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the massless gross - neveu ( gn ) model @xcite is the 1 + 1 dimensional quantum field theory of @xmath0 flavors of massless dirac fermions , interacting through a scalar - scalar contact interaction . suppressing flavor labels as usual , its lagrangian reads @xmath1 the physics phenomena inherent in this simple looking lagrangian are particularly rich and accessible in the t  hooft limit ( @xmath2 const . ) , to which we restrict ourselves from here on . the gn model can be thought of as relativistic version of particles moving along a line and interacting via an attractive @xmath3-potential . however , it exhibits many non - trivial features characteristic for relativistic quantum fields such as covariance , renormalizability , asymptotic freedom , dimensional transmutation , spontaneous symmetry breaking , interacting dirac sea . it is also one of the few models known where most of the non - perturbative questions of interest to strong interaction physics can be answered in closed analytical form . such calculations have turned out to be both challenging and instructive , generating a continued interest in this particular  toy model " over several decades , see e.g. the review articles @xcite .    in the present paper we address the problem of time - dependent scattering of multi - fermion bound states in full generality . as will be recalled in more detail in the next section , the gn model possesses bound states which can be viewed as  baryons " , with fermions bound in a dynamically created  bag " of the scalar field @xmath4 @xcite . there are even multi - baryon bound states which might be identified with nuclei " @xcite . standard large @xmath0 arguments tell us that all of these bound states can be described adequately within a relativistic version of the hartree - fock ( hf ) approach .    turning to the baryon - baryon scattering problem , the tool of choice is the time - dependent version of hartree - fock ( tdhf ) , as originally suggested by witten @xcite . the basic equations in that case are easy to state , @xmath5 but hard to solve , even in 1 + 1 dimensions . one of the reasons is the fact that the sum over occupied states includes the dirac sea , so that one is dealing with an infinite set of coupled , non - linear partial differential equations . no systematic , analytical method for solving such a complicated problem is known . nevertheless , the exact solution for the time - dependent scattering problem of two baryons has recently been found in closed analytical form by means of a joint ansatz for @xmath6 and @xmath7 @xcite . it provides us with a microscopic solution of the scattering of two composite , relativistic objects , exact in the large @xmath0 limit . the necessary details will be briefly summarized below . this result encourages us to go on and try to solve more ambitious scattering problems involving any number of bound states , including  nuclei " in addition to the  nucleons " considered so far . the paper is organized as follows . in sec . [ sect2 ] we briefly summarize what is known about multi - fermion bound states and their interactions in the gn model . we also remind the reader how the baryon - baryon scattering problem has been solved recently , since we shall use the same strategy in the present work . [ sect3 ] is devoted to the dirac equation and the ansatz for scalar potential and continuum spinors . [ sect4 ] and [ sect5 ] contain the central results of this work , namely the coefficients entering the ansatz , presented in the form of an algorithm . in sec . [ sect6 ] , we explain the extent to which the general result has been checked so far . [ sect7 ] deals with the bound state spinors which are then used in sec . [ sect8 ] to discuss the issue of self - consistency and the fermion density . [ sect9 ] addresses scattering observables like time delays or deformations of bound states . sec . [ sect10 ] contains a few illustrative examples , followed by a short summary and outlook in sec . [ sect11 ] . to put this study into perspective , we summarize what is known about multi - fermion bound states and their mutual interactions in the massless gn model , eq .  ( [ 1.1 ] ) . static multi - fermion bound states have been derived systematically with the help of inverse scattering theory and resolvent methods @xcite . the best known examples are the callan - coleman - gross - zee kink ( cited in @xcite ) and the dashen - hasslacher - neveu ( dhn ) baryon @xcite , both of which can accommodate up to @xmath0 fermions . the kink is topologically non - trivial , reflecting the @xmath8 chiral symmetry of the massless gn model . its shape ( shown in fig . [ fig1 ] ) and mass are independent of its fermion content . the dhn baryon is topologically trivial and stabilized by the bound fermions which affect its shape and mass , as illustrated in fig . [ fig2 ] . multi - baryon bound states have been constructed systematically by feinberg @xcite . they possess continuous parameters related to the position of the baryon constituents on which the mass of the bound state does not depend (  moduli " ) . they may be topologically trivial like the dhn baryon or non - trivial like the kink , depending on the ( spatial ) asymptotic behavior of @xmath6 . some examples are shown in figs . [ fig3 ] and [ fig4 ] . a common feature of all static solutions is the fact that the scalar potential is transparent , i.e. , the fermion reflection coefficient vanishes for all energies . consequently the self - consistent , static solutions of the gn model coincide with the transparent scalar potentials of the dirac equation , investigated independently by nogami and coworkers @xcite . since the static dirac equation can be mapped onto a pair of ( supersymmetric ) schrdinger equations , this also yields a bridge between static , self - consistent dirac - hf solutions on the one hand and transparent potentials of the schrdinger equation on the other hand , a problem solved long ago by kay and moses @xcite . the non - relativistic limit of the topologically trivial , static gn solutions are well - known multi - soliton solutions of coupled non - linear schrdinger ( nls ) equations , arising in the hartree approximation to particles in 1d with attractive @xmath3-interactions @xcite .    