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a common challenge for quantum field theorists consists in finding accurate methods in regimes where existing expansions break down . in the rg language , this amounts to find acceptable interpolations for the rg flows in intermediate regions between fixed points . in the case of scalar field theory , the weak coupling expansion is unable to reproduce the exponential suppression of the large field configurations operating at strong coupling .
this problem can be cured by introducing a large field cutoff @xmath2 which eliminates dyson s instability .
one is then considering a slightly different problem , however a judicious choice of @xmath2 can be used to reduce or eliminate the discrepancy with the original problem ( i. e. , the problem with no field cutoff ) .
this optimization procedure can be approximately performed using the strong coupling expansion and naturally bridges the gap between the weak and strong coupling expansions .
the workshop on qcd in extreme environments was held right after the lattice 2004 conference .
the talk of k. wilson about the early days of lattice gauge theory was a very inspirational moment of lattice 2004 .
he stressed the importance of , in his own words , `` butchering field theory '' in the development of the rg ideas and recommended that we keep doing it . in the following
, we will be butchering field theory in the space of field configurations .
we are interested in the effects of a large field cutoff on observables ( we expect the effect to be small if the field cutoff is large enough and the observable is not a product of too many fields ) and on the coeffients of the perturbative series for the observables ( we expect the effect to be drastic for the large order behavior ) . for scalar fields ,
there are many ways to accomplish this task .
the configurations can be ranked according to the largest absolute value of the field or according to the average over the sites of even powers of the field .
the larger the power is , the more emphasis is put on the configurations with the largest field values . as one may suspect , there exists correlations among the results obtained with different cutoff procedures . for gauge theory , we can define the concept of small or large field configurations in the landau gauge and in a gauge invariant way .
this was one of the questions that we discussed at lattice 2004 and an account can be found in ref .
@xcite . instead of duplicating
, we will rather give an elementary discussion of our motivations and existing results in the scalar case and explain how we expect to extend them in the gauge case .
recent progress are briefly discussed at the end .
the best way to understand why the perturbative series of @xmath4 problems in various dimensions generically have a zero radius radius of convergence is to consider the integral @xmath5 the peak of the integrand of the r.h.s .
moves too fast when the order increases .
more precisely , @xmath6 is maximum at @xmath7 such that @xmath8 .
the truncation of @xmath9 at order @xmath10 is accurate provided that @xmath11 . a good accuracy in the region where the integrand is maximum requires @xmath12 , which implies @xmath13 .
note also that the exponential function converges uniformly over a finite interval but _ not _ over @xmath14 and consequently one can not interchange the sum and the integral .
on the other hand , if we introduce a field cutoff , the peak moves outside of the integration range and we get a converging expansion @xmath15 in general we expect that for a finite lattice , the partition function @xmath16 calculated with a field cutoff is convergent and @xmath17 has a finite radius of convergence .
the problem with the field cut differs from the original problem but the difference can be made exponentially small .
the method works well in nontrivial examples .
this has been checked @xcite for the hierarchical model and in the continuum for quantum mechanical problems ( the anharmonic oscillator and the double - well potential ) .
in this section , we describe graphical representations of the accuracy reached at different orders in perturbation theory .
we typically want to know the number of ( correct ) significant digits that can be obtained at a given order for a given coupling .
three situations are represented in fig .
[ fig : sd ] . for @xmath18
, the accuracy always increases when we increase the order . in the case of @xmath19 ,
this is the case only if @xmath20 and the lines have a focus point at @xmath21 . on the other hand , for the integral discussed in the previous section ,
the lines move left as they rotate and one sees an envolope that delimitates a `` forbidden region '' for the accuracy of perturbation theory . at fixed and not too large coupling , the accuracy first increases and then decreases with the order .
the `` rule of thumb '' consists in stopping when the accuracy is optimal , in other words , when we reached the envelope discussed above .
note also that the lines flatten near 14 on the left of the graph .
this simply reflects that we have only 14 digits of accuracy in our numerical calculation of the integral .
( upper left ) and @xmath22 ( upper right ) , and orders 1 , 2 , ... , 20 for the integral discussed in section 2 ( lower ) . as the order increases , the lines `` rotate '' clockwise.,title="fig:",width=259 ]    ( upper left ) and @xmath22 ( upper right ) , and orders 1 , 2 , ... , 20 for the integral discussed in section 2 ( lower ) . as the order increases , the lines `` rotate '' clockwise.,title="fig:",width=259 ]    ( upper left ) and @xmath22 ( upper right ) , and orders 1 , 2 , ... , 20 for the integral discussed in section 2 ( lower ) . as the order increases , the lines `` rotate '' clockwise.,title="fig:",width=307 ]    when a field cut is introduced , the series apparently become convergent and we can make a significant , but localized in coupling , incursion in the forbidden region of perturbation theory .
this is illustrated for three different field cuts in fig .
