Inhomogeneous tachyon dynamics and the zipper

Niko Jokela; Matthew Lippert

doi:10.1088/1126-6708/2009/08/024

Inhomogeneous tachyon dynamics and the zipper

Jokela, Niko; Lippert, Matthew 2009-08-07 00:00:00 JHEP08(2009)024 Published by IOP Publishing for SISSA Received: June 18, 2009 Accepted: July 16, 2009 Published: August 7, 2009 Niko Jokela and Matthew Lippert Department of Physics, Technion, 32000 Haifa, Israel Department of Mathematics and Physics, University of Haifa at Oranim, 36006 Tivon, Israel E-mail: [email protected], [email protected] Abstract: We study the process of inhomogeneous tachyon condensation in an intersect- ing D1- and anti-D1-brane system using an eﬀective tachyon DBI action. By switching to the Hamiltonian formalism, we numerically solve for the dynamical evolution of the system at a small intersection angle. We ﬁnd that the decay proceeds indeﬁnitely and resembles the action of two zippers moving away from the intersection point at the speed of light, zipping the branes together and leaving inhomogeneous tachyon matter behind. We also discuss the range of validity of our analysis and discuss the relation of the D1-anti-D1 description of the system to one in terms of an intersecting D1-D1-brane pair. Keywords: Tachyon Condensation, Intersecting branes models, D-branes SISSA 2009 doi:10.1088/1126-6708/2009/08/024 JHEP08(2009)024 Contents 1 Introduction 1 2 Review of previous analysis 3 3 Derivation of the D1-D1 action 4 3.1 EM tensor for tachyon matter 6 3.2 Mode analysis 6 4 Set-up 7 4.1 Hamiltonian formalism 7 4.2 Boundary conditions 8 5 Analysis 9 6 Relationship between D-D and D-D systems 15 7 Summary and discussion 18 1 Introduction The decay of branes via tachyon condensation is one of the few tractable string theory systems with true nontrivial, nonperturbative dynamical evolution. The process by which branes decay or annihilate is of fundamental relevancy and reveals important connections between open and closed string sectors. While much recent progress has been made in understanding open string tachyon physics, the system is still not well understood and many open questions remain. In particular, the annihilation of parallel D-branes and D-branes is suﬃciently simple to allow both a limited worldsheet analysis [1] as well as an eﬀective description in terms of a single homogeneous tachyon ﬁeld [2, 3]. The inhomogeneous case, however, is both more interesting and more complicated. For example, chiral symmetry breaking in the Sakai-Sugimoto model is an example of a physically relevant localized tachyon decay [4, 5]. Much of the eﬀort so far has been limited to studying time-independent inhomogeneous soliton solutions and marginal deformations [6] rather than on the dynamical evolution. Perhaps the simplest inhomogeneous system with a localized tachyon is where two D-branes intersect at an angle θ. The tachyonic mode is localized at the intersection, and as it condenses, the two branes reconnect. This recombination process plays a central role in realizing the Higgs mechanism in Standard Model on intersecting D-branes [7] and cosmological models of brane inﬂation [8]. In addition, this type of localized condensation – 1 – JHEP08(2009)024 may serve as a toy model for the reconnection transition found, for example, in the Sakai- Sugimoto model, in non-trivial curved backgrounds. For any nonzero value of the angle θ, as mentioned above, the open strings stretching between the branes have tachyonic modes in the low-lying mass spectrum. As is well- known from worldsheet calculations, the negative mass squared of the tachyon in the NS (Neveu-Schwarz) sector increase linearly with |θ| [9]. It is important to notice that when the angle θ becomes large, these are no longer the lowest mass excited states of the system. In particular, when the intersection angle θ passes through (perpendicular branes) a new tower of states becomes increasingly light and constitute the low-lying mass spectrum of the system [9–17]. Indeed, for the most extreme case θ ≃ π, it is more natural to view the system as a D-brane and a D-brane intersecting at an angle ϕ ≡ π − θ. Ideally, one would like to study the intersecting D-D system directly on the worldsheet by calculating the string scattering amplitudes in the rolling tachyon background [2, 18]. However, this requires turning on both the inhomogeneous tachyon deformation and a transverse scalar at the same time, a task which is currently out of reach (see [1, 19, 20] for work toward this direction in diﬀerent set-ups). We therefore resort to describing the inhomogeneous tachyon condensation from an eﬀective ﬁeld theory point of view. For economic reasons we will consider here intersecting D1-branes. The generalization to other Dp-branes and multiple angles is straightforward. In both ﬂat and curved space, there are known constructions for coincident D- D eﬀective actions [3, 6] and the consequent work on understanding their homogeneous decay [21]. However, when D-branes are even slightly oﬀ from being parallel to each other, little is known about their decay process in real time. This is mostly because initially the branes meet at an angle at a single point in spacetime making the tachyon condensation process highly inhomogeneous. To derive an eﬀective action for a D1-D1 system, we will start from a known action for a coincident D2-D2-brane pair with appropriate gauge ﬁelds and tachyons turned on. Then, after T-duality, we will be led to an action describing a D1-D1-brane pair which is initially intersecting at an angle ϕ in ﬂat spacetime background. An analysis of the spectrum of small ﬂuctuations around the false vacuum will give a tachyon mass which, by construction, is exact for small ϕ. Our main objective is to solve for the explicit temporal evolution, one which we will have to approach numerically. The inhomogeneous decay is described by two coupled two- dimensional ﬁelds whose equations of motion are not amenable to analytic methods. While we can not entirely trust the quantitative results, because we are solving an eﬀective theory and in addition doing so only approximately, our solutions will hopefully correctly capture the qualitative features of the decay. The branes decay in roughly two steps. First, starting from the intersection point, the separated D1 and D1 are attracted to each other in a manner that resembles being zipped together with the zipper traveling asymptotically at the speed of light. Once the branes are parallel, the tachyon begins to roll with constant velocity toward its vacuum, and, just as in the homogenous case, the dynamics of the branes can then be described in terms of a pressureless, non-interacting tachyon matter. – 2 – JHEP08(2009)024 Our results qualitatively match the expected evolution of straight intersecting branes reconnecting into curved separating branes, as has been seen in previous analyses in terms of D1 pairs [15, 17], except that in this D1-D1 description the separating branes are connected by a parallel D1-D1 pair with a rolling tachyon. We will argue, using a slightly modiﬁed example, that from the open string point of view, this parallel D1-D1 pair is equivalent to the vacuum, and so that the ﬁnal states of the D1-D1 and D1-D1 systems in fact diﬀer only by a change of variables. This paper is organized as follows. In section 2 we will review previous work on tachyon condensation in intersecting D-brane systems, and in section 3 we will turn to deriving an eﬀective action for D1-D1-branes. In addition, we will compute the false vacuum modes and the tachyon mass. Section 4 contains some steps necessary for numerical evaluation of the dynamics including a Legendre transformation to the Hamiltonian formalism and a discussion of the boundary conditions. In section 5 we will present and describe the results of the numerical computations. We will then explain in section 6 the relationship between describing the decay as that of two D-branes rather than as a D-D pair. Finally, section 7 comprises our summary and ideas for possible extensions of this work. 2 Review of previous analysis Let us ﬁrst recall some basic facts from the worldsheet. For simplicity, let us start with two parallel D1-branes in Type IIB superstring theory. Then consider dialing the intersection angle θ to some ﬁnite value. The open strings which connect the diﬀerent D1-branes will then get conﬁned about the intersection point because the tensions of the strings tend to minimize. By analyzing the NS sector of the energy spectrum of these localized strings one ﬁnds that the mass squared of the modes behaves linearly with the angle [9], 1 |θ| m = N − , N = 0, 1, 2, . . . . (2.1) WS 2 πα In the rest of the paper we shall only focus on positive angles, θ > 0 and ϕ = π − θ > 0, and will henceforth drop the absolute value symbols. We will be interested in the lowest, tachyonic excitation N = 0. Notice that the tachyonic mode exists at all (nonzero) angles θ and thus signals the instability of the conﬁguration. The part of the spectrum in (2.1) is reproduced to order O(θ) by the spectrum of ﬂuctuations around the intersecting D1-D1-brane pair background in a non-Abelian Yang- Mills (YM) theory [15, 17]. An intrinsic feature of the eﬀective ﬁeld theory approach is that the mass of the lowest mode behaves as tan 2 2 m = − , (2.2) πα rather than linearly as in (2.1). From (2.2) it is evident that the spectra only match at small angles, but the situation becomes increasingly worse at larger angles. This reﬂects the fact that the YM description is only viable for small angles. The negative mass squared of the tachyon means that the eigenfunction blows up exponentially in time [15, 17], tan ( ) −i m t − x πα T ∼ e e . (2.3) – 3 – JHEP08(2009)024 From (2.3) we also see that the tachyon ﬂuctuation modes are localized around the intersection point x = 0, agreeing with the worldsheet. In [15] it was further argued, that the geometric realization of the tachyon condensation is a D-brane recombination process. This was explicitly shown by diagonalizing the ﬂuctuations through a local gauge transformation of the brane-coordinates. Here, however, we shall postpone discussing this phenomenon to section 6. ′2 4 ′ One can also use the spectrum (2.1) to check the α F and higher order α corrections in the expansion of the non-Abelian Dirac-Born-Infeld (DBI) action and ﬁnd agreement [11, 16, 17]. However, equipped even with the full DBI action, thus far one has not been able to reproduce the mass spectrum (2.1) exactly. In the current paper, we are mostly interested in the behavior near θ ≃ π. Given the shortcomings of matching with the mass spectrum (2.1) it is better to look for alternative routes. If one dials the intersection angle all the way to θ → π, one ﬁnally arrives to a paral- lel D1-D1 conﬁguration, which has been investigated in numerous articles. We will therefore ﬁnd it promising to begin with a well-motivated tachyon DBI action for a D1-D1-brane pair, which, by construction, will be valid at small angles ϕ = π−θ ≈ 0, in contrast to the usual eﬀective action for the D1-D1-brane pair. Derivation of the D1-D1 action shall follow below. 3 Derivation of the D1-D1 action We begin by deriving the eﬀective action for the intersecting D1-D1 system. This can be acquired from the action for a coincident D2-D2-brane pair with equal and opposite magnetic ﬁelds. We then perform a T-duality along one of the directions longitudinal to the D2 and D2 which yields the desired D1-D1 action. The action for a pair of coincident D2-D2 is the usual tachyon DBI type action [6], (1) (1) 3 −φ S = −μ d x V (|τ|) e − det A + (1 ↔ 2) , (3.1) D2 2 ab where 2πα (i) (i) (i) (i) ′ ∗ ∗ A = η + ∂ Z ∂ Z + 2πα F + (D τ(D τ) + (D τ) D τ) (3.2) ab a b a b a b ab ab (i) (i) (i) F = ∂ A − ∂ A (3.3) a b ab b (1) (2) D τ = ∂ τ − i(A − A )τ . (3.4) a a a a By choosing to work in unitary gauge, we can set Im τ = 0 and will denote Re τ = T . With a view towards an upcoming T-duality in the y-direction, let us choose an ansatz (1) (2) where T and A depend on t and x but not y. We further set A = −A ≡ A. In y y y (i) φ (i) addition, we ﬁx the dilaton to be constant e = g and set the other scalars Z and the In [11] it was attempted to ﬁx the discrepancy by considering a symmetric trace prescription of [22] in the non-Abelian DBI action. Though this procedure did ﬁx the behavior for the lowest excitation, the spacing of the mass spectrum turned out to be incorrect. – 4 – JHEP08(2009)024 other components of the gauge ﬁeld to zero. The dynamical variables are then: F = ∂ A = A (3.5) ty t y F = ∂ A = A (3.6) xy x y D T = ∂ T = T (3.7) t 0 D T = ∂ T = T (3.8) x x D T = −2iAT (3.9) With this ansatz, D2 and D2 actions are identical and combine into ′ ′ S = −2 d x V (T ) − det (η + 2πα F + 2πα D TD T ) . (3.10) D2 ab ab a b Evaluating the determinant and integrating over the circle in the y-direction of radius R yields 2 ′ 2 ′ ′ 2 2 S = −2(2πR) d xV (T ) 1 + 2πα −T + T 1 + 8πα A T D2 ! 