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Level Truncation and Rolling the Tachyon in the Lightcone Basis for Open String Field Theor


Preprint typeset in JHEP style - HYPER VERSION

Level Truncation and Rolling the Tachyon in the Lightcone Basis for Open String Field Theory

arXiv:hep-th/0409179v1 16 Sep 2004

Theodore G. Erler
University of California, Santa Barbara Santa Barbara, CA 93106, U.S.A E-mail:terler@physics.ucsb.edu

Abstract: A recent paper by Gross and Erler (hep-th/0406199) showed that by making a certain well-de?ned, unitary transformation on the mode basis for the open bosonic string—one that identi?es the lightcone component of position with the string midpoint— it is possible to render the action for cubic string ?eld theory local in lightcone time. In this basis, then, cubic string ?eld theory possesses a well-de?ned initial value formulation and a conserved Hamiltonian. With this new understanding it seems natural to study time dependent solutions representing the the decay of an unstable D-branes. In this paper we study such solutions using level truncation of mode oscillators in the lightcone basis, ?nding both homogenous solutions by perturbatively expanding the string ?eld in modes ent , and inhomogenous solutions by integrating the equations of motion on a lattice. Truncating the theory to level (? ? in α+ oscillators, we ?nd time dependent solutions 2, 4) whose behavior seems to converge to that of earlier solutions constructed in the center of mass basis, where the cubic action contains an in?nite number of time derivatives. We further construct time-dependent inhomogeneous solutions including all ?elds up to level (? ? These solutions at the outset display rather erratic behavior due to an unphysical 2, 4). instability introduced by truncating the theory at the linear level. However upon truncating away the ?eld responsible for the instability, we ?nd more reasonable solutions which may possibly represent an approximation to tachyon matter. We conclude with some discussion of future directions. Keywords: String Field Theory, Tachyon Condensation.

Contents
1. Introduction 2. Review 3. The Linear Problem 4. Truncation in the α+ oscillators ? 4) 5. Solutions at level (2, ? 6. Conclusion 1 4 7 11 18 27

1. Introduction
String ?eld theory has seen a remarkable revival following the conjectures of Sen[1], which have revealed (upon extensive numerical work) that string ?eld theory provides a framework where quite distinct string backgrounds can be formulated in terms of a single set of underlying degrees of freedom. The most famous example of this is the closed string vacuum solution of open bosonic string ?eld theory[2]. This solution describes the state of the open string after all unstable branes in the bosonic theory are allowed to decay, and as such must represent the vacuum of the closed bosonic string without any open string excitations. Indeed, numerical analysis in the level truncation scheme has both revealed the absence of open string states[3, 4] and veri?ed that the di?erence in energies between the unstable D25-brane vacuum and the stable vacuum is precisely the energy density of the space ?lling D25-brane, within a fraction of a percent[5, 6]. For a useful review of open bosonic string ?eld theory and related work see ref.[7]. String theory must eventually be able to address cosmological questions, and hence it seems crucial to understand the role of time and dynamical solutions in the theory. String ?eld theory, incorporating some measure of background independence, seems to be a natural framework in which to contemplate these questions. One particularly simple problem is the nature of the time-dependent decay of the unstable D25-brane in the bosonic theory. Sen[8] proposed a remarkably simple boundary conformal ?eld theory describing such a process, whereby the tachyon rolls homogenously o? the unstable maximum towards the closed string vacuum, but does not cross over in ?nite time. At late times, the resulting “tachyon matter” describes a pressureless gas[9] whose cosmological implications have been explored quite extensively in the literature[?]. Surprisingly, open bosonic string ?eld theory has proven to be quite inept at recovering this physically interesting solution. Studies of the rolling tachyon process in string

–1–

?eld theory have approached the problem from many perspectives: inverse Wick rotating a marginal solution[8, 10], perturbative expansion of the ?eld equations in modes[11], and studies of time-dependence in p-adic models[10, 12]. All of this work has revealed a consistent picture, though one drastically di?erent from Sen’s: the tachyon rolls o? the unstable maximum, speeds quickly through the closed string vacuum and then far up the steep side of the potential, in fact quite a bit further than the height of the unstable vacuum, after which a sequence of oscillations of diverging amplitude ensues. Strangely, in this process the tachyon can roll arbitrarily far into the negative, unbounded side of the cubic-like potential, yet somehow feel the urge to turn around and roll back up towards the unstable vacuum! The pressure of the string ?eld does not seem to vanish at late times, but rather oscillates with diverging amplitude[12]. This type of time dependent behavior is at the very least strange, if not catastrophic. It is not di?cult to see where the problem lies. As is well known, the cubic vertex describing the interaction of the open bosonic string contains an in?nite number of derivatives in both time and space. As a result, the kinetic energy for the truncated open string theory (as well as for p-adic models) is not positive de?nite; this explains in particular how the tachyon can roll up to a height even greater than that of the unstable vacuum where it originated, yet apparently manage to conserve energy. The type of instability we’re witnessing is in fact generic in any theory whose Lagrangian depends nontrivially on any more than ?rst time derivatives, as was discovered over 1.5 centuries ago by Ostrogradski[13]. To see the problem, consider a Lagrangian which depends on coordinates, velocities, accelerations etc. up to some order N , L(q, q, ..., q (N ) ). Ostrogradski tells us we can ?nd the Hamiltonian by ˙ de?ning phase space variables,
N

Qn = q (n?1)

Pn =
k=n

?

d dt

k?n

?L ?q (k)

n = 1, 2, ...N

representing the 2N initial conditions necessary to specify a solution to the Euler-Lagrange equations. The Hamiltonian is then1 , H=
N ?1 n=1

˙ ˙ Pn Qn+1 + PN QN ? L(Q1 , ..., QN , QN )

Since Pn and Qn+1 for n < N are independent phase space variables, and nothing else in the Hamiltonian depends on Pn , it is clear that the ?rst term in this equation can be made arbitrarily negative and the Hamiltonian is unbounded from below. So truncated open string ?eld theory and p-adic models are probably sick theories, and the erratic behavior of their solutions should be no surprise2 . A recent paper by Gross and the author[16] proposed an alternative. There, we showed that by making a certain well-de?ned, unitary transformation on the mode basis for the
If the theory is nondegenerate, we can invert the relation for PN to write q (N) = Q˙N (Q1 , ..., QN , PN ) There has been some hope that the N → ∞ higher derivative limit may fail in truncated string ?eld theory and p-adic string theory, and that requirements of analyticity may constrain the initial value problem to the point where the theory has a more or less the usual canonical structure and a stable Hamiltonian. This possibility does not seem to be borne out by the analyzes of references [14, 15].
1 2

–2–

0.7

0.6

0.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 12 10 8 6 4 2 0 -2 2 4 6 0.1 8 10 0.2

0.4

0.3

a)

b)

0 -4 -2 0 2 4 6 8 10

Figure 1: a) φ graphed vertically, as a function of (x+ , x? ). x? increases to the left, and “time” x+ increases to the right. b) The pro?le of φ as a function of x? at x+ = 12. The horizontal line represents the closed string vacuum. open bosonic string—one that identi?es the lightcone component of position with the string midpoint—it is possible to render the action for cubic string ?eld theory local in lightcone time. (In the usual mode basis, both space and time are identi?ed with the open string center of mass.) Thus, it would seem that the exact string ?eld theory has both a wellde?ned initial value formulation and a Hamiltonian free of higher derivative instabilities. This suggests that the unfortunate time-dependent behavior we’re witnessing may be an unphysical artifact of the level truncation scheme, and the exact theory admits more reasonable time-dependent solutions, perhaps even ones resembling Sen’s tachyon matter. In this paper we investigate time-dependent solutions by truncating string ?eld theory in this new “lightcone” basis. To get a feeling for what is involved, consider the action truncated to level 0 in the lightcone basis: S0 =
2 dx φ(??+ ?? + 1 ?⊥ ? 1)φ + 2

2gκ ? 1 V00 ? 2 ⊥φ e 2 3

3

(1.1)

1 1 where x+ = √2 (x0 + x1 ) x? = √2 (x0 ? x1 ) and x⊥ = {x2 , x3 , ...}, g is the open string coupling, and κ, V00 are numerical constants. The ?eld φ here is a linear combination of the tachyon and lightcone derivatives of an in?nite number of massive ?elds. There are two things to notice about this action. First, the Lagrangian is ?rst order in lightcone time derivatives ?+ , which appear only in the kinetic term. Second, the action is not Lorentz invariant, as can be seen by the derivative dependence in the interaction. Lorentz invariance, in fact, is only preserved in the level truncation scheme in the in?nite level limit. The curious absence of Lorentz invariance, at the deepest level, is really connected to the fact that string theory does not admit a formulation in terms of spacetime ?elds with completely local interactions[16]; we can only achieve locality along a single null direction. Thus, the loss of Lorentz invariance at ?nite level is the price we have to pay for having a formulation of string ?eld theory which is not at the outset rendered pathological by higher derivative instabilities and the absence of a well-posed initial value problem. We can easily imagine ?nding a dynamical solution to eq.1.1 representing the decay of a D-25 brane. In fact, looking for solutions independent of x⊥ , eq.1.1 reduces to that

