if a given feynman integral depends on kinematic invariants and masses which essentially differ in scale , a natural idea is to expand it in ratios of small and large parameters . as a result ,the integral is written as a series of simpler quantities than the original integral itself and it can be substituted by a sufficiently large number of terms of such an expansion . for limits typical of euclidean space ( for example , the off - shell large - momentum limit or the large - mass limit ) ,one can write down the corresponding asymptotic expansion in terms of a sum over certain subgraphs of a given graph  . this prescription of expansion by subgraphs has been mathematically proven ( see and appendix b.2 of ) .moreover , there is an automated tool  where such an expansion by subgraphs is implemented .  for limits typical of minkowski space ( i.e. which can not be formulated in euclidean space ) the universal strategy of expansion by regions  is available .it consists of the following prescriptions :  * divide the space of the loop momenta into various regions and , in every region , expand the integrand in a taylor series with respect to the parameters that are considered small there . *integrate the integrand , expanded in the appropriate way in every region , over the _ whole integration domain _ of the loop momenta . *set to zero any scaleless integral .  as shown in  ,this prescription can also be applied to parametric representations of feynman integrals , i.e. alpha parameters ( or generalized feynman parameters ) integrated from 0 to  , eventually restricted by a delta function .then the regions are specified by scaling relations between the parameters .there is no mathematical proof that this prescription is correct in all situations . butalso no examples are known where a proper application of the expansion by regions leads to wrong results .an indirect proof exists for limits typical of euclidean space because here the strategy of regions is equivalent to the mathematically proven expansion by subgraphs .a systematic study of the expansion by regions was presented recently in  .there it was shown explicitly and illustrated using various one - loop examples that one can start from a decomposition of a given integral into non - intersecting domains and arrive at an expansion by regions in the above sense .this requires certain conditions on the choice and completeness of the considered regions which were derived in  .as pointed out there , the appearance of additional overlap contributions can be avoided by adequate choices of the regions and regularization parameters .while these findings provide some hints on the proper choice of the regions , it remains a non - trivial task to actually reveal the typical regions for a given limit .usually , one starts from considering one - loop examples , checks the results against known analytical results , then proceeds in two loops etc .one can also use the second version of the code fiesta to obtain numerically several first terms of a given asymptotic expansion .recently an algorithm for an automatic search of regions was suggested and implemented on a computer as the open source mathematica code ` asy.m `  .the algorithm uses a geometric approach based on finding the convex hull of a set of points determined from a parametric representation of the feynman integral . in this wayall possible sets of scalings for the ( feynman ) parameters are found which lead to non - vanishing ( because non - scaleless ) integrals .these regions may then be used to expand the parametric integral , or they can be translated into regions for expanding the integral in loop - momentum space .this code works successfully for a large variety of limits , at least in cases where the function  in the corresponding parametric representation which depends on kinematic invariants and masses is only composed of terms with the same sign .moreover , it was shown in that in this case there are no regions except for the ones produced by the code ( all others result in scaleless integrals ) . in particular, the code asy.m works for sudakov - type limits which are typical of minkowski space .as it was pointed out in the code does not reveal potential regions in threshold expansions , and similarly it fails to detect the so - called glauber regions .our goal in the present paper is to provide an algorithm and the corresponding update asy2.m of the code ` asy.m ` which automatically identifies all regions relevant for a given integral , including potential and glauber regions .we start in section [ sec : preliminaries ] by introducing the parametric representations of loop integrals which we use later and by explaining how the contribution of a given region is obtained in the language of such parametric integrals .then we elaborate our algorithm for revealing potential and glauber regions and explain how the code ` asy2.m ` is applied in such cases .this is done in section [ sec : potential ] for an example with a potential region , and section [ sec : glauber ] deals with the more complicated problem of revealing glauber regions . in both sections [ sec : potential ] and[ sec : glauber ] , we first formulate simple changes of variables and decompositions of a given feynman integral , using instructive one - loop examples , which lead to integrals where asy2.m is able to detect the relevant regions and print their list in terms of the scalings of the parameters .then , for both cases , we explain how to use the new features of asy2.m to perform these algorithmic steps automatically .  in the case of section [ sec : glauber ] with glauber regions , the structure of the regions differs depending on whether the expansion is performed in loop - momentum space or at the level of the parametric integral .we show in section [ sec : glauber_l ] how to disentangle and match the various regions arising in this problem by using generic propagator powers , and how asy2.m can be employed to automate such an analysis .a summary of the new features and the syntax of ` asy2.m ` ( together with a download link ) is provided in section [ sec : syntax ] . in section [ sec : conclusion ] we conclude by discussing the mathematical problem of proving the expansion by regions for a simple example which is not related to feynman integrals , but where ` asy2.m ` works successfully .we are dealing with dimensionally regularized feynman integrals where is the number of loops , the indices are general powers of the propagators , the dimension is and the denominators are given by i.e. they are quadratic or linear functions of the external momenta and the loop momenta with the usual infinitesimal imaginary part .monomials in the numerator are taken into account as denominators raised to negative powers .the alpha representation of ( [ eqbn - d1 ] ) takes the form where .the functions and depend polynomially on the alpha parameters  . furthermore , and are homogeneous functions of the alpha parameters with the homogeneity degrees and , respectively .the function  is linear in the kinematic invariants and/or squared masses which we denote by , while the function  is independent of the .if ( [ eqbn - d1 ] ) is an integral with standard propagators associated with the lines of a graph , then the functions and are called symanzik polynomials and are given by the well - known formulae in terms of trees and -trees . for a general feynman integral of the form ( [ eqbn - d1 ] ) one can obtain these functions using the simple public code ` uf.m `` from ` uf.m `  is called with three arguments : the list of loop momenta , the list of denominators of the propagators and a list of replacement rules for all kinematic invariants .the output is a list with the following entries : the function  , the function  and the number of loops . in order to obtain and with the correct sign, denominators have to be specified with the opposite sign as in ( [ denom - d1 ] ) , i.e. corresponding to a negative imaginary part : ` uf[{k1,k2 , ... } , { -e1,-e2 , ... } , { } ] ` . ] which is also part of the codes ` asy.m ` and ` asy2.m ` .if some of the indices  are negative integers , i.e. they correspond to numerators instead of denominators of the integral ( [ eqbn - d1 ] ) , the alpha representation ( [ alpha - d ] ) is to be understood in the limit where these indices tend to their negative integer values .effectively , the integration over the corresponding parameters  is replaced by differentiating with respect to these parameters and setting them to zero .  closely related to ( [ alpha - d ] )is the ( generalized ) feynman parametric representation where in the argument of the delta function is an arbitrary non - empty subset of , and the functions and are the same as those in ( [ alpha - d ] ) with the parameters replaced by .it is well known that the formula ( [ alpha - d - mod ] ) holds for any choice of the subset  in the argument of the delta function .this feature is related to the above - mentioned homogeneity properties of the functions and . ) , ( [ delta_int_hom ] ) and footnote [ fn : delta_scaling ] ( p. ) for a general proof . ] if one chooses , the standard feynman parametrization is recovered .let us suppose that we have to study the asymptotic behaviour in a one - scale limit , i.e. every mass and kinematic invariant has a certain scaling , , expressed in powers of the small parameter of the problem , .the strategy of expansion by regions formulated in terms of parametric integrals ( [ alpha - d ] ) or ( [ alpha - d - mod ] ) states that the asymptotic expansion in such a limit is given by a sum over regions which are specified by the scalings of the parameters or expressed in powers  of the expansion parameter  .each region  is labelled by the list of its scaling powers .the contribution of the region  is obtained by scaling the masses and kinematic invariants according to the given limit as specified above , by substituting , in the integrand of ( [ alpha - d ] ) or in the integrand of ( [ alpha - d - mod ] ) and by expanding the integrand in powers of  . here the product of the differentials or provides another factor to the power counting .explicitly , the contribution of the region  is given by the prefactor in ( [ alpha - d - mod ] ) times times the integral with the integrand expanded