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Rolling tachyons and decaying branes

Rolling tachyons and decaying branes JHEP02(2003)039 Published by Institute of Physics Publishing for SISSA/ISAS Received: February 11, 2003 Revised: February 19, 2003 Accepted: February 19, 2003 Finn Larsen Michigan Center for Theoretical Physics, Randall Lab., University of Michigan Ann Arbor, MI 48109, USA E-mail: [email protected] Asad Naqvi Department of Physics and Astronomy, David Rittenhouse Laboratories University of Pennsylvania Philadelphia, PA 19104, USA E-mail: [email protected] Seiji Terashima Institute for Theoretical Physics, University of Amsterdam 1018 XE Amsterdam, The Netherlands E-mail: [email protected] Abstract: We present new rolling tachyon solutions describing the classical decay of D- branes. Our methods are simpler than those appearing in recent works, yet our results are exact in classical string theory. The role of pressure in the decay is studied using tachyon pro¯les with spatial variation. In this case the ¯nal state involves an array of codimension one D-branes rather than static, pressureless tachyon matter. Keywords: D-branes, Tachyon Condensation. ° SISSA/ISAS 2003 http://jhep.sissa.it/archive/papers/jhep022003039 /jhep022003039.pdf JHEP02(2003)039 Contents 1. Introduction and summary 1 2. Rolling tachyons 3 2.1 Generalities 4 2.2 Spatially homogenous decay 5 2.2.1 B(x ) 5 ¹º 2.2.2 A 6 3. Spatially inhomogenous decay 8 4. Rolling tachyons in superstring theory 10 4.1 Generalities 10 4.2 The simple decay 11 4.3 Spatial inhomogeneity 13 5. Symmetries and boundary states 14 5.1 The group of time-dependent marginal deformations 14 5.2 Spatial variation 16 5.3 The full boundary states 17 1. Introduction and summary An appealing way to generate time dependent con¯gurations in string theory is to consider the classical decay of unstable systems of D-branes, pictured as a tachyon ¯eld rolling down a potential, towards a stable minimum. This system is promising from the viewpoint of studying time-dependence in string theory since the non-trivial dynamics is con¯ned to the open string sector. Moreover, this setting provides a natural arena for discussing important cosmological ideas, such as in°ation, and the beginning of time. The quantitative study of rolling tachyons was initiated recently by A. Sen [1]{[5]. It involves deforming the world sheet conformal ¯eld theory (CFT) of the unstable D-brane by an exactly marginal, time dependent tachyon pro¯le. The deformed CFT is an exact classical background in string theory, interpreted as the classical decay of the unstable D- brane. Although several key results have been obtained, this approach is still in its infancy and central questions remain: { 1 { JHEP02(2003)039 ² In the classical approximation, the string coupling constant is strictly vanishing. However, it is not clear that the implied limit is smooth. The system with a small, but non-zero string coupling may di®er qualitatively from the system with vanishing coupling. In such a scenario, the classical approximation is misleading. This concern is fueled by the somewhat mysterious role of closed strings in tachyon condensation: after the decay of the unstable D-brane only closed strings remain in the spectrum; yet the brane cannot decay into closed strings if the coupling is strictly vanishing. 1 2 2 ~ ~ ² A tachyon pro¯le with spatial momentum k has e®ective mass m = ¡1 + k and so is unstable for any jkj < 1, indicating that all wave lengths play some role in the decay. Indeed, in quantum ¯eld theory it is well understood that tachyon conden- sation is a process where the longest wave lengths dominate, but all wave lengths participate, and the decay is de¯nitely inhomogeneous. These results are best known in the context of cosmological in°ation [7, 8], but they are valid also for tachyon condensation in string theory [9]. They indicate that spatially inhomogenous modes are important also for rolling tachyons. In this paper we study several new tachyon pro¯les with the goal of shedding light on these questions. The deformations we consider are actually technically simpler than those previously studied. We are therefore able to avoid the full machinery of boundary states and instead carry out the computations using elementary methods. The approach taken here complements the one taken in previous papers on rolling tachyons and may o®er some conceptual advantages. As discussed in section 5, the new spatially homogenous pro¯le, given below as (1.1), has a topology that is di®erent from previous examples. In addition to this decay, in which the rolling tachyon has no spatial dependence, we study an exactly soluble example with spatially varying tachyon pro¯le, where we can follow the decay to its inhomogeneous ¯nal state. The simplest of the new pro¯les studied in this paper is T (X) = ¸e : (1.1) This can be interpreted as simultaneously displacing and giving a velocity to the tachyon, 0 0 i.e. imposing the initial condition T (X = 0) = @ T (X = 0) = ¸. An alternative, 0 3 and better, spacetime interpretation is that of a perturbation at X = ¡1. Since this disturbance is automatically in¯nitesimal, the pro¯le (1.1) seems to be a particularly clean example of a rolling tachyon. The pro¯le (1.1) is also particularly simple from a technical point of view. The simplest exactly marginal deformations of a world sheet CFT are generated by the vertex operators ikX 2 2 2 V (X) = e : k = k ¡ k = ¡1 : (1.2) Of course these operator cannot usually be added to the world sheet action because they correspond to complex potentials. The standard remedy is to add also the conjugate 1 0 We use units such that ® = 1. Other discussions of spatial variation include [4] and [10]{[12]. This was the point of view taken in the recent talk by Strominger [13]. { 2 { JHEP02(2003)039 operator and so consider perturbations of the form ~ ~ ~ ~ ik¢X ¡ik¢X 2 ~ ~ ~ ~ T (X) = ¸ cos(k ¢ X) = (e + e ) ; k = 1 : (1.3) After analytical continuation this leads to Sen's pro¯le T (X) = ¸ cosh(X ). An alternative procedure, exploited in this paper, is to note that in the special case k = ¡i, k = 0 the vertex operator (1.2) is in fact real, and so we can consider (1.1) directly, without adding the complex conjugate. This is much simpler because, in the case of (1.3), complications arise from the cross-terms between the two exponentials. As explained above, it is important to study rolling tachyons with spatially varying pro¯les. Such pro¯les are generally quite complicated to analyze in the full CFT; but in the case of the pro¯le 0 1 X = 2 2 ~ ~ ~ ~ T (X) = ¸e cos(k ¢ X) ; k = ; (1.4) the study simpli¯es dramatically (similar, but more complicated tachyon pro¯les were discussed in [4]). Indeed, this pro¯le is a linear combination of two vertex operators of the form (1.2). Crucially, these vertex operators commute, in contrast to those appearing in (1.3). Thus the theory with the pro¯le (1.4) essentially reduces to two copies of (1.1). Having solved the theory with the tachyon pro¯le (1.4) we ¯nd the coupling to the energy momentum tensor for all times X . The energy momentum tensor exhibits qualita- tively di®erent behavior from the spatially homogenous case. It develops codimension one singularities in ¯nite time. These singularities can be interpreted as an array of (excited) D-branes. This result is consistent with the expectation that ¯nal states will be spatially inhomogeneous for generic decay channels. Of course, the pro¯le that we can actually solve (1.4) is actually quite special. Presumably that is why the ¯nal state in this example, although spatially inhomogeneous, is as ¯nely tuned as a perfect array of unstable branes. In a realistic, semi-classical, analysis one would choose as initial state for the brane some wave-packet localized near the top of the tachyon potential and the full decay process would be described as an average, in a precise sense, of all the initial conditions represented by this wave packet. One would expect this ¯nal state to be dominated by generic, spatially dependent, con¯gurations which, to the extent they can be described as a perfect °uid, certainly would have pressure. This paper is organized as follows. In section 2 we explain our approach to rolling tachyons and carry out the details for the pro¯le (1.1). In section 3 we consider the spatially inhomogeneous case (1.4) and discuss the lessons for the full decay process, when all spatial variations are included. In section 4, we extend these results to the superstring case. Finally, in section 5, we discuss the topology of our pro¯les, their relations to previous works, and construct the boundary states corresponding to our solutions. 2. Rolling tachyons The strategy for treating rolling tachyons is to deform the world sheet CFT of an unsta- ble D-brane by an exactly marginal operator and interpret the deformed CFT as a time dependent solution to the classical string equations of motion. Instead of studying the { 3 { JHEP02(2003)039 system in terms of a boundary state of the closed string theory (which was Sen's approach in [1, 2, 4]), we will primarily work with open strings, equating the disk partition function with space-time action, following the analysis of static tachyon con¯guations in boundary string ¯eld theory [14]. 