We reexamine the oscillator construction of the D25brane solution and the tachyon fluctuation mode of vacuum string field theory given previously. Both the classical solution and the tachyon mode are found to violate infinitesimally their determining equations in the level cutoff regularization. We study the effects of these violations on physical quantities such as the tachyon mass and the ratio of the energy density of the solution relative to the D25brane tension. We discuss a possible way to resolve the problem of reproducing the expected value of one for the ratio.
1 Introduction and summary
Recently there has been considerable progress in the oscillator formalism analysis [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] of vacuum string field theory (VSFT) [13, 1, 14, 15], which is a candidate string field theory expanded around the tachyon vacuum of bosonic open string theory. The action of VSFT reads
(1.1) 
with the BRST operator given by the following purely ghost one, respecting that there is no open string excitations around the tachyon vacuum:
(1.2) 
In the oscillator formulation of VSFT, there appear infinite dimensional matrices and vectors in the threestring vertex defining the product of two string fields. Explicitly, the matter part of the vertex reads [16, 17]
(1.3) 
with and (we adopt the convention ).^{*}^{*}* In this paper, we adopt a different convention concerning the centerofmomentum from that used in [2, 4, 18]. The state is the eigenstate of the momentum operator with eigenvalue with the normalization , and at the same time it is the Fock vacuum annihilated by with . We define the Neumann matrices and vectors () by
(1.4) 
where is the twist matrix. It is also convenient to define the following matrix and vector which are both twistodd:
(1.5) 
These matrices and vectors satisfy various linear and nonlinear identities [16, 17] which are summarized in appendix A. In particular, the matrices are all commutative to each other. Similar matrices and vectors appear also in the ghost part of the three string vertex. We denote them by adding tilde to the corresponding one in the matter part.
In order for VSFT to really make sense, we have to show the following two:

There is a classical solution of VSFT describing a D25brane, namely, the perturbative open string vacuum with tachyonic mode. The energy density of this solution relative to that of the trivial one must be equal to the D25brane tension , and the fluctuation modes around must reproduce the open string spectrum.

VSFT expanded around describes pure closed string theory despite that the dynamical variable of VSFT is a open string field.
Challenges toward the first problem have been done both by the oscillator method using the Neumann matrices and the geometric method using the boundary conformal field theory (BCFT) [14, 19, 20, 21, 22, 23, 24]. In the oscillator approach, the classical solution is assumed to be a squeezed state with its matter part given by . Then the equation of motion
(1.6) 
is reduced to an algebraic equation for the matrix , which is solved by using the identities of appendix A [25, 1]. The solution obtained this way has been identified [1, 10] as the sliver state constructed in the BCFT approach [26]. Then, our next task is to construct fluctuation modes around . This is necessary also for the potential height problem, namely, the problem of showing , since the tension is given in terms of the open string coupling constant (three tachyon onshell coupling) by . In [2], the tachyon fluctuation mode has been constructed as a momentumdependent deformation of : which is parameterized by a vector . This vector as well as the tachyon mass is determined by the wave equation
(1.7) 
with being the BRST operator around . Owing again to the identities among the matrices, the equation for can explicitly be solved.
Once and have been found, we can answer the question of whether the tachyon mass is the expected one, , and whether we have . These two physical quantities (we call them observables) are expressed in terms of the matrices and the vectors . In [4], a crucial finding has been made concerning these observables. Both and are given in terms of quantities (denoted and in [2]) which vanish if we naively use the nonlinear identities of appendix A, implying absurd results and . The vanishing of these quantities and can be ascribed to the fact that the eigenvalues of the matrix are doubly degenerate between twisteven and odd eigenvectors, and the cancellation occurs between the contributions of degenerate eigenvalues. However, since this degeneracy is violated at the end of the eigenvalue distribution , and in addition since and are singular at , a careful treatment by using the level number cutoff regularization leads to finite and nonvanishing values of and . This phenomenon that a quantity vanishing naively due to twist symmetry (eigenvalue degeneracy) can in fact gain a nonvanishing value has been called “twist anomaly” in [4].
Another important progress concerning the observables in VSFT is that the eigenvalue problem of the matrices has been solved in [7]. They found that the matrices and are expressed in terms of a simpler matrix ,
(1.8) 
which is the matrix representation of the Virasoro algebra :
(1.9)  
(1.10) 
The matrix is symmetric and twistodd: and . The eigenvalue distribution of is uniform extending from to . The eigenvalues of in the level cutoff regularization have also been found.