by boosting any static solution , one can trivially generate solutions of the tdhf equation @xcite . this kind of solution enters in the asymptotic states of the scattering problem which we are going to study . the breather is a time - dependent , oscillating solution of kink - antikink type . it was found by dhn , using the analogy with the sine - gordon breather @xcite . since it is neither a conventional bound state nor a scattering state , it has no analogue in real particle physics , but is reminiscent of collective , vibrational excitations of heavy nuclei or molecules . this underlines the classical character of the large @xmath0 limit . we shall not consider scattering of breathers in the present work . following a suggestion in ref . @xcite , kink - antikink scattering was solved in tdhf by analytic continuation of the breather @xcite . since the fermions do not react back , it is possible to map this problem rigorously onto the problem of kink - antikink scattering in sinh - gordon theory . if we set @xmath9 , then @xmath10 satisfies the classical sinh - gordon equation @xmath11 ( in natural units ) , as first noticed by neveu and papanicolaou @xcite . this mapping can be generalized . the known multi - soliton solutions of the sinh - gordon equation yield the self - consistent scalar potential for scattering of any number of kinks and antikinks @xcite . a poor man s simulation of nuclear interactions was the scattering of  trains " of solitons moving with almost the same speed in ref . @xcite ( there are no multi - soliton bound states ) . kink dynamics has no non - relativistic analogue since the internal structure of kink is ultrarelativistic , as evidenced by a zero - energy bound state . time - dependent kink - antikink scattering is illustrated in fig . [ fig5 ] . a crucial ingredient in proving the correspondence between kink dynamics and sinh - gordon solitons is the fact that kink solutions satisfy the self - consistency mode - by - mode . they are of  type i " in the classification of @xcite , i.e. , @xmath12 with constant @xmath13 for every single particle state @xmath14 . this is also the basis for an interesting geometrical interpretation of tdhf solutions , relating time - dependent solutions of the gn model to the embedding of surfaces of constant mean curvature into 3d spaces @xcite . scattering of dhn baryons is significantly more involved than kink - antikink scattering . presumably because the fermions react back , it does not seem possible to map this problem onto any known soliton equation . the exact tdhf solution for baryon - baryon scattering was found recently in a different way , namely by ansatz @xcite . a specific example is illustrated in fig . [ fig6 ] . since we shall follow the same strategy in the present paper , we briefly recall the main ideas behind the ansatz , referring the reader to ref . @xcite for technical details . the ansatz can best be described as follows . we start from the scalar mean field of a single ( boosted ) dhn baryon with label @xmath15 . it can be cast into the form of a rational function of an exponential @xmath16 , @xmath17 here , @xmath18 is a parameter governing the size of the baryon and related to its fermion number @xmath19 via @xmath20 @xmath21 denotes the baryon velocity , @xmath22 , and @xmath23 is an arbitrary real factor expressing the freedom of choosing the initial baryon position . the dirac components of the continuum spinor have the same rational form with different coefficients in the numerator only and an additional plane wave factor , @xmath24 the asymptotic behavior at fixed @xmath25 is @xmath26 for @xmath27 , showing that the potential is transparent . in order to solve the scattering problem for baryons @xmath15 and @xmath28 , we start by multiplying @xmath29 and @xmath30 and expand the numerator and denominator , @xmath31 this may be viewed as scalar potential for non - interacting baryons . the ansatz for interacting baryons proposed in @xcite now consists in assuming that the only effect of the interaction is to change the coefficients in the numerator and denominator of ( [ 2.5 ] ) , keeping the polynomial dependence on @xmath32 the same . likewise , the ansatz for the spinor is obtained by multiplying the rational factors of @xmath33 for baryons @xmath15 and @xmath28 and allowing for changes in the coefficients only . the overall exponential factor is kept unchanged , since it is expected that the potential is reflectionless also in the interacting case . it turns out that most of the coefficients in @xmath6 and @xmath33 are in fact determined by the asymptotic in- and out - states . only 4 coefficients remain to be determined , namely the factors in front of the monomials @xmath34 in the three numerators and the common denominator . inserting this ansatz into the dirac equation determines the missing coefficients and confirms that this simple idea yields the exact solution of the 2-baryon problem . so far , we have