Rolling tachyon solution in vacuum string field theory

PHYSICAL REVIEW D, VOLUME 70, 086010

Rolling tachyon solution in vacuum string field theory

Masako Fujita* and Hiroyuki Hata†

Department of Physics, Kyoto University, Kyoto 606-8502, Japan(Received 6 May 2004; published 26 October 2004)

*Electronic†Electronic

1550-7998=20

We construct a time-dependent solution in vacuum string field theory and investigate whether thesolution can be regarded as a rolling tachyon solution. First, compactifying one space direction on acircle of radius R, we construct a space-dependent solution given as an infinite number of � products ofa string field with center-of-mass momentum dependence of the form e�bp2=4. Our time-dependentsolution is obtained by an inverse-Wick rotation of the compactified space direction. We focus on oneparticular component field of the solution, which takes the form of the partition function of a Coulombsystem on a circle with temperature R2. Analyzing this component field both analytically andnumerically using Monte Carlo simulation, we find that the parameter b in the solution must be setequal to zero for the solution to approach a finite value in the large time limit x0 ! 1. We also explorethe possibility that the self-dual radius R �

��0

pis a phase transition point of our Coulomb system.

DOI: 10.1103/PhysRevD.70.086010 PACS numbers: 11.25.Sq

I. INTRODUCTION

The rolling tachyon process represents the decay ofunstable D-branes in bosonic and superstring theories[1,2]. This process is described in the limit of vanishingstring coupling constant by an exactly solvable boundaryconformal field theory (BCFT). Study of this process hasrecently evolved into various interesting physics includ-ing open-closed duality at the tree level, a new under-standing of c � 1 matrix theory and Liouville fieldtheory, and the rolling tachyon cosmology (see [3–5]and the references therein). However, there still remainmany problems left unresolved; in particular, the closedstring emission and its back reaction [6].

One may think that such problems can be analyzedusing string field theory (SFT), which is a candidate ofnonperturbative formulation of string theory and hasplayed critical roles in the study of static properties oftachyon condensation (see [7,8] and the referencestherein). However, SFT has not been successfully appliedto the time-dependent rolling tachyon process. The mainreason is that no satisfactory classical solution represent-ing the rolling process has been known in SFT, thoughthere have appeared a number of approaches toward theconstruction of the solutions [9–15]. Among such ap-proaches, Refs. [9,14] examined time-dependent solutionsin cubic string field theory (CSFT) [16] by truncating thestring field to a few lower mass component fields andexpanding them in terms of the modes enx

0

(n � 0;�1;�2; . . . ).Let us summarize the result of our previous paper [14]

(we use the unit of �0 � 1). We expanded the tachyoncomponent field tx0 as

address: [email protected]: [email protected]

04=70(8)=086010(20)$22.50 70 0860

tx0 �X1n�0

tn coshnx0; (1.1)

and solved the equation of motion for the coefficients tnnumerically (by treating t1 as a free parameter of thesolution). Our analysis shows that the n dependence of tnis given by

tn � ��n2� n; (1.2)

up to a complicated subleading n dependence. Here, � is aconstant 39=2=26 and is a parameter related to t1. Fromthe effective field theory analysis, the rolling tachyonsolution is expected to approach the stable nonperturba-tive vacuum at large time x0 ! 1 [17]. If tn behaves like(1.2), however, the profile of the tachyon field tx0 cannotbe such a desirable one: It oscillates with rapidly growingamplitude [see also (3.1) and (3.2)]:

tx0 � ex02=4 ln� � oscillating term: (1.3)

Since the radius of convergence with respect to x0 of theseries (1.1) is infinite for tn of (1.2), we cannot expect thatanalytic continuation gives another tx0 which convergesto a constant as x0 ! 1.

In order for the series (1.1) to reproduce a desirableprofile, it is absolutely necessary that the fast dumpingfactor ��n2 of (1.2) disappears. If this were the case and,in addition, if tn were exactly given by

tn � � n; (1.4)

analytic continuation of the series (1.1) would lead to

tx0 � �1 1

1 ex0

1

1 e�x0; (1.5)

which approaches monotonically a constant as x0 ! 1.This particular tx0 has another desirable feature where it

10-1 2004 The American Physical Society

MASAKO FUJITA AND HIROYUKI HATA PHYSICAL REVIEW D 70 086010

becomes independent of x0 when � 0 and 1, which maycorrespond to sitting on the unstable vacuum and thestable one, respectively. Since CSFT should reproducethe rolling tachyon process, it is expected that the behav-ior (1.2) is an artifact of truncating the string field tolower mass component fields and that some kind of moresensible analysis would effectively realize � � 1.

The purpose of this paper is to study the rollingtachyon solution in vacuum string field theory (VSFT)[18–21], which has been proposed as a candidate SFTexpanded around the stable tachyon vacuum. The action ofVSFT is simply given by that of CSFT with the Becchi-Rouet-Stora-Tyutin (BRST) operator QB in the kineticterm replaced by another operator Q consisting only ofghost oscillators. Owing to the purely ghost nature of Q,the classical equation of motion of VSFT is factorizedinto the matter part and the ghost one, each of which canbe solved analytically to give static solutions representingDp-branes. In fact, analysis of the fluctuation modesaround the solution has successfully reproduced theopen string spectrum at the unstable vacuum althoughthere still remain problems concerning the energy densityof the solution [22–24].1 If we can similarly constructtime-dependent solutions in VSFT without truncation ofthe string field, we could study more reliably whether SFTcan reproduce the rolling tachyon processes and, further-more, the unresolved problems mentioned at the begin-ning of this section.

Our strategy of constructing a time-dependent solutionin VSFT is as follows. First, we prepare a lump solutiondepending on one space direction which is compactifiedon a circle of radius R. Then, we inverse-Wick rotate thisspace direction to obtain a time-dependent solution fol-lowing the BCFT approach [1]. The lump solution ofVSFT localized in uncompactified directions has beenconstructed in the oscillator formalism by introducingthe creation/annihilation operators for the zero mode inthis direction [19]. In the compactified case, however, wecannot directly apply this method. We instead constructthe matter part �m of a lump solution as an infinitenumber of � products of a string field �b; �m � �b ��b � � � � ��b (the ghost part is the same as that in thestatic solutions). Since the equation of motion of �m issimply �m ��m � �m, this gives a solution if the limitof an infinite number of � product exists [27]. As theconstituent �b, we adopt the one which is the oscillatorvacuum with respect to the nonzero modes and has theGaussian dependence e�bp2=4 on the zero-mode momen-tum p in the compactified direction. Finally, our time-dependent solution is obtained by making the inverse-Wick rotation of the compactified direction X ! �iX0 or�iX0 �R.

1See also [25,26] for recent attempts to this problem.

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After constructing a time-dependent solution in VSFT,our next task is to examine whether it represents therolling tachyon process. Our solution consists of an infi-nite number of string states, and we focus on one particu-lar component field tx0 (we adopt the same symbol asthe tachyon field in the CSFT analysis). This tx0 has theexpansion (1.1) with coshnx0 replaced by coshnx0=R.What is interesting about tx0 is that it takes the form ofthe partition function of a statistical system of charges atsites distributed with an equal spacing on a unit circle.The temperature of this system is R2. The charges interactthrough Coulomb potential and they also have a self-energy depending on the parameter b of�b. The partitionfunction is obtained by summing over the integer value ofthe charge on each site keeping the condition that thetotal charge be equal to zero.

What we would like to know about tx0 are particu-larly the following:

(i) W

-2

hether tx0 has a profile which converges to aconstant as x0 ! 1.

(ii) W
hether the critical radius R � 1 in the BCFTapproach [28,29] is required also in our solution.
That we have to put R � 1 in our solution is also naturalin view of the fact that the correct value �1 of thetachyon mass squared is reproduced from the fluctuationanalysis around the D25-brane solution of VSFT [22–24].For these two problems, we carry out analysis using bothanalytic and numerical methods. In particular, we canapply the Monte Carlo simulation since tx0 is the par-tition function of a Coulomb system on a circle. We findthat the coefficient tn of our VSFT solution has a similar ndependence to (1.1) with � depending on the parameter b.This implies that the profile of tx0 is again an unwel-come one for a generic value of b: It is an oscillatingfunction of x0 with growing amplitude. However, we canrealize � � 1 by putting b � 0 and taking the number of�b in �m � �b � � � � ��b to infinity by keeping thisnumber even. These properties seem to hold for any valueof R. In order to see whether R � 1 has a special meaningfor our solution, we study the various thermodynamicproperties of the Coulomb system tx0. First, we argueusing a naive free energy analysis that there could be aphase transition at temperature R2 � 1. Below R2 � 1,only the excitations of neutral bound states of charges areallowed, but above R2 � 1 excitations of isolated chargesdominate the partition function. We carry out a MonteCarlo study of the internal energy and the specific heat ofthe system, but cannot confirm the existence of this phasetransition. However, we find that the correlation functionof the charges shows qualitatively different behaviorsbetween the large and small R2 regions when b � 0,possibly supporting the existence of the phase transition.