1 ′ 2 2 ′ ′ 3 ′ ′ ˙ ˙ ˙ +(2πα ) −A +A −(2πα ) AT −A T . (3.11) We now T-dualize in y to ﬁnd the action for intersecting D1-D1. Under the T-duality, ′ ′ α α 1 R → , g → g , and A → y. This yields the action: s s y R R 2πα S = −2 d x V (T ) (3.12) D1 2 2 4T y ′ 2 ′ ′2 2 ′2 ′ ′ ′ 2 ˙ ˙ × 1 − 2πα T + 2πα T 1 + − y˙ + y − 2πα (y T − y˙T ) , 2πα where μ = 2π α μ . The scalar y represents half the distance between the D1 and D1, 1 2 so the initial angle between the branes is then given by tan ϕ/2 = . (3.13) We can rescale the coordinates t and x and the ﬁeld y by (t, x, y) → 2πα (t, x, y), so we have an action in terms of dimensionless quantities: ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 S = −N d x V (T ) 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) , (3.14) D1 μ 2 ′ 1 where we deﬁne the normalization N = 2(2πα ) = . Motivated by [23], we take the g g s s tachyon potential to be V (T ) = , (3.15) cosh(βT ) where β = π for superstrings. The vacua for this potential are at T = ±∞, which implies that to reach them would require an inﬁnite amount of time. – 5 – JHEP08(2009)024 3.1 EM tensor for tachyon matter For later purposes let us record the energy-momentum tensor for the action (3.14). Let us generalize the above action (3.14) to curved space as follows ab S = −N dtdx −g V (T ) 1 + g (∂ y∂ y + ∂ T∂ T ) curved a b a b ! 1 2 2 ab ab cd ac bd +4T y (1+g ∂ T∂ T )+(g g −g g )∂ y∂ y∂ T∂ T .(3.16) a b a b c d The energy-momentum tensor is then extracted as 2 δS curved T = −√ (3.17) ab ab −g δg g=η N V (T ) = η L + ab ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) 2 2 2 × ∂ y∂ y + ∂ T∂ T + 4T y ∂ T∂ T + ∂ y∂ y(∂T ) a b a b a b a b +∂ T∂ T (∂y) − ∂ y∂ T∂y · ∂T . (3.18) a b (a b) Explicitly, ′2 ′2 2 2 ′2 1 + y + T + 4T y (1 + T ) T = N V (T ) r (3.19) 2 ′2 2 2 2 ′2 ′ ′ 2 ˙ ˙ 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) ′ ′ 2 2 ′ ˙ ˙ y˙y + TT + 4T y TT T = N V (T ) (3.20) 2 ′2 2 2 2 ′2 ′ ′ 2 ˙ ˙ 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) 2 2 2 2 2 −1 + y˙ + T − 4T y (1 − T ) T = N V (T ) . (3.21) ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) 3.2 Mode analysis As in the previous studies of D-D systems [15, 17], we can analyze the spectrum of ﬂuc- tuations of the action (3.14) around the initial state to identify the tachyonic mode. We begin at the maximum of the tachyon potential T = 0 and will also allow for the pos- sibility of small x-dependent corrections to the relative position of the D1 and D1, so y(x) = x tan(ϕ/2) + δy(x), where δy ≪ 1. The action (3.14) expanded to second order in T and δy is then 2 ′2 T T 1 2 2 2 2 2 S ∼ N dx − + + 4x tan(ϕ/2) − β (1 + tan(ϕ/2) ) T 2 2 2 2 2 + tan(ϕ/2)δy − (δy) + (δy) . (3.22) – 6 – JHEP08(2009)024 The position ﬂuctuations δy are free and decouple, so at this order the position of the branes is uncorrected. The equation of motion for the tachyon is ′′ 2 2 2 2 − T + T = 4x tan(ϕ/2) − β (1 + tan(ϕ/2) ) T . (3.23) Now, decompose T into modes of deﬁnite frequency Ω as −iΩ t T (t, x) = C e T (x) . (3.24) n n n=0 Normalizability of the modes implies a discrete spectrum, and therefore n is a nonnegative integer. The equation for the tachyon (3.23) now becomes just the Schro¨dinger equation for a harmonic oscillator with mass m and frequency ω, where mω = 2 tan(ϕ/2) and 2 2 2 2mE = β (1 + tan(ϕ/2) ) + Ω . Imposing the boundary condition T (∞) = 0, the n n solution for the modes is therefore −x tan(ϕ/2) T (x) = H x 2 tan(ϕ/2) e , (3.25) n n where H is the Hermite polynomial of order n. These modes are localized, as expected, near the intersection point at x = 0. The frequencies are given by 2 2 2 Ω = 4 tan(ϕ/2) n + − β 1 + tan(ϕ/2) . (3.26) 2 2 Focusing on the lowest mode n = 0 and plugging in β = π, we see that Ω < 0 for all values of ϕ, giving an exponentially-growing tachyonic mode. For small ϕ, Ω ≈ ϕ − π (3.27) which matches with the worldsheet calculation (2.1) of the tachyon mass. However, as with the D-D calculations in [15, 17], for larger angles the mass computed via the eﬀective action increasingly deviates from the exact result. 4 Set-up Having arrived at the eﬀective action (3.14), we now need to derive the Hamiltonian equa- tions of motion. In addition, we will also discuss the appropriate boundary conditions, particularly the modiﬁcations required for implementing numerical computations. 4.1 Hamiltonian formalism We will perform a Legendre transformation on the action (3.14) and work in the Hamil- tonian formalism [24–26]. Of course, the Lagrange equations of motion derived from the 2 ′ Recall that we are measuring dimensionful quantities in units of 2πα . – 7 – JHEP08(2009)024 action (3.14) are in principle equivalent, but for implementing a numerical solution, solv- ing four coupled ﬁrst-order diﬀerential equations was found to be easier than two coupled second-order diﬀerential equations. In addition, using the Hamiltonian framework facili- tates a description of the tachyon vacuum, V (T = ±∞) = 0. The canonical momenta are given by n o 2 2 ′ ′ ′ ˙ ˙ N V (T ) T 1 + 4y T + y y˙T − y T ∂L Π = = r (4.1) ∂T ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) n o ′ ′ ′ N V (T ) y˙ + T y˙T − y T ∂L Π = = r . (4.2) ∂y˙ ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) The Hamiltonian density is then H = Π T + Π y˙ − L (4.3) T y 2 2 2 ′ 2 2 2 ′ ′ ′ = Π (1 + T ) + Π 1 + 4y T + y + 2y T Π Π y T T y ! 1 2 2 2 2 ′ ′ 2 2 +N V (T ) y + (1 + T )(1 + 4y T ) (4.4) from which we derive the equations of motion: ′2 ′ ′ Π 1 + T + y T Π T y T = (4.5) n o 2 2 2 2 ′2 2 V ′2 ′2 2 2 −4Π y T − N V (T ) 4 1 + T y T + y + 1 + T 1 + 4T y Π = 2 ′ ′ 2 2 ′ 2 2 Π T + y Π Π + N V (T ) T 1 + 4T y T y +∂ (4.6) 2 2 ′2 ′ ′ Π 1 + 4y T + y + y T Π y T y˙ = (4.7) 2 2 2 2 ′2 2 −4Π T y − 4N V (T ) 1 + T T y Π = 2 ′ ′ 2 2 ′ Π y + T Π Π + N V (T ) y T y +∂ . (4.8) 4.2 Boundary conditions For describing the dynamical evolution, we begin at t = 0 with a straight, static D1 and D1 intersecting at an angle ϕ. This translates to y(0, x) = x tan(ϕ/2) and y˙(0, x) = 0. In order to initiate the tachyon rolling, we start it at rest, T (0, x) = 0, but slightly displaced from the maximum at T = 0 near x = 0. We do not consider a homogenous tachyon, both because the decay is localized near the intersection and, more importantly, as we will – 8 – JHEP08(2009)024 describe, for performing the numerical calculations the tachyon must be zero at the spatial cutoﬀ x . The initial tachyon proﬁle max 2 2 −x −x max T (0, x) = T e − e (4.9) is somewhat arbitrary, but we have checked that the subsequent evolution is largely in- sensitive to its shape. At the level of small perturbations at least, we can decompose the initial proﬁle into modes, and while the tachyon mode grows exponentially, the massive modes will just oscillate, contributing some small wiggles. Changing the overall constant T simply adjusts the timescale of the decay, as we will discuss in the next section. Since we will be solving the Hamiltonian equations of motion, we need to translate our initial conditions for T and y˙ into conditions for Π and Π . From (4.1) and (4.2) we ﬁnd T y that Π (t = 0, x) = 0 and Π (t = 0, x) = 0. T y In addition to initial conditions, we need to impose conditions at the spatial boundaries. Ideally, we would like to require that at spatial inﬁnities the system is in the false vacuum for all times. However, for numerical calculations, we employ a ﬁnite spatial cutoﬀ, x ∈ [−x , x ]. At these boundaries we need to ﬁx the ﬁelds, and they must satisfy the max max equations of motion there. We therefore ﬁx the ﬁnite boundary to be in the false vacuum, i.e., T (t,±x ) = 0 and y(t,±x ) = ±x tan(ϕ/2). Of course, we must be sure max max max to choose x large enough that the spatial cutoﬀ does not aﬀect the dynamics. Since max boundary contributions can only propagate at the speed of light, the region outside the future lightcone of the boundary should be insensitive to the cutoﬀ. We will be focussed on the relevant physics near the intersection point, so if we consider only times t < x max we will be free of spurious boundary eﬀects. As a check, we have varied x and shown max that for suﬃciently large values, our results are independent of the choice. 5 Analysis We solve the Hamiltonian equations (4.5), (4.6), (4.7), and (4.8) using Mathematica. For the results presented below, we choose the following representative parameters: N = 1 (5.1) ϕ = (5.2) x = 30 (5.3) max −3 T = 10 . (5.4) We ﬁnd that the numerical integration is only trustworthy up to some value of t. After this point numerical errors begin to grow large; for example, the stress tensor is no longer even approximately conserved. For these parameters, we found that the numerics broke down around t > 10. The solution for the tachyon as a function of t and x is shown in ﬁgure 1. Initially, the system remains close to the false vacuum. But then, near x = 0, the tachyon begins to grow, and the region of condensation spreads outward roughly at the speed of light. The corresponding evolution of y is shown in ﬁgure 2. For clarity, because y is antisymmetric, – 9 – JHEP08(2009)024 Figure 1. The tachyon as a function of t and x. Figure 2. y as a function of t and x. y(t, x) = −y(t,−x), we only plot positive values of x. Initially, the system again stays near the false vacuum and the branes are straight. Once the decay process begins, for x . t the branes are coincident with some small oscillations around y = 0. We can more easily illustrate certain features of the evolution by considering cross-sections of ﬁgures 1 and 2. In ﬁgure 3 we plot T (t, 0). There are three clearly distinguishable parts of the curve; at early time, for t . 3, the system remains close to the false vacuum T ≈ 0, then there is a brief transitional region where T begins to condense, rolling down the potential, and then for t & 4, the tachyon rolls at a constant velocity T ≈ 1 toward the vacuum at T = ∞. In addition, we plot in ﬁgure 4 the proﬁle of y(x) for various ﬁxed times, again only for x > 0. For t = 4, the branes are just beginning to pull toward each other. At later times, the proﬁle assumes a waterfall shape, again with three regions. Far from the intersection the branes retain their initial conﬁguration. But, at around x ≈ t, a point is reached where y → 0 rapidly, and then, in an increasingly large region around x = 0, the branes have – 10 – JHEP08(2009)024 Figure 3. The tachyon at x = 0 as a function of t. Figure 4. y as a function of x, for t = 2 (black), 4 (dotted red), 6 (dot-dashed blue), and 8 (dashed green). Figure 5. The position of the positive-velocity zipper z as a function of t. become coincident except for some small ﬂuctuations. The process resembles the action of two zippers moving away from the intersection point, zipping the branes together. We will deﬁne z(t) to be the position of the right-moving zipper and by symmetry the left-moving zipper is at −z(t). More speciﬁcally, for a given time, z equals the largest zero of y(x). Figure 5 presents a graph of z(t). For t . 4, z = 0 while at late time z˙ ≈ 1 with a sharp transition in between. – 11 – JHEP08(2009)024 Figure 6. The energy density T as a function of t and x. Notice the large spikes corresponding along the worldlines of the zippers. Figure 7. The momentum density T as a function of t and x. The upward spike corresponds to the worldline of the zipper with negative velocity while the trough corresponds to the zipper with positive velocity. We can further illustrate the physical properties of the evolution with plots of the com- ponents of the stress tensor. Figure 6 shows the energy density T . We have normalized the false vacuum energy so that T = 1. The motion of the zippers matches roughly with the large spikes in the energy density due to the large kinetic and gradient energy as the branes come together very rapidly. The position of the zippers is also clearly visible in ﬁgure 7 which plots the momentum density T ; the zipper moving to the left is represented by the spike and the one going to the right by the trough. The pressure T is shown in ﬁgure 8. The false vacuum has negative pressure, and again there are spikes corresponding to the worldlines of the zippers. Once the tachyon has begun to condense, T → 0. In addition, we checked numerically that T was conserved to very good accuracy by the evolution. μν We will ﬁrst focus on the transitional region between the false vacuum and the rolling – 12 – JHEP08(2009)024 Figure 8. The pressure T as a function of t and x. Note that the pressure vanishes in the condensing region. region, where the decay ﬁrst begins. We can identify three processes which characterize this region, and we can see that they are all related. Intuitively, the tachyon is massive where the branes are separated by more than the string length, so it cannot really begin rolling until the branes have been zipped together. However, it is the displacement of the tachyon from zero which gives a potential to y and pulls the branes together. Consequently, both of these processes must occur together. The timescale for the onset of the decay is controlled by the initial displacement parameter T ; the smaller T is, the longer it takes ǫ ǫ −3 for the decay to begin. For our choice of T = 10 , the tachyon starts rolling near the intersection at around t ∼ 3. At later times T (t, x) starts to grow when t ∼ x. Similarly, the zipper begins moving at t ∼ 4. As the zipper’s velocity approaches the speed of light, a given position on the brane begins to deviate from the false vacuum at around t ∼ x. The late time behavior, once the zipper has passed, very much resembles the decay of parallel D1-D1-branes but is inhomogeneous. We actually do not ﬁnd that