discussed only the fermion continuum states . bound states can be obtained by analytic continuation in a spectral parameter ( a function of @xmath35 ) and subsequent normalization . self - consistency can then be checked explicitly , confirming that the ansatz solves the tdhf problem . the solution is found to be of type iii , i.e. , the scalar density of any single particle orbit can be expressed as a linear combination of 3 distinct functions of ( @xmath36 ) . we have no a priori argument why the ansatz should be successful , but its simple form is most certainly a large-@xmath0 manifestation of the quantum integrability of finite-@xmath0 gn models . the result for the non - trivial coefficients is rather complicated , but by a proper choice of variables and light cone coordinates , one manages to keep all coefficients in rational form . unlike in the kink - antikink case , the non - relativistic limit is now accessible , since the dhn baryon goes over into the soliton of the nls equation in the limit of small fermion number . starting from the two - baryon solution , one then recovers the time - dependent solutions of the multi - component nls equation of nogami and warke for @xmath37 @xcite . this completes the overview of the present state of the art . here we propose to extend the two - baryon tdhf scattering solution of ref . @xcite to an arbitrary number of composite colliding particles , including multi - baryon bound states (  nuclei " ) in addition to baryons . the central idea is to use an ansatz for the scalar potential inspired by the product of @xmath0 single baryon potentials , assuming that only the coefficients of the resulting rational function of @xmath38 will be affected by the interactions . a convenient choice of the dirac matrices in 1 + 1 dimensions is @xmath39 together with light cone coordinates @xmath40 this simplifies the dirac - tdhf equation to @xmath41 here , @xmath42 is the upper , left - handed , @xmath43 the lower , right - handed spinor component . we posit the following ansatz for the scalar tdhf potential , @xmath44 as motivated in the preceding section , @xmath6 is assumed to be a rational function of @xmath0 exponentials @xmath16 , where @xmath0 is the number of baryons , @xmath45 each summation index @xmath46 runs over the values 0,1,2 , and the coefficients @xmath47 are real . the basic exponential @xmath16 has the form inferred from the single dhn baryon in flight , @xmath48 the parameter @xmath18 specifies the size ( or , equivalently , fermion number ) of the @xmath15-th baryon . @xmath49 is related to the baryon rapidity @xmath50 and velocity @xmath21 via @xmath51 for @xmath18 we shall use the parametrization @xmath52 to avoid the appearance of square roots . apart from the @xmath53 parameters @xmath54 , the baryon constituents are characterized by @xmath0 arbitrary , real scale factors @xmath23 needed to specify their initial positions . the @xmath16 must be ordered according to baryon velocities . we choose the convention that @xmath55 if @xmath56 . we now turn to the ansatz for the continuum spinors , assuming from the outset that the tdhf potential is reflectionless , @xmath57 here , @xmath58 denotes the light cone spectral parameter related to ordinary momentum and energy via @xmath59 @xmath60 are multivariate polynomials in the @xmath16 of the same degree as @xmath61 , @xmath62 but now with complex coefficients @xmath63 . in eq . ( [ 3.9 ] ) we have factored out the free dirac spinor @xmath64 to ensure that all polynomials start with a  1 " . the denominator @xmath65 in the spinor , eq . ( [ 3.9 ] ) , is assumed to be the same as in the scalar potential , eq . ( [ 3.4 ] ) . inserting this ansatz into the dirac equation ( [ 3.3 ] ) yields @xmath66 actually , we can eliminate the variable @xmath58 by rescaling @xmath67 via @xmath68 . this transforms @xmath16 into @xmath69 the final form of the dirac equation can then be obtained by setting @xmath70 in eq . ( [ 3.13 ] ) , @xmath71    the numerator and denominator functions ( @xmath72 ) are polynomials in the @xmath16 . since the @xmath16 are eigenfunctions of @xmath73 , the dirac equation ( [ 3.15 ] ) gets converted into the condition that 2 polynomials vanish identically . thus each coefficient of the monomials @xmath74 must vanish separately . the number of terms in each of the polynomials , eqs . ( [ 3.5 ] ) and ( [ 3.11 ] ) , is @xmath75 for @xmath0 baryons , as @xmath16 can appear with powers 0,1,2 . in the final dirac equation , @xmath16 appears with powers @xmath76 , so that eq . ( [ 3.15 ] ) is altogether equivalent to @xmath77 algebraic equations for the coefficients @xmath78 of our ansatz . in this and the following section , we present our results for the coefficients entering the scalar potential and the continuum spinors for @xmath0 baryons , i.e. , the coefficients of the polynomials @xmath72 introduced above . they fall naturally into 2 classes : reducible " coefficients which can be related to the @xmath79 baryon problem , and  irreducible " ones which can not . the reducible coefficients are the subject of this section , the irreducible ones will be discussed in the next section . there are two distinct ways of reducing the @xmath0 baryon problem to the @xmath79 baryon problem , either by letting @xmath80 or by letting @xmath81 .    