The rest of this paper is organized as follows. In Sec. II,first briefly reviewing VSFT and its classical solutionsrepresenting various Dp-branes, we construct time-

ROLLING TACHYON SOLUTION IN VACUUM STRING . . . PHYSICAL REVIEW D 70 086010

dependent solutions following the strategy mentionedabove. In Sec. III, we investigate the profile of the com-ponent field tx0 both analytically and numerically. InSec. IV, we argue that our solution with b � 0 could give arolling tachyon solution. In Sec. V, we study a possiblephase transition at R2 � 1 through various thermody-namic properties of the system. The final section(Sec. VI) is devoted to a summary and discussions. Inthe appendix, we present a proof concerning the mini-mum energy configuration of the Coulomb system.

II. CONSTRUCTION OF A TIME-DEPENDENTSOLUTION IN VSFT

In this section, we shall construct a time-dependentsolution in VSFT. As stated in Sec. I, we first construct alump solution which is localized in one spatial directioncompactified on a circle of radius R. This solution is givenas an infinite number of � products of a string field �b;�b ��b � � � � ��b. Our time-dependent solution is ob-tained by inverse-Wick rotating the spatial direction tothe time one. Throughout this paper, we use the conven-tion �0 � 1.

A. Dp-brane solutions in VSFT

In this subsection, we briefly review the construction oflump solutions in VSFT describing various Dp-branes inthe uncompactified space [19]. VSFT is a string fieldtheory around the nonperturbative vacuum where thereare only closed string states. Its action is written asfollows using the open string field �:

S � �12� � Q�� 1

3� � � ��

� �12h�jQj�i � 1

3 0h�j1h�j2h�jV3i012: (2.1)

The BRST operator Q of VSFT consists of only ghostoperators, and it has no nontrivial cohomology. The three-string vertex jV3i represents the midpoint interaction ofthree strings, and it factorizes into the direct product ofthe matter part and the ghost one. More generically, thematter part of the N-string vertex jVNi representing thesymmetric midpoint interaction of N-strings (N �3; 4; . . . ) is given by [30,31]

jVmN i01��N�1 �Zd26p0 � � �

Zd26pN�1 �26

XN�1

r�0

pr

!

� exp

"��

XN�1

r;s�0

1

2

X1n;m�1

Vrsnmar�yn as�y

m

X1n�1

Vrsn0ar�yn p�s

1

2Vrs00p

�r p�s

!#

ON�1

r�0

j0;prir; (2.2)

where j0;pri is the Fock vacuum of the rth string carry-

086010

ing the center-of-mass momentum pr (the index r spec-ifying the N strings runs from 0 to N � 1). Here we usethe same convention as [18–21]. a�r

n are the matteroscillators of nonzero modes normalized so that theircommutation relations are

�ar�n ; as�y

m � � ��nm�rs; n;m � 1: (2.3)

The coefficients Vrsnm are called the Neumann coefficients.In particular, Vrs00 is given by

Vrs00 �

8><>:� ln

2 sin�r�sN

; r � s

2 ln�N4

�; r � s:

(2.4)

Note that Vrsnm depends on N although we do not write itexplicitly.

The action (2.1) leads to the equation of motion,

Q� � �� : (2.5)

Assuming that the solution is given as a product of thematter part and the ghost one, � � �m ��g, the equa-tion of motion is reduced to

�m � �m �m �m; (2.6)

Q�g � ��g �g �g; (2.7)

where �m ( �g ) is the � product in the matter (ghost)sector. In this paper, we assume that the ghost part �g

is common to the various solutions, and focus on thematter part Eq. (2.6).

Classical solutions of (2.6) which represent the variousDp-branes in spacetime are given in [19]. Let us reviewthe two ways of constructing classical solutions represent-ing the translationally invariant D25-brane. One way is toassume that �m is given in the form of a squeezed state,the exponential of an oscillator bilinear acting on thevacuum:

j�mi � N exp

�1

2��

X1m;n�1

Smna�ym a�y

n

!j0; 0i; (2.8)

where N is a normalization factor. The equation ofmotion (2.6) is reduced to an algebraic equation for theinfinite dimensional matrix Smn, which, under a certaincommutativity assumption and using the algebraic rela-tions among the Neumann coefficients Vrsmn [30], can besolved to give Smn in terms of Vrsmn [32]:

S � CT; T �1

2X

�1 X �

��1 3X1� X

p �;

(2.9)

with the matrices C and X given by

Cmn � �1m�mn; X � CV11: (2.10)

-3


Another way is to construct�m as the sliver state [27].Defining the wedge states as

jNi0 � j0; 0i � j0; 0i � � � � � j0; 0i|��{z��}N�1

� 1h0; 0j2h0; 0j � � � N�1h0; 0jVNi01��N�1; (2.11)

they satisfy the following property

jNi � jMi � jN M� 1i: (2.12)

Taking the limit N;M ! 1, we have

j1i � j1i � j1i: (2.13)

Namely, the state j1i (sliver state) is a solution to (2.6). Ithas been proven that the two solutions, (2.8) and j1i, areidentical with each other [33].

Lump solutions localized in spatial directions can beconstructed in the same way as the D25-brane solution.Let us denote the directions transverse to the brane by x�.In the squeezed state construction [19], we introduce theannihilation and the creation operators for the zero modein the transverse directions by

a�0 �

��b

p

2p̂� �

i��b

p x̂�; a�y0 �

��b

p

2p̂�

i��b

p x̂�;

(2.14)

where b is an arbitrary positive constant. Since the zeromodes a�0 satisfy the same commutation relation (2.3) asthe nonzero modes, we define the new Fock vacuum j�biby

ar�n j�bi � 0; n � 0: (2.15)

The new vacuum j�bi with the normalization h�bj�bi �1 is expressed in terms of the momentum eigenstates as

j�bi �Y�

�b2�

�1=4 Z 1

�1dp� e�b=4p�2 j0;p�i: (2.16)

With these oscillators and the coordinate-dependent vac-uum j�bi, the transverse part of the three-string vertexjV3i can be written as

exp

�1

2

Xr;s�0;1;2

X1m;n�0

ar�ym V 0rs

mnas�yn

!j�bi012; (2.17)

in terms of the new coefficients V 0rsnm, which satisfy the

same algebraic relations as Vrsnm. Therefore, we can con-struct the Dp-brane solutions just in the same way as theD25-brane solution:

086010

j�mp i � j�m

ki � j�m

?i

� exp

�1

2��

X1m;n�1

Smna�ym a�y

n

!j0;pi

� exp

�1

2

X1m;n�0

S0mna

�ym a�y

n

!j�bi; (2.18)

where the indices �, � run the directions tangential to thebranes (�; � � 0; 1; . . . ; 25� p), and S0

mn is given by (2.9)with V11 replaced by V 011. This lump solution containsone arbitrary parameter b, the physical meaning of whichis not known. It has been shown that the ratio of thetensions of Dp-brane solutions is independent of b [34].Later we will argue that we must choose b � 0 to obtain atime-dependent solution with the desirable rolling profile.

Finally, note that, since the modified Neumann coef-ficients V 0rs

mn satisfy the same algebra as the original Vrsmn,the transverse part j�m

?i of the lump solution (2.18) can bewritten as a sliver state:

j�m?i � lim

N!11h�bj2h�bj � � � N�1h�bjVN?i01��N�1;

(2.19)

where jVN?i is the transverse part of the N-string vertex.

B. Time-dependent solution in VSFT

Now let us construct a time-dependent solution inVSFT which possibly represents the process of rollingtachyon. This consists of the following two steps:

(i) F

-4

irst we construct a lump solution of VSFT local-ized in one space direction which is compactifiedon a circle of radius R.

(ii) T
hen we inverse Wick rotate the compactifiedspace direction to the time one on this lump solu-tion to obtain a time-dependent solution in VSFT.
Since both the solution and the string vertices have fac-torized forms with respect to the spacetime directions,we shall focus only on this transverse direction of thebrane in the rest of this paper.

First, we shall construct a lump solution on a circle.The squeezed state construction explained in the previoussubsection, however, cannot be directly applied to thecompactified case since the zero-mode creation/annihila-tion operators of (2.14) are ill defined due to the period-icity x̂� � x̂� 2�R. Therefore, we shall adopt the sliverstate construction of the lump solution. Namely, let usconsider

limN!1

j�bi�� j�bi|��{z��}N�1

� limN!1

1h�bj��N�1h�bjVNi01��N�1;

(2.20)

with a suitably chosen j�bi. If the limit N ! 1 of (2.20)exits, it gives a solution of VSFT. Taking into account thatthe momentum zero mode p in the compactified direction


takes discrete values p � n=R, we adopt as the state j�biin (2.20) the following one which is a natural compacti-fied version of (2.16):

j�bi �X1

n��1

e�b=4n=R2 j0;n=Ri; (2.21)

where j0; n=Ri is the momentum eigenstate (and the Fockvacuum of the nonzero modes) with the normalizationh0;m=Rj0; n=Ri � �n;m. The N-string vertex jVmN i for the

086010

compactified direction is given by (2.2) with the replace-ments:

Zdp !

