Dynamics of revolving D-branes at short distances

Published for SISSA by Springer Received: September 27, 2019 Revised: December 11, 2019 Accepted: December 22, 2019 Published: January 29, 2020 Dynamics of revolving D-branes at short distances a Theory Center, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan b Department of Particle and Nuclear Physics, The Graduate University for Advanced Studies, SOKENDAI, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan c Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan E-mail: , , , Abstract: We study the behavior of the effective potential between revolving Dp-branes at all ranges of the distance r, interpolating r  ls and r  ls (ls is the string length). Since the one-loop open string amplitude cannot be calculated exactly, we instead employ an efficient method of partial modular transformation. The method is to perform the modular transformation partially in the moduli parameter and rewrite the amplitude into a sum of contributions from both of the open and closed string massless modes. It is nevertheless free from the double counting and can approximate the open string amplitudes with less than 3% accuracy. From the D-brane effective field theory point of view, this amounts to calculating the one-loop threshold corrections of infinitely many open string massive modes. We show that threshold corrections to the ω 2 r2 term of the moduli field r cancel among them, where ω is the angular frequency of the revolution and sets the scale of supersymmetry breaking. This cancellation suggests a possibility to solve the hierarchy problem of the Higgs mass in high scale supersymmetry breaking models. Keywords: D-branes, p-branes ArXiv ePrint: 1909.10717 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2020)182 JHEP01(2020)182 Satoshi Iso,a,b Noriaki Kitazawa,c Hikaru Ohtaa,b and Takao Suyamaa Contents 1 2 String threshold corrections in D-brane models 2.1 Why are the string threshold corrections important? 2.2 Partial modular transformation 2.3 Another example: D3-branes at angle 2.4 General recipe 3 4 6 7 8 3 Gauge theory calculations in revolving Dp-branes 3.1 SYM in a general background BI 3.2 One-loop amplitude of SYM in revolving Dp-branes 3.3 One-loop effective potential Ṽo (2r) from SYM 11 12 13 15 4 Supergravity calculations in revolving Dp-branes 4.1 Potential between D-branes mediated by supergravity fields 4.2 Supergravity potential of revolving Dp-branes 16 16 18 5 One-loop effective potential at all ranges of r 5.1 Shape of the one-loop contributions of the effective potential 5.2 Can the revolving D3-branes form a bound state? 19 20 22 6 Conclusions and discussions 23 A Supergravity potential between Dp-branes A.1 General formula for supergravity potential A.2 Supergravity potential in D1-branes at angle A.3 Supergravity potential between revolving branes 25 25 27 27 B ω expansion of SYM potential Ṽo (r) 29 C r expansion of SYM potential Ṽo (r) 32 1 Introduction D-branes in superstring theory play various essential roles, not only theoretically but also phenomenologically in particle physics and cosmology, and have been intensively and extensively investigated (see for example [1–3] for phenomenological works and [4–7] for cosmological works). But the dynamics has not yet been fully understood. Suppose two –1– JHEP01(2020)182 1 Introduction 1 Here, we set the string scale 2πα0 = 1 and the distance r has dimension of mass. –2– JHEP01(2020)182 Dp-branes are set in a target space-time. If they are at rest in parallel, it is a BPS configuration and stable. When they move at a relative velocity v, very weak attractive force is induced [8, 9]. Furthermore, due to the parametric resonance associated with the open string modes connecting Dp-branes [8, 10] and also due to the closed string emission [11, 12], the configuration loses its energy. What is the fate of these Dp-branes? They may be either separated apart or may be attracted to combine into a stack of Dp-branes with an enhanced gauge symmetry [13, 14]: beauty is attractive. If an appropriate initial condition is given, they may start revolving around each other and form a bound state. The motivation of the present work is to investigate such a possibility. The mechanism we search for is similar to the Coleman-Weinberg mechanism in the following sense. For a revolving motion, there is repulsive centrifugal potential. Thus, if there are no other attractive forces, revolving motion cannot be a solution. When two D-branes relatively move, the configuration generally violates the BPS condition and attractive force arises radiatively. Thus the question is whether the classical centrifugal potential can be balanced by the attractive force generated by one-loop radiative corrections of massive open string modes stretched between revolving D-branes. In order to answer whether such a stationary state exists or not, we calculate one-loop corrections to the interaction between two D3-branes revolving with each other. At large distances r > ls , we cannot expect a bound (resonant) state, since the induced attractive force is too weak compared to the centrifugal repulsive force. Thus we focus on the behavior of the potential at shorter distances r . ls . At short distances, the closed string picture is no longer valid and replaced by its dual open string picture [15]. Then the open string massless modes dominantly contribute to the potential between D-branes whose effective action is given by the supersymmetric Yang-Mills (SYM) theory. We first calculate the one-loop Coleman-Weinberg potential in a background corresponding to the two D3-branes revolving with each other with the radius r and the angular frequency ω. The U(2) gauge symmetry in D3-brane worldvolume is spontaneously broken to U(1) × U(1) by the Higgs mechanism, where the vacuum expectation value is given by the diameter 2r.1 Open strings stretched between D3-branes acquire mass ∼ 2r due to the Higgs mechanism. In addition, the revolution with the angular frequency ω breaks supersymmetry and the masses are split by an amount of ω between bosons and fermions. The one-loop Coleman-Weinberg potentials m(r, ω)4 log m(r, ω)2 generate the effective potential for the moduli field r. Since the potential must vanish at ω = 0, we expect that the potential between revolving D3branes is given by V ∝ ω 2 r2 in the leading order of r and ω expansions. Therefore, the moduli field r is expected to acquire mass proportional to the supersymmetry breaking scale ω. This is, however, not the end of the story since we have infinitely many massive open p string modes whose masses M are dominantly given by the string scale ms := 1/2πα0 , with additional r and ω corrections; M = M (ms , r, ω). One-loop corrections of the massive modes to the potential of r are given by the Coleman-Weinberg form M 4 log M 2 . Due to the supersy (...truncated)