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the bosonic d - brane has an unstable tachyonic mode on the perturbative vacuum . it has been conjectured @xcite that there is a local minimum of the tachyon potential ( non - perturbative vacuum ) , where the tension of the d - brane and the negative contribution from the tachyon potential exactly cancel each other . on this vacuum , there should be no d - branes and hence no open string excitations at all , leaving only the closed string sector . it has been also conjectured that the solutions on this non - perturbative vacuum represent various lower dimensional d - branes ( descent relations ) .    for studying these conjectures , string field theories have proved to be very useful , and there have appeared many interesting works in this direction @xcite@xcite . in particular , witten s cubic open string field theory ( csft ) @xcite combined with the level truncation approximation @xcite has been successful in verifying the conjecture on the potential height to a very high degree of accuracy @xcite . though most of these conjectures , the potential height problem and the descent relation , have later been proved @xcite in another formulation of open string field theory called boundary string field theory ( bsft ) @xcite , there still remains an important problem to be understood : how the open string excitations disappear around the non - perturbative vacuum . note that for analyzing this problem we have to treat all the modes in open string theory other than the tachyon mode . therefore , csft whose action is given in a closed form seems to be rather suited for this problem than bsft which has concise expression only for the tachyon field . one of possible ways to prove the absence of physical open string excitations is to show that they all have infinite masses @xcite . another and roughly an equivalent way is to show that the physical components of the open string field fluctuations lack their kinetic terms in the sft action and hence become non - dynamical @xcite ( kinetic term means the quadratic term of the fluctuations containing the derivatives with respect to the center - of - mass string coordinate ) . in relation to this , an interesting deformed version of csft has recently been proposed in @xcite as a possible sft model around the non - perturbative vacuum . the brst operator in this model is simply equal to the ghost zero - mode , @xmath0 , and hence the quadratic term @xmath1 contains no kinetic term at all . what we have to do for proving the absence of open string excitations on the non - perturbative vacuum is to derive this kind of sft from the ordinary csft . the purpose of this paper is to explore , using the level truncation approximation , the possibility that csft action expanded around the non - perturbative vacuum contains no kinetic terms for physical fluctuations . let the action of csft be given by @xmath2 , and let us expand the original string field @xmath3 as @xmath4 with @xmath5 being the classical solution representing the non - perturbative vacuum . then the term in @xmath6 quadratic in the fluctuation @xmath7 is @xmath8 . since the three - string interaction vertex defining the star - product @xmath9 contains any powers of the derivative @xmath10 with respect to the center - of - mass @xmath11 of the string coordinate @xmath12 , the kinetic term for the fluctuation @xmath7 consists of terms with any number of derivatives . our expectation is that these kinetic terms all vanish for the physical components of @xmath7 , where `` physical '' means being brst invariant . we are going to study this possibility in the level ( 2,6 ) truncation by taking as the fluctuations only the scalar modes at levels zero and two . as the kinetic terms we take the quadratic and the quartic ones in the derivatives @xmath10 . as the simplest and encouraging example , let us quote our result in the level @xmath13 approximation , namely , let us take only the level zero scalar field @xmath14 ( see sec . the kinetic term for @xmath14 quadratic in the derivative is @xmath15 \left({\partial}_\mu t\right)^2 , \label{eq : skin}\ ] ] where @xmath16 is the classical value of @xmath14 at the non - perturbative vacuum . this kinetic term vanishes at @xmath17 , which is close to the value of @xmath16 obtained as the stationary point of the level ( 0,0 ) potential , @xmath18 . even this simplest example supports our expectation that the kinetic term for the physical fluctuations totally vanish .    