1

R

Xn

; �p ! R�n;0;

j0;pi !��R

pj0; n=Ri:

(2.22)

Then the state j�bi � � � � � j�bi in the x representationfor the center-of-mass dependence is given by

hxjj�bi � � � � � j�bi|��{z��}N�1

�X1

n0��1

X1n1��1

� � �X1

nN�1��1

�PN�1r�0

nr;0exp

�in0xR

�

� exp

�1

2

X1n;m�1

V00nmayna

ym �

XN�1

s�0

X1n�1

V0sn0aynnsR

�1

2R2XN�1

r;s�0

Vrs00nrns �b

4R2XN�1

r�1

n2r

!j0i;

(2.23)

which in the limit N ! 1 should give a lump solution ona circle. In (2.23), nr=R (r � 1; 2; . . . ; N � 1) is thecenter-of-mass momentum carried by the rth constituent�b, and we have Fourier transformed the momentumn0=R carried by the whole j�bi � � � � � j�bi to the coor-dinate x. In this paper we are interested only in the time

dependence of the solution and, hence, ignore the overallconstant factor multiplying the solution.

Our construction of a time-dependent solution of VSFTis completed by making the inverse-Wick rotation X !

�iX0, namely, x ! �ix0 and ayn ! �iay

n , on this lumpsolution:

j�x0i � limN!1

X1n0��1

X1n1��1

� � �X1

nN�1��1

�PN�1r�0

nr;0exp

�n0x0

R

�

� exp

1

2

X1n;m�1

V00nmayna

ym i

XN�1

s�0

X1n�1

V0sn0aynnsR

�1

2R2XN�1

r;s�0

Qrsnrns

!j0i; (2.24)

where Qrs is defined by

Qrs � Vrs00 b2�r;s�r;0 � 1 �

8>>>>><>>>>>:

�2 ln

2 sin�r�sN

; r � s

2 ln�N4

� b

2 ; r � s � 0

2 ln�N4

�; r � s � 0:

(2.25)

Taylor expansion of expiPsPn V

0sn0a

ynns=R in (2.24)

gives an expression of j�x0i as an infinite summation,j�x0i �

P�j�i’�x0, where j�i are the static string

states of the form ay � � � ay exp12PV00mna

yma

yn j0i, and

’�x0 are the corresponding time-dependent componentfields. In this paper, we shall, for simplicity, focus on thecomponent field of the pure squeezed state exp12 �PV00mna

yma

yn j0i. Since we have limN!1V00mn � Smn [33],

this component field is that for the state representing theunstable vacuum. Denoting this component field by tx0,we have

tx0 �X1

n0��1

en0x0=Rtn0 ; (2.26)

where tn0 is given by

tn0 �X1

n1 ;...;nN�1��1n0 n1 �� nN�1�0

exp��1

R2Hnr; n0

�; (2.27)

with

Hnr; n0 �1

2

XN�1

r;s�0

Qrsnrns: (2.28)

Note that the coefficient tn0 can be regarded as the parti-tion function of a statistical system with Hamiltonian Hand the temperature R2. In this statistical system, we haveN charges nr on a unit circle at an equal interval (Fig. 1).The charges nr take integer values from �1 to 1, andthey have the self-interaction Qrr and the two-

-5

n

nnn

n

nN−1

0

1

24

rs

s

rn

n3

Q

FIG. 1. There are N charges nr on the unit circle. Each chargenr takes integer values, and the total charge must be equal tozero. The charges nr and ns interact via the Coulomb potentialQrs.


dimensional Coulomb interaction Qrs (r � s) betweeneach other. In tn0 , the charge n0 at r � 0 is fixed, andthere is a constraint that the total charge be equal to zero.Note that tn0 is positive definite, tn0 > 0, and is even undern0 ! �n0:

t�n0 � tn0 : (2.29)

On the other hand, tx0 itself is interpreted as the parti-tion function of the statistical system in the presence ofthe external source x0=R for n0.

Solving the constraintPN�1r�0 nr � 0 to eliminate nN�1,

tn0 and tx0 are rewritten using independent variableswithout constraint:

tn0 �X1

n1;...;nN�2��1

exp

�

1

2R2XN�2

r;s�0

Q̂rsnrns

!; (2.30)

where Q̂rs is a N � 1 � N � 1 matrix given by

Q̂rs � Qrs �Qr;N�1 �QN�1;s QN�1;N�1;

r; s � 0; 1; . . . ; N � 2: (2.31)

We have checked numerically that the matrix Q̂rs is apositive definite matrix.

In addition to the above j�x0i obtained by theinverse-Wick rotation X ! �iX0, we have another time-dependent solution via a different inverse-Wick rotation,X ! �iX0 �R. This new solution is obtained simplyby inserting �1n0 into (2.24) and (2.26), and satisfiesthe Hermiticity condition. As we shall explain in the nextsection, we expect that this new solution with �1n0

represents the rolling process to the stable tachyon vac-uum, while the original solution given by (2.24) repre-

086010

sents the rolling in the direction where the potential isunbounded from below.

III. ANALYSIS OF THE COMPONENT FIELD tx0

In this section, we study the profile of the componentfield tx0 given by (2.26) both analytically and numeri-cally. If our VSFT solution (2.24) represents the rollingtachyon solution, the component field tx0 as well as thewhole j�x0i should approach zero, namely, the tachyonvacuum, as x0 ! 1 (note that the tachyon vacuum inVSFT is at � � 0).

Let us mention the expected n0 dependence of thecoefficient tn0 , (2.27) and (2.30), necessary for tx0 tohave a rolling tachyon profile. Suppose that the n0 depen-dence of tn0 is given by

tn0 � e�an20~tn0 ; (3.1)

where ~tn0 has a milder n0 dependence than the leadingfactor e�an20 . We expect that limn!1~tn=~tn 1 is finite andlarger than 1, namely, that the series (2.26) with tn0replaced by ~tn0 has a finite radius of convergence withrespect to x0. A typical example is ~tn0 � e�bjn0j. Such tn0actually appeared in the time-dependent solution inCSFT in the level-truncation approximation [9,14]. Forthis tn0 , we have

tx0 � ex02=4a~tx0; ~tx0 �X1

n��1

~tne�x0�2na2=4a:

(3.2)

If ~tn0 does not depend on n0, ~tx0 is a periodic function ofx0 with period 2a, and the whole tx0 cannot have adesired profile: It oscillates with blowing up amplitudeex02=4a as x0 ! 1. Even if ~tn0 has a mild n0 dependencesuch as ~tn0 � e�bjn0j, it seems very unlikely that tx0approaches a finite value in the limit x0 ! 1. Theseproperties persist in the alternating sign solution�1n0tn0 obtained by another inverse-Wick rotation men-tioned at the end of Sec. II. Therefore, it is necessary thatthe leading term e�an20 in (3.1) is missing, namely, wemust have a � 0. If this is the case, the series tx0 (2.26)would have a finite radius of convergence, and the ana-lytic continuation would give a globally defined tx0 suchas (1.5). Since tn0 is positive definite, tx0 diverges at theradius of convergence and corresponds to the rolling inthe direction of the unbounded potential. On the otherhand, another tx0 with alternating sign coefficients�1n0tn0 is expected to be finite at the radius of conver-gence and represent the rolling to the tachyon vacuum.

In the rest of this section, we shall study whether thecondition a � 0 is satisfied for the present solution. Weshall omit the indices 0 of n0 and x0 unless confusionoccurs.

-6

20 40 60 80 100

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

20 40 60 80 100

-0.25

-0.2

-0.15

-0.1

-0.05

(a)

(b)

nCr

nCr

b = 0.1

b = 10

r

r

FIG. 2. The minimum energy noninteger configurations fnCr gwith n � 1 in the case N � 100. The value of b is b � 0:1 in (a)and b � 10 in (b).


