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the pioneering work of weinberg  @xcite launched the basis for the effective field theory ( eft ) of nuclear forces starting from the expansion of an effective chiral lagrangian . it gives a nucleon - nucleon ( nn ) interaction which is , in leading order , the one - pion - exchange potential ( opep ) plus a dirac - delta . the program of applying effective field methods to the nn system was pursued by many authors with significative results in few - nucleon systems ( see e.g.  @xcite ) . in a more general context of few - body systems , short range interactions have many applications which are discussed in detail in ref . @xcite . more recently , the authors of @xcite and @xcite treat the one - pion - exchange potential ( opep ) plus derivative dirac - delta interactions using dimensional and boundary condition regularizations , respectively . in particular , when comparing the theoretical approach with the data  @xcite , it was shown in ref . @xcite that the leading order interaction , opep plus a dirac - delta , renormalized using subtracted scattering equations , dominates the results obtained for the @xmath1 phase - shifts and mixing parameter . however , considering only the leading - order term , the results obtained for the @xmath0 phase shift are not satisfactory . this is an evidence that higher order terms in the effective interaction are important in this channel ( see e.g. refs . @xcite ) . the fit of the @xmath0 phase - shift up to laboratory momentum @xmath2  mev / c requires an effective nn interaction with the addition of a term with second order derivatives of the dirac - delta . in the relative momentum space , we have @xmath3 where the @xmath4 s are unregulated strengths and @xmath5 is a matrix element of the one - pion - exchange potential . the motivation of the second term of ( [ veft ] ) is to simulate effects of heavy particle exchanges by a sum of a dirac - delta interaction and its derivatives , while keeping the ope as the long range part of the interaction . in the renormalization method described in @xcite , we include @xmath6 , as the method is based on a kernel - subtraction procedure that generates terms of the type @xmath7 in the scattering matrix elements even when one considers only @xmath8 and @xmath9 in the interaction . in fact , the method also generates higher - order derivatives terms when ope is considered . in the weinberg counting rule scheme , the derivative contact interactions we are considering comes along with the two - pion - exchange ( tpe )  @xcite . for a recent calculation in the nn system considering tpe see e.g. ref .  @xcite . as an exploratory calculation , in the present work we consider only the contact terms , leaving tpe for a future work .    in the scattering equation , the effective bare potential  ( [ veft ] ) produces integrals that diverge as much as @xmath10 . therefore it is necessary at least three subtractions in the kernel of the lippman - schwinger ( ls ) equation , since each subtraction introduces a factor of @xmath11 . differently from the recent works @xcite and @xcite , we implement the method of subtracted scattering equations  @xcite to deal with this problem . the subtraction method has also been shown to be practical in providing renormalization group invariant solutions for three - body scattering equations with contact interactions  @xcite , and also proved to be useful in describing the halo structure of weakly bound exotic light nuclei  @xcite . the one - subtracted scattering equation used in our previous work  @xcite was generalized in  @xcite to allow any order of subtraction , permitting the inclusion of derivatives of the contact interaction in the effective two - body potential . the driving term of the n - subtracted ls equation is constructed recursively , renormalizing the model at each subtraction order , while keeping renormalization group invariance of the approach ( for details see  @xcite ) .    in the present work , we obtain the @xmath0 nucleon - nucleon amplitude from the effective interaction ( [ veft ] ) , using a three - times subtracted scattering equation , as demanded by the higher divergent term of the type @xmath10 . we perform an analysis of the physical contribution coming from each order term in the recursive order - by - order renormalization procedure . it this way we access their significance in the parameterization of the effective interaction to obtain the desired observables . we should note that the contact interactions that we are considering are obviously meaningless without regularization and renormalization due to the generated ultraviolet divergences in the lippman - schwinger equation . this work is organized as follows . in section 2 , we discuss the subtraction method applied to opep . in section 3 , we present the main formulas of the recursive subtraction method to treat the scattering equation , as well as our strategy to solve it . in section 4 , we show numerical results for the singlet phase - shift with our conclusion . the partial wave decomposition of opep for the @xmath0 state is  @xcite : @xmath12 where @xmath13 and @xmath14 are the relative momentum . the regular part of ope has a finite scattering