PHYSICAL REVIEW D 67, 086002 ~2003!

Butterfly tachyons in vacuum string field theory

Peter Matlock*Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

~Received 4 December 2002; published 17 April 2003!

We use geometrical conformal field theory methods to investigate tachyon fluctuations about the butterflyprojector state in vacuum string field theory. We find that the on-shell condition for the tachyon field isequivalent to the requirement that the quadratic term in the string-field action vanish on shell. This furthermotivates the interpretation of the butterfly state as a D-brane. We begin a calculation of the tension of thebutterfly, and conjecture that this will match the case of the sliver and further strengthen this interpretation.

DOI: 10.1103/PhysRevD.67.086002 PACS number~s!: 11.25.Sq

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I. INTRODUCTION

There are now several known solutions@1–4# to the equa-tions of motion of vacuum string field theory~VSFT! @5#.These include the sliver state@2,3,6# and the butterfly state@1#. The sliver state was conjectured to represent a Dbrane, and subsequent calculations of its tension basethis assumption yielded the correct brane tension. The slstate construction was used also to build solutions cosponding to Dp-branes of arbitrary dimension@7#, and ratiosof tensions were found@8,9# which were in agreement withthe known results from string theory. These calculatiowere based on a field expansion of fluctuations aboutclassical solution, using a tachyon field to find the bratension. One requirement for the consistency of the interptation of the sliver brane as a D-brane is that the equatiomotion for the tachyon must be a consequence of the stfield equation of motion; an on-shell string field must corspond to an on-shell tachyon. This means that the quadterm in the resulting tachyon action must vanish, as iscase for the sliver state@10#. While it has already been assumed in the literature that the butterfly can be interpretea D-brane, this may have been premature, as the above perties had not been verified. Were the tachyon field nosatisfy these requirements, it would mean that the buttestate, although known to be a brane~i.e. localized! solutionand a rank-one projector, could not be viewed as a D-braWe show in the present paper that the butterfly does indsupport a tachyon field with vanishing on-shell quadraterm.

We now turn our attention to the brane tension. The tsion of the brane may be obtained from the cubic term intachyon action@8–10#, and this has been completed for thcase of the sliver by Okawa@10#. In that calculation, theevaluation of the cubic terms turned out to be very tedioand lengthy, and in the present case of the butterfly wethat it is much more so. Thus rather than attempting a calation of the entire cubic term, we content ourselves wmotivating the procedure and conjecture how the reshould obtain. In this way, we will see how the correct bratension should arise.

This paper is structured as follows. In Sec. II we revie

*Electronic address: pwm@sfu.ca

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the geometrical construction of the butterfly state as a surstate@5#, along with the regularization required for any cocrete calculations@1#. We then in Sec. III turn to the expansion of deformations of a VSFT solution@10#. In Sec. IV weinvestigate the quadratic term in the tachyon action. Tsurface-state calculation involves the construction of conmal mappings in order to perform star multiplication. In SeV we begin the calculation of the ratios of tensions of diffeent butterflies, suggest how this could be completed,comment on the preliminary results. We conclude in Sec.with a discussion of surface states and the roˆle of regulariza-tion and conformal invariance with respect to deformatioof string fields and definition of fields.

II. THE BUTTERFLY STATE

We use the geometrical, surface representation of stfields; thus we specify states using a BPZ product witharbitrary stateuf&. For details of this representation anmethods thereof, we refer the reader to@5#.

The butterfly stateuB& is a factorizable state, so thatmay be decomposed into the product of a left-string futional and a right-string functional. The surfaceS, definedby 2p/2,Rez,p/2 andImz.0, used to define the butterfly is shown in Fig. 1. The unshaded region is the lopatch,2p/4,Rez,p/4, Imz.0. The dashed lines whichborder the local patch are the left- and right-string bounaries, and the solid line is the boundary of the surfacewhich we impose the standard open string boundary cotion. In the center of the local patch is the punctureP wherewe insert the operatorf, transformed to this coordinate system from the canonical half-disk via the mappingf. Thus the

FIG. 1. The butterfly, defined on the surfaceS.

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PETER MATLOCK PHYSICAL REVIEW D67, 086002 ~2003!

Belavin-Polyakov-Zamolodchikov~BPZ! product of the but-terfly with the arbitrary state represented by the operatorf is

^Buf&5^ f +f~P!&S , ~1!

where the combination of the mappingf and the surfaceSreally only need be defined up to conformal equivalenThis is not strictly true of the states we will define in subsquent sections, due to regularization of operator shdistance singularities.

