Open string pair production from the interacting dressed-angled D-strings

We calculate the rate of the open string pair production from the interaction of two non-intersecting D-strings at angle. The D-strings have been equipped with the U(1) gauge potentials in the presence of the Kalb–Ramond field. This rate will be discussed when the D-strings intersect each other. Our computations is in the framework of the bosonic string theory.


Introduction
The D-branes are essential objects for various descriptions of the string theory [1,2,3].
For computing the lowest order stringy interaction amplitude of the D-branes in the closed string channel, one can apply the boundary state formalism [4]- [17].Equivalently, it is possible to calculate the same amplitude via the one-loop annulus of open string [1,14,18].By adding dynamics, background fields and different internal gauge potentials to the Dbranes, one can reveal more properties of them [11]- [15], [19]- [23].For the branes with a large separation, the interaction occurs via the exchange of the massless closed strings, which is attractive [14,20,23].However, in the small distance limit, the nature of the branes interaction (attraction or repulsion) completely is ambiguous.As we shall see, the open string channel provides the most accurate description for such configurations.
The electric field on a D-brane provides the necessary energy for creating the open string pairs, analogous to the Schwinger pair production [24].Thus, in the presence of the electric fields, the interaction amplitude of the D-branes possesses an imaginary part, which leads to the open string pair production.In fact, the production rate of the open string pairs is infinitesimally small.However, when the magnetic fields are applied to the system of the branes this rate is extremely modified [25]- [29].
The open string pair production, similar to the other investigations of the branes, has been usually studied for the parallel branes.However, the branes at angles have been also applied in various subjects, e.g. in the super Yang-Mills theories [30,31], in the modeling of the black holes [32], in the non-chiral supersymmetric field theories [33]- [35], and so on.Therefore, we shall study the open string pair production via two angled-dressed Dstrings (D1-branes) which are not intersecting.The D-strings have been dressed with the different U(1) gauge potentials and live in a nonzero Kalb-Ramond background field.The production rate will be also investigated for two intersecting D-strings.Our calculations is in the context of the bosonic string theory This paper is organized as follows.In Sec. 2, we obtain the boundary state, associated with a skew-dressed D-string in the presence of the Kalb-Ramond field.In Sec. 3, at first, we calculate the interaction amplitude of a system of two D-strings at angle with a nonzero distance.Then, we compute the pair production rate of the open strings from the interaction of this system.In the subsection 3.1, we investigate this rate for the intersecting D-strings.The conclusions will be given in Sec. 4.