in both cases , @xmath82 drops out of the expressions for @xmath6 and @xmath83 . since this can be done for any label @xmath84 , one gets a large number of recursion relations . as explained in greater detail in ref . @xcite , one has to take into account time delays and ( in the case of the spinors ) transmission amplitudes for final states , depending on whether the eliminated baryon @xmath84 has been scattered from the remaining @xmath79 baryons or not . let us consider the scalar potential first . starting point are the following basic relations , @xmath85 @xmath82 is missing on the right hand side , which therefore refers to @xmath79 baryons . the @xmath86 are ( real ) time delay factors satisfying @xcite @xmath87 it is important to keep track of the ordering of the baryon labels ( @xmath55 if @xmath56 ) when applying these formulas . relations ( [ 4.1 ] ) imply the following recursion relations for the coefficients in ( [ 3.5 ] ) , @xmath88 we use the convention that barred indices have to be omitted . the factors @xmath89 appear here because relations ( [ 4.1 ] ) determine only the ratio @xmath90 . they can be fixed as follows . we normalize the lowest and highest coefficients of @xmath61 to 1 for any number of baryons , @xmath91 this is always possible since we must recover the vacuum potential @xmath92 in the limit where all @xmath16 go to 0 or @xmath93 , and the @xmath16 contain arbitrary scale factors @xmath23 , see eq . ( [ 3.6 ] ) . specializing relations ( [ 4.3 ] ) to the cases where all indices are 0 or all indices are 2 and using eq . ( [ 4.2 ] ) , we then find @xmath94 this yields the following final recursion relations for the coefficients entering @xmath6 , @xmath95 they determine all @xmath0-baryon coefficients containing at least one 0 or one 2 in their subscripts in terms of @xmath96-baryon coefficients , leaving only the two irreducible coefficients @xmath97 in front of @xmath98 undetermined . for the spinors , we have to take into account transmission amplitudes in addition to the time delay factors . consequently the general reduction formulas ( [ 4.1 ] ) have to be replaced by @xmath99 where @xmath100 is the transmission amplitude of baryon @xmath84 @xcite @xmath101 it is unitary ( @xmath102 ) due to the reflectionless potential . using a normalization analogous to ( [ 4.4 ] ) , i.e. , @xmath103 we arrive at the recursion relations @xmath104 for the coefficients in @xmath105 . once again this leaves only the two irreducible coefficients @xmath106 of @xmath98 undetermined . altogether , there are @xmath107 coefficients in the ansatz for @xmath6 and @xmath83 for @xmath0 baryons . all but the 4 irreducible ones are determined by normalization and recursion relations .    the first step towards solving the @xmath0 baryon problem is to eliminate all reducible coefficients , expressing the 4 polynomials in terms of irreducible coefficients , time delay factors and transmission amplitudes only . the above recursion scheme enables us to do just this . the result can most conveniently be cast into the form of an algorithm . we first formulate the algorithm and subsequently illustrate it with the explicit results for @xmath108 and point out its advantages . the algorithm will be stated separately for the 4 polynomials @xmath109 .    1 .   denominator @xmath65 of @xmath6 1 . write down the product @xmath110 and expand it . 2 .   if a term contains between 2 and @xmath0 factors @xmath111 , replace it by @xmath112 3 . substitute @xmath113 and expand again . 4 .   if any term contains @xmath114 ( @xmath115 ) , replace it by @xmath116 5 .   set @xmath117 2 . numerator @xmath118 of @xmath6 0.5 cm the numerator @xmath118 of @xmath6 can be obtained from the denominator @xmath65 of @xmath6 by replacing all @xmath119-coefficients by @xmath120-coefficients , @xmath121 3 . numerator @xmath122 of @xmath42 0.5 cm to get @xmath122 , start from @xmath65 and perform the following steps : 1 . replace @xmath123 where @xmath124 is the transmission amplitude of baryon @xmath15 . 2 .   replace all @xmath119-coefficients by @xmath125-coefficients , @xmath126 4 . numerator @xmath127 of @xmath43 0.5 cm to get @xmath127 , start from @xmath122 and replace all @xmath125-coefficients by @xmath128-coefficients , @xmath129    to avoid misunderstandings , we illustrate the outcome of the algorithm with a few explicit examples . for @xmath37 ( 9 terms ) , one finds @xmath130 these results are fully consistent with ref . @xcite . for @xmath131 ( 27 terms ) the algorithm yields @xmath132 @xmath133    inspection of these examples shows the following advantages of presenting results in the form of an algorithm . first , the recursion relations relate @xmath0-baryon coefficients to @xmath96-baryon coefficients , cf . ( [ 4.6],[4.9 ] ) . the algorithm gives directly the iterated result where everything is expressed in terms of irreducible coefficients for 1,2, ... ,@xmath0 baryons . secondly , the number of terms in the explicit expressions increases like @xmath75 , so that writing down the explicit expressions like in ( [ 4.19],[4.20 ] ) becomes quickly prohibitive . the algorithm on the other hand has been stated concisely for arbitrary @xmath0 . it can also easily be implemented in maple , so that it is never necessary to deal manually with lengthy expressions .    