We will also have a need for the regularized butteruBh&, and we borrow the formulation from@1#. The surfaceSh , shown in Fig. 2, is obtained fromS by identifying theleft and right edges, above some heighth. This is the regu-larization parameter, and the limith→` will be taken toobtain the surfaceS. The state is now defined by

^Bhuf&5^ f h+f~P!&Sh. ~2!

The regularized stateuBh& does not satisfy the string-fielequation of motion for finiteh.

III. TACHYON FLUCTUATIONS

In @10# an elegant proposition was made regarding bthe parametrization of string-field fluctuations by fields athe construction of states representing the coefficientsthese fields. We here present this briefly, referring the reato that paper for details.

The string-field action is given by

S521

2^CuQuC&2

1

3^CuC* C&. ~3!

Since the VSFT operatorQ is purely ghost, there are factoizable solutionsC5Cm^ Cg satisfying

QCg1Cg* Cg50 ~4!

Cm* Cm5Cm . ~5!

As the ghost part of the solution is thought to be in sosense universal@11,12#, attention has mainly been given tthe matter part of the solution. From@10#, a finite deforma-tion of the matter solution parametrized by fields$w i% isgiven by

FIG. 2. The regularized butterfly, defined onSh .

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^C$w i %uf&5NK expF2E

]̃SdzE dk

3(i

w i~k!Ow i (k)~z!G f +f~P!LS

, ~6!

where]̃S refers to the portion of the boundary ofS belong-ing to the stateuC$w i %

&, as opposed to the reference stateuf&.$w i% are fields which parametrize the deformation, whOw i (k) are the corresponding vertex operators. The integrasuch a vertex operator, which is of conformal dimension ofor on-shell physical states, is thus conformally invariant.

In the case of a tachyon deformation, we have

^e2Tuf&

5NK expF2E]̃S

dzE dkT~k!e2 ik•X~z!G f +f~P!LS

.

~7!

Expanding in powers of the tachyon fieldT, we have

^e2Tuf&5(j

^Tj uf&, ~8!

where

^Tj uf&5 K 1

j ! S 2E dzE dkT~k!e2 ik•X~z! D j

f +f~P!LS

.

~9!

One must take care to regularize these states when taBPZ products, as short-distance singularities will obtain.

uT0& is nothing but the classical solutionuCm&. The nextterm in the series~8! is

uT1&52E dkT~k!uxT~k!&, ~10!

where the tachyon stateuxT(k)& is given by the integral ofthe tachyon vertex operator along the boundary of the sface. The linearized equation of motion for the tachyon stis then

uxT~k!&5uxT~k!* Cm&1uCm* xT~k!&. ~11!

It is shown in @10# that in the case of the sliver stateCm5Jm this equation is satisfied on-shell not only for thtachyonxT(k) but for all physical string statesuxw i

(k)&. Inthe case of the butterfly, it is easy to see that the lineariequation of motion Eq.~11! is satisfied.

Inserting the expansion~8! into the action~3!, the termquadratic in the tachyon field is immediately given by

S(2)52^CguQCg&S 1

2^T1uT1&1^T2uT0&

2^T1uT1* T0&2^T2uT0* T0& D ~12!

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BUTTERFLY TACHYONS IN VACUUM STRING FIELD THEORY PHYSICAL REVIEW D67, 086002 ~2003!

52K2

~2p!26E dkK~k2!T~k!T~2k!, ~13!

whereK(k2) consists of four pieces coming from the abofour contributions:

1

2K~k2!5

1

2K111K202K1102K200. ~14!

In @10#, K(k2) was found to be identically zero in the casethe sliver state. In the following section we performanalogous calculation for the case of the butterfly.

Since we wish to deal with tension in Sec. V, we will alneed the part of the action cubic in the tachyon, given b

S(3)52^CguQCg&S ^T3uT0&2^T3uT0* T0&1^T2uT1&

2^T2uT1* T0&2^T2uT0* T1&21

3^T1uT1* T1& D ~15!

52K3

~2p!26E dk1dk2dk3d~k11k2

1k3!V~k1 ,k2 ,k3!T~k1!T~k2!T~k3!, ~16!

whereV is a function containing contributions from eachthe six terms in the action. We consider the on-shell case,write this quantity as

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3V5V301V212V3002V2102V2012

1

3V111. ~17!

For the case of the sliver, Okawa@10# showed that the firsfive terms together cancel; the cubic action is given solelythe sixth termV111 and thus so is the brane tension. We wdiscuss the case of the butterfly in Sec. V.