[ fig : cutanh ] in the case of the ground state of the anharmonic oscillator .    in an ideal world
, we would pick a field cut large enough to reduce the errors ( due to the field cut ) to an acceptable level .
we would then calculate enough terms in order to reach an accuracy consistent with this level . in practice ,
we are usually limited to calculations up to a certain order .
the field cut can then be chosen in order to minimize the error at that order . from fig .
[ fig : cutanh ] , one can see that it is possible to pick a cut that makes the accuracy optimal in the neighborhood of some given value of the coupling @xmath23 . when the numerical answer is known , it is easy to adjust the field cut in order to optimize the accuracy .
when the answer is not known , one can use an approximation . in ref .
@xcite , we showed that for the integral eq .
( 1 ) , the strong coupling expansion can be used to determine approximately the optimal field cut . this way strong and weak coupling expansions can be combined coherently .    up to now
, we have defined the field cut locally in configuration space ( at each lattice site ) .
it is however possible to proceed differently and to use the average of even powers of the fields to sort the configurations .
there exist correlations among these indicators @xcite .
we now report our attempts to extend the modification of perturbation theory discussed above in the scalar case to lgt .
perturbation theory for lgt has been developed almost 20 years ago @xcite .
exact calculations up to 3 loops @xcite and numerical calculations for 8 @xcite and 10 loops @xcite are available .
it proceeds in 3 steps .    1 .   with the convention @xmath24 , we set @xmath25 at every link .
we extend the range of integration of @xmath26 to @xmath27 ( anything else would be unpratical ! ) 3 .
we then expand in powers of @xmath28    in step 2 , we added the integration `` tails '' .
this presumably makes the series nonconvergent ( asymptotic ) as one can observe in the case of large argument expansions of bessel functions . in step 3 ,
one needs to expand the haar measure in power of @xmath28 .
as the original haar measure is compact and provides a natural field cut , we would like to see what happens when the integral get decompactified in step 2 .
for this purpose , we consider the simple example of a one link @xmath29 integral .
we use the parametrization of @xmath29 @xmath30 with @xmath31 covering the 2-sphere and @xmath32 . with this parametrization
we cover the @xmath29 manifold exactly once .
alternatively , we could extend the range to @xmath33 , but then we need to identify opposite points on the sphere if we want to avoid a double coverage . in these coordinates ,
the invariant haar measure reads @xmath34 after we switched to the @xmath35 measure @xmath36 , we obtain the haar correction @xmath37}$ ] .
note that @xmath38 $ ] has a radius of convergence @xmath39 , but all the coefficients are negative .
so when we expand _ in the exponential _ , the large negative contributions in the region @xmath40 effectively cutoff these contributions .
this is illustrated in fig .
[ fig : haar1 ] .
however , in perturbation theory we have @xmath41 and we need to expand the exponential in powers of @xmath28 .
the measure then `` periodicizes '' and we obtain a multiple coverage of the manifold as shown in fig .
[ fig : haar2 ] .
the logarithm of the haar measure is also used in the context of gluon equations of state @xcite .    .
the ground state of the double - well potential @xmath42 can be expanded in powers of @xmath43 . except for the zeroth order contribution ,
all the coeffients of the series are negative and their magnitude grow factorially with the order .
the borel transform has poles on the positive real axis .
the difference between the beginning of the perturbative series and the numerical values is bounded by the instanton effect @xmath44 qualitatively similar features are expected for the perturbative expansion of @xmath45 in pure gauge @xmath46 defined as @xmath47 with @xmath48 the comparison between the numerical values and successive orders are shown in fig .
[ fig : num ] .     for the double well potential ( left ) .
the blue / green lines represent the first orders of perturbation theory .
as the order increases , the curve gets more green .
the dots represent the numerical values . a similar graph ( except for the colors ) for @xmath45 in @xmath0 is shown on the right.,title="fig:",width=278]0.5 in   for the double well potential ( left ) .
the blue / green lines represent the first orders of perturbation theory .
as the order increases , the curve gets more green .
the dots represent the numerical values . a similar graph ( except for the colors ) for @xmath45 in @xmath0 is shown on the right.,title="fig:",width=268 ]    the accuracy of successive orders in perturbation theory
are shown in fig .