A. Analysis for R2 � 1 and R2 � 1

In this subsection, we shall consider the n dependenceof the coefficient tn of (2.30) for R2 � 1 and R2 � 1.First, for the analysis in the region R2 � 1 and also forlater use, we present another expression of tn obtained byapplying the Poisson’s resummation formula,

X1n��1

gn �X1

m��1

Z 1

�1dy gye2�imy; (3.3)

to the nr summations in (2.30):

tn � RN�2det ^̂Q�1=2 exp��

1

2Q̂�100

n2

R2

�

�X1

m1;...;mN�2��1

exp

(�2�2R2

XN�2

r;s�1

mr^̂Q

�1rsms

2�inXN�2

r�1

nCr mr

); (3.4)

where the matrix ^̂Q is the lower right N � 2 � N � 2part of Q̂,

^̂Q rs � Q̂rs; r; s � 1; 2; . . . ; N � 2; (3.5)

and nCr is defined by

nCr � �XN�2

s�1

^̂Q�1

rsQ̂s0; r � 1; 2; . . . ; N � 2:

(3.6)

In obtaining the expression (3.4), we have used

Q̂ 00 �XN�2

r;s�1

Q̂0r^̂Q

�1rsQ̂s0 �

1

Q̂�100; (3.7)

which is valid for any matrix Q̂ and its N � 2 � N � 2

submatrix ^̂Q.Some comments on the formula (3.4) are in order. First,

fnCr g in (3.6) is nothing but the configuration which,without the constraint that nCr be integers and keepingn � 1 fixed, minimizes the Hamiltonian (2.28),

Hnr; n �1

2

XN�2

r;s�1

^̂Qrsnrns nXN�2

r�1

Q̂0rnr 1

2Q̂00n

2:

(3.8)

Figure 2 shows the configurations fnCr g in the cases of b �0:1 [2(a)] and b � 10 [2(b)] for N � 100. As seen fromthe figure, fnCr g is localized around r � 0 (modN) toscreen the charge n � 1.2

2As seen from Fig. 2, the charges nCr near r � 0 all haveopposite sign to n0 for larger values of b, while nCr havealternating signs for smaller b.

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Our second comment is on the termexpf��2Q̂�100�

�1n2=R2g in (3.4). The exponent isequal to the value of the Hamiltonian H for the configu-ration fn � nCr g, namely, the (noninteger) configurationminimizing H for a given n:

Hn � nCr ; n �n2

2Q̂�100: (3.9)

As was analyzed in [19], 1=Q̂�100 is finite in the limitN ! 1.3 Figure 3 shows limN!1�2Q̂�100�

�1 as a func-tion of lnb. It is a monotonically increasing function of band, as we shall see in Sec. IV, approaches zero as b ! 0.

Now let us consider tn for R2 � 1. Since ^̂Q is a positivedefinite matrix, the configuration mr � 0 for all r �1; 2; . . . ; N � 2 dominates the mr summation in (3.4) forR2 � 1. Namely, we have

tn ’ RN�2det ^̂Q�1=2 exp��

1

2Q̂�100

n2

R2

�;

R2 � 1: (3.10)

31=Q̂�100 is related to S000 in [19] by

1=Q̂�100 � b�1=2 S000=1� S0

00�.

-7

1000 2000 3000 4000

10

20

30

40

50

n

∆H(n)

FIG. 4. Hn vs n for N � 2048 and b � 0:1. For obtaining Hn, we approximate the integer-valued charge nIrn by theinteger nearest to n � nCr . However, this fnIrng does not neces-sarily satisfy the constraint n

PN�1r�1 n

Irn � 0. In the figure,

only the points which satisfy the constraint are plotted.Distribution of the points is insensitive to the value of N if itis large enough.

-4 -2 2 4 6

1

2

3

4

ln b

[2(Q-1)00

]-1

FIG. 3. limN!1�2Q̂�100��1 as a function lnb. The dots rep-

resent limN!1�2Q̂�100��1 at b � 1=100, 1=10, 1, 10, 100, and

1000 obtained by evaluating its values for N � 50, 100, 200,300, 400, 500, and 600 and then extrapolating them to N � 1by using the fitting function of the form

P3k�0 ck=N

k. The curveinterpolating these six points is 0:429 214 0:216 204� lnb 0:037 972 9� lnb2 0:001 862 25� lnb3.

10 20 30 40 50

5

10

15

20

dT(x) / dx

x

FIG. 5. The numerical results of dTx=dx at x �0:5; 1:0; 1:5; . . . ; 50:0. Here we have taken b � 10, R2 � 1,and N � 256. These points are well fitted by a linear function0:4530x� 0:1713, and the slope 0:4530 is close to Q̂�100 �0:4497 for the present b and N. The small oscillatory behaviorcan be better observed in d2Tx=dx2 shown in Fig. 6.


This expression can also be obtained by simply replacingthe nr summations in (2.30) with integrations over con-tinuous variables pr � nr=R. Since the coefficient of n2

in the exponent is nonvanishing for b > 0, this tn cannotlead to a desirable rolling profile.

On the other hand, tn (2.30) for R2 � 1 can be approxi-mated by

tn ’ exp��1

R2H�nIrn; n�

�; R2 � 1; (3.11)

where fnIrng is the integer configuration which mini-mizes the Hamiltonian for a given n. This configurationfnIrng is in general different from but is close to fn � nCr g,the configuration minimizing Hnr; n without the inte-ger restriction. Therefore, rewriting (3.11) as

tn ’ exp��

1

2Q̂�100

n2

R2

�exp

��1

R2 Hn

�; (3.12)

with Hn defined by

Hn � H�nIrn; n� �Hn � nCr ; n; (3.13)

the second factor of (3.12) is expected to have a milder ndependence than the first one. Figure 4 shows Hn forN � 2048 and b � 0:1. We see that Hn is in factroughly proportional to n. Therefore, we cannot obtaina rolling profile in the case R2 � 1 either.

Summarizing this subsection, for both R2 � 1 andR2 � 1, the coefficient tn of the component field txhas the leading n dependence of the form

tn � exp��

1

2Q̂�100

n2

R2

�: (3.14)

Then, tx itself shows the behavior

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tx � exp�12Q̂�100x2� � oscillating part; (3.15)

and cannot approach the tachyon vacuum as x ! 1. Thisis the case even if we adopt the alternating sign solution�1ntn.

B. Numerical analysis using Monte Carlo simulation

For studying tn and tx for intermediate values of R2,we shall carry out Monte Carlo simulation of theCoulomb system with partition function (2.27),Hamiltonian H, and temperature R2. We have adoptedthe Metropolis algorithm. Since the total charge must be

-8

TABLE I. The slope of the linear function of x obtained by fitting the Monte Carlo results ofhnix=R with b � 10 and N � 256. They are almost independent of R2 and close to Q̂�100.

R2 0.3 0.5 1.0 1.5 2.0 3.0 5.0 Q̂�100

Slope 0:4557 0:4540 0:4530 0:4502 0:4498 0:4498 0:4497 0:4497


kept zero, a new configuration is generated from the oldone fnrg by randomly choosing two points r and s on thecircle and making the change nr; ns ! nr 1; ns � 1.This new configuration is accepted/rejected according tothe standard Metropolis algorithm.

1. Time derivatives of lntx

First, we investigate the time derivatives of the loga-rithm of tx. Defining Tx by

tx � eTx; (3.16)

we have

dTxdx

�1

Rhnix; (3.17)

d2Tx

dx2�1

R2hn2ix � hni2x; (3.18)

where the average hOix for a given x is defined by

hOix �1

tx

X1n;n1 ;...;nN�1��1

n P

N�1r�1

nr�0

Oe�Hnr;n=R2 nx=R: (3.19)

The numerical results of the ‘‘velocity’’ (3.17) versus xfor b � 10, R2 � 1, and N � 256 are shown in Fig. 5. Wefind that dTx=dx is almost linear in x. The slope of thefitted linear function for the various R2, b � 10 and N �

256, as well as the value of Q̂�100 for the present b andN, are shown in Table I. These results show that thebehavior of (3.15) obtained in the regions R2 � 1 andR2 � 1 is valid also in the intermediate region of R2.

The Monte Carlo results of the ‘‘acceleration’’ (3.18) atvarious x are shown in Fig. 6 in the case of b � 10, R2 �1, and N � 256. The acceleration oscillates with periodroughly equal to 8. The center-of the oscillation is aroundQ̂�100 � 0:4497, which is consistent with (3.15).4 Wehave studied the acceleration for other values of R2 andfound that it oscillates around Q̂�100 for any R2.

4In this case, the parameter a of (3.1) is a ��2Q̂�100R

2��1 � 1:112. If the subleading part ~tn of (3.1) isindependent of n, the oscillation period of tx should be 2a �2:224. The fact that the period of Fig. 6 is nearly equal 8, whichis 4 times the naive period, suggests that ~t4k 1, ~t4k 2, and ~t4k 3are negligibly small compared with ~t4k.

