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Closed superstring emission from rolling tachyon backgrounds

Closed superstring emission from rolling tachyon backgrounds JHEP01(2005)037 Published by Institute of Physics Publishing for SISSA/ISAS Received: December 16, 2004 Accepted: January 24, 2005 Closed superstring emission from rolling tachyon backgrounds Jessie Shelton Center for Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A. E-mail: [email protected] Abstract: We compute the lowest components of the type-II Ramond-Ramond boundary X = 2 state for the tachyon pro¯le T (X) = ¸e by direct path integral evaluation. The calculation is made possible by noting that the integrals involved in the requisite disk one-point functions reduce to integrals over the product group manifold U(n) £ U(m). We further note that one-point functions of more general closed string operators in this background can also be related to U(n) £ U(m) group integrals. Using this boundary state, we compute the closed string emission from a decaying unstable Dp-brane of type- II string theory. We also discuss closed string emission from the tachyon pro¯le T (X) = ¸ cosh(X = 2). We ¯nd in both cases that the total number of particles produced diverges for p = 0, while the energy radiated into closed string modes diverges for p · 2, in precise analogy to the bosonic case. Keywords: Superstrings and Heterotic Strings, Tachyon Condensation. ° SISSA/ISAS 2005 http://jhep.sissa.it/archive/papers/jhep012005037 /jhep012005037.pdf JHEP01(2005)037 Contents 1. Introduction 1 2. Disk one-point functions in the rolling tachyon background 4 2.1 NS-NS sector 6 2.2 Relation to fermionized computation 7 3. General closed string one-point function as a matrix integral 9 4. Closed string production 11 5. Conclusions 15 A. An alternate derivation of the boundary state: the SU(2) method 16 A.1 Bouncing tachyon pro¯le 18 A.2 Rolling tachyon pro¯le 20 B. Relations between spinor vacua 21 1. Introduction The worldsheet approach to studying tachyon condensation was introduced by Sen in [1, 2]. In this approach, we give the tachyon a space-time-dependent expectation value by deform- ing the worldsheet conformal ¯eld theory with the exactly marginal boundary operator 0 ¸ 0 0 T (X) = ¸ cosh X , or, in the supersymmetric theory, the operator ´Ã sinh(X = 2), which corresponds to the tachyon pro¯le T (X) = ¸ cosh(X = 2). Another tachyon pro¯le 0 0 X X = 2 T (X) = ¸e and its supersymmetric analogue T (X) = ¸e was introduced in [3]. These conformal ¯eld theories are related by Wick rotation to theories where the boundary deformation is taken from a local SU(2) current algebra. Worldsheet conformal ¯eld theory is a perturbative approach to string theory and can likely provide only a partial answer to the deep nonperturbative problems posed by D-brane decay. Working out conformal ¯eld theory descriptions of D-brane decay is nonetheless of interest. Perhaps most importantly, the conformal ¯eld theory approach provides a clear computation of the coupling of the decaying D-brane to closed strings, a problem which remains somewhat obscure in the open string ¯eld theory description of tachyon condensation. In the worldsheet approach, the ¯nal closed string state produced by the decaying brane can be readily computed [4]. { 1 { JHEP01(2005)037 The rolling tachyon background is also a good starting point for exploring some of the technical issues involved in working with time-dependent perturbative solutions to the string equations of motion, as the time dependence is con¯ned to the open string sector, and thereby poses fewer new conceptual challenges than working with a time-dependent space- time background. One such interesting technical issue is the relationship proposed in [5] between the mathematical ambiguities involved in de¯ning the correlators in a nontrivial wrong sign CFT and the physical ambiguities involving choice of vacua in time-dependent ¯eld theories. Although it appears that the perturbative worldsheet description of D-brane decay breaks down in most examples, as the number of emitted particles and amount of emitted energy becomes in¯nite [4], there are examples where this divergence is brought under control, either due to a higher Hagedorn temperature [6], or an ability to work with a dual, non-perturbative description of the system [7]. In type-II theories, the space-time stress energy tensor for a decaying brane has been studied in [2, 3]. In this note we present a discussion of closed string production from a decaying type-II Dp-brane. The marginal deformations we study are the so-called \rolling tachyon" dt 0 0 X = 2 ¢S = ¡ 2¼¸ ´Ã e (1.1) 2¼ and the \bouncing tachyon" Z µ ¶ dt X ¢S = ¡ 2¼¸ ´Ã sinh p : (1.2) 2¼ Here ´ is a fermionic degree of freedom living on the boundary of the worldsheet; it has trivial worldsheet dynamics, and may also be understood as a Chan-Paton factor. It appears in the vertex operators of all open string states with wrong sign GSO projection. The coupling ¸ is related to the initial position and velocity of the tachyon at the top of its potential [1]. For most of our calculations, we will assume that we are able to choose a gauge in which no timelike oscillators are excited. Then a general closed string vertex operator takes the form iE X ? i i i V = e V (X ; à ; à ) (1.3) in the NS-NS sector and iE X ? i i i s e e V = e £ £ 0 V (X ; à ; à ) (1.4) s s s 0 s There has been some confusion in the literature as to the possible appearance of a term of the form T in the boundary action. If the boundary action is written in the manifestly supersymmetric form S = dtdµ(¡D¡ + ¡T (Y )) where ¡ = ´ + µF , D = @ + µ@ , and Y = X + µÃ, a term of the form T bdy µ arises after elimination of the auxiliary ¯eld F . This form of the boundary action has been instrumental in studies of tachyon condensation using boundary string ¯eld theory. However, when the boundary theory is de¯ned through deformation by the operators above, the T term does not arise; its contribution to the boundary action vanishes on shell. It is simple to check that the supersymmetry Ward identities for the deformed SCFT are satis¯ed when the boundary action only contains the term ´ÃT (x). { 2 { JHEP01(2005)037 is H 0 0 in the R-R sector. Here the £ are spin ¯elds, £ = e . With this gauge ¯xing, s s the problem of computing general closed string one point functions in the rolling tachyon background is greatly simpli¯ed. We only need to know the expectation values ¡¢S iE X he e i disk in the NS-NS sector, and ¡¢S iE X he e £ £ i s disk 0 s in the R-R sector. Equivalently, we only need to know the lowest component of the bound- ary state describing the deformed X CFT, 0 0 jBi = g(x )j0i + ¢¢¢ : In the NS-NS sector, the function g(x ) for the tachyon pro¯le (1.2) was worked out in [2] and for the tachyon pro¯le (1.1) in [3]. The computation of g(x ) for both tachyon pro¯les (1.1) and (1.2) in the R-R sector is one of the aims of this paper. We will, however, spend more time on the rolling tachyon pro¯le, (1.1) above, as a direct perturbative calculation of the closed string disk one-point amplitudes in that background reveals an interesting U(p) £ U(q) structure, which allows the correlators to be easily calculated. This structure is an interesting generalization of the U(n) symmetry that was found in the bosonic case in [3, 9]. We will also compute the NS- NS one-point functions for this background in order to demonstrate a similar U(p) £ U(p) structure appearing in the correlators there. These calculations are in section 2. In section 3, we discuss how the relation between correlation functions in the rolling tachyon background and integrals of invariants over the group U(p)£U(q) may be extended to one-point functions of more general operators. In order to compute the function g(x ) for the bouncing tachyon, we must use some more sophisticated CFT machinery, namely the SU(2) current algebra of the euclidean version of the theory. This technique is the one originally used by Sen. We have placed this computation in appendix A and use only the result, equation (A.2), in the main body of the paper. Also in this appendix, we present an alternate derivation of the function g(x ) for the rolling tachyon. Next, we consider the closed string production from the decaying D-brane, for both tachyon pro¯les. As in the bosonic case [4], both the number and the energy of emitted particles diverge for low dimensional D-branes. Since the tachyon pro¯les (1.1) and (1.2) describe homogenous tachyon condensation in the p spatial dimensions of the brane, the condensation process is unphysical if p 6= 0. A more physical decay process for a higher- dimensional D-brane will likely proceed in several causally disconnected patches, each of which behaves like a decaying D-brane of lower dimension [3]. Thus we expect that in a physical decay process, the number and energy of particles emitted from decaying Dp- branes will diverge for any p. This suggests that all the energy of the D-brane is converted into closed strings. See also [8]. { 3 { JHEP01(2005)037 Finally, appendix B summarizes the relationship between two di®erent bases for the spinor vacua, the states jsijs~i in terms of which the R-R spin ¯elds have a simple descrip- tion, and the states j§i in terms of which the boundary state has a simple description. These relationships will be useful when we compute the R-R sector boundary state. 2. Disk one-point functions in the rolling tachyon background In this section, we compute disk one-point functions of the form D E D E ¡¢S e e £ £ 0 = e £ £ 0 s s s s deformed free by Taylor-expanding the exponential and evaluating the correlators that appear at each 0 0 order in ¸. We do not carry out the path integral over the X zero mode x in the expectation values here, but interpret the resulting amplitudes as a function of x . We x = 2 therefore de¯ne ¸ ´ ¸e , and ¿ µ Z ¶ À ¿µZ ¶ À p n p X ~ dt 0 ( 2¼¸) dt 0 0 X = 2 0 X = 2 e e exp 2¼¸ ´Ã e £ £ 0 = ´Ã e £ £ 0 s s s s 2¼ n! 2¼ ´ (i¼¸) A : (2.1) In the R-R sector, the boundary fermion ´ has a zero mode; thus, only odd numbers of insertions of the boundary perturbation give non-vanishing contributions. Since the world- sheet dynamics of the boundary fermion are trivial, it is simply a matter of convenience whether we choose to formally integrate it out, as in [3], or include it in the vertex operator, as in [2]. We will ¯nd it convenient to include ´ in the vertex operator, since in this case the vertex operators on the boundary all commute with each other. The structure of this computation is most evident when using the bosonized represen- tation of the fermion ¯elds. The correlators that appear in (2.1) are then 2k+1 D E 2k+1 Y p £ ¤ (¼¸) dt 0 ~ iH ¡iH X = 2 is¢H is ¢H 0 0 0 e (t ) ¡ e (t ) e (t ) e (0)e (0) : i i i (2k + 1)! 2¼ i=1 This expectation value factors into three parts. The contribution from the transverse fermions is ± 0, which we will drop for ease of notation. The bosonic contribution is ~s;¡~s simply * + µ ¶ 2k+1 Y Y Y t ¡ t i j X = 2 it it 0 i j e (t ) = je ¡ e j = 2 sin : (2.2) i=1 i<j i<j £ ¤ 2k+1 iH ¡iH is H is H 0 0 0 0 0 Now consider h e (t ) ¡ e (t ) e (0)e (0)i. From now on we will drop i i i=1 iH ¡iH the label 0 to clean up the notation. When we expand the product (e (t )¡e (t )), i i i=1 2k+1 i² H we will ¯nd a sum of terms, each of which has the form e (t ), where ² = §1. i i i=1 The path integral over the H zero mode imposes H momentum conservation, giving the condition s + s + ² = 0 : i=1 { 4 { JHEP01(2005)037 0 0 Since s; s = §1=2, we have two possible ways to satisfy the condition above: let s = s = 2k+1 iH ¡iH §1=2 and retain only terms in the expansion of the product (e (t )¡e (t )) where i i i=1 the H momentum sums to ¨1. There will be a sign di®erence between the expectation values in the two di®erent cases, as when we choose s = s = 1=2, we pick up an odd ¡iH 0 number of factors of ¡e , and an even number when we choose s = s = ¡1=2. As we will see, this sign di®erence gives us the correct GSO projection. 