–3–

of an ordinary φ3 theory in two dimensions. To let the D25-brane decay, all we have to do is set boundary conditions on the initial like-like surface x+ = 0 which displace φ 1 o? of the unstable maximum φ = 0 towards the “closed string vacuum” at φ = gκ , and then integrate the ?eld equations. A numerical solution for Gaussian initial conditions is plotted in ?gure 1. The ?eld simply ?ows o? of the unstable maximum and oscillates in a completely innocuous way around the stable vacuum. Of course, this is simply a solution to φ3 theory, and we wouldn’t expect much uniquely stringy physics to emerge, but as a ?rst pass it isn’t bad. This paper is organized as follows. In section 2 we review some features of the lightcone formalism of ref.[16] which will be relevant for our analysis. In section 3 we study the e?ect of level truncation on the linear term in the string ?eld theory action. We ?nd that when truncating the linear term we both loose linear solutions to the exact theory and introduce some spurious solutions associated with α? excitations. These unstable solutions result from the fact that the truncated equations of motion “drive themselves” at their resonant frequency. In section 4 we investigate time dependent solutions representing D25-brane decay using a modi?ed level expansion where we include only α+ excitations. 2n The motivation for this particular truncation scheme is that in the in?nite level limit the solutions should behave similarly to those constructed in the center of mass basis, which contains an in?nite number of time derivatives. Indeed, in this truncation we ?nd that the tachyon quickly passes through the closed string vacuum and then up the steep side of the potential to a height greater than that of the unstable vacuum where it originated. We explain in detail how in this truncation it is possible to approximate solutions “natrual” in one basis by ?elds in the other. In section 5 we study D25-brane decay including all ?elds up to level (? ? We ?nd that, due to a spurious linear instability introduced by level 2, 4). truncation, the solutions at this level behave somewhat erratically. However, by setting the ?eld responsible for this instability to zero, we ?nd much more reasonable solutions. The picture that emerges seems to indicate that, in comparison to solutions generated by including only scalars at level ? the tachyon rolls faster towards the closed string vacuum 2, initially but then rapidly decelerates and does not become as large before turning around for the ?rst time. We end with some concluding remarks.

2. Review
Let us review the basics of the lightcone basis introduced in ref.[16, 17]. A string ?eld |Ψ is an element of the state space HBCFT of a combined matter-ghost boundary conformal ?eld theory describing an open bosonic string living on a space-?lling D-25 brane. The usual basis for HBCFT is given by the mode oscillators α? , bn , cn acting on the vacuum |k n describing the open string tachyon at momentum k (α? = p? ). Consider a change of basis 0 generated by the unitary transformation, U = exp ?p+
∞ n=1

(?1)n + (α2n ? α+ ) . ?2n 2n

(2.1)

–4–

Under this change of basis the matter oscillators and zero-modes transform as, α? ≡ U α? U ?1 = α? + cos nπ p+ ?n n=0 n n 2 ∞ n √ (?1) (α+ ? α+ ) = x+ ( π ). x+ ≡ U x? U ?1 = x+ i 2 ? 2n ?2n 2 2n n=1

(2.2)

The ghosts and other Lorentz components of α? are una?ected by this transformation. To simplify notation, we will denote all components of the matter oscillators in this basis using a tilde α? , even though only the minus component of the even oscillators is a?ected. The ? vacuum |k transforms into a state |k ′ : |k ≡ exp k+


(?1)n + α |k , 2n ?2n n=1



(2.3)

Since the transformation is unitary, this basis satis?es the usual properties, [? ? , αν ] = mη ?ν δmn αm ? ?n α? |k ′ ?n p? |k ′ (? n )+ α


= k? |k = α?n ?


= bn |k = cn |k ′ = 0


[bm , c?n ] = δmn n>0 (cn )+ = c?n
∞ n=1

(bn )+ = b?n

b0 |k ′ = 0

(2.4)

The zeroth matter Virasoro generator can be expressed, L0 =
1 2

p2 + p+

? = L0 |0 +

n=1 + π 1 2 2 p⊥ + p+ P ( 2 )

(?1)n (? + + α+ ) + α2n ? ?2n

α?n · αn ? ? (2.5)

? where P ( π ) is the momentum of the string midpoint, and L0 is simply L0 with the re2 placement α → α, |0 means we evaluate the operator at zero momentum. The zeroth ghost ? Virasoro takes the usual form, Lgh = 0
∞ n=1

n(b?n cn + c?n bn )

The BPZ inner product , satis?es the familiar relations: α? Ψ, Φ = (?1)m+1 Ψ, α? Φ ? ?m ?m

c?m Ψ, Φ = (?1)m+1 (?1)Ψ Ψ, cm Φ

b?m Ψ, Φ = (?1)m (?1)Ψ Ψ, bm Φ

(2.6)

where (?1)Ψ denotes the Grassmann parity of Ψ. The two string vertex V2 | is as in the old basis after the replacement of the oscillators and vacua with their tilded counterparts. String interactions are de?ned by the cubic vertex V3 | ∈ HBCF T ? HBCF T ? HBCF T which in the matter sector takes the form, V3m | = κ dk1 dk2 dk3 δ(k1 + k2 + k3 ) +, k1 |′ +, k2 |′ +, k3 |′
1 2 AB Vmn aA · aB ?m ?n

AB A B AB AB exp ? 1 V00 k⊥ · k⊥ ? Vm0 aA · k⊥ ? Vm0 a?,A k? ? ?m B ?m B 2

(2.7)

–5–

α ? where am = √m , the capital indices A, B refer to the state space and range from 1 to 3, and ? m the mode number indices m, n range from 1 to ∞, all repeated indices summed. Note that the exponential has no dependence on k+ , meaning that the vertex contains no derivatives with respect to x+ and is therefore local in lightcone time. The ghost component of the ? vertex in Siegel gauge takes the usual form, which we write down for reference: AB V3gh |b0 b0 b0 = ?| ?| ?| exp Xmn cA bB m n (1) (2) (3)

Of course, V3 | = V3m | V3gh |. Explicit expressions for the constants κ, V , and X (the latter two are called “Neumann coe?cients”) were calculated in ref.[18] and can be found for convenient reference for example in ref.[7]. At zero momentum, the lightcone basis is identical to the center of mass basis. Therefore, since the oscillator expression for the closed string vacuum is identical in either basis, at this level either basis would seem equally appropriate for investigating dynamics about the vacuum. For a real string ?eld |Ψ in Siegel gauge b0 |Ψ = 0, the string ?eld theory action takes the form, 2g V3 ||Ψ |Ψ |Ψ (2.8) S = Ψ|c0 (L0 + Lgh ? 1)|Ψ + 0 3 The assumption of reality amounts to the condition, Ψ, A = Ψ|A for any string ?eld |A . The equations of motion derived from eq.2.8 are,
1 2

? p2 + p+ P + ( π ) + L0 |0 + Lgh ? 1 |Ψ + gb0 Ψ| Ψ||V3 = 0 ⊥ 0 2

(2.9)

In this paper we will always be looking for time dependent solutions in Siegel gauge, so eq.2.9 will be su?cient for our purposes. These equations specify the dynamics for an in?nite number of local spacetime ?elds, so it seems impossible to generate a numerical solution which includes all of them. So we are forced to truncate the theory, including only a few of the lightest mass ?elds and setting the remaining ones to zero by ?at. The usual approach is to expand the equations in the mode basis αn , bn , cn where position x corresponds to the string center of mass, and keep ?elds in the ?rst few low-lying levels. As we’ve seen, the resulting approximate equations of motion contain an in?nite number of time derivatives, and the dynamical solutions behave very strangely. Instead, we will expand the string ?eld in the lightcone basis αn , bn , cn , and keep only the ?rst few levels. ? ? to denote the truncation of the theory to include As a matter of notation we will use (? , i) n ?elds up to level n and interactions up to level i in the lightcone basis, and reserve the notation (n, i) (no tildes) for the corresponding truncation in the center of mass basis. The equations of motion are then ?rst order on lightcone time, and we can hope that the solutions will be more reasonable. One further comment: A solution to eq.2.9 does not, in itself, necessarily represent a full solution to the equations of motion in open string ?eld theory. Eq.2.9 must be supplemented by constraints on the initial conditions, which ensure that dynamics only proceeds on

–6–

a physical submanifold in phase space where the Hamiltonian is positive (modulo the tachyon). Unfortunately, the consistency of the constraints with time evolution at ?nite level turns out to be a somewhat di?cult issue. We will discuss this more in section 5.