IV. QUADRATIC TACHYON ACTION

Here we calculate the quadratic term in the tachyontion, and show that it vanishes on-shell. From Eq.~12! wehave four pieces.

For the termK11, we use the mapping shown in Fig. 3map the surface obtained by gluing together two copiesSh onto a cone of circumferencep. The boundary of the firsand second copies ofSh will be denotedg andg*. They are

FIG. 3. Mapping for two-state BPZ product.

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shown in the figure as heavy dashed and solid lines. Fthe Appendix, the mapping is given by

cos 2u51

hcos 2z, ~18!

whereh5cosh2h, so that

dz

du[s~u!5

«~u!sin 2u

A 1

h22cos22u

, ~19!

where «(u) denotes the sign ofReu. From the previoussection, we have

K11;K EgdzE

g*dz8eik•X~z!eik8•X~z8!L , ~20!

where by; we indicate that there this requires regulariztion; we see there will be short-distance singularities atz56p/4. We thus regularize the state by leaving a gap of 2dzat each of these points. The limitdz→0 will be taken at theend of the calculation. Transforming to theu coordinate, wehave

K115Eu31du

u72duduH E

u71du

u9du81E

u1

u32dudu8J

3^eik•X~u!eik8•X~u8!&Cp, ~21!

where Cp is a cone with opening anglep, and du is ourregulatordz , transformed to the theta system:

du51

s~u!U

u56p/4

dz51

hdz . ~22!

The propagator on the boundary of the cone is given by

^eik•X~u!eik8•X~u8!&Cnp5d~k1k8!Un sin

u2u8

n U2kk8,

~23!

and considering the on-shell case,k251, we have

K115Eu31du

u72duduH E

u71du

u9du81E

u1

u32dudu8J csc2~u2u8!.

~24!

Performing the integrals, we obtain

K11522 log sin 2du ~25!

→22 log 222 logdz12 logh, ~26!

where we have made use of the limitdz→0. We see thatthere is a finite and a divergent part toK11 for finite h, andalso that there is a divergence ash→`. We expect the partdivergent indz to cancel betweenK11 andK110 in Eq. ~14!,since products ofuT1& states must be regularized at the e

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PETER MATLOCK PHYSICAL REVIEW D67, 086002 ~2003!

FIG. 4. Mapping for three-state BPZ product.

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points of vertex operator integration, while theuT2& statecontains a different, independently regularized divergenc

Turning our attention now toK110, we use the mappingshown in Fig. 4. We may again write

K110;K EgdzE

g*dz8eik•X~z!eik8•X~z8!L . ~27!

Recalling the mapping in the Appendix, the relation~18! andthe derivative~19! remain unchanged. The mapping is noto a cone of angle 3p/2, so substitutingn53/2 into thetwo-point function~23!, we may write

K11054

9Eu31du

u72duduE

u71du

u112dudu8csc2

2

3~u2u8!. ~28!

Again, performing the integrals, we have

K11053

2log 323 log 212 logdz1 logh, ~29!

so that the combination

1

2K112K11052 log 22

3

2log 3 ~30!

is finite, but non-zero.In order to calculate theK20 and K200 contributions,

greater care must be taken with the regularization procedThe uT2& state from Eq.~9! involves the a double integrasince

K S 2E dze2 ik•X~z! D 2L 5E dzdz8^e2 ik•X~z!e2 ik•X~z8!&,

~31!

but this will be divergent whenz;z8.Looking at both Fig. 3 and Fig. 4, let us calculateK20 and

K200 simultaneously. We follow Okawa@10# and regularizethe double integral~31! as

Eg;z31ez

z7dzE

g;z3

z2ezdz8^e2 ik•X~z!e2 ik•X~z8!&, ~32!

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where thez integral is to be taken along the contourg from‘‘just after’’ the beginning until the end atz7, and thez8integral is taken along theg from the beginning atz3 until‘‘just before’’ the point z. That is, the small quantityez isequal toe, i e, 2 i e ande for each segment ofg respectively,where reale is then the regularization parameter. Using thformulation in theu system, we write

K20(0)5Eu31eu

u7duE

u3

u2eudu8v2csc2v~u2u8!, ~33!

where

eu51

s~u!ez ~34!

so that in the inner integral, the upper limit of integratiodepends onu. In order to do both calculations at once, whave introduced the constantv; for the cases ofK20 andK200, v51 andv52/3, respectively. Let us first performthe inner integration, giving

K20(0)5Eu31eu

u7duS 1

eu2v cotv~u2u3! D ~35!

→F 1

ezE dus~u!2 log sinvuG

u31eu

u7

. ~36!