The values of the observables in VSFT were calculated first numerically [2, 4], and later analytically [18] by using the results of [7]. They were also calculated using the BCFT method in [19]. The result is that is equal to the expected value of , but we have a strange value for the ratio; . Concerning this problem, a critical observation has been made in [19]. In the analysis of [2, 4, 18], the wave equation for the tachyon mode are implicitly considered in the Fock space of firstquantized string states, namely, the inner product of the wave equation with any Fock space elements of the form are demanded to vanish to give . However, the inner product , which is a quantity constituting the kinetic term of the tachyon field, no longer vanishes at . They claimed that this is the reason why we get a wrong value for the ratio . Later, the expected value of the ratio, , was successfully derived in [24] by introducing a nonlinear component expansion of the string field. It is interesting to clarify whether the ratio problem can still be resolved by using the conventional linear expansion.
The purpose of this paper is to reexamine the construction of the classical solution and the tachyon wave function by considering their equations (1.6) and (1.7) in the sliver space, namely, the space of the states of the form . This is necessary since the tachyon and vector fluctuation modes around are in the sliver space [2, 19]. First, we shall consider how precisely our classical solution and the tachyon wave function satisfy (1.6) and (1.7), respectively. In the analysis of twist anomaly, it has been important to refrain from freely commuting the matrices and using nonlinear identities. Such manipulations lead to wrong results. However, in solving (1.6) and (1.7) for and , we had to carry out these potentially invalid operations. We find that both (1.6) and (1.7) are “infinitesimally violated” by the present and in the level cutoff regularization.
Then, we study the effects of the violations. The violation of the equation of motion (1.6) is invisible so long as we consider (1.6) in the Fock space. However, we meet nonvanishing effects of the violation once we consider the inner product of (1.6) with the sliver space elements of the form . First, the normalization factor of determined before by no longer works for
(1.11) 
Second, there is no unique normalization factor of common to all . We shall carry out similar analysis also for the tachyon wave equation. We see how the infinitesimal violation of (1.7), which is invisible in , gives finite effect on and makes it nonvanishing at the expected tachyon onshell ; a phenomenon pointed out by [19] using the BCFT arguments.
We also reexamine the potential height problem by considering both (1.6) and (1.7) in the sliver space. We present an argument which is rather kinematical and needs no explicit calculation of twist anomalies. We find a kind of nogo theorem that the ratio is again an undesirable value even if we consider the equations in the sliver space.
Our findings in this paper do not directly help resolve the problem of the wrong value of the ratio in the oscillator approach [2, 4, 18]. Rather the problem has become even larger and more complicated by the present analysis. However, our observation here that the equation of motion in the sliver space, eq. (1.11), cannot be satisfied for all reminds us of an interesting proposal of [24] that the string field in VSFT needs a kind of nonlinear representation.^{†}^{†}† There is another different point between the expression of fluctuation modes of ours and that of [24]. As has been shown in [19], our tachyon wave function corresponds to a local insertion of a tachyon vertex operator on the sliver surface states in the BCFT language. On the other hand, the vertex operator is integrated along the surface in the tachyon wave function of [24]. Namely, the space of is restricted to the sliver space and expanded around the classical solution as
(1.12) 
where the summation is running over the sliver space states and the component fields are not all independent; some of are expressed nonlinearly in terms of other and independent (the component fields , , of the excitations , , depend on one another in [24]). If we could choose the normalization factor of in such a way that (1.11) holds for all corresponding to independent , the problem concerning (1.11) would be resolved by adopting the nonlinear representation. This could at the same time resolve the problem of since the kinetic term of the tachyon field has additional contributions from terms linear in which are not independent. This proposal may be interpreted as the string field being a constrained one. If this is the case, it is interesting to consider its relevance to the problem of whether VSFT around describes a pure closed string theory.
The organization of the rest of this paper is as follows. In sec. 2, we examine the infinitesimal violation of the equation of motion of and its effects in the sliver space. In sec. 3, we present a similar analysis for the wave equation of the tachyon mode . In sec. 4, the potential height problem is studied by considering both the equation of motion and the wave equation in the sliver space. In appendix A, B and C, we present various formulas and technical details used in the text.
2 Reexamining classical solution
In this section, we shall first summarize the oscillator construction of the D25brane classical solution of VSFT and then examine whether can be a solution even if we consider the inner product of the equation of motion with sliver space states.