3. The Linear Problem
The ?rst step in constructing time dependent solutions in the lightcone basis is to understand what level truncation does to the theory at the linear level. To facilitate discussion, ? let’s introduce a little notation. Consider a projection operator, Pn , projecting onto states at level n (in the lightcone basis). It satis?es some basic properties, ? ? ? ? ? ? Pm αn = αn Pm+n Pm bn = bn Pm+n Pm cn = cn Pm+n ? ? ? ? ? ? ? ? Pn |k ′ = δn0 |k ′ [Pn , x] = [Pn , p] = [Pn , b0 ] = [Pn , c0 ] = 0

(3.1)

Similarly we have Pn (without a tilde) which projects onto level n in the center of mass basis. Now de?ne,
n

? Πn =
i=?∞

? Pi

? Πn truncates any state up to level n in the lightcone basis (or for Πn in the center of mass ? basis). Truncating the theory up to level (? , 3n) amounts to ?nding some state |Ψ = Πn |Ψ n ? which satis?es the approximate ?eld equations, ? Πn (L0 + Lgh ? 1)|Ψ + gb0 Ψ| Ψ||V3 0 =0

In principle, sending n → ∞ we recover an exact solution. Let’s consider level truncation in the free theory (g = 0). In the center of mass basis, this amounts to solving the equations, Πn (L0 + Lgh ? 1)|Ψ = (L0 + Lgh ? 1)|Ψ = 0 0 0 (3.2)

where above we noted that Πn commutes with L0 , Lgh . Of course, eq.3.2 is just the exact 0 equations of motion. Apparently, level truncation at the free level is trivial in the center of mass basis. However, this is not what happens in the lightcone basis. In this case, the truncated linear equations are,
1 2

? ? p2 + L0 |0 + Lgh ? 1 |Ψ + p+ Πn 0



n=1

(?1)n (? + + α+ )|Ψ = 0 α2n ? ?2n

(3.3)

? Since the operator multiplying |Ψ in the second term does not preserve level number, Πn does not commute with it, and eq.3.3 is not equivalent to the exact equations of motion. Since eq.3.2 and eq.3.3 are not the same, their solutions are inequivalent. This would seem problematic, since if we can’t even get the correct time dependent solutions at the linear level, we certainly shouldn’t expect to ?nd reliable ones once we turn on the coupling.

–7–

20 "Bm_inst.dat" 15

10

5

0

-5

-10

-15

-20 -20

-15

-10

-5

0

5

10

15

20

Figure 2: Spurious instability in B+ at level 2. To see the nature of this phenomenon, consider the ?rst nontrivial case, level ? There, we 2. have the string ?eld, |Ψ = i α?2 α?2 dk φ(k) + √ (B+ (k)? + + B? (k)? ? ) |k 2


(3.4)

Of course, there are other ?elds at level 2, but they are less interesting since they only couple to ?elds at higher level. Plugging eq.3.4 into eq.3.3 for n = 2, we ?nd the equations, √ 1 0 = (? 2 ? 2 ? 1)φ + 2?+ B? √ 0 = (? 1 ? 2 + 1)B+ + 2?+ φ 2 0 = (? 1 ? 2 + 1)B? 2 (3.5) For comparison, the analogous equations of motion in the center of mass basis for φcm (the cm cm tachyon) and B+ , B? are,
cm 0 = (? 1 ? 2 + 1)B+ 2 cm 0 = (? 1 ? 2 + 1)B? 2 1 0 = (? 2 ? 2 ? 1)φcm

(3.6)

Indeed, these equations are similar except for the presence of extra linear couplings. So, what are the solutions of eq.3.5? Consider for simplicity spatially homogenous time 1 dependent solutions (t = √2 (?+ + x? )). Making a plane wave ansatz for B? , we can plug x in and ?nd the general solution, B? (t) = Aei φ(t) = Ce
√ 2t

+ Be?i + De?

√ 2t 2t

√ √ i + √ Aei 2t ? Be?i 2t 2 √ √ √ √ √ √ i 1 B+ (t) = Eei 2t + F e?i 2t ? √ Ce 2t + De? 2t ? √ t Aei 2t ? Be?i 2t 2 2 2t √



(3.7) where A, B, ... are arbitrary constants, subject to reality. The ?rst pair of terms in each of these equations is the homogenous solution, corresponding to the oscillation (or decay, as

–8–

it were) of the ?elds at their natural frequency. The remaining terms above come from the linear couplings in eq.3.5 which can be interpreted as external driving forces. Speci?cally, the ?eld φ blows up exponentially at late and/or early times—corresponding to the decay of the unstable D-25 brane—but also oscillates because of its coupling to B? . B+ is more interesting. In addition to natural oscillation and tachyonic decay because of coupling to φ, we also see an awkward component where B+ oscillates with linearly diverging amplitude (see ?gure 2). This behavior is directly because of B? , which drives B+ at its natural frequency. The appearance of this solution is interesting, since there does not seem to be any analogous behavior in the center of mass basis. Are these solutions correct? To answer this question, we have to describe some mechanism for mapping these solutions to the center of mass basis, and then ask whether they satisfy the exact linear equations of motion there. A natural mapping is, ? |Ψ = Π2 |Ψcm where |Ψcm is the analogy of eq.3.4 in the center of mass basis. Expanding this out with the help of the transformation laws eqs.2.2-2.3 and matching terms, we ?nd the relations3 , 1 φcm = φ ? √ ?+ B? 2 1 cm B+ = B+ + √ ?+ φ ? 2 cm B? = B? Plugging in our solutions eq.3.7,
cm B? = Aei √ 2t √ 2t √

1 2

2 ?+ B?

(3.8)

+ Be?i + De?

2t

φcm = Ce



2t 1 + (F ? 2 B)e?i √ 2t √ √ i ? √ t Aei 2t ? Be?i 2t 2

cm 1 B+ = (E ? 2 A)e

√ i 2t

(3.9)

cm We see that B? behaves like a massive ?eld and φcm behaves like the tachyon, as they cm should. However, B+ still has the diverging oscillatory part noted earlier. This piece obviously does not satisfy the equations of motion, so it must be an unphysical solution introduced by level truncation. The only way to get rid of this unwanted solution is to set B? = 0, so the exact solution describing the natural oscillation of B? has been lost. What happens when we include higher levels? Basically, extra linear couplings emerge and both spurious and lost solutions proliferate. To see what happens, it is useful to proceed more abstractly. We make a two claims: Claim 1: Given an exact solution to the linear equations at level n in the center of mass ? basis, Πn |Ψcm = |Ψcm , the ?eld |Ψ = Πn |Ψcm satis?es

? Πn (L0 + Lgh ? 1)|Ψ = 0 0
Note that we could have de?ned the mapping to be Π2 |Ψ = |Ψcm . This turns out to give slightly di?erent relations, di?ering from eq.3.8 by some extra terms proportional to B? . However, the discussion goes through similarly regardless of these de?nitions.
3

–9–

if an only if,
n

? p+ Πn
j=1

cos

πj + α |Ψ = 0 ? 2 j

? Claim 2: Given an solution Πn |Ψ = |Ψ to the truncated equations of motion in the lightcone basis, ? Πn (L0 + Lgh ? 1)|Ψ = 0
0

the ?eld |Ψcm = Πn |Ψ satis?es the exact equations of motion if and only if,
n

? p+ Πn (1 ? Πn )

cos
j=1

πj + α |Ψ = 0 ? 2 ?j

The proof of both of these statements is straightforward. Claim 1 tells us about lost solutions in the lightcone truncation scheme, and claim 2 tells us about spurious solutions. The ?rst thing to notice about these claims is that the constraints are automatically satis?ed if |Ψ contains no excitations proportional to α? ; hence for such states there are ?2n no spurious or lost solutions. One might then wonder whether these constraints have any solutions with α? excitations at all. Surprisingly, the answer is no. While it is fairly ?2n clear to us that this is the case, we do not know of a demonstration simple enough to be worth presenting here since for our purposes level ? will be su?cient. 2 If at no level in the truncation scheme do the correct α? solutions emerge, in what ?2n sense can we say that the truncated equations of motion converge to the exact ones in the in?nite level limit? The answer is that spurious solutions are always associated with ?elds at the highest levels in the particular truncation, and that by proceeding to a higher truncation these spurious solutions disappear, only to be replaced by new ones at even higher levels. To see how this happens, consider level ? There, we ?nd that B? sources 4. the equations of motion of a new ?eld associated with α+ α? , and this ?eld in turn ? ?2 ? ?2 sources B+ . These new couplings conspire to cancel the unwanted diverging oscillations we found in B+ earlier. However, a new unphysical solution is introduced at level ? by the 4 + coupling of B? , via φ, to the ?eld associated with α?4 . This unwanted solution, then, only ? disappears at level ? upon introduction of α+ α? . Clearly, at any level the oscillation of 6 ? ?4 ? ?2 B? will introduce unphysical solutions for the ?elds at the highest levels, so at no stage is it consistent to turn on B? . However, it is still clear that the correct solutions are, in some sense, emerging as the level of truncation is increased. While at level ? it seems okay to simply set B? = 0, at higher levels ?elds with α? 2 ? ?2n excitations proliferate and their e?ect on the equations of motion cannot be ignored. In this case, it is probably not adequate to do a straightforward level truncation. Rather, if one wanted physically correct linear equations up to level n, one would need to include select ? additional ?elds at higher levels, with their additional linear couplings, so that spurious solutions at lower levels would cancel out. At level ? for example, this could be achieved 2, + ? by simply including the ?eld associated with α?2 α?2 . ? ?

