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instantons are believed to be an essential component of the long - distance physics of the qcd vacuum , and the lattice provides a unique opportunity to gain insight into their role . in this study , we will for the first time quantitatively examine the extent to which instanton degrees of freedom , contained within standard monte - carlo generated gauge - field configurations , can maintain the characteristic features of the mass and renormalisation functions of the non - perturbative quark propagator .    in order to isolate the effects of instanton degrees of freedom a uv filter is required to remove topologically non - trivial fluctuations . a variety of filters have been used , including cooling@xcite , ape smearing@xcite , hyp smearing @xcite and stout link smearing @xcite , among others . these algorithms can suffer from destruction of the instanton content of the vacuum , and so in this work we use over - improved stout - link smearing @xcite , a form of smearing tuned to preserve instantons . section [ sec : smearing ] briefly describes these smearing methods . we then seek to quantify the effects of smearing on the lattice gauge fields in section [ sec : smearingeffect ] . we produce configurations dominated by instanton - like objects , and compare to the phenomenological instanton liquid model @xcite , which models the vacuum as composed of a constant number of instantons and anti - instantons of constant radius . we will then briefly introduce the fat link irrelevant clover(flic ) overlap action @xcite in section [ sec : overlap ] , an improved fermion action with a lattice deformed version of chiral symmetry , which removes the problem of additive mass renormalisation of the quark propagator . results will be compared for smeared and unsmeared configurations in section [ sec : results ] , and conclusions summarised in section [ sec : conclusion ] . it has long been known @xcite that smearing the lattice reveals objects which approximate classical instantons . however , it has been known for nearly as long that excessive smearing can destroy the same instanton - like objects or distort their structure . we can understand this behaviour by explicitly considering the single instanton solution , @xcite @xmath0 where @xmath1 is the instanton radius , @xmath2 the pauli matrices and @xmath3 the center of the instanton . the wilson gauge action can be expanded in powers of @xmath4 as@xcite @xmath5\mathrm { , } \nonumber \\ \quad \end{aligned}\ ] ] where @xmath6 $ ] for arbitrary @xmath7 .    inserting the instanton solution of eq . ( [ instsol ] ) into eq . ( [ wilsonactexp ] ) , we acquire @xmath8.\ ] ] the source of the problem is clear ; although at first order this is equal to the continuum instanton action , the leading order error term is strictly negative and inversely proportional to @xmath1 . the use of a plaquette - based smearing algorithm will reduce the size of instantons in decreasing the action and ultimately enable lattice artifacts to spoil and remove the instanton . this situation is not ameliorated by using improved actions such as the symanzik action @xcite    this issue can be mitigated using over - improved stout - link smearing , a form of smearing designed to preserve instantons . this was first implemented in ref . @xcite , although here we follow the work ref . @xcite , which uses a slightly modified combination of links . one introduces a new parameter , @xmath9 , defining @xmath10,\ ] ] with @xmath11 and @xmath12 the @xmath13 and @xmath14 plaquettes . we note that a value of @xmath15 gives the wilson plaquette action and @xmath16 gives the symanzik improved action @xcite . again substituting the instanton solution into this , one acquires @xcite @xmath17.\ ] ] a negative value of @xmath9 will preserve instanton - like objects by making the first order error term positive . this has however , simultaneously introduced a new problem ; we have removed the possibility of shrinking objects and replaced it with that of enlarging , and so smearing can distort the topological structure of the lattice if used excessively with large negative values of @xmath9 . following ref . @xcite , we adopt the small value @xmath18 . explicitly , a sweep of over - improved stout - link smearing is implemented by replacing all links on the lattice with a smeared link , defined by @xmath19 where @xmath20 with @xmath21 then we define @xmath22,\end{aligned}\ ] ] where @xmath1 is a parameter controlling the level of smearing and @xmath23 are staples ; the 3 links in the @xmath24 plane forming the @xmath11 plaquette with @xmath25 removed , and @xmath26 are analogously defined for @xmath27 rectangles , illustrated in fig . [ pic : stapledef ] .     