2.1 Oscillator construction of the solution
The D25brane solution in VSFT is a translationally and Lorentz invariant classical solution to the equation of motion of VSFT:
(2.1) 
It has been claimed that such solution takes a form of squeezed state in the Siegel gauge [25, 1, 2]:
(2.2) 
where is the normalization factor and the squeezed state is given by
(2.3) 
The matrices and are both twisteven, and . For the sake of notational simplicity and since we are mainly interested in the matter part, we have omitted as an argument of and . Since the threestring vertex is also of the squeezed state form, the star product is again a squeezed state and is given explicitly by
(2.4) 
where the new matrix on the RHS is defined by
(2.5) 
with
(2.6) 
The quantity on the RHS of (2.4) acting on is
(2.7) 
with the coefficient vector given by
(2.8) 
Therefore, our task of obtaining the solution has been reduced to first solving the matrix equation
(2.9) 
for , and then determining the normalization factor and the coefficient in the BRST operator (1.2) in such a way that the equation of motion (2.1) holds. Assuming the commutativity and using the nonlinear relations among given in appendix A, in particular, using the formula
(2.10) 
valid only when the nonlinear identities are used, the matrix equation (2.9) is reduced to [25, 1]
(2.11) 
As a solution to (2.11), the following with a finite range of eigenvalues has been taken
(2.12) 
The equations and the solution for the ghost part matrix are quite parallel to those for [2]. Finally, the normalization factor is determined to be given by
(2.13) 
and the coefficient in , which is arbitrary for the gauge invariance of VSFT alone, is fixed to [2]
(2.14) 
The reason of the superscript “Fock” in (2.13) will become clear in the next subsection.
2.2 Equation of motion in the sliver space
One might think that the algebraic construction of the solution summarized in the previous subsection is quite perfect. It is, however, a nontrivial problem in what sense the equation of motion (2.1) holds. The equation of motion holds in the Fock space, namely, we have
(2.15) 
for any Fock space states of the form . However, the inner product of the equation of motion with the solution itself or with the states of the form (we call such space the sliver space hereafter) is a nontrivial quantity. First, let us consider
(2.16) 
Taking into account that the sliver state on the RHS of (2.4) is that associated with the matrices and ,^{‡}^{‡}‡ The product for the ghost part matrix is defined by (2.5) with all the matrices including replaced by the tilded ones. and forgetting (2.9) for the moment, we see that the normalization factor for (2.16) to hold must be related to (2.13) by
(2.17) 
Before examining whether the ratio (2.17) is equal to one, let us next consider the inner product of the equation of motion with other sliver space elements. Defining the state by
(2.18) 
and choosing as , we obtain
(2.19) 
From (2.19), the inner product of the equation of motion with, for example, the sliver space state of the form is given by
(2.20) 
where and are arbitrary vectors in the level number space. Recall that we have chosen in (2.20).
Of course, the ratio (2.17) is equal to one and the RHS of (2.20) vanishes if we naively use and , which are the equations determining and . However, a careful analysis is necessary for them since the matrix (2.12) has eigenvalue at the end of its eigenvalue distribution [4, 5, 7], and each term in (2.17) and (2.20) is singular there. This implies that we have to introduce a regularization which lifts the smallest eigenvalue of from the dangerous point .
Here, let us adopt the regularization of cuttingoff the size of the infinite dimensional matrices and the vectors to finite size () ones. This is the regularization we used in calculating observables (twist anomaly) of VSFT [2, 4, 18], and the following analysis for (2.17) and (2.20) is quite similar to those for twist anomaly. As given in (1.9) and (1.10), and are expressed in terms of a single infinite dimensional matrix , and the dangerous eigenvalue () corresponds to . In our regularization, we regard and as primary and hence cut off (1.9) and (1.10) after evaluating them by using infinite dimensional .^{§}^{§}§ If we adopt the regularization of replacing in (1.9) and (1.10) by a finite size one , and remain commutative to each other and hence all twist anomalies in VSFT vanish identically. Since the neighborhood of is important for (2.17) and (2.20), let us expand in our regularization in powers of :
(2.21)  
(2.22) 
where in (2.21) stands for the cutoff of the infinite dimensional matrix . Similarly, (2.12) in our regularization is expanded as follows:
(2.23) 
By the present regularization, the dangerous eigenvalue of is lifted by an amount of order [7, 18].
