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this work was done together with georges ripka and wojciech broniowski . solitons corresponding to baryons have been found in several chiral quark models . many of these solutions turn out to be unstable against collapse unless additional constraints are introduced in the model . the well known examples are the linear njl model with proper time regularization @xcite and the linear @xmath0-model with sea quarks @xcite . even in the linear @xmath0-model with only valence quarks the energy of the soliton becomes too low for any choice of model parameters if one goes beyond the mean field approximation . in all these models the instability occurs because it is energetically favorable for the chiral field to acquire arbitrary ( or very ) high gradients . this suggests that cutting off high momenta in the interaction may prevent the collapse and stabilize the soliton . a simple sharp cut - off does not yield a stable solution while a smooth behavior of the regulator ( usually interpreted as a @xmath1-dependent quark mass ) can indeed lead to solitons which are stable against the decay into free quarks as well as against collapse . such a regularization has a physical justification in qcd calculations of the quark propagation in an instanton liquid which predict a non - local effective interaction between quarks with a 4-momentum cut - off @xmath2  mev @xcite . further physical implications of the non - local regularization are discussed in the contributions to this workshop by george ripka and wojciech broniowski  @xcite . the non - local regularization of the quark - quark interaction can be implemented in the njl type models by replacing the contact term @xmath3 , @xmath4 by a non - local form . usually one introduces a regulator @xmath5 diagonal in 4-momentum space such that @xmath6 . the qcd derivation of the quark propagation in a dilute instanton gas predicts the following functional dependence for @xmath7 @xcite : @xmath8 where @xmath9 is the instanton size of the order @xmath10 . as we shall see in the following it is necessary to analytically continue the regulator to negative @xmath11 in order to be able to treat the valence orbit . this is not possible with the form ( [ instanton ] ) since it has a cut along the negative real axis starting at @xmath12 . we use instead a gaussian shape of the regulator : @xmath13 or a `` monopole '' shape : @xmath14 which has the proper behavior for large @xmath11 where one gluon exchange dominates . the expression for the energy of the soliton and the self - consistency equations can be derived from the bosonized euclidean action @xmath15 where @xmath16 and @xmath17 are the chiral fields and are the dynamical variables of the system . the main difficulty is the presence of time in the regulator . in order to evaluate the trace in ( [ action ] ) it is convenient to introduce energy dependent basis states , which are solutions of the dirac equation : @xmath18 with @xmath19    from ( [ action ] ) the following expression for a stationary configuration can be derived @xcite : @xmath20 note that when no regulator ( i.e. @xmath21 ) or a time - independent regulator is used , the energies @xmath22 are independent of @xmath23 and the integration can be carried out using the cauchy theorem . closing the contour from below yields the well known expression for the energy of the dirac sea : @xmath24 . ( note that the energies of occupied orbits lie on the negative imaginary @xmath23-axis . ) when the soliton describes a baryon , the energy of three valence quarks is added to the energy of the dirac sea . the same result can be formally obtained by deforming the contour in ( [ energy ] ) in such a way as to encircle the valence orbit ( for detailed discussion on this point see wojciech broniowski contribution to this workshop ) . such a prescription gives the expected result provided the orbits do not depend on @xmath23 . however , when the regulator depends on time ( or @xmath23 ) , this may not lead to the correct result since the regulator generates additional poles scattered in the whole complex @xmath23-plane . it may still work well for an isolated pole on the positive imaginary axis close to 0 as is the case of the @xmath25 orbit in the soliton with the hedgehog form of the background chiral field @xcite . this pole can then be treated separately , yielding the valence contribution to the soliton energy @xmath26 , where the energy of the valence orbit is determined from @xmath27 the soliton energy can now be written as : @xmath28 the sea contribution is @xmath29\ ] ] with @xmath30 and is evaluated by direct numerical integration along the real @xmath23-axis . the term @xmath31 is given by the last integral in ( [ energy ] ) ( with the integrand @xmath32 ) . the above prescription is further supported by the fact that it gives an exact result for the baryon number , which can be expressed as @xcite : @xmath33    the self - consistent equations derived from ( [ action ] ) take the form ( the hedgehog ansatz , @xmath34 , for the pion field is assumed ) : @xmath35\ ; , \label{sceq}\end{aligned}\ ] ] where @xmath36 and @xmath37 is the residue of the valence pole . a necessary condition for a stable soliton configuration is that the energy ( [ energy ] ) is lower than the energy of three free quarks . when the regulator depends on time , the free quark mass , @xmath38 , is not simply the vacuum value of the chiral field , @xmath39 , but is determined by the position of the pole of the quark propagator in the vacuum @xcite , _ i.e. _ it corresponds to the solution of @xmath40 . the solution for real @xmath11 exists only below a critical value of @xmath39 ( see figure  [ figep ] ) ; above this point no stable free quarks exist . however , a stable solution can always be found beyond this point provided the quarks dress in a spatially non - uniform background chiral field .    ' '' ''    height 0pt depth 0pt the model ( [ action ] ) possesses 4 parameters : the vacuum value of the chiral field @xmath39 , the current quark mass @xmath41 , the coupling constant @xmath42 and the cut - off parameter @xmath43 . the coupling constant @xmath42 is fixed from the stationarity condition in the vacuum , @xmath43 is fitted to the pion decay constant @xmath44 , and @xmath41 to @xmath45 . we are left with one free parameter @xmath39 .    in this section we analyze how different shapes of the regulator affect the result . we compare the gaussian ( [ gaussian ] ) , the monopole ( [ pole ] ) and a modified version of the instanton ( [ instanton ] ) regulator . as we have mentioned it is not possible to continue ( [ instanton ] ) to negative @xmath11 , which is needed in order to obtain the valence contribution . we have instead introduced a form which is identical to the instanton regulator for @xmath46 while its behavior for @xmath47 is replaced by a real function which approximately follows the real part of ( [ instanton ] ) for small negative @xmath11 ( `` extended instanton '' ) . the gaussian and the monopole shapes lead to practically identical results . we have also tried other shapes and found very similar results . the reason is that several properties ( including the integral that determines @xmath44 ) depend mostly on the behavior of the regulator for small value of @xmath11 . if this behavior is @xmath48 then @xmath49 is almost uniquely determined by the value of @xmath44 . hence , all shapes with this type of behavior lead to very similar results . the situation is quite different if @xmath50 , _ e.g. _ for regulators that depend only on @xmath51 . we do not find stable solutions for such regulators ; the energy of the dirac sea is always higher than the gain due to the lowering of the valence energy . the form ( [ instanton ] ) has neither of the above behaviors for @xmath52 but behaves as @xmath53 . it is therefore interesting to check whether stable solitons can also be obtained for such a particular form . we indeed find solitons with very similar properties to those obtained using ( [ gaussian ] ) or ( [ pole ] ) .    since @xmath44 sets the scale , it is also interesting to study how a higher value for @xmath44 would affect the results . .properties of the self - consistent soliton solutions for different shapes of the regulator , `` extanton '' stands for the extended instanton regulator with @xmath54 , gaussian@xmath55 means a gaussian regulator with @xmath43 fitted to @xmath56  mev , @xmath57 is the one - flavor quark condensate . [ cols="<,^,^,^,^,^,^,^,^,^ " , ]     ' '' ''    height 0pt depth 0pt    ' '' ''    height 0pt depth 0pt    a )    ' '' ''    height 0pt depth 0pt    ' '' ''    height 0pt depth 0pt    b )    [ com ] the calculation can be made much simpler and the problem of analytic continuation to negative @xmath11 avoided if we take a regulator that does not depend on time . then the states do not depend on @xmath23 and the integration over @xmath23 in the above expressions can be carried out analytically . all quantities still remain finite and the soliton remains stable against collapse . the parameters used are the same as for the 4-momentum regulator . however , the explicit breaking of lorentz invariance leads to serious problems which we discuss in this section . the energy of the soliton reduces to @xmath58 and the self - consistency equation to @xmath59\;. \nonumber\end{aligned}\ ] ]    the simplest form of the regulator is a sharp cut - off limiting the 3-momenta to @xmath60 . since one usually uses basis states with good @xmath61 , this is equivalent to restricting the hilbert space . such a form is very easy to implement since the regulator does not appear explicitly in the calculation . unfortunately , no stable soliton solution is found in this case ; the contribution of the sea quarks is always larger than the gain due to the lowering of the valence energy . stable solutions can be obtained if one takes a smooth form of the regulator , e.g. a gaussian form @xmath62 . the results are displayed in figure  [ fige3d ] . the threshold value of @xmath39 below which the soliton does not exist is somewhat larger than for the 4-momentum gaussian regulator . one notices a rather striking feature that the soliton does not exist _ beyond _ a certain value of @xmath39 . the reason for such behavior is the following : the energy of a free quark can be written as @xmath63 . for sufficiently small @xmath64 the minimum of @xmath65 is not at @xmath66 but rather at some @xmath67 , and the energy of a free quark becomes smaller than @xmath39 . increasing @xmath39 the cut - off @xmath43 decreases and at a certain value it becomes energetically favorable for the quarks in the soliton to acquire sufficiently high momenta and leave the soliton . this happens when @xmath68 . such an unphysical behavior is a clear consequence of breaking the lorentz invariance in the interaction . the value of @xmath69 can be easily determined if we choose a gaussian regulator and @xmath70 . for the values of @xmath43 below @xmath71 the minimum of @xmath72 occurs for @xmath73 and the free quark energy at the minimum is : @xmath74 . for @xmath75 this value is slightly modified : @xmath76 ^ 2\;.