and @xmath26 ]    following the work in ref . ( @xcite ) , we choose @xmath29 and @xmath18 , as these provide values of @xmath30 close to 1 , and preserve instanton - like objects on the lattice with size above the dislocation threshold of @xmath31 . we note however , that small objects can still be destroyed by the smearing process . pair annihilation can also remove them from the lattice . we wish to find a smearing level such that the configurations are dominated by topological objects as similar to continuum instantons as possible . at the same time , one needs to be wary of distorting their topological structure by enlarging or pair annihilating these objects . we will quantify the effects of smearing in order to choose an optimal balance between these two effects .    the work in ref . @xcite suggests that after just 20 sweeps of over - improved stout link smearing , topological objects found on the lattice closely approximate instantons . here we adopt a similar approach to that taken in ref . @xcite , searching the lattice for sites which are local maxima of the action in their surrounding hypercube @xcite . these are then taken as the approximate centre of an ( anti-)instanton , around which we fit the classical instanton action density , @xmath32 where @xmath33 , @xmath1 and @xmath3 are fit parameters , noting that @xmath3 is not restricted to lattice sites . the parameter @xmath33 is introduced as lattice topological objects are expected to have a higher action than classical instantons . we wish to determine the fit by using the shape of the action density around a local maximum , rather than the height . we can then compare data obtained for the radii , @xmath1 , of instanton candidates from this to the relationship between the radius and topological charge at the centre of an ( anti-)instanton , @xmath34 where @xmath35 for an ( anti-)instanton . @xmath36 at the fitted values of @xmath3 are found using linear interpolation from neighbouring hypercubes to find an extremum inside the hypercube containing @xmath3 . this data will provide the basis for our investigation of the effects of smearing . simulations are performed on @xmath37 dynamical flic @xmath38 configurations , with a lattice spacing of @xmath39 fm , corresponding to a spatial extent of @xmath40 fm . periodic boundary conditions are used . up to 300 sweeps of smearing are investigated . to calculate the action and topological charge densities used in our fits we use the @xmath41 five - loop improved action and charge densities defined in ref . @xcite    it should be noted that at low levels of smearing we expect to fit a large number of false positives ; local maxima of the action corresponding to noise . the degree to which fitted results concur with eq . ( [ topcharge ] ) will thus be a key first test . this correspondence is graphed for a single configuration in fig . [ fig : rhovsq ] for various smearing levels . the number of instanton candidates starts out large and distributed fairly evenly around sizes of 2 - 8 lattice units , with little correlation to the predicted charge lines of eq . ( [ topcharge ] ) . this quickly changes as the number of smearing sweeps increases , eventually leading to a very close fit . the number of instanton candidates also drops off rapidly at first , then steadily decreases . by the 50 sweep mark we can be confident that almost all local maxima found closely approximate an instanton near the centre . notably , instanton candidates with low radii have systematically lower topological charge at the centre than predicted by eq . [ topcharge ] . this is due to their proximity to the dislocation threshold of @xmath42 , under which objects are shrunk . we define the squared distance from the theoretical relationship of eq . ( [ topcharge ] ) , @xmath43 , as the minimum distance of each point from the theoretical relationship , i.e. for an instanton candidate with radius @xmath44 and topological charge at the centre @xmath45 , @xmath46\mathrm{.}\ ] ] the average value of this is plotted for data from 10 configurations in fig .  [ fig : dist ] , as a function of the number of sweeps of overimproved smearing , @xmath47 . this confirms our earlier observations ; before 50 sweeps , @xmath43 decreases very rapidly as noise is removed from the lattice . after 50 sweeps , the decrease is characteristically slower . we have seen in eq . ( [ eq : slsmac ] ) that smearing carries the risk of distorting the vacuum structure by enlarging topological objects . this concern is confirmed in fig . [ fig : radius ] , illustrating the average radius of instanton candidates as a function of smearing . after an initial rapid drop , attributable to false positives being rapidly removed by the smearing algorithm , there is a small but steady increase in the average radius of the instanton candidates found .        