data have been used already in a scaling test on the @xmath9 scattering length@xcite .
no cut - off effects were seen there , but it is difficult to measure the scattering length precisely .
we need reliable information on the cut - off effects in the quenched approximation since these results help to choose the parameters of future full qcd simulations the bgr collaboration is moving towards .
in addition , this information might be helpful when comparing the results from different actions for consistency .
indeed , there exist unclarified inconsistencies between different works in light hadron spectroscopy . on the ape plot , for example ,
the cp - pacs@xcite continuum extrapolated numbers lie significantly below the data obtained with most of the different improved actions . for a recent comparison of numerical data obtained with different actions see ref .
@xcite .    the parametrized fp dirac operator lives on the hypercube , has 81 offsets , a large number of paths , uses all the elements of the clifford algebra@xcite and satisfies the ginsparg - wilson relation up to parametrization errors .
the setup of the present simulation and the analysis follows closely those applied in ref.@xcite .
we generated 180 gauge configurations separated by 500 alternating metropolis and pseudo over - relaxation sweeps at gauge coupling @xmath10 .
at this coupling the lattice unit is @xmath6fm as determined from the sommer parameter @xmath11fm@xcite .
as it is well known , it is difficult to give an estimate on the systematic errors of @xmath12 if the lattice is not fine . at this resolution
, the systematic error should be on the percent level , while the statistical error is smaller than @xmath13@xcite .
the spatial extension of the lattice is @xmath14fm . in our earlier results obtained at @xmath15fm and @xmath8fm at resolution
@xmath16fm we saw no finite size effects beyond the statistical errors .
our scaling analysis might be influenced by small finite size effects , however , and these physical effects might mix with the topological finite size artifacts as we discuss below .    before summarizing our results ,
let us make two general remarks concerning scaling analyses in light hadron spectroscopy .
although the most interesting part of the spectrum for physics is where the quark masses go to their small physical values , the heavier quarks bring more information for scaling studies : the hadrons are more compact , they are more difficult to resolve and the cut - off effects are expected to be larger .
in addition , the statistical and systematic errors are smaller , so the analysis is more conclusive .
the second remark refers to a special artifact of _ quenched _ light hadron spectroscopy : the topological finite size artifacts .
the discrete zero modes of the dirac operator at @xmath17 are not suppressed in the quenched approximation and , in a finite volume at sufficiently small quark masses , they will corrupt the hadron propagators@xcite . in ref.@xcite we eliminated / reduced the topological finite size artifacts by using special hadron correlators in the pseudoscalar ( ps ) , nucleon ( n ) and delta ( @xmath18 ) channels .
unfortunately , in the vector channel we did not find a natural solution for this problem and the artifacts turned out to be larger at @xmath6fm than at @xmath19fm in the same physical volume @xmath15fm .
the effect is seen in the vector meson mass at the lightest quark masses in fig .
[ fig : rawdata ] , where the dimensionless hadron masses are shown as the function of the dimensionless quark mass .
this feature ( namely , the bending down of the vector meson curve at small quark masses ) is very similar to the artifacts we saw earlier appearing in smaller volumes , and then disappearing in our largest box with @xmath8fm@xcite .
these quenching artifacts should be separated from the cut - off effects in a scaling test .
this is relevant in particular in the vector channel , since the vector meson mass is used traditionally ( unfortunately ) as a unit for other hadron masses .
the problem occurs at sufficiently light quark masses only , which gives an additional argument to avoiding this region in scaling studies .
the hadron masses as the function of the quark mass in lattice units at @xmath20fm in a box of size @xmath15fm.,scaledwidth=80.0% ]    we give here a brief summary of our conclusions . considering mass ratios ( like in the ape plot , or measuring the masses in units of a well determined fixed hadron mass ) we see small cut - off effects in the baryon channel when comparing the results at @xmath16 and 0.102fm resolutions .
the effect is seen for relative heavy quarks beyond the statistical errors . in the @xmath21 channel
this is a several standard deviation effect and has a size of up to @xmath22 in the ratio @xmath23 , where @xmath24 is the vector meson mass . in the nucleon and delta channels the cut - off effects are smaller , but for heavy quarks they are also visible beyond the statistical errors
this small shift brings the fp results on the ape plot closer to and essentially consistent with the cp - pacs continuum extrapolated results with wilson fermions@xcite .