– 10 –

4. Truncation in the α+ oscillators
As a ?rst look into level truncation in the lightcone basis, it is natural to ask whether we can ?nd time dependent solutions resembling the known (though admittedly problematic) solutions constructed in the center of mass basis, where the interaction vertex contains an in?nite number of time derivatives. Indeed, it seems curious that a simple change of basis could radically alter the apparent solution space of the theory and the qualitative behavior of the dynamics. Of course, there is no real paradox here, since string ?eld theory truncated in the center of mass basis is not equivalent to the theory truncated in the lightcone basis. However, since both truncations are an approximation to the same exact theory, truncated solutions in one basis should emerge from a suitable approximation in the other basis. It is interesting to see how this comes about, and may help us appreciate the degree to which the erratic time dependent behavior in the center of mass basis should, or should not, be taken seriously. As a start we investigate the rolling of the pure tachyon in the center of mass basis. The tachyon in the lightcone basis is an in?nite linear combination of states excited by α+ oscillators: ??2n |φcm = dkφcm (k)|k = dkφcm (k) exp ?k+
∞ n=1

(?1)n + α ? |k 2n ?2n



It is natural to suppose that if we truncate the theory in the lightcone basis in such a way that only ?elds corresponding to α+ excitations are included, we should recover solutions ? ?2n resembling the old solution in the center of mass basis. More precisely, the in?nite level limits of α+ truncation in the center of mass and lightcone bases should agree with each ?2m other. The ?rst step is to ?nd an appropriate time dependent solution for the pure tachyon at level 0 in the center of mass basis, which will serve as a standard for comparison when truncating α+ excitations in the lightcone basis. Since the level 0 action for the pure ? ?2n tachyon contains an in?nite number of time derivatives, it does not have a well-de?ned initial value formulation and it is not a priori clear how to construct generic time dependent solutions. In the past[11, 10], a useful approach has been to start with a homogenous time dependent solution in the free theory, whose equations of motion are only second order in time derivatives, and add nonlinear corrections to this perturbatively. The level 0 equations of motion for the tachyon are,
1 (? 2 ? 2 ? 1)φcm + gκe 2 V00 ?
9/2

1

2

e 2 V00 ? φcm

1

2

2

=0

(4.1)

1 where κ = 326 , V00 = 2 ln 27 and g is the open string coupling, which we will set to one. 16 At the linear level, an interesting homogenous time dependent solution is,

φcm = φcm e 1



2t

This corresponds to placing the tachyon in?nitesimally close to the unstable vacuum at t → ?∞ and then letting it roll o? the unstable maximum in the approximation that

– 11 –

1.8

1000

1.6 800 1.4 600 1.2 400

1

0.8

200

0.6 0 0.4 -200 0.2

a)

0 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

b)

-400 0 0.5 1 1.5 2 2.5 3 3.5

Figure 3: Time evolution of the pure tachyon at level 0 in the center of mass basis, from a) t = ?3, .6 and b) t = 0, 3.35. These two pictures capture the ?rst two oscillations of the tachyon about the closed string vacuum. The two horizontal lines in ?gure 3a represent the closed string vacuum and the classical turning point in φ3 theory. the potential is quadratic. Di?erent choices of φcm corresponds to time translations of 1 the solution; we will set it to a convenient value in a moment. To solve the full nonlinear √ equations of motion, we make a perturbative ansatz, expanding the tachyon in modes en 2t : φ
cm

=

∞ n=1

φcm en n



2t

.

Plugging this in to eq.4.1 implies a recursive formula for the φn s, φcm n
1 2

gκe? 2 V00 n =? n2 ? 1

1

2

n?1 k=1

φcm e? 2 V00 k φcm e? 2 V00 (n?k) k n?k

1

2

1

2

(4.2)

The factors e? 2 V00 n , originate from the in?nite number of time derivatives in the interaction. The suppression of these factors will make the higher φn s much smaller than they would be in, say, φ3 theory. Since the higher modes are responsible for turning the tachyon around after it rolls through the closed string vacuum and the ensuing oscillations, the fact that the φn s are small means that the tachyon will turn around in each oscillation comparatively late; so we would expect the tachyon to roll back and forth about the vacuum in a sequence of diverging oscillations. This is indeed what we ?nd upon explicit summation of the series as shown in ?gure 3. In ?gure 3a, we see that at the ?rst peak of oscillation the tachyon rolls on the static potential V (φ) = ? 1 φ2 + gκ φ3 to a height 64 times the energy 2 3 di?erence ?E = .0347 between the stable and unstable vacua. Figure 3b shows the peak of the second oscillation of the tachyon, which reaches a staggering height of 1.6 × 1010 times ?E! Clearly, if such time dependent behavior was characteristic of string theory we would be in big trouble. In ref.[11] a similar time dependent solution for the tachyon was √ constructed by expanding in cosh n 2t modes. This is slightly more complicated but has the advantage of giving control over the position of the tachyon (and the velocity, which vanishes) at t = 0. The solution eq.4.2, however, will be su?cient for our purposes.

– 12 –

Now we turn towards constructing analogous time dependent solutions in the lightcone basis. First consider level ? where we have only the tachyon-like ?eld φ, 0, |Ψ0 = dkφ(k)|k


As explained in the introduction, the action for φ at level zero is simply that of φ3 theory if we restrict to spatially constant solutions. In this case, we can actually ?nd an exact solution for the decay process, 3 1 φ(level 0) = (4.3) t 2gκ cosh2 √ 2 This solution describes φ rolling o? the unstable maximum, bouncing o? the steep side of the cubic potential, and then approaching the unstable maximum again at late times. We have placed the “bounce” at t = 0 corresponding to the choice, 6 φ1 = φcm = 1 gκ ?xing the arbitrary constant encountered earlier. To ?nd a solution more closely resembling eq.4.2 in the lightcone basis, we need to include additional ?elds corresponding to higher α+ mode excitations. Keeping α+ ? ?2n ? ?2n excitations up to level ? requires us to include the extra ?eld B+ , which we met in eq.3.4: 2 i dkB+ (k)? + |k ′ α?2 |Ψ2 = |Ψ0 + √ 2 What are the relevant truncated equations of motion for φ, B+ ? The correct prescription is to write the cubic action including all ?elds up to level ? (including B? ). Then, we 2 cm , B cm , and then set everything vary the action with respect to the center of mass ?elds φ ? in the resulting equations of motion to zero except φ, B+ . It is crucial that we vary the action with respect to the center of mass ?elds since ultimately we are interested in the approximate equations of motion for the pure tachyon, which is obtained by varying the action with respect to φcm . Thus, the relevant equations of motion are, 0= = δS δφcm (x)
φ,B+ =0;B? ...=0

δS δS 1 + √ ?+ δφ(x) 2 δB+ (x) =
φ,B+ =0;B? ...=0

φ,B+ =0;B? ...=0

δS 0= cm δB? (x)

δS δB? (x)

(4.4)
φ,B+ =0;B? ...=0

Here we used the chain rule and the transformation law eq.3.8 relating ?elds in the center of mass basis to the lightcone basis. Explicitly including interactions up to level ? and 4 setting the transverse momenta to zero, these equations of motion are,
2 0 = (?+ ?? ? 1)φ + gκ φ2 + GφφB+ φ?? B+ + G1 + B+ ?? B+ B+ + G2 + B+ (?? B+ )2 φB φB

1 2 2 + √ ?+ GB+ φφ φ?? φ + G1 + φB+ ?? φB+ + G2 + φB+ ?? φ?? B+ + G3 + φB+ φ?? B+ B B B 2 √ 0 = ?(?+ ?? + 1)B+ ? 2?+ φ + gκGB? φB+ φB+ (4.5)

– 13 –

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

Figure 4: The level ? approximation to the rolling of the pure tachyon in the center of 2 mass basis, sandwiched between the level ? pro?le in the lightcone basis and the level 0 0 pro?le in the center of mass basis. The perturbative series eq.4.6 does not converge past t ≈ ?.22. The horizontal lines represent the level 0 closed string vacuum and the turning point in the φ3 potential. The eight Gs above are various powers and linear combinations of the Neumann coe?cients. We can easily solve these equations by making a perturbative ansatz: φ=
∞ n=1

φn e

√ n 2t

B+ =

∞ n=1

B+,n en



2t

(4.6)

where for consistency at the linear level we have, 1 B+,1 = ? √ φ1 2 (4.7)

Plugging this ansatz into eq.4.5 we derive recursive formulas for the various coe?cients in the usual way. Summing the series, we ?nd the tachyon pro?le shown in ?gure 4, which for comparison we have graphed alongside the level ? pro?le of φ3 theory and the level 0 pro?le 0 in the center of mass basis. We see that, already at level ? the rolling of the tachyon looks 2, much more like it does at level 0 in the center of mass basis. Strictly speaking, however, we cm should be comparing this result to the analogous computation including B+ in the center cm of mass basis. However, at least in the region of convergence of eq.4.6, including B+ has negligible e?ect on the level 0 pro?le. We emphasize that the equations of motion we have just solved, 0= δS δφcm (x)
cm cm φcm ,B+ =0;B? ...=0

=

δS cm δB? (x)

= ...
cm cm φcm ,B+ =0;B? ...=0

(4.8)

– 14 –

are not equivalent to 0= δS δφ(x) =
φ,B+ =0;B? ...=0

δS δB? (x)

= ...
φ,B+ =0;B? ...=0

(4.9)

even when both are expressed in terms of ?elds in the same basis. The second set of equations, in particular, contain no ?+ derivatives in the interaction when written in the lightcone basis. These equations di?er from each other by terms proportional the B? equation of motion—and, at higher levels, those for other α? ?elds. If we allow the α? ? ?2n ? ?2n ?elds to be nonzero and separately solve their equations of motion, then eq.4.8 and eq.4.9 become equivalent. However, this suggests the following: Since the second set of equations, eq.4.9, contain no lightcone time derivatives in the interaction, we would expect their solutions to be more nicely behaved— resembling that of φ3 theory at level ? This suggests that it might be 0. possible to approximate these more well-behaved solutions by truncating α+ modes in ?2n cm the center of mass basis. At level 2 we must consider φcm , B+ . By a similar prescription to eq.4.4, the relevant equations of motion are, 0= = 0= δS δφ(x)
cm cm φcm ,B+ =0;B? ...=0