in powers of  .this expansion also involves the argument of the delta function in ( [ alpha - d - mod - scaled ] ) , such that , under the expansion , certain parameters drop out of the argument of the delta function and are integrated from 0 to  . for this reasonthe upper integration limit of all feynman parameters should be kept at infinity and not switched to  even if , before the expansion , their integration is restricted by the delta function .one may avoid expanding the delta function by choosing the original subset  in ( [ alpha - d - mod ] ) sufficiently small .let us write down the leading - order ( lo ) contribution of a given region in a more explicit way .for the two basic functions in ( [ integrand ] ) we have where the arguments of the polynomials and on the left - hand side indicate that they are expressed in terms of the scaled parameters and , while their expansion coefficients on the right - hand side are expressed in terms of and .according to the prescription formulated above , the lo contribution of the region  is represented as in principle , the argument of the delta function in ( [ alpha - d - mod0 ] ) contains a sum over only those scaled parameters with the minimal scaling power .but by rescaling , the delta function is transformed into its standard form without powers of  , while the rest of the integral remains invariant due to the homogeneity of the polynomials and . finally , as for the original feynman parametric representation ( [ alpha - d - mod ] ), one can choose again an arbitrary non - empty subset  for the sum in the argument of the delta function in ( [ alpha - d - mod0 ] ) .the list of scalings of a region is only determined up to adding the same arbitrary real number  to each entry , because the corresponding contribution stays the same under .in particular , the lo behaviour presented in ( [ alpha - d - mod0 ] ) is independent of such a shift  , because and , due to the homogeneity properties of and  .the terms of the expansion come from various regions and can be ordered according to accompanying powers of  . after keeping some first terms of the expansion one can set andwrite down the given feynman integral as these selected first terms plus a remainder which vanishes sufficiently fast in the given limit .it turns out that only a limited number of regions contribute to the expansion because for the rest of the regions one obtains integrals without scale which are set to zero .it is the subject of this paper and the task of the code asy.m and its updated version asy2.m to find all relevant regions for a given integral .let us consider the one - loop propagator diagram with two massive lines in the threshold limit , i.e. when with being the external momentum : where the usual is implied in all the propagators . within the strategy of expansion by regions ,the hard and the potential regions give contributions to the expansion .the previous version of the code asy.m reported only about the hard region .the reason for this can be seen in the corresponding parametric representation , ^{{\varepsilon}}}\ ; ,  \label{prop1_alpha}\end{aligned}\ ] ] where the parameters  are integrated from 0 to ( restricted by the delta function ) . as it was pointed out in , it is the region where ( more precisely ) which causes problems . in other words ,the polynomial in the square brackets in ( [ prop1_alpha ] ) ( considered at positive and ) has terms of different sign , such that cancellations occur because of the presence of the negative term .  to reveal the missing potential contribution ,let us perform a simple trick .we decompose the integration domain into two subdomains , and .the two resulting integrals are equal to each other , but such an equality will not generally take place for any integral . in the first domain we turn to new variables by ,remove the primes at and obtain the integral ( again from 0 to with the usual restrictions via the delta function ) ^{{\varepsilon}}}\;.  \label{prop1_alpha_1}\ ] ] the goal of this trick was to make the line ( in the old variables ) the border of an integration domain which turned out to be ( in the new variables ) .now we can run the code asy2.m . since this is a parametrical integral rather than a feynman integral we use the newly introduced command wilsonexpand [ ] for integrals where all parameters are integrated from 0 to :  .... wilsonexpand[q^2/4*x2 ^ 2 + y*(x1 + x2)^2 ,  x1 + x2 , { x1 , x2 } , { q - > 1 , y - > x } ,  delta - > true ] ....  the first two arguments of ` wilsonexpand [ ] ` are the polynomials and , respectively , as defined in section [ sec : preliminaries ] .they can easily be determined from the square brackets in the denominator of the parametric integral ( [ prop1_alpha_1 ] ) and from the round brackets in the numerator .the third argument is the list of integration parameters , and the fourth argument specifies the scaling of the kinematic quantities with respect to the small parameter which is labelled by the global symbol ` x ` . here by ` y - > x ` we tell the code that is the small expansion parameter , and by ` q - > 1 ` we specify that the momentum  scales as .the option ` delta - > true ` tells ` wilsonexpand [ ] ` that , under the integration , the sum over an arbitrary non - empty subset of the integration parameters is restricted to 1 by a delta function .note that ` wilsonexpand [ ] ` can only take into account such a delta function if the specific choice of the sum over parameters in the argument of the delta function is irrelevant .this is the case for the generalized feynman parametric integral ( [ alpha - d - mod ] ) introduced in section [ sec : preliminaries ] .the integrals ( [ prop1_alpha ] ) and ( [ prop1_alpha_1 ] ) are special cases of ( [ alpha - d - mod ] ) such that we could e.g. replace by without changing the value of the integrals .if , however , a specific form of the delta function is assumed , e.g. by replacing under the integral , then the option ` delta ` of ` wilsonexpand [ ] ` does not apply ( see section [ sec : syntax ] for details ) . alternatively , ` wilsonexpand [ ] ` can be used without the option ` delta ` after eliminating one of the integrations , e.g. via .the output of the above - mentioned call of ` wilsonexpand [ ] ` is a list of regions specified by the scaling of the parameters  in powers of the small parameter  :  .... { { 0 , 0 } , { 0 , 1/2 } } ....  this specification of a region corresponds to the list of scalings introduced in section [ sec : preliminaries ] .the first entry in the output , , refers to the hard region with the scaling , which has already been found by ` asy.m ` .but now also the region is found with the scaling , which provides the potential contribution .the contribution of the hard region starts with the order .every term of the expansion can be evaluated in terms of gamma functions for general  . according to the prescriptions for writing down the contribution of a region formulated in section [ sec : preliminaries ] , the contribution of the -th order expansion of ( [ prop1_alpha_1 ] ) in the potential region reads only the leading order ( ) , yields a non - vanishing contribution which is evaluated in terms of gamma functions at general . taking into account that we have two identical integrals after our decomposition , we arrive at the following result for the potential contribution which is of order : in agreement with  .  in fact , such a trick of making manifest squares of some linear combination of the integration parameters was already used in the code fiesta in order to evaluate numerically feynman integrals at a threshold . using the implementation of this procedure in fiesta it turned out to be possible to automate the above trick for a general feynman integral . in the present version asy2.m, the user may call the command ` alpharepexpand [ ] ` with the additional option ` preresolve ` enabled which automatically looks for the change of variables described above :  .... alpharepexpand[{k } ,  { k^2 - m^2 , ( k - q)^2 - m^2 } ,  { q^2 - > qq , m^2 - > qq/4 + y } ,  { qq - > 1 , y - > x } ,  preresolve - > true ] ....  as in the previous version of ` asy.m ` , the arguments of ` alpharepexpand [ ] ` are the list of loop momenta , the list of denominators of the loop integral , a list of replacements for the kinematic quantities , and the list of scalings with respect to the small parameter ` x ` .the output is a list of entries which indicate for each region the changes of variables , the jacobian of the integral transformation , and the scalings of the new variables :  .... { { { x[1 ] - > y[1 ] + y[2]/2 , x[2 ] - > y[2]/2 } ,   2 , { 0 , -1/2 } } ,  { { x[1 ] - > y[1 ] + y[2]/2 , x[2 ] - > y[2]/2 } ,   2 , { 0 , 0 } } ,  { { x[1 ] - > y[1]/2 , x[2 ] - > y[1]/2 + y[2 ] } ,   2 , { 0 , 0 } } ,  { { x[1 ] - > y[1]/2 , x[2 ] - > y[1]/2 + y[2 ] } ,   2 , { 0 , 1/2 } } } ....  the original feynman parameters are labelled ` x[1 ] ` , ` x[2 ] ` , , while the new parameters are labelled ` y[1 ] ` , ` y[2 ] ` , .the scaling relations which determine the regions are specified for the new parameters .  