Pair production of the open superstrings from the parallel-dressed D3-branes in the compact spacetime

We employ the boundary state formalism to compute the pair creation rate of the open superstrings from the interaction of two parallel D3-branes. The branes live in the partially compact spacetime. In addition, they have been dressed with the internal gauge potentials and the Kalb–Ramond field.


Introduction
The D-branes are effective tools to study the string theory [1].For investigating the D-branes interaction an appropriate approach is the boundary state formalism [2]- [14].Since the boundary state encodes all properties of the corresponding D-brane, it is a powerful tool for studying the various configurations of the D-branes.For example, a phenomenon similar to the Casimir effect, which is known as the open string pairs creation from the branes system, is one of these setups.The pair production of the open strings in the bosonic and superstring theories has been studied in various papers [15]- [25].The fundamental strings creation through the branes interaction demonstrates the attractive force between the D-branes [19]- [22].
Using the boundary state formalism, we shall compute the rate of the open superstring pair creation from the interaction of two parallel D3-branes.The branes have been embedded in the partially compact spacetime.Besides, the electric and magnetic fields live on them.Therefore, at first we shall compute the closed superstring amplitude for a general branes dimension p, and then, we rewrite the interaction amplitude for the case p = 3.For the small separation of the branes, the nature of the interaction (repulsion or attraction) completely is ambiguous.However, the annulus open superstring amplitude is an appropriate tool for describing such cases.Thus, from the cylinder amplitude we shall obtain the latter one.The one-loop annulus amplitude enables us to extract the rate of the open superstring pair creation.In fact, production of the superstring pairs obviously indicates the decay of the underlying system.As a special case, the pair production rate for the particular electromagnetic fields on the branes will be calculated.We shall see that the compactification, similar to the magnetic field, drastically increases the pair production rate.This paper is organized as follows.In Sec. 2, the boundary state, associated with a partially compact Dp-brane in the presence of the background fields, will be introduced.
In Sec. 3, the interaction amplitude of a system, which consists of two parallel-dressed Dpbranes, will be calculated.In Sec. 4, the decay rate of a system of two D3-branes via the open superstring pair production will be obtained.Sec. 5 is devoted to the conclusions.

The associated boundary state to the Dp-brane
We begin with the following sigma-model action for the closed string We consider a constant Kalb-Ramond background field B µν .For the internal U(1) gauge potential A α , we apply the Landau gauge A α = − 1 2 F αβ X β whit the constant field strength F αβ .We employ the flat worldsheet Σ with the metric h ab = η ab = diag(−1, 1), which lives in the flat spacetime with the metric Some of the brane directions and some of the normal directions to it are compacted on tori.Therefore, the subscript "n" and "c" refer to the non-compact and compact directions, respectively.For the perpendicular directions to the brane we shall use The bosonic portions of the boundary state equations are obtained by the variation of above action with respect to X µ (σ, τ ), where shows the total field strength.On the closed string boundary we applied (δX i ) τ =0 = (X i − y i ) τ =0 .The parameters y i s represent the location of the Dp-branes.The second equation implies that ∂ σ X i vanishes on the boundary.
Hence, the last term of the first equation is removed.
The solution of the equation of motion of closed string is For the non-compact and compact directions L µ is zero and N µ R µ , respectively.If the x µ -direction is compact it has the radius R µ .Thus, around this direction, the closed string possesses a winding number N µ .Note that we assume the time direction x 0 is non-compact.
By combining Eqs.(2.2) and (2.3) the boundary state equations take the features The last equation implies that the closed string cannot wind around the compact directions which are perpendicular to the brane.
The action (2.1) is invariant under the global worldsheet supersymmetry.Therefore, by using the replacements 2) we acquire the fermionic parts of the boundary state equations where η = ±1 is preserved for applying the GSO projection.In terms of the fermions oscillators these boundary conditions take the following forms where the index q is integer (half-integer) for the R-R (NS-NS) sector.The orthogonal Now we solve Eqs.(2.4) and (2.6).For the bosonic part, the solution of equations (2.4) is given by in which S µ ν = (Q α β , −δ i j ), and T p refers to the tension of the Dp-brane.The factor det (1 − F ) is originated from the path integral in the presence of the Gaussian boundary action [26]- [28].
Eqs. (2.6) give the boundary state of the NS-NS and R-R sectors as in the following ) R . (2.9) In comparison with the bosonic equation (2.7), the factor det (1 − F ) has been reversed.This is because of the presence of the Grassmannian variables in computing the path integral.The zero-mode part of the R-R sector possesses the form The conventional notation ; ; implies that we should expand the exponential with the convention that all Γ-matrices anticommute.Thus, for each value of p, we receive a finite number of terms.
The total boundary state, corresponding to the Dp-brane, in each sector is given by the product of the matter-and ghost-parts (2.12) The GSO-projected boundary states in the NS-NS and R-R sectors have the forms Since the conformal ghosts and super-conformal ghosts are not influenced by the background fields we shall apply the standard form of their boundary states.