The boundary state corresponding to an obliquedressed D-string
In order to compute the open string pair production, it is necessary to calculate the interaction amplitude.The angled D-strings (D1-branes) of the setup have been dressed with the U(1) gauge potentials in the presence of a constant background B-field.The interaction will be computed in the closed string channel.Thus, we use the boundary state formalism.Hence, at first we construct the boundary state, corresponding to a D-string along the x 1 -direction.So, we begin with the following string action with the constant field strength F αβ .In fact, this gauge dedicates a squared structure to the action (1).
Setting the action variation to zero yields the equation of motion and also the following boundary state equations where E ≡ F 01 − B 01 , the parameters y i indicate the position of the D-string, and |B ′ represents the corresponding boundary state of it.Now consider a D-string parallel to the x 1 x 2 -plane, which makes the angle φ with the x 1 -direction.By utilizing the well-known closed string mode expansion, the zero-mode part of the boundary state equations for the oblique D-string become where is the tension of the D-string.
The oscillating parts of the boundary state equations can be written in a unified form, i.e., By employing the coherent state method, we acquire the solution The prefactor √ 1 − E 2 comes from disk partition function [9].For the future purposes, we should note that the matrix M is orthogonal and hence Ω also is an orthogonal matrix.
In the bosonic string theory, the direct product |B tot = |B osc ⊗ |B 0 ⊗ |B g exhibits the total boundary state, associated with the D-string, where |B g is the well-known boundary state of the conformal ghosts.
3 Open string pair production from the non-intersectingangled D-strings In this section, we extract the rate of the open string pair creation via the interaction amplitude of two D-strings at angles.For preserving the generality, assume that the electric fields and angles of the D-strings with the x 1 -direction are different.Thus, the subscripts ( 1) and ( 2) will be used to show these differences.
In the closed string channel, the interaction of two D-branes takes place by exchanging closed strings between the branes.The geometry of the worldsheet of the exchanged closed string is a cylinder with τ as the coordinate along the length of the cylinder, 0 ≤ τ ≤ t, and σ as the periodic coordinate, i.e. 0 ≤ σ ≤ π.The interaction amplitude can be computed via the overlap of the two boundary states, through the closed string propagator , where H is the total Hamiltonian of the closed string.It includes the ghost and matter parts.After long calculations one finds 1−E 2 (2) with Φ ≡ φ (2) − φ (1) is the angle between the D-strings, and is the square distance between them.
Let us analyze the amplitude (6).The factor e (d−2)πt/12 comes from the Hamiltonian regularization, and V 1 = 2πδ(0) is the length of the D-string.The advent of | sin Φ| in the denominator, originates from the property δ(ax) = |a| −1 δ(x).This elucidates that the amplitude ( 6) is valid only for Φ = 0, and the parallel case is different and cannot be obtained from this by setting Φ = 0.In the latter configuration, for example, the Y is (d − 2)-dimensional and V 1 should be substituted by V 2 = (2π) 2 δ 2 (0).The position-dependent factor clarifies that the interaction amplitude is exponentially damped by increasing the distance between the D-strings.The infinite product is the contributions of the oscillatory part and the conformal ghosts portion.Precisely, the contribution of the conformal ghosts is (1 − e −2nπt ) 2 , and the contribution of the transverse directions to both The expression in the denominator of the infinite product can be rewritten in the form (1 ), where γγ ′ = 1.Therefore, we receive By writing γ ≡ e 2iπν , Eq. ( 7) takes the feature .
In the case of parallel D-strings there is cos Φ = 1.For this configuration we have 1 − E 2 (2) , e.g.see [25]- [29].This inequality elaborates that ν is a pure imaginary quantity.This is an essential element for calculating the production rate of the open string pairs.Hence, we utilize ν = iν in which 0 < ν < ∞.Consequently, Eq. ( 8) takes the feature We observe that the perpendicularity of the two D-strings is not possible unless for the case This condition imposes some restrictions on the angle Φ and the electric fields.Precisely, for the given electric fields E (1) and E (2) , for occurring the open string pair production the possible range of the angle Φ is defined by cos Φ > x + or cos Φ < x − , in which 1 − E 2 (2) .In fact, the electric field of a D-string is parallel to that D-string.Thus, the angle Φ represents the angle between the electric fields too.
Hence, for happening the pair creation of open strings, the electric fields should possess special alignments with each other.
By utilizing the Jacobi Θ-and Dedekind η-functions, the amplitude finds the following feature This amplitude is real.For the large D-strings separation, which is equivalent to t → ∞, the dominant contribution to the interaction arises from the massless closed strings.In this case we receive A closed | t→∞ > 0, which represents an attractive force.
We now consider the small "t"-integration.In this limit, the sign of the denominator in the infinite product of Eq. ( 6) becomes negative.Hence, the sign of the infinite product is unclear.This implies that any statement regarding the nature of the interaction (attraction or repulsion) is ambiguous.In fact, for the infinitesimal value of "t", the open string amplitude is more appropriate.Thus, let us employ the Jacobi transformation t → 1/t to convert the closed string amplitude (10) to the open string annulus amplitude.
With the use of we obtain For the next purposes, we write the amplitude (12) in the form where the effective mass of the open string, stretched between the D-strings, is given by When Y < π α ′ d−2 6 − 2ν 2 we receive the tachyonic shift.In this case, the integrand of Eq. ( 13) for t → ∞ diverges, which indicates the tachyonic instability.Therefore, a

Conclusions
We utilized the boundary state formalism to compute the pair production rate of the open strings from the interaction of two non-intersecting D1-branes at angle.The distance between the D-strings is arbitrary.The D-strings have been dressed with the different U(1) gauge potentials.Besides, the Kalb-Ramond field as a background was introduced.As a special case, we obtained the production rate of the photon pairs from the intersecting D-strings.
We observed that the production rate of the heavy open strings is very small, as expected.This is due to the fact that the required energy for the production of the heavy string pairs is large.Note that by increasing the distance between the D-strings and also by enhancing of the spacetime dimension we receive the heavy open strings.
We observed that for acquiring the open string pair creation, at least one of the Dstrings should be dressed with a nonzero electric field.Precisely, at least an electric field is obviously required to polarize the region between the branes.Otherwise, the pair production cannot occur.
where µ, ν are the spacetime indices and h ab , with a, b ∈ {τ, σ}, represents the metric of the closed string worldsheet.The indices α, β ∈ {0, 1} indicate the worldsheet of the D-string.For the next purposes the indices i, j ∈ {2, • • • , d − 1} will represent transverse directions to the D-string.We shall apply the flat metric G µν = η µν = diag(−1, 1, • • • , 1) for the spacetime, a constant Kalb-Ramond field B µν , and the Landau gauge and |B shows the boundary state of the oblique D-string.We employed the decomposition |B = |B 0 ⊗ |B osc .Using the quantum mechanical technics, the zero-mode portion of the boundary state finds the following solution