2.1 Generalities According to the ¾-model approach to string theory, the space-time action is given by the partition function of the world-sheet theory, with the world-sheet couplings interpreted as spacetime ¯elds [15]. Thus, in the open string sector, ¹ ¡I ¡I bulk bndy S[¸ ] / Z (¸ ) = [dX ]e ; (2.1) i disk i where Z is the disk partition function, ¸ are exactly marginal couplings for the bound- disk i ary operators, and 2 ¹ º I = d z ´ @X @X ; (2.2) bulk ¹º 2¼ I = dt T (X) + ¢¢¢ ; (2.3) bndy @D where D is the unit disk and @D is its boundary. T is the tachyon ¯eld and the ¢¢¢ indicate other marginal boundary perturbations. This procedure is similar in spirit to the boundary string ¯eld theory approach to tachyon condensation [14], although here we limit ourselves to marginal perturbations. Our formulae above are written for euclidean spacetimes, as well as euclidean world- sheets. The time-dependence is then taken into account by including the minkowskian ¹º 0 metric ´ = (¡; +;¢¢¢ ; +) when contracting the temporal ¯elds X . This procedure is motivated by analytical continuation, as in Sen's computations using boundary states and cubic string ¯eld theory. This type of analytical continuation is clearly not completely satisfying; indeed, the precise relation between lorentzian and euclidean signature is one of the main unsettled questions facing most approaches to time dependence in string theory. On the other hand, the analytical continuation gives physically reasonable results in many examples, including those considered here; so it presumably captures important aspects of the problem. It will be useful to write the action S as a space-time integral over a lagrangian density. ¹ ¹ ¹ 0¹ To this end, we split X into a constant and a varying part, X = x + X and write ¹ ¡I ¡I bulk bndy S / [dX ]e Z Z 1 2 0¹ 0º p 0¹ ¡ d z g @X @X ¡I ¹º bndy 2¼ D = d x ¡g [dX ]e ; (2.4) where, in the second line, we have made the obvious generalization of the expression to curved space. In this paper we are primarily considering the coupling to gravity, to explore the time-evolution of an unstable D-brane in a °at background. From the spacetime action It would be interesting to apply our methods for couplings to more massive strings as well, in an e®ort to illluminate the problems discussed in [16]. See also [17] for the coupling to the closed string. { 4 { JHEP02(2003)039 ± ¡g 2 ±S 1 we form the energy-momentum tensor T = ¡ and use = ¡ ¡gg to ¹º ¹º ¹º ¹º ±g ±g 2 ¡g ¯nd T (x) = K (B(x) ´ + A (x)) ; (2.5) ¹º ¹º ¹º in °at space. Here K is an overall normalization constant and 2 0¹ ¹ 0º 0¹ ¡ d z ´ @X @X ¡I ¹º bndy 2¼ D B(x) = [dX ]e ; (2.6) Z Z d z 1 2 1 0¹ 0º ¹º 0¹ ¹ º ¡ d z ´ @X @X ¡I ¹º ¹ bndy 2¼ D 2¼ A (x) = 2 [dX ] @X @X e 2¼ 2 0¹ ¹ 0º 0¹ ¹ º ¡ d z ´ @X @X ¡I ¹º ¹ bndy 2¼ D = 2 [dX ] @X (0)@X (0)e : (2.7) d z In the second line of (2.7) we ¯xed the position of the vertex operator and used = 2¼ A(D) = 1 for the unit disc. The expression (2.5) for the energy-momentum tensor was previously derived by Sen [1] using BRST invariance of the corresponding boundary state. In the following we consider various T (X), corresponding to tachyon pro¯les with speci¯c space-time dependence. To determine the energy-momentum tensor for each pro¯le we ¹º need to compute A and B. 2.2 Spatially homogenous decay In this section we study the tachyon pro¯le T (X) = ¸e : (2.8) This is an exactly marginal deformation of the CFT and so an exact solution to the classical string equations of motion. It is interpreted in spacetime as a perturbation at X = ¡1, displacing the tachyon in¯nitesimally from the unstable maximum of the potential. Alternatively, this pro¯le corresponds to kicking the tachyon from T (X = 0) = ¸, with velocity @ T (X = 0) = ¸. To determine the stress tensor T for this tachyonic pro¯le, we need to compute the ¹º 0 0 functions B(x ) and A (x ). ¹º 2.2.1 B(x ) The function B(x ) is the disk partition function, except that zero modes remain uninte- grated. Using h¢¢¢i as symbols for expectation values on the disc we have the perturbative expansion 0 00 0 x X 0 ¡I (x+X ) ¡¸e dt e bndy B(x ) = he i = he i ; (2.9) 0 Z x n (¡2¼¸e ) dt dt 00 00 1 n X (t ) X (t ) 1 n = ¢¢¢ he ¢¢¢ e i ; (2.10) n! 2¼ 2¼ n=0 The Green's function on the unit disk with Neumann boundary conditions is ¹º 0 ¹ º 0 ¹º 0 0 G (z; z ) = hX (z)X (z )i = ´ (¡ logjz ¡ z j ¡ logjzz¹ ¡ 1j) ; (2.11) { 5 { JHEP02(2003)039 it so, taking z = e , we ¯nd µ ¶ Y Y 00 00 t ¡ t i j X (t ) X (t ) it it 2 n(n¡1)=2 2 1 n i j he ¢¢¢ e i = je ¡ e j = 4 sin : (2.12) i<j i<j The integrals in (2.10) give Z µ ¶ dt dt t ¡ t 1 n i j n(n¡1)=2 2 ¢¢¢ 4 sin = n! ; (2.15) 2¼ 2¼ 2 i<j and the ¯nal result for B(x ) becomes x n (¡2¼¸e ) 0 0 B(x ) = n! = f(x ) ; (2.16) n! n=0 where f(x ) ´ : (2.17) 1 + 2¼¸e The summation of the perturbative series is clearly justi¯ed for couplings within the radius 1 ¡x of convergence j¸j < e . The regime of validity may be extended by analytical contin- 2¼ uation to include all positive ¸. The precise justi¯cation for this extension is an interesting question that deserves further study. ¹º 2.2.2 A ¹º ¹ º The A are proportional to expectation values of graviton vertex operators : @X @X :, where the normal ordering symbol : : indicate that the divergent pieces have been sub- tracted as z ! z ¹ º 0 ¹ º 0 ¹º 0 0 ¹ ¹ ¹ : @X (z)@X (z ) : = @X (z)@X (z ) + ´ @@ logjz ¡ z j : (2.18) For the purpose of our calculation, it is useful to de¯ne another kind of normal ordering ± ± symbol where we subtract the full Green's function (2.11), viz ± ± ± ¹ º 0 ± ¹ º 0 0 ¹º 0 ¹ ¹ ¹ @X (z)@X (z ) = @X (z)@X (z ) ¡ @@ G (z; z ) ; (2.19) ± ± including the contribution from the image charge. The two normal orderings are related as ¹ º ± ¹ º ± ¹º ¹ ¹ : @X (0)@X (0) : = @X (0)@X (0) + ´ : (2.20) ± ± These integrals can be derived by considering an integration over U(n) matrices and using the known result that the U(n) Haar measure dU, when expressed in terms of the eigenvalues, becomes Z Z 1 1 dt i 2 dU = ¢ (t) ; (2.13) vol U(n) n! 2¼ where ¢(t) is the relevant Vandermonde determinant for U(n) matrices µ ¶ t ¡ t i j ¢(t) = 2 sin : (2.14) i<j By noticing that the LHS in (2.13) is 1, the integral in (2.15) follows immediately. { 6 { JHEP02(2003)039 ¹º Now we are ready to calculate A . For i; j 6= 0 we have 0 00 x X ij i j ¡¸e dt e A (x) = 2h: @X (0)@X (0) : e i ¿µ ¶ À ij 0 00 ± x X ± i j ± ¡¸e dt e = 2 @X (0)@X (0) + e ± ± ij 0 = ± f(x ) : (2.21) ± ± In going from the second to the third line, the normal ordered term between gives ± ± ij 0 no contribution, and the term proportional to ± is exactly the same as B(x ) computed earlier. The calculation of A is a bit more involved. 00 0 0 ¡¸ dte A = 2h: @X (0)@X (0) : e i 0 00 x X ± 0 0 ± ¡¸e dte 0 = 2h @X (0)@X (0) e i ¡ f(x ) ± ± 0 Z x n X Y (¡2¼¸e ) dt 0 it i i ± 0 0 ± X (e ) 0 = 2 h @X (0)@X (0) e i ¡ f(x ) : (2.22) ± ± n! 2¼ i=1 The correlation function in (2.22) yields µ ¶µ ¶ Y Y X 1 1 ± 0 0 ± X(w ) 2 h @X (z)@X (z¹) : e :i = jw ¡ w j ± ; (2.23) ± ± i j n>0 z ¡ w z¹¡ w¹ i j i=1 i<j i;j it which, for z = 0 and w = e gives Z Z n n n Y Y Y Y X dt it dt i i i ± 0 0 ± X(e ) it it 2 ¡i(t ¡t ) i j i j h @X (0)@X (0) : e :i = je ¡ e j e ± ± 2¼ 2¼ i=1 i=1 i=1 i<j i;j µ ¶ Y Y dt t ¡ t i i j = 2 sin £ 2¼ 2 i=1 i<j 0 1 @ A £ n + 2 cos(t ¡ t ) : i j i<j The integral is 0 1 Z µ ¶ Y Y X dt t ¡ t i j @ A 2 sin n + 2 cos(t ¡ t ) = n! ; (2.24) i j 2¼ 2 i=1 i<j i<j and (2.22) becomes 00 0 A = f(x ) ¡ 2 : (2.25) Collecting the various results we ¯nd the stress tensor for the tachyon pro¯le (2.8) 0 0 0 0 0 T = K(¡B(x ) + A (x )) = ¡T ; T = K(B(x ) + A (x )) = ± T f(x ) : (2.26) 00 00 p ij ij ij p We have determined the normalization constant K = T by comparison with the static limit ¸ = 0. As expected, T is independent of x , which is just the statement of conser- 00 0 vation of energy. Moreover, T ! 0 as x ! 1, so the pressure vanishes in this limit, i.e. ij the decay product is pressureless tachyon matter, as in [2]. { 7 { JHEP02(2003)039 3. Spatially inhomogenous decay We will now investigate the spatially inhomogenous decay. A spatially inhomogenous pro¯le !X 2 2 ~ ~ ~ T (X) = 2¸e cos(k¢X) is marginal for ! +k = 1. This can be written as a sum of two ³ ´ 0 ~ ~ 0 ~ ~ !X +ik:X !X ¡ik:X vertex operators, each of which is exactly marginal: T (X) = ¸ e + e . For generic !, this is not an exactly marginal deformation because of the singular OPE between the two vertex operators. Hence it does not yield a solution to the classical string equations of motion. However, for ! = , this perturbation is exactly marginal. Without any loss of generality, we can keep only one component of k to be non-zero, denoting the ~ ~ corresponding direction Y ´ 2k ¢ X. Thus we have µ ¶ ³ ´ p 0 0 X +iY X ¡iY 0 Y p p X = 2 2 2 T (X) = 2¸e cos = ¸ e + e (3.1) U V = ¸(e + e ) ; (3.2) where we have de¯ned new variables 0 0 X + iY X ¡ iY p p U = ; V = ; (3.3) 2 2 such that hU(z)U(w)i = hV (z)V (w)i = logjz ¡ wj + logjzw¹ ¡ 1j ) ; (3.4) U(z)V (w) » regular : (3.5) Thus U and V behave as commuting time-like coordinates, with no mixing between U and V . ¹º Using the variables U and V , the calculation of B and A proceeds very similarly to the calculation in the last section. To compute B, we need to compute 0 0 u+U v+V ¸ dt(e +e ) B(u; v) = he i ; (3.6) where u and v are linear combinations of the zero modes of X and Y 0 0 x + iy x ¡ iy u = p ; v = p ; (3.7) 2 2 0 0 and U and V are non-zero modes of U and V . Since there is no mixing between U and V and (3.6) factorizes as R R 0 0 u U v V ¸e dte ¸e dte B(u; v) = he ihe i = f(u) f(v) ; (3.8) where, as in the previous section, we have de¯ned f(u) as f(u) = : (3.9) 1 + 2¼¸e { 8 { JHEP02(2003)039 ¹º The calculations for A , with the insertion of a graviton vertex operator factorize similarly. For example 0 0 u+U v+V uu ¸ dt(e +e ) A (x) = 2h: @U(0)@U(0) : e i ¿ À ³ ´ R R 0 0 1 U U v V ± ± ¸e dte ¸e dte = 2 @U(0)@U(0) ¡ e he i ± ± ³ ´ = f(u) ¡ 2 f(v) : (3.10) The remaining components give, uv A (u; v) = 0 ; vv A (u; v) = f(u) (f(v) ¡ 2) ij A (u; v) = ± f(u) f(v) : ij Furthermore, from the relations 1 1 1 A = (A ¡ A + 2iA ); A = (A + A ); A = (A ¡ A ¡ 2iA ) ; (3.11) uu 00 yy 0y uv 00 yy vv 00 yy 0y 2 2 2 and using (2.5), the stress tensor for this spatially inhomogenous decaying solution can be calculated to be x = 2 1 + 2¼¸e cos(y= 2) p p T (x ; y) = ¡T p ; (3.12) 00 p 0 0 x = 2 2 2 2x 1 + 4¼¸e cos(y= 2) + 4¼ ¸ e x = 2 ¡2¼¸e sin(y= 2) T (x ; y) = T p p ; (3.13) 0y p 0 0 x = 2 2 2 2x 1 + 4¼¸e cos(y= 2) + 4¼ ¸ e 0 0 T (x ; y) = ¡T (x ; y) : (3.14) yy 00 This stress tensor is conserved, i.e. 00 y0 @ T ¡ @ T = 0 ; (3.15) 0 y The form of the stress tensor and its late time behavior is qualitatively di®erent from that obtained in the spatially homogenous case in section 2.2. Certainly at large times x ! 