as explained in section [ sec : preliminaries ] , a region remains invariant if all of its scalings are shifted by the same amount , i.e. if all parameters are rescaled simultaneously .so the scaling shown above is equivalent to , i.e. the first new parameter is suppressed by with respect to the second one . for both parts of the integral decompositionthe hard and potential regions are found .the preresolution algorithm implemented in asy2.m ( and switched on with the option ` preresolve ` ) tries to eliminate factorized combinations of terms in the function which potentially cancel each other , like in the example above .it checks all pairs of variables ( say , and ) which are part of monomials with opposite sign . for all those pairsthe code tries to build a linear combination of and such that in the variables and or and this monomial disappears .the code checks whether in the new variables the number of monomials with opposite sign decreases .for all such pairs the code recursively repeats the initial procedure in the new variables . as a resultit creates a tree of possible bisections and corresponding replacements of variables .a leaf of this tree is a set of sectors and functions such that one can not decrease the number of monomials with opposite sign any longer .ideally it means that all monomials now have the same sign .the code analyzes all leafs and chooses one of those with the minimal number of opposite - sign monomials ( or the minimal number of sectors if the numbers of monomials with opposite sign coincide ) . after finishing with the preresolution, the code performs the replacements and looks for regions in all those sectors , using the algorithm of the original code ` asy.m ` described in  .note that the algorithm can only find the necessary variable transformations if it is able to determine the relative signs of all terms in the polynomial  . as the signs of symbols are unknown to ` asy2.m ` , the substitution rules of the third and fourth arguments of ` alpharepexpand [ ] ` must replace all kinematic quantities by numbers ( integers or fractions of integers ) or powers of the small parameter ` x ` .we have checked that the updated version asy2.m works in various examples of the threshold expansion ( considered in  ) : a triangle , a box , the two - loop propagator diagram ( with the masses ) , a two - loop vertex diagram . because of the decomposition of a given integration domain into subdomains , the number of resulting integrals for various regions increases a little bit .for example , the ( hard - hard ) region for the two - loop propagator diagram is described by six integrals , the ( potential - ultrasoft ) region is also described by six integrals , etc .however , the ( potential - hard ) region is described by four integrals with some regions ( with scalings composed of powers 1 , 1/2 and 0 ) , and four more integrals with a set of regions of a different type ( composed of 1 and 0 ) .let us finally mention that the preresolution algorithm also works for threshold expansions with unequal masses .returning to the one - loop example ( [ prop1_integral ] ) , but now with different masses and in the propagators and the expansion parameter defined as , we call :  .... alpharepexpand[{k } ,  { k^2 - m1 ^ 2 , ( k - q)^2 - m2 ^ 2 } ,  { q^2 - > ( m1 + m2)^2 - 4*y } ,  { y - > x , m1 - > 2 , m2 - > 3/2 } ,  preresolve - > true ] ....  we have set the values of the masses to different rational numbers , and ( the actual values are irrelevant ) , thus permitting the preresolution algorithm to know about their sign without assuming any equality or other relation between them . the polynomial  in the parametric representation of the feynman integral reads , and cancellations occur where .these cancellations are automatically made explicit by adequate changes of variables . the output  .... { { { x[1 ] - > y[1 ] + 3*y[2]/7 ,   x[2 ] - > 4*y[2]/7 } , 7/4 , { 0 , -1/2 } } ,  { { x[1 ] - > y[1 ] + 3*y[2]/7 ,   x[2 ] - > 4*y[2]/7 } , 7/4 , { 0 , 0 } } ,  { { x[1 ] - > 3*y[1]/7 ,   x[2 ] - > 4*y[1]/7 + y[2 ] } , 7/3 , { 0 , 0 } } ,  { { x[1 ] - > 3*y[1]/7 ,   x[2 ] - > 4*y[1]/7 + y[2 ] } , 7/3 , { 0 , 1/2 } } } ....  shows that ` asy2.m ` detects both regions .the variable transformations are always normalized such that sums of the parameters ( here ) remain invariant .so , we now have a manifestly lorentz - invariant treatment of threshold expansion and a code that automatically provides the set of relevant regions .let us consider the one - loop five - point integral in fig .[ fig : glauber ] , where two initial - state partons both perform a collinear splitting into two partons each with momenta and , respectively . while two partons , one of each pair , collide with a large centre - of - mass energy , the two remaining partons exchange a particle with the small mass  .we will use the simplified kinematics and with and in the limit :   before we search for regions using ` asy.m ` , we notice that this five - point integral is similar to the sudakov form factor example treated in section 6 of  . from the viewpoint of the convergence of the expansions , the second and third propagators of ( [ 5pt_integral ] ) are equivalent , and so are the fourth and fifth propagators .effectively , the five - point integral has only three different types of propagators which are equivalent to the ones of the three - point integral in  .so the integral ( [ 5pt_integral ] ) can be expanded in loop - momentum space employing the regions and convergence domains known from  ( and using generic propagator powers as analytic regulators where necessary ) :  * a hard region where , * a 1-collinear region where and , * a 2-collinear region where and , * a glauber region where , and the components of  perpendicular to the plane spanned by scale as .the collinear - plane region mentioned in  yields only scaleless contributions .but , in contrast to the three - point integral , the five - point integral has a non - vanishing glauber contribution .the glauber region even provides the leading contribution scaling as , whereas the collinear contributions start with and the hard contribution starts with .the five - point integral ( [ 5pt_integral ] ) can be represented in terms of an integral over feynman parameters , ^{3+{\varepsilon } }   } \ ; ,  \label{fp_integral}\end{aligned}\ ] ] where one can choose the sum in the argument of the delta function in an appropriate way , i.e. restrict only the sum over a subset of the parameters to 1 and extend the integration over the rest of the parameters to the whole domain .  applying the strategy of expansion by regions in feynman - parameter space and trying to reveal regions relevant to the given limit with the help of the code asy.m , we call :  .... alpharepexpand[{k } ,  { k^2 - m^2 , k^2 - 2*p*k , k^2 + 2*p*k ,        k^2 - 2*q*k , k^2 + 2*q*k } ,  {p^2 - > 0 , q^2 - > 0 , p*q - > q^2/2 } ,  { q - > 1 , m^2 - > x } ] ....  the output states the following set of three regions :  .... { { 0 , 0 , 0 , 0 , 0 } , { 0 , 0 , 0 , 1 , 1 } ,  { 0 , 1 , 1 , 0 , 0 } } ....  as before , the regions are specified by the scaling of the feynman parameters in terms of powers of the small parameter  . for example , for the second region we have , .the first region is hard ; its contribution starts with .the second and third regions start with order .they correspond to the two collinear regions stated for the momentum - space expansion above .but ` asy.m `does not find anything corresponding to the glauber region ; in particular , none of the regions found by ` asy.m ` provides the leading contribution .we notice that , as in the previous section about potential contributions , the polynomial in the square brackets of ( [ fp_integral ] ) has terms of different sign .the missing glauber contribution stems from the parameter region where either or .so let us decompose the parametric integral into four parts corresponding to the domains where the two factors and are either positive or negative and then introduce new variables in such a way that this product takes the form .for example , in the domain , we change the variables by and by , similarly to our example in the previous section .however , in the threshold expansion the cancelling terms appeared in squared form such that a transformation between one pair of variables was sufficient . heretwo separate factors involve cancellations , which requires a twofold change of variables .  removing the primes from the variables  ,the parametric integral reads with ^{3+{\varepsilon } }   } \ ; ,  \label{fp_integral_a}\end{aligned}\ ] ] where we have chosen the argument of the delta function as , so that we may also write ^{3+{\varepsilon } }   } \;.\label{fp_integral_b}\end{aligned}\ ] ] it is sufficient to consider the expansion of and obtain a result for by analytically continuing , taking into account that the dependence on is power - like .now we can apply asy2.m to the integral  using either the command  .... wilsonexpand[x1*(x1 + x2 + x3 + x4 + x5)*m^2   + x2*x4*q^2 ,  x1 + x2 + x3 + x4 + x5 ,  { x1 , x2 , x3 , x4 , x5 } ,  { q^2 - > 1 , m^2 - > x } , delta - > true ] ....  for integrals ( [ fp_integral_a ] ) restricted by a delta function ( see section [ sec : syntax ] ) , yielding the output :  .... { { 0 , 0 , 0 , 0 , 0 } , { 0 , 1 , 0 , 0 , 0 } ,  { 0 , 0 , 0 , 1 , 0 } } ....  or we use the command  .... wilsonexpand[(1 + x2 + x3 + x4 + x5)*m^2   + x2*x4*q^2 ,  1 + x2 + x3 + x4 + x5 ,  { x2 , x3 , x4 , x5 } ,  { q^2 - > 1 , m^2 - > x } ] ....  for integrals ( [ fp_integral_b ] ) over variables from to without any restriction , and obtain the output :  .... { { 0 , 0 , 0 , 0 } , { 1 , 0 , 0 , 0 } , { 0 , 0 , 1 , 0 } } ....  the two results are equivalent , as is implied in the second case .so we obtain again three regions .we will see in a moment that this list of regions is indeed correct and complete .  butfirst , let us emphasize that the new preresolution algorithm in ` asy2.m ` is capable of performing the transformation of the integral from ( [ fp_integral ] ) to ( [ fp_integral_a ] ) automatically :  .... alpharepexpand[{k } ,  { k^2 - m^2 , k^2 - 2*p*k , k^2 + 2*p*k ,        k^2 - 2*q*k , k^2 + 2*q*k } ,  {p^2 - > 0 , q^2 - > 0 , p*q - > q^2/2 } ,  { q - > 1 , m^2 - > x } ,  preresolve - > true ] ....  the output of this command lists four different variable transformations according to the twofold decomposition described above . for each of the integrals over new parameters ,the regions , and are found , up to permutations in the order of the parameters from different changes of variables .the evaluation of the contributions to each region , as found by ` wilsonexpand [ ] ` or ` alpharepexpand [ ] ` ( including the ` preresolve ` option ) , is straightforward .the first region is the hard one .the contributions of the second and third regions are not individually regularized by dimensional regularization , as it often happens for sudakov - type limits .we use an auxiliary analytic regularization by introducing additional powers of the new variables into the integrand of ( [ fp_integral_b ] ) , taking the limit in the end . the leading - order ( lo )contribution of the second and third regions to the integral reads this agrees with the leading contribution of the glauber region in the momentum - space expansion .we have found the leading glauber contribution of order .but we seem to have lost the two collinear regions with the scalings and found before the change of variables .in fact , we can evaluate the contributions from these two regions by expanding the integral ( [ fp_integral_b ] ) .the resulting integrals are scaleless and regularized by the parameters , so they vanish , and ` asy2.m ` is right in omitting these two regions .we are also able to solve the integral ( [ fp_integral_b ] ) including the auxiliary analytic regularization factor in terms of a onefold mellin barnes representation : the asymptotic expansion of  in the limit is obtained by taking the residues of the poles of the functions .the poles of correspond to the hard region , while the poles of the two functions provide the contributions of the second and third regions .so ` asy2.m ` has found all contributing regions .  