2k+1 0 iH Let us consider the s = s = 1=2 for de¯niteness. Then in the product [e (t )¡ i=1 ¡iH e (t )] we need to keep the terms with k + 1 minus signs, that is, terms with ² = ¡1. i i The H correlators become µ ¶ ² ² i j P Y t ¡ t i j ¡k ¡i t =2 (i) e 2 sin : i<j Including the phases coming from the transformation from z to t thus yields simply µ ¶ ² ² i j t ¡ t i j (¡1) i 2 sin : (2.3) i<j Combining equations (2.2) and (2.3), we have (adjusting the overall phase for simplic- ity) 2 3 µ µ ¶¶ 2k+1 1+² ² Y X Y ¾(i) ¾(j) t ¡ t 1 dt ¾(i) ¾(j) 4 5 A = 2 sin ; (2.4) 2k+1 (2k + 1)! 2¼ 2 i=1 ¾2P ¾(i)<¾(j) where P denotes the set of all distinct ways to distribute k + 1 values of ¡1 and k values 2k+1 of 1 among the ² . Since the boundary operators are not path ordered, each of the ( ) terms in the sum above contribute equally. Thus, we may choose one ordering of the ² , namely ² = ¡1 ; 1 · i · k + 1 ; ² = 1 ; k + 1 < i · 2k + 1 ; i i and write the correlator in terms of that ordering alone: Z µ µ ¶¶ 2k+1 1+² ² i j Y Y 1 dt t ¡ t i i j A = 2 sin : (2.5) 2k+1 (k + 1)!k! 2¼ 2 i=1 i<j Now, the Haar measure for the group U(n) is Z Z 1 dt [dU] 1 = ¢ (t) = 1 ; n! 2¼ U(n) i=1 where ¢(t) is the Vandermonde determinant, µ ¶ t ¡ t i j ¢(t) = 2 sin : i<j It is now easy to see that (2.5) is simply the integral of unity over the group manifold of the product group U(k + 1) £ U(k): Z Z A = [dU] 1 [dU] 1 = 1 : 2k+1 U(k+1) U(k) { 5 { JHEP01(2005)037 Since A = 1 for all k, the sum over all orders (2.1) becomes simply 2k+1 x = 2 i¼¸ e 2k+1 R 0 (i¼¸) = ´ g (x ) : (2.6) 2 2 2x 1 + ¼ ¸ e It remains to restore the dependence on the spin. The amplitudes that we have calculated are built on spin vacua of the form (j1=2ij1=2i ¡ j¡1=2ij¡1=2i) where we have labeled states by their s quantum numbers, suppressing the transverse spin eigenvalues. The R-R boundary state, however, is built on 0 0 1 1 e e spinors j§§i that satisfy (à § ià )j§§i = (à § ià )j§§i = 0. Thus to write down the 0 0 0 0 boundary state, we need to know the relationship between the two bases jsijs i and j§§i, which we work out in appendix B. We ¯nd from (B.4) j1=2ij1=2i ¡ j¡1=2ij¡1=2i = j+¡i + j¡+i : We may now easily restore the transverse spin quantum numbers to ¯nd that the spin vacuum should be j+ ¡ ¢¢¢¡i + j¡ + ¢¢¢ +i. Note that this is the correct GSO projection for type IIA, and has the right Lorentz properties to source C °ux. The boundary state for the supersymmetric rolling tachyon pro¯le is therefore R 0 jBi = g (x ) (j+ ¡ ¢¢¢¡i + j¡ + ¢¢¢ +i) + oscillators: (2.7) The generalization to other Dp-branes of either type-II string theory is very simple; we simply need to change the dependence on the transverse spin degrees of freedom, exactly as we would in the absence of the tachyon expectation value. 2.1 NS-NS sector In the NS-NS sector, the expectation value that we need to compute is ­ ® ¡¢S h1i = e deformed free This expectation value was discussed in [3]; our principal interest in this section will be to demonstrate an interesting U(n) £ U(n) structure that allows the integrals to be easily performed. As before, we will evaluate this correlator by expanding in powers of the boundary perturbation, and de¯ne ¿ µ ¶ À dt 0 0 X = 2 n exp 2¼¸ ´Ã e 1 = (i¼¸) A : 2¼ The boundary fermion ´ has no zero mode in the NS-NS sector, so only terms in the expansion with n even will contribute. Using bosonized fermions, A is 2k 2k D E £ ¤ 1 dt iH ¡iH X = 2 A = e (t ) ¡ e (t ) e (t ) : (2.8) 2k i i i (2k)! 2¼ i=1 { 6 { JHEP01(2005)037 The H-momentum conservation condition now yields ² = 0 : The expectation value (2.8) is thus 2 3 Z µ µ ¶¶ 2k 1+² ² ¾(i) ¾(j) Y X Y t ¡ t 1 dt i ¾(i) ¾(j) 4 5 A = 2 sin ; 2k (2k)! 2¼ 2 i=1 ¾2P ¾(i)<¾(j) where P denotes the set of all distinct ways to distribute k values of 1 and k values of ¡1 among the ² . Again we choose a representative ordering, ² = ¡1 ; 1 · i · k ; ² = 1 ; k < i · 2k ; i i and write the correlator in terms of that ordering alone: µ µ ¶¶ 2k 1+² ² i j Y Y 1 dt t ¡ t i i j A = 2 sin : (2.9) (k!) 2¼ 2 i=1 i<j This can easily be written as the integral of unity over the group manifold of the product group U(k) £ U(k): " # A = [dU] 1 = 1 2k U(k) for all k. Summing all orders, the functional form of the time-dependent NS-NS source is then 2k NS 0 (i¼¸) = p ´ g (x ) : (2.10) 2 2 2x 1 + ¼ ¸ e The boundary state in the NS-NS sector is therefore NS 0 jBi = g (x ) (j0; +i ¡ j0;¡i) + oscillators : (2.11) NS This answer di®ers from that found in [3] by ¸ = 2¸ . This di®erence is not here there meaningful as ¸ can be set to any value by a time translation. 2.2 Relation to fermionized computation The U(p) £ U(q) structure that allowed us to easily carry out the integrals in (2.5) and (2.9) is obscured when the fermions are not bosonized. It is interesting to perform the same calculation without bosonizing the fermions, and then compare the two representations of the calculation. In the R-R sector, the fermionic correlators are equivalent to the expectation value of 2k + 1 (holomorphic) fermions on a sphere with £ at 0 and £ at in¯nity | that is, the expectation values of 2k + 1 holomorphic Ramond-sector fermions on a sphere. The contractions between two timelike fermions are therefore given by µ ¶ 1 t ¡ t 1 2 (R) D (t ; t ) = ¡ cot ; 1 2 2 2 { 7 { JHEP01(2005)037 where we have included the phase factors necessary to transform the propagator to the an- gular variables t . Because we have an odd number of fermions, all terms in the correlation function will have one uncontracted fermion. The expectation value of this last fermion is hÃ(t)i = (2s )± 0 ± 0 ; 0 s ;s ~s;¡~s where again we have included the phase factor coming from the conformal transformation of Ã. 0 0 The contribution from s again imposes that either s = s = 1=2 or s = s = 0 0 0 0 0 ¡1=2. The factor of 2s provides the relative sign between the two choices; this can be 0 + ¡ thought of as coming from the fact that à = ( ¡  )= 2, where  are the raising and lowering operators for the s basis. To simplify our notation, we will drop the spin structure 0 0 contributions, (2s )± ± , to the amplitude. We ¯nd that A is given in fermionic 0 2k+1 s ;s ~s;¡~s language by " # Z µ ¶ µ ¶ 2k+1 k 2¼ Y Y X Y t ¡t 1 dt t ¡t i i j ¾(2l¡1) ¾(2l) A = 2 sin sign (¾) cot ; 2k+1 (2k+1)! 2¼ 2 2 i=1 i<j ¾2Q l=1 (2.12) where we have again made an overall choice of phase to simplify the notation. Here Q denotes the set of all distinct contractions. In the NS-NS sector, the contractions between the fermions appearing in the boundary perturbation are µ ¶ 1 t ¡ t 1 2 (NS) ¡1 D (t ; t ) = ¡ sin ; 1 2 2 2 and (2.8) becomes " # ¡1 µ ¶ µ ¶ 2k k Y Y X Y t ¡ t 2 dt t ¡ t ¾(2l¡1) ¾(2l) i i j A = 2 sin 2 sin ; (2.13) 2k (2k)! 2¼ 2 2 i=1 i<j ¾2Q l=1 where again Q is the set of all distinct contractions. The equivalence between the two representations of the à ¯elds guarantees that the integrands of (2.12) and (2.5) are equal, it as are the integrands of (2.13) and (2.9). By rewriting these equalities in terms of x = e , we arrive at the following identities between two symmetric polynomials. The NS-NS correlation functions lead to the identity " # 2k k Y X Y k ¡1 2 (x ¡ x ) sign (¾) (x ¡ x ) = i j ¾(2l¡1) ¾(2l) i<j ¾2Q l=1 2 3 X Y Y 2 2 4 5 = (x ¡ x ) (x ¡ x ) : (2.14) ½(i) ½(j) ½(i) ½(j) ½2P ½(i)<½(j)·k k<½(i)<½(j)·2k The R-R correlation functions lead to the identity " # 2k+1 k Y X Y x + x ¾(2l¡1) ¾(2l) (x ¡ x ) sign (¾) = (2.15) i j x ¡ x ¾(2l¡1) ¾(2l) i<j ¾2Q l=1 { 8 { JHEP01(2005)037 " # X Y Y Y 2 2 = (x ¡x ) (x ¡x ) x : ½(i) ½(j) ½(i) ½(j) ½(i) ½2P ½(i)<½(j)·k+1 k+1<½(i)<½(j)·2k+1 k+1<½(i)·2k+1 In these forms, the identities hold for any abstract set of variables fx g. We may check these results by multiplying out the terms at ¯nite order. For example, when k = 2, equation (2.14) becomes 4 (x ¡ x )(x ¡ x )(x ¡ x )(x ¡ x ) ¡ (x ¡ x )(x ¡ x )(x ¡ x )(x ¡ x ) + 1 3 1 4 2 3 2 4 1 2 1 4 2 3 3 4 + (x ¡ x )(x ¡ x )(x ¡ x )(x ¡ x ) = 1 2 1 3 2 4 3 4 £ ¤ 2 2 2 2 2 2 = 2 (x ¡ x ) (x ¡ x ) + (x ¡ x ) (x ¡ x ) + (x ¡ x ) (x ¡ x ) : 1 3 2 4 1 4 2 3 2 1 3 4 This can be seen to be true by multiplying out the terms. Likewise, when k = 1, eq. (2.15) becomes (x + x )(x ¡ x )(x ¡ x )¡(x ¡ x )(x + x )(x ¡ x )+(x ¡ x )(x ¡ x )(x + x )= 1 2 1 3 2 3 1 2 1 3 2 3 1 2 1 3 2 3 2 2 2 = (x ¡ x ) x +(x ¡ x ) x +(x ¡ x ) x : 1 2 3 1 3 2 2 3 1 3. General closed string one-point function as a matrix integral In this section, we extend the results of the previous section to sketch how more general one-point functions in the rolling tachyon background can be evaluated using matrix inte- grals. This is a generalization of the work that was done for the bosonic rolling tachyon background in [9]. Since the group structure is most evident using the bosonized representation of the à ¯elds, we will work in bosonized language. A general closed string operator built out of the à ¯eld will take the form µ ¶ µ ¶ N N i j Y Y 1 1 0 º º~ i(pH+p H) i ¹ j V = @ H @ H e ; (º ¡ 1)! (º~ ¡ 1)! i j i j 0 0 where p; p 2 Z in the NS-NS sector, and p; p 2 Z+1=2 in the R-R sector. A general closed string operator built out of the X ¯eld will take the form à ! à ! p M p M i j Y Y 2 2 ¹ 0 ¹~ 0 i ¹ j V = @ X @ X : (¹ ¡ 1)! (¹~ ¡ 1)! i j i j We have chosen here a convenient normalization for the operators. We want to evaluate the amplitude ¿ µ ¶ À dt 0 0 X n exp 2¼¸ ´Ã e V V (0) = (i¼¸) A : (3.1) f b n 2¼ disk We note immediately that due to the boundary fermion ´, n must be even in the NS-NS sector, and odd in the R-R sector. Consider ¯rst the bosonic portion of this correlator. The propagator for a timelike boson on the disk is 1 1 2 2 G (z; w) = lnjz ¡ wj + lnjzw¹ ¡ 1j : 2 2 { 9 { JHEP01(2005)037 The bosonic correlators in (3.1) are then à ! à ! µ ¶ N N k j n n Y Y X Y X t ¡t 1 1 i j º it º~ it k l l 2 sin @ G (0; e ) @ G (0; e ) ; b b 2 (º ¡1)! (º~ ¡1)! k j i<j k l=1 j l=1 or à ! à ! µ ¶ N N k j n n Y Y X Y X t ¡ t i j ¡it º it º~ l k l 2 sin e e : (3.2) i<j k l=1 j l=1 Meanwhile, the propagator for the H ¯eld on the disk is G (z; w) = ln(z ¡ w) and the propagator between a H ¯eld and a H ¯eld is G (z; w) = ln(zw¹ ¡ 1) : The fermionic correlators in (3.1) are thus Y Y ¡n=2 it it ² ² it (2p² +1)=2 i j i j i i 2 (e ¡ e ) e £ i<j i à ! à ! N N j k Y X Y X 1 1 º º~ ¹ k £ ² @ G (0; w ) ² @ G (0; w ) ; l f l l ~ l (º ¡ 1)! (º~ ¡ 1)! j k j l k l or à ! à ! ~ N N µ ¶ j j Y Y Y X Y X it it ² ² it (2p² +1)=2 ¡it º it (º~ ¡1) i j i j i i l j l k p (e ¡ e ) e ² e ² e : l l i<j i j l k l Here, as in the previous section, ² = §1, and the conservation of H momentum leads to the condition ² + p + p = 0 : When this condition cannot be ful¯lled, that is, if jp + p j > n, the correlator vanishes. When the correlator does not vanish, we will need to sum over all possible groupings of the ² into a group of k with ² = ¡1 and a group of n ¡ k with ² = 1, where i i i k = (n + p + p ) : There are ( ) such terms, all of which contribute equally after integration. Let us choose the ordering ² = ¡1 ; 1 · i · k ² = 1 ; k < i · n : i i We ¯nd that the fermionic correlators become µ µ ¶¶ n n ² ² ¡n=2 i j Y Y 2 n! t ¡ t 0 i j it ² (p¡p ) i i 2 sin e £ k!(n ¡ k)! 2 i<j i { 10 { JHEP01(2005)037 à ! à ! N N Y X Y X ¡it º it (º~ ¡1) l j l k £ ² e ² e : (3.3) l l j l k l We are now ready to put together equations (3.2) and (3.3) and write down the amplitude for a general closed string vertex operator and n insertions of the boundary action. De¯ning ¡ ¢ it it U = Diag e ; : : : ; e and ¡ ¢ it it k+1 V = Diag e ; : : : ; e ; we ¯nd Z Z µ ¶ 0 p ¡p det U A = [dU] [dV ] £ det V U(k) U(n¡k) ³ ´ Y Y M ¡ ¢ ~ y ¹ y ¹ ¹~ ¹~ i i j j £ Tr(U ) + Tr(V ) Tr U ¡ Tr V £ i j ³ ´ Y Y ¡ ¢ ~ y º y º º~ ¡1 º~ ¡1 k k l l £ Tr(U ) ¡ Tr(V ) Tr U ¡ Tr V : (3.4) k l This is indeed a matrix integral with product structure U(k) £ U(n ¡ k). In principle, the integrals over the unitary groups may be performed, as in [9]. However, in practice the expansions of the products above must become prohibitive, as the exponents M ; N can i j become arbitrarily large, as can the cardinality of the index sets fig;fjg. 4. Closed string production In this section, we calculate closed string emission from the decaying D-branes described by the boundary state we derived in section 2. Analogous computations were done for the bosonic string in [4, 6]. The computation for the superstring does not di®er greatly from that done for the bosonic string, and our discussion will therefore be largely parallel to that in [4]. We begin with the expression for the cylinder diagram ¿ ¯ ¯ À ¯ ¯ 1 1 ¯ ¯ W = B B : (4.1) ¯ ¯ L + L + i² 0 0 This expression picks up an imaginary part whenever particles go on shell. Writing jBi = U(! )jsi ; the imaginary part of (4.1) is 1 1 ImW = jU(! )j : (4.2) 2 2! The optical theorem relates this to particle production, so that N=V = 2 Im W . The volume V here is the p-dimensional spatial volume of the decaying brane. { 11 { JHEP01(2005)037 We ¯rst need to calculate U(!). This reduces essentially to choosing a prescription for integrating the zero mode of the bosonic coordinate. Assuming the factorization (1.3) and (1.4) for a general closed string vertex operator, the computation of U(!) factorizes ? i i i into one part from the expectation value of V (X ; à ; à ) in the transverse CFT, and the part of interest from the time-dependent components of the boundary state. As in [4], the transverse CFT contributes only a phase, which drops out of the computations below. 