1 δS δS ? √ ?+ cm δφcm (x) δB+ (x) 2 δS δB? (x) =
cm cm φcm ,B+ =0;B? ...=0

cm cm φcm ,B+ =0;B? ...=0

δS cm δB? (x)

(4.10)
cm cm φcm ,B+ =0;B? ...=0

As usual, we solve these equations by making a perturbative ansatz. The resulting tachyon pro?le is shown in ?gure 5a, along with the pro?les at level 0, eq.4.2, and at level ? with 2 cm makes the tachyon turn around φ, B+ in the lightcone basis. We can see that including B+ well before it does at level 0. Figure 5b shows the evolution of φcm at later times. While the subsequent oscillations are still pronounced, they are much less virulent then they were at level 0; the peak of the second oscillation reaches a potential height of only 2× 107 × ?E, about 1000 times smaller than the height of the second peak at level 0. Remarkably, even computing in the center of mass basis, truncating the theory in a particular way can yield time dependent solutions whose qualitative behavior is substantially improved over the level 0 solution. Thus it seems possible to approximate solutions in one basis by making an appropriate truncation in the other. However, this brings up a natural question: Precisely in what sense does level truncation in α+ modes in the lightcone basis converge to the equations ? ?2n of motion eq.4.1 for the tachyon in the center of mass basis? To explain this, consider the level ? equations of motion for φ, B+ , eq.4.5. To compare this with eq.4.1, we substitute 2 1 cm in center of mass ?elds via the transformation law φ = φcm , B+ = B+ ? √2 ?+ φcm and cm suppose B+ can be ignored. Then eq.4.5 results in the following approximate equations for the pure tachyon φcm : 1 0 = (?+ ?? ? 1)φcm + gκ (φcm )2 ? √ (GφφB+ ? GB+ φφ )φcm ?+ ?? φcm 2 (4.11)

– 15 –

1.8

300

1.6 200 1.4 100 1.2 0

1

0.8

-100

0.6 -200 0.4 -300 0.2

a)

0 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

b)

-400 0 0.5 1 1.5 2 2.5 3

Figure 5: a) The level 2 approximation to the rolling of φ in the lightcone basis, graphed in comparison to the φ, B+ solution at level ? at the bottom, and the level 0 tachyon solution 2, the center of mass basis, at the top. b) The level 2 approximation to φ (dotted line) for later times 0 < t < 2.7, graphed in comparison to level 0 (solid line). We have not graphed the φ, B+ solution out here since the series does not converge. 1 + √ GB+ φφ ?+ φcm ?? φcm + ... 2 (4.12)

where +... are nonlinear terms with two ?+ derivatives, which we will not need for the sake of the present discussion. Since the nonlinear terms in eq.4.12 contain only up to 2 lightcone derivatives in the interaction, whereas eq.4.1 contains an in?nite number, it is clear that they must be related by some sort of polynomial expansion of the in?nite di?erential operator exp(?V00 ?+ ?? ). Expanding the derivatives in the interaction term out to ?rst order in ?+ , eq.4.2 becomes, 0 = (?+ ?? ? 1)φcm + gκ (φcm )2 ? 4V00 φcm ?+ ?? φcm ? 2V00 ?+ φcm ?? φcm + ... This leads to the identi?cations, 1 1 4V00 ? √ (GφφB+ ? GB+ φφ ) 2V00 ? ? √ GB+ φφ 2 2 Noting that 1 33 13 23 GφφB+ = √ (2V02 ? V02 ? V02 ) 2 these equations amount to the identi?cation, GB+ φφ = ?GφφB+

1 33 13 23 V00 ? √ (2V02 ? V02 ? V02 ) 2 2
33 13 23 Noting that V0n + V0n + V0n = 0, this can be rewritten,

ln

1 33 27 ? ?3 ? √ V02 16 2


This is the ?rst approximation to the exact identity[16], ln 27 = ?3 16 (?1)n 33 √ V0,2n 2n n=1 (4.13)

– 16 –

n, (φcm ) n level 2k

2 4

3 8

4 14

5 22

6 32

7 44

... ...

n n(n ? 1) + 2

Table 1: This table shows what level in α+ excitations we need to include to get φcm s, ? ?2k n up to some n, resembling those in the center of mass basis within 10%. So it seems there are two levels of approximation going on. The ?rst is polynomial approximation to the in?nite order di?erential operator exp(?V00 ?+ ?? ); the second is an approximation to the coe?cients of this polynomial through eq.4.13. Using this knowledge we can get an estimate for how a solution including α+ modes up to level n converges to ??2n ? the center of mass solution, eq.4.1. De?ne Pk (x) to be the kth order polynomial approx? imation to ex , expanded around x = 0. Roughly speaking, the level 2k approximation to eq.4.1 is, 0 = (?+ ?? ? 1)φcm + gκPk (?V00 ?+ ?? ) [Pk (?V00 ?+ ?? )φcm ]2 + ... Here +... includes interactions with ?elds at higher levels and some higher derivative (φcm )2 nonlinear terms, which we will ignore. Also, we have assumed that V00 has been adequately approximated by the identity eq.4.13. Making a perturbative ansatz, the solution is, φcm n gκPk (? 1 V00 n2 ) 2 =? n2 ? 1
n?1 j=1

φcm Pk (? 1 V00 j 2 )φcm Pk (? 1 V00 (n ? j)2 ) n?j j 2 2

It is clear that for large enough n the coe?cients in this solution will look very di?erent 1 from eq.4.2. Only for the ?rst few modes, where Pk (? 2 V00 n2 ) is a good approximation 1 2 to e 2 V00 n do the coe?cients look similar. So, if we want a good approximation to φn up to some integer n, how many α+ modes do we need to include in the lightcone basis? ? ?2n Assuming we want to reproduce the φcm to an accuracy of 10%, the answer is shown n in Table 1. Roughly speaking, as we consider higher modes we must increase the level of truncation quadratically to get answers similar to those in the center of mass basis. Similar comments apply if we want to approximate solutions in the lightcone basis by level truncation in the center of mass basis. So we have studied two types of time dependent solutions describing the decay of an unstable D-brane: those arising naturally in the center of mass basis and those arising naturally from the lightcone basis. We have seen that, with a suitable truncation and de?nition of the equations of motion, it is possible to approximate the “natural” solution in one basis by ?elds in the other. However, this brings up an obvious question: which solution is more representative of the actual time dependent behavior of the string ?eld as the D-brane decays? At this stage that we have seen no reason to favor one type of solution over the other. The di?erence between the lightcone and center of mass solutions really comes down to the fact that they solve di?erent equations of motion, despite the fact that the ?eld content of the solutions is identical. Indeed there is an intrinsic ambiguity in de?ning the equations of motion in the α+ truncation scheme. In order to derive ??2n + equations of motion for α?2n ?elds, we must vary the action with respect to α? ?elds, ? ? ?2n

– 17 –

but we do not separately account for the equations that the α? ?elds must satisfy— ? ?2n ? rather, we have simply set α?2n ?elds to zero. This means that the equations of motion ? + in the α?2n truncation scheme are really only de?ned up to terms proportional to the ? α? equations of motion. This ambiguity accounts of the profound di?erence between the ??2n solution at level 0 in the center of mass basis and that at level ? in the lightcone basis. 0 To resolve this ambiguity one might propose to include α? ?elds and solve their ? ?2n equations of motion. We will do essentially this (also including other ?elds out to level ? in the next section. However, we do not believe this is likely to give a more realistic 2) picture of D-brane decay. The reasons are twofold: First, α? excitations create spurious ? ?2n solutions at the linear level in the lightcone basis—as we saw in section 3. Second, more fundamentally, α? ?elds infect solutions with negative energies. We will have more to ? ?2n say on this in the next section, but it su?ces to remark that these unphysical negative energies can only be projected out with a consistent imposition of constraints. That being said, we believe that the φ3 solution at level ? probably more closely resembles the true 0 decay process. The reason is that eq.1.1 possesses two features we know must be true of the exact theory: it is local in lightcone time and has a Hamiltonian free of all instabilities except the physical one associated with the tachyon.

5. Solutions at level (? ? 2, 4)
We now turn towards ?nding time dependent solutions representing the decay of the D-25 brane vacuum, including all relevant ?elds up to level (? ? For our purposes it will be 2, 4). su?cient to restrict to solutions which depend only on x+ (time) and x? , but not on the ? remaining 24 transverse directions. To ?nd the most e?cient consistent truncation at level ? we require that the string ?eld |Ψ satisfy the following properties: 2, 1) 2) 3) 4) |Ψ is real and has ghost number 1. b0 |Ψ = 0 |Ψ is twist even, i.e. only has excitations at even levels. |Ψ is a singlet under the transverse subgroup SO(24) of the Lorentz group.