y = 0 past the zipper but rather that y overshoots and oscillates around zero. As T increases, the masses of these oscillations grow, so their frequencies and amplitudes diminish. However, we should not really trust our eﬀective action to describe features such as these wiggles which are small compared with the string length. In the approximation that at late times y = 0, the dynamics becomes that of parallel branes, with the action for the tachyon becoming just 2 2 2 ′ S = −N d x V (T ) 1 − T + T . (5.5) D1 At late time, T is suﬃciently large that we may neglect V (T ). The Hamiltonian becomes just 2 2 H = Π (1 + T ) , (5.6) and the equations of motion (4.5) and (4.6) reduce to ˙ ′ T = 1 + T (5.7) – 13 – JHEP08(2009)024 Figure 9. To determine the extent to which the late-time tachyon obeys (5.7), we plot (5.11) as a function of x at ﬁxed t = 6. This function approximately vanishes in the zipped region, showing that the dust description is valid. Π T Π = ∂ . (5.8) T x 1 + T This is the well-studied pressureless tachyon dust [24–28] (see also [29]) where the velocity of the dust is given by v = −∂ T (5.9) μ μ and the local rest energy density is ǫ = p . (5.10) 1 + T We ﬁnd that the numerical solution accurately reproduces the tachyon dust once the branes have been zipped together. At x = 0 where T = 0, we saw in ﬁgure 3 that T → 1 in agreement with (5.7). More generally, we plot in ﬁgure 9 the function 2 ′ 1 − T + T (5.11) ˙ 2 ′ 1 + T + T which provides a normalized measure of the degree to which (5.7) is satisﬁed numerically. To good accuracy, (5.11) is zero in the condensing region around x = 0, implying that in fact (5.7) is approximately obeyed. Furthermore, we see from ﬁgure 8 that the condensing region behind the zipper is almost exactly pressureless. If this zipped region is essentially a parallel D1-D1, we arrive at the following inter- pretation for the ﬁnal conﬁguration of the system. At late times there are two curved D1-branes separating from each other and straightening but connected to each other by the parallel D1-D1. Based on the results from the D-D description, the two separating D1-branes is expected. However, rather than being devoid of branes, the region between them contains this decaying D1-D1 pair. In the next section we will discuss the relation between these two descriptions. – 14 – JHEP08(2009)024 U smooth fuzz time time increases decreases U singular Figure 10. The process of D-D decay and recombination is shown in two gauges. On the left, the intersection of two straight branes develops oﬀ-diagonal elements and becomes fuzzy. After a gauge transformation U, the branes have reconnected and are pulling apart. However, because U is singular at the beginning of the decay, the initial conditions are inequivalent. 6 Relationship between D-D and D-D systems We have been analyzing the system of intersecting D1-branes using a description in terms of a D-D pair. This choice of variables, at the level of an exact worldsheet description, amounts to an arbitrary gauge choice; a D1 is simply a D1 with opposite orientation, and a pair of D1-branes intersecting with an angle θ is equivalent to a D1 crossing a D1 at an angle ϕ = π − θ. However, the two eﬀective ﬁeld theory descriptions of these systems are inequivalent; each integrates out a diﬀerent set of massive modes, resulting in diﬀerent tachyonic and low-mass modes remaining. For example, using the YM approximation to the non-abelian DBI for the D-D pair, the tachyon’s mass (2.2) matches the worldsheet value (2.1) only for θ ∼ 0, while in the tachyon DBI description of the D-D system, the tachyon’s mass (3.27) matches for θ ∼ π. Even though the two eﬀective descriptions are not identical and the results need not to match a priori, as we will argue, they actually do produce dynamics which are the same, at least qualitatively. However, because the change of variables relating the two descriptions is nontrivially complicated, seeing that the results are physically similar can be a bit subtle. Before considering the relation between the D-D and D-D systems, we will begin with a simple illustrative example, a case where the two eﬀective descriptions are in fact exactly equivalent and the change of variables between them is explicit. In the intersecting D-D system, how one describes the decay process depends on one’s choice of gauge. In addition to choosing brane or anti-brane labels, the way the ends of the branes are connected is also to some degree gauge-dependent. The multiple equivalent descriptions can be easily seen from the non-abelian YM analysis of a D1-D1 decay [15], as illustrated in ﬁgure 10. – 15 – JHEP08(2009)024 In the initial variables, the matrix y giving the brane position is diagonal, corresponding to two straight D1-branes. But, as the decay proceeds, oﬀ-diagonal elements begin to grow, fuzzing out the intersection point. However, y can be rediagonalized by a gauge transformation U which exchanges the brane ends [15, 17]. In the new variables, the branes are disconnected, curved, and separating with time. Evolving back in time to the beginning of the decay, these two bent branes touch at a point. This odd conﬁguration is, of course, physically inequivalent to the initial straight branes, because at t = 0 the gauge transformation U relating them is singular. The important feature, which also appears in the D-D picture, is that in the origi- nal variables the decaying branes are not seen to annihilate at the intersection point, and the ends remain connected as in the initial state. Instead, the decay region is character- ized by an unusual, non-intuitive state such as the stringy fuzz, and only after a gauge transformation does the system appear as two disconnected branes. For the description in terms of an intersecting D-D pair, we showed in section 5 that the branes do not annihilate or reconnect into a disconnected pair of D1-branes. Such an event is, in fact, outside the range of the tachyon DBI eﬀective theory we used. Instead, the branes became zipped together, and where the D1 and D1 were parallel, the rolling tachyon was identiﬁed as inhomogeneous tachyon matter. However, although we do not have an explicit gauge transformation analogous to U in the D-D case, there is at least in principle a change of variables such that the parallel D1-D1 are replaced by a gap between a pair of disconnected D1-branes. Furthermore, in a more complete theory the parallel branes with tachyon matter could be alternatively described by a gas of closed strings and D0-branes (which themselves would decay) between the disconnected D1-branes. Although we can not perform the explicit change of variables for the intersecting D- D pair, we can illustrate how it works in a similar but physically distinct system. For the initially straight intersecting D1-D1 considered in section 5, the relative position of the branes y was always antisymmetric, y(x) = −y(−x). We can just as easily solve the equations of motion but with y(x) = y(−x) instead, although for numerical computation we have to smooth out the branes at x = 0 so as to avoid singular ﬁrst derivatives there. This symmetric boundary condition corresponds to two angled branes with opposite orientations which touch at the point where they are bent. In addition, we will choose the intersection angle to be ϕ = π/2. This system is unstable, and there are two diﬀerent directions in which it can decay corresponding to the two ways the branes can recombine, as shown in ﬁgure 11. One decay channel, mode (a) in ﬁgure 11, closely resembles the late-time state of intersecting D1-branes, except that instead of ending up with two separating D1-branes, the orientation of one of the D1-branes reversed. The numerical solution, presented in ﬁgure 12, shows the D1 and D1 growing further apart while the tachyon ﬁeld stays very close to zero. The other decay mode corresponds to the other way of recombining the ends of the branes and is (b) in ﬁgure 11. In the ﬁnal state there is still one curved D1 and one curved D1, but now they are connected. Numerically solving the equations of motion yields an For the numerical computations, which decay mode the system takes depends on how exactly the corner at x = 0 is resolved. The details, however, are not that relevant to our discussion. – 16 – JHEP08(2009)024 (a) (b) Figure 11. The decay of two angled D1-branes which touch at a point, such that y(x) is symmetric, can proceed in two ways. Final state (a) consists of a disconnected D1 and D1. Final state (b) also contains a separating D1 and D1, but they are connected by a parallel D1-D1. Figure 12. y as a function of x, for t = 0 (black), 2 (dotted red), 4 (dot-dashed blue), and 6 (dashed green). evolution qualitatively very similar to those of section 5; the branes zip together and then the tachyon rolls towards inﬁnity on the parallel D1-D1 pair connecting the two separating D1- and D1-branes. Initially, these two ﬁnal states appear quite diﬀerent, but in fact, they must be equiv- alent due to the symmetry of the initial state which is invariant under the exchange of x and y. This operation should map the ﬁnal states into each other, and it does up to the parallel D1-D1. However, if we also exchange the roles of x and y in the tachyon DBI action, writing x and T functions of y, the parallel D1-D1 would be present in decay (a) and not in mode (b). We conclude that the two modes are, up to this exchange of directions, diﬀerent de- scriptions of the same physical state. In this example the change of variables is particularly simple; whether a parallel D1-D1 connects the separating D1 and D1 or not depends on the The exchange corresponds to a π/2 rotation and an orientation reversal. – 17 – JHEP08(2009)024 choice of the worldvolume coordinates. At least at the level of open strings where we are working, both the parallel D1-D1 with a rolling tachyon and empty space are equivalent. In particular, since at late time the parallel D1-D1 with rolling tachyon does not support open string modes, it is eﬀectively as if there are no branes there at all. Although the change of variables involved in the intersecting D1-D1 system is much more complicated, the intuition gained from this symmetric example should hold. Because they are diﬀerent approximations of the same system, the non-Abelian DBI D-D eﬀective theory and the tachyon DBI D-D eﬀective theory do not give identical quantitative results. However, since we should regard the parallel D1-D1 as essentially the vacuum, both the D-D and D-D are qualitatively the same. 7 Summary and discussion In this paper we studied the D1-D1 conﬁguration, initially intersecting at an angle ϕ. This conﬁguration is unstable because tachyonic modes are present at all angles (except for the parallel D-D-brane pair ϕ = π). We modeled the evolution of this system by deriving the equations of motion for the tachyon T (t, x) and the separation ﬁeld y(t, x) from the tachyon DBI action and solved them numerically. We found that, at the very beginning of the evolution, the tachyon, which was localized at the intersection point, slowly rolled away from the maximum point of the eﬀective potential. Then, after some time value, which could be interpreted as the (local) lifetime of the brane system [30], the tachyon began to grow linearly with time and thus induced the dynamics for the separation ﬁeld y(t, x). As time passed, the branes were pulled toward each other such that the point where they ﬁrst met moved from 0 to larger ±x values. The process resembled that of two zippers moving at opposite directions with speeds of light, zipping the branes together, and continuing indeﬁnitely. The region between the zippers had the behavior of a decaying parallel D1-D1 pair, but an inhomogeneous one. At late time, we were able to capture the essential features of the remnant by the well-studied inhomogeneous pressureless tachyon dust. Although the tachyon DBI is by construction accurate for θ ∼ π while the YM de- scription is valid for θ ∼ 0, both these results in terms of a D1-D1 pair and the description in terms of intersecting D1-branes give qualitatively the same evolution. The ﬁnal state in both pictures contains two reconnected separating D1-branes. We have argued that the parallel D1-D1 connecting the branes in the tachyon DBI picture is an artifact of the choice of variables and is, in fact, equivalent to the gap between the branes in the YM description. In both of these eﬀective theories the trace prescriptions for the actions are somewhat ambiguous. In the case of D-D, the proposed action of [31] gives an alternative way of performing the trace and potentially gives diﬀerent results from those found here. The non- Abelian DBI eﬀective description of the D-D system should be valid not just at small angles. If the correct trace prescription could be found, for example, the worldsheet formula (2.1) for the mass at all angles may be reproduced. Another potential extension of these results is to other systems featuring localized tachyon condensation. One may have hoped this simple ﬂat-space system could serve as – 18 – JHEP08(2009)024 a toy model for inhomogeneous brane decays in non-trivial curved backgrounds. In the Sakai-Sugimoto model, for example, a parallel D8-D8 pair decays into a single U-shaped D8 via the condensation of a tachyon