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2. Numerical analysis of tnThe n dependence of tn itself can be directly measured

using Monte Carlo simulation as follows [35]. Here, weuse instead of R2 the inverse temperature � 1=R2, andmake explicit the dependence of tn to write it as tn; .Let us define the average hOin; with subscript n and by

hOin; �1

tn;

X1n1 ;...;nN�1��1

n P

N�1r�1

nr�0

Oe� Hnr;n: (3.20)

Integrating the relation

@@

lntn; � h�Hin; ; (3.21)

with respect to , we obtain

tn; tn; �0

� exp

Z

0d 0h�Hin; 0

!; (3.22)

and, hence,

tn; tn�0;

�tn; �0

tn�0; �0

� exp

"Z

0d 0h�Hin; 0 � h�Hin�0; 0

#:

(3.23)

Equation (3.10) implies that tn; �0 is independent of n andtherefore tn; �0=tn�0; �0 � 1. Thus, we obtain the for-mula

tn; tn�0;

� exp

"�Z

0d 0hHin; 0 � hHin�0; 0

#: (3.24)

This allows us, in principle, to directly evaluate the ndependence of tn using the expectation values of H ob-tained by Monte Carlo simulation. Note that hHin; inhigh temperature region � 1 is given using (3.10) by

hHin; ’N � 2

2

1

2Q̂�100� n2; � 1: (3.25)

Figure 7 shows the Monte Carlo result of hHin; �

hHin�0; � n2=�2Q̂�100�, namely, the deviation ofhHin; � hHin�0; from the high temperature value, forthe various values of n (b � 10 and N � 256). As isdecreased, the data for each n approach zero. On the other

-9

10 20 30 40 50

0.25

0.35

0.4

0.45

0.5

0.55

d2T(x) / dx2

(Q-1)00

x

FIG. 6. d2Tx=dx2 at x � 0:5; 1:0; 1:5; . . . ; 50:0. Here we havetaken b � 10, R2 � 1, and N � 256. The horizontal line showsthe value of Q̂�100 � 0:4497 for the present b and N.

0.5 1.5 2 2.5

5

10

15

20

25

30

35

n = 10

n = 20

n = 40

n = 100

β

H n,β− n=0,βH − n2 / (2(Q-1)00

)

FIG. 7 (color online). hHin; � hHin�0; � n2=�2Q̂�100� forn � 10 (red points), n � 20 (green), n � 40 (blue), and n �100 (purple) from bottom to top. Here we have taken N � 256and b � 10.


hand, they seem to approach a positive constant as isincreased. Since the asymptotic value at large grows nofaster than linearly in n, the results of Fig. 7 together withthe formula (3.24) seems consistent with our low tem-perature analysis using (3.12) and Fig. 4.

From our analyses in this subsection, we have foundthat the behaviors (3.14) of tn and (3.15) of tx in theR2 � 1 and R2 � 1 regions are valid for any R2.Concerning these behaviors, R � 1 does not seem to bea special radius.

IV. POSSIBLE ROLLING SOLUTION WITH b � 0

Our analysis in the previous section implies that rollingsolutions with a desirable profile can never be obtainedunless the coefficient of n2 in the exponent of (3.14)vanishes. Namely, we must have

1

Q̂�100� 0: (4.1)

This condition can in fact be realized by putting b � 0 ascan be inferred from Fig. 3, though the precise way howthis condition is satisfied differs between the N � evenand the odd cases. For an even (and finite) N, the condi-tion (4.1) is satisfied by simply putting b � 0. This isbecause Q̂ with N � even and b � 0 has a zero modef�1rg:XN�2

s�0

Q̂rs�1s � 0;

r � 0; 1; . . . ; N � 2;N � even; b � 0:

(4.2)

This zero mode is at the same time the minimum energyconfiguration nCr (3.6).5

5Note that the condition for the minimum energy configura-tion is

PN�2s�0 Q̂rsn

Cs � 0 for r � 1; 2; . . . ; N � 2, while the zero

mode of Q̂ should satisfy this equation for r � 0; 1; . . . ; N � 2including r � 0.

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On the other hand, in the N � odd case, the condition(4.1) is realized by putting b � 0 and in addition takingthe limit N ! 1. In fact, numerical analysis shows that

1

Q̂�100jb�0�1:705N

O�1

N2

�; N � 1; N � odd:

(4.3)

This property is related to the fact that the matrix Q̂ withb � 0 has an approximate zero mode for large and odd N.This approximate zero mode, which is at the same timethe minimum energy configuration nCr (3.6), is given by

nCr ’ �1r�1�

2jrjN 1

�;

�r � 0;�1;�2; . . . ;�N � 1=2�;(4.4)

where the index r should be understood to be definedmodN; nr � nr N . This configuration satisfiesPN�1r�0 n

Cr � 0 and

XN�2

s�0

Q̂rsnCs � O�1

N

�;

r � 0; 1; . . . ; N � 2;N � 1; b � 0:

(4.5)

A proof of (4.5) is given in the appendix.In the rest of this section, we shall consider only the

case of N � odd since (2.12) is closed among the jN �oddi states [jNi is the state given by (2.20) in the presentcase]. Here we shall just make a comment on the case ofN � even. When N � even and b � 0, the coefficient tnis independent of n as can be seen from the expression(3.4) with nCr � �1r. Therefore, we have tx �

t0P

1n��1�1nenx=R. A naive summation of this series

gives tx � 0, or the Poisson’s resummation formulagives tx � 2�t0

Pm�even=odd�ix=R� �m [36].

-10

2500 5000 7500 10000 12500 15000

1000

2000

3000

4000

5000

∆H(n)

n

FIG. 8. Hn vs n for N � 8191 and b � 0.

2000

3000

4000

5000

∆H(n)


We would like to investigate whether the solution withb � 0 and N� odd ! 1 can be regarded as a rollingsolution. However, it is not an easy matter to repeat theMonte Carlo analysis of Secs. III B1 and III B 2 in thepresent case. First, since the coefficient Q̂�100 of theleading x-dependent term of (3.15) blows up as N ! 1,6

so do the slope of the velocity curve of Fig. 5 and thecentral value of the acceleration curve of Fig. 6. Therefore,it is hard to read off the subleading x dependence whichshould become the leading one in the limit N ! 1.Second, the direct evaluation of the coefficient tn usingthe formula (3.24) is not an easy task because of a badstatistics problem. Namely, the difference hHin; �

hHin�0; , which is of order n2=�2Q̂�100� [see (3.25)], isvery small compared with the leading bulk term N �2=2 of hHi when b � 0.

Here we shall content ourselves with the analysisof tn in the low temperature region R2 � 1 usingthe expression (3.12). Since the leading termexpf�n2=�2Q̂�100R

2�g of (3.12) disappears in the limitN ! 1, we study the n dependence of Hn in the sameway as we did in Fig. 4. The result forN � 8191 is plottedin Fig. 8.

Note that the data of Fig. 8 has an approximate periodicstructure with respect to n with period of about 4100.This periodicity can be understood from (3.4) for tn andthe expression (4.4) for nCr independently of the lowtemperature approximation. In fact, (3.4) without theleading term expf�n2=�2Q̂�100R2�g is invariant underthe shift n ! n N 1=2 since the change of theexponent in the mr summations under this shift is aninteger multiple of 2�i for nCr of (4.4). Although Fig. 8shows data only for positive values of n, recall that tn iseven under n ! �n; (2.29). This parity symmetry and theperiodicity lead to the structure shown in Fig. 8.

The periodicity stated above,

tn N 1=2 � tn; (4.6)

is not an exact one for a finite N since both the condition(4.1) and Eq. (4.4) for nCr are only approximately satisfied.If the periodicity (4.6) were exact, tx given by (2.26)could be rewritten as

tx � tone-periodxX1

k��1

exp�N 1

2Rkx�; (4.7)

where tone-periodx is defined by

6This divergence is only an apparent one coming fromapplying the evaluation of tx given in (3.2) to the case withan infinitesimally small a. If the leading term (3.14) of tn ismissing from the start, we have to adopt a different way ofestimating the summation (2.26).

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tone�periodx �X�N 1=4�

n��N=4�

tnenx=R; (4.8)

with �c� being the largest integer not exceeding c.Namely, tone-periodx is the one-period part around n �

0 in the summation of tx. The geometric series multi-plying (4.7) is formally summed up to give an unwelcomeresult; it is equal to zero or the summation of deltafunctions for pure imaginary values of x. However, sincethe periodicity (4.6) is not exact for a finite N, we have tocarry out more precise analysis taking into account theviolation of the periodicity to obtain the profile tx in thelimit N ! 1. Here we would like to propose another wayof defining tx which could lead to a desirable rollingprofile. It is the N ! 1 limit of the one-period summa-tion (4.8):

tx � limN!1

tone-periodx: (4.9)

This is also formally equal to the original summation(2.26) with N � 1.

500 1000 1500 2000

1000

n

FIG. 9 (color online). Hn for n 2 �02048� in the case N �8191. The red (straight) line is an auxiliary one with slope 3=2.

-11


Let us return to the low temperature analysis withR2 � 1. For tx given by (4.9), it is sufficient to studythe n dependence of Hn only in the half period regionn 2 �0; �N 1=4��, which is shown in Fig. 9 for N �8191. As shown in Fig. 9, Hn has a complicatedstructure with peaks and valleys. However, if we neglectsuch local structures and see Fig. 9 globally, we find that Hn grows almost linearly in n; Hn / n [the red(straight) line in Fig. 9 is the line with slope 3=2]. Thiscould give a desirable tn with the behavior (1.4) up tocomplicated local structures. However, it is a nontrivialproblem whether the N ! 1 limit of (4.9) really existseven when we take into account the local structures. Herewe shall point out a kind of self-similarity of Hn and,hence, of tn; 2 HnjN ’ H2nj2N . Figure 10 shows Hn for N � 8191 (blue asterisks) and that for N �4095 (red diamonds). Both the horizontal and the verticalscales are doubled for the N � 4095 points. For example,the real coordinate of the red diamond 1500; 1600 in thefigure is actually 750; 800. Note that the red diamondsand the blue asterisks have overlapping local structures. Itis our future subject to study whether tx of (4.9) canexist for tn with such self-similarity.