Condition 1 is simply a property of a physical string ?eld. Condition 2 is our choice of gauge. Condition 3 follows from the well-known fact that twist odd ?elds only appear quadratically in the string ?eld theory action, so setting them equal to zero solves their equations of motion identically. Condition 4 follows from symmetry; if we require |Ψ to be a singlet initially, it will remain a singlet under time evolution since the solutions we study have zero transverse momentum. Imposing these conditions, we ?nd that the string ?eld at level ? takes the form, 2 |Ψ = 1 1 α?1 ? ?1 α?1 ? ?1 α?1 ? ?1 dk φ(k) + D (k)? + α? + √ D⊥ (k)? i αi + √ (D++ (k)? + α+ + D?? (k)? ? α? ) α?1 ? ?1 48 2 i + √ (B+ (k)? + + B? (k)? ? ) + β(k)c?1 b?1 |k ′ α?2 α?2 (5.1) 2

– 18 –

The eight spacetime ?elds above are expressed in the momentum representation and are real in the momentum space sense, e.g. φ(k) = φ(?k). Actually, there is another simpli?cation coming from the fact that we include interactions only up to level ? we can consistently set 4: D++ = 0, and as a result D?? decouples from the dynamics. We can choose D++ = 0 since the only nonlinear term in the D++ equation of motion is proportional to φD++ ; the other possible term B+ B+ only enters at level (? ? By simple matching of Lorentz indices, it 2, 6). is easy to see that all other couplings are ruled out by the absence of ?+ derivatives in the vertex. Also by matching Lorentz indices it is possible to see that D?? decouples when D++ = 0, since the only way D?? e?ects the dynamics is through a D?? D++ term in the equation of motion for φ; other couplings do not enter until (? ? Hence we can set 2, 6). D++ = 0 and ignore D?? , bringing the total to six relevant spacetime ?elds at √ level (? ? 2, 4). A natural approach to ?nding a solution is to expand the ?elds in modes e 2nt , as we did in the last section. This method has the advantage of giving a spatially homogenous solution, like Sen’s rolling tachyon[1]. However, we have found that the radius of convergence of this solution at level (? ? is not su?ciently large to follow the interesting dynamics—the 2, 4) series diverges before the tachyon turns around for the ?rst time. One might attempt to perform some sort of Pad? resummation, but we have found this to be unreliable. e However, in the lightcone basis we have an obvious alternative: we can make use of the fact that the theory possesses an initial value formulation. In particular, we can specify initial conditions for the ?elds at x+ = 0, and then integrate the equations of motion to ? generate a full time dependent solution. To see how this is done, let us write the equations of motion for the eight ?elds eq.5.1 in an abbreviated notation, √ 0 = (?+ ?? ? 1)φ + 2?+ B? + gφint [φ, B+ , Fi ] √ int 0 = (?+ ?? + 1)B+ + 2?+ φ + gB+ [φ, B+ , Fi ] 0 = (?+ ?? + 1)Fi + gFiint [φ, B+ , Fi ] (5.2) where Fi = B? , D⊥ , D , β, D?? , D++ , and φint ... denote the nonlinear terms in the equations of motion. There are over a hundred of these terms at level (? ? 2, 4)—we will not write them explicitly—but we emphasize that they contain no lightcone time derivatives ?+ . We can integrate the equations of motion eq.5.2 stepwise on a lattice, de?ning, x+ = n? ? φn (x? ) = φ(n?, x? ) B+,n (x? ) = B+ (n?, x? ) Fi,n (x? ) = Fi (n?, x? )

where ? is a su?ciently small number. We can then use eq.5.2 to generate the ?elds at the n + 1st time from the ?elds at the nth time: Fi,n+1 (x? ) = Fi,n (x? ) ? ? φn+1 (x? ) = φn (x? ) ? ?
? ? x? x? 0 x? x? 0

da Fi,n (a) + gF int [φn , B+,n , Fi,n ](a) √ B?,n+1 (a) ? B?,n (a) 2 + gφint [φn , B+,n , Fi,n ](a) ? φn (a) ? √ φn+1 (a) ? φn (a) int + gB+ [φn , B+,n , Fi,n ](a) + B+,n (a) 2 ? (5.3)

da
x? x? 0

B+,n+1 (x ) = B+,n (x ) ? ?

da

– 19 –

In these expressions we have already ?xed a boundary condition which requires the ?elds to be constant in time x+ along the light-like surface x? = x? . This choice of boundary ? 0 condition corresponds to the familiar p? = 0 ambiguity occurring in any relativistic ?eld theory whose initial value formulation is constructed in the lightcone frame. For most of our discussion we will take x? to be su?ciently far in the past that we can assume that 0 the ?elds vanish. Note that these formulas must be solved in the order presented: The ?rst formula gives the Fi at n + 1st time given the ?elds at n; the second gives φ at n + 1 given the ?elds at n and B? at n + 1 (which we found in the ?rst formula); the third gives B+ at n + 1 given the ?elds at n and φ at n + 1 (which we found from the second formula). Thus, taking ? to zero, we generate a solution by specifying the ?eld con?guration at x+ = 0. ? This technique gives us a way of generating a wide class of solutions from generic initial conditions. Unfortunately, except for very special cases, the resulting solutions are not homogenous, i.e. depend on both x+ and x? . Nevertheless, we have found that ? generally these types of solutions look similar to the homogenous (perturbative) solution when the radius of convergence of the latter allow them to be usefully compared—at least for initial conditions that are a su?ciently “small” perturbation of the open string vacuum. For de?niteness, in most of our discussion we will choose the initial conditions, φ(x? ) = H 4 cosh2 1 x? 2 H 1 B+ (x? ) = ? √ (1 + tanh 2 x? ) 2 2 Fi (x? ) = 0 (5.4)

where H is “su?ciently small” (for our purposes we choose H = .1). For most purposes, this choice of initial conditions seems irrelevant; any small perturbation of the D-brane generates a similar decay as long as the tachyon is pushed towards the closed string vacuum. Of course, if the initial conditions are too “large,” the resulting solution will depend more sensitively on the exact pro?le of the ?elds at x+ = 0. For the most part, however, we are ? interested in generic features of the decay resulting from small perturbations of the D-25 brane, and not on detailed features of the time evolution of certain large ?eld excitations. Therefore “small” initial conditions will serve us well. That being said, eq.5.4 has some noteworthy advantages as a choice of initial conditions. 1 x First, for early times t = √2 (?+ + x? ) eq.5.4 generates a homogenous solution in the linear theory, and as a result the ?rst wave-crest in the nonlinear solution coincides roughly with t = 0. If, for example, we had chosen gaussian initial conditions, the ?rst wave crest would advance at the speed of light along the line x? = 0. Still, however, the solution would possess all of the principal qualitative features we will discuss. The second advantage of eq.5.4 is that they solve the truncated constraints for string ?eld theory in Siegel gauge. As mentioned earlier, a time dependent solution of the Siegel gauge equations of motion eq.2.9 does not a priori represent a full solution to the string ?eld equations QB Ψ+gΨ?Ψ = 0. The other “half” of the string ?eld equations, not accounted for by the Siegel gauge equations of motion, amount to a set of constraints which the string ?eld must satisfy at all times (particularly x+ = 0), like Gauss’s law in electrodynamics. In ref.[16] the constraints were ? found to take the form, 0 = b0 (c( π )P + ( π ) ? iπb ( π )x+ ( π )′ )QB Ψ 2 2 2 2 (5.5)

– 20 –

where c( π ), P ( π ), πb ( π ), x( π )′ are the c ghost, string momentum, b ghost momentum, and 2 2 2 2 worldsheet derivative of the string position evaluated at the string midpoint, respectively. In principle, the constraints eq.5.5 play an extremely important role in the theory: they restrict dynamics to a submanifold in phase space where the Hamiltonian is positive (modulo the tachyon). This essentially amounts to a ?eld-theoretic restatement of the no-ghost theorem of ?rst quantized strings. It is not di?cult to see that constraints are necessary to give physically acceptable solutions. Even at the linear level and ignoring the tachyon, we can see from the Siegel gauge action (at level ? 2), Sfree = dx
1 2 1 φ( 2 ? 2 ? 1)φ + 1 D ( 1 ? 2 + 1)D + 1 D⊥ ( 1 ? 2 + 1)D⊥ 2 2 2 2 1 2

1 +D++ ( 1 ? 2 + 1)D?? ? B+ ( 2 ? 2 + 1)B? ? 2

β( 1 ? 2 + 1)β + 2

√ 2φ?+ B?