1 all components of the stress tensor (3.12){(3.14), including the energy density, approach zero. However, this result probably cannot be trusted since, at a ¯nite critical time x ´ 2 ln(1=(2¼j¸j)), the stress energy tensor exhibits singularities at the spatial loci y = 2 2n¼; n 2 Z ; (¸ < 0) ; (3.16) y = 2 2(n + )¼; n 2 Z ; (¸ > 0) : (3.17) These singularities are what one would expect since, for some values of y, we are per- turbing with a negative tachyon and, according to (2.26), such perturbations give rise to singularities at ¯nite time already in the spatially homogenous setting. In [4] Sen proposed { 9 { JHEP02(2003)039 that these singularities should be interpreted as codimension one D-branes. To see this, we introduce the auxiliary variable 0 0 x ¡x ¢ = e ; (3.18) and, for either sign of ¸, we write the energy density ½ = ¡T as " # (1 ¡ ¢) + 2¢ sin ((y ¡ y )=2 2) 1 ¢!1 0 0 ½ = T » T + 2¼ sgn(x ¡ x )±(y ¡ y ) : p p n (1 ¡ ¢) + 4¢ sin ((y ¡ y )=2 2) n2Z (3.19) The limit was computed using lim = ¼±(®) close to each singular locus. The ²!0 2 2 ² +® corresponding average energy density changes discontinuously from T to 0 as we pass x . The form of the limiting energy density (3.19) suggests that the missing energy forms co- dimension one defects at the y . As we pass the critical time x , the loss in energy at each y is 2 2¼T . This result for the defect energy can be veri¯ed using energy conservation, n p noting that the defects are ¢y = 2 2¼ apart, and no bulk energy remains after they form. A co-dimension one D-brane has tension T = 2¼T and so the defect has additional p¡1 p energy, beyond that needed to form a D-brane. Nevertheless it is plausible that these defects are indeed related to D-branes since the spatial potential T (X) / cos(Y= 2) tends to con¯ne the ends of the strings, much as in the corresponding o®-shell discussion in [18]). 4. Rolling tachyons in superstring theory The purpose of this section is to generalize the results of the previous sections to the superstring. In each step of the computation details are modi¯ed, and so must be repeated; but the ¯nal results are closely analogous to the bosonic case. 4.1 Generalities In the superstring case world-sheet fermions must be included in a manner consistent with world-sheet supersymmetry. A convenient way implement this is to introduce world-sheet super¯elds. Thus the string coordinates (on the boundary of the disk) are represented by ¹ ¹ ¹ X = X + µÃ ; (4.1) and the Chan-Paton index of the brane is encoded in the boundary fermions I I I ¡ = ´ + µF : (4.2) We will consider only the simplest case of a single non-BPS D-brane. Since this corresponds to a single boundary fermion we can omit the index I. In this formalism the boundary action for a general tachyon pro¯le T (X) is I = dtdµ[¡D¡ + ¡T (X)] ; (4.3) bndy { 10 { JHEP02(2003)039 where D denotes the derivative in superspace D = @ + µ@ . A single boundary fermion ¡ can be integrated out with the result ¿ · ¸ À ¿ · ¸ À Z Z ¡I bndy he ¢¢¢i = P exp ¡ dtdµ¡T (X) ¢¢¢ = P cosh dtdµT (X) ¢¢¢ ; ¡¡even (4.4) within correlators. Here P is the standard path-ordering operator. Note that this path- ordering operator is not trivial in the above espression because dtdµT (X) is fermionic and then it does not commute with itself. The boundary fermions serve to make world-sheet supersymmetry manifest but, in the present context, they play the role of Chan-Paton matrices ¾ , for which the restriction to even terms arises from the overall trace. Following the bosonic example, our main interest is in inserting the identity operator ¿ µZ ¶À B(x) = P cosh dtdµT (X) ; (4.5) and the gravity vertex operator ¿ µ ¶À ¹º ¹º A = V (0; 0)P cosh dtdµT (X) ; (4.6) where, in the present case, ¹º ¹ º V (0; 0) = 2 dµdµ[DX DX ] (4.7) z=z¹=0 The energy momentum tensor still follows from (2.5). In (4.5) and (4.6) the brackets h¢¢¢i denote averaging with respect to the non-zero mode part of the bosonic ¯elds, as before. In concrete examples, we can evaluate these expressions in perturbation theory using the two point function [20] ¹ º ¹º 2 hX X i = ¡´ logjz j ; (4.8) where z = z ¡ z ¡ i z z µ µ : (4.9) 12 1 2 1 2 1 2 The form (4.8) of the two point function is valid when both coordinates are on the boundary of the disk. This will su±ce for our applications. 4.2 The simple decay We consider ¯rst the supersymmetric version of the pro¯le (3.1), i.e. X = 2 T (X) = ¸e : (4.10) 6 ¡ dtdµT (X) We have ignored the contact term e which appear in (4.4) for general tachyon pro¯les. This term is important in BSFT discussions of tachyon condensation [14], as well as in the time dependent case [19]. Here we follow Sen [2] and regard the right hand side of (4.4) as the starting point of our discussions. { 11 { JHEP02(2003)039 Expanding (4.5) in the parameter ¸ and using (4.8) yields 1 2n ³ ´ X 2n Y dt 0 n x = 2 B(x ) = (¡1) 2¼¸e dµ £(t ¡ t )£(t ¡ t )¢¢¢ £(t ¡ t ) £ i 1 2 2 3 2n¡1 2n 2¼ n=0 i=1 it it (t +t ) i j i j £ je ¡ e ¡ ie µ µ j : (4.11) i j i<j (t +t ) it it i j it it i j i j Noticing je ¡ e ¡ ie µ µ j = je ¡ e j + sign(t ¡ t )µ µ , the integrals can be i j i j i j evaluated with the result 2n Y Y dt i 1 it it (t +t ) i j i j dµ £(t ¡ t )£(t ¡ t )¢¢¢ £(t ¡ t ) je ¡ e ¡ ie µ µ j = ; i 1 2 2 3 2n¡1 2n i j 2¼ 2 i=1 i<j (4.12) so, after summation of the series, we ¯nd B(x ) = : (4.13) 2 2 2x 1 + 2¼ ¸ e The insertion of a graviton operator brings a few more complications, as in the bosonic case. The proper normal ordering again gives ¹º ± ¹º± ¹º : V : = V + ´ ; (4.14) ± ± and thus, without any further e®ort, ij 0 0 ij A (x ) = B(x )± : (4.15) The graviton with two temporal indices is evaluated in perturbation theory starting from (4.6). In addition to a term B´ = ¡B from the normal ordering (4.14), we ¯nd an integral over the correlator * + µ ¶ 2n ± ¹ º ± X(z )= 2 ¹ ¹ dµdµ DX (w)DX (w¹) e = ± ± i=1 Y X p 1 1 1 = jz ¡ z ¡ i z z µ µ j ; (4.16) i j i j i j 2 z ¡ w z¹ ¡ w¹ k l i<j k;l with w = w¹ = 0. The expressions can then be combined as 00 n x = 2 2n A = ¡B + 2 (¡1) (2¼¸e ) I ; (4.17) 2n n=0 where the integrals 2n Y Y dt it it (t +t ) i j i j I = dµ £(t ¡ t )¢¢¢ £(t ¡ t ) je ¡ e ¡ ie µ µ j £ 2n i 1 2 2n¡1 2n i j 2¼ i=1 i<j 1 1 i(t ¡t ) k l £ e µ µ = ± : (4.18) k l n>0 2 2 k;l We checked (4.12) for n · 3. We also checked (4.18) for n · 3. { 12 { JHEP02(2003)039 The ¯nal result thus becomes 00 0 0 A (x ) = p ¡ 2 = B(x ) ¡ 2 ; (4.19) 2 2 2x 1 + 2¼ ¸ e as in the bosonic case. The non-vanishing components of the energy momentum tensor now read 0 0 0 0 0 T = K(¡B(x ) + A (x )) = ¡2K ; T = K(B(x )± + A (x )) = 2K± B(x ) ; 00 00 ij ij ij ij (4.20) where B was given in (4.13). The overall constant K again is identi¯ed as K = T . As in the bosonic case, the energy density is constant ½ = T throughout the decay. The pressure p = 2B(x ) = ; (4.21) 2 2 2x 1 + 2¼ ¸ e is equal to the energy density p = ½ for the unstable brane at x = ¡1; but it decays exponentially to zero at large times. The main di®erence with the bosonic case is that now the decay is symmetric under ¸ ! ¡¸. In the supersymmetric case there is no singularity for either sign, as expected since the tachyon potential is symmetric, with both directions sloping down to the stable closed string vacuum. All these results are closely analogous to Sen's discussions, based on the potential T (X) = ¸ cosh(X). 4.3 Spatial inhomogeneity We also want to consider a spatially inhomogeneous pro¯le for the superstring. The simplest example is µ ¶ 1 1 T (X) = 2¸ e cos Y ; (4.22) where Y is one of the spatial directions. As in the bosonic case, this example is factorizable 1 1 0 0 (X +iY) (X ¡iY) 2 2 T (X) = ¸e + ¸e ; (4.23) 0 0 where, crucially, X +iY and X ¡iY have regular OPEs. Thus the example is essentially two copies of the pro¯le (4.10). From (4.13) we immediately ¯nd 1 1 B(x ; y) = T ; (4.24) p 0 2 2 x +iy 2 j1 + 2¼ ¸ e j ij ij while (4.15) gives A = ± B for i; j 6= y, and (4.17) combines with (4.13) to give µ ¶ 1 1 1 A 1 1 = T ¡ 2 : (4.25) 0 0 p p 0 0 (x +iy); (x +iy) 2 2 x +iy 2 2 x ¡iy 2 2 2 1 + 2¼ ¸ e 1 + 2¼ ¸ e The factorization could be imperfect in the superstring case, due to the fermionic nature of the boundary fermions [4]. This may spoil the exact marginality of the pro¯le. However, we expect that it indeed factorizes since there is a path-ordered operator P in the de¯nition of the correators and this P may recover the the marginality of the pro¯le. { 13 { JHEP02(2003)039 The expressions, along with the complex conjugate of the last equation, yields the energy momentum tensor 2 2 x 1 1 + 2¼ ¸ e cos y T = ¡T Re = ¡T ; (4.26) 00 p p 0 0 0 2 2 x ¡iy 2 2 x 4 4 2x 1 + 2¼ ¸ e 1 + 4¼ ¸ e cos y + 4¼ ¸ e T = ¡T ; (4.27) yy 00 2 2 x 1 2¼ ¸ e sin y T = T Im = T : (4.28) 0y p p 0 0 0 2 2 x ¡iy 2 2 x 4 4 2x 1 + 2¼ ¸ e 1 + 4¼ ¸ e cos y + 4¼ ¸ e The energy momentum tensor again exhibits singularities. They appear at the critical time 0 2 2 x = ¡ log(2¼ ¸ ) and at the loci y = (2n + 1)¼ ; n 2 Z : (4.29) The energy density ½ = ¡T behaves as " # 0 0 ½ ! T + 2¼ sgn(x ¡ x )±(y ¡ y ) ; (4.30) p n n2Z as the critical time is approached, essentially like the bosonic case (3.19). In the superstring case the interpretation is on a ¯rmer footing since the tachyon pro¯le (4.22) amounts to the rolling down either of two sides of a symmetric, and regular, potential. Given the topology of this situation it is not at all surprising that codimension one defects result. Additionally, defects interpolating between the two sides of the potential are known to couple to RR- ¯elds such that, in the present case, consecutive branes have opposite signs, D-branes and anti-D-branes. They are su±ciently separated that the low energy °uctuation spectrum contains no tachyons; so the con¯guration is classically stable. Nevertheless, the energy density of the defect is larger by a factor of 2 than that of a BPS D-brane. As in the bosonic case we can verify this result using energy conservation. 5. Symmetries and boundary states The purpose of this section is to reconsider the tachyon pro¯les in the previous sections using the methods and results from Sen's recent work on related pro¯les [1]. 5.1 The group of time-dependent marginal deformations A natural set of spatially homogeneous tachyon pro¯les in bosonic string theory is 0 0 T (X) = ¸ cosh X + ¸ sinh X ; (5.1) 1 2 0 0 for general (¸ ; ¸ ). These tachyon pro¯les are invariant under X ! X + c; (¸ § ¸ ) ! 1 2 1 2 ¨c (¸ § ¸ )e ; so time translations act on the parameters (¸ ; ¸ ) as the group SO(1; 1). 1 2 1 2 2 2 This means group invariants ¸ ¡ ¸ and sign(¸ § ¸ ) classify the possible perturbations, 1 2 1 2 in the sense that any two tachyon pro¯les of the form (5.1) with identical values of these invariants are physically equivalent. { 14 { JHEP02(2003)039 2 2 As a representative of the elliptic equivalence class ¸ ¡ ¸ > 0, we can take T (X) = 1 2 0 0 0 ¸ cosh X . Since T (X = 0) = ¸ , @ T (X = 0) = 0 this corresponds to there being a 1 1 0 time, chosen without loss of generality as X = 0, where the tachyon is displaced from the top of potential, but its velocity vanishes. Physically, we can thus think of the elliptic equivalence class as having negative energy. The tachyon ¯eld starts at the bottom of the potential, reaches a maximum at an intermediate time taken as X = 0, and it returns to its starting point at large times. The sign of ¸ determines which side of the potential the entire trajectory takes place, with positive lambda corresponding to the stable side for bosonic strings. 