[ fig : acc ] .
note that unlike the scalar case , the weak coupling ( large @xmath49 ) is now displayed on the right of the figure .    .
for the double - well ( left ) , the significant digits are bounded by the instanton effect @xmath50 ( red curve).,title="fig:",width=288]0.4 in . for the double - well ( left ) , the significant digits are bounded by the instanton effect @xmath50 ( red curve).,title="fig:",width=288 ]    appropriate field cuts can restore the instanton effects in the perturbative series @xcite .
this is illustrated in fig .
[ fig : dwcut ] where the modified series allows us to go above the instanton envelope .
we expect to be able to achieve similar results for @xmath0 .
in particular , we expect to be able to use the strong coupling expansion to obtain an optimal choice of field cut , since the validity of this expansion seems to extend close to the scaling window .
[ fig : strong ] indicates that the radius of convergence of the strong coupling expansion is between 4 and 6 for @xmath49 .
this series was calculated using the expansion of the free energy of ref .
at the time of the workshop , we presented results related to the questions discussed below .
since our understanding has evolved in the meantime , we will give a brief summary and refer to recent preprints for more details .
we have attempted to follow the same procedure as for the scalar models , for gauge models using the landau gauge where @xmath51 should play a role analogous to @xmath52 in scalar models .
we found correlations between the lattice average of this quantity and the average action .
however , we found no correlations between the average and the maximum value .
these results are explained in more detail in the proceedings of lattice 2004 @xcite .
the lack of correlation is due to the imperfect way the landau gauge condidtion is implemented numerically .
this is being remedied @xcite .
we considered wilson s @xmath46 lattice gauge theory ( without fermions ) at negative values of @xmath53 and for @xmath54=2 or 3 .
we showed that in the limit @xmath55 , the path integral is dominated by configurations where links variables are set to a nontrivial element of the center on selected non intersecting lines . for @xmath56 ,
these configurations can be characterized by a unique gauge invariant set of variables , while for @xmath57 a multiplicity growing with the volume as the number of configurations of an ising model is observed . in general , there is a discontinuity in the average plaquette when @xmath43 changes its sign which prevents us from having a convergent series in @xmath43 for this quantity . for @xmath56 ,
a change of variables relates the gauge invariant observables at positive and negative values of @xmath49 . for @xmath57
, we derived an identity relating the observables at @xmath49 with those at @xmath49 rotated by @xmath58 in the complex plane and showed numerical evidence for a ising like first order phase transition near @xmath59 .
so far we see no obvious connections to the known singularities @xcite .
these results are discussed in more detail in a recent preprint @xcite . for another approach of problems at negative coupling
@xcite .
we revisited the question of the convergence of lattice perturbation theory for a pure @xmath0 lattice gauge theory in 4 dimensions . using the most recent calculation of the weak coupling expansion of the plaquette average
, we showed that the extrapolated ratio and the extrapolated slope suggest a nonanalytical power behavior at @xmath60 with an exponent @xmath61 in agreement with an existing analysis @xcite .
we found indications for a possible singularity in the third derivative of the free energy on @xmath62 and @xmath63 lattices .
as the lattice size increases , the statistical errors become large and a significantly larger number of independent configurations is needed in order to draw definite conclusions .
this will be discussed in a forthcoming preprint @xcite .
[ [ a - proposal - for - a - perfect - field - cut - in - lattice - gauge - perturbation - theory ] ] a proposal for a  perfect " field cut in lattice gauge perturbation theory ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~    we considered the effects of a field cutoff on the weak coupling series of a one plaquette @xmath29 lattice gauge theory .
it possible to pick a the ( perfect ) field cutoff in such a way that the series _ converges _ toward the _ correct _ answer .
we are considering the implementation of the method with a langevin equation and its extension for four dimensional lattice gauge theory .
this will be discussed in a forthcoming preprint @xcite .
in pure gauge @xmath0 near @xmath1 , weak and strong coupling expansions break down and the mc method seems to be the only practical alternative . 
 we discuss the possibility of using a modified version of perturbation theory which relies on a large field cutoff and has been successfully applied to the double - well potential ( y. m. , prl 88 141601 ) . 
 generically , in the case of scalar field theory , the weak coupling expansion is unable to reproduce the exponential suppression of the large field configurations . 
 this problem can be solved by introducing a large field cutoff @xmath2 . 
 the value of @xmath3 can be chosen to reduce the discrepancy with the original problem . 
 this optimization can be approximately performed using the strong coupling expansion and bridges the gap between the two expansions . 
 we report recent attempts to extend this procedure for @xmath0 gauge theory on the lattice . 
 we compare gauge invariant and gauge dependent ( in the landau gauge ) criteria to sort the configurations into `` large - field '' and `` small - field '' configurations . 
 we discuss the convergence of lattice perturbation theory and the way it can be modified in order to obtain results similar to the scalar case .