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in order to test the theory of strong interactions ( qcd ) it has long been common practice to confront experimental data with perturbative predictions and then extract fundamental parameters such as the strong coupling @xmath2 . a consistent extraction of @xmath2 from different experiments and different observables points at the correctness of assuming qcd as the underlying theory . since qcd is a theory where non - perturbative effects play a clear role one has to first question the feasibilty of such an approach and review the role of perturbative qcd ( pqcd ) . the fact that one can make perturbative predictions to any extent is indeed a blessing and in the case of many observables factorisation theorems allow the separation of pqcd from non - perturbative effects . simply stated a factorisation theorem implies for an observable @xmath3 @xmath4 in the above we indicate by @xmath5 any set of phase space variables on which the observable may depend and by @xmath6 the hard momentum scale . @xmath7 is typically a universal operator matrix element which contains non - perturbative information and whose value can be extracted from experiment and tabulated , for example a parton distribution measure . @xmath8 and @xmath9 are perturbative coefficients calculated by using feynman graphs to represent the hard process . the dots indicate the limitations of the perturbative approach . the dots in the brackets tell us that there are missing terms in the perturbative series since any perturbative predictions ( even resummed ones ) are missing higher order pieces which can not be easily computed . while this is just a technological limitation in the sense that higher order calculations become rapidly prohibitive computationally , it is a common misconception that given adequate computing power one could have a well defined perturbative answer that would be arbitrarily accurate .    that this belief is ill - founded can most easily be seen by examining a particular type of contribution ( renormalon graphs ) ( see for examples and further references @xcite ) which give that for large enough @xmath10 the coefficients @xmath11 diverge factorially . this does not mean that the concept of a perturbative expansion is useless . what it implies is that pqcd gives an asymptotic approximation to the observable @xmath3 . the error made in truncating the perturbative expansion is of the order of the first omitted term . therefore one should carry out the perturbative expansion till the smallest term is reached . the optimal error which is of the order of this smallest term is thus an unvoidable inherent ambiguity in the perturbative expansion and one finds @xmath12 where @xmath13 and @xmath14 depend on the observable and @xmath15 represents the uncertainty of the perturbative estimate . the coefficient @xmath13 may contain dependence on one or more phase space variables on which @xmath3 depends .    also in eq . ( 1 ) there are dots to indicate that the simple factorisation may not be correct at some level , e.g higher twist . these contributions are also typically power behaved and the expectation is that in order to obtain an unambigous answer the ambiguity of the perturbative ( pt ) part is cancelled by an ambiguity in the definition of the non - perturbative ( np ) higher twist operators . assuming further that the size of the ambiguity is proportional to value of np terms ( condensates ) one can use the ambiguity in the pt piece to model np effects which are of a power law type . in other words where there is a large power behaved ambiguity in the pt series it can be treated as a signal that np effects are damaging in proportion . this assumption is known as ultraviolet dominance of higher twist . having arrived at the notion that the ambiguity of the perturbative expansion is relevant to the study of power corrections , one is still faced with the task of computing it and comapring with experimental data . this is conveniently done in the framework of the dispersive approach we shall next discuss . this method was formulated by dokshitzer , marchesini and webber @xcite and relies on the assumption that the power correction to an observable is given by evaluating the perturbative renormalon graphs and estimating the ensuing ambiguity of the pt series . to investigate this method one observes that inserting renormalon bubbles on gluon lines in a lowest order feynman graph for a given observable @xmath3 results in the appearance of a running coupling constant . if this gluon is an internal line inserting renormalon bubbles gives for an observable @xmath3 @xmath16 with @xmath17 representing the gluon virtuality . the role of the renormalon bubbles is simply to give rise to the running coupling in the integrand above . strictly within a perturbative context the above integral gives a divergent result due to the divergence of @xmath2 at the landau pole . this is just the factorial renormalon behaviour discussed previously .    