2 2 As representative of the hyperbolic equivalence class ¸ ¡ ¸ < 0, we can take T (X) = 1 2 0 0 0 ¸ sinh X , also considered in [1]. For this pro¯le T (X = 0) = 0, @ T (X = 0) = ¸ ; so 2 0 2 this corresponds to there being a time where the tachyon is on top of the potential, with a non-vanishing velocity. Physically we can think of the hyperbolic trajectories as having positive energy, with the tachyon ¯eld starting at the bottom on one side, reaching the maximum the top of the potential at the time chosen as X = 0, and then rolling to the bottom of the potential on the other side. The sign of ¸ determines which side of the potential the motion starts from. 2 2 The main focus in this paper is the parabolic equivalence class ¸ = ¸ . Taking 1 2 ¸ = ¸ (= ¸) in (5.1) gives 1 2 T (X) = ¸e : (5.2) A good physical characterization of the parabolic case is vanishing energy; the tachyon starts at the top of the potential, reaching the bottom of the potential at late times. Having no energy in the initial state, except for the tension of the unstable brane itself, this pro¯le realizes the intuition of a spontaneously decaying brane. Time translations can be absorbed in the magnitude of the parameter ¸ which is thus inconsequential. Taking ¸ = ¡¸ (= ¸) would be the time-reversed trajectory, and the sign of ¸ corresponds to the 1 2 two sides of the potential. The parabolic tachyon pro¯le is called a half S-brane in [13], with the hyperbolic one being an S-brane. The parabolic pro¯les can be obtained as limiting cases of the elliptic ones. Indeed, starting from T (X) = ¸ cosh(X ) and taking the limit ¸ ! 0; c ! 1 with ¯xed 1 1 1 c ¸ = ¸ e , we recover (5.2). The limit corresponds to tuning the displacement of the tachyon at X = 0 to zero, while moving the time at which the maximum is reached from from X = 0 to the in¯nite past. Since the parabolic pro¯les can be represented as limits of the elliptic ones, the corre- sponding energy-momentum tensors can be determined from those computed by Sen [1]. For (5.1), with ¸ = 0, Sen found the stress tensor T = (1 + cos 2¼¸ ) and T = 2 00 1 ij ¡± T f(x ), where ij 00 1 1 f(x ) = + ¡ 1 : (5.3) 0 0 x ¡x 1 + sin(¸ ¼)e 1 + sin(¸ ¼)e 1 1 Generalizing this result to arbitrary elliptic (¸ ; ¸ ), using the symmetry under time trans- 1 2 { 15 { JHEP02(2003)039 lation, we ¯nd · ¸ 2 2 T = 1 + cos(2¼ ¸ ¡ ¸ ) ; 1 2 T = ¡± T f(x ) ; (5.4) ij ij 00 where 1 1 p p f(x ) = + ¡ 1: (5.5) 2 2 2 2 sin(¼ ¸ ¡¸ ) sin(¼ ¸ ¡¸ ) 0 0 1 2 1 2 x ¡x p p 1 + (¸ + ¸ ) e 1 + (¸ ¡ ¸ ) e 1 2 1 2 2 2 2 2 ¸ ¡¸ ¸ ¡¸ 1 2 1 2 Taking the limit ¸ ! ¸ (´ ¸) to the parabolic case this gives T = T and T = 1 00 p ij ¡± T f(x ) with ij p f(x ) = ; (5.6) 1 + 2¼¸e in agreement with our explicit computations. The corresponding limit for the superstring similarly lead to the results found in the previous section. 5.2 Spatial variation We can also consider the pro¯le with spatial variation µ ¶ 0 Y X = 2 T (X) = 2¸e cos p ; (5.7) as a limit of the pro¯le µ ¶ X Y T (X) = ¸ cosh p cos p ; (5.8) 2 2 0 0 considered previously by Sen. The procedure is to replace X ! X + c and then taking 1 c= 2 the limit ¸ ! 0; c ! 1 with ¯xed ¸ ´ ¸ e . The pro¯le (5.8) was found to yield 1 1 the energy momentum tensor T = T Ref ; 00 p T = T Imf ; 0y p T = ¡T Ref = ¡T ; yy p 00 T = ¡T ± jfj ; (5.9) ij p ij where 1 1 f = p + p ¡ 1 ; (5.10) 0 0 (x +iy)= 2 ¡(x +iy)= 2 ~ ~ 1 + sin(¸¼=2) e 1 + sin(¸¼=2) e so we can simply take the limit and ¯nd f = : (5.11) (x +iy)= 2 1 + 2¼¸ e Then (5.9) agrees with the results of our explicit computations (3.12){(3.14). { 16 { JHEP02(2003)039 5.3 The full boundary states Since the parabolic case can be obtained as a suitable limit of other cases we can also use previous works to obtain the full boundary state. Let us ¯rst review the strategy following [1]. Starting with the general pro¯le (5.1), 0 0 performing the Wick rotation X = iX and rede¯ning ¸ = ¡i¸ , we ¯nd the tachyon pro¯le T (X) = ¸ cos X + ¸ sin X. This action contains only modes with integer mo- mentum modes; so we can consider the theory compacti¯ed on a self-dual radius R = 1, instead of the uncompacti¯ed theory. At this radius there is an SU(2) current algebra with zero-modes I I dz dz § §2iX (z) 3 J = e ; J = i@X (z) : (5.12) 2¼i 2¼i The tachyon pro¯le T (X) is precisely a linear combination of these generators and we see that the (¸ ; ¸ ) can be represented as SU(2) parameters as µ µ ¶¶ µ ¶ 0 ¸ + i¸ a b R = exp i¼ = ; (5.13) 0 ¤ ¤ ¸ ¡ i¸ 0 ¡b a where µ ¶ µ ¶ q q ¸ + i¸ 2 2 2 2 0 2 0 a = cos ¼ ¸ + ¸ ; b = i q sin ¼ ¸ + ¸ : (5.14) 1 2 1 2 2 0 ¸ + ¸ 1 2 The boundary state for the unperturbed D-brane, with X kept explicit, can be written as [21] X X Neumann jBi = jj; m; mii ; (5.15) m=j j=0; ;¢¢¢ where jj; m; mii is the Virasoro-Ishibashi state [22] for the discrete Virasoro primary jj; m; mi ´ jj; mijj; mi. It is simply a sum of all Virasoro descendants of the primary jj; m; mi. Since the tachyon pro¯le T (X) is an element of the SU(2) algebra, and jj; mi transforms in the (j; m) representation of SU(2) algebra, the non-trivial part of the boundary state becomes simply [21, 23, 24] X X jBi 0 = D (R)jj; m; mii ; (5.16) x m;¡m m=j j=0;1=2;¢¢¢ where D (R) is the spin j representation matrix of the rotation R in J eigenbasis m;¡m (see [24] for an explicit form) . To ¯nd the boundary state for the time-dependent tachyon pro¯le (5.1), we then apply the appropriate inverse Wick rotation noting that, after the inverse Wick rotation, b is not the complex conjugate of b. In the parabolic limit ¸ ! ¸ ´ ¸ the "rotation" matrix becomes µ ¶ 1 0 R = : (5.17) 2¼i¸ 1 { 17 { JHEP02(2003)039 These considerations indeed give the correct stress tensor, for general (¸ ; ¸ ). Writing 1 2 the boundary state as 0 0 jBi 0 = f(x )j0i + g (x )® ® j0i + ¢¢¢ ; (5.18) ¹º ¡1 ¡1 0 0 0 the stress tensor is given by T (x ) = f(x )´ + g (x ), as in section 2. The boundary ¹º ¹º ¹º j j 2j state jBi 0 » (D jj; j; jii + D jj;¡j;¡jii), with jj;§j;§jii = (i) £ j=0;1=2;¢¢¢ j;¡j ¡j;j j j §2ijX(0) ¤ 2j 2j 0 e j0i + ¢¢¢ and D = (¡b ) ,D = b indeed lead to the f(x ) given in (5.5). j;¡j ¡j;j The g (x ) is similarly reproduced correctly. From the boundary state point of view the ¹º simpli¯cation o®ered by the parabolic case is that the representation matrices take the simple form (j + m)! 2m D = (2¼i¸) ± ; (5.19) m¸0 ¡m;m (j ¡ m)!(2m)! rather than the complex formula given in [24]. Acknowledgments We thank V. Balasubramanian, M. Einhorn, Y. He, M. Huang, T. Levi, and B. Zwiebach for discussions. F.L. is supported in part by DOE grant and A.N. is supported by DOE grant DOE-FG02-95ER40893. A.N. and S.T. thank the Michigan Center for theoretical physics for hospitality during portions of this work. References [1] A. Sen, Rolling tachyon, J. High Energy Phys. 04 (2002) 048 [hep-th/0203211]. [2] A. Sen, Tachyon matter, J. High Energy Phys. 07 (2002) 065 [hep-th/0203265]. [3] A. Sen, Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143]. [4] A. Sen, Time evolution in open string theory, J. High Energy Phys. 10 (2002) 003 [hep-th/0207105]. [5] A. Sen, Time and tachyon, hep-th/0209122. [6] A. Sen, Rolling tachyon, J. High Energy Phys. 04 (2002) 048 [hep-th/0203211]; Tachyon matter, J. High Energy Phys. 07 (2002) 065 [hep-th/0203265]; Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143]; Time and tachyon, hep-th/0209122. [7] A.H. Guth and S.-Y. Pi, The quantum mechanics of the scalar ¯eld in the new in°ationary universe, Phys. Rev. D 32 (1985) 1899. [8] E.J. Weinberg and A.-q. Wu, Understanding complex perturbative e®ective potentials, Phys. Rev. D 36 (1987) 2474. [9] B. Craps, P. Kraus and F. Larsen, Loop corrected tachyon condensation, J. High Energy Phys. 06 (2001) 062 [hep-th/0105227]. [10] G.N. Felder, L. Kofman and A. Starobinsky, Caustics in tachyon matter and other Born-Infeld scalars, J. High Energy Phys. 09 (2002) 026 [hep-th/0208019]. { 18 { JHEP02(2003)039 [11] S. Mukohyama, Inhomogeneous tachyon decay, light-cone structure and D-brane network problem in tachyon cosmology, Phys. Rev. D 66 (2002) 123512 [hep-th/0208094]. [12] M. Berkooz, B. Craps, D. Kutasov and G. Rajesh, Comments on cosmological singularities in string theory, hep-th/0212215. [13] A. Strominger, Open string creation by s-branes, hep-th/0209090. [14] A.A. Gerasimov and S.L. Shatashvili, On exact tachyon potential in open string ¯eld theory, J. High Energy Phys. 10 (2000) 034 [hep-th/0009103]; D. Kutasov, M. Marino ~ and G.W. Moore, Remarks on tachyon condensation in superstring ¯eld theory, hep-th/0010108; Some exact results on tachyon condensation in string ¯eld theory, J. High Energy Phys. 10 (2000) 045 [hep-th/0009148]; P. Kraus and F. Larsen, Boundary string ¯eld theory of the dd-bar system, Phys. Rev. D 63 (2001) 106004 [hep-th/0012198]; T. Takayanagi, S. Terashima and T. Uesugi, Brane-antibrane action from boundary string ¯eld theory, J. High Energy Phys. 03 (2001) 019 [hep-th/0012210]. [15] E.S. Fradkin and A.A. Tseytlin, Quantum string theory e®ective action, Nucl. Phys. B 261 (1985) 1; C.G. Callan Jr., E.J. Martinec, M.J. Perry and D. Friedan, Strings in background ¯elds, Nucl. Phys. B 262 (1985) 593. [16] T. Okuda and S. Sugimoto, Coupling of rolling tachyon to closed strings, Nucl. Phys. B 647 (2002) 101 [hep-th/0208196]. [17] B. Chen, M. Li and F.L. Lin, Gravitational radiation of rolling tachyon, J. High Energy Phys. 11 (2002) 050 [hep-th/0209222] [18] J.A. Harvey, D. Kutasov and E.J. Martinec, On the relevance of tachyons, hep-th/0003101. [19] S. Sugimoto and S. Terashima, Tachyon matter in boundary string ¯eld theory, J. High Energy Phys. 07 (2002) 025 [hep-th/0205085]. [20] O.D. Andreev and A.A. Tseytlin, Partition function representation for the open superstring e®ective action: cancellation of mobius in¯nities and derivative corrections to Born-Infeld lagrangian, Nucl. Phys. B 311 (1988) 205. [21] C.G. Callan, I.R. Klebanov, A.W. Ludwig and J.M. Maldacena, Exact solution of a boundary conformal ¯eld theory, Nucl. Phys. B 422 (1994) 417 [hep-th/9402113]. [22] N. Ishibashi, The boundary and crosscap states in conformal ¯eld theories, Mod. Phys. Lett. A 4 (1989) 251. [23] J. Polchinski and L. Thorlacius, Free fermion representation of a boundary conformal ¯eld theory, Phys. Rev. D 50 (1994) 622 [hep-th/9404008]. [24] A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of D-branes, Nucl. Phys. B 545 (1999) 233 [hep-th/9811237]. { 19 { http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of High Energy Physics IOP Publishing