In our regularization, the matrices and no longer satisfy the nonlinear identities given in appendix A. This implies that the equation (2.9) determining is not exactly satisfied by the regularized . For studying the effects of this violation of (2.9), let us consider the expansion of in powers of . Namely, we substitute (2.21), (2.22) and
(2.24) 
with being of order into the original definition (2.5) of , and calculate it to order by keeping the ordering of the matrices. The details of the calculation is given in appendix B, and the result is
(2.25) 
where and should be regarded as cutoff ones. Eq. (2.25) is valid for any of order , not restricted to corresponding to (2.23). Eq. (2.23), namely,
(2.26) 
and (2.25) implies that the equation determining , (2.9), is indeed not satisfied by near since we have .
Although the violation of (2.9) we have found here is infinitesimal since we are considering the range of infinitesimal eigenvalues of of order [7, 18], it can give finite effects on (2.17) and (2.20). First, the RHS of (2.20) is nothing but of the form of twist anomaly in VSFT discussed in [4, 18]. It vanishes naively and has degree of singularity three if the vectors and are twistodd ones with degree of singularity equal to one such as .^{¶}^{¶}¶ See [4, 18] for the degree of singularity in the calculation of twist anomalies. We present it in table 1 for various quantities. If the degree of singularity of a quantity is less than three, naive manipulations using the nonlinear identities are allowed. Twist anomaly is a quantity which has degree of divergence equal to three and vanishes if we use the nonlinear identities.
Therefore, the value of the RHS of (2.20) is correctly calculated by substituting the expansions
(2.27) 
with defined by
(2.28) 
and keeping only the most singular part. We have
(2.29) 
As we experienced in the analysis of twist anomaly, this is indeed finite and nonvanishing for generic twistodd and with degree of singularity equal to one (for example, ).
As for the ratio (2.17) we do not know whether a similar treatment of taking only the most singular part, for example,
(2.30) 
can correctly reproduce the original value. However, there is no reason to believe that the ratio can keep the value one even though it is nontrivial near . Numerical analysis of the ratio also supports a value largely deviated from one [2].
Let us summarize the observations made in this subsection. The sliver state solution with given by (2.12) and by (2.13) satisfies the equation of motion in the Fock space, (2.15). However, this does not satisfy the equation of motion in the sliver space. Even if we choose another , given by (2.17), for which the inner product of the equation of motion with itself vanishes, the inner products with other sliver space states fail to vanish in general. Namely, there exists no universal for which the equation motion holds in the whole sliver space.
The origin of this trouble concerning the classical solution is the fact that infinitesimally violates the basic equation (2.9) in our level truncation regularization. Namely, of (2.26) does not satisfy the part of (2.9) obtained from (2.24) and (2.25):
(2.31) 
One might think that the troubles we have seen above are resolved once we find an ideal which does satisfy (2.31). We shall see in the next section that, even if there is such satisfying (2.31), it leads to completely uninteresting results concerning the observables in VSFT. Here we point out that (2.31) cannot completely fix the matrix . To see this, let us move to the representation of matrices in the level number space where the odd indices are in the upper/left block and the even ones in the lower/right block. Taking into account that is twisteven while is twistodd (both are symmetric), and in this representation are expressed as follows:
(2.32)  
(2.33) 
Substituting these expressions into (2.31), we obtain only one independent equation relating and :
(2.34) 
In the particular case that the size of the regularized matrices is an odd integer, , the relation (2.34) shows that (2.31) is selfcontradictory. In this case, is a rectangular matrix and necessarily has a zeromode. Eq. (2.34) implies that this zeromode is at the same time a zeromode of and hence the inverse does not exist.
3 Reexamining the tachyon wave function
In the previous section, we saw how the infinitesimal violation of the basic equation (2.9) leads to difficulties of the equation of motion (2.1) in the sliver space. In this section we shall examine the same kind of violation of the wave equation for the tachyon fluctuation mode.
3.1 Tachyon wave function in the Fock space
The tachyon (in general a physical state) fluctuation mode should satisfy the wave equation, namely, the linearized equation of motion of the fluctuation:
(3.1) 
where is the BRST operator around the classical solution which we expect to describe a D25brane. In this subsection, we shall recapitulate the construction of given in [2]. Like in the case of the classical solution , it is a nontrivial matter in which space the wave equation (3.1) holds. As we shall see in the next subsection, the tachyon mode constructed here satisfies (3.1) in the Fock space without any problem:
(3.2) 
However, there will arise subtle issues if we consider (3.1) in the sliver space.
