Effective potential for revolving D-branes

Published for SISSA by Springer Received: January 8, Revised: March 14, Accepted: April 11, Published: April 29, 2019 2019 2019 2019 Satoshi Iso,a,b Hikaru Ohtaa,b and Takao Suyamaa a Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan b Graduate University for Advanced Studies (SOKENDAI), Tsukuba, Ibaraki 305-0801, Japan E-mail: , , Abstract: We quantize an open string stretched between D0-branes revolving around each other. The worldsheet theory is analyzed in a rotating coordinate system in which the worldsheet fields obey simple boundary conditions, but instead the worldsheet Lagrangian becomes nonlinear. We quantize the system perturbatively with respect to the velocity of the D-branes and determine the one-loop partition function of the open string, from which we extract the short-distance behavior of the effective potential for the revolving D0-branes. It is compared with the calculation of the partition function of open strings between D0-branes moving at a constant relative velocity. Keywords: Bosonic Strings, D-branes ArXiv ePrint: 1812.11505 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2019)151 JHEP04(2019)151 Effective potential for revolving D-branes Contents 1 Introduction 1 2 Open strings stretched between revolving D-branes 2.1 Open strings in the rotating coordinate system 2.2 Perturbative Hamiltonian with respect to v 4 4 6 7 8 8 10 12 4 Effective potential at short distance 4.1 Massive contributions: Vn≥1 (r, ω) 4.2 Massless contributions; V0 (r, ω) 12 13 14 5 D0-branes at constant velocities 16 6 Conclusions and discussions 20 A Improved perturbation theory 22 B Mode expansions 23 C Calculation of [H2 ]D C.1 Diagonal parts C.2 [V2 ]D C.3 [V1 V1 ]D 23 25 26 27 D Useful formulae for traces and derivation of the partition function (3.21) 30 E Matrix elements for low lying states 31 F Integration for V02 and V03 33 (l) G [H2 ]D for the linear system 1 33 Introduction D-branes in the superstring theory have played pivotal roles in understanding nonperturbative behaviors of string theory. They are also widely used in the string phenomenology and cosmology (for reviews see [1–3] and references therein). In many cases, static configurations of D-branes are considered, especially those with a fraction of supersymmetry –1– JHEP04(2019)151 3 One-loop partition function 3.1 Improved perturbation 3.2 Perturbative calculations of the trace 3.3 Diagonal elements: [H1 ]D , [H12 ]D and [H2 ]D 3.4 One-loop open string partition function ∞ Z Z= 0 ds  −2πsL0  Tr e =T 2s ∞ Z 0 y2 1 ds (8π 2 α0 s)− 2 e− 2πα0 s η(is)−24 2s (1.1) Q −2πms ) and T is the time duration of the configuration. where η(is) = e−2πs/24 ∞ m=1 (1 − e √ 0 If y  α , we can use the modular transformation η(is) = s−1/2 η(i/s) and the expansion η(i/s)−24 = exp(2π/s) + 24 + · · · to obtain the effective potential V(y) ∝ 24/y 23 that is dominated by the closed string massless modes such as a graviton, if the closed string √ tachyon is neglected. In the present paper, we are interested in the opposite limit y  α0 where low energy open string modes dominate the potential. The Dedekind η-function can 1 This is certainly different from a freely rotating classical open string which can be readily analyzed. –2– JHEP04(2019)151 preserved. It is partially because these configurations are of particular interest in mathematical settings but also because the analysis is simple and exact calculations can be performed. However, in many phenomenologically or cosmologically interesting situations, D-branes are moving and no supersymmetries are preserved. For example, if our universe is described by the brane-world scenario [4–7], those branes may have experienced irregular motions in the very early universe. In particular, if D-branes are accelerating to each other, they would emit closed string radiation [8–10], and particle creation of open strings would occur [11]. Then we may ask [12, 13]: what are the final configurations of such moving D-branes? Do D-branes collapse or scatter away from each other? In the D-brane scenario of universe and in the D-brane constructions of the standard model of particle physics, answering these questions will be relevant to study stability of our universe as well as the moduli stabilization, which may include the hierarchy problem of the electroweak scale against various UV scales. As a first step towards answering these questions, we study a pair of D0-branes of bosonic string theory which revolve around each other in the flat space-time and calculate potential between them. In this paper, we will not discuss the underlying mechanism of the rotation, but instead, we analyze properties of open strings stretched between such a pair of D0-branes.1 In particular, potential between D0-branes is read from the one-loop partition function of the open strings. If D0-branes are far from each other, the system is more appropriately described by the closed strings and the potential is given by the gravitational potential. In [14], the amplitude for the exchange of a single closed string between two D0-branes was obtained. The result is for D0-branes moving along arbitrary trajectories with small accelerations, and includes contributions from all massive closed string modes. See also [15]. On the other hand, if the distance is shorter than the string scale, massless open string modes dominate and the effective dynamics of D0-branes are described by the DBI action, or the Yang-Mills action if we neglect higher derivative terms [11, 16, 17]. In this paper, in order to take into account massive open string states as well as the massless states, we calculate one-loop partition function of open strings stretched between revolving D0-branes. We are interested in the behavior when the relative distance of D0-branes is shorter than the string length [16]. Let us here recall the well-known result of the one-loop open string partition function Z = −VT between D0-branes at rest with relative distance y. It is given [18] by P −2nπs where c be expanded as η(is)−24 = ∞ −1 = 1, c0 = 24, c1 = 324, c2 = 3200 n=−1 cn e and so on. Then the effective potential V can be calculated as a sum   ∞ y2 − 2nπ+ 2πα ds X 0 s V(y) = − √ cn e 3/2 2 0 8π α 0 2s n=0 r Z ∞ ∞ X cn y2 ds − 1 −s √ =− 2πn + s 2e , 0 2 0 2πα 0 2s 8π α 1 Z ∞ (1.2) n=0 12 V0 (y) = √ α0 162 V1 (y) = √ α0 r y2 4π 2 α0 1 1+ 2 (1.3)  y2 4π 2 α0  1 − 8  y2 4π 2 α0 ! 2 + ··· . (1.4) Here V0 (y) is the massless mode contribution and V1 (y) is the first excited massive mode contribution. What we would like to study in the present paper is the behavior of the effective potential V(y, ω) near y = 0 when D0-branes are revolving around each other with angular velocity ω and distance y = 2r. Since the potential is an analytic function of the velocity v = ωr of each D0-brane, it is naturally ex (...truncated)