in our analysis in the level ( 2,6 ) approximation , we consider the kinetic terms quadratic and quartic in the derivatives for the three scalar fluctuation modes at levels zero and two . then the coefficients of the kinetic terms form @xmath19 matrices in the space of fluctuations . since there is only one physical component among the three scalar fluctuations , the two matrices , one for the kinetic term with two derivatives and the other with four , must have a common zero - mode ( namely , the eigenvector corresponding to the eigenvalue zero ) in order for the physical fluctuation to disappear from the kinetic terms . we examine numerically whether this condition is satisfied in the neighborhood of the non - perturbative vacuum , and obtain results which support our expectation . the organization of the rest of this paper is as follows . in sec .  2 , after making a setup for the level truncation calculation in csft , we present the derivative expansion of the kinetic term of the action and the brst transformation at a generic translation invariant background . then , in sec . 3 we carry out the numerical analysis of the kinetic terms and the brst transformations of the fluctuations to examine whether the kinetic terms for the physical fluctuation can disappear in the neighborhood of the non - perturbative vacuum . finally in sec .  4 , we discuss future problems . in this section , as a preparation for the analysis in sec .  3 , we fix our conventions for the csft action , and then present the derivative expansion of the kinetic terms and the brst transformation for the scalar fluctuation modes in the level ( 2,6 ) approximation . we shall first recapitulate the gauge - fixed action and the brst transformation of csft in the siegel gauge . they are both obtained from the gauge invariant action : @xmath20 where @xmath21 is the open string coupling constant , and @xmath22 denotes the integrations over the anti - ghost zero - mode @xmath23 and the center - of - mass coordinate @xmath11 of the @xmath24-th string . the string field @xmath25 in ( [ eq : sinv ] ) is a state in the fock space of the first quantized string and carry ghost number @xmath26 . it is expanded in terms of the anti - ghost zero mode @xmath23 as @xmath27 where @xmath28 and @xmath29 carry ghost number @xmath30 and @xmath26 , respectively . the ghost zero - mode structure of the brst operator @xmath31 is @xmath32 where we have @xmath33 with @xmath34 . the three - string interaction vertex @xmath35 is given in the momentum representation for @xmath11 as @xmath36{|\ , 0\ , \rangle_{123 } } { \nonumber}\\ & \hspace*{5cm}\times ( 2\pi)^{26}\delta^{26}\left(p_1+p_2+p_3\right ) , \end{aligned}\ ] ] where @xmath37 is the fock vacuum satisfying @xmath38 ( @xmath39 ) , @xmath40 is a constant @xmath41 @xcite , and the neumann coefficients , @xmath42 and @xmath43 , are defined in @xcite . in particular , for @xmath44 we have @xmath45 with @xmath46 . then the gauge - fixed action @xmath6 in the siegel gauge , @xmath47 , is obtained from @xmath48 ( [ eq : sinv ] ) simply by putting @xmath49 : @xmath50 where the three - string vertex for the gauge - fixed action is @xmath51 . this action has an invariance under the brst transformation @xmath52 : @xmath53 physical quantities in the present sft are required to be invariant under the brst transformation .      as mentioned in sec .  