Comments on new multiple-brane solutions based on Hata-Kojita duality in open string field theory

Recently, Hata and Kojita proposed a new energy formula for a class of solutions in Witten's open string field theory based on a novel symmetry of correlation functions they found. Their energy formula can be regarded as a generalization of the conventional energy formula by Murata and Schnabl. Following their proposal, we investigate their new ansatz for the classical solution representing double D-branes. We present a regularized definition of this solution and show that the solution satisfies the equation of motion when it is contracted with the solution itself and when it is contracted with any states in the Fock space. However, the Ellwood invariant and the boundary state of the solution are the same as those for the perturbative vacuum. This result disagrees with an expectation from the Ellwood conjecture.


Introduction
Since the seminal study of Murata and Schnabl [1,2], solutions for multiple D-branes in Witten's open string field theory [3] have been intensively considered. Very recently, there has appeared an interesting paper by Hata and Kojita [4]. They proposed a new way to construct multiple-brane solutions. Their work can potentially reform our conventional understanding on this subject.
The starting point for the discussion is estimation of the energy density of the Okawa-type solution, (1.1) Here K, B and c are symbols introduced to conveniently express a class of wedge states with operator insertions [5]. These three symbols satisfy simple algebraic relations called the KBc subalgebra. We (1.2) Here n a denotes the order of the pole (or the multiplicity of the zero times minus one) of G(z) at z = a. If we admit a formal object 1/K n and drop some surface terms, the anomalous term does not appear; however, it is not zero in the calculation of [2], except for the case n 0 = ±1, 0. Despite several efforts, fully acceptable definition of multiple-brane solutions based on (1.2) is not yet obtained. 1 In [4], Hata and Kojita argued that the pole at K = −∞ is in a sense equivalent to the pole at K = 0, and it also contribute to the energy density. This argument arise from a novel symmetry of correlation functions in the KBc subspace. This observation, together with (1.2), lead us to the following energy formula 2 E(Ψ F ) ∼ 1 2π 2 g 2 o (n 0 + n −∞ ) + (anomalous term) . (1.3) From this formula, we can make new ansatzes for multiple-brane solutions.
We here note that some previous studies [6][7][8][9][10] do not appear to be consistent with (1.3). Still, it might be premature to dismiss (1.3). The pole at K = −∞ is not fully considered so far, while it is related to the singularity of the identity string field. We empirically know that the identity string field requires quite careful treatment. Further investigation might resolve these apparent conflicts.
In this paper, we study the following ansatz of a classical solution presented in [4]: 3 This expression is singular in the sense that the energy density of (1.4) is indefinite. We then define a double-brane solution as a limit of sequence of regular string fields as follows: Here 1 ǫ is a regularized identity string field, defined by (1.6) The δ-sequence on the right-hand side of (1.6) has some special property, which is essential for our calculation. We will describe it in § 3.1. As an example of δ ǫ (x), we can take the following: . (1.7) We summarize properties of the solution (1.5) below: (1) The solution satisfies the equation of motion when it is contracted with any state in the Fock space.
(2) The solution satisfies the equation of motion when it is contracted to the solution itself. 1 In appendix C, we summarize our attempt to construct the double-brane solution based on (1.2). Although it is not successful, the regularization method obtained there is essential in the present work.
2 To be precise, the discussion of [4] is based on a particular regularization scheme, and the explicit form of the anomalous term is also derived. 3 Notation used in [4] is different from ours. See appendix D for details.
(3) The solution reproduces the energy density for double D-branes.
This is the first multiple-brane solution which satisfies both (1) and (2); however, it is still not clear whether (1) and (2)  Since our calculation is completely independent of the argument of [4], this agreement is interesting.
We note that regularization of the multi-brane solution is also claimed in [4] (K ǫη regularization).
Yet, if one calculates the energy density of (1.4) under the K ǫη regularization, one needs to use an analytic continuation method called the s-z trick [1] to obtain a finite value. Indeed, without the s-z trick, one needs to drop singular terms by hand. This fact is explained in § 2.5 of [4]. This means that if one uses the s-z trick, then the result is not equal to the original expression in some cases.
Since the s-z trick drastically simplifies calculation in many cases, and it is used in several studies, it is important to clarify that when it can be used as an identical transformation and when it cannot.
This paper is organized as follows: In § 2, we briefly introduce some preliminary materials and sketch a derivation of the formula (1.3). In § 3, we introduce the delta sequences appearing in (1.6), which is essential for our regularization. In § 4, we present the definition of the double-brane solution and check the equation of motion; and then we also calculate some physical quantities including the energy density, the Ellwood invariant and the boundary state. In § 5, we summarize our results. field theory, using a class of wedge states with operator insertions [11]. The KBc subalgebra was introduced by Okawa to express this class of wedge states with operator insertions [5]. Here K is a grassman even object, and the wedge state |n+1 is represented as e nK . The object B is grassman odd, and it represents line integral of the anti-ghost. The object c is also grassman odd, and it represents insertion of the c-ghost at the boundary. Together with the usual BRST operator Q, they satisfy the following algebraic relations: As described by Erler in [12], we can regard K, B and c as identity-based string fields. The commutation relations (2.1) then can be written using the star product: [K, B] = K * B − B * K, etc. 5 . The 4 In this paper, we use the word solution to refer to Ψ in (1.5) for simplicity; however, it is not precise in this sense. 5 The product symbol * is usually omitted when we express string fields using K, B and c.
space of string fields which can be written by K, B and c is closed under the star multiplication and the action of Q. Thus we can use the KBc subalgebra to find solutions to the equation of motion.
Consider the following formal solution to the equation of motion: Here F (K) is a function of K. Choice of F (K) determines physical properties of the solution. For instance, the choice F (K) = e K/2 corresponds to Schnabl's original tachyon vacuum solution, while the choice F (K) = (1 − K) −1/2 corresponds to the simple tachyon vacuum solution by Erler and Schnabl [13]. Each of these two solutions reproduces the energy density of the tachyon vacuum. We can also construct solutions with zero energy density with a suitable choice of F (K).
It is remarkable that the Okawa-type solution (2.3) can formally be written as a pure-gauge form: Since U or U −1 might be singular in general, Ψ F is not necessarily be a pure-gauge solution . For example, if we take F (K) = (1 − K) −1/2 , which corresponds to the simple tachyon vacuum solution, then the string field U has a factor 1/K. Generalizing the Okawa-type solution, a class of formal solutions which can formally be written as a pure-gauge form is presented in [9].
We mostly follow the convention of [5], except for the overall factors of K, B and c. See appendix D for details. For introduction to these topics including the KBc subalgebra, see reviews [14,15].