Substitutings(u) from Eq. ~19!, we have

E dus~u!52«~u!

2tan21

sin 2u

A 1

h22cos22u

. ~37!

We may now evaluate Eq.~35!, obtaining

K20(0)521

etan21

sin 2u4

A 1

h22cos22u4

2 log sinvp

21 log

ve

2.

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BUTTERFLY TACHYONS IN VACUUM STRING FIELD THEORY PHYSICAL REVIEW D67, 086002 ~2003!

We thus find the contribution

K202K200522 log 213

2log 3 ~39!

to Eq. ~14! to be again finite and non-zero, precisely cancling the contribution~30! to the quadratic term~14!.

V. CUBIC TACHYON ACTION AND BRANE TENSION

We have verified in Sec. IV that the butterfly does indesupport a tachyon field. In order to further motivate its intpretation as a D-brane state, the brane tension should alscalculated. This was properly done for the sliver in@10#, andwe consider the same procedure here for the butterfly.

We recall that the tachyon coupling constant can be idtified by first canonically normalizing the tachyon field usinthe quadratic term, and then extracting the coefficient ofcubic term. This coupling will be related to the tension, athe tension may be expressed in terms of the energy den

The cubic term in the action contains the quantity froEq. ~17!,

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3V5V301V212V3002V2102V2012

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3V111. ~40!

These may be calculated as follows. The three-point funcon the boundary of the coneCnp is given by

Pn~w1 ,w2 ,w3!d~k11k21k3!

[^eik1•X~w1!eik2•X~w2!eik3•X~w3!&Cnp~41!

51

n3 Ucscuw12w2u

ncsc

uw22w3un

cscuw32w1u

n U3d~k11k21k3!. ~42!

In the following, as before, we imply the use of Fig. 3 aFig. 4 for two- and three-state products, respectively. Wethe same regularization parametersdu5(1/h)dz and ef[@1/s(f)#ez as in the preceding section.

We begin withV111 and write

V111;K Egdw1E

g*dw2E

g̃dw3eik•X~w1!

3eik•X~w2!eik•X~w3!L . ~43!

Regularization may be carried out as before, so that

V1115Eu31df

u72dfdf1E

u71df

u112dfdf2H E

u111df

u13df3

1Eu1

u32dfdf3J S 2

3D 3

csc2

3~f12f2!csc

2

3

3~f22f3!csc2

3~f32f1!. ~44!

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The three integrals in this case are independent~i.e. the in-tegration bounds do not depend on the integration variab!and we find that this is identical toV111 in the case of thesliver, calculated in@10#, that is

V1115p2

3. ~45!

Moving on toV21 andV210 we may write the unregularized quantity as

V21(0);Egdw1dw2E

g*dw3P1/v~w1 ,w2 ,w3!, ~46!

where we have written both states simultaneously usingv,as we did in Sec. IV. We regularize as we did withK20(0) :

V21(0)

52Eg;z31ez3

z7dw1E

g;z3

w12ew1dw2E

g*dw3P1/v~w1 ,w2 ,w3!.

~47!

In the u system, this becomes

V2152Eu31eu3

u7df1E

u3

f12ef1df2

3H Eu71du

u9df31E

u1

u32dudf3J csc~f12f2!

3csc~f22f3!csc~f32f1!. ~48!

Here we are using the regulatore just as we did in the pre-vious section.

V2105V201 may similarly be expressed as

V21052Eu31eu3

u7df1E

u3

f12ef1df2E

u71du

u112dudf3S 2

3D 3

3csc2

3~f12f2!csc

2

3~f22f3!csc

2

3~f32f1!.

~49!

Finally, V30 andV300 may be written

V30(0);Egdw1dw2dw3P1/v~w1 ,w2 ,w3!, ~50!

which may be regularized and written as

V30(0)54Eu312tu3

u7df1E

u31tu3

f12tf1df2E

u3

f22tf2df3v3

3cscv~f12f2!cscv~f22f3!cscv~f32f1!.

~51!

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PETER MATLOCK PHYSICAL REVIEW D67, 086002 ~2003!

Here,t is a new regulator, which is used exactly as ise inprevious expressions. This means thattf5@1/s(f)#tz , andtz ‘‘follows the contour’’ in thez system, as explained forezjust before Eq.~33!.