localized at small radii [4]. However, one important qualitative diﬀerence between this and intersecting branes is that in the ﬂat-space case the decay process continues without end and the condensing region does not stay localized but instead grows without bound. Acknowledgments We thank O. Bergman, M. Ja¨rvinen, R.G. Leigh, G. Lifschytz, and S. Nowling for useful discussions. This work was supported in part by the Israel Science Foundation under grant no. 568/05. In addition, N.J. has been supported in part at the Technion by a fellowship from the Lady Davis Foundation. References [1] A. Bagchi and A. Sen, Tachyon condensation on separated brane-antibrane system, JHEP 05 (2008) 010 [arXiv:0801.3498] [SPIRES]. [2] A. Sen, Rolling tachyon, JHEP 04 (2002) 048 [hep-th/0203211] [SPIRES]. [3] P. Kraus and F. Larsen, Boundary string ﬁeld theory of the DD system, Phys. Rev. D 63 (2001) 106004 [hep-th/0012198] [SPIRES]; T. Takayanagi, S. Terashima and T. Uesugi, Brane-antibrane action from boundary string ﬁeld theory, JHEP 03 (2001) 019 [hep-th/0012210] [SPIRES]; N.T. Jones and S.H. Henry Tye, An improved brane anti-brane action from boundary superstring ﬁeld theory and multi-vortex solutions, JHEP 01 (2003) 012 [hep-th/0211180] [SPIRES]; M.R. Garousi, D-brane anti-D-brane eﬀective action and brane interaction in open string channel, JHEP 01 (2005) 029 [hep-th/0411222] [SPIRES]; On the eﬀective action of D-brane-anti-D-brane system, JHEP 12 (2007) 089 [arXiv:0710.5469] [SPIRES]. [4] O. Bergman, S. Seki and J. Sonnenschein, Quark mass and condensate in HQCD, JHEP 12 (2007) 037 [arXiv:0708.2839] [SPIRES]. [5] N. Jokela, M. J¨arvinen and S. Nowling, Winding eﬀects on brane/anti-brane pairs, JHEP 07 (2009) 085 [arXiv:0901.0281] [SPIRES]. [6] A. Sen, Dirac-Born-Infeld action on the tachyon kink and vortex, Phys. Rev. D 68 (2003) 066008 [hep-th/0303057] [SPIRES]. [7] C. Bachas, A way to break supersymmetry, hep-th/9503030 [SPIRES]; C. Angelantonj, I. Antoniadis, E. Dudas and A. Sagnotti, Type-I strings on magnetised orbifolds and brane transmutation, Phys. Lett. B 489 (2000) 223 [hep-th/0007090] [SPIRES]; R. Blumenhagen, L. G¨orlich, B. K¨ors and D. Lu¨st, Noncommutative compactiﬁcations of type-I strings on tori with magnetic background ﬂux, JHEP 10 (2000) 006 [hep-th/0007024] [SPIRES]; G. Aldazabal, S. Franco, L.E. Iba´n˜ez, R. Rabada´n and A.M. Uranga, Intersecting brane worlds, JHEP 02 (2001) 047 [hep-ph/0011132] [SPIRES]; – 19 – JHEP08(2009)024 G. Aldazabal, S. Franco, L.E. Iba´n˜ez, R. Rabada´n and A.M. Uranga, D = 4 chiral string compactiﬁcations from intersecting branes, J. Math. Phys. 42 (2001) 3103 [hep-th/0011073] [SPIRES]; R. Blumenhagen, B. K¨ors and D. Lu¨st, Type I strings with F- and B-ﬂux, JHEP 02 (2001) 030 [hep-th/0012156] [SPIRES]; L.E. Iba´n˜ez, F. Marchesano and R. Rabada´n, Getting just the standard model at intersecting branes, JHEP 11 (2001) 002 [hep-th/0105155] [SPIRES]; R. Blumenhagen, B. K¨ors, D. Lu¨st and T. Ott, The standard model from stable intersecting brane world orbifolds, Nucl. Phys. B 616 (2001) 3 [hep-th/0107138] [SPIRES]; M. Cvetiˇc, G. Shiu and A.M. Uranga, Three-family supersymmetric standard like models from intersecting brane worlds, Phys. Rev. Lett. 87 (2001) 201801 [hep-th/0107143] [SPIRES]; Chiral four-dimensional N = 1 supersymmetric type IIA orientifolds from intersecting D6-branes, Nucl. Phys. B 615 (2001) 3 [hep-th/0107166] [SPIRES]; D. Cremades, L.E. Iba´n˜ez and F. Marchesano, Intersecting brane models of particle physics and the Higgs mechanism, JHEP 07 (2002) 022 [hep-th/0203160] [SPIRES]; R. Blumenhagen, V. Braun, B. K¨ors and D. Lu¨st, Orientifolds of K3 and Calabi-Yau manifolds with intersecting D-branes, JHEP 07 (2002) 026 [hep-th/0206038] [SPIRES]; C. Kokorelis, Exact standard model structures from intersecting D5-branes, Nucl. Phys. B 677 (2004) 115 [hep-th/0207234] [SPIRES]; R. Blumenhagen, L. G¨orlich and T. Ott, Supersymmetric intersecting branes on the type IIA T /Z orientifold, JHEP 01 (2003) 021 [hep-th/0211059] [SPIRES]; A.M. Uranga, Chiral four-dimensional string compactiﬁcations with intersecting D-branes, Class. Quant. Grav. 20 (2003) S373 [hep-th/0301032] [SPIRES]. [8] C. Herdeiro, S. Hirano and R. Kallosh, String theory and hybrid inﬂation/acceleration, JHEP 12 (2001) 027 [hep-th/0110271] [SPIRES]; J. Garcia-Bellido, R. Rabada´n and F. Zamora, Inﬂationary scenarios from branes at angles, JHEP 01 (2002) 036 [hep-th/0112147] [SPIRES]; R. Blumenhagen, B. K¨ors, D. Lu¨st and T. Ott, Hybrid inﬂation in intersecting brane worlds, Nucl. Phys. B 641 (2002) 235 [hep-th/0202124] [SPIRES]; K. Dasgupta, C. Herdeiro, S. Hirano and R. Kallosh, D3/D7 inﬂationary model and M-theory, Phys. Rev. D 65 (2002) 126002 [hep-th/0203019] [SPIRES]; N.T. Jones, H. Stoica and S.H. Henry Tye, Brane interaction as the origin of inﬂation, JHEP 07 (2002) 051 [hep-th/0203163] [SPIRES]; S. Sarangi and S.H. Henry Tye, Cosmic string production towards the end of brane inﬂation, Phys. Lett. B 536 (2002) 185 [hep-th/0204074] [SPIRES]; M. Gomez-Reino and I. Zavala, Recombination of intersecting D-branes and cosmological inﬂation, JHEP 09 (2002) 020 [hep-th/0207278] [SPIRES]; R. Brandenberger, G. Geshnizjani and S. Watson, On the initial conditions for brane inﬂation, Phys. Rev. D 67 (2003) 123510 [hep-th/0302222] [SPIRES]; M. Hindmarsh and H. Li, Perturbations and moduli space dynamics of tachyon kinks, Phys. Rev. D 77 (2008) 066005 [arXiv:0711.0678] [SPIRES]; W. Nelson and M. Sakellariadou, Tachyonic decay of unstable Dirichlet branes, Phys. Rev. D 78 (2008) 104012 [arXiv:0809.3958] [SPIRES]; Space-time dimensionality from brane collisions, Phys. Lett. B 674 (2009) 210 [arXiv:0810.0363] [SPIRES]; M. Hindmarsh and H. Li, Inhomogeneous tachyon condensation, JHEP 06 (2009) 050 [arXiv:0903.2019] [SPIRES]. [9] M. Berkooz, M.R. Douglas and R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265 [hep-th/9606139] [SPIRES]. – 20 – JHEP08(2009)024 [10] H. Arfaei and M.M. Sheikh Jabbari, Diﬀerent D-brane interactions, Phys. Lett. B 394 (1997) 288 [hep-th/9608167] [SPIRES]. [11] A. Hashimoto and W. Taylor, Fluctuation spectra of tilted and intersecting D-branes from the Born-Infeld action, Nucl. Phys. B 503 (1997) 193 [hep-th/9703217] [SPIRES]. [12] M.M. Sheikh Jabbari, Classiﬁcation of diﬀerent branes at angles, Phys. Lett. B 420 (1998) 279 [hep-th/9710121] [SPIRES]. [13] N. Ohta and P.K. Townsend, Supersymmetry of M-branes at angles, Phys. Lett. B 418 (1998) 77 [hep-th/9710129] [SPIRES]. [14] T. Kitao, N. Ohta and J.-G. Zhou, Fermionic zero mode and string creation between D4-branes at angles, Phys. Lett. B 428 (1998) 68 [hep-th/9801135] [SPIRES]. [15] K. Hashimoto and S. Nagaoka, Recombination of intersecting D-branes by local tachyon condensation, JHEP 06 (2003) 034 [hep-th/0303204] [SPIRES]. [16] S. Nagaoka, Fluctuation analysis of non-Abelian Born-Infeld action in the background intersecting D-branes, Prog. Theor. Phys. 110 (2004) 1219 [hep-th/0307232] [SPIRES]. [17] F.T.J. Epple and D. Lu¨st, Tachyon condensation for intersecting branes at small and large angles, Fortsch. Phys. 52 (2004) 367 [hep-th/0311182] [SPIRES]. [18] F. Larsen, A. Naqvi and S. Terashima, Rolling tachyons and decaying branes, JHEP 02 (2003) 039 [hep-th/0212248] [SPIRES]; N.D. Lambert, H. Liu and J.M. Maldacena, Closed strings from decaying D-branes, JHEP 03 (2007) 014 [hep-th/0303139] [SPIRES]; V. Balasubramanian, E. Keski-Vakkuri, P. Kraus and A. Naqvi, String scattering from decaying branes, Commun. Math. Phys. 257 (2005) 363 [hep-th/0404039] [SPIRES]; J. Shelton, Closed superstring emission from rolling tachyon backgrounds, JHEP 01 (2005) 037 [hep-th/0411040] [SPIRES]; N. Jokela, E. Keski-Vakkuri and J. Majumder, On superstring disk amplitudes in a rolling tachyon background, Phys. Rev. D 73 (2006) 046007 [hep-th/0510205] [SPIRES]; N. Jokela, M. J¨arvinen, E. Keski-Vakkuri and J. Majumder, Disk partition function and oscillatory rolling tachyons, J. Phys. A 41 (2008) 015402 [arXiv:0705.1916] [SPIRES]; N. Jokela, M. J¨arvinen and E. Keski-Vakkuri, N-point functions in rolling tachyon background, Phys. Rev. D 79 (2009) 086013 [arXiv:0806.1491] [SPIRES]; The energy-momentum tensor of perturbed tachyon matter, Phys. Rev. D 79 (2009) 106005 [arXiv:0901.3368] [SPIRES]. [19] S.-J. Rey and S. Sugimoto, Rolling tachyon with electric and magnetic ﬁelds: t-duality approach, Phys. Rev. D 67 (2003) 086008 [hep-th/0301049] [SPIRES]. [20] S.-J. Rey and S. Sugimoto, Rolling of modulated tachyon with gauge ﬂux and emergent fundamental string, Phys. Rev. D 68 (2003) 026003 [hep-th/0303133] [SPIRES]. [21] A. Sen, Tachyon dynamics in open string theory, Int. J. Mod. Phys. A 20 (2005) 5513 [hep-th/0410103] [SPIRES]. [22] A.A. Tseytlin, On non-abelian generalisation of the Born-Infeld action in string theory, Nucl. Phys. B 501 (1997) 41 [hep-th/9701125] [SPIRES]. [23] D. Kutasov and V. Niarchos, Tachyon eﬀective actions in open string theory, Nucl. Phys. B 666 (2003) 56 [hep-th/0304045] [SPIRES]. [24] G.W. Gibbons, K. Hori and P. Yi, String ﬂuid from unstable D-branes, Nucl. Phys. B 596 (2001) 136 [hep-th/0009061] [SPIRES]. – 21 – JHEP08(2009)024 [25] A. Sen, Fundamental strings in open string theory at the tachyonic vacuum, J. Math. Phys. 42 (2001) 2844 [hep-th/0010240] [SPIRES]. [26] P. Yi, Membranes from ﬁve-branes and fundamental strings from Dp branes, Nucl. Phys. B 550 (1999) 214 [hep-th/9901159] [SPIRES]; U. Lindstr¨om and M. Zabzine, Strings at the tachyonic vacuum, JHEP 03 (2001) 014 [hep-th/0101213] [SPIRES]; A. Sen, Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143] [SPIRES]; G. Gibbons, K. Hashimoto and P. Yi, Tachyon condensates, Carrollian contraction of Lorentz group and fundamental strings, JHEP 09 (2002) 061 [hep-th/0209034] [SPIRES]. [27] O. Bergman, K. Hori and P. Yi, Conﬁnement on the brane, Nucl. Phys. B 580 (2000) 289 [hep-th/0002223] [SPIRES]. [28] K. Hashimoto, Dynamical decay of brane-antibrane and dielectric brane, JHEP 07 (2002) 035 [hep-th/0204203] [SPIRES]. [29] A. Sen, Open and closed strings from unstable D-branes, Phys. Rev. D 68 (2003) 106003 [hep-th/0305011] [SPIRES]; Open-closed duality at tree level, Phys. Rev. Lett. 91 (2003) 181601 [hep-th/0306137] [SPIRES]; H.-U. Yee and P. Yi, Open/closed duality, unstable D-branes and coarse-grained closed strings, Nucl. Phys. B 686 (2004) 31 [hep-th/0402027] [SPIRES]. [30] D. Gaiotto, N. Itzhaki and L. Rastelli, Closed strings as imaginary D-branes, Nucl. Phys. B 688 (2004) 70 [hep-th/0304192] [SPIRES]. [31] M.R. Garousi, On modiﬁed tachyon DBI action, Nucl. Phys. B 817 (2009) 252 [arXiv:0811.4334] [SPIRES]. – 22 – http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of High Energy Physics IOP Publishing http://www.deepdyve.com/lp/iop-publishing/inhomogeneous-tachyon-dynamics-and-the-zipper-KH7icU4h2i

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References (96)

M. Garousi (2007)
On the effective action of D-brane-anti-D-brane system
Nuclear Physics, 809
(2001)
J. Math. Phys
G. Gibbons, K. Hori, Piljin Yi (2000)
String fluid from unstable D-branes
Nuclear Physics, 596
W. Nelson, M. Sakellariadou (2008)
Space–time dimensionality from brane collisions
Physics Letters B, 674
arXiv:hep-th/0009061
A. Sen (2002)
Field Theory of Tachyon Matter
Modern Physics Letters A, 17
C. Kokorelis (2002)
Exact standard model structures from intersecting D5-branes
Nuclear Physics, 677
Piljin Yi (1999)
Membranes from five-branes and fundamental strings from Dp-branes
Nuclear Physics, 550
F. Larsen, A. Naqvi, S. Terashima (2002)
Rolling tachyons and decaying branes
Journal of High Energy Physics, 2003
R. Blumenhagen, B. Kors, D. Lust, T. Ott (2002)
Hybrid inflation in intersecting brane worlds
Nuclear Physics, 641
R. Blumenhagen, B. Kors, D. Lust (2000)
Type I Strings with F- and B-Flux
Journal of High Energy Physics, 2001
Arjun Bagchi, A. Sen (2008)
Tachyon condensation on separated brane-antibrane system
Journal of High Energy Physics, 2008
Takuhiro Kitao, N. Ohta, Jian-Ge Zhou (1998)
Fermionic zero mode and string creation between D4-branes at angles
Physics Letters B, 428
(2004)
Fortsch
S. Nagaoka (2003)
Fluctuation analysis of the non-abelian Born-Infeld action in background intersecting D-branes
Progress of Theoretical Physics, 110
R. Blumenhagen, V. Braun, B. Kors, D. Lust (2002)
Orientifolds of K3 and Calabi-Yau manifolds with intersecting D-branes
Journal of High Energy Physics, 2002
K. Hashimoto (2002)
Dynamical Decay of Brane-Antibrane and Dielectric Brane
Journal of High Energy Physics, 2002
A. Sen (2003)
Open-closed string duality at tree level.
Physical review letters, 91 18
Keshav Dasgupta, C. Herdeiro, S. Hirano, R. Kallosh (2002)
D3-D7 inflationary model and M theory
Physical Review D, 65
R. Brandenberger, G. Geshnizjani, Scott Watson (2003)
Initial conditions for brane inflation
Physical Review D, 67
N. Jones, S. Tye (2002)
An Improved Brane Anti-Brane Action from Boundary Superstring Field Theory and Multi-Vortex Solutions
Journal of High Energy Physics, 2003
D. Gaiotto, N. Itzhaki, L. Rastelli (2003)
Closed strings as imaginary D-branes
Nuclear Physics, 688
G. Aldazabal, S. Franco, L. Ibáñez, R. Rabadán, Á. Uranga (2000)
D=4 chiral string compactifications from intersecting branes
Journal of Mathematical Physics, 42
J. Ellis, N. Mavromatos, D. Nanopoulos (2005)
Int. J. Mod. Phys.