0 0 The contribution from the (X ; à ) CFT takes the form iEX iEt U(E) = he i = i dt g(t)e : (4.3) disk Here g(t) is the function appearing in the lowest component of the boundary state, which we computed for the rolling tachyon pro¯le in section 2. This function is given by (2.10) in the NS-NS sector, and (2.6) in the R-R sector. For the bouncing tachyon pro¯le, g(t) is given by (A.3) in the NS-NS sector, and (A.2) in the R-R sector. There are two contours of interest, namely C , which runs along the real axis and real is closed in the upper half plane, and the Hartle-Hawking contour C , which runs along HH the imaginary axis from t = i1 to t = 0, then runs along the real axis [4]. For the rolling tachyon pro¯le, both contours yield the same result. For the bouncing tachyon pro¯le, the contour C yields the same result as for the rolling tachyon, while the contour C yields HH real a di®erent result. We will consider the rolling tachyon pro¯le ¯rst. For this pro¯le, the contour integral (4.3) yields in the NS-NS sector 1 ¼= 2 i!t ¡i 2! U (!) = i dt p e = (¼¸) : NS ¼! 2 2 2t p sinh C 1 + ¼ ¸ e real In the R-R sector, the contour integral calculation yields t= 2 i¼¸e ¼= 2 i!t ¡i 2! U (!) = i dt e = ¡(¼¸) : ¼! 2 2 2t p cosh 1 + ¼ ¸ e real We now use these results for U(!) to calculate closed string emission from the decaying branes, using (4.2). Let us consider ¯rst the portion of the cylinder diagram coming from the overlap hB; +j jB; +i . We have argued that the imaginary part of this amplitude is NS e NS L +L +i² 0 0 given by X X 1 ¼ 1 2 Im W = jU(! )j = p : NS++ s 2! 4! s s sinh (¼! = 2) s s We now go through a series of manipulations to bring this into a more familiar form. Using X p ¡ 2¼n! p = 4 n e sinh (¼! = 2) n=0 and 1 1 1 ¡ 2¼! n 2¼ik n s 0 e = dk e 2 1 2¼ 2! k + ! 0 s { 12 { JHEP01(2005)037 Z Z 1 2 1 2 ¡t(k + ! ) 2¼ik n 0 2 = dk dt e e 2¼ 1 ¼ 2 2 2 ¡¼ n =t ¡t! =2 = dt e e ; (4.4) 2¼ t the contribution to particle production is X X 2 2 2 ¡1=2 ¡¼ n =t ¡t! =2 2 Im W = C n dt t e e : NS++ p n s Here C is a numerical constant coming from the normalization of the boundary state. 2 ¡2t We now write ! = 4N + 4N ¡ 2 and set q = e . The sum over states can now be s à performed in the usual manner, so that µ ¶ 2 2 f (q) ¡1=2 ¡¼ n =t 2 Im W = C n dt t e : NS++ p ´(q) In a very similar fashion, one may calculate the contribution to particle production from 1 1 the overlaps hB; +j jB;¡i and hB; +j jB; +i . The total number NS R e NS e R L +L +i² L +L +i² 0 0 0 0 of emitted particles is then ¹ X ¡ ¢ N 2 2 ¡1=2 ¡¼ n =t ¡8 8 8 n 8 = C n dt t e [´(q)] f (q) ¡ f (q) ¡ (¡1) f (q) : 3 4 2 This expression diverges for small t. This divergence is easier to examine after making a ¡2¼s modular transformation to s = . De¯ning w = e , we ¯nd Z Z ¹ X ¡ ¢ N ds 2 2 9 ¡(k +n )=2 ¡8 8 n 8 8 = n d k w [´(w)] f (w) ¡ (¡1) f (w) ¡ f (w) : 3 4 2 V s For a decaying Dp-brane, the answer generalizes in the obvious fashion, Z Z ¡ ¢ N ds 2 2 p ¡(k +n )=2 ¡8 8 n 8 8 = n d k w [´(w)] f (w) ¡ (¡1) f (w) ¡ f (w) : (4.5) 3 4 2 V s When s ! 1, we may expand (4.5) for small w to ¯nd Z Z Z ds 2 ds p k =2 ¡1=2 ¡p=2 d k w (1 + O(w )) » s : s s It is now easy to see that this diverges for p = 0. As noted in [4], this has the following open string interpretation. Let µ ¶ À X p jB i = B ; (¡1) ; x = 2 n + ¼ + p n=0 ¯ p ® where B ; (¡1) ; x = 2(n + 1=2)¼ denotes the boundary state for a euclidean D-instan- ton with p spatial dimensions, located at the position x = 2(n + 1=2)¼ in the euclidean { 13 { JHEP01(2005)037 time direction, sourcing positive or negative R-R °ux according to the sign (¡1) . Analo- gously, we de¯ne µ ¶ À X p jB i = B ;¡(¡1) ; x = ¡ 2 n + ¼ : ¡ p n=0 The imaginary part of the partition function (4.5), is then ¯ ¯ ¿ À ¯ ¯ N 1 ¯ ¯ = B B : ¡ + ¯ ¯ V e L + L 0 0 In other words, the expression (4.5) is the partition function that arises from a collection of alternating euclidean Dp-branes and anti-Dp-branes located at regular positions along the imaginary time axis, provided we only include the open strings that stretch across the line x = 0 [4]. This open string description of the closed string radiation arises from writing the closed string ¯elds produced by the decaying D-brane as radiation sourced by D-instantons in euclidean time. From this point of view, the euclidean path integral over x < 0 in the presence of the source jB i prepares a state in the closed string Hilbert space at x = 0 which is closely related to the ¯nal closed string state; see [4] for a complete discussion. In this open string language, the divergence in equation (4.5) comes from the existence of a massless string state stretching between the instantons nearest to the imaginary time axis. This divergence comes from the IR of the open string channel, and thus from the UV of the closed string channel. It re°ects the fact that the amplitude for emission of a very massive string state does not fall o® quickly enough as a function of energy to o®set the Hagedorn divergence in the number of states. While this divergence only occurs for p = 0, the homogenous solution studied here should not be trusted for larger p, as higher dimensional D-branes should rapidly break up into causally disconnected patches that decay locally as unstable D0-branes. (This discussion is complicated slightly in the superstring by the R-R couplings, but it is easy to see that the NS-NS contributions to particle production are divergent on their own for p = 0.) Therefore the divergence that occurs when p = 0 should be thought of as the generic tree level result in the superstring as well as the bosonic string. This seems to signal the breakdown of string perturbation theory. We may also consider the energy that is radiated into these closed string modes, E 1 = jU(! )j : V 2 Since from (4.4) we may write · ¸ Z ³ ´ p p 3=2 1 @ 1 1 1 ¼ ¡ 2¼! (n+a) ¡ 2¼! n ¡t! =2 s s ¡ p e = e = dt ne ; 2¼ @a 2 2¼ t 2! a=0 the energy radiated into closed strings is given by Z Z ¡ ¢ E ds 2 2 2 p (k +n )=2 ¡8 8 n 8 8 = n d k s w [´(w)] f (w) ¡ (¡1) f (w) ¡ f (w) : (4.6) 3 4 2 V s { 14 { JHEP01(2005)037 The divergence at small w has now been worsened, ¡p=2 » ds s ; so that the total energy emitted into closed strings is divergent for p · 2. This is precisely the expected supersymmetric generalization of the results of [4] for the bosonic string, and suggests that all the energy of the D-brane is converted into closed strings. The story is slightly more complicated for the bouncing tachyon pro¯le when the contour C is used in equation (4.3), but the fundamental physical picture does not real change. With this contour, we ¯nd ¡ 2¼ U(!) = p sin( 2!·) ; NS sinh(¼!= 2) 2¼ U(!) = p cos( 2!·) ; cosh(¼!= 2) where · ´ ln sin ¼¸. The number of emitted particles is now Z Z ¹ X ¡ ¢ N ds 2 2 p k =2 ¡8 n =2 8 n 8 8 = n d k w [´(w)] 2w f (w) ¡ (¡1) f (w) ¡ f (w) ¡ 3 4 2 V s n=0 ¡ ¢ 2 2 ¡(n+2i·=¼) =2 ¡(n+2i·=¼) =2 ¡ w + w £ ¡ ¢ 8 n 8 8 £ f (w) + (¡1) f (w) ¡ f (w) : 3 4 2 Note that the signs of the GSO projection are reversed in the second term of the above equation. Due to this change of sign, the second term is ¯nite. However, the ¯rst line is identical (up to an overall factor of 2) to the calculation performed for the rolling tachyon pro¯le, and in particular diverges in the same way. In the superstring, the tachyon potential is bounded from below, so it is a sensible physical question to ask about the closed string emission from the tachyon pro¯le T (X ) = 0 3 ¸ sinh(X = 2). We ¯nd in this case that the functions U(!) di®er from those for the bouncing tachyon pro¯le only by an overall phase and the rede¯nition · = sinh(¼¸), and therefore that the divergence in the closed string radiation from this tachyon pro¯le is identical to the divergence in the radiation from the bouncing tachyon pro¯le. 5. Conclusions We have presented a basic analysis of closed string one-point functions in rolling tachyon backgrounds in type-II string theories. We noted the appearance of an interesting U(p) £ U(q) structure that allowed us to easily evaluate the integrals appearing in the correlation functions, and pointed out how this structure could be generalized to compute one-point 0 0 functions involving excitations of the X and à oscillators. We con¯rmed that the total number and energy of emitted particles diverge for low-dimensional D-branes. Thanks to N. Lambert for this point. { 15 { JHEP01(2005)037 The computations of closed string expectation values in the rolling tachyon back- ground could be extended to multi-point amplitudes of several closed string operators on the disk. These amplitudes should also be computable in terms of matrix integrals. It would be interesting to understand if there is any deeper meaning to the matrix in- tegral structure appearing in the closed string correlators in the rolling tachyon back- ground. One of the biggest outstanding problems in this approach to D-brane decay is the proper way to handle the divergences that appear in the calculation of the number and energy of the closed strings radiated from the brane. Naively, it seems as if the open string physics must be modi¯ed at late times. The divergence in equations (4.5) and (4.6) comes from the IR of the open string channel, suggesting that the proper way to address the divergence is to shift the open string background. The parts of the rolling tachyon 0 0 boundary state that contain X ; à oscillators all grow exponentially at late times [10], a divergence which does not appear at tree level, but a®ects physical quantities through open string loop diagrams. This suggests that the late time behavior of the open string solution as represented in the boundary state must be modi¯ed. The correct way to modify the open string solution remains unclear; perhaps the matrix model may o®er some guidance. Also, since we have a clear picture of the coherent closed string state that the brane decays into, it is natural to ask whether there is an open string description of this state. This is certainly true in two-dimensional string theory [7], and it would be of great interest to work out such a description in the full 10-dimensional theory. Once this open string description is found, the necessary steps to remove the divergence may well be more evident. These are all issues deserving of further study. Acknowledgments The author wishes to thank I. Ellwood, S. Robinson, W. Taylor, and especially H. Liu for useful conversations. Thanks to H. Liu for suggesting this topic, and to H. Liu and W. Taylor for comments on the manuscript. Research supported in part by the U.S. Department of Energy under cooperative research agreement #DF-FC02-94ER40818. A. An alternate derivation of the boundary state: the SU(2) method In this section we will extend Sen's calculations of [2] to obtain the lowest components of the boundary states for both the rolling and bouncing tachyon backgrounds in the Ramond Ramond sector by computing the boundary state in the euclidean theory ¯rst, and then Wick rotating. This approach to tachyon condensation was introduced by Sen in [1]. Free conformal ¯eld theories deformed by a boundary perturbation taken from a local SU(2) current algebra were studied in [11, 12]. The results appearing in this section are not necessarily new (the result (A.2) appeared as a guess in [8]), but there is no derivation of these results in the literature. { 16 { JHEP01(2005)037 In our conventions, the SU(2) current algebra is given by p p J (z) = ¡ 2à sin( 2X (z)) p p J (z) = 2à cos( 2X (z)) J (z) = i 2@X(z) : We will suppress Lorentz indices whenever possible. Here only the holomorphic part of the ¯eld X is used to de¯ne the SU(2) generators. We are interested in a set of closed string disk one-point functions in the tachyon backgrounds (1.1) and (1.2). The key step in the argument we will use to evaluate these one-point functions is the reinterpretation of a tachyon insertion integrated around the boundary of the disk as a holomorphic contour integral of one of the elements of the SU(2) algebra above. When the contour surrounds a vertex operator inserted in the bulk, performing the contour integral reduces to rotating the vertex operator in a sense we will dt make precise below. Concretely, if the perturbation is given by ¢S = ¡µ J, the 2¼ one-point function of the operator O(z)O(z¹) on the disk is dt µ J e e 2¼ hO(z)O(z¹)i = he O(z)O(z¹)i : perturbed unperturbed We may now imagine deforming the J integration contours away from the boundary one by one. If the OPE of O and J is well-de¯ned, then O will have a well-de¯ned transformation property with respect to the symmetry algebra to which J belongs. Thus we may conclude e e hOOi = hR (O)Oi ; perturbed µ unperturbed iµJ where jR (O)i = e jOi. For a complete discussion of these ideas, we refer the reader to [12]. There is some subtlety involved in deforming the integration contours of boundary perturbations o® the boundary. However, in our case there is no ambiguity involved, as all operators O that we need to consider are well-de¯ned with respect to the boundary perturbation. In order to carry out this program, our perturbation must be self local; in particular, it must commute with itself. It is simple to check that, with ´ included in the vertex operator, our deformations satisfy this criterion. In the NS-NS sector, the operators which have nonvanishing one-point functions in the p p in 2X in 2X e L R rolling tachyon background are of the form V(z; z¹) = O (z)O (z¹) ´ e e [2]. n n in 2X The holomorphic part O (z) = e of such vertex operators transforms in the j = jnj; m = n representation of SU(2). In the our our sector, we will be interested in operators of the form V = ik X ik X e n L e n R 0 0 O (z)O (z¹) ´ £ e £ e , where n;s n;s s s µ ¶ k ´ n + 2 : The operators O transform in spinor representations of SU(2), j = jn+1=2j, m = n+1=2. n;s 1 2 There is a branch cut in the OPE of J ; J with O due to the half integral momentum, which o®sets the branch cut the spin ¯eld puts in the fermion ¯eld; thus all contour integrals that appear are well-de¯ned. { 17 { JHEP01(2005)037 Because we need to group the fermions into pairs in order to de¯ne the £ , we will need to carry around a \spectator dimension"; in other words, while we are perturbing only one spatial direction ¹, we must always consider both the ¹ and the ¹ + 1 directions in order to be able to write down the spin ¯elds. We will see that in the end, however, the dependence of the boundary state on the fermion zeromodes will indeed factorize. A.1 Bouncing tachyon pro¯le We ¯rst consider the tachyon pro¯le T (X) = ¸ cos( 2X). For this tachyon pro¯le, the boundary action is proportional to J : dt ¢S = ¡¼¸ J : 2¼ Using the state-operator correspondence, we may write dt 1 1 ¼¸ J i¼¸J 2¼ 0 e O = e jO i ; n;s n;s so the boundary perturbation becomes a rotation of jOi by ¼¸ around the 1 axis; we denote the group element of SU(2) corresponding to this rotation as ¡. Now J / J + J , so + ¡ action of J will, in general, mix O with all the other operators in its SU(2) multiplet. n;s However, once we take the expectation value, only operators whose bosonic part is of the ik (X ¡X ) R L form e will contribute. This convenient fact is due to conservation of momentum on the disk with Neumann boundary conditions [2]. For our purposes, we therefore only need to know that a rotation by J through an angle of ¼¸ will take the highest (lowest)- ik X ¡ik X n L n L weight state £ e to the lowest (highest)-weight state £ e multiplied by some s ¡s calculable coe±cient plus operators whose expectation values vanish. The coe±cient we j=jn+1=2j 2jn+1=2j need is given by D (¡) = (i sin(¸¼)) , [12]. n+1=2;¡n¡1=2 The lowest component of the boundary state jB; +i will take the form jB; +i = f jk ;¡+i + oscillators ; n n where the f are c-numbers. The periodicity of the potential ensures that the restriction to momenta of the form k is legitimate. The choice of spinor vacuum j¡+i is what we expect for a decaying unstable D9-brane, as it has the right Lorentz properties to source C °ux. We will demonstrate below that j¡+i is indeed the correct choice. To solve for f , we have ik X ik X n n f = hk ;¡ + jB; +i = hO e i = hR (O e )i : n n ¡+ perturbed ¡ ¡+ unperturbed Here £ is the combination of spin ¯eld operators corresponding to the state j¡+i. We ¡+ can use our spinor dictionary (B.3) to ¯nd j=jn+1=2j ik X ¡ik X n L n L R (£ e ) = D (¡)£ e + ¢¢¢ ¡ ¡+ ++ n+1=2;¡n¡1=2 j=jn+1=2j ik X ¡ik X n n L L R (£ e ) = D (¡)£ e + ¢¢¢ : ¡ +¡ ¡¡ n+1=2;¡n¡1=2 { 18 { JHEP01(2005)037 Note that the rotated operator has the right quantum numbers to have nonvanishing one point function on the disk. Note also that the e®ect of the perturbation on the spinor vacuum has indeed factorized as advertised; we will drop the spectator dimension from this point on. Therefore, D E 2jn+1=2j ik (X ¡X ) n R L f = (i sin(¸¼)) £ e : (A.1) n ++ unperturbed ik (X ¡X ) n R L The expectation value h£ e i contributes only an overall constant, and the ++ boundary state now becomes, up to normalization, h i 2jn+1=2j ik X jB; +i = (i sin(¸¼)) ²(n) e j0;¡i + ¢¢¢ : Here ²(n) is a phase factor which we must include because the basis for which the coe±cient j=jn+1=2j D is calculated may not be the same basis that we are using [2]. This phase n+1=2;¡n¡1=2 factor may be ¯xed by asking that when ¸ = 1=2, the boundary state reduce to that of alternating D-branes and anti-D-branes placed at a = 2¼(n+1=2). Using this criterion, we ¯nd ²(n) = sign (n). We may now perform the sum on n: " # p p h i iX= 2 ¡iX= 2 e e 2jn+1=2j ik X p p (i sin(¸¼)) ²(n)e = sin(¸¼) ¡ : 2 2 i 2X ¡i 2X 1 + sin (¸¼)e 1 + sin (¸¼)e At this point we may Wick rotate: setting " # p p 0 0 X = 2 ¡X = 2 e e p p f (X ) = sin(¸¼) ¡ ; (A.2) 0 0 2 2 2X ¡ 2X 1 + sin (¸¼)e 1 + sin (¸¼)e the boundary state for the X direction becomes jB; +i = f (X )j0;¡i + ¢¢¢ ; while jB;¡i = f (X )j0; +i + ¢¢¢ : Our result for f (X ) agrees with that written down in [8]. The corresponding formula for the NS-NS sector was found in [2]. Here jB; ´i = f (X )j0i + oscillators ; NS NS where 1 1 p p f (X ) = + ¡ 1 : (A.3) NS 0 0 2 2 2X ¡ 2X 1 + sin (¸¼)e 1 + sin (¸¼)e { 19 { JHEP01(2005)037 A.2 Rolling tachyon pro¯le iX= 2 We now turn our attention to the tachyon pro¯le T (X) = ¸e . The perturbing bound- ary action for this pro¯le is proportional to the raising operator of the SU(2) algebra, dt S = ¡¼¸ J : boundary 2¼ We obtain the lowest components of the boundary state in both the NS-NS and the R-R sectors. This serves as a check on our computations in section 2. Once again, matters are simpli¯ed by the fact that only the lowest and highest weight states have nonvanishing expectation values, as a consequence of Neumann boundary con- ditions. Therefore we only need to consider operators of the form V = O O, and the j;m=¡j e®ect of the boundary perturbation is simple: dt + i¼¸ J 2j 2¼ e O = (i¼¸) O : j;m=¡j j;m=j Therefore, the lowest component of the NS-NS sector boundary state is X p 2n (i¼¸) ¯k = ¡n 2 : ¡n n=1 We can perform the sum to ¯nd jB; ´i = p j0i + ¢¢¢ : NS 2 2 ¡i 2X 1 + ¼ ¸ e Upon Wick rotation we ¯nd jB; ´i = j0i + ¢¢¢ ´ g (X )j0i + ¢¢¢ : NS NS 2 2 2X 1 + ¼ ¸ e Similarly, the lowest component of the Ramond sector boundary state is ¯ µ ¶ À ¯ p 2n¡1 jB; +i = (i¼¸) k = ¡ n + 2;¡ : ¡n R ¯ n=1 The dependence on the spin structure is precisely the same as that in the bouncing tachyon case above. We can perform the sum to ¯nd ¡iX= 2 i¼¸e jB; +i = p j0;¡i + oscillators : 2 2 ¡i 2X 1 + ¼ ¸ e Upon Wick rotation we ¯nd X = 2 i¼¸e jB; +i = j0;¡i + ¢¢¢ ´ g (X )j0;¡i + ¢¢¢ : 2 2 2X 1 + ¼ ¸ e These results agree precisely with those we found in (2.6) and (2.10) by directly evaluating the correlators. { 20 { JHEP01(2005)037 B. Relations between spinor vacua ¯ E In this appendix, we summarize the relationships between the spin states jsi ¯s , in terms of which the Ramond Ramond spin ¯elds have a simple description, and the spin states j§§i, in terms of which the boundary state has a simple description. We de¯ne two di®erent sets of raising operators, 2a¡1 y 2a  = p (à + ià ) (B.1) a 0 0 2a 2a 2a ³ = (à + ià ) ; (B.2) + 0 0 where the directions 2a ¡ 1 and 2a are spatial. Their hermitean conjugates are simi- larly de¯ned. The  are raising and lowering operators for the jsi basis, de¯ned so that  j¡1=2i = 0, while  j¡1=2i = j1=2i. The ³s are raising and lowering operators for the a a 2a 2a j§i basis, ³ j¡i = 0, ³ j¡i = j+i. ¡ + We may express the ³s in terms of the Âs; for example, 2a¡1 y y ³ = ( +  + i(Âe + Âe )) : a a + a a ¯ E ~0 The last piece of information we need concerns the action of the Âes on the states jsi s . Since the operators Âe need to pass through the holomorphic states jsi before they can act ~0 on ¯s , we need to know if they pick up a sign in doing so. In other words, we need to know when to count the state jsi as fermionic, which amounts to knowing the action F F of (¡1) on the state jsi. On the zero modes the action of (¡1) reduces to that of the chirality matrix ¡, which in the basis we are using for the ten-dimensional spinors is simply ¡js ; s ; : : : ; s i = sign ( s )js ; s ; : : : ; s i. Thus we assign (¡1) j¡1=2i = ¡j¡1=2i, 0 1 5 a 0 1 5 while (¡1) j1=2i = j1=2i. We are now ready to write down the relationship between the two bases. We have µ¯ À¯ À ¯ À¯ À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j++i = ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 ¯ ¯ ¯ ¯ µ À À À À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡¡i = p ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 ¯ ¯ ¯ ¯ µ À À À À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j+¡i = p ¡ i ¡ ¡ ¯ ¯ ¯ ¯ 2 2 2 2 ¯ ¯ ¯ ¯ µ À À À À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡+i = p ¡ ¡ ¡ i (B.3) ¯ ¯ ¯ ¯ 2 2 2 2 The phase of j++i has been determined arbitrarily, which ¯xes the phases of the other three states. When a = 0, the relations are altered slightly, becoming instead: à ! ¯ À¯ À ¯ À¯ À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j++i = p ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 à ! ¯ ¯ ¯ ¯ À À À À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡¡i = p ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 { 21 { JHEP01(2005)037 à ! ¯ ¯ ¯ ¯ À À À À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j+¡i = p i ¡ ¡ ¡ ¯ ¯ ¯ ¯ 2 2 2 2 à ! ¯ À¯ À ¯ À¯ À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡+i = ¡ i ¡ ¡ : ¯ ¯ ¯ ¯ 2 2 2 2 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] F. Larsen, A. Naqvi and S. Terashima, Rolling tachyons and decaying branes, J. High Energy Phys. 02 (2003) 039 [hep-th/0212248]. [4] N. Lambert, H. Liu and J. Maldacena, Closed strings from decaying D-branes, hep-th/0303139. [5] M. Gutperle and A. Strominger, Timelike boundary Liouville theory, Phys. Rev. D 67 (2003) 126002 [hep-th/0301038]. [6] J.L. Karczmarek, H. Liu, J. Maldacena and A. Strominger, UV ¯nite brane decay, J. High Energy Phys. 11 (2003) 042 [hep-th/0306132]. [7] I.R. Klebanov, J. Maldacena and N. Seiberg, D-brane decay in two-dimensional string theory, J. High Energy Phys. 07 (2003) 045 [hep-th/0305159]. [8] A. Sen, Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143]. [9] N. Constable and F. Larsen, The rolling tachyon as a matrix model, J. High Energy Phys. 06 (2003) 017 [hep-th/0305177]. [10] T. Okuda and S. Sugimoto, Coupling of rolling tachyon to closed strings, Nucl. Phys. B 647 (2002) 101 [hep-th/0208196]. [11] C.G. Callan Jr., I.R. Klebanov, A.W.W. Ludwig and J.M. Maldacena, Exact solution of a boundary conformal ¯eld theory, Nucl. Phys. B 422 (1994) 417 [hep-th/9402113]. [12] A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of D-branes, Nucl. Phys. B 545 (1999) 233 [hep-th/9811237]. { 22 { http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of High Energy Physics IOP Publishing

Closed superstring emission from rolling tachyon backgrounds