The inversion symmetry
In this subsection, we summarize a derivation of the formula (1.3). Let us start from the following homomorphisms of the KBc subalgebra [10,9], Bc . (2.5) These hatted objects,K,B andĉ, satisfy the same algebraic relations as the original K, B and c : Transformation law of the Okawa-type solution under these homomorphisms is simple, Let us concentrate on the special case f (K) = 1/K. We defineK,B andc as Hata and Kojita proved the following symmetry of the correlation function (the inversion symmetry): We call this relation the Hata-Kojita duality. From (2.11) and (1.2), we are lead to (1.3). 6 Note that we used the symbol ∼ = rather than = in (2.10). The reason is that the left-hand side has some singular terms, and the value is indefinite in the usual sense; if we define the string field e x/K as where 0 F 1 (; a; z) denotes a confluent hypergeometric function, then the left-hand side of (2.10) contains some identity-based terms, such as tr[BcK 2 cK 2 c 2 K 2 ]. To maintain the equivalence, we need to drop these terms. They are naturally dropped when we use the s-z trick.

Defining the solution as a limit
In this paper, we define the double-brane solution as a limit of a sequence of regular string fields (see (1.5)). That is, we consider one parameter family of string fields Ψ ǫ with a small parameter ǫ > 0, and regard the limit Ψ ≡ lim ǫ→0 Ψ ǫ as a solution to the equation of motion. We would like to clarify this point in the following.
For nonzero ǫ, the string field Ψ ǫ does not satisfy the equation of motion: We would like to require that the contraction of eom(Ψ ǫ ) and any state ϕ in the state space of the open string field theory vanishes as ǫ approaches 0: However, we do not know how to define the state space of the open string field theory. We then only require that lim ǫ→0 ϕ eom(Ψ ǫ ) vanishes when ϕ is any state in the Fock space and when ϕ is the solution lim ǫ→0 Ψ ǫ itself. Note that there is no relationship between these two requirements in general, for the state lim ǫ→0 Ψ ǫ usually lies outside the Fock space.
When we calculate the physical quantities from the solution, we take the limit ǫ → 0 at the end of the calculation. For example, the energy density of Ψ is defined as follows: We define the Ellwood invariant 7 and the boundary state in a similar fashion. See §4.4 and §4.5 for details. Above treatment of the equation of motion and physical quantities reflect an expectation that the state space of the open string field theory is complete with respect to some norm.