Rolling tachyons and decaying branes

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JHEP02(2003)039 Published by Institute of Physics Publishing for SISSA/ISAS Received: February 11, 2003 Revised: February 19, 2003 Accepted: February 19, 2003 Finn Larsen Michigan Center for Theoretical Physics, Randall Lab., University of Michigan Ann Arbor, MI 48109, USA E-mail: [email protected] Asad Naqvi Department of Physics and Astronomy, David Rittenhouse Laboratories University of Pennsylvania Philadelphia, PA 19104, USA E-mail: [email protected] Seiji Terashima Institute for Theoretical Physics, University of Amsterdam 1018 XE Amsterdam, The Netherlands E-mail: [email protected] Abstract: We present new rolling tachyon solutions describing the classical decay of D- branes. Our methods are simpler than those appearing in recent works, yet our results are exact in classical string theory. The role of pressure in the decay is studied using tachyon pro¯les with spatial variation. In this case the ¯nal state involves an array of codimension one D-branes rather than static, pressureless tachyon matter. Keywords: D-branes, Tachyon Condensation. ° SISSA/ISAS 2003 http://jhep.sissa.it/archive/papers/jhep022003039 /jhep022003039.pdf JHEP02(2003)039 Contents 1. Introduction and summary 1 2. Rolling tachyons 3 2.1 Generalities 4 2.2 Spatially homogenous decay 5 2.2.1 B(x ) 5 ¹º 2.2.2 A 6 3. Spatially inhomogenous decay 8 4. Rolling tachyons in superstring theory 10 4.1 Generalities 10 4.2 The simple decay 11 4.3 Spatial inhomogeneity 13 5. Symmetries and boundary states 14 5.1 The group of time-dependent marginal deformations 14 5.2 Spatial variation 16 5.3 The full boundary states 17 1. Introduction and summary An appealing way to generate time dependent con¯gurations in string theory is to consider the classical decay of unstable systems of D-branes, pictured as a tachyon ¯eld rolling down a potential, towards a stable minimum. This system is promising from the viewpoint of studying time-dependence in string theory since the non-trivial dynamics is con¯ned to the open string sector. Moreover, this setting provides a natural arena for discussing important cosmological ideas, such as in°ation, and the beginning of time. The quantitative study of rolling tachyons was initiated recently by A. Sen [1]{[5]. It involves deforming the world sheet conformal ¯eld theory (CFT) of the unstable D-brane by an exactly marginal, time dependent tachyon pro¯le. The deformed CFT is an exact classical background in string theory, interpreted as the classical decay of the unstable D- brane. Although several key results have been obtained, this approach is still in its infancy and central questions remain: { 1 { JHEP02(2003)039 ² In the classical approximation, the string coupling constant is strictly vanishing. However, it is not clear that the implied limit is smooth. The system with a small, but non-zero string coupling may di®er qualitatively from the system with vanishing coupling. In such a scenario, the classical approximation is misleading. This concern is fueled by the somewhat mysterious role of closed strings in tachyon condensation: after the decay of the unstable D-brane only closed strings remain in the spectrum; yet the brane cannot decay into closed strings if the coupling is strictly vanishing. 1 2 2 ~ ~ ² A tachyon pro¯le with spatial momentum k has e®ective mass m = ¡1 + k and so is unstable for any jkj < 1, indicating that all wave lengths play some role in the decay. Indeed, in quantum ¯eld theory it is well understood that tachyon conden- sation is a process where the longest wave lengths dominate, but all wave lengths participate, and the decay is de¯nitely inhomogeneous. These results are best known in the context of cosmological in°ation [7, 8], but they are valid also for tachyon condensation in string theory [9]. They indicate that spatially inhomogenous modes are important also for rolling tachyons. In this paper we study several new tachyon pro¯les with the goal of shedding light on these questions. The deformations we consider are actually technically simpler than those previously studied. We are therefore able to avoid the full machinery of boundary states and instead carry out the computations using elementary methods. The approach taken here complements the one taken in previous papers on rolling tachyons and may o®er some conceptual advantages. As discussed in section 5, the new spatially homogenous pro¯le, given below as (1.1), has a topology that is di®erent from previous examples. In addition to this decay, in which the rolling tachyon has no spatial dependence, we study an exactly soluble example with spatially varying tachyon pro¯le, where we can follow the decay to its inhomogeneous ¯nal state. The simplest of the new pro¯les studied in this paper is T (X) = ¸e : (1.1) This can be interpreted as simultaneously displacing and giving a velocity to the tachyon, 0 0 i.e. imposing the initial condition T (X = 0) = @ T (X = 0) = ¸. An alternative, 0 3 and better, spacetime interpretation is that of a perturbation at X = ¡1. Since this disturbance is automatically in¯nitesimal, the pro¯le (1.1) seems to be a particularly clean example of a rolling tachyon. The pro¯le (1.1) is also particularly simple from a technical point of view. The simplest exactly marginal deformations of a world sheet CFT are generated by the vertex operators ikX 2 2 2 V (X) = e : k = k ¡ k = ¡1 : (1.2) Of course these operator cannot usually be added to the world sheet action because they correspond to complex potentials. The standard remedy is to add also the conjugate 1 0 We use units such that ® = 1. Other discussions of spatial variation include [4] and [10]{[12]. This was the point of view taken in the recent talk by Strominger [13]. { 2 { JHEP02(2003)039 operator and so consider perturbations of the form ~ ~ ~ ~ ik¢X ¡ik¢X 2 ~ ~ ~ ~ T (X) = ¸ cos(k ¢ X) = (e + e ) ; k = 1 : (1.3) After analytical continuation this leads to Sen's pro¯le T (X) = ¸ cosh(X ). An alternative procedure, exploited in this paper, is to note that in the special case k = ¡i, k = 0 the vertex operator (1.2) is in fact real, and so we can consider (1.1) directly, without adding the complex conjugate. This is much simpler because, in the case of (1.3), complications arise from the cross-terms between the two exponentials. As explained above, it is important to study rolling tachyons with spatially varying pro¯les. Such pro¯les are generally quite complicated to analyze in the full CFT; but in the case of the pro¯le 0 1 X = 2 2 ~ ~ ~ ~ T (X) = ¸e cos(k ¢ X) ; k = ; (1.4) the study simpli¯es dramatically (similar, but more complicated tachyon pro¯les were discussed in [4]). Indeed, this pro¯le is a linear combination of two vertex operators of the form (1.2). Crucially, these vertex operators commute, in contrast to those appearing in (1.3). Thus the theory with the pro¯le (1.4) essentially reduces to two copies of (1.1). Having solved the theory with the tachyon pro¯le (1.4) we ¯nd the coupling to the energy momentum tensor for all times X . The energy momentum tensor exhibits qualita- tively di®erent behavior from the spatially homogenous case. It develops codimension one singularities in ¯nite time. These singularities can be interpreted as an array of (excited) D-branes. This result is consistent with the expectation that ¯nal states will be spatially inhomogeneous for generic decay channels. Of course, the pro¯le that we can actually solve (1.4) is actually quite special. Presumably that is why the ¯nal state in this example, although spatially inhomogeneous, is as ¯nely tuned as a perfect array of unstable branes. In a realistic, semi-classical, analysis one would choose as initial state for the brane some wave-packet localized near the top of the tachyon potential and the full decay process would be described as an average, in a precise sense, of all the initial conditions represented by this wave packet. One would expect this ¯nal state to be dominated by generic, spatially dependent, con¯gurations which, to the extent they can be described as a perfect °uid, certainly would have pressure. This paper is organized as follows. In section 2 we explain our approach to rolling tachyons and carry out the details for the pro¯le (1.1). In section 3 we consider the spatially inhomogeneous case (1.4) and discuss the lessons for the full decay process, when all spatial variations are included. In section 4, we extend these results to the superstring case. Finally, in section 5, we discuss the topology of our pro¯les, their relations to previous works, and construct the boundary states corresponding to our solutions. 2. Rolling tachyons The strategy for treating rolling tachyons is to deform the world sheet CFT of an unsta- ble D-brane by an exactly marginal operator and interpret the deformed CFT as a time dependent solution to the classical string equations of motion. Instead of studying the { 3 { JHEP02(2003)039 system in terms of a boundary state of the closed string theory (which was Sen's approach in [1, 2, 4]), we will primarily work with open strings, equating the disk partition function with space-time action, following the analysis of static tachyon con¯guations in boundary string ¯eld theory [14]. 