Shelton, Jessie
Journal of High Energy Physics , Volume 2005 (01) – Feb 10, 2005
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JHEP01(2005)037 Published by Institute of Physics Publishing for SISSA/ISAS Received: December 16, 2004 Accepted: January 24, 2005 Closed superstring emission from rolling tachyon backgrounds Jessie Shelton Center for Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A. E-mail: [email protected] Abstract: We compute the lowest components of the type-II Ramond-Ramond boundary X = 2 state for the tachyon pro¯le T (X) = ¸e by direct path integral evaluation. The calculation is made possible by noting that the integrals involved in the requisite disk one-point functions reduce to integrals over the product group manifold U(n) £ U(m). We further note that one-point functions of more general closed string operators in this background can also be related to U(n) £ U(m) group integrals. Using this boundary state, we compute the closed string emission from a decaying unstable Dp-brane of type- II string theory. We also discuss closed string emission from the tachyon pro¯le T (X) = ¸ cosh(X = 2). We ¯nd in both cases that the total number of particles produced diverges for p = 0, while the energy radiated into closed string modes diverges for p · 2, in precise analogy to the bosonic case. Keywords: Superstrings and Heterotic Strings, Tachyon Condensation. ° SISSA/ISAS 2005 http://jhep.sissa.it/archive/papers/jhep012005037 /jhep012005037.pdf JHEP01(2005)037 Contents 1. Introduction 1 2. Disk one-point functions in the rolling tachyon background 4 2.1 NS-NS sector 6 2.2 Relation to fermionized computation 7 3. General closed string one-point function as a matrix integral 9 4. Closed string production 11 5. Conclusions 15 A. An alternate derivation of the boundary state: the SU(2) method 16 A.1 Bouncing tachyon pro¯le 18 A.2 Rolling tachyon pro¯le 20 B. Relations between spinor vacua 21 1. Introduction The worldsheet approach to studying tachyon condensation was introduced by Sen in [1, 2]. In this approach, we give the tachyon a space-time-dependent expectation value by deform- ing the worldsheet conformal ¯eld theory with the exactly marginal boundary operator 0 ¸ 0 0 T (X) = ¸ cosh X , or, in the supersymmetric theory, the operator ´Ã sinh(X = 2), which corresponds to the tachyon pro¯le T (X) = ¸ cosh(X = 2). Another tachyon pro¯le 0 0 X X = 2 T (X) = ¸e and its supersymmetric analogue T (X) = ¸e was introduced in [3]. These conformal ¯eld theories are related by Wick rotation to theories where the boundary deformation is taken from a local SU(2) current algebra. Worldsheet conformal ¯eld theory is a perturbative approach to string theory and can likely provide only a partial answer to the deep nonperturbative problems posed by D-brane decay. Working out conformal ¯eld theory descriptions of D-brane decay is nonetheless of interest. Perhaps most importantly, the conformal ¯eld theory approach provides a clear computation of the coupling of the decaying D-brane to closed strings, a problem which remains somewhat obscure in the open string ¯eld theory description of tachyon condensation. In the worldsheet approach, the ¯nal closed string state produced by the decaying brane can be readily computed [4]. { 1 { JHEP01(2005)037 The rolling tachyon background is also a good starting point for exploring some of the technical issues involved in working with time-dependent perturbative solutions to the string equations of motion, as the time dependence is con¯ned to the open string sector, and thereby poses fewer new conceptual challenges than working with a time-dependent space- time background. One such interesting technical issue is the relationship proposed in [5] between the mathematical ambiguities involved in de¯ning the correlators in a nontrivial wrong sign CFT and the physical ambiguities involving choice of vacua in time-dependent ¯eld theories. Although it appears that the perturbative worldsheet description of D-brane decay breaks down in most examples, as the number of emitted particles and amount of emitted energy becomes in¯nite [4], there are examples where this divergence is brought under control, either due to a higher Hagedorn temperature [6], or an ability to work with a dual, non-perturbative description of the system [7]. In type-II theories, the space-time stress energy tensor for a decaying brane has been studied in [2, 3]. In this note we present a discussion of closed string production from a decaying type-II Dp-brane. The marginal deformations we study are the so-called \rolling tachyon" dt 0 0 X = 2 ¢S = ¡ 2¼¸ ´Ã e (1.1) 2¼ and the \bouncing tachyon" Z µ ¶ dt X ¢S = ¡ 2¼¸ ´Ã sinh p : (1.2) 2¼ Here ´ is a fermionic degree of freedom living on the boundary of the worldsheet; it has trivial worldsheet dynamics, and may also be understood as a Chan-Paton factor. It appears in the vertex operators of all open string states with wrong sign GSO projection. The coupling ¸ is related to the initial position and velocity of the tachyon at the top of its potential [1]. For most of our calculations, we will assume that we are able to choose a gauge in which no timelike oscillators are excited. Then a general closed string vertex operator takes the form iE X ? i i i V = e V (X ; à ; à ) (1.3) in the NS-NS sector and iE X ? i i i s e e V = e £ £ 0 V (X ; à ; à ) (1.4) s s s 0 s There has been some confusion in the literature as to the possible appearance of a term of the form T in the boundary action. If the boundary action is written in the manifestly supersymmetric form S = dtdµ(¡D¡ + ¡T (Y )) where ¡ = ´ + µF , D = @ + µ@ , and Y = X + µÃ, a term of the form T bdy µ arises after elimination of the auxiliary ¯eld F . This form of the boundary action has been instrumental in studies of tachyon condensation using boundary string ¯eld theory. However, when the boundary theory is de¯ned through deformation by the operators above, the T term does not arise; its contribution to the boundary action vanishes on shell. It is simple to check that the supersymmetry Ward identities for the deformed SCFT are satis¯ed when the boundary action only contains the term ´ÃT (x). { 2 { JHEP01(2005)037 is H 0 0 in the R-R sector. Here the £ are spin ¯elds, £ = e . With this gauge ¯xing, s s the problem of computing general closed string one point functions in the rolling tachyon background is greatly simpli¯ed. We only need to know the expectation values ¡¢S iE X he e i disk in the NS-NS sector, and ¡¢S iE X he e £ £ i s disk 0 s in the R-R sector. Equivalently, we only need to know the lowest component of the bound- ary state describing the deformed X CFT, 0 0 jBi = g(x )j0i + ¢¢¢ : In the NS-NS sector, the function g(x ) for the tachyon pro¯le (1.2) was worked out in [2] and for the tachyon pro¯le (1.1) in [3]. The computation of g(x ) for both tachyon pro¯les (1.1) and (1.2) in the R-R sector is one of the aims of this paper. We will, however, spend more time on the rolling tachyon pro¯le, (1.1) above, as a direct perturbative calculation of the closed string disk one-point amplitudes in that background reveals an interesting U(p) £ U(q) structure, which allows the correlators to be easily calculated. This structure is an interesting generalization of the U(n) symmetry that was found in the bosonic case in [3, 9]. We will also compute the NS- NS one-point functions for this background in order to demonstrate a similar U(p) £ U(p) structure appearing in the correlators there. These calculations are in section 2. In section 3, we discuss how the relation between correlation functions in the rolling tachyon background and integrals of invariants over the group U(p)£U(q) may be extended to one-point functions of more general operators. In order to compute the function g(x ) for the bouncing tachyon, we must use some more sophisticated CFT machinery, namely the SU(2) current algebra of the euclidean version of the theory. This technique is the one originally used by Sen. We have placed this computation in appendix A and use only the result, equation (A.2), in the main body of the paper. Also in this appendix, we present an alternate derivation of the function g(x ) for the rolling tachyon. Next, we consider the closed string production from the decaying D-brane, for both tachyon pro¯les. As in the bosonic case [4], both the number and the energy of emitted particles diverge for low dimensional D-branes. Since the tachyon pro¯les (1.1) and (1.2) describe homogenous tachyon condensation in the p spatial dimensions of the brane, the condensation process is unphysical if p 6= 0. A more physical decay process for a higher- dimensional D-brane will likely proceed in several causally disconnected patches, each of which behaves like a decaying D-brane of lower dimension [3]. Thus we expect that in a physical decay process, the number and energy of particles emitted from decaying Dp- branes will diverge for any p. This suggests that all the energy of the D-brane is converted into closed strings. See also [8]. { 3 { JHEP01(2005)037 Finally, appendix B summarizes the relationship between two di®erent bases for the spinor vacua, the states jsijs~i in terms of which the R-R spin ¯elds have a simple descrip- tion, and the states j§i in terms of which the boundary state has a simple description. These relationships will be useful when we compute the R-R sector boundary state. 2. Disk one-point functions in the rolling tachyon background In this section, we compute disk one-point functions of the form D E D E ¡¢S e e £ £ 0 = e £ £ 0 s s s s deformed free by Taylor-expanding the exponential and evaluating the correlators that appear at each 0 0 order in ¸. We do not carry out the path integral over the X zero mode x in the expectation values here, but interpret the resulting amplitudes as a function of x . We x = 2 therefore de¯ne ¸ ´ ¸e , and ¿ µ Z ¶ À ¿µZ ¶ À p n p X ~ dt 0 ( 2¼¸) dt 0 0 X = 2 0 X = 2 e e exp 2¼¸ ´Ã e £ £ 0 = ´Ã e £ £ 0 s s s s 2¼ n! 2¼ ´ (i¼¸) A : (2.1) In the R-R sector, the boundary fermion ´ has a zero mode; thus, only odd numbers of insertions of the boundary perturbation give non-vanishing contributions. Since the world- sheet dynamics of the boundary fermion are trivial, it is simply a matter of convenience whether we choose to formally integrate it out, as in [3], or include it in the vertex operator, as in [2]. We will ¯nd it convenient to include ´ in the vertex operator, since in this case the vertex operators on the boundary all commute with each other. The structure of this computation is most evident when using the bosonized represen- tation of the fermion ¯elds. The correlators that appear in (2.1) are then 2k+1 D E 2k+1 Y p £ ¤ (¼¸) dt 0 ~ iH ¡iH X = 2 is¢H is ¢H 0 0 0 e (t ) ¡ e (t ) e (t ) e (0)e (0) : i i i (2k + 1)! 2¼ i=1 This expectation value factors into three parts. The contribution from the transverse fermions is ± 0, which we will drop for ease of notation. The bosonic contribution is ~s;¡~s simply * + µ ¶ 2k+1 Y Y Y t ¡ t i j X = 2 it it 0 i j e (t ) = je ¡ e j = 2 sin : (2.2) i=1 i<j i<j £ ¤ 2k+1 iH ¡iH is H is H 0 0 0 0 0 Now consider h e (t ) ¡ e (t ) e (0)e (0)i. From now on we will drop i i i=1 iH ¡iH the label 0 to clean up the notation. When we expand the product (e (t )¡e (t )), i i i=1 2k+1 i² H we will ¯nd a sum of terms, each of which has the form e (t ), where ² = §1. i i i=1 The path integral over the H zero mode imposes H momentum conservation, giving the condition s + s + ² = 0 : i=1 { 4 { JHEP01(2005)037 0 0 Since s; s = §1=2, we have two possible ways to satisfy the condition above: let s = s = 2k+1 iH ¡iH §1=2 and retain only terms in the expansion of the product (e (t )¡e (t )) where i i i=1 the H momentum sums to ¨1. There will be a sign di®erence between the expectation values in the two di®erent cases, as when we choose s = s = 1=2, we pick up an odd ¡iH 0 number of factors of ¡e , and an even number when we choose s = s = ¡1=2. As we will see, this sign di®erence gives us the correct GSO projection. 