the improved staggered ( asqtad@xcite ) data at @xmath25fm@xcite with heavier quarks lie a few standard deviations higher and are close to the cp - pacs results at @xmath26fm . in addition , unlike at @xmath16fm , we see a discrepancy in the overall scale obtained in the gauge sector from the sommer parameter @xmath27 and the scale from the spectrum of light hadrons .
the hadron sector prefers a scale of @xmath28fm as opposed to the scale of @xmath6fm . in this case
, however , it is difficult to estimate the systematic errors .
concerning the details of the analysis we refer to ref.@xcite . in order to avoid very busy figures we shall use a few selected results only from other works in this paper .
one can get a more complete picture by combining the figures here with those in ref.@xcite .
the measured new data are collected in tables 1 and 2 .
[ fig : overview ] gives a quick overview on the cut - off effects in the different channels . in this figure
the hadron masses are measured in @xmath29 units , where @xmath29 is the vector meson mass at @xmath30 .
this figure indicates already that the change due to cut - off effects is small in @xmath31fm .    .
the hadron masses for different quark masses in lattice units .
[ table:1 ] [ cols="^,^,^,^,^,^",options="header " , ]      the energy shift of the two pion state in the @xmath32 box for the three different lattice spacings.,scaledwidth=80.0%,scaledwidth=60.0% ]     the dimensionless scattering length in the @xmath33 box for two different lattice spacings , @xmath34 and @xmath35 .
chiral perturbation theory curves ( @xcite ) are shown for @xmath36.,scaledwidth=80.0%,scaledwidth=60.0% ]     summary of all the intermediate and large volume results together with the the wilson action results ( @xcite ) for the ratio @xmath37 .
nnlo chiral perturbation theory results ( @xcite ) and nlo quenched chiral p.t .
curves ( @xcite ) are shown with the measured values for @xmath38 .
the behaviour expected from the scattering of hard spheres are shown at large @xmath39.,scaledwidth=80.0%,scaledwidth=60.0% ]
in this work we discussed the cut - off effects in simulations with the parametrized fp action . for this purpose going deep in the chiral limit is not very important .
it is good to see , nevertheless , that with this action one can reproduce cleanly the divergence @xmath40 in @xmath41 predicted by quenched chiral perturbation theory@xcite as shown in fig .
[ fig : logdiv ] .
here @xmath42 , where @xmath43 is the quark mass in the simulation and @xmath44 is the residual additive mass renormalization due to the parametrization error of the fp dirac operator .     the ratio @xmath41 at fixed @xmath20fm as the function of the quark mass @xmath45 , @xmath46 , where @xmath44 is the residual additive renormalization due to the parametrization error of the fp dirac operator .
the continuous line is a fit of the form @xmath47 suggested by quenched chiral perturbation theory.,scaledwidth=80.0% ]    * acknowledgements * we thank christine davies for the correspondence on the interpretation of the ape plot .
we thank also the discussions with anna hasenfratz and the members of the bgr collaboration , and the support from the swiss center for scientific computing in manno , where the numerical simulations were done .
k.j.j . would like to thank colin morningstar for some of the fitting routines used in the analysis of the scattering length .
this work was supported by the schweizerischer nationalfonds .
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* 105 * ( 2001 ) 37
our study on the cut - off effects in quenched light hadron spectroscopy and pion scattering length with the fixed point action is extended by results obtained at a lattice spacing a=0.102 fm in a box of size l=1.8 fm . 
 the cut - off effects are small , but clearly seen as the resolution is increased from a=0.153 fm to a=0.102 fm . in the quark mass region where the errors are small and under control , 
 our results on the ape plot lie close to the extrapolated numbers of the cp - pacs collaboration .    # 1 januaryfebruarymarch aprilmayjunejulyaugustseptemberoctober novemberdecember    new results on cut - off effects in spectroscopy with the fixed point action peter hasenfratz@xmath0 , k.  jimmy juge@xmath1 and ferenc niedermayer@xmath2 + bgr 
 ( bern - graz - regensburg ) collaboration @xmath0 institute for theoretical physics , university of bern + sidlerstrasse 5 , ch-3012 bern , switzerland    @xmath1 school of mathematics , trinity college + dublin 2 , ireland @xmath3 on leave of absence from etvs university , has research group , budapest , hungary