in the mellin barnes integral ( [ fp_integral_mb ] ) we can safely take the limit , add up and , and arrive at the mellin barnes representation the lo contribution to is obtained from the residue of the single pole at , in agreement with ( [ fp_lo ] ) .the next - to - leading - order ( nlo ) contribution stems from the residue of the double pole at and reads this agrees with the nlo contributions of the second and third regions .  at next - to - next - to - leading order ( nnlo ) there is a contribution from the residue of the single pole at which reads and agrees with the nnlo contributions of the second and third regions .the second nnlo contribution comes from the residue of the single pole at .it is given by and agrees with the lo contribution of the hard region .so indeed all contributions to the five - point integral up to nnlo are correctly reproduced by the contributions of the three regions found by ` asy2.m ` after the decomposition of the integral and the change of variables .  when revealing glauber regions for a general diagram , the preresolution algorithm of ` asy2.m ` tries to eliminate monomials with opposite sign in the polynomial  by automatically separating the integration into domains and performing changes of variables .if the option ` preresolve ` is enabled for ` alpharepexpand [ ] ` , the code warns the user if the elimination of monomials with opposite sign has not been successful , such that possibly not all regions are revealed .this is the case if some monomials of opposite sign remain in the polynomial  after tries to eliminate them , or if symbols with unknown signs are present in the polynomial .we are therefore convinced that ` alpharepexpand [ ] ` , with the preresolution enabled , either reveals all relevant regions or issues a warning .  as for the threshold expansion of the previous section , ` asy2.m ` can also treat more complicated kinematical situations , e.g. the five - point integral depicted in fig .[ fig : glauber ] with and ( retaining and such that and with ) :  .... alpharepexpand[{k } ,  { k^2 - m^2 , k^2 - 2*p1*k , k^2 + 2*p2*k ,        k^2 - 2*q2*k , k^2 + 2*q1*k } ,  { p1 ^ 2 - > 0 , p2 ^ 2 - > 0 , p1*p2 - > 0 ,   q1 ^ 2 - > 0 , q2 ^ 2 - > 0 , q1*q2 - > 0 ,   p1*q1 - > p1*q1/2 , p1*q2 - > p1*q2/2 ,   p2*q1 - > p2*q1/2 , p2*q2 - > p2*q2/2 } ,  { m^2 - > x , p1 - > 1 , p2 - > 3 ,        q1 - > 2 , q2 - > 3/2 } ,  preresolve - > true ] ....all kinematic invariants are replaced by rational numbers in order to enable the preresolution algorithm to work .the code correctly decomposes the integral into four pieces and finds the three regions for each of them .let us finally discuss the reason why , besides the hard region which is always present , the expansion in loop - momentum space requires two collinear regions and one glauber region , whereas the expansion of the parametric integrals ( [ fp_integral_b ] ) has two regions providing the leading glauber contribution and no further collinear regions .in fact , the momentum - space expansion is also valid for loop integrals ( [ 5pt_integral ] ) where each propagator is raised to an arbitrary , even non - integer power . for the decomposition of the parametric integral and the change of variables ,however , we have assumed the specific form ( [ 5pt_integral ] ) with each propagator present exactly once . in this case , the loop integrand can be expanded into partial fractions as follows : expanding this product of terms , one obtains twelve three - point integrals , which are well known . because they only depend on , we recognize from the last two factors the structure , where and are related by as before .the three - point integrals with the massless propagators or only have a hard region . the massive three - point integral with propagator known to possess a hard and two collinear regions .its lo and nlo hard contributions are cancelled by the massless three - point integrals , such that the uncancelled hard contributions start with , as for the five - point integral .the lo collinear contributions of the three - point integrals , enhanced by the prefactor , scale as .so , in the special case when all propagator powers are equal to 1 , the five - point integral reduces to a linear combination of three - point integrals revealing the same structure of regions as found from the expansion of the parametric integral ( [ fp_integral_b ] ) .this picture changes when generic propagator powers are introduced as analytic regulators , which is done in the next section .in the previous section we have seen different patterns of regions arising when expanding either in loop - momentum space or in parametric space .the individual contributions can be disentangled more easily when the dependence on the propagator powers is retained . instead of ( [ 5pt_integral ] ) ,let us consider the integral where the analytic regularization parameters  make the propagator powers different from the previous case . the asymptotic expansion in loop - momentum space yields contributions from the four regions listed in the beginning of section [ sec : glauber ] .the lo hard contribution still scales as , but now the lo 1-collinear contribution scales as , the lo 2-collinear contribution as and the lo glauber contribution as .so all regions are characterized by a distinct scaling .when performing the expansion in parametric space for generic  , we are able to disentangle the contributions from each region and match them with the regions obtained in loop - momentum space .  for generic  , the parametric integral corresponding to ( [ fp_integral ] ) reads ^{-(3+{\varepsilon}+{\lambda}_1+\ldots+{\lambda}_5 ) }  \;.\label{fp_integral_l}\end{aligned}\ ] ] the change of variables performed in section [ sec : glauber ] , e.g.  , for , is complicated by the presence of the factors , where parts of the monomials will change into polynomials .if we want to keep the simple form from the previous section , then we have to require and . under this restriction of the parameters , the parametric integrals can be written as ^{3+{\varepsilon}+{\lambda}_1 + 2{\lambda}_2 + 2{\lambda}_4 }   }  \;.\label{fp_integral_l_a}\end{aligned}\ ] ] in order to find the regions for the asymptotic expansion of ( [ fp_integral_l_a ] ) , we have to provide the additional polynomial factors and to ` wilsonexpand [ ] ` .we can do this by multiplying the new polynomials to the second argument of the command :  .... wilsonexpand[x1*(x1 + x2 + x3 + x4 + x5)*m^2   + x2*x4*q^2 ,  ( x1 + x2 + x3 + x4 + x5 ) * ( 2*x2 + x3 )   * ( 2*x4 + x5 ) ,  { x1 , x2 , x3 , x4 , x5 } ,  { q^2 - > 1 , m^2 - > x } , delta - > true ] ....  the output of this command reads :  .... { { 0 , 0 , 0 , 1 , 0 } , { 0 , 0 , 0 , 1 , 1 } ,  { 0 , 1 , 0 , 0 , 0 } , { 0 , 1 , 1 , 0 , 0 } ,  { 0 , 0 , 0 , 0 , 0 } } ....  in addition to the regions present in the analysis of section [ sec : glauber ] , we retrieve the two collinear regions with scalings and .the updated code ` asy2.m ` is capable of taking into account generic propagator powers automatically when told through the additional option ` genericpowers ` :  .... alpharepexpand[{k } ,  { k^2 - m^2 , k^2 - 2*p*k , k^2 + 2*p*k ,        k^2 - 2*q*k , k^2 + 2*q*k } ,  {p^2 - > 0 , q^2 - > 0 , p*q - > q^2/2 } ,  { q - > 1 , m^2 - > x } ,  preresolve - > true , genericpowers - > true ] ....  when the option ` genericpowers ` is enabled , the polynomial  obtained from the loop integral is multiplied by the product of all feynman parameters , , before the changes of variables are performed .some of these additional factors then turn into polynomials through replacements in the preresolution algorithm , while others remain monomials and are therefore irrelevant for the analysis of ` asy2.m ` .the call of ` alpharepexpand [ ] ` stated above correctly yields all five regions for each of the variable transformations .  for the evaluation of the glauber contributions with scalings and ,an additional analytic regularization is needed , and we choose to multiply the integrand of ( [ fp_integral_l_a ] ) by .( the parameters from section [ sec : glauber ] are not needed here due to the presence of . )the two glauber contributions are individually singular in the limit , but this singularity cancels in the sum of the two contributions .  the leading contribution to the integral originates from the sum of the lo glauber contributions .it reads this agrees with the lo contribution from the one glauber region in the momentum - space expansion of ( [ 5pt_integral_l ] ) for and . here , in the expansion of the parametric integrals , we have two regions producing the glauber contribution .this is possible because the contributions from both regions have the same scaling ( for ) , starting with . for ,the result ( [ fp_l_lo ] ) reproduces ( [ fp_lo ] ) .  