2k+1 0 iH Let us consider the s = s = 1=2 for de¯niteness. Then in the product [e (t )¡ i=1 ¡iH e (t )] we need to keep the terms with k + 1 minus signs, that is, terms with ² = ¡1. i i The H correlators become µ ¶ ² ² i j P Y t ¡ t i j ¡k ¡i t =2 (i) e 2 sin : i<j Including the phases coming from the transformation from z to t thus yields simply µ ¶ ² ² i j t ¡ t i j (¡1) i 2 sin : (2.3) i<j Combining equations (2.2) and (2.3), we have (adjusting the overall phase for simplic- ity) 2 3 µ µ ¶¶ 2k+1 1+² ² Y X Y ¾(i) ¾(j) t ¡ t 1 dt ¾(i) ¾(j) 4 5 A = 2 sin ; (2.4) 2k+1 (2k + 1)! 2¼ 2 i=1 ¾2P ¾(i)<¾(j) where P denotes the set of all distinct ways to distribute k + 1 values of ¡1 and k values 2k+1 of 1 among the ² . Since the boundary operators are not path ordered, each of the ( ) terms in the sum above contribute equally. Thus, we may choose one ordering of the ² , namely ² = ¡1 ; 1 · i · k + 1 ; ² = 1 ; k + 1 < i · 2k + 1 ; i i and write the correlator in terms of that ordering alone: Z µ µ ¶¶ 2k+1 1+² ² i j Y Y 1 dt t ¡ t i i j A = 2 sin : (2.5) 2k+1 (k + 1)!k! 2¼ 2 i=1 i<j Now, the Haar measure for the group U(n) is Z Z 1 dt [dU] 1 = ¢ (t) = 1 ; n! 2¼ U(n) i=1 where ¢(t) is the Vandermonde determinant, µ ¶ t ¡ t i j ¢(t) = 2 sin : i<j It is now easy to see that (2.5) is simply the integral of unity over the group manifold of the product group U(k + 1) £ U(k): Z Z A = [dU] 1 [dU] 1 = 1 : 2k+1 U(k+1) U(k) { 5 { JHEP01(2005)037 Since A = 1 for all k, the sum over all orders (2.1) becomes simply 2k+1 x = 2 i¼¸ e 2k+1 R 0 (i¼¸) = ´ g (x ) : (2.6) 2 2 2x 1 + ¼ ¸ e It remains to restore the dependence on the spin. The amplitudes that we have calculated are built on spin vacua of the form (j1=2ij1=2i ¡ j¡1=2ij¡1=2i) where we have labeled states by their s quantum numbers, suppressing the transverse spin eigenvalues. The R-R boundary state, however, is built on 0 0 1 1 e e spinors j§§i that satisfy (à § ià )j§§i = (à § ià )j§§i = 0. Thus to write down the 0 0 0 0 boundary state, we need to know the relationship between the two bases jsijs i and j§§i, which we work out in appendix B. We ¯nd from (B.4) j1=2ij1=2i ¡ j¡1=2ij¡1=2i = j+¡i + j¡+i : We may now easily restore the transverse spin quantum numbers to ¯nd that the spin vacuum should be j+ ¡ ¢¢¢¡i + j¡ + ¢¢¢ +i. Note that this is the correct GSO projection for type IIA, and has the right Lorentz properties to source C °ux. The boundary state for the supersymmetric rolling tachyon pro¯le is therefore R 0 jBi = g (x ) (j+ ¡ ¢¢¢¡i + j¡ + ¢¢¢ +i) + oscillators: (2.7) The generalization to other Dp-branes of either type-II string theory is very simple; we simply need to change the dependence on the transverse spin degrees of freedom, exactly as we would in the absence of the tachyon expectation value. 2.1 NS-NS sector In the NS-NS sector, the expectation value that we need to compute is ­ ® ¡¢S h1i = e deformed free This expectation value was discussed in [3]; our principal interest in this section will be to demonstrate an interesting U(n) £ U(n) structure that allows the integrals to be easily performed. As before, we will evaluate this correlator by expanding in powers of the boundary perturbation, and de¯ne ¿ µ ¶ À dt 0 0 X = 2 n exp 2¼¸ ´Ã e 1 = (i¼¸) A : 2¼ The boundary fermion ´ has no zero mode in the NS-NS sector, so only terms in the expansion with n even will contribute. Using bosonized fermions, A is 2k 2k D E £ ¤ 1 dt iH ¡iH X = 2 A = e (t ) ¡ e (t ) e (t ) : (2.8) 2k i i i (2k)! 2¼ i=1 { 6 { JHEP01(2005)037 The H-momentum conservation condition now yields ² = 0 : The expectation value (2.8) is thus 2 3 Z µ µ ¶¶ 2k 1+² ² ¾(i) ¾(j) Y X Y t ¡ t 1 dt i ¾(i) ¾(j) 4 5 A = 2 sin ; 2k (2k)! 2¼ 2 i=1 ¾2P ¾(i)<¾(j) where P denotes the set of all distinct ways to distribute k values of 1 and k values of ¡1 among the ² . Again we choose a representative ordering, ² = ¡1 ; 1 · i · k ; ² = 1 ; k < i · 2k ; i i and write the correlator in terms of that ordering alone: µ µ ¶¶ 2k 1+² ² i j Y Y 1 dt t ¡ t i i j A = 2 sin : (2.9) (k!) 2¼ 2 i=1 i<j This can easily be written as the integral of unity over the group manifold of the product group U(k) £ U(k): " # A = [dU] 1 = 1 2k U(k) for all k. Summing all orders, the functional form of the time-dependent NS-NS source is then 2k NS 0 (i¼¸) = p ´ g (x ) : (2.10) 2 2 2x 1 + ¼ ¸ e The boundary state in the NS-NS sector is therefore NS 0 jBi = g (x ) (j0; +i ¡ j0;¡i) + oscillators : (2.11) NS This answer di®ers from that found in [3] by ¸ = 2¸ . This di®erence is not here there meaningful as ¸ can be set to any value by a time translation. 2.2 Relation to fermionized computation The U(p) £ U(q) structure that allowed us to easily carry out the integrals in (2.5) and (2.9) is obscured when the fermions are not bosonized. It is interesting to perform the same calculation without bosonizing the fermions, and then compare the two representations of the calculation. In the R-R sector, the fermionic correlators are equivalent to the expectation value of 2k + 1 (holomorphic) fermions on a sphere with £ at 0 and £ at in¯nity | that is, the expectation values of 2k + 1 holomorphic Ramond-sector fermions on a sphere. The contractions between two timelike fermions are therefore given by µ ¶ 1 t ¡ t 1 2 (R) D (t ; t ) = ¡ cot ; 1 2 2 2 { 7 { JHEP01(2005)037 where we have included the phase factors necessary to transform the propagator to the an- gular variables t . Because we have an odd number of fermions, all terms in the correlation function will have one uncontracted fermion. The expectation value of this last fermion is hÃ(t)i = (2s )± 0 ± 0 ; 0 s ;s ~s;¡~s where again we have included the phase factor coming from the conformal transformation of Ã. 0 0 The contribution from s again imposes that either s = s = 1=2 or s = s = 0 0 0 0 0 ¡1=2. The factor of 2s provides the relative sign between the two choices; this can be 0 + ¡ thought of as coming from the fact that à = ( ¡  )= 2, where  are the raising and lowering operators for the s basis. To simplify our notation, we will drop the spin structure 0 0 contributions, (2s )± ± , to the amplitude. We ¯nd that A is given in fermionic 0 2k+1 s ;s ~s;¡~s language by " # Z µ ¶ µ ¶ 2k+1 k 2¼ Y Y X Y t ¡t 1 dt t ¡t i i j ¾(2l¡1) ¾(2l) A = 2 sin sign (¾) cot ; 2k+1 (2k+1)! 2¼ 2 2 i=1 i<j ¾2Q l=1 (2.12) where we have again made an overall choice of phase to simplify the notation. Here Q denotes the set of all distinct contractions. In the NS-NS sector, the contractions between the fermions appearing in the boundary perturbation are µ ¶ 1 t ¡ t 1 2 (NS) ¡1 D (t ; t ) = ¡ sin ; 1 2 2 2 and (2.8) becomes " # ¡1 µ ¶ µ ¶ 2k k Y Y X Y t ¡ t 2 dt t ¡ t ¾(2l¡1) ¾(2l) i i j A = 2 sin 2 sin ; (2.13) 2k (2k)! 2¼ 2 2 i=1 i<j ¾2Q l=1 where again Q is the set of all distinct contractions. The equivalence between the two representations of the à ¯elds guarantees that the integrands of (2.12) and (2.5) are equal, it as are the integrands of (2.13) and (2.9). By rewriting these equalities in terms of x = e , we arrive at the following identities between two symmetric polynomials. The NS-NS correlation functions lead to the identity " # 2k k Y X Y k ¡1 2 (x ¡ x ) sign (¾) (x ¡ x ) = i j ¾(2l¡1) ¾(2l) i<j ¾2Q l=1 2 3 X Y Y 2 2 4 5 = (x ¡ x ) (x ¡ x ) : (2.14) ½(i) ½(j) ½(i) ½(j) ½2P ½(i)<½(j)·k k<½(i)<½(j)·2k The R-R correlation functions lead to the identity " # 2k+1 k Y X Y x + x ¾(2l¡1) ¾(2l) (x ¡ x ) sign (¾) = (2.15) i j x ¡ x ¾(2l¡1) ¾(2l) i<j ¾2Q l=1 { 8 { JHEP01(2005)037 " # X Y Y Y 2 2 = (x ¡x ) (x ¡x ) x : ½(i) ½(j) ½(i) ½(j) ½(i) ½2P ½(i)<½(j)·k+1 k+1<½(i)<½(j)·2k+1 k+1<½(i)·2k+1 In these forms, the identities hold for any abstract set of variables fx g. We may check these results by multiplying out the terms at ¯nite order. For example, when k = 2, equation (2.14) becomes 4 (x ¡ x )(x ¡ x )(x ¡ x )(x ¡ x ) ¡ (x ¡ x )(x ¡ x )(x ¡ x )(x ¡ x ) + 1 3 1 4 2 3 2 4 1 2 1 4 2 3 3 4 + (x ¡ x )(x ¡ x )(x ¡ x )(x ¡ x ) = 1 2 1 3 2 4 3 4 £ ¤ 2 2 2 2 2 2 = 2 (x ¡ x ) (x ¡ x ) + (x ¡ x ) (x ¡ x ) + (x ¡ x ) (x ¡ x ) : 1 3 2 4 1 4 2 3 2 1 3 4 This can be seen to be true by multiplying out the terms. Likewise, when k = 1, eq. (2.15) becomes (x + x )(x ¡ x )(x ¡ x )¡(x ¡ x )(x + x )(x ¡ x )+(x ¡ x )(x ¡ x )(x + x )= 1 2 1 3 2 3 1 2 1 3 2 3 1 2 1 3 2 3 2 2 2 = (x ¡ x ) x +(x ¡ x ) x +(x ¡ x ) x : 1 2 3 1 3 2 2 3 1 3. General closed string one-point function as a matrix integral In this section, we extend the results of the previous section to sketch how more general one-point functions in the rolling tachyon background can be evaluated using matrix inte- grals. This is a generalization of the work that was done for the bosonic rolling tachyon background in [9]. Since the group structure is most evident using the bosonized representation of the à ¯elds, we will work in bosonized language. A general closed string operator built out of the à ¯eld will take the form µ ¶ µ ¶ N N i j Y Y 1 1 0 º º~ i(pH+p H) i ¹ j V = @ H @ H e ; (º ¡ 1)! (º~ ¡ 1)! i j i j 0 0 where p; p 2 Z in the NS-NS sector, and p; p 2 Z+1=2 in the R-R sector. A general closed string operator built out of the X ¯eld will take the form à ! à ! p M p M i j Y Y 2 2 ¹ 0 ¹~ 0 i ¹ j V = @ X @ X : (¹ ¡ 1)! (¹~ ¡ 1)! i j i j We have chosen here a convenient normalization for the operators. We want to evaluate the amplitude ¿ µ ¶ À dt 0 0 X n exp 2¼¸ ´Ã e V V (0) = (i¼¸) A : (3.1) f b n 2¼ disk We note immediately that due to the boundary fermion ´, n must be even in the NS-NS sector, and odd in the R-R sector. Consider ¯rst the bosonic portion of this correlator. The propagator for a timelike boson on the disk is 1 1 2 2 G (z; w) = lnjz ¡ wj + lnjzw¹ ¡ 1j : 2 2 { 9 { JHEP01(2005)037 The bosonic correlators in (3.1) are then à ! à ! µ ¶ N N k j n n Y Y X Y X t ¡t 1 1 i j º it º~ it k l l 2 sin @ G (0; e ) @ G (0; e ) ; b b 2 (º ¡1)! (º~ ¡1)! k j i<j k l=1 j l=1 or à ! à ! µ ¶ N N k j n n Y Y X Y X t ¡ t i j ¡it º it º~ l k l 2 sin e e : (3.2) i<j k l=1 j l=1 Meanwhile, the propagator for the H ¯eld on the disk is G (z; w) = ln(z ¡ w) and the propagator between a H ¯eld and a H ¯eld is G (z; w) = ln(zw¹ ¡ 1) : The fermionic correlators in (3.1) are thus Y Y ¡n=2 it it ² ² it (2p² +1)=2 i j i j i i 2 (e ¡ e ) e £ i<j i à ! à ! N N j k Y X Y X 1 1 º º~ ¹ k £ ² @ G (0; w ) ² @ G (0; w ) ; l f l l ~ l (º ¡ 1)! (º~ ¡ 1)! j k j l k l or à ! à ! ~ N N µ ¶ j j Y Y Y X Y X it it ² ² it (2p² +1)=2 ¡it º it (º~ ¡1) i j i j i i l j l k p (e ¡ e ) e ² e ² e : l l i<j i j l k l Here, as in the previous section, ² = §1, and the conservation of H momentum leads to the condition ² + p + p = 0 : When this condition cannot be ful¯lled, that is, if jp + p j > n, the correlator vanishes. When the correlator does not vanish, we will need to sum over all possible groupings of the ² into a group of k with ² = ¡1 and a group of n ¡ k with ² = 1, where i i i k = (n + p + p ) : There are ( ) such terms, all of which contribute equally after integration. Let us choose the ordering ² = ¡1 ; 1 · i · k ² = 1 ; k < i · n : i i We ¯nd that the fermionic correlators become µ µ ¶¶ n n ² ² ¡n=2 i j Y Y 2 n! t ¡ t 0 i j it ² (p¡p ) i i 2 sin e £ k!(n ¡ k)! 2 i<j i { 10 { JHEP01(2005)037 à ! à ! N N Y X Y X ¡it º it (º~ ¡1) l j l k £ ² e ² e : (3.3) l l j l k l We are now ready to put together equations (3.2) and (3.3) and write down the amplitude for a general closed string vertex operator and n insertions of the boundary action. De¯ning ¡ ¢ it it U = Diag e ; : : : ; e and ¡ ¢ it it k+1 V = Diag e ; : : : ; e ; we ¯nd Z Z µ ¶ 0 p ¡p det U A = [dU] [dV ] £ det V U(k) U(n¡k) ³ ´ Y Y M ¡ ¢ ~ y ¹ y ¹ ¹~ ¹~ i i j j £ Tr(U ) + Tr(V ) Tr U ¡ Tr V £ i j ³ ´ Y Y ¡ ¢ ~ y º y º º~ ¡1 º~ ¡1 k k l l £ Tr(U ) ¡ Tr(V ) Tr U ¡ Tr V : (3.4) k l This is indeed a matrix integral with product structure U(k) £ U(n ¡ k). In principle, the integrals over the unitary groups may be performed, as in [9]. However, in practice the expansions of the products above must become prohibitive, as the exponents M ; N can i j become arbitrarily large, as can the cardinality of the index sets fig;fjg. 4. Closed string production In this section, we calculate closed string emission from the decaying D-branes described by the boundary state we derived in section 2. Analogous computations were done for the bosonic string in [4, 6]. The computation for the superstring does not di®er greatly from that done for the bosonic string, and our discussion will therefore be largely parallel to that in [4]. We begin with the expression for the cylinder diagram ¿ ¯ ¯ À ¯ ¯ 1 1 ¯ ¯ W = B B : (4.1) ¯ ¯ L + L + i² 0 0 This expression picks up an imaginary part whenever particles go on shell. Writing jBi = U(! )jsi ; the imaginary part of (4.1) is 1 1 ImW = jU(! )j : (4.2) 2 2! The optical theorem relates this to particle production, so that N=V = 2 Im W . The volume V here is the p-dimensional spatial volume of the decaying brane. { 11 { JHEP01(2005)037 We ¯rst need to calculate U(!). This reduces essentially to choosing a prescription for integrating the zero mode of the bosonic coordinate. Assuming the factorization (1.3) and (1.4) for a general closed string vertex operator, the computation of U(!) factorizes ? i i i into one part from the expectation value of V (X ; à ; à ) in the transverse CFT, and the part of interest from the time-dependent components of the boundary state. As in [4], the transverse CFT contributes only a phase, which drops out of the computations below. 