as a result of this section , we have reduced @xmath6 and @xmath83 to those coefficients @xmath78 whose subscripts contain only 1 s and which refer to 1,2, ... ,@xmath0 baryons with all permutations of labels . these irreducible coefficients have to be determined algebraically from the dirac equation ( [ 3.15 ] ) and are the subject of the following section . we denote those @xmath0-baryon coefficients of the polynomials @xmath72 which can not be determined recursively from the @xmath79 baryon problem as irreducible . as explained above , there are only 4 such coefficients for given @xmath0 , namely the coefficients of the monomials @xmath134 in each of the 4 polynomials , @xmath135 . they encode the dynamical information about the situation where all @xmath0 baryons overlap and have to be determined by means of the dirac equation . for reasons to be discussed later in more detail , this is a difficult task for computer algebra programs like maple , once the baryon number gets too large . we have therefore determined the irreducible coefficients for low baryon numbers analytically , analyzed their structure and extrapolated the formulas to arbitrary @xmath0 . in this section we present our conjectured results for the 4 irreducible coefficients and general @xmath0 . in the next section , we will describe in detail the extent to which these conjectured results have actually been checked so far .    given the complexity of the coefficients , it is once again easier for us to communicate our results in the form of an algorithm , rather than a closed expression . the algorithm is actually a very simple one . let us define a combinatorial expression @xmath136 through the following two steps :    1 . write down the product @xmath137 where @xmath138 is a @xmath139 matrix , and expand it . 2 .   for each of the @xmath140 terms in the sum and each index @xmath141 , denote by @xmath19 the number of indices @xmath15 appearing in this term ( @xmath142 ) . then , if @xmath143 is odd , multiply the term by @xmath144    by way of example , we write down the explicit result for @xmath37 ( 2 terms ) , @xmath145 and @xmath131 ( 8 terms ) , @xmath146 after this preparation , the irreducible coefficients can be expressed in compact form as follows , @xmath147 with @xmath148 all what remains to be done is to define exactly the various symbols appearing in ( [ 5.5],[5.6 ] ) . we divide them into two categories . the first category comprises those symbols which can be deduced from the single dhn baryon problem @xcite , @xmath149}{(z_i^2 + 1 ) ( \zeta_i - z_i)(\zeta_i z_i + 1 ) } , \nonumber \\ d_1^i & = &   \frac{2 [ 2 z_i^2 - \zeta_i^2(z_i^4 + 1)]}{(z_i^2 + 1)(\zeta_i - z_i)(\zeta_i z_i + 1)}. \label{5.7}\end{aligned}\ ] ] they enter in the prefactor of the combinatorial expression @xmath136 in eq . ( [ 5.5 ] ) and are the same as in eqs . ( [ 2.2],[2.3 ] ) , up to trivial normalization factors in @xmath150 and @xmath151 . the 2nd category consists of symbols which can be deduced from the two - baryon problem if one applies these formulas to @xmath37 and compares them with the results of ref . @xcite , @xmath152 we have used everywhere the spectral parameter @xmath153 boosted into the rest frame of baryon @xmath15 , introduced in eq . ( [ 3.14 ] ) . note however that @xmath153 could be replaced by @xmath49 in @xmath154 and @xmath138 , so that the @xmath58-dependence of these quantities is spurious . by using the variable @xmath155 rather than @xmath18 and @xmath153 rather than @xmath21 and @xmath84 , we have achieved that all the basic expressions are rational functions of the 2@xmath0 arguments ( @xmath156 ) . the same holds true for @xmath86 , eq . ( [ 4.2 ] ) , and @xmath100 , eq .  ( [ 4.7a ] ) .    a noteworthy property of this construction is the fact that the algorithm leading to @xmath136 is based on a factorization in terms of quantities @xmath138 referring to 2 baryons @xmath157 only , see eq . ( [ 5.1 ] ) . this implies that the solution of the two - baryon scattering problem is sufficient to determine completely @xmath0 baryon scattering . this observation is behind the phrase evidence for factorized scattering " in the title of this paper . it goes beyond the usual factorization of the fermion scattering matrix , which holds trivially in our case ( see sec . [ sect9 ] ) . it teaches us that even when all @xmath0 baryons overlap , there is nothing new going on as compared to having two overlapping baryons only . in this sense , factorization does not only hold for the on - shell scattering matrix , but also off - shell . in the preceding sections , we have provided rules for explicitly constructing the scalar potential @xmath6 and the continuum spinors @xmath83 for the @xmath0-baryon tdhf problem in the gn model . let us summarize where we stand . the main ingredients in @xmath6 and @xmath83 are 4 polynomials in @xmath0 exponentials @xmath16 , consisting of @xmath75 terms each . the coefficients in these polynomials can all be expressed through a set of irreducible coefficients multiplying @xmath158 in the @xmath159 baryon problem , time delay factors @xmath86 and fermion transmission amplitudes @xmath124 , using the algorithm of sec . [ sect4 ] . the irreducible coefficients in turn can be constructed starting from 1- and 2-baryon input only , using the algorithm of sec . [ sect5 ] . since the dirac equation reduces to a set of algebraic equations and all ingredients are known rational functions , one would not expect any particular difficulties in checking that the spinor satisfies the dirac equation , using computer algebra programs like maple . however , the complexity of the resulting expressions increases rapidly with increasing baryon number , quickly exceeding the capabilities of maple due to storage and computation time problems . thus , for @xmath37 and @xmath131 , we could still check all @xmath160 algebraic equations analytically with maple in a straightforward way . for @xmath161 or larger , the maximum size of expressions which maple can handle is exceeded and we have only been able to check our results numerically , for random values of the input parameters @xmath156 . this test has been carried out successfully for @xmath162 . by increasing the number of digits , one can find out whether the floating point result is exact or approximate . since the number of operations increases faster than exponentially with @xmath0 , it is actually necessary to run maple with very high accuracy for large @xmath0 values . thus for example , during a full @xmath163 calculation , 40 digits get lost , so that one has to start out with 50 digits precision to be sure that the dirac equation is solved exactly . clearly , there must be a way of proving our results in full generality . the complexity of the solution and the intricate way in which @xmath0 baryon scattering is related to the scattering problem of fewer baryons have prevented us so far from finding such a proof . therefore , strictly speaking , our result still has the status of a conjecture . in the meantime , we shall restrict all applications shown below to problems with low values of @xmath0 for which we have established the validity beyond any doubt . we are confident that the results hold for arbitrary @xmath0 , but this has to await a complete mathematical proof .    up to this point , we have only dealt with the dirac equation for continuum spinors . this still leaves open other aspects of the full tdhf problem like bound states , self - consistency , and fermion density . in some sense , all we have achieved so far is to find time - dependent , transparent potentials for the dirac equation , which look asymptotically like boosted static potentials . this solves in part another open problem which has been raised in the literature @xcite , namely to classify all time - dependent , transparent potentials of the 1 + 1 dimensional dirac equation . how general is our result in this respect ? all static transparent potentials are well known ( see the discussion in sec . [ sect2 ] ) . we can now construct all time - dependent transparent potentials which asymptotically consist of an arbitrary number of such static solutions , boosted to arbitrary velocities . this can not be the complete set of all transparent potentials though , as evidenced by the example of the breather which does not fit into this scheme . evidently , there must be another set of solutions where boosted breathers appear as asymptotic states , in addition to boosted static bound states . we do not know yet whether our ansatz will be capable of describing this more general class of solutions . all we have checked is that the single breather can indeed be reproduced with our ansatz , provided we allow for complex valued @xmath16 s . scattering problems involving breathers are interesting in their own right , but will be left for future studies . in the @xmath0 baryon problem , one expects @xmath0 positive and @xmath0 negative energy bound states . as discussed in ref . @xcite , the bound state spinors can be obtained from the continuum spinors by analytic continuation in the spectral parameter @xmath58 . to this end we first re - introduce the @xmath58 dependence of the coefficients ( [ 5.5][5.8 ] ) by using @xmath164 . only the coefficients @xmath165 are @xmath58-dependent . for positive energy bound states for example , @xmath166 develop a single pole at @xmath167 . the bound state spinor associated with baryon @xmath15 can then be obtained from the residue of @xmath83 at the pole , @xmath168 the result is a normalizable solution of the dirac equation . the normalization factor @xmath169 can readily be determined for times @xmath25 when the @xmath15-th baryon is isolated , with the result @xmath170 for this value of @xmath169 , the bound state spinor ( [ 7.1 ] ) is normalized according to @xmath171 this method has been checked analytically for @xmath37 in ref . @xcite and numerically for @xmath131 by us . the situation in the @xmath0-baryon problem is the same as in the 2-baryon problem @xcite . the scalar density for a continuum state can be decomposed as @xmath172 where @xmath173 is the perturbative piece which gives self - consistency by itself . the 2nd part is cancelled against the discrete state contribution , @xmath174 if one makes use of the self - consistency conditions in the asymptotic in- and out - states . we can deduce @xmath175 by subtracting the expression ( [ 8.2 ] ) from the full scalar density and can then check eq . ( [ 8.3 ] ) numerically , since we know the discrete state spinors and the integral is convergent . this test has been performed analytically for @xmath37 in ref . @xcite and numerically for @xmath131 in the present work . likewise , the fermion density can be dealt with in the same manner as for 1 or 2 baryons . the basic identity is @xmath176 relating the continuum and bound state densities @xcite . the integral is convergent owing to the vacuum subtraction . we have checked this identity here numerically for @xmath131 . from this and the self - consistency relation , one can again express the total , subtracted fermion density through the bound state densities as @xmath177 generalizing the @xmath37 results @xcite . the fermion transmission amplitude for the @xmath0-baryon problem factorizes , since it can be evaluated when all baryons are far apart , @xmath178 with @xmath100 from eq . ( [ 4.7a ] ) . this fact has actually already been used in the normalization conditions ( [ 4.8 ] ) . the more interesting question is how to characterize the outcome of the scattering process in terms of the baryon or multi - baryon bound states . comparing the asymptotics for @xmath179 , we find that the exponential @xmath16 acquires the following factor during an arbitrary @xmath0-baryon collision , @xmath180 the @xmath86 have been given in eq . ( [ 4.2 ] ) . if @xmath181 for one or several @xmath28 s , there is no shift factor because baryons @xmath15 and @xmath28 belong to the same compound state (  nucleus " ) and do not scatter from each other .    