V. THERMODYNAMIC PROPERTIES OF tx

In the BCFT analysis, the rolling tachyon solution isobtained by the inverse-Wick rotation of one space direc-tion compactified on a circle at the self-dual radius R � 1[1,28]. Therefore, also in our VSFT construction of therolling tachyon solution, it is natural to expect that themeaningful solution exists only at R � 1. This is alsosupported by the fact that the correct tachyon masssquared �1 has been successfully reproduced in theanalysis of the fluctuation modes around the D25-branesolution in VSFT [22–24]. If the tachyon mass squared isequal to �1, the natural mode of the expansion in (2.26)

500 1000 1500 2000

1000

2000

3000

4000

5000

∆H(n)

n

FIG. 10 (color online). Hn for N � 8191 (blue asterisks)and that for N � 4095 (red diamonds). Both the horizontal andthe vertical scales are doubled for the red diamonds.

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is enx with R � 1 [9,14]; in particular, the n � �1 modese�x are the massless modes at the unstable vacuum.

In this section, we shall study how this critical radiusR � 1 appears in our construction of time-dependentsolution in VSFT, especially in the component field tx.One would naively expect that the N ! 1 limit of oursolution (2.24) can exist only at R � 1. Here, we do notpursue this possibility directly, but instead address theproblem from a statistical mechanics point of view.Recall that tx (2.26) and tn (2.27) can be interpretedas the partition functions of a statistical system of chargeslocated on a unit circle with temperature R2. One possiblemechanism of R2 � 1 being a special point for thisstatistical system is that it undergoes some kind of phasetransition at R2 � 1. In this section, we first claim, on thebasis of a simple energy-entropy argument, the presenceof a phase transition at R2 � 1. Then, we study in moredetail the thermodynamic properties of the system bothanalytically and numerically. Our results here suggest butdo not definitely confirm the presence a phase transition atR2 � 1.

A. Boundstate phase and dissociated state phase

Let us consider the statistical system with partitionfunction tx � 0:7

t0 �X1

n0 ;n1 ;...;nN�1��1n0 n1 �� nN�1�0

exp��1

R2Hnr; n0

�: (5.1)

We would like to argue that this system has a possiblephase transition at temperature R2 � 1. In the low tem-perature region R2 � 1, configurations with lower energycontribute more to the partition function. The lowestenergy configuration of the Hamiltonian H (2.28) is ofcourse that with all nr � 0. Because of the self-energypart 2 lnN=4 of Qrr (2.25), the energy of a genericconfiguration with zero total charge can be lnN divergent.Finite energy configurations are those where the chargesare confined in finite size regions and the sum of chargesin each region is equal to zero. Namely, they consist ofneutral boundstates of charges. The simplest among themis the configuration of a pair of 1 and �1 charges with afinite separation. Let us consider the configuration fnk;

r gwith nk � 1, nk � �1, and all other nr � 0 for agiven position k and separation . The energy of thisconfiguration is

Hfnk; r g �

b2

2 ln�N4

� 2 ln

2 sin�� N

� : (5.2)

This is approximated in the close case � O1 (modN)and in the far separated case � ON (modN) by

7Although we consider here t0 with x � 0, tx � 0 and tnhave the same bulk thermodynamic properties since the dif-ference is only the local one at r � 0.

-12


Hfnk; r g �

8><>:b2 2 ln

��j j2

�; � O1

2 lnN; � ON:(5.3)

The energy in the finite separation case � O1 is in-deed free from the lnN divergence. Other neutral bound-states with finite energy are, for example, a pair ofcharges q and �qwith q > 1, and a chain of alternatingcharges

0; 0 . . . ; 0; q;�q; q;�q; . . . ; q;�q; 0; � � � ; 0: (5.4)

Configurations with isolated charges havelnN-divergent energy as seen from the � ON caseof (5.3). However, this does not imply that such configu-rations do not contribute at all to the partition function:We have to take into account their entropy. Let us considera naive free energy argument of an isolated charge. Asseen from Qrs (2.25) or (5.3) for � ON, the energy ofan isolated charge is E � lnN, while the entropy of thischarge is S � lnN since there are N points where it cansit. Therefore the free energy of this isolated charge isgiven by

F isolated charge � E � R2S � 1� R2 lnN: (5.5)

This means that, for R2 > 1, the free energy becomeslower as more isolated charges are excited. Namely, R2 �1 could be a phase transition point separating the bound-state phase in R2 < 1 and the dissociated state phase inR2 > 1. To confirm the existence of this phase transition,more precise analysis is of course necessary.

B. Dilute pair approximation

Before carrying out numerical studies of the system(5.1) for the possible phase transition at R2 � 1, we shallin this subsection present some analytic results valid inlow temperature. In the low temperature region R2 � 1, itshould be a good approximation to take into account onlythe pairs of charges as configurations contributing to thepartition function (5.1). This approximation is better forlarger b since more complicated boundstates such as (5.4)have larger energy coming from the b=2 term of Qrr(2.25). The partition function of a pair of charges is givenby summing over the position k and the separation ofthe configuration fnk;

r g:

Z1-pair �XN�1

k�0

XN�1

�1

e�Hfnk; r g=R2 : (5.6)

In the low temperature region where the number of pairexcitations is small and the pairs are far separated fromeach other, we can exponentiate Z1-pair to obtain

t0 � expZ1-pair: (5.7)

We call this ‘‘dilute pair approximation.’’ Using (5.2),Z1-pair is calculated as follows:

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Z1-pair �XN�1

k�0

XN�1

�1

e�b=2R2�N2

��2=R2

sin� N �2=R2

� e�b=2R2�N2

��2=R2 N2

�

Z �1�1=N

�=Ndysiny�2=R

2

�

8>>>><>>>>:2N

2=R2�1

��eb=42

��2=R2

; R2 < 2

N2�2=R2

�eb=42

��2=R2 (12�

1

R2��

�p(1� 1

R2; R2 > 2:

(5.8)

In the second line of (5.8), we changed the summationto the integral with respect to y � � =N for N � 1.This integral is divergent (convergent) at the edges asN ! 1 in the region R2 < 2 (R2 > 2). The final resultof (5.8) in R2 < 2 has been obtained by taking the con-tribution near the edges y� �=N and �1� 1=N, whilethat in R2 > 2 by extending the y-integration region to�0; ��.

As we raise R2, the more pairs are excited and theseparation of the two charges of a pair becomes larger,leading to the breakdown of the dilute pair approxima-tion. From (5.8), one might think that this breakdownoccurs at R2 � 2. However, let us estimate more preciselythe limiting temperature above which the dilute pairapproximation is no longer valid. The criterion for thevalidity of the dilute pair approximation is given by

hpi � & N; (5.9)

where hpi and are the average number of pairs and theaverage separation between 1 and the �1 charges of apair, respectively. First, hpi is given by

hpi �1

t0

X1p�0

pp!

Z1-pairp � Z1-pair: (5.10)

The average separation is calculated as follows:

�N�

�

R�1�1=N�=N dyminy;�� ysiny�2=R

2

R�1�1=N�=N dysiny�2=R

2

�

8>><>>:1�R2=21�R2

; R2 < 1�N�

�2�2=R2

�1R2

� 12

� ��

p(1� 1

R2

(32�1

R2; 1<R2 < 2;

(5.11)

where the separation is the smaller one between andN � . We are considering only the region R2 < 2 sincewe have hpi � N2�2=R

2� N in the other region R2 > 2.

Note that the critical R2 below which the y integration inthe numerator of (5.11) diverges at the edges has changedto R2 � 1 due to the presence of miny;�� y.

As seen from (5.11), the average separation divergesas R2 " 1. This is consistent with the boundstate-dissoci-ated-state transition which we claimed to occur at R2 �1. From (5.8), (5.10), and (5.11), the condition (5.9) for the

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0.2 0.4 0.6 0.8 1 1.2 1.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.6

0.8

E

Cv

R2


validity of the dilute pair approximation (5.11) is nowexplicitly given by

1

1=R2 � 1

��eb=4

2

��2=R2

& 1; R2 < 1: (5.12)

The breakdown temperature of the dilute pair approxi-mation determined by (5.12) becomes lower as b is de-creased. Moreover, for smaller b we have to take intoaccount also longer neutral boundstates such as (5.4) and,hence, the dilute pair approximation becomes worse thanthe above estimate. However, we would like to emphasizethat the free energy argument of an isolated charge using(5.5) holds independently of the value of b.

C. Monte Carlo analysis of internal energyand specific heat

In order to study whether the boundstate-dissociated-state phase transition which we predicted in Sec. VAreally exists, we have calculated using Monte Carlomethod the internal energy E and the specific heat CV

0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

R2

FIG. 12. E and CV vs R2 for N � 512, b � 1, and x � 0.