1 , we are interested in the kinetic term of the string field fluctuation around the non - perturbative vacuum . to analyze it in the simplest framework , we adopt the level ( 2,6 ) truncation and take the scalar fields at levels zero and two , both as the vacuum coordinates and the fluctuations . namely , we restrict the @xmath54 component field sector of @xmath28 to the following form : @xmath55 then , reexpressing the three component field @xmath56 as @xmath57 with @xmath58 being a ( translation invariant ) vacuum value and @xmath59 the fluctuation around @xmath58 , we carry out the derivative expansion of the kinetic term of the fluctuation @xmath59 obtained from the action @xmath6 ( [ eq : s ] ) : @xmath60    here we are making the simplification of considering only the scalar fields @xmath61 . in particular , we took only the trace part of the tensor fields @xmath62 which appear in the expansion of @xmath28 in the form @xmath63 . we have also discarded the vector field at level two which appear as the coefficient of @xmath64 . complete treatment in the ( 2,6 ) truncation should take into account the mixing among all the scalar , vector and tensor fields at levels zero , one and two . however , such analysis would be so complicated since the size of the matrices @xmath65 and @xmath66 become much larger , and there are many physical fields that should be confined . therefore , we shall carry out the following analysis in the smallest fluctuation space @xmath61 with a hope that the mixing with other components is small . now each component of the @xmath19 matrix @xmath67 is a linear function of @xmath68 and is given explicitly by ( hereafter we omit the subscript 0 on @xmath68 unless confusion occurs ) @xmath69 note that in the present convention we have @xmath70 at the perturbative vacuum ( the minus sign for the @xmath71 field is due to the negative norm property of the ghosty state @xmath72 ) . the matrix @xmath73 for the four - derivative kinetic term have contributions only from the interaction term of ( [ eq : s ] ) . it is given by @xmath74 ^ 2}\left ( 27\,t + 11\,u - 5\sqrt{13}\,v\right ) , \\ h_{uu}=&-\frac{19}{54\sqrt{39}}{\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)}v + \frac{\sqrt{3}}{64}{\left[\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)\right]^2}\left ( \frac{19}{3}\,t + u - \frac{95\sqrt{13}}{81}\,v\right ) , \\ h_{vv}=&\frac{1}{\sqrt{3}}\left(\frac{1}{78}\,t + \frac{11}{2106}\,u -\frac{5}{18\sqrt{13}}\,v\right ) + \frac{1}{156\sqrt{3}}{\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)}\left ( 49\,t + \frac{539}{27}\,u -\frac{2779}{3\sqrt{13}}\,v\right ) { \nonumber}\\ & + \frac{1}{192\sqrt{3}}{\left[\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)\right]^2}\left ( 581\,t + \frac{6391}{27}\,u -\frac{20951}{9\sqrt{13}}\,v\right ) , \\ h_{tu}= & -\frac{11}{6\sqrt{39}}{\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)}v + \frac{\sqrt{3}}{64}{\left[\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)\right]^2}\left ( 33\,t + \frac{19}{3}\,u -\frac{55\sqrt{13}}{9}\,v\right ) , \\ \frac{22}{1053\sqrt{3}}\,v + \frac{1}{24\sqrt{3}}{\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)}\left(-\frac{11}{\sqrt{13}}\,t -\frac{19}{9\sqrt{13}}\,u + \frac{4279}{351}\,v\right ) { \nonumber}\\ & + \frac{1}{192\sqrt{3}}{\left[\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)\right]^2}\left ( -55\sqrt{13}\,t - \frac{95\sqrt{13}}{9}\,u + \frac{6391}{27}\,v \right ) , \\ h_{vt}= & \frac{2}{39\sqrt{3}}\,v + \frac{\sqrt{3}}{8}{\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)}\left(-\frac{3}{\sqrt{13}}\,t -\frac{11}{9\sqrt{13}}\,u + \frac{389}{117}\,v\right ) { \nonumber}\\ & + \frac{\sqrt{3}}{64}{\left[\ln\!\left(\frac{3 ^ 3}{2 ^ 4}\right)\right]^2}\left(-15\sqrt{13}\,t -\frac{55\sqrt{13}}{9}\,u + \frac{581}{9}\,v\right ) .\end{aligned}\ ] ]    for completeness we present the potential @xmath75 in the ( 2,6 ) truncation : @xmath76      in the analysis of sec .  3 , we also need the derivative expansion of the brst transformation of the fields @xmath61 for identifying their physical ( brst invariant ) combination at the non - perturbative vacuum . since the ghost field in @xmath28 at level one does not contribute to @xmath77 , we have only to consider the ghost fields at level two , @xmath78 and @xmath79 , which appear in @xmath28 as @xmath80 then , @xmath77 can be calculated using the formula ( [ eq : delb ] ) . similarly to the kinetic term ( [ eq : expands ] ) , we carry out the derivative expansion for the contribution from the last term of ( [ eq : delb ] ) and express the result as @xmath81 where we have