\ ] ]    figure  [ fige3d ] shows that the soliton exists for @xmath77  mev . at @xmath78 mev the cut - off @xmath43 reaches the critical value and the lowest free quark state has @xmath79 . nonetheless , the soliton remains bound since @xmath80 . at @xmath81 mev the condition is no longer fulfilled and no bound solution exists beyond this point . between these two values the energy of the soliton differs only little from the energy of three free quarks . the soliton is weakly bound ; the chiral field stays close to its vacuum value while the soliton radius is large .    ' '' ''    height 0pt depth 0pt it is well known that the njl model can be transformed into the form of an equivalent linear @xmath0-model . this transformation is explained in @xcite , chapter 5 . if one assumes a sharp cut - off and sufficiently large @xmath43 ( compared to @xmath39 ) the lagrangian density of the equivalent @xmath0-model takes the familiar form @xmath82 the fields @xmath0 and @xmath83 are related to the two components of the chiral field of the bosonized njl model as @xmath84 , @xmath85 , where @xmath86 . the parameter @xmath87 and the mass of the @xmath0 meson are related to the parameters of the njl model by here is a factor 2 smaller than the one in @xcite . ] @xmath88 , @xmath89 and @xmath90 the question remains whether the above assumptions are met in the model described in section 2 . is close to 1 while the smooth form of the regulator may and does generate additional terms in the lagrangian ( [ elsm - fam ] ) and modifies the values of the parameters . a work that will take into account the additional terms is in progress . the conclusions in this section remain valid but the results should be considered only as qualitative . ] we assume that the lagrangian ( [ elsm - fam ] ) describes the sea quarks while the valence quarks are treated separately . this is in the spirit of the approaches used in @xcite . however , in all these approaches the valence orbit is not regularized and  as we shall see in this section  this brings a qualitative difference with respect to the situation where it is regularized . if we agree that a regulator in the quark - quark or , equivalently , quark - chiral field interaction has a well grounded physical origin , the valence orbit should be regularized in the same way as the orbits in the dirac sea . the valence orbit is determined as in ( [ eval ] ) . the soliton energy then becomes ( @xmath91 ) : @xmath92    + g \left [ \sigma j^\sigma + \pi_a j_a^\pi \right ]                                                        \right )    + e_{s.i.}\;,\ ] ] where the meson self - interaction term ( the `` mexican hat '' ) is given by @xmath93 here we have introduced the source terms which explicitly contain the regulator ( see ( [ sceq ] ) ) : @xmath94 the mean - field solution is obtained by solving the self - consistency equations ( the hedgehog ansatz @xmath95 and @xmath96 is assumed ) : @xmath97\ ; , \nonumber\\ & & \left({\d^2\over\d r^2 } + { 2\over r}\,{\d\over\d r } - { 2\over r^2 } - m_\pi^2\right)\pi(r )    = j^\pi_a(\vec{r})\hat{r}_a +    \lambda^2\left [    { \sigma'(r)}^2 + 2f_\pi{\sigma'(r ) } + \pi(r)^2\right]\pi(r)\;.       \label{sigmaeq}\end{aligned}\ ] ]    in figures  [ figelsm ] and [ mflsm ] we compare the properties of the soliton when the valence orbit is regularized and when it is not . close to the threshold the two solutions do not differ much  as it should be  since here the cut - off @xmath43 is relatively large . for higher values of @xmath39 ( or @xmath98 ) the behavior is qualitatively quite different ; in particular , the energy of the unregularized valence orbit soon becomes negative which never happens for the regularized one . let us finally mention why the regularization is important also in the linear @xmath0-model with only valence quarks . even if there is no instability as in the regularized linear @xmath0-model - propagator @xcite . ] , one encounters serious difficulties when going beyond the mean field approximation . in the @xmath0-model it is possible to use the peierls - yoccoz projection of good linear momentum , spin and isospin in order to obtain physical nucleon states . it turns out that the energy of the soliton after projection is strongly reduced already when the mean - field solution is used for the chiral fields @xcite . if in addition one allows a variation of the chiral field profiles , one obtains a solution with an energy considerably lower than the nucleon mass . the energy gain is mostly due to a strongly localized chiral field which lowers the valence energy . such a strong localization is not allowed when the regulator is used since it puts a physical limit on the gradients of the field ( through the source term ( [ source ] ) ) .
we describe how the non - local regularization can be implemented in the calculation of solitons in the nambu jona - lasinio model as well as in the equivalent linear @xmath0-model . 
 we investigate different forms of regulators and show that the 3-momentum cut - off leads to serious conceptual difficulties .