1 2 3 4

0.5

1

1.5

2

1 2 3 4

0.2

0.4

0.6

0.8

1

E

R2

Cv

R2

FIG. 11. E and CV vs R2 for N � 512, b � 10, and x � 0. Thecurves in the smaller region of R2 and the straight lines in thelarger R2 region have been obtained by the dilute pair approxi-mation and the high temperature approximation, respectively.

086010

of the system tx � 0:

E � �@@

1

Nlnt0; (5.13)

CV � 2@2

@ 21

Nlnt0: (5.14)

Figures 11 and 12 show E and CV for b � 10 and b � 1,respectively, (N � 512 in both figures). The curves in thesmaller R2 region have been obtained by using the dilutepair approximation (5.7):

1

Nlnt0 ’

2

2 � 1

��eb=4

2

��2

; R2 � 1: (5.15)

The straight lines in the larger R2 region are from thehigh temperature approximation [cf. (3.10)]:

1

Nlnt0 ’ �

1

2ln ; R2 � 1; (5.16)

giving

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2 4 6 8 10

0.5

1

1.5

2

2 4 6 8 10

0.2

0.4

0.6

0.8

1

E

Cv

β

βFIG. 13. E and CV for b � 0 and N � 511. Here, the hori-zontal axis is . The curve of E in the dilute pair approxima-tion in the larger region is invisible since it almost overlapswith the axis.

. PHYSICAL REVIEW D 70 086010

E ’ 12R2; CV ’ 1

2; R2 � 1: (5.17)

The curves of the dilute pair approximation fit better withthe data in the b � 10 case than in the b � 1 one. This isconsistent with our analysis in Sec. V B using (5.12).

Figures 11 and 12 show no sign of phase transitionaround R2 � 1. Note that there is a peak structure inCV . The R2 of the peak becomes larger as the parameterb is increased.We have carried out simulations for smallervalues of N, and found that CV , and, in particular, theheight of the peak are almost independent of N.Therefore, the peak in CV is not a signal of a secondorder phase transition.8

We claimed in Sec. IV that we have to set b � 0 andN � odd for obtaining a sensible rolling solution in ourconstruction. Figure 13 shows E and CV of t0 for b � 0and N � 511. The high temperature approximation (5.17)fits well with the data in the region & 3. However, thedilute pair approximation is not a good approximationeven in the region of the largest in the figure. As wementioned in Sec. V B, we have to take into accountlonger neutral boundstates besides the simple pair forobtaining a better low temperature approximation.9 Inany case, we cannot observe any sign of lower orderphase transitions from the figure.

D. Correlation function

Next we shall investigate the correlation functions ofnr in the low and the high temperature regions. Here, weconsider the two-point correlation function hnrnr i inthe system with partition function tx � 0 for a largedistance j j � 1 (modN):

hnknk i �1

t0

X1n0 ;...;nN�1��1

n0 �� nN�1�0

nknk e� Hnr;n0: (5.18)

Let us consider the low temperature region R2 � 1.There are two candidate configurations with lower energywhich mainly contribute to (5.18). One is the configura-tion of a pair excitation, f�nk;

r g, which we defined inSec. VA; nk; nk � �1;!1 and all the other nr � 0.The other configuration is the chain of alternating signcharges (5.4) with ends at r � k and k ;nk; nk 1; . . . ; nk � �1;�1; . . . ;�1; 1 and all othernr � 0. We call it the chain excitation. This chain exci-tation exists only in the case of odd . In the case of even

ROLLING TACHYON SOLUTION IN VACUUM STRING . .

8The global structure of CV with a peak and the asymptoticvalue of 1=2 can roughly be reproduced from a simple partitionfunction

P1n��1 exp� bn

2=4 neglecting the Coulomb inter-actions although the position and the height of the peak differfrom those obtained by simulations.

9In fact, incorporation of the chain excitation (5.4) with q �1, whose energy is given by (5.19) in the next subsection forsufficiently large length , seems to considerably improve thelow temperature approximation for b � 0.

086010

, there are similar alternating sign configurations withzero total charge. Here, for simplicity, we consider onlythe case of odd .

The energy of the pair excitation is given by (5.2). Wecalculated numerically the energy of the chain excitationHchain. The dots in Fig. 14 show Hchain for the variouslengths (N � 1023 and b � 0). These points are wellfitted by the curve Hchain � 3:42 0:500 ln�sin� =N�.We have carried out this analysis for various values of N,and found that the dependence of Hchain on (sufficientlylarge) and N is given by

Hchainjb�0 �1

2lnN

1

2ln�sin�� N

�� const: (5.19)

In particular, the coefficient of lnsin� =N is 1=2 andindependent of N. In the case of b � 0, Hchain has anadditional self-energy contribution b=4 1. Thus,we find that the contributions of the two kinds of excita-tions to the correlation function are

-15


hnknk i �

8>><>>:N�2=R2

sin�� N

� �2=R2

: pair excitation;

N�1=2R2e�b =4R2

sin�� N

� �1=2R2: chain excitation:

(5.20)

Equation (5.20) is valid in a wide range of including both � ON=2 and 1 � � N=2. In particular, in the region1 � � N=2, the N dependence cancels out and we obtain

hnknk i �1� �N=2

8><>:�

1

2=R2

�: pair excitation;

e�b =4R2 1

1=2R2

: chain excitation:(5.21)

200 400 600 800 1000

2.25

2.5

2.75

3

3.25

Hchain

∆

From(5.20) and (5.21), we see the following. In the case ofb � 0, hnknk i is given by the contribution from the pairexcitation, the upper ones in (5.20) and (5.21), for a largedistance � 1. Contribution of the chain excitation issuppressed by e�b =4R2. On the other hand, in the case ofb � 0, hnknk i is given by the contribution from thechain excitation, the lower ones in (5.20) and (5.21).

Next, let us consider the high temperature region R2 �1. For R2 � 1, approximating the summations over nr in(5.18) by integrations over continuous variables pr �nr=R, we obtain

hnknk i � R2Q̂�1k;k : (5.22)

Incidentally, Q̂�1k;k with k � 0 is related to nCr (3.6)by10

Q̂�10 �det ^̂Q

detQ̂nC : (5.23)

We calculated Q̂�1k;k numerically and found that it isindependent of the position k; namely, the translationalinvariance holds for large N despite that the self-energyb=4 is missing for n0 [see (2.25)]. Therefore, Q̂�1k;k isessentially equal to nC up to a -independent factor.

The dependence of nC is shown in Fig. 2 in the casesof b � 0:1 and 10. As we mentioned in the footnote there,nC and hence Q̂�1k;k have quite different depen-dences in the smaller region: for larger b, Q̂�1k;k isnegative definite, while it has alternating sign structurefor smaller b. However, for � ON=2 in the midre-gion, Q̂�1k;k is negative definite and has the followinguniversal dependence for any nonvanishing b as weshall see below:

Q̂�1k;k � N�2

sin�� N

� �2; � � ON=2�:

(5.24)

10Numerical analysis shows that det ^̂Q= detQ̂ is finite in thelimit N ! 1 for b � 0, while we have det ^̂Q= detQ̂ ’ 0:59N forb � 0.

086010

First, for larger b, Q̂�1k;k is well fitted by (5.24) forany (see Fig. 15). We have confirmed the N dependenceof (5.24) by the fitting for various N. Next, let us considerQ̂�1k;k for a smaller b. Figure 16 shows Q̂�10 in thecase of b � 0:01 and N � 511 in all the region of [16(a)], and lnjQ̂�10 j in the midregion 180 " "

332 where Q̂�10 is negative definite [16(b)]. As wesee from 16(b), the dependence of (5.24) holds wellin the midregion of . By carrying out this analysis forvarious N, we have confirmed the N dependence of (5.24)also for smaller b.

The behavior of Q̂�10 in the case of b � 0 and N �odd is quite different from the nonzero b case above. nCr inthis case is given by (4.4), and Q̂�10 is shown in Fig. 17for N � 511 and b � 0. Q̂�10 is a linear function of with alternating sign for all .

From the above analysis of hnknk i in both the R2 �1 and R2 � 1 regions, we find the following. In the caseof b � 0, hnknk i is given by the upper one in (5.20) inthe R2 � 1 region, and by (5.24) in the R2 � 1 region.Namely, hnknk i has the same kind of dependencej sin� =Nj�/ although the exponent / differs in thetwo regions. On the other hand, hnknk i in the b � 0case has entirely different dependences in the two

FIG. 14. The energy of the chain excitation at various in thecase of N � 1023 and b � 0 (dots). The curve is Hchain �3:42 0:500 ln�sin� =N� obtained by fitting.

-16

100 200 300 400 500

-300

-200

-100

100

200

300

∆

(Q−1)0∆

FIG. 17. Q̂�10 for N � 511 and b � 0.

