Winding Strings and Decay of D-Branes with Flux

We study the boundary state associated with the decay of an unstable D-brane with uniform electric field, 1 > e > 0 in the string units. Compactifying the D-brane along the direction of the electric field, we find that the decay process is dominated by production of closed strings with some winding numbers; closed strings produced are such that the winding mode carries precisely the fraction e of the individual string energy. This supports the conjecture that the final state at tree level is composed of winding strings with heavy oscillations turned on. As a corollary, we argue that the closed strings disperse into spacetime at a much slower rate than the case without electric field. 1 [email protected] 2 [email protected] 1. Closed Strings and Unstable D-Branes with Electric Field Decay of unstable D-branes [1] has been studied extensively in the classical limit of open string theory yet our understanding remains limited. Most notably, whether and precisely how closed string degrees of freedom emerge from the decay remains an issue. For some of effort in searching for fundamental strings in the low energy pictures can be found in [2][3][4][5]. Only in the limited setting of 1 + 1 dimensions, a successful recovery of closed strings has been seen, thanks to reinterpretation of old c = 1 matrix model as theory of many unstable D-branes [6][7][8][9]. For more generic string theories, an important breakthrough came about two years ago, when A. Sen wrote down time-dependent boundary states describing decay of unstable D-branes [10][11]. This set a stage for more precise questions about evolution of unstable D-branes in fully stringy setting [12][13][14][15][16][17][18][19]. So far, however, many successful efforts in this regard have been aimed at understanding the homogeneous final state. For questions related to dynamical objects, such as lower dimensional D-branes, the low energy approach still proves much more powerful and also somewhat unexpectedly accurate. One unusual property of this tachyonic system is that all fundamental degrees of freedom one starts with, namely those associated with open strings, will have to disappear eventually. This seems to raise a paradox since the dynamics involves no dissipation: all conserved quantities such as energy, momentum, and gauge charges must survive somehow. In particular, it is important to note that this has to be true even in the classical limit, gs → 0. We are accustomed to the idea that open string theory is self-contained at classical level and does not require presence of closed strings, however, and we are thus faced with quandary of where to attribute these conserved quantities after disappearance of all dynamical degrees of freedom available at tree level. Through study of low energy approximation [4][21][22], and also study of boundary states [10][11], both at tree level, some effective degrees of freedom have been found to carry these conserved quantities. They are called tachyon matter [11] and string fluid [4]. Both carry energy and momentum, while the latter carry in addition the fundamental string charge induced by the electric field on the worldvolume. They can both be viewed 3 Perhaps one of the more beautiful examples of this can be found in the construction of low energy dynamics of solitonic D-branes, starting with the severely truncated low energy approximation of unstable D-branes [20]. as fluid in that their state is specified completely by the distribution and the flow of energy, momentum, and electric flux. Their equation of states are such that no pressure is present other than the tension along the flux lines [4][22][23]. For instance, the Hamiltonian for the tachyon matter (in the absence of string fluid) collapses to [21] H = √ (π2 T )(1 + ( ~ ∂T )2), (1.1) where πT is the conjugate momentum to T . The resulting dynamics is that of a perfect fluid with density distribution πT and the rotationless velocity field ∂iT , i = 1, . . . , p, on the unstable Dp-brane worldvolume. While tachyon matter is intriguing on its own, life becomes more interesting when electric field is turned on. The final state of the decay encodes fundamental string charge in the conjugate momentum πi of the gauge field, which is nothing but the conserved electric flux obeying the Gauss law, ∂iπi = 0. The Hamiltonian with electric field is also pretty simple [4], H = √ (~π)2 + π2 T + ( ~ P )2 + (πi∂iT )2, (1.2) with the Poynting vector Pi = Fikπk + ∂iTπT . Again the equation of motion from this is fluidic, where the tachyon matter density, πT , interacts with the string fluid density, πi [23]. As the name suggests, the string fluid behaves remarkably like a continuum of noninteracting Nambu-Goto strings [4][5]. This prompted the speculation that the string fluid represents seed for formation of closed string as a coherent state of unstable open strings [4][24][16]. Irrespective of such effort, on the other hand, a more conservative interpretation is possible in the classical limit. Note that the total energy of an unstable D-brane scales as ∼ 1/gs, so with finite electric field, the amount of fundamental string charge per unit volume is enormous. This fact allows a macroscopic interpretation of the classical string fluid as a collection of large number of long winding strings [4][25]. At the end of decay process, nothing but closed strings would be left in full string theory, so this is interpretation is both natural and quite satisfying. Once this macroscopic interpretation of string fluid is accepted, it still remains to attribute the energy and momentum of tachyon matter. Again in real string theory, the only degrees of freedom at the end are closed strings, so it is tempting to view tachyon matter again as large number of highly massive closed string modes. From inspection of low energy dynamics, furthermore, it is suggested that these high frequency modes of strings are not independent of winding strings but rather must be on top of the latter. Here, one must imagine the closed strings carrying both the winding mode and additional high frequency oscillation modes with the latter moving up and down along the length of strings [25]. Note that the conserved electric flux πi is different from the electric field strength F0i in general. Assuming a static configuration, Pi = 0, and also πi∂iT = 0 for the sake of simplicity, we have the relationship [22][23], πi = HF0i (1.3) and πT = H √