2.1 Generalities According to the ¾-model approach to string theory, the space-time action is given by the partition function of the world-sheet theory, with the world-sheet couplings interpreted as spacetime ¯elds [15]. Thus, in the open string sector, ¹ ¡I ¡I bulk bndy S[¸ ] / Z (¸ ) = [dX ]e ; (2.1) i disk i where Z is the disk partition function, ¸ are exactly marginal couplings for the bound- disk i ary operators, and 2 ¹ º I = d z ´ @X @X ; (2.2) bulk ¹º 2¼ I = dt T (X) + ¢¢¢ ; (2.3) bndy @D where D is the unit disk and @D is its boundary. T is the tachyon ¯eld and the ¢¢¢ indicate other marginal boundary perturbations. This procedure is similar in spirit to the boundary string ¯eld theory approach to tachyon condensation [14], although here we limit ourselves to marginal perturbations. Our formulae above are written for euclidean spacetimes, as well as euclidean world- sheets. The time-dependence is then taken into account by including the minkowskian ¹º 0 metric ´ = (¡; +;¢¢¢ ; +) when contracting the temporal ¯elds X . This procedure is motivated by analytical continuation, as in Sen's computations using boundary states and cubic string ¯eld theory. This type of analytical continuation is clearly not completely satisfying; indeed, the precise relation between lorentzian and euclidean signature is one of the main unsettled questions facing most approaches to time dependence in string theory. On the other hand, the analytical continuation gives physically reasonable results in many examples, including those considered here; so it presumably captures important aspects of the problem. It will be useful to write the action S as a space-time integral over a lagrangian density. ¹ ¹ ¹ 0¹ To this end, we split X into a constant and a varying part, X = x + X and write ¹ ¡I ¡I bulk bndy S / [dX ]e Z Z 1 2 0¹ 0º p 0¹ ¡ d z g @X @X ¡I ¹º bndy 2¼ D = d x ¡g [dX ]e ; (2.4) where, in the second line, we have made the obvious generalization of the expression to curved space. In this paper we are primarily considering the coupling to gravity, to explore the time-evolution of an unstable D-brane in a °at background. From the spacetime action It would be interesting to apply our methods for couplings to more massive strings as well, in an e®ort to illluminate the problems discussed in [16]. See also [17] for the coupling to the closed string. { 4 { JHEP02(2003)039 ± ¡g 2 ±S 1 we form the energy-momentum tensor T = ¡ and use = ¡ ¡gg to ¹º ¹º ¹º ¹º ±g ±g 2 ¡g ¯nd T (x) = K (B(x) ´ + A (x)) ; (2.5) ¹º ¹º ¹º in °at space. Here K is an overall normalization constant and 2 0¹ ¹ 0º 0¹ ¡ d z ´ @X @X ¡I ¹º bndy 2¼ D B(x) = [dX ]e ; (2.6) Z Z d z 1 2 1 0¹ 0º ¹º 0¹ ¹ º ¡ d z ´ @X @X ¡I ¹º ¹ bndy 2¼ D 2¼ A (x) = 2 [dX ] @X @X e 2¼ 2 0¹ ¹ 0º 0¹ ¹ º ¡ d z ´ @X @X ¡I ¹º ¹ bndy 2¼ D = 2 [dX ] @X (0)@X (0)e : (2.7) d z In the second line of (2.7) we ¯xed the position of the vertex operator and used = 2¼ A(D) = 1 for the unit disc. The expression (2.5) for the energy-momentum tensor was previously derived by Sen [1] using BRST invariance of the corresponding boundary state. In the following we consider various T (X), corresponding to tachyon pro¯les with speci¯c space-time dependence. To determine the energy-momentum tensor for each pro¯le we ¹º need to compute A and B. 2.2 Spatially homogenous decay In this section we study the tachyon pro¯le T (X) = ¸e : (2.8) This is an exactly marginal deformation of the CFT and so an exact solution to the classical string equations of motion. It is interpreted in spacetime as a perturbation at X = ¡1, displacing the tachyon in¯nitesimally from the unstable maximum of the potential. Alternatively, this pro¯le corresponds to kicking the tachyon from T (X = 0) = ¸, with velocity @ T (X = 0) = ¸. To determine the stress tensor T for this tachyonic pro¯le, we need to compute the ¹º 0 0 functions B(x ) and A (x ). ¹º 2.2.1 B(x ) The function B(x ) is the disk partition function, except that zero modes remain uninte- grated. Using h¢¢¢i as symbols for expectation values on the disc we have the perturbative expansion 0 00 0 x X 0 ¡I (x+X ) ¡¸e dt e bndy B(x ) = he i = he i ; (2.9) 0 Z x n (¡2¼¸e ) dt dt 00 00 1 n X (t ) X (t ) 1 n = ¢¢¢ he ¢¢¢ e i ; (2.10) n! 2¼ 2¼ n=0 The Green's function on the unit disk with Neumann boundary conditions is ¹º 0 ¹ º 0 ¹º 0 0 G (z; z ) = hX (z)X (z )i = ´ (¡ logjz ¡ z j ¡ logjzz¹ ¡ 1j) ; (2.11) { 5 { JHEP02(2003)039 it so, taking z = e , we ¯nd µ ¶ Y Y 00 00 t ¡ t i j X (t ) X (t ) it it 2 n(n¡1)=2 2 1 n i j he ¢¢¢ e i = je ¡ e j = 4 sin : (2.12) i<j i<j The integrals in (2.10) give Z µ ¶ dt dt t ¡ t 1 n i j n(n¡1)=2 2 ¢¢¢ 4 sin = n! ; (2.15) 2¼ 2¼ 2 i<j and the ¯nal result for B(x ) becomes x n (¡2¼¸e ) 0 0 B(x ) = n! = f(x ) ; (2.16) n! n=0 where f(x ) ´ : (2.17) 1 + 2¼¸e The summation of the perturbative series is clearly justi¯ed for couplings within the radius 1 ¡x of convergence j¸j < e . The regime of validity may be extended by analytical contin- 2¼ uation to include all positive ¸. The precise justi¯cation for this extension is an interesting question that deserves further study. ¹º 2.2.2 A ¹º ¹ º The A are proportional to expectation values of graviton vertex operators : @X @X :, where the normal ordering symbol : : indicate that the divergent pieces have been sub- tracted as z ! z ¹ º 0 ¹ º 0 ¹º 0 0 ¹ ¹ ¹ : @X (z)@X (z ) : = @X (z)@X (z ) + ´ @@ logjz ¡ z j : (2.18) For the purpose of our calculation, it is useful to de¯ne another kind of normal ordering ± ± symbol where we subtract the full Green's function (2.11), viz ± ± ± ¹ º 0 ± ¹ º 0 0 ¹º 0 ¹ ¹ ¹ @X (z)@X (z ) = @X (z)@X (z ) ¡ @@ G (z; z ) ; (2.19) ± ± including the contribution from the image charge. The two normal orderings are related as ¹ º ± ¹ º ± ¹º ¹ ¹ : @X (0)@X (0) : = @X (0)@X (0) + ´ : (2.20) ± ± These integrals can be derived by considering an integration over U(n) matrices and using the known result that the U(n) Haar measure dU, when expressed in terms of the eigenvalues, becomes Z Z 1 1 dt i 2 dU = ¢ (t) ; (2.13) vol U(n) n! 2¼ where ¢(t) is the relevant Vandermonde determinant for U(n) matrices µ ¶ t ¡ t i j ¢(t) = 2 sin : (2.14) i<j By noticing that the LHS in (2.13) is 1, the integral in (2.15) follows immediately. { 6 { JHEP02(2003)039 ¹º Now we are ready to calculate A . For i; j 6= 0 we have 0 00 x X ij i j ¡¸e dt e A (x) = 2h: @X (0)@X (0) : e i ¿µ ¶ À ij 0 00 ± x X ± i j ± ¡¸e dt e = 2 @X (0)@X (0) + e ± ± ij 0 = ± f(x ) : (2.21) ± ± In going from the second to the third line, the normal ordered term between gives ± ± ij 0 no contribution, and the term proportional to ± is exactly the same as B(x ) computed earlier. The calculation of A is a bit more involved. 00 0 0 ¡¸ dte A = 2h: @X (0)@X (0) : e i 0 00 x X ± 0 0 ± ¡¸e dte 0 = 2h @X (0)@X (0) e i ¡ f(x ) ± ± 0 Z x n X Y (¡2¼¸e ) dt 0 it i i ± 0 0 ± X (e ) 0 = 2 h @X (0)@X (0) e i ¡ f(x ) : (2.22) ± ± n! 2¼ i=1 The correlation function in (2.22) yields µ ¶µ ¶ Y Y X 1 1 ± 0 0 ± X(w ) 2 h @X (z)@X (z¹) : e :i = jw ¡ w j ± ; (2.23) ± ± i j n>0 z ¡ w z¹¡ w¹ i j i=1 i<j i;j it which, for z = 0 and w = e gives Z Z n n n Y Y Y Y X dt it dt i i i ± 0 0 ± X(e ) it it 2 ¡i(t ¡t ) i j i j h @X (0)@X (0) : e :i = je ¡ e j e ± ± 2¼ 2¼ i=1 i=1 i=1 i<j i;j µ ¶ Y Y dt t ¡ t i i j = 2 sin £ 2¼ 2 i=1 i<j 0 1 @ A £ n + 2 cos(t ¡ t ) : i j i<j The integral is 0 1 Z µ ¶ Y Y X dt t ¡ t i j @ A 2 sin n + 2 cos(t ¡ t ) = n! ; (2.24) i j 2¼ 2 i=1 i<j i<j and (2.22) becomes 00 0 A = f(x ) ¡ 2 : (2.25) Collecting the various results we ¯nd the stress tensor for the tachyon pro¯le (2.8) 0 0 0 0 0 T = K(¡B(x ) + A (x )) = ¡T ; T = K(B(x ) + A (x )) = ± T f(x ) : (2.26) 00 00 p ij ij ij p We have determined the normalization constant K = T by comparison with the static limit ¸ = 0. As expected, T is independent of x , which is just the statement of conser- 00 0 vation of energy. Moreover, T ! 0 as x ! 1, so the pressure vanishes in this limit, i.e. ij the decay product is pressureless tachyon matter, as in [2]. { 7 { JHEP02(2003)039 3. Spatially inhomogenous decay We will now investigate the spatially inhomogenous decay. A spatially inhomogenous pro¯le !X 2 2 ~ ~ ~ T (X) = 2¸e cos(k¢X) is marginal for ! +k = 1. This can be written as a sum of two ³ ´ 0 ~ ~ 0 ~ ~ !X +ik:X !X ¡ik:X vertex operators, each of which is exactly marginal: T (X) = ¸ e + e . For generic !, this is not an exactly marginal deformation because of the singular OPE between the two vertex operators. Hence it does not yield a solution to the classical string equations of motion. However, for ! = , this perturbation is exactly marginal. Without any loss of generality, we can keep only one component of k to be non-zero, denoting the ~ ~ corresponding direction Y ´ 2k ¢ X. Thus we have µ ¶ ³ ´ p 0 0 X +iY X ¡iY 0 Y p p X = 2 2 2 T (X) = 2¸e cos = ¸ e + e (3.1) U V = ¸(e + e ) ; (3.2) where we have de¯ned new variables 0 0 X + iY X ¡ iY p p U = ; V = ; (3.3) 2 2 such that hU(z)U(w)i = hV (z)V (w)i = logjz ¡ wj + logjzw¹ ¡ 1j ) ; (3.4) U(z)V (w) » regular : (3.5) Thus U and V behave as commuting time-like coordinates, with no mixing between U and V . ¹º Using the variables U and V , the calculation of B and A proceeds very similarly to the calculation in the last section. To compute B, we need to compute 0 0 u+U v+V ¸ dt(e +e ) B(u; v) = he i ; (3.6) where u and v are linear combinations of the zero modes of X and Y 0 0 x + iy x ¡ iy u = p ; v = p ; (3.7) 2 2 0 0 and U and V are non-zero modes of U and V . Since there is no mixing between U and V and (3.6) factorizes as R R 0 0 u U v V ¸e dte ¸e dte B(u; v) = he ihe i = f(u) f(v) ; (3.8) where, as in the previous section, we have de¯ned f(u) as f(u) = : (3.9) 1 + 2¼¸e { 8 { JHEP02(2003)039 ¹º The calculations for A , with the insertion of a graviton vertex operator factorize similarly. For example 0 0 u+U v+V uu ¸ dt(e +e ) A (x) = 2h: @U(0)@U(0) : e i ¿ À ³ ´ R R 0 0 1 U U v V ± ± ¸e dte ¸e dte = 2 @U(0)@U(0) ¡ e he i ± ± ³ ´ = f(u) ¡ 2 f(v) : (3.10) The remaining components give, uv A (u; v) = 0 ; vv A (u; v) = f(u) (f(v) ¡ 2) ij A (u; v) = ± f(u) f(v) : ij Furthermore, from the relations 1 1 1 A = (A ¡ A + 2iA ); A = (A + A ); A = (A ¡ A ¡ 2iA ) ; (3.11) uu 00 yy 0y uv 00 yy vv 00 yy 0y 2 2 2 and using (2.5), the stress tensor for this spatially inhomogenous decaying solution can be calculated to be x = 2 1 + 2¼¸e cos(y= 2) p p T (x ; y) = ¡T p ; (3.12) 00 p 0 0 x = 2 2 2 2x 1 + 4¼¸e cos(y= 2) + 4¼ ¸ e x = 2 ¡2¼¸e sin(y= 2) T (x ; y) = T p p ; (3.13) 0y p 0 0 x = 2 2 2 2x 1 + 4¼¸e cos(y= 2) + 4¼ ¸ e 0 0 T (x ; y) = ¡T (x ; y) : (3.14) yy 00 This stress tensor is conserved, i.e. 00 y0 @ T ¡ @ T = 0 ; (3.15) 0 y The form of the stress tensor and its late time behavior is qualitatively di®erent from that obtained in the spatially homogenous case in section 2.2. Certainly at large times x ! 1 all components of the stress tensor (3.12){(3.14), including the energy density, approach zero. However, this result probably cannot be trusted since, at a ¯nite critical time x ´ 2 ln(1=(2¼j¸j)), the stress energy tensor exhibits singularities at the spatial loci y = 2 2n¼; n 2 Z ; (¸ < 0) ; (3.16) y = 2 2(n + )¼; n 2 Z ; (¸ > 0) : (3.17) These singularities are what one would expect since, for some values of y, we are per- turbing with a negative tachyon and, according to (2.26), such perturbations give rise to singularities at ¯nite time already in the spatially homogenous setting. In [4] Sen proposed { 9 { JHEP02(2003)039 that these singularities should be interpreted as codimension one D-branes. To see this, we introduce the auxiliary variable 0 0 x ¡x ¢ = e ; (3.18) and, for either sign of ¸, we write the energy density ½ = ¡T as " # (1 ¡ ¢) + 2¢ sin ((y ¡ y )=2 2) 1 ¢!1 0 0 ½ = T » T + 2¼ sgn(x ¡ x )±(y ¡ y ) : p p n (1 ¡ ¢) + 4¢ sin ((y ¡ y )=2 2) n2Z (3.19) The limit was computed using lim = ¼±(®) close to each singular locus. The ²!0 2 2 ² +® corresponding average energy density changes discontinuously from T to 0 as we pass x . The form of the limiting energy density (3.19) suggests that the missing energy forms co- dimension one defects at the y . As we pass the critical time x , the loss in energy at each y is 2 2¼T . This result for the defect energy can be veri¯ed using energy conservation, n p noting that the defects are ¢y = 2 2¼ apart, and no bulk energy remains after they form. A co-dimension one D-brane has tension T = 2¼T and so the defect has additional p¡1 p energy, beyond that needed to form a D-brane. Nevertheless it is plausible that these defects are indeed related to D-branes since the spatial potential T (X) / cos(Y= 2) tends to con¯ne the ends of the strings, much as in the corresponding o®-shell discussion in [18]). 