among the nlo contributions to , we expect nlo glauber contributions and lo collinear contributions. however , the nlo glauber contributions vanish exactly for general , and , due to non - trivial cancellations between the pieces which contribute to the nlo expansion of ( [ fp_integral_l_a ] ) for either of the scalings and .the same happens in loop - momentum space , where the nlo glauber contribution is proportional to , thus vanishing in the case considered here .so the nlo contribution to is made up entirely from the lo collinear contributions .the 1-collinear region provides in agreement with the momentum - space expansion .the 2-collinear contribution is obtained from this by exchanging .adding the two collinear contributions together and performing the limit , the result ( [ fp_nlo ] ) is reproduced .considering finally the nnlo contributions to , we expect nnlo glauber contributions , nlo collinear contributions and a lo hard contribution .but here the nlo collinear contributions vanish exactly for and , both in loop - momentum space and when expanding the parametric integrals .so we are left with the nnlo glauber contributions yielding and the lo hard contribution , both consistent between the expansions of the loop integral and of the parametric integrals .setting all in ( [ fp_l_nnlo_g ] ) and ( [ fp_l_nnlo_h ] ) , we recover the results from ( [ fp_nnlo_g ] ) and ( [ fp_nnlo_h ] ) , respectively .we may also evaluate the parametric integral ( [ fp_integral_l_a ] ) in terms of a onefold mellin barnes representation : the relevant regions can easily be determined from the gamma functions in the last two lines .in particular , the squared gamma function indicates that the expansion of the parametric integrals  requires _ two _ regions for the glauber contribution , both scaling as at leading order .when combining , one of these gamma functions is cancelled , and we obtain from this representation the contributions to the asymptotic expansion in the limit can be extracted : the hard contributions stem from the residues of the poles at , the 1- and 2-collinear contributions from and , respectively , the glauber contributions from ( with ) . all lo , nlo and nnlo contributions reported in ( [ fp_l_lo])([fp_l_nnlo_h ] ) are confirmed by the corresponding residue contributions from ( [ fp_integral_l_mb2 ] ) . in particular, the structure of the poles in ( [ fp_integral_l_mb2 ] ) clearly shows that the glauber region does not contribute to the nlo result , and that the collinear contributions are absent at nnlo , as obtained before .the results reported in this section show that the use of generic propagator powers helps disentangling the glauber and collinear regions from each other , making all regions contribute in the same way to the asymptotic expansion in loop - momentum space and to the expansion of the parametric integrals . keeping the dependence of the contributions and their scalings on the propagator powers also facilitates the identification of regions found by ` asy2.m ` in parametric space for a subsequent expansion at the level of the loop integration .the updated version of the code , ` asy2.m ` , can be downloaded from the known web site , where further installation instructions are found .the ` mathematica ` code is loaded using ` < < asy2.m `.  the main function ` alpharepexpand [ ] ` identifies all regions which contribute to the asymptotic expansion of a given loop integral :  .... alpharepexpand[{k1 , k2 , ... } ,  { ( k1 + p1)^2 - m1 ^ 2 , ( k2 + p2)^2 - m2 ^ 2 ,   ... } ,  { p1 ^ 2 - > q1 , p2 ^ 2 - > q2 , p1*p2 - > q3 , ... } ,  { m1 ^ 2 - > x , m2 ^ 2 - > x^2 ,   q1 - > 1 , q2 - > 3/2 , ... } ,  options ] ....  the first argument is the list of loop momenta .the second argument lists the denominators of the propagators .the third argument contains replacement rules for all kinematic invariants .in particular , all external momenta appearing in the denominators must be replaced here , otherwise they are not identified correctly as vectors .the fourth argument sets the scaling of the parameters by replacing all symbols with powers of the expansion parameter , labelled by the global symbol ` x ` , and rational numbers ( integers or explicit fractions of integers ) .the output is a list of regions , specified by the scaling ( in powers of the expansion parameter ) of the feynman parameters corresponding to the propagators in the order stated in the second argument .e.g. the output  .... { ... , { 0 , 2 , 1 , ... } , ... } ....  indicates that there is a region specified by the scaling , , , of the feynman parameters , where is the small parameter of the problem .possible options of ` alpharepexpand [ ] ` are :  * ` preresolve - > true ` : try to eliminate cancellations between terms in the parametric representation by decomposing the integral and performing changes of variables .the output contains for each region a list of three entries , i.e. a region is e.g. specified by : + .... { { x[1 ] - > y[1]/2 , x[2 ] - > y[1]/2 + y[2 ] } ,  2 , { 0 , 1/2 } }   .... + the first entry in this list is the transformation between the original feynman parameters ` x[1 ] ` , ` x[2 ] ` , and the new variables ` y[1 ] ` , ` y[2 ] ` , .the second entry is the jacobian of the integral transformation ( here `` 2 '' , i.e.  ) .the last entry specifies the region in the usual form with the scalings of the new variables in powers of the small parameter , here , .+ when the option ` preresolve ` is enabled , the code warns if it fails to eliminate all possible cancellations in the parametric representation .when no warning is issued , all regions are found . without this option , however , regions occurring at such cancellations will not be revealed . * ` genericpowers - > true ` : take into account generic ( in particular non - integer ) powers of the propagators , e.g. when these powers are used as analytic regulators . without this option , the preresolution algorithm triggered by the option ` preresolve ` only finds all regions for integrals with propagators raised to positive integer powers . *` verbose - > true ` : print verbose internal information . * ` scalar - > true ` : permit more complex structures of the denominators by specifying scalar products of momenta via the function ` scalar[k , p ] ` instead of simple products ` k*p ` or ` scalar2[k ] ` instead of ` k^2 ` , e.g. : + .... alpharepexpand[{k } ,  { scalar[k , k ] - m^2 ,   scalar2[k - q ] - m^2 } ,  { scalar2[q ] - > qq , m^2 - > qq/4 + y } ,  { qq - > 1 , y - > x } ,   preresolve - > true , scalar - > true ]   ....  for expanding more general integrals , which need not originate from feynman diagrams , the command ` wilsonexpand [ ] ` may be used :  .... wilsonexpand[f , u , { x1 , x2 , ... } ,  { ... - > x , ... } , options ] ....  traditionally , the first two arguments are the polynomials and , respectively , from the alpha parametric representation ( [ alpha - d ] ) of the feynman integral or from the feynman parametric representation ( [ alpha - d - mod ] ) , ( [ integrand ] ) .but , more generally , ` wilsonexpand [ ] ` reveals regions for integrals over parameters , integrated from 0 to  each , where all non - trivial polynomials of the parameters  occurring in the integrand are specified either in the first or the second argument .the third argument of ` wilsonexpand [ ] ` is the list of integration parameters .the fourth argument specifies the scaling of all quantities with the expansion parameter ` x ` , in the same way as in the fourth argument of ` alpharepexpand [ ] ` .  the output is a list of regions , specified by the scaling of the integration parameters in powers of the expansion parameter , exactly as described for the output of ` alpharepexpand [ ] ` ( without the ` preresolve ` option ) .possible options of ` wilsonexpand [ ] ` are :  * ` delta - > true ` : under the integral , the sum over an arbitrary non - empty subset of the integration parameters is restricted to 1 via a delta function .the specific choice of the sum in this delta function must be irrelevant for the integral , which is the case for the generalized feynman parametric representation ( [ alpha - d - mod ] ) .* ` verbose - > true ` : print verbose internal information .more generally , the option ` delta ` of ` wilsonexpand [ ] ` works correctly for all integrals of the form where the linear combination in the argument of the delta function has no negative and at least one positive coefficient ( and ) , and where the function  scales homogeneously with the parameters  as the degree of homogeneity being equal to minus the number of integration parameters .it can be shown that for such integrals ( [ delta_int ] ) the specific choice of the coefficients  is irrelevant . to see this ,multiply the integrand of ( [ delta_int ] ) by 1 in the form , where is any of the integration parameters . then , inside the -integration , transform the integration variables as , , and use the homogeneity relation ( [ delta_int_hom ] ) with . finally evaluate the -integration first , yielding , independent of the coefficients  , as long as the linear combination is positive .the integral ( [ delta_int ] ) is given by .] the integrand ( [ integrand ] ) of the feynman parametric representation ( [ alpha - d - mod ] ) and the integrands of all parametric representations of feynman integrals used in this paper ( without the additional analytic regularization factors ) fulfill the homogeneity condition ( [ delta_int_hom ] ) , so they do not depend on the specific choice for the arguments of their delta functions .we have presented an algorithm for identifying all regions which are relevant for the asymptotic expansion of a given loop integral at the level of its parametric representation .in contrast to the previous version ` asy.m ` of the code , also potential regions and glauber regions are found now .the necessary decompositions and variable transformations of the integral are automated by the updated ` mathematica ` code ` asy2.m `  . when the command ` alpharepexpand [ ] ` is used with its option ` preresolve ` enabled , we are convinced that it either reveals all relevant regions or issues a warning . in particular , regions corresponding to cancellations between large positive and negative terms in the parametric representation of the loop integrals ( such as potential and glauber regions ) will now be found .let us emphasize that to prove expansion by regions at least for some specific limit typical of minkowski space is a natural mathematical problem .perhaps , this problem is not specifically related to feynman integrals .let us present an example of a one - dimensional parametric integral , without any relevance to feynman integrals , and show that expansion by regions works successfully . to do this, we will use asy2.m .let us consider the integral with a complex parameter , in the limit .we assume that is in the domain re in order to have an absolute convergence of the integral which then can be continued analytically to the whole complex plane as an analytic function of . running   we obtain the two regions ` { { 1 } , { 0 } } ` .the leading - order terms from each region can be evaluated analytically in terms of gamma functions at general , with the results and they can be checked easily by deriving the onefold mellin barnes representation and evaluating the first terms of the asymptotic expansion in the limit by shifting the contour to the right and taking residues at the poles of the two gamma functions in the integrand .this work is supported by the deutsche forschungsgemeinschaft sonderforschungsbereich / transregio 9 `` computergesttzte theoretische teilchenphysik '' .the work of a.s . and v.s .is also supported by the russian foundation for basic research through grant 11 - 02 - 01196 .the authors thank m. beneke and a. pak for helpful discussions .chetyrkin , theor .phys .  * 75 * , 346 ( 1988 ) [ teor . mat .* 75 * , 26 ( 1988 ) ] k.g .chetyrkin , theor .phys .  * 76 * , 809 ( 1988 ) [ teor . mat .76 * , 207 ( 1988 ) ] s.g .gorishny , nucl .b * 319 * , 633 ( 1989 ) v.a .smirnov , commun .phys .  * 134 * , 109 ( 1990 ) v.a .smirnov , mod .lett . a * 10 * , 1485 ( 1995 ) [ hep - th/9412063 ] v.a .smirnov , _ applied asymptotic expansions in momenta and masses _ , springer tracts in modern physics 177 ( springer , berlin , 2002 ) t. seidensticker , hep - ph/9905298 r. harlander , t. seidensticker , m. steinhauser , phys .b * 426 * , 125 ( 1998 ) [ hep - ph/9712228 ] m. beneke , v.a .smirnov , nucl .b * 522 * , 321 ( 1998 ) [ hep - ph/9711391 ] v.a .smirnov , e.r .rakhmetov , theor .* 120 * , 870 ( 1999 ) [ teor. mat .* 120 * , 64 ( 1999 ) ] [ hep - ph/9812529 ] v.a .smirnov , phys .b * 465 * , 226 ( 1999 ) [ hep - ph/9907471 ] b. jantzen , jhep * 12 * , 076 ( 2011 ) [ arxiv:1111.2589 [ hep - ph ] ] a.v .smirnov , v.a .smirnov , m. tentyukov , comput .commun .  * 182 * , 790 ( 2011 ) [ arxiv:0912.0158 [ hep - ph ] ] a.v .smirnov , m.n .tentyukov , comput .commun .* 180 * , 735 ( 2009 ) [ arxiv:0807.4129 [ hep - ph ] ] a. pak , a. smirnov , eur .j. c * 71 * , 1626 ( 2011 ) [ arxiv:1011.4863 [ hep - ph ] ] a.v .smirnov , http://science.sander.su/tools-uf.htm

Abstract: when performing asymptotic expansions using the strategy of expansion by regions , it is a non - trivial task to find the relevant regions . the recently published mathematica code ` asy.m ` automates this task , but it has not been able to detect potential regions in threshold expansions or glauber regions . in this work we present an algorithm and its implementation in the update ` asy2.m ` which also reveals potential and glauber regions automatically .


in this paper , we address a single question : is there a numerically - viable coordinatization of a schwarzschild black hole spacetime foliated by hypersurfaces of constant ( not necessarily zero ) mean extrinsic curvature ?in other words , can we coordinatize the schwarzschild spacetime with constant mean extrinsic curvature ( ) hypersurfaces so as to bound the growth of metric components and their gradients ?we demonstrate here that the single shift freedom yields a spacetime metric that is static , and therefore bounds the growth in time of such gradients. a more complete analysis of the stability of our coordinatization , and a more thorough canvassing of the parameter space , will appear elsewhere . our foliation is consistent with that of iriondo _ et al . _ , who provided a generic constant mean curvature ( cmc ) foliation of the reissner - nordstrm spacetime for the purpose of finding trapped surfaces . in this paperwe focus on the utility of cmc slicings for the numerical simulation of black holes , in support of the emerging field of gravity - wave astrophysics .the trace of the extrinsic curvature tensor ( ) at a point on a spacelike hypersurface measures the fractional rate of contraction of 3-volume along a unit normal to the surface .it represents the amount of `` crunch '' the 3-surface is experiencing at the point , at a given time .if all the observers throughout a spacelike hypersurface moving in time orthogonal to the surface experience the same amount of contraction per unit proper time , we say that the surface is a -surface or a `` constant crunch '' surface . in this paperwe examine foliations of a single spherically - symmetric , static black hole where each spacelike hypersurface has the same constant value of the extrinsic curvature , .generic -surface foliations have found great utility in the numerical simulation of cosmological spacetimes . in addition to decoupling the three momentum constraint equations from the hamiltonian constraint , these surfaces ( in the case of compact or w - model universes ) provide a convenient cosmological time parameter ( , or york , time ) . furthermore , for such cosmological spacetimes one has powerful existence and uniqueness theorems . extensive work into the characteristics of these surfaces for schwarzschild spacetimes has been done by brill _et al . _ ,  and foundational work into their use in numerical relativity was done by d. eardley _et al._.  more recently pervez _ et al . _ provided a foliation partially covering the schwarzschild spacetime with -surfaces , with ranging from to , and iriondo _ et al . _ provided a generic constant mean curvature foliation of the reissner - nordstrm spacetime for the purpose of finding trapped surfaces . in this paperwe build upon the work of these investigators by examining the utility of these surfaces for numerical relativity in support of gravity - wave detectors .although surfaces of constant were thoroughly investigated decades ago , their use in current numerical simulations of black holes is conspicuously absent ( apart from the use of maximal ( ) surfaces ) . one reason why these slicing methods have not been more fully developed is that they lag in time close in , to avoid crashing into the singularity , while they simultaneously evolve forward normally at the outer edge of the grid to allow for wave extraction .this tension , many fear , will unavoidingly lead to unbounded growth in the metric and extrinsic curvature components in the intermediate region , as is indeed found in maximal slicing .this computational concern has been referred to by the numerical relativity community as `` grid stretching '' or `` grid sucking . ''we show in this paper that a proper choice of radial shift can yield a constant crunch foliation of a spherically - symmetric black hole without such pathologies .in fact , we foliate a schwarzschild black hole such that the 3-metric and extrinsic curvature are both bounded and static ( i.e. unchanging in time ) .  to numerically evolve a black hole 3-space in time it is desirable to have a foliation , and its coordinatization , which satisfy the following four properties :  1 .avoids black hole singularities or facilitates their excision .2 .  possesses asymptotically null hypersurfaces to aid in radiation extraction .3 .  minimizes steep gradients in the lapse , shift , 3-metric and extrinsic curvature tensor .4 .  maximizes the future development of the initial data for the purpose of gravity - wave extraction .  as a first step towards achieving these goals for systems containing multiple black holes, we explore the families of -surfaces in the schwarzschild spacetime , and find a cmc foliation satisfying the above properties .  in the next sectionwe construct the -surfaces outside and inside the horizon . in section [ sec:3 ]we explore the properties of the -surfaces , dwelling in particular on their approaches to the singularity .we also examine and illustrate the three families of -surfaces . in section [ sec:4 ]we derive a metric for schwarzschild whose constant time slices are -surfaces . we restrict our attention to a subfamily of -surfaces surfaces which , when generalized to the colliding black hole spacetimes , support the gravity wave detection problem .we also present some preliminary numerical simulations using constant - crunch coordinates .we conclude with general comments on the applicability of -surfaces to numerical calculations of more general black hole spacetimes .over three decades ago eardley and smarr  carried out a generic classification of the spacetimes that could be simulated numerically , and investigated the limitations that the presence of singularities would impose . in their paperthey argued that cmc slicings are particularly useful for numerical purposes .in particular , they demonstrated this explicitly by constructing numerical solutions to a wide array of dust collapse models . in a similar vein , brill et al . explored the nature of cmc slices of the schwarzschild spacetime , and they also presented some numerical examples . in the present work ,we explore the numerical utility of cmc slicings in the case of single black hole spacetimes . for the sake of clarity, we will commence with a re - derivation of the equations governing the cmc surfaces starting from schwarzschild coordinates .from there , we will explore specific properties of the surfaces , paying particular attention to implications for numerical relativity .we wish to find a spacelike hypersurface in the schwarzschild spacetime such that every point on the surface has the same constant value of the trace of the extrinsic curvature tensor .we have at our disposal the specification of the initial - value data , as well as the freedom to choose the lapse and shift throughout the evolution .to begin , let us take the standard coordinate system of a single black hole spacetime of mass in schwarzschild coordinates : with and .it will be convenient to treat separately the regions inside and outside of the horizon .we will find that the two are related by an isometry .= 4.0truein   outside the horizon , cmc surfaces will be labeled by ( fig .[ fig1 ] ) .the requirement that the trace of the extrinsic curvature be constant throughout this surface yields a first order differential equation for , determined by examining the behavior of the normals to the surface .the normal to the spacelike hypersurface is given by , where is a normalization constant and primes denote differentiation with respect to .the normalization is fixed by demanding that therefore , and the contravariant components of the normal are given by the trace of the extrinsic curvature is the fractional rate of contraction of 3-volume per unit proper time along the normal , namely substitution of into eq .