matrix , solution of the partial - wave projected ls equation : @xmath15 where @xmath16 is the second term in the right - hand side of ( [ sing ] ) . the t - matrix of the one - pion - exchange plus the dirac - delta potential  @xcite is obtained here using distorted wave theory  @xcite as suggested in @xcite : @xmath17 t_\delta(e ) \left[1+g^{(+)}_0(e)t_{\pi}(e ) \right ] \ , \label{tpivw1}\end{aligned}\ ] ] where @xmath18^{-1}$ ] is the free green s function with @xmath19 in the two - nucleon rest - frame , and @xmath20 the green s function for the regular part of the pion - exchange interaction is @xmath21    the direct solution of ( [ tpivw2 ] ) is plagued by infinities in the momentum integration . in order to get finite solution within our scheme we reformulate ( [ tpivw2 ] ) in a subtracted fashion , where only one subtraction is enough  @xcite . let us briefly explain the subtraction method  @xcite discussing the treatment of potentials where the matrix elements behave as a constant in the ultraviolet momentum region , like in ( [ tpivw2 ] ) , which needs one subtraction to allow finite results for the t - matrix .    using the lippmann - schwinger equation @xmath22 = [ 1+t(e)g^{(+)}_0(e)]v   \ , \label{rtv1}\end{aligned}\ ] ] the potential @xmath23 can be formally written in terms of the t - matrix at a certain energy @xmath24 as @xmath25^{-1}t(-\mu^2 ) \ . \label{rv}\end{aligned}\ ] ] for convenience we use a negative energy for the subtraction such that the green s function is real . > from ( [ rv ] ) and ( [ rtv1 ] ) we obtain the one - subtracted scattering equation : @xmath26    by introducing a convenient notation , which is useful for more subtractions , the driving term of ( [ rt1v0 ] ) is denoted as @xmath27 . a subtracted free green function is written as @xmath28 which allows to rewrite the renormalized t - matrix equation  ( [ rt1v0 ] ) as @xmath29 for a dirac - delta potential , ( [ rt10 ] ) produces finite results once @xmath30 is given . in this simple case , the matrix elements of @xmath30 in the relative momentum representation is just the renormalized coupling constant . the method of subtracted equations is easily applied to ( [ tpivw2 ] ) , and one should note that the free green s function in ( [ rt10 ] ) is being replaced by the interacting one , @xmath31 . we obtain the t - matrix for an arbitrary energy @xmath32 in terms of the t - matrix at a given energy scale @xmath24 : @xmath33    the renormalized strength of the interaction defines @xmath34 at the subtraction point , @xmath35 , which is enough to get a finite amplitude for the opep plus a dirac - delta at this energy : @xmath36\lambda_{\mathcal { r}00 } \left[1+\frac{2}{\pi}\int^\infty_0dq'~{q'}^2 \frac{t_{\pi}(q',p;-\mu^2)}{-\mu^2-{q'}^2 } \right ] \ , \label{tpivw3}\end{aligned}\ ] ] where the t - matrix for the regular part of the one - pion - exchange potential in the @xmath0 channel is given by eq . ( [ tpifinal ] ) . the scattering t - matrix is not finite for the bare effective potential ( [ veft ] ) , which imply that regularization and renormalization are required to define the scattering amplitude . the method we use consists in constructing regularized and renormalized scattering equations with propagators subtracted at certain scales , which are convenient for introducing the physical inputs . our motivation in using the subtracted method was to treat a bare potential that includes a regular part plus the dirac - delta interaction and its derivatives , for which the t - matrix has a definite form for a certain @xmath37 . that given , no approximations are performed in getting the scattering amplitude . let us consider point - like interactions that include up to four terms , a contact interaction plus its derivatives . after partial - wave decomposition to the singlet @xmath38wave state , the bare potential is given by @xmath39 where @xmath40 corresponds to the regular part of opep plus a dirac - delta interaction @xmath41 .    in order to calculate the scattering amplitude for the bare potential ( [ vefts ] ) with subtracted equations , the strategy is the following . first one has to define the starting point of the iterative procedure to get the driving term of the subtracted equation . in the present case , at least three subtractions are required to allow a finite result for the the t - matrix , and two iterations are consequently needed to get the driving term . the iterative process begins with the t - matrix of ope plus dirac - delta potentials ( [ tpivw3 ] ) , leading to one subtraction in the kernel ( @xmath42 ) , as shown in section 2 . next , as we show in the following subsection , two iterations are performed and added the amplitude @xmath43 at the subtraction scale ( @xmath44 and @xmath45 are the renormalized strengths of the interaction ) . a three - fold subtracted t - matrix equation has to be used to define the scattering amplitude of the effective potential ( [ vefts ] ) , which in operator form and for a general number of subtractions @xmath46 reads  @xcite : @xmath47 where @xmath48^{-1}v^{(n-1)}(-\mu^2;e)\ ; , \label{vn } \\ g^{(+)}_n(e;-\mu^2)&\equiv&\left[(-\mu^2-e)g_0(-\mu^2)\right]^n g^{(+)}_0(e ) . \label{gn}\end{aligned}\ ] ] one should note that the form of the above