to generate power corrections from the above contribution the assumption one makes is that the qcd coupling @xmath2 instead of diverging is a well behaved universal function down to small scales , i.e infrared finite . further knowledge of the exact behaviour of the coupling is not required for us to proceed . this is certainly not an assumption that can be justified from first principles . however it leads to simple results that can be easily tested against a wealth of experimental data . this exercise has been carried out for different observables and so far it seems that experimental data are in support of this notion of a universal ir finite coupling . assuming that the coupling is analytic except a branch cut across the negative real axis which is a requirement of causality , it follows that one can write the formal dispersion relation @xmath18 where the spectral function @xmath19 represents the discontinuity across the cut , @xmath20    non - perturbative effects at long distances are expected to give rise to a modification in the strong coupling at low scales , @xmath21 , which generates a corresponding modification in the spectral function : @xmath22    from eqs . ( [ master][deltas ] ) it follows that the non - perturbative contribution to @xmath3 , trigerred by a virtual gluon with renormalon insertions and an np modified coupling can be written as @xmath23\ ] ] a similar equation holds for the real emission contribution and hence in what follws we shall remove the superscript and talk about the sum of real and virtual pieces .    the quantity @xmath24 represents the @xmath25 virtual calculation performed with a finite gluon mass @xmath26 . the gluon mass is just the dispersive variable in eq . [ alphas ] and does not imply that one works with a massive gluon field and spoils gauge invariance . after some further manipulation one gets the following non - perturbative contribution to @xmath3 @xmath27 where , setting @xmath28 , @xmath29 since @xmath30 is limited to low values of @xmath31 , the asymptotic behaviour of @xmath32 at large @xmath6 is controlled by the behaviour of @xmath33 as @xmath34 . we see from eq . [ gdef ] that no terms analytic at @xmath35 can contribute to @xmath32 . on the other hand for a square - root behaviour at small @xmath36 , @xmath37 where we defined the coupling moment @xmath38 we can express @xmath39 in terms of the average value of @xmath40 in the infrared region , as follows . we substitute for @xmath41 in eq . ( m^2 ) ( m^2 ) - ( m^2 ) , where @xmath42 represents the expression for @xmath40 corresponding to the part already included in the perturbative prediction . if the perturbative calculation is carried out to second order in the  renormalization scheme , with renormalization scale @xmath43 , then we have ( m^2 ) = ( ^2 ) + [ b(^2/m^2)+k]^2(^2 ) where for @xmath44 active flavours ( @xmath45 ) b = , k= . the constant @xmath46 comes from a change of scheme from  to the more physical scheme @xcite in which @xmath40 is preferably defined at low scales . then above some infrared matching scale @xmath47 we assume that @xmath48 and @xmath49 approximately coincide , so that [ a1exp ] _ 1 & & _ 0^ dm((m^2)- ( ^2 ) - [ b(^2/m^2)+k]^2(^2 ) ) + & = & ( 0()- ( ^2 ) - [ b(^2/^2)+k+2b]^2(^2 ) ) , where [ a0def ] 0 ( ) _ 0^(m^2)dm . studies of event shapes in @xmath50 annihilation suggest that @xmath51 for @xmath52 gev , which translates into a value of @xmath53 gev for @xmath54 gev . here we shall concentrate on the phenomenology of power corrections to event shape variables measured by the h1 and zeus collaborations . this study is carried out in the breit frame and the current hemisphere @xmath55 is used to define the variables , freeing us from contamination by the proton remnant jet . some of the observables studied are defined below    * current jet thrust this is constructed by summing the modulii of hadron components along the photon axis in the breit frame . two normalisations are possible , to @xmath56 on one hand and to the energy in the current hemisphere ( provided this exceeds a certain preset minimum ) . @xmath57 @xmath58 * current hemisphere jet mass [ rhodef ] = ( _ ah_cp_a)^2/q^2 . * @xmath59-parameter the @xmath59-parameter is [ cdef ] c=6_a , bh_c|p_a||p_b|^2_ab / q^2 .    there are still other possible definitions and variables studied in addition to the ones mentioned above ( an important one being the jet broadening ) but for illustrative purposes it is enough to restrict the discussion to the variables mentioned . the power corrections for different event shape variables are given by following the dispersive recepie . one first computes the one - loop graphs with a finite gluon mass @xmath60 for all the variables mentioned above the leading non - analytic behaviour at small @xmath61 is a @xmath62 singularity . this leads to @xmath63 power corrections of the form of eq . [ deltaf1 ]    the coefficients @xmath64 differ from variable to variable and one gets @xmath65 the constant @xmath39 is expected to be approximately universal ( to about @xmath66)as observed in @xmath67 data and with similar values to those found there . one point that needs to be made in the dis context is that the leading power corrections to most event shapes are independent of the bjorken variable @xmath5 . this comes about since the @xmath63 effects in question are trigered by a soft gluon , the emission of which does not change the momentum fraction carried by the incoming projectile . an exception to this is the jet broadening variable where an interplay between this emitted soft gluon and @xmath5 dependent perturbative radiation gives rise to an @xmath5 dependent power correction . a complete estimate for the broadening power correction