how does this translate into observables ? the scattering process at the level of the tdhf potential is classical , so that the situation is analogous to classical soliton scattering . if a single baryon is involved in the scattering process , the situation is very simple . the incoming and outgoing baryons can be associated with straight - line space - time trajectories defined by @xmath182 they have the same slope in the ( @xmath36 ) diagram , since the velocity does not change . the factor @xmath183 given in eq . ( [ 9.2 ] ) then leads to a parallel shift of the outgoing space - time trajectory , which is usually interpreted as time delay ( or advance ) . if an @xmath159-baryon bound state (  nucleus " ) is scattered , the initial state contains @xmath159 baryon constituents @xmath184 moving with the same velocity @xmath185 on parallel straight - line trajectories . such a bound state depends on the scale factors @xmath23 of @xmath16 (  moduli " ) , cf . ( [ 3.6 ] ) , determining the relative positions and the shape of the bound state without affecting its energy . in the final state , the @xmath159 trajectories will be displaced laterally relative to the incoming trajectories . since all @xmath186 parameters within one composite state must be chosen differently , according to ( [ 9.2 ] ) , the displacement will be different for each trajectory . therefore the net result can not be interpreted anymore as a mere time delay , but is always accompanied by a change in moduli space , resulting in different relative baryon positions and a corresponding deformation of the scalar potential . in this sense , the scattering process is not really elastic and the composite bound states undergo a change in their internal structure . a time delay of the full composite object could be defined , but this is neither unambiguous , nor necessary . the full asymptotic information about the scattering process is contained in eq . ( [ 9.2 ] ) . since we have verified the above formulas analytically or numerically with high precision for up to 8 baryons , we now present some illustrative results for smaller values of @xmath0 . depending on the choice of velocity parameters , the same formalism can describe a variety of physical problems .    for @xmath37 , there are two distinct possibilities . if the velocities are chosen to be equal , we obtain a boosted 2-baryon bound state , provided that the @xmath186 parameters are different . if the velocities are different , there is no restriction on the @xmath186 parameters and we describe scattering of baryon ( @xmath187 ) on baryon ( @xmath188 ) . in both cases , this yields nothing new as compared to refs . @xcite , but has been used to test our formulas .    for @xmath131 , we have to distinguish 3 cases . if @xmath189 and all @xmath18 s are different , we are dealing with a boosted 3-baryon bound state . if two velocities are equal and the corresponding @xmath186-parameters are different , the formalism describes scattering of a baryon on a 2-baryon bound state , analogous to @xmath190-scattering in nature . an example of this process is shown in fig . [ fig7 ] , where the time evolution of the scalar tdhf potential during the collision is displayed . as announced above , the internal structure of the bound state necessarily changes during such a collision . to emphasize this point , we compare in fig . [ fig8 ] the first and last time slice of fig . [ fig7 ] , i.e. , the incoming and outgoing states .    1.0 cm if all 3 velocities are different , the formalism describes a 3-baryon scattering process with 3 baryons in the initial and final state . since scattering processes with more than 2 incident particles are somewhat academic from the particle physics point of view , we do not show any example . with increasing @xmath0 , the number of scattering channels increases . the next number of baryons is @xmath161 , describing one boosted 4-baryon bound state , scattering of a baryon on a 3-baryon bound state , scattering of two 2-baryon bound states , scattering of 3 particles ( 2 baryons and a 2-baryon bound state ) or of 4 particles ( 4 individual baryons ) . the most interesting and new process out of these is the scattering of 2 bound states , the analogue of @xmath191-scattering  the simplest case of nucleus - nucleus scattering . this is illustrated in fig . the change in structure of the bound state is exhibited more clearly in fig . [ fig10 ] .    1.0 cm    finally , we give an example with 5 baryons . out of the many possibilities , we have chosen scattering of a single baryon on a 4-baryon bound state , the analogue of @xmath192-scattering in the real world , see fig . [ fig11 ] . we refrain from showing any results with larger number of baryons , since we have not yet checked our formulas thoroughly beyond @xmath193 . however , we have no doubt that we could describe correctly scattering processes with any number of baryons . all of these examples involve topologically trivial bound states only . there is no difficulty in applying the same formulas to topologically non - trivial scatterers as well . as already demonstrated in ref . @xcite , all one has to do is let one or several @xmath186 s go to 1 . then , the corresponding baryon becomes a kink - antikink pair at infinite separation . this diverging separation has to be compensated by a change of the scale parameter @xmath23 in the @xmath16 factor , so that half of the baryon disappears at infinity . in this way one can describe scattering of any number of topologically trivial or non - trivial bound states , without