that the Hamiltonian is unbounded from below, containing unphysical “ghost” degrees 1 of freedom like β and √2 (B+ + B? ) whose kinetic and potential energies are negative de?nite. Some extra conditions are necessary to ensure that the dynamics generated by the Siegel gauge equations of motion is not completely unstable. Thus it seems important that the initial conditions eq.5.4 satisfy the constraints and therefore set the solution on the right track. Unfortunately, however, we will ?nd that the qualitative behavior of our solutions, in particular the unphysical negative energy instabilities we will encounter, are largely insensitive to the fact that eq.5.4 satis?es the constraints. The reason is that, at ?nite level, the constraints are inconsistent with the equations of motion; even if the string ?eld satis?es the constraints initially, it will in general fail to upon time evolution at ?nite level. Indeed, the consistency of the constraints eq.5.5 with time evolution in the exact theory is a subtle issue relying on certain nontrivial identities between the Neumann coe?cients[16] which only emerge properly in the in?nite level limit. Later, we will have to deal with unphysical instabilities in a more direct fashion, “freezing out” problematic negative energy ?elds by hand. So let us present the level (? ? time dependent solution derived from these initial 2, 4) conditions. Figure 6a shows the dynamics of φ as a function of x+ , x? , and ?gure 6b shows ? more speci?cally the pro?le at x+ = 2.5. At early times the solution does the expected ? thing; it rolls o? the open string vacuum, bounces o? the φ3 potential wall and turns around. What happens after this is more disturbing: the tachyon rolls back over the open string vacuum into the unbounded side of the φ3 potential, but somehow feels the urge to turn around again and shoots quickly back towards the open string vacuum. At later times the solution becomes increasingly unstable, oscillating back and forth with increasing frequency and amplitude as shown in ?gure 6c. After a certain point the gradients become too large to follow the solution reliably, but we have no reason to believe that the dynamics ever stabilizes. We hope, of course, that this solution does not in any way approximate the dynamics of D-brane decay. This erratic behavior clearly indicates that we are dealing with a system of equations which couple to unbounded negative energies. What mechanism is responsible for this erratic behavior? The problem seems closely related to the unphysical linear instability we discovered in section 3, where the linear motion of B? drives B+ at its

– 21 –

1 "fin_phi.dat" "phi.dat" 0.8

0.6 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 10 8 6 0 4 0.5 1 2 1.5 0 2 -2 -4 2.5 -0.6 -0.4

0.4

0.2

0

-0.2

a)
50 40

b)

-0.8 -4 -2 0 2 4 6 8 10

"fin_phi.dat"

30

20

10

0

c)

-10 -4 -2 0 2 4 6 8 10

Figure 6: a) The level (? ? approximation to the rolling of φ in the lightcone basis, graphed 2, 4) + , x? for ?4 < x? < 10 and 0 < x+ < 2.5. b) The pro?le of φ at x+ = 2.5, as a function of x ? ? ? c) at x+ = 3.6. ? resonant frequency. In ?gure 7 we have graphed the pro?les of B+ , φ, B? at x+ = 2.5; at its ? peak B+ acquires a value approximately ten times that of other ?elds, strongly suggesting that B+ is the principal victim of the instability. The fact that the linear motion of B? is the cause of the instability is supported by two bits of evidence. First, if the linear coupling of B? to the φ equations of motion is turned o?, B+ is not driven to large values and the resulting solution is much more well-behaved. Second, if the nonlinear couplings in the B? equations of motion are turned o?, but we set nonvanishing initial conditions for B? , the resulting purely linear motion of B? still causes the instability. However, nonlinear e?ects do seem to play an important role. For example, we saw in section 3 that including an extra ?eld at level ? α+ α? |k ′ , makes the linear instability of B+ disappear. However, adding 4, ? ?2 ? ?2 + ? ′ onto our solution does not make it more well-behaved. The reason, we think, α?2 α?2 |k ? ? is that by the time the instability sets in, the tachyon is far o? shell and the cancellation of the unphysical B+ instability no longer occurs. Another important nonlinear e?ect comes from the φ?? B+ coupling in the φ equation of motion. If this term is set to zero, the instability does not occur in its present form. Since the linear instability introduced by B? is an artifact of the level truncation scheme, it seems clear that the instability of the level ? solution is also an unphysical 2 artifact of level truncation. However, limited at present to level ? it seems important to 2, explore possible modi?cations of the equations of motion at this level which have a chance of painting a more realistic picture of D-brane decay. We propose to set all nonlinear

– 22 –

9 8 7 6 5 4 3 2 1 0 -1 -4 -2 0 2 4 6 8 10

Figure 7: The level (? ? pro?les for B+ (dashed line), φ (solid line), and B? (dotted line) 2, 4) at x+ = 2.5. ? couplings in the B? equation of motion to zero. Then, given our initial conditions eq.5.4, this has the e?ect of setting B? = 0. Remarkably, with this modi?cation alone the solutions we ?nd will paint a consistent and believable picture of D-brane decay beyond level ? 0. + = 2 generated from With B? = 0, we have plotted in ?gure 8 the pro?le of φ at x ? the initial conditions eq.5.4. For the purposes of comparison, we have graphed it alongside another pro?le generated from the equations of motion obtained upon setting B+ = 0. When setting B+ = 0, the equations of motion simplify to that of an ordinary cubic theory coupling three scalars φ, β, A = ? 1 D = √1 D⊥ with no derivative dependence 2 48 in the interaction. The linear and nonlinear couplings of B+ can be thought of as more characteristic of string theory, so the B+ = 0 pro?le provides a useful gauge for identifying uniquely stringy behavior. We see two characteristic features emerge from ?gure 8: First, initially φ rolls somewhat more quickly towards the closed string vacuum relative to the B+ = 0 pro?le. Second, φ does not become as large before turning around. Relative to the B+ = 0 pro?le, the ?rst peak is 6% closer to the closed string vacuum. Both of these features are fairly insensitive to the choice of initial conditions. However, at this point it is worth mentioning that even having set B? = 0 there is another point of possible concern with the equations of motion: The ?eld β has negative kinetic energy and may introduce further unphysical instabilities into solutions. It is worth seeing the nature of these instabilities, though we will see that they do not signi?cantly e?ect the qualitative features we identi?ed in ?gure 8. Suppose that we set initial conditions so that the string ?eld is very close to the closed string vacuum, φvac = .542, Dvac = ?.0519
vac D⊥ = .180

β vac = ?.173

(5.6)

but we displace φ away from the vacuum by a small amount ?φ = .01. The resulting time dependent solution is plotted in ?gure 9. The tachyon oscillates back and forth around the stable vacuum, as expected, but the oscillations grow with time. This behavior is directly caused by the dynamics of β; ?xing β to its value at the closed string vacuum, the oscillations no longer grow with time. We can try two things in an e?ort to deal with the negative energy instabilities of β. First, we can simply ?x β = 0, which is valid at least in the linear approximation.

– 23 –

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 -8 -6 -4 -2 0 2 4 6 8 10

Figure 8: Pro?le for φ (solid line) at x+ = 2 including all ?elds except B? starting from ? the initial conditions eq.5.4. The dashed line shows the analogous pro?le generated upon setting B+ = 0. The horizontal lines denote the vev of φ at the closed string vacuum.
0.7

0.65

0.7 0.65 0.6 0.55 0.5 0.45 0.4 10 8 6 0 4 1 2 2 3 4 0 5 6 -2 7 8 -4 9 -6 0.5 0.55 0.6

a)

b)

0.45 -6 -4 -2 0 2 4 6 8 10

Figure 9: Dynamics of φ at level (? ? (excluding B? ) if we place the string ?eld close to 2, 4) the closed string vacuum, plotted a) as a function of 0 < x+ < 9, ?4 < x? < 10 and b) at ? x+ = 9. The horizontal line denotes the vev of φ at the closed string vacuum. ? Truncating away β however changes the location of the closed string vacuum, φ = .674 D⊥ = .174 D = ?.0503

and the vacuum energy density at this point is ?1.37 times that of the D-25 brane tension. Starting from the initial conditions eq.5.4 we have generated a solution with β = B? = 0 as shown in ?gure 10a. Figure 10b shows the pro?le of φ at x+ = 6, graphed for comparison ? to the corresponding pro?le with B+ = 0. Again we see that φ initially rolls faster to the closed string vacuum, but turns around 17% closer to the closed string vacuum than it does with B+ = 0. The second approach to dealing with β is to set β = ?.173, its value at the closed string vacuum, and look at small time dependent ?uctuations around

– 24 –

1 0.9 0.8 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 8 10 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

1

2

3

4

5

6 -8

-6

-4

-2

0

2

4

6

a)

b)

0 -8 -6 -4 -2 0 2 4 6 8 10

Figure 10: Dynamics of φ at level (? ? excluding B? and β a) as a function of 0 < x+ 6, 2, 4) ? ?8 < x? < 10 and b) at x+ = 6 (shown as solid line). The dashed line shows the same ? pro?le generated from the equations after setting B+ = 0. The horizontal line denotes the vev of φ at the closed string vacuum.

0.546

0.544

0.542

0.54

0.538

0.536

0.534

0.532 0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 11: Fixing β = ?.173, B? = 0, the solid line shows small time dependent ?uctuations around the closed string vacuum graphed at x+ = 4 starting from the initial conditions ? eq.5.6. The dashed line shows the analogous pro?le obtained upon setting B+ = 0.

the vacuum. Using the initial conditions eq.5.6, at x+ = 4 we get the φ pro?le shown in ? ?gure 11, alongside the analogous pro?le with B+ = 0. Again φ falls faster to the closed string vacuum, but turns around early. The fact that these qualitative features emerge from three inequivalent truncations strongly suggests that this behavior not an artifact of our approximation scheme, and is largely independent of the negative energies of β. Since φ turns around early in all of these solutions, it is tempting to hope that in the in?nite level limit φ may not cross the closed string vacuum at all. Thus, the picture of Dbrane decay in open string ?eld theory would resemble Sen’s “rolling tachyon” boundary conformal ?eld theory solution, where the tachyon falls homogeneously o? the unstable maximum towards the closed string vacuum, but does not cross over in ?nite time. Indeed, there are preliminary indications that at higher levels the height of the ?rst peak of φ is further suppressed. For example, in ?gure 12 we have graphed the pro?le of φ at x+ = 6 ?