The Dp-branes interaction
The overlap of the boundary states, via the closed string propagator, gives the treelevel interaction amplitude between two parallel-dressed branes The closed string propagator is as follows where L 0 and L0 are the right-and left-moving total Virasoro generators, including the bosonic, fermionic, conformal ghosts and super-conformal ghosts parts.The total interaction amplitude, via the exchange of the closed superstring, is the cylinder amplitude . Thus, we acquire where l αc = N αc R αc is the eigenvalue of the operator L αc , and d in is the dimension of the transverse non-compact directions.The transverse compact directions induce the Θ 3function in the second line.Besides, the Θ 2 -function comes from the R-R sector, and the Θ 3 -and Θ 4 -functions originate from the NS-NS sector.In addition, the quantities They obviously depend on the magnetic and electric fields on the branes.The integer part of (p±1)/2 is denoted by [(p±1)/2].The branes distance along the non-compact directions . Note that in the third line, the momentum components are extracted from the second equation of Eq. (2.4), i.e., p α 1(2) = − 1 α ′ F α 1(2) βc N βc R βc .From now on, for simplification, we restrict our setup to the case p = 3.This dimension also is appropriate for relating the D-branes to the real world.However, the amplitude for two parallel D3-branes takes the feature in which we employed the following identity for the theta-functions [29], According to the orthogonality of the matrix Λ and the characteristics of its eigenvalues, one of the ν a s is pure imaginary, e.g.ν 0 = iυ 0 .The range of the real ν a s and υ 0 are ν a ∈ [0, 1) and υ 0 ∈ (0, ∞), respectively.Now consider small distance of the branes (which is corresponding to the small value of "t").Thus, the infinite product form of Θ 1 (ν a |it) in the denominator of Eq. (3.3) gives the factor ∞ n=1 [1 − cosh(2πυ 0 )].Therefore, the sign of the amplitude will be proportional to (−1) ∞ , which is ambiguous.This demonstrates that the nature of the interaction (repulsion or attraction) obviously is ambiguous.This ambiguity indicates a new phenomenon.Precisely, the exponential factor exp (−Y 2 n /4α ′ t) for the small t and also the foregoing ambiguity play a significant role for occurrence of this new phenomenon.In fact, this new phenomenon implies the decay of the system via the open superstring pair production.

The rate of open superstring pair creation
The Jacobi transformation t → 1/t ′ will be used to calculate the open superstring one-loop interaction amplitude from the cylinder one.This enables us to understand the nature of the foregoing physical phenomenon.In fact, this new phenomenon comprises the open superstring pair production, which we will be found later.However, the annulus amplitude possesses the feature where C m (t ′ ) is given by To acquire this feature of Eq. ( 4.1), we applied the infinite product form of the Θ 1 -and Dedekind η-functions, and also their Jacobi transformation formulas, e.g.see Ref. [29].Now, by applying ν 0 = iυ 0 in the amplitude (4.1) we can obtain its imaginary part, which exhibits the pair production of the open superstrings, and hence the decay of the underlying system.Presence of the factor sin (πυ 0 t ′ ) in the denominator indicates the simple poles in the positive direction of the t ′ -axis, i.e., t ′ k = k/υ 0 where k is any positive integer number.Hence, each pole separately leads to the creation of an open superstring pair, and consequently deterioration of the branes system.The decay rate per unit volume of the D3-branes is defined by W 3,3 = −2ImA annulus /V 4 .Thus, we obtain in which the worldvolume of each D3-brane indicated by V 4 , and C ′ k (υ 0 , ν 1 ) has the definition

An special case
As we see, the above result is very complicated to accurately express the influence of the parameters of the setup on the decay rate.Thus, as an example, we shall rewrite the foregoing rate for the special matrices F 1 and F 2 .Therefore, let us choose the matrix F 1 (F 2 ), with the nonzero electric field E (E ′ ) and magnetic field B (B ′ ) for the first (the second) D3-brane, as in the following The eigenvalues λ 0 and λ 1 of the matrix Λ satisfy the equations Thus, υ 0 and ν 1 possess the forms tanh(πυ For this configuration, we receive the decay rate as in the following The Ω-function comes from the third line of Eq. (4.1).According to the delta-functions we applied E = E ′ and B = B ′ .Hence, the second equation of Eqs.(4.7) gives ν 1 = 0.

.11)
We observe that the pair creation rate has been prominently enhanced by the magnetic field.Though the two factors which contain the electric field are very small, by increasing the magnetic field the rate can be adjusted to any desirable value.In this case, the compactification radii R 2 and R 3 should be sufficiently small such that |E|R The exponential factors of Eqs.(4.11) and (4.12) elaborate that by compactifying some of the spatial directions the pair production rate possesses an enhancement.

Conclusions
In the context of the superstring theory we introduced a boundary state, associated with a Dp-brane, in which some of the longitudinal and transverse directions have been compacted on tori.The brane has been dressed with electric and magnetic fields.The closed superstring amplitude was also presented.Afterward, we changed the cylinder amplitude to the annulus one.
The 1-loop annulus amplitude enabled us to compute the decay rate of two parallel D3branes system through the pair production of the open superstrings.Since the background fields on the branes are arbitrary and different, the foregoing rate found a generalized form.However, for clarity we chose special electric and magnetic fields on the branes.Thus, we observed that for small electric fields the pair production rate is independent of the branes distance in the compact subspace.In addition, in comparison with the noncompact spacetime, this specific configuration demonstrated that the compactification enhances the value of the pair creation rate.Besides, the magnetic field also enhances the rate.Moreover, for acquiring a nonzero pair production rate, the electric fields are clearly needed to induce a polarized region between the branes.
) where A and B represent the 32-dimensional indices for the spinors, Γ µ s show the Dirac matrices in the 10-dimensional spacetime, |A | B is the vacuum of the R-R zero modes d µ 0 and dµ 0 , C defines the charge conjugation matrix, and the matrix U has the definition

2 2 , 3 ( 1 +
B 2 ) ≪ 1.As we see, the presence of the electric fields drastically plays the main role in the production of open superstring pairs, as expected.Let us rewrite this equation in the non-compact spacetime.Therefore, we should replaceY 2 n → Y 2 = 9 i=4 (y i 1 − y i 2 )2 and d in → 6. Accordingly, we receive W