In [2] the following form has been assumed for the tachyon mode carrying the centerofmass momentum :
(3.3) 
where is a normalization factor (which is irrelevant for (3.1) alone) and the state is a dependent deformation of the sliver state :
(3.4) 
The state depends on a vector which is even, , since the tachyon state is twisteven. Then the following formula holds for the product of the sliver state and the present tachyon mode:
(3.5) 
where the constant and a new vector are defined respectively by
(3.6)  
(3.7) 
Therefore, the wave equation (3.1) holds at
(3.8) 
provided is given by (2.13), the matrix satisfies (2.9), and the vector is a solution to
(3.9) 
In [2], by adopting and freely using the nonlinear relations among the matrices, the following vector was taken as a solution to (3.9):
(3.10) 
In (3.10), the second expression is due to from (A.7), and the last approximate expression has been obtained by substituting the expansions (2.21), (2.22) and (2.23) and keeping only the leading term with degree of singularity one. It has been shown numerically in [2, 4] and analytically in [18] that the expected value of the tachyon mass, is obtained for . Namely, we have
(3.11) 
3.2 Reexamination of the tachyon mode
Let us reexamine how precisely the equation determining , eq. (3.9), is satisfied by (3.10) in the level cutoff regularization. Using the formulas in appendix B, in particular, (B.6) and (B.10), we can show that the degree of singularity one part of is given for a generic with the expansion (2.24) and a generic with degree of singularity one by
(3.12)  
(3.13) 
Therefore, the even part and the odd one of (3.9) restricted to the degree of singularity one part read respectively
(3.14)  
(3.15) 
Taking (2.26), we see that the last expression of (3.10) satisfies neither (3.14) nor (3.15), though of course they are satisfied if we are allowed to carry out naive calculations by forgetting that the matrices are the regularized ones.
The violation of the degree of singularity one part of (3.9) observed above, does not invalidate the fact that the wave equation in the Fock space, (3.2), is satisfied by and at .^{∥}^{∥}∥ In order for (3.2) to hold for any Fock space state , must hold for any normalizable vector . Since the degree of singularity of such is at most one and hence the degree of the whole inner product is at most two, naive manipulations are allowed for calculating this inner product to give zero. In this sense, the violation of (3.9) found here is infinitesimal. However, let us consider whether a better choice of is possible which fully satisfies both (3.14) and (3.15). The degree one part of such an ideal should be given by (3.14) for a chosen . In order for the second equation (3.15) to be consistent, must satisfy (2.31), which is the part of (2.9). Therefore, an ideal exists for an ideal . Unfortunately, we have for such an ideal solution, implying that . This is seen as follows. Keeping in (3.6) only those terms with degree of singularity equal to three by the help of the formulas (B.6) and (B.10), we obtain the following concise expression of valid for any and which are both twisteven and carry degree of singularity one:
(3.16) 
where the last term with degree less three should be determined from the requirement that the whole of the RHS of (3.16) vanishes by naive manipulations. Plugging (3.14) into (3.16) and using (2.31), we get
(3.17) 
Therefore, we do not obtain physically sensible results if both the equations (2.9) and (3.9) hold rigorously.^{**}^{**}** If we respect only (3.14) and take , the corresponding value of no longer reproduces . Namely, we have
Let us return to generic and which do not necessarily satisfy (2.9) and (3.9). In [2, 4, 18], the tachyon wave equation is implicitly considered in the Fock space. However, recalling that the original string field in VSFT is expanded around a classical solution as
(3.18) 
with and being the fluctuation wave function and the corresponding component field (dynamical variable) for the open string mode ( tachyon, massless vector, etc.), it is necessary to examine the wave equation in the sliver space, or more specifically the inner product if we can choose for other modes so that the mixing with vanishes, . In the rest of this section we shall show how the tachyon mass in the sliver space determined by
(3.19) 
can differ from that in the Fock space, (3.8). We shall see that this discrepancy is again due to the infinitesimal violations of (2.9) and (3.9).
To identify the tachyon mass from (3.19), let us calculate its LHS by first calculating , in particular, , and then taking its inner product with .^{††}^{††}†† If we calculate directly like in the evaluation of the threetachyon coupling [2], we obtain the same result without referring to nor . This way of calculation is essentially carried out in sec. 4. The tachyon mass squared (4.5) obtained there agrees with (3.24) since we have [2, 4, 18]. Note first the following equation obtained by taking the inner product between and (3.5):
(3.20) 
with defined by
(3.21) 
Using (3.20) and (2.17), we find that the LHS of (3.19) is given by
(3.22) 
where is