relies on a resummed perturbative calculation which shall be published shortly @xcite . another point that needed to be addressed was one first raised by nason and seymour @xcite . they pointed out that gluon decay can only be mimimcked by a massive gluon provided one can integrate inclusively over the gluon offspring . in other words the machinary of the dispersive approach should only strictly be applied to inclusive observables which do not register offspring gluons . event shapes are not such variables as the offspring gluon momenta affect the event shape . hence one needs an improved approach where one explicitly treats gluon decay properly . the two - loop calculations that take gluon decay into proper account were first performed for @xmath50 observables @xcite following this similar work was performed for dis observables @xmath68 . the conclusions could at first sight be described as surprising . taking two loop effects into account meant a simple rescaling of the old one - loop coefficients by a universal factor @xmath69 for @xmath70 . hence the ratios of power corrections to different observables are still given by the ratios of the one - loop coefficents @xmath71 . this surprising result came about as a result of    * universality of soft gluon radiation . * common geometrical properties of event shapes they are all linear in the emitted transverse momenta . [ hum ]    [ mbell ]    another area of investigation at hera is the study of event shape differential distributions @xmath72 and power corrections . here while data has been available for a while there was a lack of theoretical estimates . although there are monte carlo programs to make fixed order estimates at @xmath73 this is not sufficient in the above case . the presence of large logarithms as big as @xmath74 at @xmath75 perturbative order means that a fixed order expansion is not very useful in the small @xmath3 region .    as in @xmath50 one needs to resum all orders till next - to  leading logarithmic accuracy to describe the data . this resummation program has been started @xcite and results for many variables are soon to be published @xcite . however a preliminary comparison to data has already begun . the power corrections to different event shape distributions result in simple shifts of the perturbative spectra , the amount of the shift is exactly equal to the @xmath63 correction to the corresponding mean values . @xmath76 where @xmath64 for the different variables are quoted . again the jet broadening is different and does not show a simple shift of the pt spectrum giving instead a variable dependent shift as in @xmath67 . the experimental results from h1 and zeus ( see figs .  1 and 2 ) have been available for quite some time for mean values whereas fits to distributions are on the way . for the mean values the fits are best summarised by plotting error ellipses in the @xmath77 plane where @xmath78 is the parameter introduced in eq . [ a1exp ] . some brief comments about the results are in order . in the h1 results one should note that the jet mass ellipse is shifted and falls in line with expectations from other variables provided one neglects hadron masses ( the relevant ellipse is labelled @xmath79 ) in keeping with the theoretical approximations made . for a fuller discussion of this point see @xcite . in the zeus plot one notices the effect of large systematic errors . zeus have also studied the @xmath5 dependence of power corrections and as noted before in this article the power correction to the jet broadening variable displays a clear @xmath5 dependence which comes from the anomalous dimension matrix @xcite . in addition one remarks that that the values of the @xmath80 parameter obtained from dis event shapes has similar values ( 0.40.5 ) as for @xmath67 variables and this seems to support the hypothesis of a universal infrared finite coupling . the spread in @xmath2 values however is still uncomfortably large and is a potential area for improvement .    finally turning to differential distributions one can use a nominal value of @xmath81 and @xmath82 and the resummation of @xcite . the curves in fig . 3 are produced by binning like the h1 data which is responsible for their non - smooth nature . preliminary indications are positive as shown by fig .  3 but detailed fits are awaited . 1 beneke m 1999 _ phys . rept  _ * 317 * p  1 . dokshitzer yu l , marchesini g and webber b r 1996 _ nucl . * b 469 * p  93 . catani s , marchesini g and webber br 1991 _ nucl . * b 349 * p  635 . dasgupta m and salam g p in preparation . nason p and seymour m h 1995 _ nucl . phys .  _ * b 454 * p  291 . dokshitzer yu l , lucenti a , marchesini g and salam gp 1998 _ jhep  _ * 9805 * p  001 . dasgupta m and webber b r 1998 _ jhep  _ * 9810 * p  003 . martyn h u 2000 talk given at _ 30th international conference on high - energy physics ( ichep 2000 ) _ , osaka , japan . mccance g j 2000 talk given at _ 35th rencontres de moriond : qcd and high energy hadronic interactions _ , les arcs , savoie , france . antonelli v , dasgupta m and salam g p 2000 _ jhep  _ * 0002 * p  001 . salam g p and wicke d 2001 , _ jhep  _ * 0105 * p  061 .
this article provides a review of progress made in understanding inverse power law corrections @xmath0 where @xmath1 is the hard scale involved in a given qcd process . 
 particular attention will be paid to hera results .