Regularization
In this section, we describe the regularization method used in this paper. In §3.1, we describe our delta sequence δ ǫ (x) . In §3.2, we describe the regularized identity state 1 ǫ .

A class of δ-sequence
Consider one parameter family of positive functions {δ ǫ (x)} with a small parameter ǫ > 0 . We require the following conditions on δ ǫ (x):

2.
∞ 0 δ ǫ (x)dx = 1 for ∀ ǫ > 0. Note that the lower limit of the integral in the condition 2 is zero. Since δ ǫ (x) is positive, We further require the following special condition on λ ǫ (t): 7 It is also commonly referred to as the gauge-invariant observable or the gauge-invariant overlap.

For 0 <
These three conditions characterize our delta sequence δ ǫ (t).

(3.4)
As a simple example of δ ǫ (x), we may take .
Note that this choice satisfies the following stronger condition: Then, it follows that even if we change the ratio of two small parameters on the left-hand side of (3.1), the right-hand side does not change:

Regularization of the identity state
Using the delta sequence δ ǫ (x) in (3.5), we define the regularized identity state 1 ǫ , Some correlators in the KBc subalgebra are singular, and their singularity is related to the identity string field. We can use the object 1 ǫ to regularize some of these correlators. For example, we regularize the correlator tr[cKcKcK] as follows: where we used (3.4). An explicit form of the correlation function tr[ce 2). Assuming (3.6), we can change the ratios of small parameters, Taking the opportunity, we comment on the regularization of the identity-based solution in the We define a regularized solution as follows: Similarly, we find That is, Ψ satisfies the equation of motion when it is contracted with the solution itself, and its energy density is zero. This result agrees with (1.3). We can also show that Ψ satisfies the equation of motion when it is contracted with any state in the Fock space. The energy density of Ψ is the same as that of the perturbative vacuum, yet we are not sure whether it is a pure-gauge solution. In this case, the formula (2.4) does not give a regular U , for it contains a negative power of K. In terms of the winding number [16], this is a question whether winding numbers around K = 0 are canceled by those of opposite sign around K = −∞.

Definition of the double-brane solution
In this section, we give a definition of the double-brane solution. According to the energy formula (1.3), the following ansatz for the solution is expected to have the energy density of double D-branes: The solution (4.1) is the symmetric counterpart of the following under the Hata-Kojita inversion: This is the familiar ansatz for the double-brane solution [1]. Since it contains the factor 1/K, the expression (4.2) itself is clearly singular . In contrast, the singularity of (4.1) is not so clear at first glance. However, there does exist an unobtrusive singularity, as essentially explained in [4], and the energy density of the solution is indefinite without suitable regularization. This seems to be consistent with the discussion by Erler [10], for the highest level in the dual L − level expansion of Ψ is zero.
Now, let us present the regularized definition of the solution in question: We also calculate the Ellwood invariant and the boundary state in § 4.4 and in § 4.5, respectively. We will see that both of them are the same as those for the perturbative vacuum.
If we change the position where 1 ǫ is inserted, then the properties of the solution drastically change.
This means that we can make several distinct solutions with different properties from the ansatz (4.1).
Note that this situation also occurs when we consider other ansatzes for solutions. We will discuss this subject in § 4.6.