100 200 300 400 500

-10

-5

5

10(a)

(b)

ln |(Q-1)0∆|

(Q-1)0∆

∆

200 220 240 260 280 300 320

-13.1

-13.05

-12.95

-12.9

∆

FIG. 16 (color online). Part (a) shows Q̂�10 for b � 0:01and N � 511 in all the region � 1; . . . ; N � 1. Part (b) showslnjQ̂�10 j and the fitted solid (red) curve �13:09�2:00 lnj sin� =Nj only in the midregion 180 " " 332.

100 200 300 400 500

-12

-10

-6

-4

-2

ln |(Q-1)0∆|

∆

FIG. 15 (color online). lnjQ̂�10 j for N � 511 and b � 10(dots) and the fitted solid (red) curve �12:02�2:107 lnj sin� =Nj.


regions; it is given by the lower one in (5.20) in the R2 �1 region, and by (4.4) in the R2 � 1 region. This resultsuggests the existence of a phase transition at an inter-mediate R2 at least in the b � 0 case. Further study of thecorrelation function in the intermediate region of R2

using Monte Carlo simulation is needed.

VI. SUMMARY AND DISCUSSIONS

In this paper, we constructed a time-dependent solutionin VSFT and studied whether it can represent the rollingtachyon process. Our solution is given as the inverse-Wickrotation of the lump solution on a circle of radius R whichis given as an infinite number of � products �b ��b �� b of a string field �b with Gaussian momentumdependence e�bp2=4. We focused on one particular com-ponent field in the solution, which has an interpretation asthe partition function of a Coulomb system on a circlewith temperature R2. Our finding in this paper is that, forthe solution not to diverge in the large time limit, we haveto put b � 0 and take the number of �b constituting thesolution to infinity by keeping it even. We also examinedthe various thermodynamic quantities of our solution as aCoulomb system to see whether the self-dual radius R �1 has a special meaning. We pointed out a possibility thatR � 1 is a phase transition point separating the bound-state phase and the dissociated state phase. Our analysisof the correlation function for b � 0 supports the exis-tence of the phase transition.

Many parts of this paper are still premature and needfurther study. The most important among them is to studyin more detail our solution with b � 0: whether the limitN ! 1 really exists and, if so, what the profile will be. Inthis paper we tried to find a special nature of the criticalradius R � 1 in the thermodynamic properties of thesystem. However, we have to find a more direct relevanceof R � 1 to our solution. For example, the most naturalscenario is that the limit N ! 1 can exist only at R � 1.

086010

Analysis of the whole of our solution not restricted to thecomponent tx0 is also necessary.

Originally our time-dependent solution had two pa-rameters, b and R. If we have to put b � 0 and R � 1 forobtaining a solution with a desirable rolling profile, thissolution seems to have no free parameters. However, therolling solution should have one free parameter which

-17


corresponds to the initial tachyon value at x0 � 0. It isanother problem to find the origin of this parameter. Itmight be necessary to generalize our solution to incorpo-rate this parameter.

After establishing the rolling solution, our next task isof course to apply our solution to the analysis of unre-solved problems in the rolling tachyon physics.

ACKNOWLEDGMENTS

We would like to thank H. Fukaya, M. Fukuma,Y. Kono, T. Matsuo, K. Ohmori, S. Shinomoto,S. Teraguchi, and E. Watanabe for valuable dis-cussions. The work of H. H. was supported in part by aGrant-in-Aid for Scientific Research from Ministry ofEducation, Culture, Sports, Science, and Technology(No. 12640264).

APPENDIX: PROOF OF EQ. (4.5)

In this appendix, we present a proof of (4.5) for nCrgiven by the right-hand side (rhs) of (4.4). Since this nCrsatisfies

PN�1r�0 n

Cr � 0, and for such nCr we have

XN�1

s�0

Q̂rsnCr �XN�2

s�0

QrsnCs �XN�1

s�0

QN�1;snCs ; (A1)

086010

Eq. (4.5) holds if we can show that

XN�1

s�0

QrsnCs � r-independent term O�1

N

�;

r � 0; 1; . . . ; N � 1:(A2)

Before starting the proof of (A2) for nCr of (4.4), weshall mention Eq. (4.2) in the b � 0 and N � even case.This (4.2) holds owing to a stronger equationPN�1s�0 Qrs�1

s � 0, which is rewritten explicitly as

XN�1

r�1

�1r ln

2 sin��rN

� � ln�N4

�; N � even:

(A3)

Formulas essentially equivalent to (A3) can be found inthe various tables of series and products.

Now let us consider the left-hand side of (A2) for nCrgiven by the rhs of (4.4). Taylor expanding lnj2 sin��r�

s=N�j � Re ln1� e2�ir�s=N in Qrs (2.25) in powerseries of e2�ir�s=N , we have

�1r

2

XN�1=2

s��N�1=2

QrsnCs � ln�N4

��1�

2jrjN 1

� Re

X1k�1

1

k

XN�1=2

s��N�1=2�r

�e2�ik=Nr�s�1�

2jsjN 1

�

� ln�N4

��1�

2jrjN 1

� X1k�1

1

k

�fk �

�1�

2jrjN 1

��; (A4)

with fk defined by

fk � ��1r

N 1��1�N 1=2� k�ei�k=N e�i�k=N� � 2

ei�k=N e�i�k=N2

� �e2�ik=Nr e�2�ikr=Nr�: (A5)

In obtaining the last line of (A4), we have applied theformula

XM�1

s��M 1

z�s�1�

jsjM

��1

M�zzM z�M � 2

1� z2; (A6)

to the case ofM � N 1=2 and z � �e2�ik=N, and used�e�2�ik=NN 1=2 � �1N 1=2 � �e�i�=Nk.

Note that fk defined above has the periodicity:

fk N � fk: (A7)

Introducing the cutoff LN in the k summation in (A4) andusing the periodicity to make the manipulationPLNk�1 fk=k �

PNk�1 fk

PL�1p�0 1=pN k, we have

XLNk�1

1

k

�fk �

�1�

2jrjN 1

��1

N

XNk�1

fk

� �L

kN

��

�kN

��

�1�

2jrjN 1

�� LN 1 � 1�

�L�1

1

N

XNk�1

fk

�lnL�

�kN

��

�1�

2jrjN 1

��lnLN /�; (A8)

-18


where z is the polygamma function:

z �@@zln(z � �/�

X1n�0

�1

n z�

1

n 1

�: (A9)

In obtaining the last line of (A8), we have used 1 ��/ and the asymptotic behavior of z for jzj � 1:

z ’ lnz� 1 1

2z� 1�

1

12z� 12 � � � : (A10)

Now we have to carry out the two summations in (A8),

S1 �1

N

XNk�1

fk; S2 �1

N

XNk�1

fk �kN

�;

for largeN. One way to evaluate S1 is to approximate it bya contour integration with respect to z � e2�ik=N:

S1 �N�1

1

2�i

Ijzj�1

dzzfz � Resz�0

1

zfz � 1�

2jrjN 1

;

(A11)

where fz is

fz ��zN 1

��zN 1=2 �1=zN 1=2 � 2

1 z2

� ��zr �1=zr�: (A12)

Note that fz=z is regular at z � �1.Another way of evaluating S1 is to observe the follow-

ing. fk is of O1=N except at k� N 1=2, where fk �ON. Therefore, we have only to carry out the k sum-mation around k� N 1=2. Expressing k as k � N 1=2 ‘, fk is expanded around k � N 1=2 as

fk�N 1=2 ‘ �1

N 1

�N2

�2‘ 122 N�1‘

�‘ 12

O1�

� cos�2�rN

�‘

1

2

��; (A13)

and we regain the same result as (A11):

S1 ’1

N

X1‘��1

fk�N 1=2 ‘ �1

N 1

�N�1�

2jrjN

� 1

�

� 1�2jrjN 1

;

(A14)

086010

where we have used the formulas

X1n�0

cos2n 1

2n 12��4

��2

� jxj�; (A15)

X1n�0

�1ncos2n 1x2n 1

��4;

��2< x<

�2

�:

(A16)

The other summation S2 is evaluated in the samemanner:

S2 ’X1

‘��1

fk�N 1=2 ‘

� �1

2

� 1

N

�‘

1

2

� 0

�1

2

��

� �/ ln4�1�

2jrjN 1

�;

(A17)

where we have used 1=2 � �/� ln4, and that thesummations,

X1‘��1

�1

‘ 12

; �1‘�cos

�2�rN

�‘

1

2

��;

vanish due to the oddness under ‘ ! �‘� 1. Recallingthat

�1r

2

XN�1=2

s��N�1=2

QrsnCs � ln�N4

��

�1�

2jrjN 1

�

S1 � lnL� S2

�

�1�

2jrjN 1

��lnLN /�;

(A18)

and plugging the results (A14) [or (A11)] for S1 and (A17)for S2 into (A18), we find that the terms on the rhs justcancel out. Therefore, the r-independent constant on therhs of (A2) is in fact equal to zero.

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