Ho-Ung Yee, Piljin Yi (2004)
Open/closed duality, unstable D-branes and coarse-grained closed strings
Nuclear Physics, 686
arXiv:hep-th/9703217
A. Sen (2003)
Dirac-Born-Infeld action on the tachyon kink and vortex
Physical Review D, 68
G. Aldazabal, Sebastián Franco, Luis Ibáñez, R. Rabadán, Á. Uranga (2000)
Intersecting brane worlds
Journal of High Energy Physics, 2001
(1998)
Phys. Lett. B
N. Jones, H. Stoica, S. Tye (2002)
Brane interaction as the origin of inflation
Journal of High Energy Physics, 2002
M. Berkooz, Michael Douglas, R. Leigh (1996)
Branes intersecting at angles
Nuclear Physics, 480
(2004)
Prog
V. Balasubramanian, E. Keski-Vakkuri, P. Kraus, A. Naqvi (2004)
String Scattering from Decaying Branes
Communications in Mathematical Physics, 257
H. Kim, Jihn Kim, Hyun Lee (2002)
TeV scale 5D SU(3)W unification and the fixed point anomaly cancellation with chiral split multiplets
Journal of High Energy Physics, 2002
W. Nelson, M. Sakellariadou (2008)
Tachyonic decay of unstable Dirichlet branes
Physical Review D, 78
L. Ibáñez, F. Marchesano, R. Rabadán (2001)
Getting just the standard model at intersecting branes
Journal of High Energy Physics, 2001
Niko Jokela, E. Keski-Vakkuri, J. Majumder (2005)
Superstring disk amplitudes in a rolling tachyon background
Physical Review D, 73
D. Cremades, L. Ibáñez, F. Marchesano (2002)
Intersecting Brane Models of Particle Physics and the Higgs Mechanism
arXiv: High Energy Physics - Theory
N. Ohta, P. Townsend (1997)
Supersymmetry of M-branes at angles
Physics Letters B, 418
F. Epple, Dieter Lüst (2003)
Tachyon condensation for intersecting branes at small and large angles
Fortschritte der Physik, 52
Niko Jokela, M. Jarvinen, E. Keski-Vakkuri (2008)
N-point functions in rolling tachyon background
Physical Review D, 79
M. Hindmarsh, Huiquan Li (2007)
Perturbations and moduli space dynamics of tachyon kinks
Physical Review D, 77
U. Lindstrom, M. Zabzine (2001)
Strings at the tachyonic vacuum
Journal of High Energy Physics, 2001
(2003)
106003 [arXiv:hep-th/0305011]; A. Sen
A. Tseytlin, A. Tseytlin (1997)
On non-abelian generalisation of the Born-Infeld action in string theory
Nuclear Physics, 501
H. Arfaei, M. Jabbari (1996)
Different D-brane interactions
Physics Letters B, 394
Niko Jokela, M. Jarvinen, S. Nowling (2009)
Winding effects on brane/anti-brane pairs
Journal of High Energy Physics, 2009
(2003)
Phys. Rev. D
(2007)
JHEP
O. Bergman, S. Seki, J. Sonnenschein (2007)
Quark mass and condensate in HQCD
Journal of High Energy Physics, 2007
O. Bergman, K. Hori, Piljin Yi (2000)
Confinement on the brane
Nuclear Physics, 580
(2005)
Commun. Math. Phys. Phys. Rev. D J. Phys. A
Á. Uranga (2003)
Chiral four-dimensional string compactifications with intersecting D-branes
Classical and Quantum Gravity, 20
J. García-Bellido, R. Rabadán, F. Zamora (2001)
Inflationary scenarios from branes at angles
Journal of High Energy Physics, 2002
R. Blumenhagen, B. Kors, D. Lust, T. Ott (2001)
The Standard Model from stable intersecting brane world orbifolds
Nuclear Physics, 616
P. Kraus, F. Larsen (2000)
Boundary string field theory of the D anti-D system
Physical Review D, 63
(2005)
Int
(2002)
036 [arXiv:hep-th
arXiv:hep-th/0303133
M. Cvetič, G. Shiu, Á. Uranga (2001)
Chiral Four-Dimensional N=1 Supersymmetric Type IIA Orientifolds from Intersecting D6-Branes
Nuclear Physics, 615
A. Sen (2000)
Fundamental strings in open string theory at the tachyonic vacuum
Journal of Mathematical Physics, 42
M. Garousi (2004)
D–brane–anti–D–brane effective action and brane interaction in open string channel
Physical Review D
M. Garousi (2008)
On modified tachyon DBI action
Nuclear Physics, 817
M. Gómez-Reino, I. Zavala (2002)
Recombination of Intersecting D-Branes and Cosmological Inflation
Journal of High Energy Physics, 2002
M. Jabbari (1998)
Classification of different branes at angles
Physics Letters B, 420
Akikazu Hashimoto, IV WashingtonTaylor (1997)
Fluctuation spectra of tilted and intersecting D-branes from the Born-Infeld action
Nuclear Physics, 503
arXiv:hep-th/0304045
Niko Jokela, M. Järvinen, E. Keski-Vakkuri, J. Majumder (2007)
Disk partition function and oscillatory rolling tachyons
Journal of Physics A: Mathematical and Theoretical, 41
N. Lambert, Hong Liu, J. Maldacena (2003)
Closed strings from decaying D-branes
Journal of High Energy Physics, 2007
(2004)
Nucl
T. Takayanagi, S. Terashima, Tadaoki Uesugi (2000)
Brane-antibrane action from boundary string field theory
Journal of High Energy Physics, 2001
R. Blumenhagen, Lars Goerlich, B. Kors, D. Lust (2000)
Noncommutative compactifications of type I strings on tori with magnetic background flux
Journal of High Energy Physics, 2000
M. Cvetič, G. Shiu, Á. Uranga (2001)
Three-family supersymmetric standardlike models from intersecting brane worlds.
Physical review letters, 87 20
Saswat Sarangi, S. Tye (2002)
Cosmic string production towards the end of brane inflation
Physics Letters B, 536
C. Bachas (1995)
A WAY TO BREAK SUPERSYMMETRY
arXiv: High Energy Physics - Theory
S. Rey, S. Sugimoto (2003)
Rolling of a modulated tachyon with gauge flux and an emergent fundamental string
Physical Review D, 68
(2002)
020 [arXiv:hep-th
Ashoke Sen (2002)
Rolling Tachyon
(2001)
106004 [arXiv:hep-th 019 [arXiv:hep-th
R. Blumenhagen, Lars Goerlich, T. Ott (2002)
Supersymmetric Intersecting Branes on the Type IIA T6/Bbb Z4 Orientifold
Journal of High Energy Physics, 2003
(2003)
Phys
D. Kutasov, V. Niarchos (2003)
Tachyon effective actins in open string theory
(2000)
3103 [arXiv:hep-th 030 [arXiv:hep-th
K. Hashimoto, S. Nagaoka (2003)
Recombination of intersecting D-branes by local tachyon condensation
Journal of High Energy Physics, 2003
Niko Jokela, M. Jarvinen, E. Keski-Vakkuri (2009)
Energy-momentum tensor of perturbed tachyon matter
Physical Review D, 79
C. Angelantonj, I. Antoniadis, E. Dudas, A. Sagnotti (2000)
Type-I strings on magnetised orbifolds and brane transmutation
Physics Letters B, 489
(2000)
Nucl. Phys. B
C. Herdeiro, S. Hirano, R. Kallosh (2001)
String theory and hybrid inflation / acceleration
Journal of High Energy Physics, 2001
M. Hindmarsh, Huiquan Li (2009)
Inhomogeneous tachyon condensation
Journal of High Energy Physics, 2009
(2007)
S
A. Sen (2004)
Tachyon Dynamics in Open String Theory
International Journal of Modern Physics A, 20
(1999)
014 [arXiv:hep-th/0101213]; A. Sen, Mod
J. Shelton (2004)
Closed superstring emission from rolling tachyon backgrounds
Journal of High Energy Physics, 2005
A. Sen (2003)
Open and closed strings from unstable D-branes
Physical Review D, 68
S. Rey, S. Sugimoto (2003)
Rolling tachyon with electric and magnetic fields: T-duality approach
Physical Review D, 67
Eric Cornell, C. Wieman (2021)
BOSE-EINSTEIN CONDENSATION
Structural Aspects of Quantum Field Theory and Noncommutative Geometry

Copyright: Copyright © IOP Publishing Ltd
ISSN: 1126-6708
eISSN: 1029-8479
DOI: 10.1088/1126-6708/2009/08/024
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Abstract

JHEP08(2009)024 Published by IOP Publishing for SISSA Received: June 18, 2009 Accepted: July 16, 2009 Published: August 7, 2009 Niko Jokela and Matthew Lippert Department of Physics, Technion, 32000 Haifa, Israel Department of Mathematics and Physics, University of Haifa at Oranim, 36006 Tivon, Israel E-mail: [email protected], [email protected] Abstract: We study the process of inhomogeneous tachyon condensation in an intersect- ing D1- and anti-D1-brane system using an eﬀective tachyon DBI action. By switching to the Hamiltonian formalism, we numerically solve for the dynamical evolution of the system at a small intersection angle. We ﬁnd that the decay proceeds indeﬁnitely and resembles the action of two zippers moving away from the intersection point at the speed of light, zipping the branes together and leaving inhomogeneous tachyon matter behind. We also discuss the range of validity of our analysis and discuss the relation of the D1-anti-D1 description of the system to one in terms of an intersecting D1-D1-brane pair. Keywords: Tachyon Condensation, Intersecting branes models, D-branes SISSA 2009 doi:10.1088/1126-6708/2009/08/024 JHEP08(2009)024 Contents 1 Introduction 1 2 Review of previous analysis 3 3 Derivation of the D1-D1 action 4 3.1 EM tensor for tachyon matter 6 3.2 Mode analysis 6 4 Set-up 7 4.1 Hamiltonian formalism 7 4.2 Boundary conditions 8 5 Analysis 9 6 Relationship between D-D and D-D systems 15 7 Summary and discussion 18 1 Introduction The decay of branes via tachyon condensation is one of the few tractable string theory systems with true nontrivial, nonperturbative dynamical evolution. The process by which branes decay or annihilate is of fundamental relevancy and reveals important connections between open and closed string sectors. While much recent progress has been made in understanding open string tachyon physics, the system is still not well understood and many open questions remain. In particular, the annihilation of parallel D-branes and D-branes is suﬃciently simple to allow both a limited worldsheet analysis [1] as well as an eﬀective description in terms of a single homogeneous tachyon ﬁeld [2, 3]. The inhomogeneous case, however, is both more interesting and more complicated. For example, chiral symmetry breaking in the Sakai-Sugimoto model is an example of a physically relevant localized tachyon decay [4, 5]. Much of the eﬀort so far has been limited to studying time-independent inhomogeneous soliton solutions and marginal deformations [6] rather than on the dynamical evolution. Perhaps the simplest inhomogeneous system with a localized tachyon is where two D-branes intersect at an angle θ. The tachyonic mode is localized at the intersection, and as it condenses, the two branes reconnect. This recombination process plays a central role in realizing the Higgs mechanism in Standard Model on intersecting D-branes [7] and cosmological models of brane inﬂation [8]. In addition, this type of localized condensation – 1 – JHEP08(2009)024 may serve as a toy model for the reconnection transition found, for example, in the Sakai- Sugimoto model, in non-trivial curved backgrounds. For any nonzero value of the angle θ, as mentioned above, the open strings stretching between the branes have tachyonic modes in the low-lying mass spectrum. As is well- known from worldsheet calculations, the negative mass squared of the tachyon in the NS (Neveu-Schwarz) sector increase linearly with |θ| [9]. It is important to notice that when the angle θ becomes large, these are no longer the lowest mass excited states of the system. In particular, when the intersection angle θ passes through (perpendicular branes) a new tower of states becomes increasingly light and constitute the low-lying mass spectrum of the system [9–17]. Indeed, for the most extreme case θ ≃ π, it is more natural to view the system as a D-brane and a D-brane intersecting at an angle ϕ ≡ π − θ. Ideally, one would like to study the intersecting D-D system directly on the worldsheet by calculating the string scattering amplitudes in the rolling tachyon background [2, 18]. However, this requires turning on both the inhomogeneous tachyon deformation and a transverse scalar at the same time, a task which is currently out of reach (see [1, 19, 20] for work toward this direction in diﬀerent set-ups). We therefore resort to describing the inhomogeneous tachyon condensation from an eﬀective ﬁeld theory point of view. For economic reasons we will consider here intersecting D1-branes. The generalization to other Dp-branes and multiple angles is straightforward. In both ﬂat and curved space, there are known constructions for coincident D- D eﬀective actions [3, 6] and the consequent work on understanding their homogeneous decay [21]. However, when D-branes are even slightly oﬀ from being parallel to each other, little is known about their decay process in real time. This is mostly because initially the branes meet at an angle at a single point in spacetime making the tachyon condensation process highly inhomogeneous. To derive an eﬀective action for a D1-D1 system, we will start from a known action for a coincident D2-D2-brane pair with appropriate gauge ﬁelds and tachyons turned on. Then, after T-duality, we will be led to an action describing a D1-D1-brane pair which is initially intersecting at an angle ϕ in ﬂat spacetime background. An analysis of the spectrum of small ﬂuctuations around the false vacuum will give a tachyon mass which, by construction, is exact for small ϕ. Our main objective is to solve for the explicit temporal evolution, one which we will have to approach numerically. The inhomogeneous decay is described by two coupled two- dimensional ﬁelds whose equations of motion are not amenable to analytic methods. While we can not entirely trust the quantitative results, because we are solving an eﬀective theory and in addition doing so only approximately, our solutions will hopefully correctly capture the qualitative features of the decay. The branes decay in roughly two steps. First, starting from the intersection point, the separated D1 and D1 are attracted to each other in a manner that resembles being zipped together with the zipper traveling asymptotically at the speed of light. Once the branes are parallel, the tachyon begins to roll with constant velocity toward its vacuum, and, just as in the homogenous case, the dynamics of the branes can then be described in terms of a pressureless, non-interacting tachyon matter. – 2 – JHEP08(2009)024 Our results qualitatively match the expected evolution of straight intersecting branes reconnecting into curved separating branes, as has been seen in previous analyses in terms of D1 pairs [15, 17], except that in this D1-D1 description the separating branes are connected by a parallel D1-D1 pair with a rolling tachyon. We will argue, using a slightly modiﬁed example, that from the open string point of view, this parallel D1-D1 pair is equivalent to the vacuum, and so that the ﬁnal states of the D1-D1 and D1-D1 systems in fact diﬀer only by a change of variables. This paper is organized as follows. In section 2 we will review previous work on tachyon condensation in intersecting D-brane systems, and in section 3 we will turn to deriving an eﬀective action for D1-D1-branes. In addition, we will compute the false vacuum modes and the tachyon mass. Section 4 contains some steps necessary for numerical evaluation of the dynamics including a