– 25 –

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -8 -6 -4 -2 0 2 4 6 8 10

Figure 12: Three pro?les of φ at x+ = 6 generated from the initial conditions eq.5.4. The ? solid line includes all ?elds at level ? except β, B? . The dotted line includes all ?elds at 2 level ? except β, B? , but also includes the level ? ?eld C+ . The dashed line is the pro?le 2 4 one obtains setting B+ = C+ = 0. (starting from eq.5.4) generated if we include the extra level ? ?eld, 4 i α?4 dk C+ (k)? + |k 2 and set β = B? = 0. Now φ turns around even earlier, 7% closer to the closed string vacuum than of we had set C+ = 0 and 23% closer than if we had set B+ = C+ = 0. However, before we jump to conclusions, it is worth mentioning that, from the perspective of open string ?eld theory, it is not clear what it means for the string ?eld to monotonically approach the closed string vacuum but not “cross over” in ?nite time. In string ?eld theory the closed string vacuum corresponds to a single point in an in?nite dimensional space of ?elds at zero momentum, and in the process of decay the string ?eld might oscillate many times around the closed string vacuum but never actually cross over this point. To illustrate this, we have generated two parametric plots following the path of φ, D⊥ , as a function of x? at x+ = 6, as the D-brane decays. Setting the negative energy ?elds B? , β to zero, ? ?gure 13a shows the path followed if we truncate away B+ and ?gure 13b shows the path including B+ . In 13a the string ?eld oscillates regularly around the closed string vacuum; in 13b, φ does not become as large (as we saw before) but the oscillations around the vacuum seem much more chaotic. Indeed, it is not clear from these pictures how our solutions might be approaching Sen’s “monotonic” rolling tachyon solution, or even how we would know it if they were. Still, the fact that D⊥ oscillates more wildly for our solutions could have been expected, since the kinetic energy which would otherwise allow φ to roll further up the potential wall must be absorbed by more massive ?elds, causing them to oscillate more strongly. If the rolling tachyon solution exists in open string ?eld theory, this could be the mechanism allowing the ?eld to approach the closed string vacuum asymptotically—the energy originally stored in the tachyon and lightest mass ?elds gets absorbed by ?elds of arbitrarily large mass.

– 26 –

0.4

0.5

0.35 0.4 0.3

0.25

0.3

0.2 0.2 0.15

0.1

0.1

0.05 0 0

a)

-0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b)

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 13: Plots showing the dynamics of φ (horizontal axis) and D⊥ (vertical axis) as parametric functions of x? at x+ = 6. In both plots we truncated away the negative energy ?elds B? , β. 12a shows the dynamics with B+ = 0 and 12b shows the dynamics with B+ = 0. The little circle in these ?gures shows the position of the closed string vacuum.

6. Conclusion
In this paper we have studied level truncation and time dependent solutions representing the decay of an unstable D25-brane in open string ?eld theory. We introduced a basis of spacetime ?elds, the “lightcone basis,” for which the equations of motion are local in lightcone time and therefore admit a well-de?ned initial value formulation and a Hamiltonian free of higher derivative instabilities. We studied the e?ect of level truncation in the linear theory and investigated a modi?ed level expansion which allowed us to understand the relationship between time dependent solutions in the lightcone basis and the center of mass basis, where the interaction vertex possesses an in?nite number of time derivatives. Finally, we investigated time dependent solutions including all spacetime ?elds up to level ? We found that, due to spurious instabilities introduced by truncating the theory at 2. the linear level, the solutions at level ? are unstable. However, by looking at a slightly 2 modi?ed set of equations of motion, we found much more regular solutions. The picture that emerges seems to indicate that, in comparison to solutions generated by only including the scalars at level ? the tachyon rolls faster towards the closed string vacuum initially 2, but then rapidly decelerates and does not become as large before turning around. Though these results can only be regarded as a ?rst step, they seem to indicate that it is at least possible that Sen’s rolling tachyon solution can be realized in open string ?eld theory. Of course, it would seem highly desirable to take our analysis further, perhaps by including higher levels, to see of a more re?ned picture of the D-brane decay emerges. Unfortunately, at higher levels the equations of motion become exponentially more complicated and unphysical instabilities proliferate. In principle, if the constraints are consistently imposed and we go to high enough levels, the unphysical instabilities should work themselves out, but we have no understanding of how far we need to go for this to occur. Probably it is unreasonable to hope that a brute force approach to higher levels will yield a clearer picture of the decay. What is needed is a clearer analytic understanding of the constraints

– 27 –

eq.5.5, particularly how they exorcise unphysical modes in the linear and nonlinear theory. The consistency of the constraints with time evolution is subtle, relying on nontrivial properties of the vertex in the in?nite level limit. Indeed, it is not even clear that the closed string vacuum satis?es the constraints—though, barring some sort of anomaly, naively it does. Armed with a better understanding of constraints, it may be easier to identify the relevant approximations and take our analysis further. Of course, there are other techniques which may be worth exploring. One might be to follow the approach of ref.[10], inverse Wick rotating a marginal solution interpolating 1 between a D1 and D0-brane compacti?ed on a unit circle along x1 = √2 (?+ ? x? ). Indeed, ? x constructing time independent marginal and lump solutions carrying nonzero momentum along x1 in the lightcone basis is of interest in its own right. Another possibly interesting ? approach is to ?nd time dependent solutions using the technique of ref.[19] based on the regulated butter?y state, though so far this has only been applied towards approximating the closed string vacuum solution at zero momentum. One might also try to investigate time dependent solutions in the context of vacuum string ?eld theory[20, 21], where direct analytic understanding is possible. Unfortunately, it seems doubtful that the lightcone basis could be usefully applied in this context, since naively the action would contain no ?+ derivatives at all, rendering the dynamics trivial. Still, it may be interesting to reconcile the lightcone basis with some of the time dependent solutions constructed so far in vacuum string ?eld theory. I would like to thank D.Gross and W.Taylor for some useful discussions. This work was generously supported by the National Science foundation under Grant No.PHY99-07949.

References
[1] A. Sen, “Universality of the tachyon potential,” JHEP 0003 002 (2000) hep-th/0109100. [2] E. Witten, “Noncommutative Geometry and String Field Theory,” Nucl. Phys. B268 253 (1986). [3] I. Ellwood and W. Taylor, “Gauge Invariance and Tachyon Condensation in Open String Field Theory,” hep-th/0105156. [4] I. Ellwood, B. Feng, Y. He, and N. Moeller, “The Identity String Field and the Tachyon Vacuum,” JHEP 0107 016 (2001) hep-th/010524. [5] N. Moeller and W. Taylor, “Level truncation and the tachyon in open bosonic string ?eld theory,” Nucl. Phys. B585, 105-144 (2000) hep-th/0002237. [6] D. Gaiotto and L. Rastelli, “Experimental String Field Theory,” JHEP 0204 060 (2002) hep-th/0202151. [7] W. Taylor and B. Zwiebach, “D-Branes, Tachyons, and String Field Theory,” hep-th/0311017 v2 [8] A. Sen, “Rolling Tachyon,” JHEP 0204 048 (2002) hep-th/0203211. [9] A. Sen, “Tachyon Matter,” JHEP 0207 065 (2002) hep-th/0203265 . [10] N. Moeller and B. Zwiebach, “Dynamics with In?nitely Many Time Derivatives and Rolling Tachyons,” JHEP 0210 034 (2002) hep-th/0207107.

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[11] M. Fujita and H. Hata, “Time Dependent Solution in Cubic String Field Theory,” JHEP 0305 043 (2003) hep-th/0208067. [12] H. Yang, “Stress Tensors in p-adic String Theory and Truncated OSFT,” JHEP 0211 007 (2002) hep-th/0209197. [13] M. Ostrogradski, Mem. Ac. St. Petersbourg VI 4 (1850). [14] J. Gomis, K. Kamimura, and T. Ramirez, “Physical Reduced Phase Space of Non-local Theories,” Fortsch.Phys. 52 525-530 (2004) hep-th/0311184 . [15] D.A. Eliezer and R.P. Woodward, “The Problem of Nonlocality in String Theory,” Nucl. Phys. B325 389 (1989). [16] T.G. Erler and D.J. Gross, “Locality, Causality, and an Initial Value Formulation of Open Bosonic String Field Theory,” hep-th/0406199. [17] M. Maeno, “Canonical quantization of Witten’s string ?eld thoery using midpoint light-cone time,” Phys. Rev. D43 no. 12 (1991). [18] D.J. Gross and A. Jevicki, “Operator Formulation of Interacting String Field Theory, I,II” Nucl. Phys. B 283, 1 (1987), Nucl. Phys. B 283, 1 (1987). [19] Y. Okawa, “Solving Witten’s String Field Theory using the Butter?y State” Phys. Rev. D69 086001 (2004) hep-th/0311115. [20] L. Bonora, C. Maccaferri, R.J. Scherer Santos, and D.D. Tolla, “Exact time-localized solutions in Vacuum String Field Theory,” hep-th/0409063. [21] M. Fujita and H. Hata, “Rolling Tachyon Solution in Vacuum String Field Theory,” hep-th/0403031.

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