4. Rolling tachyons in superstring theory The purpose of this section is to generalize the results of the previous sections to the superstring. In each step of the computation details are modi¯ed, and so must be repeated; but the ¯nal results are closely analogous to the bosonic case. 4.1 Generalities In the superstring case world-sheet fermions must be included in a manner consistent with world-sheet supersymmetry. A convenient way implement this is to introduce world-sheet super¯elds. Thus the string coordinates (on the boundary of the disk) are represented by ¹ ¹ ¹ X = X + µÃ ; (4.1) and the Chan-Paton index of the brane is encoded in the boundary fermions I I I ¡ = ´ + µF : (4.2) We will consider only the simplest case of a single non-BPS D-brane. Since this corresponds to a single boundary fermion we can omit the index I. In this formalism the boundary action for a general tachyon pro¯le T (X) is I = dtdµ[¡D¡ + ¡T (X)] ; (4.3) bndy { 10 { JHEP02(2003)039 where D denotes the derivative in superspace D = @ + µ@ . A single boundary fermion ¡ can be integrated out with the result ¿ · ¸ À ¿ · ¸ À Z Z ¡I bndy he ¢¢¢i = P exp ¡ dtdµ¡T (X) ¢¢¢ = P cosh dtdµT (X) ¢¢¢ ; ¡¡even (4.4) within correlators. Here P is the standard path-ordering operator. Note that this path- ordering operator is not trivial in the above espression because dtdµT (X) is fermionic and then it does not commute with itself. The boundary fermions serve to make world-sheet supersymmetry manifest but, in the present context, they play the role of Chan-Paton matrices ¾ , for which the restriction to even terms arises from the overall trace. Following the bosonic example, our main interest is in inserting the identity operator ¿ µZ ¶À B(x) = P cosh dtdµT (X) ; (4.5) and the gravity vertex operator ¿ µ ¶À ¹º ¹º A = V (0; 0)P cosh dtdµT (X) ; (4.6) where, in the present case, ¹º ¹ º V (0; 0) = 2 dµdµ[DX DX ] (4.7) z=z¹=0 The energy momentum tensor still follows from (2.5). In (4.5) and (4.6) the brackets h¢¢¢i denote averaging with respect to the non-zero mode part of the bosonic ¯elds, as before. In concrete examples, we can evaluate these expressions in perturbation theory using the two point function [20] ¹ º ¹º 2 hX X i = ¡´ logjz j ; (4.8) where z = z ¡ z ¡ i z z µ µ : (4.9) 12 1 2 1 2 1 2 The form (4.8) of the two point function is valid when both coordinates are on the boundary of the disk. This will su±ce for our applications. 4.2 The simple decay We consider ¯rst the supersymmetric version of the pro¯le (3.1), i.e. X = 2 T (X) = ¸e : (4.10) 6 ¡ dtdµT (X) We have ignored the contact term e which appear in (4.4) for general tachyon pro¯les. This term is important in BSFT discussions of tachyon condensation [14], as well as in the time dependent case [19]. Here we follow Sen [2] and regard the right hand side of (4.4) as the starting point of our discussions. { 11 { JHEP02(2003)039 Expanding (4.5) in the parameter ¸ and using (4.8) yields 1 2n ³ ´ X 2n Y dt 0 n x = 2 B(x ) = (¡1) 2¼¸e dµ £(t ¡ t )£(t ¡ t )¢¢¢ £(t ¡ t ) £ i 1 2 2 3 2n¡1 2n 2¼ n=0 i=1 it it (t +t ) i j i j £ je ¡ e ¡ ie µ µ j : (4.11) i j i<j (t +t ) it it i j it it i j i j Noticing je ¡ e ¡ ie µ µ j = je ¡ e j + sign(t ¡ t )µ µ , the integrals can be i j i j i j evaluated with the result 2n Y Y dt i 1 it it (t +t ) i j i j dµ £(t ¡ t )£(t ¡ t )¢¢¢ £(t ¡ t ) je ¡ e ¡ ie µ µ j = ; i 1 2 2 3 2n¡1 2n i j 2¼ 2 i=1 i<j (4.12) so, after summation of the series, we ¯nd B(x ) = : (4.13) 2 2 2x 1 + 2¼ ¸ e The insertion of a graviton operator brings a few more complications, as in the bosonic case. The proper normal ordering again gives ¹º ± ¹º± ¹º : V : = V + ´ ; (4.14) ± ± and thus, without any further e®ort, ij 0 0 ij A (x ) = B(x )± : (4.15) The graviton with two temporal indices is evaluated in perturbation theory starting from (4.6). In addition to a term B´ = ¡B from the normal ordering (4.14), we ¯nd an integral over the correlator * + µ ¶ 2n ± ¹ º ± X(z )= 2 ¹ ¹ dµdµ DX (w)DX (w¹) e = ± ± i=1 Y X p 1 1 1 = jz ¡ z ¡ i z z µ µ j ; (4.16) i j i j i j 2 z ¡ w z¹ ¡ w¹ k l i<j k;l with w = w¹ = 0. The expressions can then be combined as 00 n x = 2 2n A = ¡B + 2 (¡1) (2¼¸e ) I ; (4.17) 2n n=0 where the integrals 2n Y Y dt it it (t +t ) i j i j I = dµ £(t ¡ t )¢¢¢ £(t ¡ t ) je ¡ e ¡ ie µ µ j £ 2n i 1 2 2n¡1 2n i j 2¼ i=1 i<j 1 1 i(t ¡t ) k l £ e µ µ = ± : (4.18) k l n>0 2 2 k;l We checked (4.12) for n · 3. We also checked (4.18) for n · 3. { 12 { JHEP02(2003)039 The ¯nal result thus becomes 00 0 0 A (x ) = p ¡ 2 = B(x ) ¡ 2 ; (4.19) 2 2 2x 1 + 2¼ ¸ e as in the bosonic case. The non-vanishing components of the energy momentum tensor now read 0 0 0 0 0 T = K(¡B(x ) + A (x )) = ¡2K ; T = K(B(x )± + A (x )) = 2K± B(x ) ; 00 00 ij ij ij ij (4.20) where B was given in (4.13). The overall constant K again is identi¯ed as K = T . As in the bosonic case, the energy density is constant ½ = T throughout the decay. The pressure p = 2B(x ) = ; (4.21) 2 2 2x 1 + 2¼ ¸ e is equal to the energy density p = ½ for the unstable brane at x = ¡1; but it decays exponentially to zero at large times. The main di®erence with the bosonic case is that now the decay is symmetric under ¸ ! ¡¸. In the supersymmetric case there is no singularity for either sign, as expected since the tachyon potential is symmetric, with both directions sloping down to the stable closed string vacuum. All these results are closely analogous to Sen's discussions, based on the potential T (X) = ¸ cosh(X). 4.3 Spatial inhomogeneity We also want to consider a spatially inhomogeneous pro¯le for the superstring. The simplest example is µ ¶ 1 1 T (X) = 2¸ e cos Y ; (4.22) where Y is one of the spatial directions. As in the bosonic case, this example is factorizable 1 1 0 0 (X +iY) (X ¡iY) 2 2 T (X) = ¸e + ¸e ; (4.23) 0 0 where, crucially, X +iY and X ¡iY have regular OPEs. Thus the example is essentially two copies of the pro¯le (4.10). From (4.13) we immediately ¯nd 1 1 B(x ; y) = T ; (4.24) p 0 2 2 x +iy 2 j1 + 2¼ ¸ e j ij ij while (4.15) gives A = ± B for i; j 6= y, and (4.17) combines with (4.13) to give µ ¶ 1 1 1 A 1 1 = T ¡ 2 : (4.25) 0 0 p p 0 0 (x +iy); (x +iy) 2 2 x +iy 2 2 x ¡iy 2 2 2 1 + 2¼ ¸ e 1 + 2¼ ¸ e The factorization could be imperfect in the superstring case, due to the fermionic nature of the boundary fermions [4]. This may spoil the exact marginality of the pro¯le. However, we expect that it indeed factorizes since there is a path-ordered operator P in the de¯nition of the correators and this P may recover the the marginality of the pro¯le. { 13 { JHEP02(2003)039 The expressions, along with the complex conjugate of the last equation, yields the energy momentum tensor 2 2 x 1 1 + 2¼ ¸ e cos y T = ¡T Re = ¡T ; (4.26) 00 p p 0 0 0 2 2 x ¡iy 2 2 x 4 4 2x 1 + 2¼ ¸ e 1 + 4¼ ¸ e cos y + 4¼ ¸ e T = ¡T ; (4.27) yy 00 2 2 x 1 2¼ ¸ e sin y T = T Im = T : (4.28) 0y p p 0 0 0 2 2 x ¡iy 2 2 x 4 4 2x 1 + 2¼ ¸ e 1 + 4¼ ¸ e cos y + 4¼ ¸ e The energy momentum tensor again exhibits singularities. They appear at the critical time 0 2 2 x = ¡ log(2¼ ¸ ) and at the loci y = (2n + 1)¼ ; n 2 Z : (4.29) The energy density ½ = ¡T behaves as " # 0 0 ½ ! T + 2¼ sgn(x ¡ x )±(y ¡ y ) ; (4.30) p n n2Z as the critical time is approached, essentially like the bosonic case (3.19). In the superstring case the interpretation is on a ¯rmer footing since the tachyon pro¯le (4.22) amounts to the rolling down either of two sides of a symmetric, and regular, potential. Given the topology of this situation it is not at all surprising that codimension one defects result. Additionally, defects interpolating between the two sides of the potential are known to couple to RR- ¯elds such that, in the present case, consecutive branes have opposite signs, D-branes and anti-D-branes. They are su±ciently separated that the low energy °uctuation spectrum contains no tachyons; so the con¯guration is classically stable. Nevertheless, the energy density of the defect is larger by a factor of 2 than that of a BPS D-brane. As in the bosonic case we can verify this result using energy conservation. 5. Symmetries and boundary states The purpose of this section is to reconsider the tachyon pro¯les in the previous sections using the methods and results from Sen's recent work on related pro¯les [1]. 5.1 The group of time-dependent marginal deformations A natural set of spatially homogeneous tachyon pro¯les in bosonic string theory is 0 0 T (X) = ¸ cosh X + ¸ sinh X ; (5.1) 1 2 0 0 for general (¸ ; ¸ ). These tachyon pro¯les are invariant under X ! X + c; (¸ § ¸ ) ! 1 2 1 2 ¨c (¸ § ¸ )e ; so time translations act on the parameters (¸ ; ¸ ) as the group SO(1; 1). 1 2 1 2 2 2 This means group invariants ¸ ¡ ¸ and sign(¸ § ¸ ) classify the possible perturbations, 1 2 1 2 in the sense that any two tachyon pro¯les of the form (5.1) with identical values of these invariants are physically equivalent. { 14 { JHEP02(2003)039 2 2 As a representative of the elliptic equivalence class ¸ ¡ ¸ > 0, we can take T (X) = 1 2 0 0 0 ¸ cosh X . Since T (X = 0) = ¸ , @ T (X = 0) = 0 this corresponds to there being a 1 1 0 time, chosen without loss of generality as X = 0, where the tachyon is displaced from the top of potential, but its velocity vanishes. Physically, we can thus think of the elliptic equivalence class as having negative energy. The tachyon ¯eld starts at the bottom of the potential, reaches a maximum at an intermediate time taken as X = 0, and it returns to its starting point at large times. The sign of ¸ determines which side of the potential the entire trajectory takes place, with positive lambda corresponding to the stable side for bosonic strings. 