Kinetic term
Let us calculate the normalized kinetic term E K for the solution (4.3), defined by Note that, if Ψ is a multiple-brane solution, the quantity E K +1 represents the multiplicity of D-branes.
Using the correlation function 10 we define the quantity E K (η, ǫ) as follows: Thanks to the relation (3.1), the regularized kinetic term can be expressed as 11 By a straightforward calculation, we find Here we changed integration variables from (u, v) to (s, v) ≡ (u + v, v). The functions c(s, ǫ, η) and s(s, ǫ, η) are given by For finite ǫ, we can change the order of the s-integral and the limit η → 0. We then obtain that Using the integration formula below, Therefore, we conclude that
As far as we keep ǫ 1 finite, we can take the limits ǫ 2 → 0 and ǫ 3 → 0 before we perform the s integral: This integral is the same as that appearing in (4.7). Therefore, we obtain

Equation of motion
So far we have confirmed that the solution reproduces the energy density for double D-branes. From Let φ be a state in the Fock space. Each term of the equation of motion can be written as follows: where we defined Above expressions are valid as far as C φ (x, y, z) is analytic around (x, y, z) = (0, u, v), (0, 0, u) and (0, u, 0). Since C φ (x, y, z) is a correlation function of three local operator insertions with a line integral of b-ghost, C φ (x, y, z) is regular for 0 ≤ x + y + z < ∞. Using integration by parts, we conclude that Ψ satisfies the equation of motion contracted with any state φ in the Fock space:

The Ellwood invariant
In [6], Ellwood conjectured that there exists a relation between the gauge-invariant observables of open string field theory which were discovered in [17,18], and the closed string tadpole on a disk. 12 In this paper, we call these gauge-invariant observables the Ellwood invariant.

The Ellwood invariant for a classical solution Ψ is defined by
Here, φ closed is a closed string vertex operator of weight (1,1) and ghost number 2; Ψ(0) is the operator corresponding to the classical solution Ψ, and f I • Ψ(0) is the conformal transformation of Ψ(0) under the map associated with the identity state, Ellwood conjectured that W(Ψ, φ closed ) is equivalent to the difference of two tadpole diagrams, Here A 0 (φ closed ) denotes the closed string tadpole on a disk with the original boundary condition, and A Ψ (φ closed ) denotes the closed string tadpole with the boundary condition corresponding to the classical solution Ψ.

In [1, 2], Murata and Schnabl calculated the Ellwood invariant for the Okawa-type solution (1.1).
Simply applying their formula to the solution (4.3), we find that the Ellwood invariant for the solution is zero, This means that the Ellwood invariant of the solution Ψ is that of the perturbative vacuum. In [8,9], the boundary states for different classical solutions in the KBc subalgebra are calculated.

Boundary states
Let | B denote the boundary state for the perturbative vacuum. The boundary state for the Okawatype solution (1.1) is given by From this formula, we find that the boundary state | B * (Ψ) for (4.3) is that of the perturbative vacuum,

Remarks on the ambiguity of classical solutions
Now, let us slightly modify the definition of the solution (4.3). We consider the following solution: It is straightforward to calculate the energy or the Ellwood invariant of Ψ. We summarize properties of this solution as follows: • The energy density of this solution is zero. 13 These expressions for x and y are valid only for the non-real solution (1.1). For more general expression, see [9].

• The equation of motion is satisfied when it is contracted to the solution itself and when it is
contracted with states in the Fock space.
• The Ellwood invariant is for the perturbative vacuum.
At least naively, both (4.11) and (4.3) can be considered as regularizations of (4.1). To define the solution without ambiguity, we need to regularize the solution and determine the order of limits. Note that these two solutions, (4.11) and (4.3), possess the same components at every level.
This kind of ambiguity is not limited to the ansatz (4.1). Take the identity-based solution −(1+K)c for example. As stated in § 3.2, it can be regularized as (3.9). On the other hand, as described in Zeze [21], we can also regularize it using the one parameter family of tachyon vacuum solutions that interpolates (1 − K)c and the simple tachyon-vacuum solution as follows: Two regularized solutions, (3.9) and (4.12), are different in physical properties. The energy density of (3.9) is zero, while that of (4.12) is −1/(2π 2 g 2 o ) . It is hoped to gain a deeper understanding of different regularization methods and be able to predict the properties of the regularized solutions without calculating the physical quantities.
Let us here state one more question about the solution (4.3). The expression (4.1) can formally be written as a pure-gauge form as follows: If we define the solution as (4.3), we expect that the solution is not true pure gauge. So, the gauge parameter U or U −1 must be singular in some sense. In particular, they must be disconnected to 1.
We need to understand in what sense it is singular and characterize the singularity.