0 0 The contribution from the (X ; à ) CFT takes the form iEX iEt U(E) = he i = i dt g(t)e : (4.3) disk Here g(t) is the function appearing in the lowest component of the boundary state, which we computed for the rolling tachyon pro¯le in section 2. This function is given by (2.10) in the NS-NS sector, and (2.6) in the R-R sector. For the bouncing tachyon pro¯le, g(t) is given by (A.3) in the NS-NS sector, and (A.2) in the R-R sector. There are two contours of interest, namely C , which runs along the real axis and real is closed in the upper half plane, and the Hartle-Hawking contour C , which runs along HH the imaginary axis from t = i1 to t = 0, then runs along the real axis [4]. For the rolling tachyon pro¯le, both contours yield the same result. For the bouncing tachyon pro¯le, the contour C yields the same result as for the rolling tachyon, while the contour C yields HH real a di®erent result. We will consider the rolling tachyon pro¯le ¯rst. For this pro¯le, the contour integral (4.3) yields in the NS-NS sector 1 ¼= 2 i!t ¡i 2! U (!) = i dt p e = (¼¸) : NS ¼! 2 2 2t p sinh C 1 + ¼ ¸ e real In the R-R sector, the contour integral calculation yields t= 2 i¼¸e ¼= 2 i!t ¡i 2! U (!) = i dt e = ¡(¼¸) : ¼! 2 2 2t p cosh 1 + ¼ ¸ e real We now use these results for U(!) to calculate closed string emission from the decaying branes, using (4.2). Let us consider ¯rst the portion of the cylinder diagram coming from the overlap hB; +j jB; +i . We have argued that the imaginary part of this amplitude is NS e NS L +L +i² 0 0 given by X X 1 ¼ 1 2 Im W = jU(! )j = p : NS++ s 2! 4! s s sinh (¼! = 2) s s We now go through a series of manipulations to bring this into a more familiar form. Using X p ¡ 2¼n! p = 4 n e sinh (¼! = 2) n=0 and 1 1 1 ¡ 2¼! n 2¼ik n s 0 e = dk e 2 1 2¼ 2! k + ! 0 s { 12 { JHEP01(2005)037 Z Z 1 2 1 2 ¡t(k + ! ) 2¼ik n 0 2 = dk dt e e 2¼ 1 ¼ 2 2 2 ¡¼ n =t ¡t! =2 = dt e e ; (4.4) 2¼ t the contribution to particle production is X X 2 2 2 ¡1=2 ¡¼ n =t ¡t! =2 2 Im W = C n dt t e e : NS++ p n s Here C is a numerical constant coming from the normalization of the boundary state. 2 ¡2t We now write ! = 4N + 4N ¡ 2 and set q = e . The sum over states can now be s à performed in the usual manner, so that µ ¶ 2 2 f (q) ¡1=2 ¡¼ n =t 2 Im W = C n dt t e : NS++ p ´(q) In a very similar fashion, one may calculate the contribution to particle production from 1 1 the overlaps hB; +j jB;¡i and hB; +j jB; +i . The total number NS R e NS e R L +L +i² L +L +i² 0 0 0 0 of emitted particles is then ¹ X ¡ ¢ N 2 2 ¡1=2 ¡¼ n =t ¡8 8 8 n 8 = C n dt t e [´(q)] f (q) ¡ f (q) ¡ (¡1) f (q) : 3 4 2 This expression diverges for small t. This divergence is easier to examine after making a ¡2¼s modular transformation to s = . De¯ning w = e , we ¯nd Z Z ¹ X ¡ ¢ N ds 2 2 9 ¡(k +n )=2 ¡8 8 n 8 8 = n d k w [´(w)] f (w) ¡ (¡1) f (w) ¡ f (w) : 3 4 2 V s For a decaying Dp-brane, the answer generalizes in the obvious fashion, Z Z ¡ ¢ N ds 2 2 p ¡(k +n )=2 ¡8 8 n 8 8 = n d k w [´(w)] f (w) ¡ (¡1) f (w) ¡ f (w) : (4.5) 3 4 2 V s When s ! 1, we may expand (4.5) for small w to ¯nd Z Z Z ds 2 ds p k =2 ¡1=2 ¡p=2 d k w (1 + O(w )) » s : s s It is now easy to see that this diverges for p = 0. As noted in [4], this has the following open string interpretation. Let µ ¶ À X p jB i = B ; (¡1) ; x = 2 n + ¼ + p n=0 ¯ p ® where B ; (¡1) ; x = 2(n + 1=2)¼ denotes the boundary state for a euclidean D-instan- ton with p spatial dimensions, located at the position x = 2(n + 1=2)¼ in the euclidean { 13 { JHEP01(2005)037 time direction, sourcing positive or negative R-R °ux according to the sign (¡1) . Analo- gously, we de¯ne µ ¶ À X p jB i = B ;¡(¡1) ; x = ¡ 2 n + ¼ : ¡ p n=0 The imaginary part of the partition function (4.5), is then ¯ ¯ ¿ À ¯ ¯ N 1 ¯ ¯ = B B : ¡ + ¯ ¯ V e L + L 0 0 In other words, the expression (4.5) is the partition function that arises from a collection of alternating euclidean Dp-branes and anti-Dp-branes located at regular positions along the imaginary time axis, provided we only include the open strings that stretch across the line x = 0 [4]. This open string description of the closed string radiation arises from writing the closed string ¯elds produced by the decaying D-brane as radiation sourced by D-instantons in euclidean time. From this point of view, the euclidean path integral over x < 0 in the presence of the source jB i prepares a state in the closed string Hilbert space at x = 0 which is closely related to the ¯nal closed string state; see [4] for a complete discussion. In this open string language, the divergence in equation (4.5) comes from the existence of a massless string state stretching between the instantons nearest to the imaginary time axis. This divergence comes from the IR of the open string channel, and thus from the UV of the closed string channel. It re°ects the fact that the amplitude for emission of a very massive string state does not fall o® quickly enough as a function of energy to o®set the Hagedorn divergence in the number of states. While this divergence only occurs for p = 0, the homogenous solution studied here should not be trusted for larger p, as higher dimensional D-branes should rapidly break up into causally disconnected patches that decay locally as unstable D0-branes. (This discussion is complicated slightly in the superstring by the R-R couplings, but it is easy to see that the NS-NS contributions to particle production are divergent on their own for p = 0.) Therefore the divergence that occurs when p = 0 should be thought of as the generic tree level result in the superstring as well as the bosonic string. This seems to signal the breakdown of string perturbation theory. We may also consider the energy that is radiated into these closed string modes, E 1 = jU(! )j : V 2 Since from (4.4) we may write · ¸ Z ³ ´ p p 3=2 1 @ 1 1 1 ¼ ¡ 2¼! (n+a) ¡ 2¼! n ¡t! =2 s s ¡ p e = e = dt ne ; 2¼ @a 2 2¼ t 2! a=0 the energy radiated into closed strings is given by Z Z ¡ ¢ E ds 2 2 2 p (k +n )=2 ¡8 8 n 8 8 = n d k s w [´(w)] f (w) ¡ (¡1) f (w) ¡ f (w) : (4.6) 3 4 2 V s { 14 { JHEP01(2005)037 The divergence at small w has now been worsened, ¡p=2 » ds s ; so that the total energy emitted into closed strings is divergent for p · 2. This is precisely the expected supersymmetric generalization of the results of [4] for the bosonic string, and suggests that all the energy of the D-brane is converted into closed strings. The story is slightly more complicated for the bouncing tachyon pro¯le when the contour C is used in equation (4.3), but the fundamental physical picture does not real change. With this contour, we ¯nd ¡ 2¼ U(!) = p sin( 2!·) ; NS sinh(¼!= 2) 2¼ U(!) = p cos( 2!·) ; cosh(¼!= 2) where · ´ ln sin ¼¸. The number of emitted particles is now Z Z ¹ X ¡ ¢ N ds 2 2 p k =2 ¡8 n =2 8 n 8 8 = n d k w [´(w)] 2w f (w) ¡ (¡1) f (w) ¡ f (w) ¡ 3 4 2 V s n=0 ¡ ¢ 2 2 ¡(n+2i·=¼) =2 ¡(n+2i·=¼) =2 ¡ w + w £ ¡ ¢ 8 n 8 8 £ f (w) + (¡1) f (w) ¡ f (w) : 3 4 2 Note that the signs of the GSO projection are reversed in the second term of the above equation. Due to this change of sign, the second term is ¯nite. However, the ¯rst line is identical (up to an overall factor of 2) to the calculation performed for the rolling tachyon pro¯le, and in particular diverges in the same way. In the superstring, the tachyon potential is bounded from below, so it is a sensible physical question to ask about the closed string emission from the tachyon pro¯le T (X ) = 0 3 ¸ sinh(X = 2). We ¯nd in this case that the functions U(!) di®er from those for the bouncing tachyon pro¯le only by an overall phase and the rede¯nition · = sinh(¼¸), and therefore that the divergence in the closed string radiation from this tachyon pro¯le is identical to the divergence in the radiation from the bouncing tachyon pro¯le. 5. Conclusions We have presented a basic analysis of closed string one-point functions in rolling tachyon backgrounds in type-II string theories. We noted the appearance of an interesting U(p) £ U(q) structure that allowed us to easily evaluate the integrals appearing in the correlation functions, and pointed out how this structure could be generalized to compute one-point 0 0 functions involving excitations of the X and à oscillators. We con¯rmed that the total number and energy of emitted particles diverge for low-dimensional D-branes. Thanks to N. Lambert for this point. { 15 { JHEP01(2005)037 The computations of closed string expectation values in the rolling tachyon back- ground could be extended to multi-point amplitudes of several closed string operators on the disk. These amplitudes should also be computable in terms of matrix integrals. It would be interesting to understand if there is any deeper meaning to the matrix in- tegral structure appearing in the closed string correlators in the rolling tachyon back- ground. One of the biggest outstanding problems in this approach to D-brane decay is the proper way to handle the divergences that appear in the calculation of the number and energy of the closed strings radiated from the brane. Naively, it seems as if the open string physics must be modi¯ed at late times. The divergence in equations (4.5) and (4.6) comes from the IR of the open string channel, suggesting that the proper way to address the divergence is to shift the open string background. The parts of the rolling tachyon 0 0 boundary state that contain X ; à oscillators all grow exponentially at late times [10], a divergence which does not appear at tree level, but a®ects physical quantities through open string loop diagrams. This suggests that the late time behavior of the open string solution as represented in the boundary state must be modi¯ed. The correct way to modify the open string solution remains unclear; perhaps the matrix model may o®er some guidance. Also, since we have a clear picture of the coherent closed string state that the brane decays into, it is natural to ask whether there is an open string description of this state. This is certainly true in two-dimensional string theory [7], and it would be of great interest to work out such a description in the full 10-dimensional theory. Once this open string description is found, the necessary steps to remove the divergence may well be more evident. These are all issues deserving of further study. Acknowledgments The author wishes to thank I. Ellwood, S. Robinson, W. Taylor, and especially H. Liu for useful conversations. Thanks to H. Liu for suggesting this topic, and to H. Liu and W. Taylor for comments on the manuscript. Research supported in part by the U.S. Department of Energy under cooperative research agreement #DF-FC02-94ER40818. A. An alternate derivation of the boundary state: the SU(2) method In this section we will extend Sen's calculations of [2] to obtain the lowest components of the boundary states for both the rolling and bouncing tachyon backgrounds in the Ramond Ramond sector by computing the boundary state in the euclidean theory ¯rst, and then Wick rotating. This approach to tachyon condensation was introduced by Sen in [1]. Free conformal ¯eld theories deformed by a boundary perturbation taken from a local SU(2) current algebra were studied in [11, 12]. The results appearing in this section are not necessarily new (the result (A.2) appeared as a guess in [8]), but there is no derivation of these results in the literature. { 16 { JHEP01(2005)037 In our conventions, the SU(2) current algebra is given by p p J (z) = ¡ 2à sin( 2X (z)) p p J (z) = 2à cos( 2X (z)) J (z) = i 2@X(z) : We