Legendre transformation to the Hamiltonian formalism and a discussion of the boundary conditions. In section 5 we will present and describe the results of the numerical computations. We will then explain in section 6 the relationship between describing the decay as that of two D-branes rather than as a D-D pair. Finally, section 7 comprises our summary and ideas for possible extensions of this work. 2 Review of previous analysis Let us ﬁrst recall some basic facts from the worldsheet. For simplicity, let us start with two parallel D1-branes in Type IIB superstring theory. Then consider dialing the intersection angle θ to some ﬁnite value. The open strings which connect the diﬀerent D1-branes will then get conﬁned about the intersection point because the tensions of the strings tend to minimize. By analyzing the NS sector of the energy spectrum of these localized strings one ﬁnds that the mass squared of the modes behaves linearly with the angle [9], 1 |θ| m = N − , N = 0, 1, 2, . . . . (2.1) WS 2 πα In the rest of the paper we shall only focus on positive angles, θ > 0 and ϕ = π − θ > 0, and will henceforth drop the absolute value symbols. We will be interested in the lowest, tachyonic excitation N = 0. Notice that the tachyonic mode exists at all (nonzero) angles θ and thus signals the instability of the conﬁguration. The part of the spectrum in (2.1) is reproduced to order O(θ) by the spectrum of ﬂuctuations around the intersecting D1-D1-brane pair background in a non-Abelian Yang- Mills (YM) theory [15, 17]. An intrinsic feature of the eﬀective ﬁeld theory approach is that the mass of the lowest mode behaves as tan 2 2 m = − , (2.2) πα rather than linearly as in (2.1). From (2.2) it is evident that the spectra only match at small angles, but the situation becomes increasingly worse at larger angles. This reﬂects the fact that the YM description is only viable for small angles. The negative mass squared of the tachyon means that the eigenfunction blows up exponentially in time [15, 17], tan ( ) −i m t − x πα T ∼ e e . (2.3) – 3 – JHEP08(2009)024 From (2.3) we also see that the tachyon ﬂuctuation modes are localized around the intersection point x = 0, agreeing with the worldsheet. In [15] it was further argued, that the geometric realization of the tachyon condensation is a D-brane recombination process. This was explicitly shown by diagonalizing the ﬂuctuations through a local gauge transformation of the brane-coordinates. Here, however, we shall postpone discussing this phenomenon to section 6. ′2 4 ′ One can also use the spectrum (2.1) to check the α F and higher order α corrections in the expansion of the non-Abelian Dirac-Born-Infeld (DBI) action and ﬁnd agreement [11, 16, 17]. However, equipped even with the full DBI action, thus far one has not been able to reproduce the mass spectrum (2.1) exactly. In the current paper, we are mostly interested in the behavior near θ ≃ π. Given the shortcomings of matching with the mass spectrum (2.1) it is better to look for alternative routes. If one dials the intersection angle all the way to θ → π, one ﬁnally arrives to a paral- lel D1-D1 conﬁguration, which has been investigated in numerous articles. We will therefore ﬁnd it promising to begin with a well-motivated tachyon DBI action for a D1-D1-brane pair, which, by construction, will be valid at small angles ϕ = π−θ ≈ 0, in contrast to the usual eﬀective action for the D1-D1-brane pair. Derivation of the D1-D1 action shall follow below. 3 Derivation of the D1-D1 action We begin by deriving the eﬀective action for the intersecting D1-D1 system. This can be acquired from the action for a coincident D2-D2-brane pair with equal and opposite magnetic ﬁelds. We then perform a T-duality along one of the directions longitudinal to the D2 and D2 which yields the desired D1-D1 action. The action for a pair of coincident D2-D2 is the usual tachyon DBI type action [6], (1) (1) 3 −φ S = −μ d x V (|τ|) e − det A + (1 ↔ 2) , (3.1) D2 2 ab where 2πα (i) (i) (i) (i) ′ ∗ ∗ A = η + ∂ Z ∂ Z + 2πα F + (D τ(D τ) + (D τ) D τ) (3.2) ab a b a b a b ab ab (i) (i) (i) F = ∂ A − ∂ A (3.3) a b ab b (1) (2) D τ = ∂ τ − i(A − A )τ . (3.4) a a a a By choosing to work in unitary gauge, we can set Im τ = 0 and will denote Re τ = T . With a view towards an upcoming T-duality in the y-direction, let us choose an ansatz (1) (2) where T and A depend on t and x but not y. We further set A = −A ≡ A. In y y y (i) φ (i) addition, we ﬁx the dilaton to be constant e = g and set the other scalars Z and the In [11] it was attempted to ﬁx the discrepancy by considering a symmetric trace prescription of [22] in the non-Abelian DBI action. Though this procedure did ﬁx the behavior for the lowest excitation, the spacing of the mass spectrum turned out to be incorrect. – 4 – JHEP08(2009)024 other components of the gauge ﬁeld to zero. The dynamical variables are then: F = ∂ A = A (3.5) ty t y F = ∂ A = A (3.6) xy x y D T = ∂ T = T (3.7) t 0 D T = ∂ T = T (3.8) x x D T = −2iAT (3.9) With this ansatz, D2 and D2 actions are identical and combine into ′ ′ S = −2 d x V (T ) − det (η + 2πα F + 2πα D TD T ) . (3.10) D2 ab ab a b Evaluating the determinant and integrating over the circle in the y-direction of radius R yields 2 ′ 2 ′ ′ 2 2 S = −2(2πR) d xV (T ) 1 + 2πα −T + T 1 + 8πα A T D2 ! 1 ′ 2 2 ′ ′ 3 ′ ′ ˙ ˙ ˙ +(2πα ) −A +A −(2πα ) AT −A T . (3.11) We now T-dualize in y to ﬁnd the action for intersecting D1-D1. Under the T-duality, ′ ′ α α 1 R → , g → g , and A → y. This yields the action: s s y R R 2πα S = −2 d x V (T ) (3.12) D1 2 2 4T y ′ 2 ′ ′2 2 ′2 ′ ′ ′ 2 ˙ ˙ × 1 − 2πα T + 2πα T 1 + − y˙ + y − 2πα (y T − y˙T ) , 2πα where μ = 2π α μ . The scalar y represents half the distance between the D1 and D1, 1 2 so the initial angle between the branes is then given by tan ϕ/2 = . (3.13) We can rescale the coordinates t and x and the ﬁeld y by (t, x, y) → 2πα (t, x, y), so we have an action in terms of dimensionless quantities: ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 S = −N d x V (T ) 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) , (3.14) D1 μ 2 ′ 1 where we deﬁne the normalization N = 2(2πα ) = . Motivated by [23], we take the g g s s tachyon potential to be V (T ) = , (3.15) cosh(βT ) where β = π for superstrings. The vacua for this potential are at T = ±∞, which implies that to reach them would require an inﬁnite amount of time. – 5 – JHEP08(2009)024 3.1 EM tensor for tachyon matter For later purposes let us record the energy-momentum tensor for the action (3.14). Let us generalize the above action (3.14) to curved space as follows ab S = −N dtdx −g V (T ) 1 + g (∂ y∂ y + ∂ T∂ T ) curved a b a b ! 1 2 2 ab ab cd ac bd +4T y (1+g ∂ T∂ T )+(g g −g g )∂ y∂ y∂ T∂ T .(3.16) a b a b c d The energy-momentum tensor is then extracted as 2 δS curved T = −√ (3.17) ab ab −g δg g=η N V (T ) = η L + ab ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) 2 2 2 × ∂ y∂ y + ∂ T∂ T + 4T y ∂ T∂ T + ∂ y∂ y(∂T ) a b a b a b a b +∂ T∂ T (∂y) − ∂ y∂ T∂y · ∂T . (3.18) a b (a b) Explicitly, ′2 ′2 2 2 ′2 1 + y + T + 4T y (1 + T ) T = N V (T ) r (3.19) 2 ′2 2 2 2 ′2 ′ ′ 2 ˙ ˙ 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) ′ ′ 2 2 ′ ˙ ˙ y˙y + TT + 4T y TT T = N V (T ) (3.20) 2 ′2 2 2 2 ′2 ′ ′ 2 ˙ ˙ 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) 2 2 2 2 2 −1 + y˙ + T − 4T y (1 − T ) T = N V (T ) . (3.21) ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) 3.2 Mode analysis As in the previous studies of D-D systems [15, 17], we can analyze the spectrum of ﬂuc- tuations of the action (3.14) around the initial state to identify the tachyonic mode. We begin at the maximum of the tachyon potential T = 0 and will also allow for the pos- sibility of small x-dependent corrections to the relative position of the D1 and D1, so y(x) = x tan(ϕ/2) + δy(x), where δy ≪ 1. The action (3.14) expanded to second order in T and δy is then 2 ′2 T T 1 2 2 2 2 2 S ∼ N dx − + + 4x tan(ϕ/2) − β (1 + tan(ϕ/2) ) T 2 2 2 2 2 + tan(ϕ/2)δy − (δy) + (δy) . (3.22) – 6 – JHEP08(2009)024 The position ﬂuctuations δy are free and decouple, so at this order the position of the branes is uncorrected. The equation of motion for the tachyon is ′′ 2 2 2 2 − T + T = 4x tan(ϕ/2) − β (1 + tan(ϕ/2) ) T . (3.23) Now, decompose T into modes of deﬁnite frequency Ω as −iΩ t T (t, x) = C e T (x) . (3.24) n n n=0 Normalizability of the modes implies a discrete spectrum, and therefore n is a nonnegative integer. The equation for the tachyon (3.23) now becomes just the Schro¨dinger equation for a harmonic oscillator with mass m and frequency ω, where mω = 2 tan(ϕ/2) and 2 2 2 2mE = β (1 + tan(ϕ/2) ) + Ω . Imposing the boundary condition T (∞) = 0, the n n solution for the modes is therefore −x tan(ϕ/2) T (x) = H x 2 tan(ϕ/2) e , (3.25) n n where H is the Hermite polynomial of order n. These modes are localized, as expected, near the intersection point at x = 0. The frequencies are given by 2 2 2 Ω = 4 tan(ϕ/2) n + − β 1 + tan(ϕ/2) . (3.26) 2 2 Focusing on the lowest mode n = 0 and plugging in β = π, we see that Ω < 0 for all values of ϕ, giving an exponentially-growing tachyonic mode. For small ϕ, Ω ≈ ϕ − π (3.27) which matches with the worldsheet calculation (2.1) of the tachyon mass. However, as with the D-D calculations in [15, 17], for larger angles the mass computed via the eﬀective action increasingly deviates from the exact result. 4 Set-up Having arrived at the eﬀective action (3.14), we now need to derive the Hamiltonian equa- tions of motion. In addition, we will also discuss the appropriate boundary conditions, particularly the modiﬁcations required for implementing numerical computations. 4.1 Hamiltonian formalism We will perform a Legendre transformation on the action (3.14) and work in the Hamil- tonian formalism [24–26]. Of course, the Lagrange equations of motion derived from the 2 ′ Recall that we are measuring dimensionful quantities in units of 2πα . – 7 – JHEP08(2009)024 action (3.14) are in principle equivalent, but for implementing a numerical solution, solv- ing four coupled ﬁrst-order diﬀerential equations was found to be easier than two coupled second-order diﬀerential equations. In addition, using the Hamiltonian framework facili- tates a description of the tachyon vacuum, V (T = ±∞) = 0. The canonical momenta are given by n o 2 2 ′ ′ ′ ˙ ˙ N V (T ) T 1 + 4y T + y y˙T − y T ∂L Π = = r (4.1) ∂T ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) n o ′ ′ ′ N V (T ) y˙ + T y˙T − y T ∂L Π = = r . (4.2) ∂y˙ ˙ 2 ′2 2 2 2 ′2 ′ ˙ ′ 2 1 − T + T (1 + 4T y ) − y˙ + y − (y T − y˙T ) The Hamiltonian density is then H = Π T + Π y˙ − L (4.3) T y 2 2 2 ′ 2 2 2 ′ ′ ′ = Π (1 + T ) + Π 1 + 4y T + y + 2y T Π Π y T T y ! 1 2 2 2 2 ′ ′ 2 2 +N V (T ) y + (1 + T )(1 + 4y T ) (4.4) from which we derive the equations of motion: ′2 ′ ′ Π 1 + T + y T Π T y T = (4.5) n o 2 2 2 2 ′2 2 V ′2 ′2 2 2 −4Π y T − N V (T ) 4 1 + T y T + y + 1 + T 1 + 4T y Π = 2 ′ ′ 2 2 ′ 2 2 Π T + y Π Π + N V (T ) T 1 + 4T y T y +∂ (4.6) 2 2 ′2 ′ ′ Π 1 + 4y T + y + y T Π y T y˙ = (4.7) 2 2 2 2 ′2 2 −4Π T y − 4N V (T ) 1 + T T y Π = 2 ′ ′ 2 2 ′ Π y + T Π Π + N V (T ) y T y +∂ . (4.8) 4.2 Boundary conditions For describing the dynamical evolution, we begin at t = 0 with a straight, static D1 and D1 intersecting at an angle ϕ. This translates to y(0, x) = x tan(ϕ/2) and y˙(0, x) = 0. In order to initiate the tachyon rolling, we start it at rest, T (0, x) = 0, but slightly displaced from the maximum at T = 0 near x = 0. We do not consider a homogenous tachyon, both because the decay is localized near the intersection and, more importantly, as we will – 8 – JHEP08(2009)024 describe, for performing the numerical calculations the tachyon must be zero at the spatial cutoﬀ x . The initial tachyon proﬁle max 2 2 −x −x max T (0, x) = T e − e (4.9) is somewhat arbitrary, but we have checked that the subsequent evolution is largely in- sensitive to its shape. At the level of small perturbations at least, we can decompose the initial proﬁle into modes, and while the tachyon mode grows exponentially, the massive modes will just oscillate, contributing some small wiggles. Changing the overall constant T simply adjusts the timescale of the decay, as we will discuss in the next section. Since we will be solving the Hamiltonian equations of motion, we need to translate our initial conditions for T and y˙ into conditions for Π and Π . From (4.1) and (4.2) we ﬁnd T y that Π (t = 0, x) = 0 and Π (t = 0, x) = 0. T y In addition to initial conditions, we need to impose conditions at the spatial boundaries. Ideally, we would like to require that at spatial inﬁnities the system is in the false vacuum for all times. However, for numerical calculations, we employ a ﬁnite spatial cutoﬀ, x ∈ [−x , x ]. At these boundaries we need to ﬁx the ﬁelds, and they must satisfy the max max equations of motion there. We therefore ﬁx the ﬁnite boundary to be in the false vacuum, i.e., T (t,±x ) = 0 and y(t,±x ) = ±x tan(ϕ/2). Of course, we must be sure max max max to choose x large enough that the spatial cutoﬀ does not aﬀect the dynamics. Since max boundary contributions can only propagate at the speed of light, the region outside the future lightcone of the boundary should be insensitive to the cutoﬀ. We will be focussed on the relevant physics near the intersection point, so if we consider only times t < x max we will be free of spurious boundary eﬀects. As a check, we have varied x and shown max that for suﬃciently large values, our results are independent of the choice. 5 Analysis We solve the Hamiltonian equations (4.5), (4.6), (4.7), and (4.8) using Mathematica. For the results presented below, we choose the following representative parameters: N = 1 (5.1) ϕ = (5.2) x = 30 (5.3) max −3 T = 10 . (5.4) We ﬁnd that the numerical integration is only trustworthy up to some value of t. After this point numerical errors begin to grow large; for example, the stress tensor is no longer even approximately conserved. For these parameters, we found that the numerics broke down around t > 10. The solution for the tachyon as a function of t and x is shown in ﬁgure 1. Initially, the system remains close to the false vacuum. But then, near x = 0, the tachyon begins to grow, and the region of condensation spreads outward roughly at the speed of light. The corresponding evolution of y is shown in ﬁgure 2. For clarity, because y is antisymmetric, – 9 – JHEP08(2009)024 Figure 1. The tachyon as a function of t and x. Figure 2. y as a function of t and x. y(t, x) = −y(t,−x), we only plot positive values of x. Initially, the system again stays near the false vacuum and the branes are straight. Once the decay process begins, for x . t the branes are coincident with some small oscillations around y = 0. We can more easily illustrate certain features of the evolution by considering cross-sections of ﬁgures 1 and 2. In ﬁgure 3 we plot T (t, 0). There are three clearly distinguishable parts of the curve; at early time, for t . 