Summary
We accepted. In particular, we need to clarify the relation of our results and the discussion by Baba and Ishibashi [19], where the authors proved the correspondence between the energy density and the

B Some limits of correlation functions
In this appendix, we explicitly calculate lim ǫ→0 E K (aǫ, bǫ) and lim ǫ→0 E C (aǫ, bǫ, cǫ) . We start from the expression (4.6). We take up the first term and consider the following limit: for 0 ≤ k ≤ 6, where we put α = 1 + b/a. For k = 0, the integral in this expression can be written as where Γ(z, ǫ) denotes the incomplete gamma function defined by We now differentiate (B.2) with respet to ǫ. Using the relations, Note that this series can be expressed in terms of trigonometric functions.
Similarly, we can derive the following expressions: for 0 ≤ k ≤ 6, lim ǫ→0 ∞ 0 ds s k (aǫ) 7−k (s + aǫ + bǫ) 8 e −s sin Using above formulae, we obtain the following expressions: From these expressions, we see that E K (x, y) satisfies the condition presented in footnote 8, and E C (x, y, z) satisfies the condition presented in footnote 9, respectively.

C On the ansatz (4.2) for the double-brane solution
Following Murata and Schnabl [1], several studies have been made to construct the multiple-brane solutions based on (1.2) [2,16]. The point here is that the expression (4.2) contains a factor 1/K, and we need to regularize it. In this appendix, we summarize our attempt to construct the double-brane solution based on the ansatz (4.2). We show that the regularized solution satisfies the equation of motion when it is contracted with the solution itself. We also show that the equation of motion is broken when it is contracted with some states in the Fock space. These results are similar as those of [2,16], where the solution is regularized using the ǫ-regularization.

C.1 Regularization
Consider a string field ϕ(Λ) with a large cutoff parameter Λ. We define a regularized string field ϕ R as follows: The following property is important for our discussion: We now would like to prove an identity which is similar to (3.1). Letf (ϕ(Λ 1 ), ϕ(Λ 2 )) be a bilinear function of two ϕ(Λ)s. For notational simplicity, we set f (Λ 1 , Λ 2 ) ≡f (ϕ(Λ 1 ), ϕ(Λ 2 )) . We assume that the function f (Λ 1 , Λ 2 ) is bounded for 0 ≤ Λ 1 , Λ 2 < ∞. We also assume that the following limits exist: lim a→∞ lim Λ→∞ f (Λ, aΛ) and lim Under these conditions, we can prove the following identity: To prove (C.4), we divide the parameter space of (s 1 , s 2 ) ∈ [0, 1] × [0, 1] into three parts: (C.5) In like manner, iff 3 is a bounded, trilinear function, we can show that Here , and we assumed that the limits on the right-hand side of (C.6) exist. To be precise, the identities (C.4) and (C.6) hold under milder conditions; however, we shall not pursue this matter here.

C.2 Regularized definition
The regularized form of the solution is given as follows: For convenience, we also define a string field Ψ cutoff (Λ) as which corresponds to the expression (C.1).

C.3 Energy density
In this subsection, we calculate the energy density of the solution (C.7).

C.3.1 Kinetic term
We start with evaluation of the normalized kinetic term E K (Ψ) for the solution (C.7), defined by Using the correlation function C K (x, y; u, v) ≡ tr e xK ce uK Bc Q(e yK ce vK Bc) , we define the quantity E K (Λ 1 , Λ 2 ) as follows: From the relation (C.4), it follows that We can carry out the differentiation with respect to u and v and the integration over x ′ and y ′ in (C.11) in a straightforward way. Since the integration over u and v in (C.10) is absolutely convergent, we can take the limit Λ → ∞ before the integration. We then find that Plugging this expression into (C.12), we obtain that

(C.16)
It is straightforward to derive the following expression: From this expression, we find that