equation with @xmath42 when applied to three - body systems results in the subtracted form used in @xcite , where the boundary condition was taken at the scattering threshold . the recursive formula to derive the driving term @xmath49 starts from @xmath50 given by ( [ tpivw3 ] ) . so far , only the opep and the dirac - delta interactions have been introduced in the calculation . the higher order derivatives of the dirac - delta potential in ( [ vefts ] ) are introduced in the driving term of the three - fold subtracted equation . the matrix element of @xmath51 for the full effective interaction of ( [ vefts ] ) in the angular momentum basis , @xmath52 get the contribution from the derivatives of the dirac - delta interaction , through the values of @xmath53 - the renormalized strengths of the corresponding terms in the potential . the integral equations for the matrix elements , @xmath54 , are @xmath55 with @xmath56 . the explicit form of the three - fold subtracted ls equation is : @xmath57 in the angular momentum basis the scattering amplitude is @xmath58 and the low - energy parameters are defined by the effective range expansion @xmath59 with @xmath60 the scattering length and @xmath61 the effective range . we observe that in our method the kernel is subtracted at a given scale which allows to perform the momentum integrals up to infinity , in that sense the subtracted equations are automatically regularized . in the cutoff regularization , the hilbert space is truncated and intermediate states goes up to a given momentum scale , such that the effective interaction includes the effect of the neglected states  @xcite . our method softens the contribution of the higher momentum intermediate states , with the partially removed physics parameterized through the driving term . of course we could move the subtraction scale , at the expense of a more complicated form for the driving term this fact also happens in a cut - off regularization when the cut - off is moved  @xcite . the nucleon - nucleon observables are invariant under the change of the arbitrary subtraction point , therefore one can start at any convenient energy scale @xmath62 . however , the form of the driving term and its coefficients in ( [ vrenfinal ] ) which define the scattering amplitude are tied to the prescription used to define the renormalized theory . the key point of the renormalization group method is to change this prescription without altering the predictions of the theory  @xcite . the invariance of the t - matrix under changes of renormalization prescriptions imposes a definite rule to modify @xmath63 in ( [ tren ] ) , which appears in a form of a non - relativistic callan - symanzik ( nrcs ) equation  @xcite : @xmath64 derived from ( [ tren ] ) and @xmath65 . equation  ( [ ss3 ] ) substantiate the invariance of the renormalized t - matrix under dislocation of the subtraction point , with the boundary condition given by ( [ vrenfinal ] ) . then , we observe that there is a non - trivial dependence on the subtraction point appearing in the driving term of the subtracted scattering equation , although the physical results of the model are kept unchanged . the solution of ( [ ss3 ] ) implies in a complicated evolution of @xmath63 as @xmath66 changes . not only the lambda s would change , but also the form of the driving term . the ultraviolet behavior of the driving term is not changed by the evolution in @xmath66 . the evolution should not be truncated as @xmath66 is varied to keep the t - matrix invariant . at different @xmath66 the potential @xmath67 has a complicate form from the solution of nrcs equation . similarly , the evolution of renormalization group equations as introduced by bogner , kuo and schwenk  @xcite for the nn scattering does not truncate on certain operators as the cutoff is varied to keep observables unchanged . the renormalized strengths @xmath68 , @xmath69 , @xmath45 and the subtracion point @xmath66 are found by fitting some low - energy phase shift data . for each set of @xmath44 , @xmath45 and @xmath66 , we fit the singlet scattering length @xmath70  fm through the value of @xmath68 . the three parameters left are adjusted to reproduce the nijgemen data  @xcite up to the center of mass momentum @xmath13 of about 300  mev / c . we get an effective range of @xmath71  fm compared with the value of 2.68  fm from ref . the parameters found are displayed in table  [ tab1 ] , where @xmath72  mev / c corresponds to a negative subtraction energy point @xmath73 - 50 mev . we emphasize that the results for the scattering length shown in the table correspond to the values obtained considering only the parameters found from the best fit of the singlet phase - shift using eq.