2 2 As representative of the hyperbolic equivalence class ¸ ¡ ¸ < 0, we can take T (X) = 1 2 0 0 0 ¸ sinh X , also considered in [1]. For this pro¯le T (X = 0) = 0, @ T (X = 0) = ¸ ; so 2 0 2 this corresponds to there being a time where the tachyon is on top of the potential, with a non-vanishing velocity. Physically we can think of the hyperbolic trajectories as having positive energy, with the tachyon ¯eld starting at the bottom on one side, reaching the maximum the top of the potential at the time chosen as X = 0, and then rolling to the bottom of the potential on the other side. The sign of ¸ determines which side of the potential the motion starts from. 2 2 The main focus in this paper is the parabolic equivalence class ¸ = ¸ . Taking 1 2 ¸ = ¸ (= ¸) in (5.1) gives 1 2 T (X) = ¸e : (5.2) A good physical characterization of the parabolic case is vanishing energy; the tachyon starts at the top of the potential, reaching the bottom of the potential at late times. Having no energy in the initial state, except for the tension of the unstable brane itself, this pro¯le realizes the intuition of a spontaneously decaying brane. Time translations can be absorbed in the magnitude of the parameter ¸ which is thus inconsequential. Taking ¸ = ¡¸ (= ¸) would be the time-reversed trajectory, and the sign of ¸ corresponds to the 1 2 two sides of the potential. The parabolic tachyon pro¯le is called a half S-brane in [13], with the hyperbolic one being an S-brane. The parabolic pro¯les can be obtained as limiting cases of the elliptic ones. Indeed, starting from T (X) = ¸ cosh(X ) and taking the limit ¸ ! 0; c ! 1 with ¯xed 1 1 1 c ¸ = ¸ e , we recover (5.2). The limit corresponds to tuning the displacement of the tachyon at X = 0 to zero, while moving the time at which the maximum is reached from from X = 0 to the in¯nite past. Since the parabolic pro¯les can be represented as limits of the elliptic ones, the corre- sponding energy-momentum tensors can be determined from those computed by Sen [1]. For (5.1), with ¸ = 0, Sen found the stress tensor T = (1 + cos 2¼¸ ) and T = 2 00 1 ij ¡± T f(x ), where ij 00 1 1 f(x ) = + ¡ 1 : (5.3) 0 0 x ¡x 1 + sin(¸ ¼)e 1 + sin(¸ ¼)e 1 1 Generalizing this result to arbitrary elliptic (¸ ; ¸ ), using the symmetry under time trans- 1 2 { 15 { JHEP02(2003)039 lation, we ¯nd · ¸ 2 2 T = 1 + cos(2¼ ¸ ¡ ¸ ) ; 1 2 T = ¡± T f(x ) ; (5.4) ij ij 00 where 1 1 p p f(x ) = + ¡ 1: (5.5) 2 2 2 2 sin(¼ ¸ ¡¸ ) sin(¼ ¸ ¡¸ ) 0 0 1 2 1 2 x ¡x p p 1 + (¸ + ¸ ) e 1 + (¸ ¡ ¸ ) e 1 2 1 2 2 2 2 2 ¸ ¡¸ ¸ ¡¸ 1 2 1 2 Taking the limit ¸ ! ¸ (´ ¸) to the parabolic case this gives T = T and T = 1 00 p ij ¡± T f(x ) with ij p f(x ) = ; (5.6) 1 + 2¼¸e in agreement with our explicit computations. The corresponding limit for the superstring similarly lead to the results found in the previous section. 5.2 Spatial variation We can also consider the pro¯le with spatial variation µ ¶ 0 Y X = 2 T (X) = 2¸e cos p ; (5.7) as a limit of the pro¯le µ ¶ X Y T (X) = ¸ cosh p cos p ; (5.8) 2 2 0 0 considered previously by Sen. The procedure is to replace X ! X + c and then taking 1 c= 2 the limit ¸ ! 0; c ! 1 with ¯xed ¸ ´ ¸ e . The pro¯le (5.8) was found to yield 1 1 the energy momentum tensor T = T Ref ; 00 p T = T Imf ; 0y p T = ¡T Ref = ¡T ; yy p 00 T = ¡T ± jfj ; (5.9) ij p ij where 1 1 f = p + p ¡ 1 ; (5.10) 0 0 (x +iy)= 2 ¡(x +iy)= 2 ~ ~ 1 + sin(¸¼=2) e 1 + sin(¸¼=2) e so we can simply take the limit and ¯nd f = : (5.11) (x +iy)= 2 1 + 2¼¸ e Then (5.9) agrees with the results of our explicit computations (3.12){(3.14). { 16 { JHEP02(2003)039 5.3 The full boundary states Since the parabolic case can be obtained as a suitable limit of other cases we can also use previous works to obtain the full boundary state. Let us ¯rst review the strategy following [1]. Starting with the general pro¯le (5.1), 0 0 performing the Wick rotation X = iX and rede¯ning ¸ = ¡i¸ , we ¯nd the tachyon pro¯le T (X) = ¸ cos X + ¸ sin X. This action contains only modes with integer mo- mentum modes; so we can consider the theory compacti¯ed on a self-dual radius R = 1, instead of the uncompacti¯ed theory. At this radius there is an SU(2) current algebra with zero-modes I I dz dz § §2iX (z) 3 J = e ; J = i@X (z) : (5.12) 2¼i 2¼i The tachyon pro¯le T (X) is precisely a linear combination of these generators and we see that the (¸ ; ¸ ) can be represented as SU(2) parameters as µ µ ¶¶ µ ¶ 0 ¸ + i¸ a b R = exp i¼ = ; (5.13) 0 ¤ ¤ ¸ ¡ i¸ 0 ¡b a where µ ¶ µ ¶ q q ¸ + i¸ 2 2 2 2 0 2 0 a = cos ¼ ¸ + ¸ ; b = i q sin ¼ ¸ + ¸ : (5.14) 1 2 1 2 2 0 ¸ + ¸ 1 2 The boundary state for the unperturbed D-brane, with X kept explicit, can be written as [21] X X Neumann jBi = jj; m; mii ; (5.15) m=j j=0; ;¢¢¢ where jj; m; mii is the Virasoro-Ishibashi state [22] for the discrete Virasoro primary jj; m; mi ´ jj; mijj; mi. It is simply a sum of all Virasoro descendants of the primary jj; m; mi. Since the tachyon pro¯le T (X) is an element of the SU(2) algebra, and jj; mi transforms in the (j; m) representation of SU(2) algebra, the non-trivial part of the boundary state becomes simply [21, 23, 24] X X jBi 0 = D (R)jj; m; mii ; (5.16) x m;¡m m=j j=0;1=2;¢¢¢ where D (R) is the spin j representation matrix of the rotation R in J eigenbasis m;¡m (see [24] for an explicit form) . To ¯nd the boundary state for the time-dependent tachyon pro¯le (5.1), we then apply the appropriate inverse Wick rotation noting that, after the inverse Wick rotation, b is not the complex conjugate of b. In the parabolic limit ¸ ! ¸ ´ ¸ the "rotation" matrix becomes µ ¶ 1 0 R = : (5.17) 2¼i¸ 1 { 17 { JHEP02(2003)039 These considerations indeed give the correct stress tensor, for general (¸ ; ¸ ). Writing 1 2 the boundary state as 0 0 jBi 0 = f(x )j0i + g (x )® ® j0i + ¢¢¢ ; (5.18) ¹º ¡1 ¡1 0 0 0 the stress tensor is given by T (x ) = f(x )´ + g (x ), as in section 2. The boundary ¹º ¹º ¹º j j 2j state jBi 0 » (D jj; j; jii + D jj;¡j;¡jii), with jj;§j;§jii = (i) £ j=0;1=2;¢¢¢ j;¡j ¡j;j j j §2ijX(0) ¤ 2j 2j 0 e j0i + ¢¢¢ and D = (¡b ) ,D = b indeed lead to the f(x ) given in (5.5). j;¡j ¡j;j The g (x ) is similarly reproduced correctly. From the boundary state point of view the ¹º simpli¯cation o®ered by the parabolic case is that the representation matrices take the simple form (j + m)! 2m D = (2¼i¸) ± ; (5.19) m¸0 ¡m;m (j ¡ m)!(2m)! rather than the complex formula given in [24]. Acknowledgments We thank V. Balasubramanian, M. Einhorn, Y. He, M. Huang, T. Levi, and B. Zwiebach for discussions. F.L. is supported in part by DOE grant and A.N. is supported by DOE grant DOE-FG02-95ER40893. A.N. and S.T. thank the Michigan Center for theoretical physics for hospitality during portions of this work. References [1] A. Sen, Rolling tachyon, J. High Energy Phys. 04 (2002) 048 [hep-th/0203211]. [2] A. Sen, Tachyon matter, J. High Energy Phys. 07 (2002) 065 [hep-th/0203265]. [3] A. Sen, Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143]. [4] A. Sen, Time evolution in open string theory, J. High Energy Phys. 10 (2002) 003 [hep-th/0207105]. [5] A. Sen, Time and tachyon, hep-th/0209122. [6] A. Sen, Rolling tachyon, J. High Energy Phys. 04 (2002) 048 [hep-th/0203211]; Tachyon matter, J. High Energy Phys. 07 (2002) 065 [hep-th/0203265]; Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143]; Time and tachyon, hep-th/0209122. [7] A.H. Guth and S.-Y. Pi, The quantum mechanics of the scalar ¯eld in the new in°ationary universe, Phys. Rev. D 32 (1985) 1899. [8] E.J. Weinberg and A.-q. Wu, Understanding complex perturbative e®ective potentials, Phys. Rev. D 36 (1987) 2474. [9] B. Craps, P. Kraus and F. Larsen, Loop corrected tachyon condensation, J. High Energy Phys. 06 (2001) 062 [hep-th/0105227]. [10] G.N. Felder, L. Kofman and A. Starobinsky, Caustics in tachyon matter and other Born-Infeld scalars, J. High Energy Phys. 09 (2002) 026 [hep-th/0208019]. { 18 { JHEP02(2003)039 [11] S. Mukohyama, Inhomogeneous tachyon decay, light-cone structure and D-brane network problem in tachyon cosmology, Phys. Rev. D 66 (2002) 123512 [hep-th/0208094]. [12] M. Berkooz, B. Craps, D. Kutasov and G. Rajesh, Comments on cosmological singularities in string theory, hep-th/0212215. [13] A. Strominger, Open string creation by s-branes, hep-th/0209090. [14] A.A. Gerasimov and S.L. Shatashvili, On exact tachyon potential in open string ¯eld theory, J. High Energy Phys. 10 (2000) 034 [hep-th/0009103]; D. Kutasov, M. Marino ~ and G.W. Moore, Remarks on tachyon condensation in superstring ¯eld theory, hep-th/0010108; Some exact results on tachyon condensation in string ¯eld theory, J. High Energy Phys. 10 (2000) 045 [hep-th/0009148]; P. Kraus and F. Larsen, Boundary string ¯eld theory of the dd-bar system, Phys. Rev. D 63 (2001) 106004 [hep-th/0012198]; T. Takayanagi, S. Terashima and T. Uesugi, Brane-antibrane action from boundary string ¯eld theory, J. High Energy Phys. 03 (2001) 019 [hep-th/0012210]. [15] E.S. Fradkin and A.A. Tseytlin, Quantum string theory e®ective action, Nucl. Phys. B 261 (1985) 1; C.G. Callan Jr., E.J. Martinec, M.J. Perry and D. Friedan, Strings in background ¯elds, Nucl. Phys. B 262 (1985) 593. [16] T. Okuda and S. Sugimoto, Coupling of rolling tachyon to closed strings, Nucl. Phys. B 647 (2002) 101 [hep-th/0208196]. [17] B. Chen, M. Li and F.L. Lin, Gravitational radiation of rolling tachyon, J. High Energy Phys. 11 (2002) 050 [hep-th/0209222] [18] J.A. Harvey, D. Kutasov and E.J. Martinec, On the relevance of tachyons, hep-th/0003101. [19] S. Sugimoto and S. Terashima, Tachyon matter in boundary string ¯eld theory, J. High Energy Phys. 07 (2002) 025 [hep-th/0205085]. [20] O.D. Andreev and A.A. Tseytlin, Partition function representation for the open superstring e®ective action: cancellation of mobius in¯nities and derivative corrections to Born-Infeld lagrangian, Nucl. Phys. B 311 (1988) 205. [21] C.G. Callan, I.R. Klebanov, A.W. Ludwig and J.M. Maldacena, Exact solution of a boundary conformal ¯eld theory, Nucl. Phys. B 422 (1994) 417 [hep-th/9402113]. [22] N. Ishibashi, The boundary and crosscap states in conformal ¯eld theories, Mod. Phys. Lett. A 4 (1989) 251. [23] J. Polchinski and L. Thorlacius, Free fermion representation of a boundary conformal ¯eld theory, Phys. Rev. D 50 (1994) 622 [hep-th/9404008]. [24] A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of D-branes, Nucl. Phys. B 545 (1999) 233 [hep-th/9811237]. { 19 {

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Published: Feb 24, 2003

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