in fig . [ fig : number ] , we plot the average density of instanton candidates , @xmath48 , where @xmath49 is the number of instanton candidates found on a configuration and @xmath50 the lattice volume . this shows a rapid decrease until around 70 sweeps in the early stages of smearing as false positives are rapidly eliminated . eventually , instanton / anti - instanton annihilation becomes the dominant factor , slowing as the population becomes sparse . again , the regime of 50 sweeps characterises this transition . also of note is the density of @xmath51 at this point . in the instanton liquid model , phenomenological constraints set the instanton radius to be around @xmath52 and density to @xmath53 , leading to a packing fraction , the proportion of the vacuum composed of instantons , of @xmath54 . here , @xmath55 is the 4-volume of an instanton of radius @xmath1 . on the lattice , we have found a density of between @xmath56 and @xmath57 , reaching an equivalent density to the instanton liquid model at approximately 160 sweeps of smearing . in the second half of this study , we use configurations with less than 100 sweeps of smearing , leading to a density much higher than used in the instanton liquid model .    , of topological charge density at the centre of ( anti-)instantons is explored as a function of smearing sweeps , @xmath58 . plotted are the fraction of lattice sites within @xmath59 ( upper ) and @xmath60 ( lower ) of the fitted radius of an instanton candidates centre with topological charge of the same sign.,title="fig : " ] , of topological charge density at the centre of ( anti-)instantons is explored as a function of smearing sweeps , @xmath58 . plotted are the fraction of lattice sites within @xmath59 ( upper ) and @xmath60 ( lower ) of the fitted radius of an instanton candidates centre with topological charge of the same sign.,title="fig : " ]    in order to investigate this further , we have plotted the topological charge coherence , the proportion of lattice sites with topological charge of the same sign as at the centre , within @xmath59 , @xmath61 , and @xmath60 , @xmath62 , of the fitted instanton radius , in fig . [ fig : cohere ] . at @xmath59 of the instanton radius , almost all sites are charge coherent after a relatively small number of smearing sweeps , whereas at @xmath60 of instanton radius , only @xmath63 of the sites are charge coherent , even after a large amount of smearing . this suggests that our fitted values of @xmath1 may be over - estimating the true radius of instantons . we thus calculate the packing fraction using the topological charge found at the centre and eq . ( [ topcharge ] ) . this is plotted in fig . [ fig : packing ] . we note that a packing fraction exceeding 1 is not an issue , as instanton candidates may overlap , particularly at low levels of smearing . we find a packing fraction of between @xmath64 and @xmath65 . once again , this is notably higher than the instanton liquid model value of @xmath66 , suggesting that we have found a far `` fuller '' vacuum on the lattice . aggregating these measures , it is clear that between 50 and 100 sweeps of smearing gives an optimal balance , whereby the gauge field is dominated by instantons without excessive distortion . we will thus calculate quark propagators at 0 , 30 , 50 , 80 and 100 sweeps of smearing . the overlap fermion operator @xcite is defined in the massless case by @xmath67 with @xmath68 the matrix sign function , and the overlap kernel , @xmath69 , any reasonable hermitian dirac operator with mass parameter @xmath70 , governing the resolution of toplogically non - trivial field structures @xcite . the overlap is an explicit solution of the ginsparg - wilson relation @xcite , @xmath71 and will thus have a lattice - deformed version of chiral symmetry . this is sufficient to prevent additive quark mass renormalisation @xcite , greatly simplifying propagator analysis @xcite . the massive overlap operator is then given by @xmath72 where the overlap mass parameter @xmath73 is defined to represent a bare quark mass @xmath74    we use the flic action @xcite as the overlap kernel , as studies have shown it to have superior spectral properties , accelerating calculation of the overlap operator , and redcued lattice discretisation errors @xcite . we project low modes of the kernel and calculate their contribution to the propagator explicitly , greatly reducing the condition number of the matrix square root . notably , we have found results consistent with the work of neuberger @xcite , who predicts that in the presence of a sufficiently smooth background field , the eigenvalues of the wilson kernel are maximally displaced from @xmath75 at @xmath76 . in fig . [ eigenflow ] , we have plotted the 50 lowest lying eigenvalues of the flic kernel on a configuration with 50 sweeps of smearing and we observe that the region where the spectrum becomes dense(i.e . ignoring the isolated low - lying topological nodes ) is indeed maximally separated from 0 near @xmath76 . we choose to perform all calculations at @xmath76 . we consider