3, the system remains close to the false vacuum T ≈ 0, then there is a brief transitional region where T begins to condense, rolling down the potential, and then for t & 4, the tachyon rolls at a constant velocity T ≈ 1 toward the vacuum at T = ∞. In addition, we plot in ﬁgure 4 the proﬁle of y(x) for various ﬁxed times, again only for x > 0. For t = 4, the branes are just beginning to pull toward each other. At later times, the proﬁle assumes a waterfall shape, again with three regions. Far from the intersection the branes retain their initial conﬁguration. But, at around x ≈ t, a point is reached where y → 0 rapidly, and then, in an increasingly large region around x = 0, the branes have – 10 – JHEP08(2009)024 Figure 3. The tachyon at x = 0 as a function of t. Figure 4. y as a function of x, for t = 2 (black), 4 (dotted red), 6 (dot-dashed blue), and 8 (dashed green). Figure 5. The position of the positive-velocity zipper z as a function of t. become coincident except for some small ﬂuctuations. The process resembles the action of two zippers moving away from the intersection point, zipping the branes together. We will deﬁne z(t) to be the position of the right-moving zipper and by symmetry the left-moving zipper is at −z(t). More speciﬁcally, for a given time, z equals the largest zero of y(x). Figure 5 presents a graph of z(t). For t . 4, z = 0 while at late time z˙ ≈ 1 with a sharp transition in between. – 11 – JHEP08(2009)024 Figure 6. The energy density T as a function of t and x. Notice the large spikes corresponding along the worldlines of the zippers. Figure 7. The momentum density T as a function of t and x. The upward spike corresponds to the worldline of the zipper with negative velocity while the trough corresponds to the zipper with positive velocity. We can further illustrate the physical properties of the evolution with plots of the com- ponents of the stress tensor. Figure 6 shows the energy density T . We have normalized the false vacuum energy so that T = 1. The motion of the zippers matches roughly with the large spikes in the energy density due to the large kinetic and gradient energy as the branes come together very rapidly. The position of the zippers is also clearly visible in ﬁgure 7 which plots the momentum density T ; the zipper moving to the left is represented by the spike and the one going to the right by the trough. The pressure T is shown in ﬁgure 8. The false vacuum has negative pressure, and again there are spikes corresponding to the worldlines of the zippers. Once the tachyon has begun to condense, T → 0. In addition, we checked numerically that T was conserved to very good accuracy by the evolution. μν We will ﬁrst focus on the transitional region between the false vacuum and the rolling – 12 – JHEP08(2009)024 Figure 8. The pressure T as a function of t and x. Note that the pressure vanishes in the condensing region. region, where the decay ﬁrst begins. We can identify three processes which characterize this region, and we can see that they are all related. Intuitively, the tachyon is massive where the branes are separated by more than the string length, so it cannot really begin rolling until the branes have been zipped together. However, it is the displacement of the tachyon from zero which gives a potential to y and pulls the branes together. Consequently, both of these processes must occur together. The timescale for the onset of the decay is controlled by the initial displacement parameter T ; the smaller T is, the longer it takes ǫ ǫ −3 for the decay to begin. For our choice of T = 10 , the tachyon starts rolling near the intersection at around t ∼ 3. At later times T (t, x) starts to grow when t ∼ x. Similarly, the zipper begins moving at t ∼ 4. As the zipper’s velocity approaches the speed of light, a given position on the brane begins to deviate from the false vacuum at around t ∼ x. The late time behavior, once the zipper has passed, very much resembles the decay of parallel D1-D1-branes but is inhomogeneous. We actually do not ﬁnd that y = 0 past the zipper but rather that y overshoots and oscillates around zero. As T increases, the masses of these oscillations grow, so their frequencies and amplitudes diminish. However, we should not really trust our eﬀective action to describe features such as these wiggles which are small compared with the string length. In the approximation that at late times y = 0, the dynamics becomes that of parallel branes, with the action for the tachyon becoming just 2 2 2 ′ S = −N d x V (T ) 1 − T + T . (5.5) D1 At late time, T is suﬃciently large that we may neglect V (T ). The Hamiltonian becomes just 2 2 H = Π (1 + T ) , (5.6) and the equations of motion (4.5) and (4.6) reduce to ˙ ′ T = 1 + T (5.7) – 13 – JHEP08(2009)024 Figure 9. To determine the extent to which the late-time tachyon obeys (5.7), we plot (5.11) as a function of x at ﬁxed t = 6. This function approximately vanishes in the zipped region, showing that the dust description is valid. Π T Π = ∂ . (5.8) T x 1 + T This is the well-studied pressureless tachyon dust [24–28] (see also [29]) where the velocity of the dust is given by v = −∂ T (5.9) μ μ and the local rest energy density is ǫ = p . (5.10) 1 + T We ﬁnd that the numerical solution accurately reproduces the tachyon dust once the branes have been zipped together. At x = 0 where T = 0, we saw in ﬁgure 3 that T → 1 in agreement with (5.7). More generally, we plot in ﬁgure 9 the function 2 ′ 1 − T + T (5.11) ˙ 2 ′ 1 + T + T which provides a normalized measure of the degree to which (5.7) is satisﬁed numerically. To good accuracy, (5.11) is zero in the condensing region around x = 0, implying that in fact (5.7) is approximately obeyed. Furthermore, we see from ﬁgure 8 that the condensing region behind the zipper is almost exactly pressureless. If this zipped region is essentially a parallel D1-D1, we arrive at the following inter- pretation for the ﬁnal conﬁguration of the system. At late times there are two curved D1-branes separating from each other and straightening but connected to each other by the parallel D1-D1. Based on the results from the D-D description, the two separating D1-branes is expected. However, rather than being devoid of branes, the region between them contains this decaying D1-D1 pair. In the next section we will discuss the relation between these two descriptions. – 14 – JHEP08(2009)024 U smooth fuzz time time increases decreases U singular Figure 10. The process of D-D decay and recombination is shown in two gauges. On the left, the intersection of two straight branes develops oﬀ-diagonal elements and becomes fuzzy. After a gauge transformation U, the branes have reconnected and are pulling apart. However, because U is singular at the beginning of the decay, the initial conditions are inequivalent. 6 Relationship between D-D and D-D systems We have been analyzing the system of intersecting D1-branes using a description in terms of a D-D pair. This choice of variables, at the level of an exact worldsheet description, amounts to an arbitrary gauge choice; a D1 is simply a D1 with opposite orientation, and a pair of D1-branes intersecting with an angle θ is equivalent to a D1 crossing a D1 at an angle ϕ = π − θ. However, the two eﬀective ﬁeld theory descriptions of these systems are inequivalent; each integrates out a diﬀerent set of massive modes, resulting in diﬀerent tachyonic and low-mass modes remaining. For example, using the YM approximation to the non-abelian DBI for the D-D pair, the tachyon’s mass (2.2) matches the worldsheet value (2.1) only for θ ∼ 0, while in the tachyon DBI description of the D-D system, the tachyon’s mass (3.27) matches for θ ∼ π. Even though the two eﬀective descriptions are not identical and the results need not to match a priori, as we will argue, they actually do produce dynamics which are the same, at least qualitatively. However, because the change of variables relating the two descriptions is nontrivially complicated, seeing that the results are physically similar can be a bit subtle. Before considering the relation between the D-D and D-D systems, we will begin with a simple illustrative example, a case where the two eﬀective descriptions are in fact exactly equivalent and the change of variables between them is explicit. In the intersecting D-D system, how one describes the decay process depends on one’s choice of gauge. In addition to choosing brane or anti-brane labels, the way the ends of the branes are connected is also to some degree gauge-dependent. The multiple equivalent descriptions can be easily seen from the non-abelian YM analysis of a D1-D1 decay [15], as illustrated in ﬁgure 10. – 15 – JHEP08(2009)024 In the initial variables, the matrix y giving the brane position is diagonal, corresponding to two straight D1-branes. But, as the decay proceeds, oﬀ-diagonal elements begin to grow, fuzzing out the intersection point. However, y can be rediagonalized by a gauge transformation U which exchanges the brane ends [15, 17]. In the new variables, the branes are disconnected, curved, and separating with time. Evolving back in time to the beginning of the decay, these two bent branes touch at a point. This odd conﬁguration is, of course, physically inequivalent to the initial straight branes, because at t = 0 the gauge transformation U relating them is singular. The important feature, which also appears in the D-D picture, is that in the origi- nal variables the decaying branes are not seen to annihilate at the intersection point, and the ends remain connected as in the initial state. Instead, the decay region is character- ized by an unusual, non-intuitive state such as the stringy fuzz, and only after a gauge transformation does the system appear as two disconnected branes. For the description in terms of an intersecting D-D pair, we showed in section 5 that the branes do not annihilate or reconnect into a disconnected pair of D1-branes. Such an event is, in fact, outside the range of the tachyon DBI eﬀective theory we used. Instead, the branes became zipped together, and where the D1 and D1 were parallel, the rolling tachyon was identiﬁed as inhomogeneous tachyon matter. However, although we do not have an explicit gauge transformation analogous to U in the D-D case, there is at least in principle a change of variables such that the parallel D1-D1 are replaced by a gap between a pair of disconnected D1-branes. Furthermore, in a more complete theory the parallel branes with tachyon matter could be alternatively described by a gas of closed strings and D0-branes (which themselves would decay) between the disconnected D1-branes. Although we can not perform the explicit change of variables for the intersecting D- D pair, we can illustrate how it works in a similar but physically distinct system. For the initially straight intersecting D1-D1 considered in section 5, the relative position of the branes y was always antisymmetric, y(x) = −y(−x). We can just as easily solve the equations of motion but with y(x) = y(−x) instead, although for numerical computation we have to smooth out the branes at x = 0 so as to avoid singular ﬁrst derivatives there. This symmetric boundary condition corresponds to two angled branes with opposite orientations which touch at the point where they are bent. In addition, we will choose the intersection angle to be ϕ = π/2. This system is unstable, and there are two diﬀerent directions in which it can decay corresponding to the two ways the branes can recombine, as shown in ﬁgure 11. One decay channel, mode (a) in ﬁgure 11, closely resembles the late-time state of intersecting D1-branes, except that instead of ending up with two separating D1-branes, the orientation of one of the D1-branes reversed. The numerical solution, presented in ﬁgure 12, shows the D1 and D1 growing further apart while the tachyon ﬁeld stays very close to zero. The other decay mode corresponds to the other way of recombining the ends of the branes and is (b) in ﬁgure 11. In the ﬁnal state there is still one curved D1 and one curved D1, but now they are connected. Numerically solving the equations of motion yields an For the numerical computations, which decay mode the system takes depends on how exactly the corner at x = 0 is resolved. The details, however, are not that relevant to our discussion. – 16 – JHEP08(2009)024 (a) (b) Figure 11. The decay of two angled D1-branes which touch at a point, such that y(x) is symmetric, can proceed in two ways. Final state (a) consists of a disconnected D1 and D1. Final state (b) also contains a separating D1 and D1, but they are connected by a parallel D1-D1. Figure 12. y as a function of x, for t = 0 (black), 2 (dotted red), 4 (dot-dashed blue), and 6 (dashed green). evolution qualitatively very similar to those of section 5; the branes zip together and then the tachyon rolls towards inﬁnity on the parallel D1-D1 pair connecting the two separating D1- and D1-branes. Initially, these two ﬁnal states appear quite diﬀerent, but in fact, they must be equiv- alent due to the symmetry of the initial state which is invariant under the exchange of x and y. This operation should map the ﬁnal states into each other, and it does up to the parallel D1-D1. However, if we also exchange the roles of x and y in the tachyon DBI action, writing x and T functions of y, the parallel D1-D1 would be present in decay (a) and not in mode (b). We conclude that the two modes are, up to this exchange of directions, diﬀerent de- scriptions of the same physical state. In this example the change of variables is particularly simple; whether a parallel D1-D1 connects the separating D1 and D1 or not depends on the The exchange corresponds to a π/2 rotation and an orientation reversal. – 17 – JHEP08(2009)024 choice of the worldvolume coordinates. At least at the level of open strings where we are working, both the parallel D1-D1 with a