need to derive separate formulas for this purpose . this paper has dealt with the large @xmath0 limit of the gn model , the quantum field theory of massless , self - interacting , flavored fermions in 1 + 1 dimensions . the fascinating aspect of lagrangian ( [ 1.1 ] ) is the fact that a single contact interaction term is able to generate a host of non - trivial phenomena . even more surprisingly , it seems that all of these can be worked out in closed analytical form , a rather exceptional situation in quantum field theory . the story begins with asymptotic freedom , the generation of a dynamical fermion mass , accompanied by spontaneous breakdown of the @xmath8 chiral symmetry , and a scalar fermion - antifermion bound state , in the original work @xcite . soon afterwards baryons were discovered @xcite , subsequently complemented by a whole zoo of multi - baryon bound states @xcite . as time evolved and computer algebra software became more powerful , ambitions were raised , leading to results like soliton crystals in the ground state and phase diagram of dense matter @xcite or time - dependent scattering processes of kinks and antikinks @xcite . the most recent result is the tdhf solution of time - dependent baryon - baryon scattering @xcite .    in the present work , we have tried to add another chapter to this progress report . by generalizing the joint ansatz for the tdhf potential and the spinors recently proposed in ref . @xcite , we have most probably found the solution to a whole class of scattering problems , namely all those where the incoming and outgoing scatterers are boosted , static multi - fermion bound states of the gn model . the word  probably " has to be used here because we have not yet been able to prove our results in full generality . the solution which we have presented is based on the analytical solution of the 2- and 3-baryon problems , followed by a tentative extrapolation to arbitrary @xmath0 . these results have then been checked numerically for @xmath162 , and all heralds well for their general validity . this method could only work because of a kind of factorization property which we have observed  scattering of any number of baryons can apparently be predicted on the basis of 1- and 2-baryon input only . this holds not only for the asymptotic scattering data , but also during the entire time evolution , where more than 2 baryons can overlap at a time . we interpret these findings as a large-@xmath0 manifestation of the quantum integrability of the gn model . the solution which we have presented is relevant for yet another problem , namely how to find transparent , time - dependent scalar potentials for the dirac equation in 1 + 1 dimensions . it is clear that unlike in the static case , we have not yet arrived at the most general time - dependent solution . at least one time - dependent solution of the gn model is already known which does not belong to our class of solutions , the breather . it also yields a reflectionless potential . this suggests that a whole class of solutions is still missing , namely the tdhf potentials of scattering processes involving breathers in the initial and final states . we know already that the single breather can be obtained with our ansatz if one admits complex valued exponentials @xmath16 . it will be interesting to see whether breather - baryon or breather - breather scattering can be solved along similar lines . one other question which we have not been able to answer yet is whether our new solution is related to the solution of some known , classical non - linear equation or system of equations . this question is a natural one , given prior experience . thus for instance , all static baryons can be related to soliton solutions of the static nls equation . higher bound states are related to the static multi - channel nls equation . all dynamical kink solutions can be mapped onto multi - soliton solutions of the sinh - gordon equation . the non - relativistic limit of baryon - baryon scattering was shown to be equivalent to solutions of the time - dependent , multi - component nls equation . the advantage of such mappings is obvious . a lot of expertise and powerful techniques have been accumulated in the field of non - linear systems over the years , which can be helpful for finding new solutions of the gn model or proving certain results in full generality . a natural candidate for the present case would be the multi - component non - linear dirac equation , i.e. , the set of classical equations @xmath194 inspection of the various condensates in sec . [ sect8 ] shows that it is indeed possible to construct solutions of eq . 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scattering of two baryons in the large-@xmath0 gross - neveu model via the time - dependent dirac - hartree - fock approach has recently been solved in closed analytical form . here , we generalize this result to scattering processes involving any number and complexity of the scatterers . 
 the result is extrapolated from the solution of few baryon problems , found via a joint ansatz for the scalar mean field and the dirac spinors , and presented in analytical form . 
 it has been verified numerically for up to 8-baryon problems so far , but a full mathematical proof is still missing . 
 examples shown include the analogue of proton - nucleus and nucleus - nucleus scattering in this toy model . 
 all the parameters of the general result can be fixed by one- and two - baryon input only . 
 we take this finding as evidence for factorized scattering , but on the level of composite multi - fermion states rather than elementary fermions .