( [ eq : trk ] ) yields a second - order ordinary differential equation for . integrating this equation , we find with an integration constant and an indefinite integral given by along the surface the rate of change of proper time , , with proper distance , , is related to the slope of the surface , from eq .( [ th ] ) we find    finding the -constant slices of eq .( [ eq1 ] ) within the horizon is similar to the calculation done in the previous section ; however , as the roles of time and space coordinates reverse within the horizon , we will find it useful to parameterize our spacelike surface as a function of coordinate , and look for -constant surfaces of the form ( fig .[ fig1 ] )   the normal to the spacelike hypersurface is given by with differentiation with respect to denoted by dots and the a normalization constant fixed by : we have , therefore , and the contravariant components of the normal are given by   from eq .( [ kequation ] ) , we once again find that fixing the trace of the extrinsic curvature gives us a second - order differential equation for , namely , which can be simplified to   paralleling the approach from the last section , we introduce an integration constant , , and an indefinite integral ( given by eq .( [ j ] ) ) , to obtain the first integral : from eq . ( [ r ] ) we find that the `` proper velocity '' along the surface , , results in the same equation both inside and outside of the horizon ( eq .( [ brill8 ] ) ) . this can be rewritten as   the spacelike -surfaces obtained from the first integrals , eqs .( [ brill8 ] ) and ( [ brill11 ] ) , differ only by an isometry , spatial metric of a -surface outside of the horizon is given by within the horizon it becomes these two expressions differ by the isometry of eq .( [ isometry ] ) . using eq .( [ brill8 ] ) we can rewrite them in terms of and : from this we arrive at the scalar curvature of the -surface : similarly , by using eqs .( [ brill8 ] ) and ( [ brill11 ] ) , the extrinsic curvature associated with observers moving on world lines orthogonal to the -slices are also expressible in terms of and : the -surfaces are therefore parametrized by two constants : the trace of the extrinsic curvature tensor , , and the constant of integration , .in addition one must fix a single point on the surface , , which amounts to setting a time translation parameter . as can be seen in eqs .( [ scalarr])([kpp ] ) , the constant controls the variation of the intrinsic and extrinsic curvatures over the -surface .  to elucidate the nature of the -surfaces , we numerically integrate eqs .( [ brill8 ] ) and ( [ brill11 ] ) .we find that within the horizon there are 3 classes of -constant surfaces , differentiated by their asymptotic behavior .the singularity - singularity surfaces ( _ ss _ ) begin at the singularity aligned with the null surface , reach up towards the horizon , and then fall back , reaching the singularity along the null cone . the horizon - horizon surfaces ( _ hh _ ) , which we have also dubbed `` horizon - hugging '' surfaces , asymptote to the horizon ( for in schwarzschild ) , dipping down towards the singularity in between .this feature was previously remarked upon by brill _the asymptotes converge toward a null surface at the horizon . finally , the horizon - singularity ( _ hs _ ) surfaces begin at the horizon , and asymptote in to the singularity .representative surfaces for the value are shown in fig .[ sshhhs ] .we integrate the _ hh _ and _ hs _ surfaces across the horizon into the region by imposing continuity of the surface and its first derivative at the horizon . because of the isometry , eq .( [ isometry ] ) , the surfaces outside of the horizon are characteristically similar to those on the inside ; in particular , both sets are null at their asymptotes .= 6.0truein  we have chosen to use the acronym _hh _ for the horizon - to - horizon hypersurfaces , in lieu of referring to them as `` regular '' hypersurfaces  , as each of the three types of -surfaces are , in a strict sense , regular .in particular , each surface asymptotes to a null surface , be it at the singularity or the horizon .observers on such a surface , or more precisely , observers that are time synchronized throughout the surface , are never seen crossing the horizon , nor do they ever reach the singularity ! outside the horizon , every _ hh _ and _ hs _ k - surface ( ) asymptotes for large to null infinity .-surfaces corresponding to positive values of asymptote to past null infinity , and asymptote to future null infinity for .  to gain a qualitative understanding of the -constant foliation , it is useful to analyze eq .( [ brill11 ] ) as an energy conservation equation for a particle of unit total energy ( ) moving in the potential by using this energy equation we can determine the closest approach to the singularity , , of a given _ hh _-constant surface .two conditions must be satisfied to determine .first , the closest approach occurs when , which is equivalent to demanding this condition leads to a sixth - order polynomial in : second , the solution for this surface at must be concave , so as to rule out the _ ss _ surfaces ( which bend towards the singularity rather than the horizon ) .this is enforced by demanding : }{r_{min}^3}\geq 0.\ ] ] the solution of these two conditions , as shown in fig .[ contour ] , gives rise to the emergence of two critical values for , namely , for a given value of .in addition , an _ hh _ surface can be made to approach arbitrarily closely to the singularity at by choosing an appropriately large positive value of .negative values of tend to `` hug the horizon . ''= 5.0truein  for fixed and we know how to compute how closely a cmc surface comes to the singularity .but , for a given value of , what value of gives the closest overall approach to the singularity ?we can determine the critical values for and , given by and respectively , by looking at the point where the first and second time derivatives of vanish . occurs along the lower boundary of the contour plot in fig .[ contour ] , and corresponds to the -surface that reaches down the furthest towards the singularity for a given value of .the vanishing of leads to the following equation for : which can be rewritten as a 4th order polynomial in  one must take care in examining the roots of this equation , as there are more solutions to eq .( [ 4poly ] ) than there are for eq .( [ rcriteqn ] ) .nevertheless this equation gives two distinct real roots , depending on the sign of : where when we see from eq .( [ rcriteqn ] ) that .the regular -surfaces are thus bounded between the horizon at and .using eq .( [ rh ] ) , and setting , we obtain , for the case of a black hole in schwarzschild coordinates with we therefore have   = 3.0truein  for large values of one can show that the critical value of , , depends upon the sign of .in particular , this in turn gives the following asymptotic values for : number of features of the -constant surfaces presented in the previous sections seem particularly well suited to the numerical analysis of generic black hole spacetimes .first , the surfaces asymptote to a null surface , making them effective for gravity wave extraction .second , the -surfaces naturally avoid the crushing singularity .finally , for large negative values of the surfaces `` hug the horizon . ''this last feature , illustrated in fig .[ rhcrit ] , allows one to focus attention on the region relevant for gravity wave generation the region outside the horizon .  in this sectionwe generate a -constant foliation for the schwarzschild black hole that , in addition to the properties just mentioned , also has regular and static metric and extrinsic curvature components . to generate this -constantslicing we use the coordinate transformation under this transformation , the metric from eq .( [ eq1 ] ) becomes the constant slices of this metric are -constant surfaces .it is to be noted that eq .( [ kmetric ] ) agrees with eq .( 53 ) of iriondo et al .( for the case of constant and vacuum ) . however , in order to regularize the metric component at the throat , we add the isotropic - like radial transformation  : with the minimum coordinate location of the throat , given by eq .( [ polynomial ] ) .this coordinate representation of a black hole spacetime provides a foliation with -constant , -constant spacelike hypersurfaces .each hypersurface is metrically equivalent to all others the surfaces are independent of , and hence static .in addition , the hypersurfaces are asymptotically null ( as ) .furthermore , the lapse , the shift , and all of the 3-metric and extrinsic curvature components are regular and well behaved , as illustrated in fig .[ components ] .= 5.0truein  in addition to restricting ourselves to the singularity - avoiding family ( _ hh _ ) of -surfaces , two additional conditions on the -surfaces are demanded by the nature of our problem the eventual simulation of the gravity - wave emission from two interacting black holes .first , the foliations must asymptote at large to future null infinity .therefore , we must restrict our attention to negative values for the trace of the extrinsic curvature tensor , .second , we require that such negative hypersurfaces enter the future singularity region .this will ensure proper coverage of the relevant region just above the future horizon , which is precisely where the gravity waves are produced .however , we expect the initial - data formulation for such surfaces to be involved , and this may guide our choices even more systemically .the two additional requirements limit us to the relatively narrow wedge of fig .[ contour ] , formed by restricting to the `` future _ hh _ and _ ss _ '' shaded region with .  a representative constant - crunch foliation generated by eq .( [ kmetric ] ) is shown in fig .[ foliate ] .the avoidance of `` grid stretching '' is accomplished by a suitable choice of shift vector . to illustrate the non - zero shift we show the , foliation of schwarzschild in fig .[ shift ] , with the explicit misalignment of the line segment and the normal vector .