omitted those terms containing the fluctuation @xmath59 on the rhs . the explicit expression of the coefficients @xmath82 are as follows : @xmath83 note that the constant terms in @xmath84 , @xmath85 , @xmath86 and @xmath87 came from @xmath88 in ( [ eq : delb ] ) , and that the tachyon field @xmath14 is brst invariant at the perturbative vacuum . in this section , we shall study numerically whether the conditions for the absence of the kinetic terms of the physical scalar field are realized near the non - perturbative vacuum . the precise expressions of the conditions will be stated below , in particular , in sec . . what we shall test are i ) location of the zero of @xmath89 and that of @xmath90 , ii ) coincidence of the zero - modes of @xmath65 and @xmath66 , and iii ) existence of the brst invariant linear combination of @xmath61 including the higher derivative terms in @xmath91 . unfortunately , these conditions are not satisfied to high precision at the non - perturbative vacuum  determined as a stationary point of the potential . therefore , we shall look at each of them globally in the space of vacuum coordinate @xmath92 to test whether there is a point satisfying the condition near the non - perturbative vacuum . let us first consider the simplest ( 0,0 ) truncation , namely , let us keep only the level zero field @xmath14 . in this approximation the field @xmath14 is physical everywhere , @xmath93 , and the potential @xmath94 ( [ eq:26v ] ) and the metric @xmath95 ( [ eq : gtt ] ) are reduced to @xmath96 the non - trivial stationary point ( non - perturbative vacuum ) of @xmath97 exists at @xmath98 , while the metric @xmath99 vanishes at @xmath100 . surprisingly these two points are very close , supporting that the kinetic term of the physical field @xmath14 is missing . as for @xmath101 for the four - derivative kinetic term , it vanishes only at @xmath102 . however , the level ( 0,0 ) approximation would be too poor for such higher derivative kinetic terms .      encouraged by the above result in the ( 0,0 ) approximation , let us proceed to the analysis of the full @xmath65 of eqs . ( [ eq : gtt ] )  ( [ eq : gvt ] ) and the first two terms of the brst transformation ( [ eq : expanddelb ] ) : @xmath104 namely , we shall examine the lowest derivative kinetic term and the lowest derivative part of the brst transformation .    before starting the analysis , let us state our expectation for the disappearance of the physical open string modes in the present situation . since we have two independent ghost fields , @xmath105 and @xmath106 , on the rhs of ( [ eq : expanddelbab ] ) , there is only one brst invariant linear combination @xmath107 of the three scalar fields @xmath61 satisfying @xmath108 : @xmath109 where @xmath110 ( @xmath111 ) is the solution of @xmath112 namely , the level zero scalar @xmath14 , which was brst invariant at the perturbative vacuum , needs a mixing of @xmath71 and @xmath113 to reconstruct a brst invariant field at the non - perturbative vacuum . a possible mechanism that confines this physical scalar @xmath107 is that @xmath107 has no kinetic term . in particular , @xmath107 should not appear in @xmath114 . for this to be realized , it is sufficient that the matrix @xmath115 has a zero - mode @xmath116 ( i.e. , the eigenvector of the eigenvalue zero ) and that it has a non - vanishing inner product with @xmath117 : @xmath118 the reason is as follows . let @xmath119 ( @xmath120 ) be the three orthonormal eigenvectors of the ( real and symmetric ) matrix @xmath65 corresponding to the eigenvalue @xmath121 : @xmath122 then , the two - derivative kinetic term is diagonalized as @xmath123 where the new field @xmath124 is defined by @xmath125 . in order for the kinetic term ( [ eq : kineticpsi ] ) not to contain the physical @xmath126 , @xmath107 should not be expressed as a linear combination of @xmath124 ( @xmath127 , and this is realized if the condition ( [ eq : betae0 ] ) holds . we would like to emphasize that the missing combination @xmath128 need not be exactly equal to @xmath107 . is not invariant under a transformation of the coordinate system @xmath59 of the field space . ] a rather