([vrenfinal ] ) as input for the three - fold subtracted scattering equation ( [ trenfinal ] ) . of course , from pionless eft , we know that one could adjust the scattering length and the singlet phase - shifts at very low - energies with only contact interactions . first , we show in figure  [ fig1 ] the results of our method for only @xmath74 for the singlet and triplet channels using the one - fold subtracted t - matrix equations as performed in detail in ref . @xcite , where we have used @xmath75 it is worth to mention that , for @xmath76 , @xmath77 . in the present framework , we are using directly as input @xmath78 ( see eqs . ( [ tpivw3 ] ) and ( [ vrenfinal ] ) ) . the large values of @xmath66 in fig . [ fig1 ] arise due to the choice given by eq . ( [ v1pd ] ) . we observe that the large @xmath66 limit converges . for the singlet channel , @xmath40 is not enough to describe the data . however , the approach works reasonably well in the spin - triplet channel when we consider eq . ( [ v1pd ] ) for the coupled channels with @xmath79 and @xmath80 . therefore , we will not consider the spin - triplet channel further in the present work where we focus on the huge discrepancy of the singlet channel . we leave a detailed analysis of the coupled channels for a future work , where we also intend to include the two - pion - exchange interaction . in figure  [ fig2 ] we present the results obtained when we include @xmath84 . in order to evaluate the contributions of the different terms in the effective potential for the parameters of the best fit displayed in table i , we calculate the @xmath0 phase shifts for the individual contributions of @xmath85 , @xmath41 and @xmath84 , as well as for the combinations @xmath86 , @xmath40 and @xmath87 . below @xmath88  mev / c the different calculations presents results that underestimate the data due to the fact that @xmath89 23.7  fm , especially for @xmath85 and @xmath40 , while for @xmath84 and @xmath86 @xmath90 is better approximated ( see table  [ tab1 ] ) . in the last two cases , as the short - range part of the effective interaction is dominated by the higher - order derivatives of the dirac - delta , the calculation is about enough to describe the phase shifts above @xmath91  mev / c . therefore , the most relevant contribution in this channel comes from @xmath86 , while the regular part of the pion exchange potential appears at low energies as it provides the long range part of the potential . although our approach is rg invariant , it is not simple to explicitly incorporate rg invariance in the driving term of the three - fold subtracted equations . in order to have an insight of the effects included by the evolution of the driving term through the cs equation , it is important to study how our fit depends on @xmath66 . for this purpose , we calculate the phase shifts for values of @xmath66 around the best - fit value ( @xmath92 ) keeping the @xmath4 s intact , as shown in fig . we observe that , at very low momenta the fit is not affected . for higher momenta , the phase shifts present a monotonic behaviour : it increases ( decreases ) as @xmath66 gets larger ( smaller ) . the use of the nrcs equation ( [ ss3 ] ) would cancel this effect in order to keep the t - matrix , and thus the observables , invariant .      in conclusion , we show in this letter how to apply the subtracted scattering equations  @xcite to the nn singlet channel , when more than one subtraction is required to renormalize the corresponding interaction . the approach consists of a regularization of the integrand of the original scattering equation by using a given subtraction procedure in the propagator at some energy scale with where the t - matrix is known . the subtraction scale can be moved without modifying the calculated observables once the inhomogeneous term of the subtracted scattering equation runs with the scale according to the non - relativistic callan - symanzik ( nrcs ) equation . in the present case , the boundary condition is determined by the renormalized coupling constants and by the t - matrix of the ope and dirac - delta potentials at the specified value of the subtraction point . as a final remark , the subtracted scattering equation method is recursive and can be easily extended when higher order derivatives of the dirac - delta are present in the two - body effective interaction . * acknowledgements * + this work was partially supported by fundao de amparo  pesquisa do estado de so paulo ( fapesp ) , conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) and fundao de amparo  pesquisa do estado do rio de janeiro ( faperj ) . v.s.t thanks to faepex / unicamp for financial support . m. t. yamashita , t. frederico , a. delfino , and l. tomio , phys . a * 66 * ( 2002 ) 052702 ; ibidem , phys . rev . a * 68 * ( 2003 ) 033406 ; f. bringas , m. t. yamashita , and t. frederico , phys . a * 69 * ( 2004 ) 040702 .
the subtracted kernel approach is shown to be a powerful method to be implemented recursively in scattering equations with regular plus point - like interactions . 
 the advantages of the method allows one to recursively renormalize the potentials , with higher derivatives of the dirac - delta , improving previous results . 
 the applicability of the method is verified in the calculation of the @xmath0 nucleon - nucleon phase - shifts , when considering a potential with one - pion - exchange plus a contact interaction and its derivatives . 
 the @xmath0 renormalization parameters are fitted to the data . 
 the method can in principle be extended to any derivative order of the contact interaction , to higher partial waves and to coupled channels . 
 pacs 03.65.ca , 11.10.hi , 05.10.cc , 03.65.nk    renormalization , renormalization group , nonrelativistic scattering theory