C.4 Equation of motion contracted with states in the Fock space
From the calculation in the proceeding subsection, we conclude that the equation of motion is satisfied when it is contracted with the solution itself. Now, let us study the equation of motion contracted with states in the Fock space. For convenience, we define the string field Ψ Λ as follows:  Here Ci(x) denotes the cosine integral function and γ denotes the Euler-Mascheroni constant. The function F (x 1 , x 2 , x 3 , x 4 ) denotes the correlation Its explicit form is presented in (D.2). We also used the notation The correspondence between φ m,n and the states in the L 0 Fock space is given bỹ We also see that the following quantity is zero, since the matter one-point function vanishes. We then find that Thus, we conclude that the equation of motion is broken when it is contracted with some states in the Fock space. The constants presented in these expressions can be gathered into a single series, Comparison to the results in [2] In [2], the remainder of the equation of motion under the ǫ-regularization [2,16] was minutely investigated. In our notation, the regularization of the ansatz (4.2) under the ǫ-regularization is written as follows: The remainder of the equation of motion eom(Ψ ǫ ) is given by Using this expression, one can derive the following result: This result is consistent with that obtained in [2] (see the paragraph including (3.15) in [2]).

D Notation
In this appendix, we summarize our notation.

D.1 Conventions of the star-product
In this subsection, we clarify conventions of the star-product. 14 Let Φ 1 and Φ 2 be open string fields.
When we calculate the star-product Φ 1 * Φ 2 , we glue the right half (0 ≤ σ ≤ π/2) of Φ 1 to the left half (π/2 ≤ σ ≤ π) of Φ 2 . We call this convention the right-handed convention. This convention is convenient when we depict pictures of wedge states in the sliver coordinates.
On the other hand, in the original definition of the star-product [3], we glue the left half of Φ 1 to the right half of Φ 2 to calculate the star-product of Φ 1 and Φ 2 . We call this convention the left-handed convention. In order to avoid possible confusion, we here write the star-product in the left-handed convention as (Φ 1 * Φ 2 ) L . Translation from one convention to the other is simple: To write this appendix, we consult the following textbook in part: N. Ishibashi and K. Murakami, "String Field Theory -for a deeper understanding of string theory (Gen no ba no riron -gen riron no yori fukai rikai no tame ni)," Rinji Bessatsu Suuri Kagaku SGC Raiburari-92, Saiensu-sha, (2012) [ISSN0386-8257] (in Japanese).
Classical solutions in the left-handed convention Ψ L and that in the right-handed convention Ψ are related as follows: (D.1) Then, Ψ L and Ψ satisfy the equation of motion as follows: QΨ L + (Ψ L * Ψ L ) L = 0 , QΨ + Ψ * Ψ = 0 .

D.2 Definition of K, B and c
The KBc subalgebra is originally introduced to represent a class of wedge states with operator insertions. Let | n + 1 denote the wedge state of width n. If n is a natural number, then | n + 1 can be written as follows: | n + 1 = | 0 * · · · * | 0 n .
Using K, it is expressed as follows: | n + 1 = e nK .
All the functions of K appearing in this paper are defined as a superposition of wedge states, except for the formal object 1/K: In the left-handed convention, definition of basic elements of the KBc subalgebra is different from that in the right-handed convention: Under this definition, K L , B L and c L satisfy the same algebraic relations as K, B and c. As an example, let us write the solution (4.1) in the left-handed convention: At the end of this subsection, we clarify the overall factors of K, B and c. Our K and B are π/2 times those of Okawa's original definition [5], which we here write as K Okawa and B Okawa , respectively; our c is 2/π times that of the original definition, which we write as c Okawa :

D.3 Correlation functions
We use the notation tr[. . . ] to represent correlation functions in the KBc subalgebra. In our notation, the four point function is expressed as follows: tr Bce x 1 K ce x 2 K ce x 3 K ce x 4 K ≡ B L c(0) c(x 1 ) c(x 1 + x 2 ) c(x 1 + x 2 + x 3 ) C x 1 +x 2 +x 3 +x 4 = − s 2 4π 3 x 3 sin Here C r denotes a semi-infinite cylinder of circumference r. The character s represents the circumference of the cylinder; B L denotes a line integral of the b-ghost −i∞ i∞ dz b(z). In this case, the path of integration is along the line Re(z) = 0− . Note that (D.2) is not smooth at (x 1 , x 2 , x 3 , x 4 ) = (0, 0, 0, 0). This is a source of the singularity of correlation functions discussed in § 3.2.
The following two correlators are used when we calculate the energy density of classical solutions: C K (x, y; u, v) ≡ tr e xK ce uK Bc Q(e yK ce vK Bc)