weak condition ( [ eq : betae0 ] ) is sufficient . therefore , what we have to test first of all is whether the determinant of the matrix @xmath115 vanishes near the non - perturbative vacuum . recall that the coordinate @xmath129 of the non - perturbative vacuumobtained as a stationary point of the level ( 2,6 ) potential ( [ eq:26v ] ) is @xcite @xmath130 first , the value of @xmath89 at the point of ( [ eq : varphi_c ] ) is @xmath131 , which is relatively close to zero compared with its value at the perturbative vacuum @xmath132 , indicating that the point ( [ eq : varphi_c ] ) is near the two - dimensional surface of @xmath133 , though not exactly on it . ( 317,148)@xmath134 ( 20,258)@xmath135 ( 290,253)(@xmath136 ) ( 310,238)(@xmath137 ) ( 310,225)(@xmath138 ) ( 290,200)@xmath139    figs . [ fig : plotdetg ] and [ fig : detgzerocurve ] would help understanding how the non - perturbative vacuum  is close to the surface @xmath133 . in fig .  [ fig : plotdetg ] we plot @xmath89 as a function of @xmath14 for various fixed values of @xmath140 . in particular , the zero of @xmath89 for @xmath141 of ( [ eq : varphi_c ] ) is at @xmath142 , while the zero for @xmath143 is at @xmath144 . , @xmath89 has other two zeros at @xmath145 and @xmath146 . when @xmath143 , the other two zeros are complex . ] on the other hand , fig . [ fig : detgzerocurve ] shows curves of @xmath133 in the @xmath140 plane for several fixed values of @xmath14 . as a concrete example , assuming the conjectured equality @xmath147 at the non - perturbative vacuum  @xcite , the value of @xmath71 satisfying @xmath148 is @xmath149 . ( 290,120)@xmath150 ( 21,230)@xmath151 ( 285,220)@xmath152 ( 285,190)@xmath153 ( 285,163)@xmath154 ( 285,140)@xmath155 ( 285,105)@xmath156    as a representative point on the @xmath133 surface which is reasonably close to the non - perturbative vacuum  of ( [ eq : varphi_c ] ) , there is a stationary point @xmath157 of the level ( 2,6 ) potential ( [ eq:26v ] ) restricted on the surface @xmath133 : @xmath158 the value of the potential ( [ eq:26v ] ) ( normalized so that the true value is equal to @xmath26 ) at this point is @xmath159 , while that at the point ( [ eq : varphi_c ] ) is @xmath160 . the above analysis tells that the non - perturbative vacuum  is indeed close to the surface @xmath133 . in particular , for @xmath161 the condition @xmath133 combined with the assumption @xmath162 predicts the expected value @xmath163 for @xmath71 and @xmath113 . we have also examined the condition ( [ eq : betae0 ] ) to find that the inner product @xmath164 on the surface @xmath133 has no zeros at least in the region @xmath165 . next let us consider the four - derivative kinetic term in ( [ eq : expands ] ) . what we expect for @xmath73 in order for the physical scalar @xmath107 ( [ eq : t ] ) to drop out from the four - derivative kinetic term is as follows :    * @xmath66 has a zero - mode @xmath167 at the non - perturbative vacuum . * the two non - zero - modes of @xmath66 span the same two - dimensional subspace as that spanned by the non - zero - modes of @xmath65 ( equivalently , @xmath167 is parallel to the zero - mode @xmath116 of @xmath65 ) . if these conditions are satisfied , then @xmath168 is also expressed only in terms of @xmath124 ( @xmath169 ) appearing in the two - derivative kinetic term ( [ eq : kineticpsi ] ) . first , let us see whether @xmath90 vanishes near the non - perturbative vacuum . fig . [ fig : plotdeth ] shows @xmath90 as a function of @xmath14 for several fixed values of @xmath140 . for @xmath143 , @xmath90 has no zeros in the range of @xmath14 of our interest , and for @xmath170 of ( [ eq : varphi_r ] ) , @xmath90 become close to zero at @xmath171 but does not cross zero . for larger values of @xmath113 compared with @xmath71 , @xmath172 and @xmath173 , @xmath90 has zeros around @xmath174 . ( 317,175)@xmath134 ( 0,258)@xmath175 ( 106,55)(@xmath136 ) ( 157,75)(@xmath176 ) ( 210,55)(@xmath177 ) ( 275,75)(@xmath178 )    ( 290,140)@xmath151 ( 18,232)@xmath150 ( 275,90)@xmath179 ( 265,58)@xmath174 ( 215,50)@xmath180    fig . [ fig : resolhu ] shows the curves of @xmath181 in the @xmath182 plane for three