= 5.0truein  = 5.0truein  finally , we describe several preliminary numerical experiments using -surfaces . a full treatment of a single black hole using -constant foliations will be presented elsewhere . herewe present several sample evolutions demonstrating the utility of constant mean curvature slicings .figures [ convergence ] and [ hamiltonian ] display results from the simplest possible test of -constant foliation of the schwarzschild geometry ; the domain is taken to be a thin shell close to the horizon ( in this case $ ] ) , analytic dirichlet conditions are applied at both boundaries of the computational domain , and analytic lapse and shift conditions obtained from eqs .( [ kmetric ] ) and ( [ iso ] ) are used .the figures represent the singularity - avoiding foliation and , for which the evolution was found to be stable and accurate over very long timescales . using grid points ,the code succesfully ran beyond while maintaining high accuracy . the fractional error in the metric components was typically .[ convergence ] shows the convergence of the mean fractional error in the metric component as a function of time .each curve has been rescaled by a factor of , where the number of grid points is given by , with . for solution is approximately second order accurate .initially , noise generated on the inner boundary causes fluctuations whose magnitude is largely independent of the number of grid points .[ hamiltonian ] shows snapshots of the hamiltonian constraint at various times in the evolution .qualitatively , a wave which is triggered by truncation error is seen to propagate outwards from the inner boundary .the amplitude reduces rapidly , before growing once more as it is reflected off the outer boundary . the magnitude and speed of propagation of the wave quickly decay as the wave moves back into the domain , leaving a static solution which is stable beyond . the numerical error which initially propagates through the domainis caused entirely by the analytic dirichlet boundary conditions , and can be largely eliminated by the use of more realistic conditions .  the numerical runs presented here evolve a singularity - avoiding hypersurface that asymptotes to null infinity , entering the past singularity region .this is not of the class of -surfaces emphasized in this paper for use in numerical relativity .ideally , we would have preferred presenting the evolution of a _ hh _ surface that enters the future singularity region .however , we were unable to find an _surface in the future singularity region that evolved stably using the naive dirichlet boundary conditions and analytic lapse and shift conditions .nevertheless , we have evolved such surfaces stably by incorporating ( 1 ) area locking shift conditions , ( 2 ) isometery conditions at the throat , and ( 3 ) an outgoing boundary condition based on the difference between the numeric and analytic solutions .these numerical results , and the corresponding stability analysis , are not within the scope of this paper and will be presented elsewhere. this paper we considered a static spacetime metric for the schwarzschild black hole , with spacelike hypersurfaces of constant ( not necessarily zero ) value of the trace of the extrinsic curvature tensor .this slicing provides a natural generalization of the maximal slicing scheme currently in use in some numerical approaches to the binary black hole problem .  an essential feature of our -constant metric is a spatially - varying radial shift vector , which allows the surfaces to avoid the singularity while evading the grid stretching problems often encountered with other metrics .work is in progress to develop a geometric handle on our shift condition , akin to that of the `` minimal distortion '' shift often discussed .the inner and outer boundary conditions are also particularly convenient . in the inner regions , the `` horizon hugging '' feature of the -slices , together with their regularity , may remove the need to excise the grid within the apparent horizon , thus providing a natural `` boundary without a boundary '' avoidance of the singularity .in addition , at the outer boundary the surfaces are asymptotically null , which may aid in gravity wave extraction .  in examining the characteristics of the -slices we reviewed the three families of -surfaces , including the horizon - to - singularity ( _ hs _ ) surfaces , as well as the more familiar horizon - to - horizon ( _ hh _ ) andsingularity - to - singularity ( _ ss _ ) surfaces .all three families of surfaces either asymptote to the singularity , or to spatial infinity along a null surface . thus the family of observers that make up such a foliationnever `` observe ''the surfaces reach the singularity .this suggests a possible payoff in using the _ hs _ surfaces in the evolution of black hole spacetimes , as there is no danger of the surface reaching the singularity .just as there are natural boundary conditions which `` freeze out '' gravitational radiation as it progresses to null infinity , so too can we expect boundary conditions where the dynamic space freezes out as it approaches the singularity .are there such natural boundary conditions at the future singularity ?it would be worthwhile to investigate such asymptotically - null boundary conditions at the singularity of an surface .the future utility of numerical simulations of black hole spacetimes hinges in large part on a suitable choice of coordinates for the initial data , and on the particular evolution of these coordinates through the four lapse and shift conditions .these conditions are the only handles by which to manage the growth of the metric and curvature components during evolution .it is through judicious choices of lapse and shift that one is able to effectively enable singularity avoidance , and allow for efficient extraction of gravitational radiation .the simplistic example presented here may provide some guidance as to how to proceed in the more general case .-surfaces in the generic two black hole problem can be expected to preserve the singularity - avoidance `` horizon hugging '' behavior , as well as remain asymptotically null at the boundaries .it remains to be seen what shift vectors are required to manage the growth of the intrinsic and extrinsic properties of the metric in the more general case , though the constant - crunch shift presented here is a natural starting point .we are currently pursuing three avenues towards furthering and generalizing the work presented here .first , we are analyzing the stability of this coordinatization to small perturbations .second , we are using the ( 1 + 1)-dimensional code discussed in the last section to examine a greater portion of the parameter space of initial data , so as to fully explore the numerical stability of the slicing . from our preliminary numerical investigations ,we expect to be able to do a full spacetime `` evolution '' of schwarzschild , and have it run stably and accurately for extended periods of time over a wide range of - parameter space .finally , we are analyzing the oppenheimer - snyder collapse in such -slicings , following the lead of eardley and smarr ,  as this will further test our slicing in a non static setting .  we wish to acknowledge the los alamos national laboratory ldrd / er program for financial support .wam wishes to thank the institute for theoretical physics at ucsb for providing a stimulating working environment in which to complete part of this research .dh was supported in part by the national science foundation under grant no .phy9907949 to the itp .pl was supported in part by nsf grants phy9800973 and phy9800970 .we wish to thank richard matzner for advice on handling the coordinate ambiguity at the throat .we are especially grateful to john a. wheeler for encouraging us to examine these slices in the context of the numerical treatment of black hole spacetimes .gentle , d.e .holz , p. laguna , w. a. miller , and d. shoemaker , in preparation .m. iriondo , e. malec and n.  murchadha , _ phys ._ * d54 * ( 1996 ) 4792 .j. centrella and j. wilson , _ ap .* 54 * ( 1984 ) 229 .j. a. wheeler , _ international j. mod .phys . _ * a3 * ( 1988 ) 2207 - 2247. f. j. tipler and j. e. marsden , _ phys .reports _ , * 66 * ( 1980 ) 109 - 139 .f. j. tipler and j. d. barrow , _ mon . not ._ , * 216 * ( 1985 ) 395 - 402 .d. r. brill , j. m. cavallo and j. a. isenberg , _* 21 * ( 1978 ) 2789 .d. eardley and l. smarr , _ phys ._ * d19 * ( 1978 ) 2239 .a. pervez , a. qadir and a. siddiqui , _ phys ._ * d51 * ( 1995 ) 4598 .m. alcubierre , g. allen , b. bruegmann , t. dramlitsch , j. a. font , p. papadopoulos , e. seidel , n. stergioulas , w. suen , r. takahashi , `` towards a stable numerical evolution of strongly gravitating systems in general relativity : the conformal treatments , '' gr - qc/0003071 .richard matzner , private communication , the university of texas at austin , march 20 , 2000 .private communication at the 16th pacific coast gravity meeting , caltech , march 24 - 25 , 2000 .

Abstract: we reinvestigate the utility of time - independent constant mean curvature foliations for the numerical simulation of a single spherically - symmetric black hole . each spacelike hypersurface of such a foliation is endowed with the same constant value of the trace of the extrinsic curvature tensor , . of the three families of -constant surfaces possible ( classified according to their asymptotic behaviors ) , we single out a sub - family of singularity - avoiding surfaces that may be particularly useful , and provide an analytic expression for the closest approach such surfaces make to the singularity . we then utilize a non - zero shift to yield families of -constant surfaces which ( 1 ) avoid the black hole singularity , and thus the need to excise the singularity , ( 2 ) are asymptotically null , aiding in gravity wave extraction , ( 3 ) cover the physically relevant part of the spacetime , ( 4 ) are well behaved ( regular ) across the horizon , and ( 5 ) are static under evolution , and therefore have no `` grid stretching / sucking '' pathologies . preliminary numerical runs demonstrate that we can stably evolve a single spherically - symmetric static black hole using this foliation . we wish to emphasize that this coordinatization produces -constant surfaces for a single black hole spacetime that are regular , static and stable throughout their evolution .