rolling tachyon and empty space are equivalent. In particular, since at late time the parallel D1-D1 with rolling tachyon does not support open string modes, it is eﬀectively as if there are no branes there at all. Although the change of variables involved in the intersecting D1-D1 system is much more complicated, the intuition gained from this symmetric example should hold. Because they are diﬀerent approximations of the same system, the non-Abelian DBI D-D eﬀective theory and the tachyon DBI D-D eﬀective theory do not give identical quantitative results. However, since we should regard the parallel D1-D1 as essentially the vacuum, both the D-D and D-D are qualitatively the same. 7 Summary and discussion In this paper we studied the D1-D1 conﬁguration, initially intersecting at an angle ϕ. This conﬁguration is unstable because tachyonic modes are present at all angles (except for the parallel D-D-brane pair ϕ = π). We modeled the evolution of this system by deriving the equations of motion for the tachyon T (t, x) and the separation ﬁeld y(t, x) from the tachyon DBI action and solved them numerically. We found that, at the very beginning of the evolution, the tachyon, which was localized at the intersection point, slowly rolled away from the maximum point of the eﬀective potential. Then, after some time value, which could be interpreted as the (local) lifetime of the brane system [30], the tachyon began to grow linearly with time and thus induced the dynamics for the separation ﬁeld y(t, x). As time passed, the branes were pulled toward each other such that the point where they ﬁrst met moved from 0 to larger ±x values. The process resembled that of two zippers moving at opposite directions with speeds of light, zipping the branes together, and continuing indeﬁnitely. The region between the zippers had the behavior of a decaying parallel D1-D1 pair, but an inhomogeneous one. At late time, we were able to capture the essential features of the remnant by the well-studied inhomogeneous pressureless tachyon dust. Although the tachyon DBI is by construction accurate for θ ∼ π while the YM de- scription is valid for θ ∼ 0, both these results in terms of a D1-D1 pair and the description in terms of intersecting D1-branes give qualitatively the same evolution. The ﬁnal state in both pictures contains two reconnected separating D1-branes. We have argued that the parallel D1-D1 connecting the branes in the tachyon DBI picture is an artifact of the choice of variables and is, in fact, equivalent to the gap between the branes in the YM description. In both of these eﬀective theories the trace prescriptions for the actions are somewhat ambiguous. In the case of D-D, the proposed action of [31] gives an alternative way of performing the trace and potentially gives diﬀerent results from those found here. The non- Abelian DBI eﬀective description of the D-D system should be valid not just at small angles. If the correct trace prescription could be found, for example, the worldsheet formula (2.1) for the mass at all angles may be reproduced. Another potential extension of these results is to other systems featuring localized tachyon condensation. One may have hoped this simple ﬂat-space system could serve as – 18 – JHEP08(2009)024 a toy model for inhomogeneous brane decays in non-trivial curved backgrounds. In the Sakai-Sugimoto model, for example, a parallel D8-D8 pair decays into a single U-shaped D8 via the condensation of a tachyon localized at small radii [4]. However, one important qualitative diﬀerence between this and intersecting branes is that in the ﬂat-space case the decay process continues without end and the condensing region does not stay localized but instead grows without bound. Acknowledgments We thank O. Bergman, M. Ja¨rvinen, R.G. Leigh, G. Lifschytz, and S. Nowling for useful discussions. This work was supported in part by the Israel Science Foundation under grant no. 568/05. In addition, N.J. has been supported in part at the Technion by a fellowship from the Lady Davis Foundation. References [1] A. Bagchi and A. Sen, Tachyon condensation on separated brane-antibrane system, JHEP 05 (2008) 010 [arXiv:0801.3498] [SPIRES]. [2] A. Sen, Rolling tachyon, JHEP 04 (2002) 048 [hep-th/0203211] [SPIRES]. [3] P. Kraus and F. Larsen, Boundary string ﬁeld theory of the DD system, Phys. Rev. D 63 (2001) 106004 [hep-th/0012198] [SPIRES]; T. Takayanagi, S. Terashima and T. Uesugi, Brane-antibrane action from boundary string ﬁeld theory, JHEP 03 (2001) 019 [hep-th/0012210] [SPIRES]; N.T. Jones and S.H. Henry Tye, An improved brane anti-brane action from boundary superstring ﬁeld theory and multi-vortex solutions, JHEP 01 (2003) 012 [hep-th/0211180] [SPIRES]; M.R. Garousi, D-brane anti-D-brane eﬀective action and brane interaction in open string channel, JHEP 01 (2005) 029 [hep-th/0411222] [SPIRES]; On the eﬀective action of D-brane-anti-D-brane system, JHEP 12 (2007) 089 [arXiv:0710.5469] [SPIRES]. [4] O. Bergman, S. Seki and J. Sonnenschein, Quark mass and condensate in HQCD, JHEP 12 (2007) 037 [arXiv:0708.2839] [SPIRES]. [5] N. Jokela, M. J¨arvinen and S. Nowling, Winding eﬀects on brane/anti-brane pairs, JHEP 07 (2009) 085 [arXiv:0901.0281] [SPIRES]. [6] A. Sen, Dirac-Born-Infeld action on the tachyon kink and vortex, Phys. Rev. D 68 (2003) 066008 [hep-th/0303057] [SPIRES]. [7] C. Bachas, A way to break supersymmetry, hep-th/9503030 [SPIRES]; C. Angelantonj, I. Antoniadis, E. Dudas and A. Sagnotti, Type-I strings on magnetised orbifolds and brane transmutation, Phys. Lett. B 489 (2000) 223 [hep-th/0007090] [SPIRES]; R. Blumenhagen, L. G¨orlich, B. K¨ors and D. Lu¨st, Noncommutative compactiﬁcations of type-I strings on tori with magnetic background ﬂux, JHEP 10 (2000) 006 [hep-th/0007024] [SPIRES]; G. Aldazabal, S. Franco, L.E. Iba´n˜ez, R. Rabada´n and A.M. Uranga, Intersecting brane worlds, JHEP 02 (2001) 047 [hep-ph/0011132] [SPIRES]; – 19 – JHEP08(2009)024 G. Aldazabal, S. Franco, L.E. Iba´n˜ez, R. Rabada´n and A.M. Uranga, D = 4 chiral string compactiﬁcations from intersecting branes, J. Math. Phys. 42 (2001) 3103 [hep-th/0011073] [SPIRES]; R. Blumenhagen, B. K¨ors and D. Lu¨st, Type I strings with F- and B-ﬂux, JHEP 02 (2001) 030 [hep-th/0012156] [SPIRES]; L.E. Iba´n˜ez, F. Marchesano and R. Rabada´n, Getting just the standard model at intersecting branes, JHEP 11 (2001) 002 [hep-th/0105155] [SPIRES]; R. Blumenhagen, B. K¨ors, D. Lu¨st and T. Ott, The standard model from stable intersecting brane world orbifolds, Nucl. Phys. B 616 (2001) 3 [hep-th/0107138] [SPIRES]; M. Cvetiˇc, G. Shiu and A.M. Uranga, Three-family supersymmetric standard like models from intersecting brane worlds, Phys. Rev. Lett. 87 (2001) 201801 [hep-th/0107143] [SPIRES]; Chiral four-dimensional N = 1 supersymmetric type IIA orientifolds from intersecting D6-branes, Nucl. Phys. B 615 (2001) 3 [hep-th/0107166] [SPIRES]; D. Cremades, L.E. Iba´n˜ez and F. Marchesano, Intersecting brane models of particle physics and the Higgs mechanism, JHEP 07 (2002) 022 [hep-th/0203160] [SPIRES]; R. Blumenhagen, V. Braun, B. K¨ors and D. Lu¨st, Orientifolds of K3 and Calabi-Yau manifolds with intersecting D-branes, JHEP 07 (2002) 026 [hep-th/0206038] [SPIRES]; C. Kokorelis, Exact standard model structures from intersecting D5-branes, Nucl. Phys. B 677 (2004) 115 [hep-th/0207234] [SPIRES]; R. Blumenhagen, L. G¨orlich and T. Ott, Supersymmetric intersecting branes on the type IIA T /Z orientifold, JHEP 01 (2003) 021 [hep-th/0211059] [SPIRES]; A.M. Uranga, Chiral four-dimensional string compactiﬁcations with intersecting D-branes, Class. Quant. Grav. 20 (2003) S373 [hep-th/0301032] [SPIRES]. [8] C. Herdeiro, S. Hirano and R. Kallosh, String theory and hybrid inﬂation/acceleration, JHEP 12 (2001) 027 [hep-th/0110271] [SPIRES]; J. Garcia-Bellido, R. Rabada´n and F. Zamora, Inﬂationary scenarios from branes at angles, JHEP 01 (2002) 036 [hep-th/0112147] [SPIRES]; R. Blumenhagen, B. K¨ors, D. Lu¨st and T. Ott, Hybrid inﬂation in intersecting brane worlds, Nucl. Phys. B 641 (2002) 235 [hep-th/0202124] [SPIRES]; K. Dasgupta, C. Herdeiro, S. Hirano and R. Kallosh, D3/D7 inﬂationary model and M-theory, Phys. Rev. D 65 (2002) 126002 [hep-th/0203019] [SPIRES]; N.T. Jones, H. Stoica and S.H. Henry Tye, Brane interaction as the origin of inﬂation, JHEP 07 (2002) 051 [hep-th/0203163] [SPIRES]; S. Sarangi and S.H. Henry Tye, Cosmic string production towards the end of brane inﬂation, Phys. Lett. B 536 (2002) 185 [hep-th/0204074] [SPIRES]; M. Gomez-Reino and I. Zavala, Recombination of intersecting D-branes and cosmological inﬂation, JHEP 09 (2002) 020 [hep-th/0207278] [SPIRES]; R. Brandenberger, G. Geshnizjani and S. Watson, On the initial conditions for brane inﬂation, Phys. Rev. D 67 (2003) 123510 [hep-th/0302222] [SPIRES]; M. Hindmarsh and H. Li, Perturbations and moduli space dynamics of tachyon kinks, Phys. Rev. D 77 (2008) 066005 [arXiv:0711.0678] [SPIRES]; W. Nelson and M. Sakellariadou, Tachyonic decay of unstable Dirichlet branes, Phys. Rev. D 78 (2008) 104012 [arXiv:0809.3958] [SPIRES]; Space-time dimensionality from brane collisions, Phys. Lett. B 674 (2009) 210 [arXiv:0810.0363] [SPIRES]; M. Hindmarsh and H. Li, Inhomogeneous tachyon condensation, JHEP 06 (2009) 050 [arXiv:0903.2019] [SPIRES]. [9] M. Berkooz, M.R. Douglas and R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265 [hep-th/9606139] [SPIRES]. – 20 – JHEP08(2009)024 [10] H. Arfaei and M.M. Sheikh Jabbari, Diﬀerent D-brane interactions, Phys. Lett. B 394 (1997) 288 [hep-th/9608167] [SPIRES]. [11] A. Hashimoto and W. Taylor, Fluctuation spectra of tilted and intersecting D-branes from the Born-Infeld action, Nucl. Phys. B 503 (1997) 193 [hep-th/9703217] [SPIRES]. [12] M.M. Sheikh Jabbari, Classiﬁcation of diﬀerent branes at angles, Phys. Lett. B 420 (1998) 279 [hep-th/9710121] [SPIRES]. [13] N. Ohta and P.K. Townsend, Supersymmetry of M-branes at angles, Phys. Lett. B 418 (1998) 77 [hep-th/9710129] [SPIRES]. [14] T. Kitao, N. Ohta and J.-G. Zhou, Fermionic zero mode and string creation between D4-branes at angles, Phys. Lett. B 428 (1998) 68 [hep-th/9801135] [SPIRES]. [15] K. Hashimoto and S. Nagaoka, Recombination of intersecting D-branes by local tachyon condensation, JHEP 06 (2003) 034 [hep-th/0303204] [SPIRES]. [16] S. Nagaoka, Fluctuation analysis of non-Abelian Born-Infeld action in the background intersecting D-branes, Prog. Theor. Phys. 110 (2004) 1219 [hep-th/0307232] [SPIRES]. [17] F.T.J. Epple and D. Lu¨st, Tachyon condensation for intersecting branes at small and large angles, Fortsch. Phys. 52 (2004) 367 [hep-th/0311182] [SPIRES]. [18] F. Larsen, A. Naqvi and S. Terashima, Rolling tachyons and decaying branes, JHEP 02 (2003) 039 [hep-th/0212248] [SPIRES]; N.D. Lambert, H. Liu and J.M. Maldacena, Closed strings from decaying D-branes, JHEP 03 (2007) 014 [hep-th/0303139] [SPIRES]; V. Balasubramanian, E. Keski-Vakkuri, P. Kraus and A. Naqvi, String scattering from decaying branes, Commun. Math. Phys. 257 (2005) 363 [hep-th/0404039] [SPIRES]; J. Shelton, Closed superstring emission from rolling tachyon backgrounds, JHEP 01 (2005) 037 [hep-th/0411040] [SPIRES]; N. Jokela, E. Keski-Vakkuri and J. Majumder, On superstring disk amplitudes in a rolling tachyon background, Phys. Rev. D 73 (2006) 046007 [hep-th/0510205] [SPIRES]; N. Jokela, M. J¨arvinen, E. Keski-Vakkuri and J. Majumder, Disk partition function and oscillatory rolling tachyons, J. Phys. A 41 (2008) 015402 [arXiv:0705.1916] [SPIRES]; N. Jokela, M. J¨arvinen and E. Keski-Vakkuri, N-point functions in rolling tachyon background, Phys. Rev. D 79 (2009) 086013 [arXiv:0806.1491] [SPIRES]; The energy-momentum tensor of perturbed tachyon matter, Phys. Rev. D 79 (2009) 106005 [arXiv:0901.3368] [SPIRES]. [19] S.-J. Rey and S. Sugimoto, Rolling tachyon with electric and magnetic ﬁelds: t-duality approach, Phys. Rev. D 67 (2003) 086008 [hep-th/0301049] [SPIRES]. [20] S.-J. Rey and S. Sugimoto, Rolling of modulated tachyon with gauge ﬂux and emergent fundamental string, Phys. Rev. D 68 (2003) 026003 [hep-th/0303133] [SPIRES]. [21] A. Sen, Tachyon dynamics in open string theory, Int. J. Mod. Phys. A 20 (2005) 5513 [hep-th/0410103] [SPIRES]. [22] A.A. Tseytlin, On non-abelian generalisation of the Born-Infeld action in string theory, Nucl. Phys. B 501 (1997) 41 [hep-th/9701125] [SPIRES]. [23] D. Kutasov and V. Niarchos, Tachyon eﬀective actions in open string theory, Nucl. Phys. B 666 (2003) 56 [hep-th/0304045] [SPIRES]. [24] G.W. Gibbons, K. Hori and P. Yi, String ﬂuid from unstable D-branes, Nucl. Phys. B 596 (2001) 136 [hep-th/0009061] [SPIRES]. – 21 – JHEP08(2009)024 [25] A. Sen, Fundamental strings in open string theory at the tachyonic vacuum, J. Math. Phys. 42 (2001) 2844 [hep-th/0010240] [SPIRES]. [26] P. Yi, Membranes from ﬁve-branes and fundamental strings from Dp branes, Nucl. Phys. B 550 (1999) 214 [hep-th/9901159] [SPIRES]; U. Lindstr¨om and M. Zabzine, Strings at the tachyonic vacuum, JHEP 03 (2001) 014 [hep-th/0101213] [SPIRES]; A. Sen, Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143] [SPIRES]; G. Gibbons, K. Hashimoto and P. Yi, Tachyon condensates, Carrollian contraction of Lorentz group and fundamental strings, JHEP 09 (2002) 061 [hep-th/0209034] [SPIRES]. [27] O. Bergman, K. Hori and P. Yi, Conﬁnement on the brane, Nucl. Phys. B 580 (2000) 289 [hep-th/0002223] [SPIRES]. [28] K. Hashimoto, Dynamical decay of brane-antibrane and dielectric brane, JHEP 07 (2002) 035 [hep-th/0204203] [SPIRES]. [29] A. Sen, Open and closed strings from unstable D-branes, Phys. Rev. D 68 (2003) 106003 [hep-th/0305011] [SPIRES]; Open-closed duality at tree level, Phys. Rev. Lett. 91 (2003) 181601 [hep-th/0306137] [SPIRES]; H.-U. Yee and P. Yi, Open/closed duality, unstable D-branes and coarse-grained closed strings, Nucl. Phys. B 686 (2004) 31 [hep-th/0402027] [SPIRES]. [30] D. Gaiotto, N. Itzhaki and L. Rastelli, Closed strings as imaginary D-branes, Nucl. Phys. B 688 (2004) 70 [hep-th/0304192] [SPIRES]. [31] M.R. Garousi, On modiﬁed tachyon DBI action, Nucl. Phys. B 817 (2009) 252 [arXiv:0811.4334] [SPIRES]. – 22 –

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Inhomogeneous tachyon dynamics and the zipper

Inhomogeneous tachyon dynamics and the zipper

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Inhomogeneous tachyon dynamics and the zipper

Inhomogeneous tachyon dynamics and the zipper

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