values of @xmath14 . from fig . [ fig : resolhu ] , we see that , for @xmath161 and @xmath183 corresponding to the non - perturbative vacuum , the @xmath113 coordinate on the @xmath181 surface is @xmath184 , which is more than 50% larger than the value of the non - perturbative vacuum . in spite of this discrepancy , it seems miraculous that the surface @xmath181 is rather close to the non - perturbative vacuum . therefore , expecting that the discrepancy would be resolved in a more precise analysis , let us proceed to the study of another condition , @xmath185 . ideally , we should test whether the two vectors @xmath116 and @xmath167 become parallel at some point on the curve of the intersections of the two surfaces @xmath133 and @xmath181 . however , though the two surfaces are separately close to the non - perturbative vacuum , their intersection curve is not so close to it in the present approximation . in the range @xmath186 , the corresponding @xmath14 and @xmath113 coordinates on the intersection curve do not change rapidly and take values @xmath187 and @xmath188 . for large values of @xmath71 , the corresponding @xmath113 coordinate become outside the range @xmath189 . ] therefore , we instead carry out an indirect analysis . as @xmath116 , we take the zero - mode of @xmath115 at a fixed @xmath190 which is on the surface @xmath133 and close to the non - perturbative vacuum . concretely , we take @xmath191 of ( [ eq : varphi_r ] ) , and the corresponding zero - mode is @xmath192 . then , we plot , on the surface @xmath181 , the absolute value of the inner products @xmath193 and @xmath194 , where @xmath195 and @xmath196 are the normalized eigenvectors of @xmath66 corresponding to non - zero eigenvalues . on the intersection curve , the condition @xmath185 is equivalent to @xmath197 . we are going to examine the behavior of the approximate quantities to the two inner products on the surface @xmath181 . ( 288,65)@xmath151    fig . [ fig : fe ] shows these two inner products as a function of @xmath113 with @xmath14 fixed to @xmath198 and @xmath71 determined by @xmath181 . note that one of the inner products vanishes at @xmath199 and the other at @xmath200 . moreover , the positions of these two zeros , in particular , @xmath199 , are fairly insensitive to the choice of @xmath116 used for taking the inner products . namely , the positions of the zeros change little even if we replace @xmath201 by almost any vector . this is ascribed to the fact that the eigenvectors @xmath195 and @xmath196 change their directions rapidly by large angle in the neighborhood of @xmath199 and that of @xmath200 , respectively ( recall that @xmath202 vanishes when @xmath203 crosses the surface perpendicular to @xmath116 ) . although the inner product @xmath204 is not invariant under the change of the field space coordinates @xmath59 , the positions of the zeros are insensitive to the coordinate change owing to the same property of @xmath205 . for example , if we adopt the coordinates of @xcite , @xmath206 , the global behavior of the inner products differs from fig .  [ fig : fe ] , but there are zeros near the corresponding points .    therefore , the presence of the two zeros of the inner products @xmath207 , in particular , the zero at @xmath199 which is closer to the non - perturbative vacuum , is very encouraging . it is expected that , as we improve the approximation , the two zeros approach to each other and to @xmath208 . we have defined the physical scalar field @xmath107 ( [ eq : t ] ) by the requirement that it be invariant under the brst transformation @xmath209 ( [ eq : expanddelbab ] ) truncated at the lowest order in the derivatives on @xmath105 and @xmath106 . however , as given in ( [ eq : expanddelb ] ) , the full brst transformation contains higher derivative terms , and we have to check whether the @xmath210 and @xmath211 terms are also cancelled in @xmath212 . namely , we shall test whether @xmath213 and @xmath214 , @xmath215 with @xmath216 satisfying ( [ eq : eqforbeta ] ) vanish near the non - perturbative vacuum . ( 288,155)@xmath150 ( 22,230)@xmath151    ( 288,130)@xmath134 ( 10,230)*@xmath217 , @xmath218 * ( 210,217)(@xmath138 ) ( 280,210)(@xmath219 ) ( 240,180)(@xmath220 )    let us first consider @xmath213 on the surface @xmath221 . fig . [ fig : zerotesty ] shows the curve in the @xmath140 plane determined by @xmath222 and @xmath221 . note that this curve passes the point @xmath223 , implying that @xmath222 holds near the non - perturbative vacuum .    