will suppress Lorentz indices whenever possible. Here only the holomorphic part of the ¯eld X is used to de¯ne the SU(2) generators. We are interested in a set of closed string disk one-point functions in the tachyon backgrounds (1.1) and (1.2). The key step in the argument we will use to evaluate these one-point functions is the reinterpretation of a tachyon insertion integrated around the boundary of the disk as a holomorphic contour integral of one of the elements of the SU(2) algebra above. When the contour surrounds a vertex operator inserted in the bulk, performing the contour integral reduces to rotating the vertex operator in a sense we will dt make precise below. Concretely, if the perturbation is given by ¢S = ¡µ J, the 2¼ one-point function of the operator O(z)O(z¹) on the disk is dt µ J e e 2¼ hO(z)O(z¹)i = he O(z)O(z¹)i : perturbed unperturbed We may now imagine deforming the J integration contours away from the boundary one by one. If the OPE of O and J is well-de¯ned, then O will have a well-de¯ned transformation property with respect to the symmetry algebra to which J belongs. Thus we may conclude e e hOOi = hR (O)Oi ; perturbed µ unperturbed iµJ where jR (O)i = e jOi. For a complete discussion of these ideas, we refer the reader to [12]. There is some subtlety involved in deforming the integration contours of boundary perturbations o® the boundary. However, in our case there is no ambiguity involved, as all operators O that we need to consider are well-de¯ned with respect to the boundary perturbation. In order to carry out this program, our perturbation must be self local; in particular, it must commute with itself. It is simple to check that, with ´ included in the vertex operator, our deformations satisfy this criterion. In the NS-NS sector, the operators which have nonvanishing one-point functions in the p p in 2X in 2X e L R rolling tachyon background are of the form V(z; z¹) = O (z)O (z¹) ´ e e [2]. n n in 2X The holomorphic part O (z) = e of such vertex operators transforms in the j = jnj; m = n representation of SU(2). In the our our sector, we will be interested in operators of the form V = ik X ik X e n L e n R 0 0 O (z)O (z¹) ´ £ e £ e , where n;s n;s s s µ ¶ k ´ n + 2 : The operators O transform in spinor representations of SU(2), j = jn+1=2j, m = n+1=2. n;s 1 2 There is a branch cut in the OPE of J ; J with O due to the half integral momentum, which o®sets the branch cut the spin ¯eld puts in the fermion ¯eld; thus all contour integrals that appear are well-de¯ned. { 17 { JHEP01(2005)037 Because we need to group the fermions into pairs in order to de¯ne the £ , we will need to carry around a \spectator dimension"; in other words, while we are perturbing only one spatial direction ¹, we must always consider both the ¹ and the ¹ + 1 directions in order to be able to write down the spin ¯elds. We will see that in the end, however, the dependence of the boundary state on the fermion zeromodes will indeed factorize. A.1 Bouncing tachyon pro¯le We ¯rst consider the tachyon pro¯le T (X) = ¸ cos( 2X). For this tachyon pro¯le, the boundary action is proportional to J : dt ¢S = ¡¼¸ J : 2¼ Using the state-operator correspondence, we may write dt 1 1 ¼¸ J i¼¸J 2¼ 0 e O = e jO i ; n;s n;s so the boundary perturbation becomes a rotation of jOi by ¼¸ around the 1 axis; we denote the group element of SU(2) corresponding to this rotation as ¡. Now J / J + J , so + ¡ action of J will, in general, mix O with all the other operators in its SU(2) multiplet. n;s However, once we take the expectation value, only operators whose bosonic part is of the ik (X ¡X ) R L form e will contribute. This convenient fact is due to conservation of momentum on the disk with Neumann boundary conditions [2]. For our purposes, we therefore only need to know that a rotation by J through an angle of ¼¸ will take the highest (lowest)- ik X ¡ik X n L n L weight state £ e to the lowest (highest)-weight state £ e multiplied by some s ¡s calculable coe±cient plus operators whose expectation values vanish. The coe±cient we j=jn+1=2j 2jn+1=2j need is given by D (¡) = (i sin(¸¼)) , [12]. n+1=2;¡n¡1=2 The lowest component of the boundary state jB; +i will take the form jB; +i = f jk ;¡+i + oscillators ; n n where the f are c-numbers. The periodicity of the potential ensures that the restriction to momenta of the form k is legitimate. The choice of spinor vacuum j¡+i is what we expect for a decaying unstable D9-brane, as it has the right Lorentz properties to source C °ux. We will demonstrate below that j¡+i is indeed the correct choice. To solve for f , we have ik X ik X n n f = hk ;¡ + jB; +i = hO e i = hR (O e )i : n n ¡+ perturbed ¡ ¡+ unperturbed Here £ is the combination of spin ¯eld operators corresponding to the state j¡+i. We ¡+ can use our spinor dictionary (B.3) to ¯nd j=jn+1=2j ik X ¡ik X n L n L R (£ e ) = D (¡)£ e + ¢¢¢ ¡ ¡+ ++ n+1=2;¡n¡1=2 j=jn+1=2j ik X ¡ik X n n L L R (£ e ) = D (¡)£ e + ¢¢¢ : ¡ +¡ ¡¡ n+1=2;¡n¡1=2 { 18 { JHEP01(2005)037 Note that the rotated operator has the right quantum numbers to have nonvanishing one point function on the disk. Note also that the e®ect of the perturbation on the spinor vacuum has indeed factorized as advertised; we will drop the spectator dimension from this point on. Therefore, D E 2jn+1=2j ik (X ¡X ) n R L f = (i sin(¸¼)) £ e : (A.1) n ++ unperturbed ik (X ¡X ) n R L The expectation value h£ e i contributes only an overall constant, and the ++ boundary state now becomes, up to normalization, h i 2jn+1=2j ik X jB; +i = (i sin(¸¼)) ²(n) e j0;¡i + ¢¢¢ : Here ²(n) is a phase factor which we must include because the basis for which the coe±cient j=jn+1=2j D is calculated may not be the same basis that we are using [2]. This phase n+1=2;¡n¡1=2 factor may be ¯xed by asking that when ¸ = 1=2, the boundary state reduce to that of alternating D-branes and anti-D-branes placed at a = 2¼(n+1=2). Using this criterion, we ¯nd ²(n) = sign (n). We may now perform the sum on n: " # p p h i iX= 2 ¡iX= 2 e e 2jn+1=2j ik X p p (i sin(¸¼)) ²(n)e = sin(¸¼) ¡ : 2 2 i 2X ¡i 2X 1 + sin (¸¼)e 1 + sin (¸¼)e At this point we may Wick rotate: setting " # p p 0 0 X = 2 ¡X = 2 e e p p f (X ) = sin(¸¼) ¡ ; (A.2) 0 0 2 2 2X ¡ 2X 1 + sin (¸¼)e 1 + sin (¸¼)e the boundary state for the X direction becomes jB; +i = f (X )j0;¡i + ¢¢¢ ; while jB;¡i = f (X )j0; +i + ¢¢¢ : Our result for f (X ) agrees with that written down in [8]. The corresponding formula for the NS-NS sector was found in [2]. Here jB; ´i = f (X )j0i + oscillators ; NS NS where 1 1 p p f (X ) = + ¡ 1 : (A.3) NS 0 0 2 2 2X ¡ 2X 1 + sin (¸¼)e 1 + sin (¸¼)e { 19 { JHEP01(2005)037 A.2 Rolling tachyon pro¯le iX= 2 We now turn our attention to the tachyon pro¯le T (X) = ¸e . The perturbing bound- ary action for this pro¯le is proportional to the raising operator of the SU(2) algebra, dt S = ¡¼¸ J : boundary 2¼ We obtain the lowest components of the boundary state in both the NS-NS and the R-R sectors. This serves as a check on our computations in section 2. Once again, matters are simpli¯ed by the fact that only the lowest and highest weight states have nonvanishing expectation values, as a consequence of Neumann boundary con- ditions. Therefore we only need to consider operators of the form V = O O, and the j;m=¡j e®ect of the boundary perturbation is simple: dt + i¼¸ J 2j 2¼ e O = (i¼¸) O : j;m=¡j j;m=j Therefore, the lowest component of the NS-NS sector boundary state is X p 2n (i¼¸) ¯k = ¡n 2 : ¡n n=1 We can perform the sum to ¯nd jB; ´i = p j0i + ¢¢¢ : NS 2 2 ¡i 2X 1 + ¼ ¸ e Upon Wick rotation we ¯nd jB; ´i = j0i + ¢¢¢ ´ g (X )j0i + ¢¢¢ : NS NS 2 2 2X 1 + ¼ ¸ e Similarly, the lowest component of the Ramond sector boundary state is ¯ µ ¶ À ¯ p 2n¡1 jB; +i = (i¼¸) k = ¡ n + 2;¡ : ¡n R ¯ n=1 The dependence on the spin structure is precisely the same as that in the bouncing tachyon case above. We can perform the sum to ¯nd ¡iX= 2 i¼¸e jB; +i = p j0;¡i + oscillators : 2 2 ¡i 2X 1 + ¼ ¸ e Upon Wick rotation we ¯nd X = 2 i¼¸e jB; +i = j0;¡i + ¢¢¢ ´ g (X )j0;¡i + ¢¢¢ : 2 2 2X 1 + ¼ ¸ e These results agree precisely with those we found in (2.6) and (2.10) by directly evaluating the correlators. { 20 { JHEP01(2005)037 B. Relations between spinor vacua ¯ E In this appendix, we summarize the relationships between the spin states jsi ¯s , in terms of which the Ramond Ramond spin ¯elds have a simple description, and the spin states j§§i, in terms of which the boundary state has a simple description. We de¯ne two di®erent sets of raising operators, 2a¡1 y 2a  = p (à + ià ) (B.1) a 0 0 2a 2a 2a ³ = (à + ià ) ; (B.2) + 0 0 where the directions 2a ¡ 1 and 2a are spatial. Their hermitean conjugates are simi- larly de¯ned. The  are raising and lowering operators for the jsi basis, de¯ned so that  j¡1=2i = 0, while  j¡1=2i = j1=2i. The ³s are raising and lowering operators for the a a 2a 2a j§i basis, ³ j¡i = 0, ³ j¡i = j+i. ¡ + We may express the ³s in terms of the Âs; for example, 2a¡1 y y ³ = ( +  + i(Âe + Âe )) : a a + a a ¯ E ~0 The last piece of information we need concerns the action of the Âes on the states jsi s . Since the operators Âe need to pass through the holomorphic states jsi before they can act ~0 on ¯s , we need to know if they pick up a sign in doing so. In other words, we need to know when to count the state jsi as fermionic, which amounts to knowing the action F F of (¡1) on the state jsi. On the zero modes the action of (¡1) reduces to that of the chirality matrix ¡, which in the basis we are using for the ten-dimensional spinors is simply ¡js ; s ; : : : ; s i = sign ( s )js ; s ; : : : ; s i. Thus we assign (¡1) j¡1=2i = ¡j¡1=2i, 0 1 5 a 0 1 5 while (¡1) j1=2i = j1=2i. We are now ready to write down the relationship between the two bases. We have µ¯ À¯ À ¯ À¯ À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j++i = ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 ¯ ¯ ¯ ¯ µ À À À À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡¡i = p ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 ¯ ¯ ¯ ¯ µ À À À À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j+¡i = p ¡ i ¡ ¡ ¯ ¯ ¯ ¯ 2 2 2 2 ¯ ¯ ¯ ¯ µ À À À À¶ ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡+i = p ¡ ¡ ¡ i (B.3) ¯ ¯ ¯ ¯ 2 2 2 2 The phase of j++i has been determined arbitrarily, which ¯xes the phases of the other three states. When a = 0, the relations are altered slightly, becoming instead: à ! ¯ À¯ À ¯ À¯ À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j++i = p ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 à ! ¯ ¯ ¯ ¯ À À À À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡¡i = p ¡ ¡ i ¡ ¯ ¯ ¯ ¯ 2 2 2 2 { 21 { JHEP01(2005)037 à ! ¯ ¯ ¯ ¯ À À À À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j+¡i = p i ¡ ¡ ¡ ¯ ¯ ¯ ¯ 2 2 2 2 à ! ¯ À¯ À ¯ À¯ À 0 0 ¯ ¯ ¯ ¯ 1 1 1 1 1 ¯ ¯ ¯ ¯ j¡+i = ¡ i ¡ ¡ : ¯ ¯ ¯ ¯ 2 2 2 2 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] F. Larsen, A. Naqvi and S. Terashima, Rolling tachyons and decaying branes, J. High Energy Phys. 02 (2003) 039 [hep-th/0212248]. [4] N. Lambert, H. Liu and J. Maldacena, Closed strings from decaying D-branes, hep-th/0303139. [5] M. Gutperle and A. Strominger, Timelike boundary Liouville theory, Phys. Rev. D 67 (2003) 126002 [hep-th/0301038]. [6] J.L. Karczmarek, H. Liu, J. Maldacena and A. Strominger, UV ¯nite brane decay, J. High Energy Phys. 11 (2003) 042 [hep-th/0306132]. [7] I.R. Klebanov, J. Maldacena and N. Seiberg, D-brane decay in two-dimensional string theory, J. High Energy Phys. 07 (2003) 045 [hep-th/0305159]. [8] A. Sen, Field theory of tachyon matter, Mod. Phys. Lett. A 17 (2002) 1797 [hep-th/0204143]. [9] N. Constable and F. Larsen, The rolling tachyon as a matrix model, J. High Energy Phys. 06 (2003) 017 [hep-th/0305177]. [10] T. Okuda and S. Sugimoto, Coupling of rolling tachyon to closed strings, Nucl. Phys. B 647 (2002) 101 [hep-th/0208196]. [11] C.G. Callan Jr., I.R. Klebanov, A.W.W. Ludwig and J.M. Maldacena, Exact solution of a boundary conformal ¯eld theory, Nucl. Phys. B 422 (1994) 417 [hep-th/9402113]. [12] A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of D-branes, Nucl. Phys. B 545 (1999) 233 [hep-th/9811237]. { 22 {

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Journal of High Energy Physics – IOP Publishing

Published: Feb 10, 2005

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