nine values of @xmath73 , given in table [ mutable ] . .the nine values of @xmath73 considered , with the corresponding bare mass in physical units [ cols="^,^",options="header " , ]     in order to define an overlap quark propagator , we naturally wish to preserve the most important properties of the continuum propagator . as a simple consequence of chiral symmetry , in the massless case the continuum quark propagator obeys @xmath77 we also note that eq . ( [ gwrel ] ) implies @xmath78 suggesting an appropriate form for the overlap propagator , @xmath79 it is also important to maintain the correct continuum limit , and so given @xmath80 it is then natural to define the massless bare overlap propagator as @xcite @xmath81 we can then construct the bare massive overlap propagator simply by adding a bare mass to the inverse of eq . ( [ eq : baremassoverlap ] ) , @xmath82 then recalling the definition of the overlap mass parameter in eq . ( [ eq : mudef ] ) , we have @xmath83 due to the lack of additive mass renormalisation , the quark propagator on the lattice will have as its general form @xmath84 with @xmath85 the non - perturbative mass function and @xmath86 containing all renormalisation information . this can be defined as having the inverse @xmath87 where @xmath88 is defined@xcite by the tree level propagator , calculated with @xmath89 , @xmath90 comparing eq . ( [ propingenform ] ) to eq . ( [ propgenform ] ) , we see @xmath91 multiplying eq . ( [ propingenform ] ) by @xmath88 and taking the trace provides @xmath92 to determine @xmath93 and @xmath94 we follow ref . @xcite and define @xmath95 and @xmath96 by @xmath97 such that @xmath98,\end{aligned}\ ] ] where @xmath99 and @xmath100 are the extent of spin and colour indices .    comparing to eq . ( [ propingenform ] ) , we find @xmath101 defining @xmath102 we thus have @xmath103 we fix to landau gauge by maximizing the @xmath104 improved gauge fixing functional@xcite @xmath105 using a fourier transform accelerated algorithm@xcite . to avoid gribov copy issues , we first gauge fix configurations with 100 sweeps of smearing and then use these as a preconditioner for the same configurations with lower levels of smearing@xcite . the matrix sign function is calculated using the zolotarev rational polynomial approximation @xcite . we average data over spatial symmetries and choose @xmath106 as the renormalization point for @xmath107 , with @xmath108 . we apply a cylinder cut @xcite to data with a radius of @xmath109 , and estimate errors using a second - order single - elimination jack - knife method . @xmath107 is plotted in fig . [ z1271 ] for the lightest quark mass , @xmath110 , for various levels of smearing . this reveals the well - known shape of the renormalization function for the unsmeared case ; dipping in the infrared region and rising in the ultraviolet limit . after just 30 sweeps of smearing however , different behaviour is seen . the renormalisation function is more tree like , dipping half the amount and approximating @xmath111 for @xmath112 . the renormalisation function rises earlier to this plateau value , with similar behaviour seen at all levels of smearing . this appears to confirm our earlier observations ; after 30 sweeps most short - range behaviour is removed , and the gauge field structure of the lattice is much simpler . the physics responsible for the drop in @xmath107 is removed early as all curves agree .    to further show this , @xmath107 is plotted as a function of @xmath113 at given values of @xmath114 in fig . [ z1271indq ] . we have chosen to plot against @xmath113 as this is proportional to the smearing radius . this shows a large change from 0 to 30 sweeps of smearing , affirming that the physics responsible for the drop in @xmath107 is removed early in the smearing process , after which smearing has relatively little effect . higher masses are illustrated for all @xmath114 in fig . [ fig : zhigh ] . remarkably little sensitivity to the quark mass is observed . we note that the lowest two @xmath114 values are purely timelike , and thus have comparatively large error bars due to the lack of symmetries for averaging .    [ h ! ! ]    we now plot the mass function for a quark mass of @xmath110 in fig . [ m1271thirty ] . the unsmeared data reveals the expected shape ; a large effective quark mass in the infrared region created through dynamical mass generation , tapering to an approximate plateau in the ultraviolet region , where logarithmic corrections produce a running quark mass higher than the input bare mass , illustrated by the dot - dash line . looking at the lowest level of smearing considered , 30 sweeps , we note that the smeared results consistently have smaller error bars , particularly in the ultra - violet region , indicating short distance physics is a significant source of noise . although the instanton content of each ensemble is different , its impact on the mass function remains similar . we also note that , as expected , in the ultraviolet region the running quark mass is now barely higher than the input bare mass , due to the spoiling of short distance perturbative physics at this range . at the lowest momenta , we find perfect agreement with the unsmeared case , but as we increase momentum there is some suppression of dynamically generated mass . we attribute this to the destruction of some topological objects by the smearing algorithm , both those smaller than the dislocation threshold and those eliminated by pair annihilation . smearing also has the effect of enlarging objects , again removing dynamically generated mass from this region . indeed one anticipates a significant smearing dependence in the regime of @xmath115 . we now plot all levels of smearing , in fig .  [ m1271hundred ] . even at high levels of smearing , we have maintained the qualitative shape of the mass function , although increasing smearing results in a larger loss of dynamically generated mass . as the level of smearing increases , the mass function becomes flat for lower values of momenta . this is in accord with the increase of the smearing radius , the distance within which physics is suppressed . the most dramatic shift is between the unsmeared case and 30 sweeps . this suggests that , to at least some extent , smearing has changed the mass renormalization , and so in order to acquire results with the same physical mass we must shift our value of @xmath73 for the smeared results .    in fig . [ mshifted ] we have plotted smeared results at a value of @xmath116 , chosen in order to match unsmeared results in the ultraviolet limit . this reveals that not all loss of dynamical mass generation is due to a changing mass renormalization . although we have retained the majority of long - range physics , it is clear that smearing , particularly after a large number of sweeps , has removed some important aspects of vacuum structure , creating a gap between unsmeared and smeared results . @xmath85 is plotted as a function of @xmath113 at fixed values of @xmath114 in fig . [ m1694indq ] , for a quark mass of @xmath117 . this shows an approximately linear dependence on @xmath113 and thus the smearing radius . this suggests that the increase in smearing radius is responsible for loss of dynamical mass generation by removing topological objects from the lattice . at higher momenta corresponding to smaller distances , smearing has removed all relevant physics from the lattice early , and so an increase in the smearing radius has a smaller effect . higher masses are shown in fig . [ fig : mhigh ] for all @xmath114 , using the same bare mass for both smeared and unsmeared configurations . these show broadly similar results at all masses , though the gap between levels of smearing becomes clearer for larger masses , which should be more sensitive to the disruption of short distance physics . this reinforces the necessity of maintaining a high density of topological objects in the qcd vacuum and reinforcing the danger of destroying topological objects with excess smearing . the largest gap , however , remains between no smearing and 30 sweeps . we have used over - improved stout - link smearing , to reveal an underlying gauge field structure resembling an instanton liquid . after around 50 sweeps we can be confident the lattice is dominated by instanton - like objects . our calculations of the non - perturbative mass function on smeared configurations reveals that it retains its qualitative shape at even high levels of smearing . there is some loss of dynamical mass generation , increasing with smearing , which can be attributed to a thinning of the vacuum through the destruction of instanton / anti - 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we quantitatively examine the extent to which instanton degress of freedom , contained within standard monte - carlo generated gauge - field configurations , can maintain the characteristic features of the mass and renormalisation functions of the non - perturbative quark propagator . 
 we use over - improved stout - link smearing to isolate instanton effects on the lattice . using a variety of measures , 
 we illustrate how gauge fields consisting almost solely of instanton - like objects are produced after only 50 sweeps of smearing . 
 we find a full vacuum , with a packing fraction more than three times larger than phenomenological models predict . 
 we calculate the overlap quark propagator on these smeared configurations , and find that even at high levels of smearing the majority of the characteristic features of the propagator are reproduced . 
 we thus conclude that instantons contained within standard monte - carlo generated gauge - field configurations are the degrees of freedom responsible for the dynamical generation of mass observed in lattice qcd .