however , the corresponding curve for @xmath214 is totally outside the region of our interest in the @xmath140 plane . therefore , we instead studied the behavior of @xmath214 as a function of @xmath14 for a number of fixed values of @xmath140 ( see fig .  [ fig : testyzall ] ) . from fig . [ fig : testyzall ] we see that @xmath214 has a zero at a rather small @xmath14 when the @xmath140 coordinate is close to that of the non - perturbative vacuum .      in this section , we have analyzed the conditions for the absence of the kinetic term of the brst invariant scalar fluctuation . the conditions can simply be stated as ( i ) @xmath65 and @xmath66 have a common zero - mode at the non - perturbative vacuum , and ( ii ) the brst invariant fluctuation component can be consistently defined there . we divided the test of the conditions into many steps and examined whether the following quantities vanish near the non - perturbative vacuum : @xmath89 ( sec . 3.2 ) , @xmath90 ( sec . 3.3 ) , @xmath224 ( sec . 3.3 ) , @xmath225 and @xmath226 ( sec . 3.4 ) . among them , we obtained fairly good results for @xmath89 , @xmath90 , @xmath227 ( the one having a zero at @xmath199 in fig . [ fig : fe ] ) and @xmath225 . however , the behaviors of @xmath228 and @xmath226 were not as we expected . they are hoped to improve in a better treatment , in particular , by taking into account the mixing with the vector and tensor components at level two which we neglected in this paper . we have carried out the same analysis also in the level ( 2,4 ) approximation , and found that the results are not qualitatively changed . our analysis in this paper is incomplete in various aspects , and further detailed studies are needed for confirming the absence of kinetic terms . we have to extend the space of fluctuations to incorporate the vector and tensor components even in the level ( 2,6 ) approximation . this is indispensable for making clear the full brst structure of the fluctuations . we have to consider the kinetic terms with more than four derivatives . and we have to raise the order of the level truncation . what is more important is of course the analytical understanding of the absence of the kinetic terms , though this must be a difficult task since the exact solution @xmath5 for the non - perturbative vacuum  would be needed . conversely , the missing kinetic terms for the fluctuations could give a hint for finding the exact solution . it is also an interesting question whether the csft expanded around the non - perturbative vacuum  is a topological theory described by a brst exact action @xcite . we have to clarify how such an open sft can support the closed string sector . we would like to thank m.  bando , i.  kishimoto , t.  kugo , s.  moriyama , t.  noguchi and s.  shinohara for valuable discussions and useful comments . we would also like to thank s.  moriyama for careful reading of the manuscript . the work of h.  h.  is supported in part by grant - in - aid for scientific research from ministry of education , science , sports and culture of japan ( # 12640264 ) .
it has been conjectured that the bosonic open string theory around the non - perturbative tachyon vacuum has no open string dynamics at all . 
 we explore , in the cubic open string field theory with level truncation approximation , the possibility that this conjecture is realized by the absence of kinetic terms of the string field fluctuations . 
 we study the kinetic terms with two and four derivatives for the lower level scalar modes as well as their brst transformation properties . 
 the behavior of the coefficients of the kinetic terms in the neighborhood of the non - perturbative vacuum supports our expectation that the brst invariant scalar component lacks its kinetic term . 
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