Explicit evaluation of the integrals forV21(0) andV30(0) inEqs.~48!, ~49! and ~51! is not impossible, but very tediousPrimarily this is due to the complicated dependence ofupper limits of the inner integrals on the variables of touter integrals. We leave this evaluation for future investition and make some comments here. We first mentionthe calculation must be done for fixed, finiteh, taking thelimit h→` only at the very end. Thed→0, e→0 and t→0 limits may be considered while performing the calcution, but care must be taken to keep all divergent termsthese regulators. In addition, in individual expressions, ththree limits do not always commute, and it is only at the ewhere the divergent parts should be seen to cancel. Gthat these two regulators are independent, we again exthat the combinationsV302V300 and V212V2102V201 willnot be divergent. We also expect that combined, they ctribute zero to the expression~15! for the cubic tachyon coupling, through Eq.~17!, leaving the tension dependent onon the termV111. Finally, to calculate the tension one mucalculate the overall normalization of the quadratic term,that it may be written canonically and the cubic term normized appropriately. This calculation also is tedious for tcase of the butterfly; we do not undertake it here. Work isprogress on the quadratic normalization and the cubic tevaluation; if reasonable, we intend to address the full cculation of the tension in a future publication.

VI. DISCUSSION

We have calculated the on-shell quadratic term intachyon action, and found that it vanishes. The structurethis calculation is the same as that for the sliver@10#. This iscompatible with the assertion that the butterfly represenD-brane state. The tension could be calculated by followthe procedure outlined in the last section, and we expectthis will also agree with the canonical value of unity forD-brane, matching the case of the sliver.

Finally, it is interesting to note that when defining surfastates with operators on the boundary, such as the defotion states in Eqs.~6!–~9!, the definition of the state dependon the geometry used. That is to say, although such specations are formally conformally invariant, the necessregularization procedure will break this symmetry. When dfined using operators requiring regularization, conformaequivalent surfaces can correspond to different states.information is contained in the function which maps tregulator from one coordinate system to another; herewas the functions(u) which maps the regulatorez in the zsystem toeu in the u system.

ACKNOWLEDGMENTS

The author would like to thank K. S. Viswanathan andC. Rashkov for enlightening discussions, and Simon FraUniversity for financial support. This work was supported

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part by the National Sciences and Engineering ReseaCouncil of Canada.

APPENDIX: CONFORMAL MAPPING TO THE CONE

Let V be the region of the complex plane given b2p/4,Rez,p/4, Imz.0. First let us construct a mafrom V to itself, which transforms the boundary]V asshown in Fig. 5. That is, the contour given by line segmefrom z52p/41 ih down to z52p/4, across toz51p/4and up toz51p/41 ih should be mapped to the segmentthe real line2p/4,z,p/4, with the rest of the boundarymapped accordingly. This map is given implicitly by the rlation

cos 2u51

hcos 2z, ~A1!

whereh5cosh 2h. Now, due to the periodicity of the functions in Eq.~A1!, this map in fact extends to arbitrarily mancopies of the surfaces, as shown in Fig. 6. Each ‘‘bucket’’width p/2 is folded down onto the real line. We will use thmap for two copies ofV for surfaces corresponding toK11andK20, and the three-copy map forK110 andK200. Explic-itly, the map~A1! can of course be written as

u51

2cos21

cos 2z

h, ~A2!

although we must be careful to note that the inverse cosfunction is not single-valued.

FIG. 5. h cos 2u5cos 2z.

FIG. 6. h cos 2u5cos 2z.

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BUTTERFLY TACHYONS IN VACUUM STRING FIELD THEORY PHYSICAL REVIEW D67, 086002 ~2003!

@1# Davide Gaiotto, Leonardo Rastelli, Ashoke Sen, and BarZwiebach, J. High Energy Phys.04, 060 ~2002!.

@2# V. Alan Kostelecky´ and Robertus Potting, Phys. Rev. D63,046007~2001!.

@3# Leonardo Rastelli, Ashoke Sen, and Barton Zwiebach, ATheor. Math. Phys.5, 393 ~2002!.

@4# Leonardo Rastelli, Ashoke Sen, and Barton Zwiebach, J. HEnergy Phys.11, 045 ~2001!.

@5# Leonardo Rastelli, Ashoke Sen, and Barton Zwieba‘‘Vacuum String Field Theory,’’ hep-th/0106010.

@6# Leonardo Rastelli and Barton Zwiebach, J. High Energy Ph

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09, 038 ~2001!.@7# Leonardo Rastelli, Ashoke Sen, and Barton Zwiebach, J. H

Energy Phys.11, 045 ~2001!.@8# P. Matlock, K.S. Viswanathan, Y. Yang, and R.C. Rashk

Phys. Rev. D66, 026004~2002!.@9# Kazumi Okuyama, J. High Energy Phys.03, 050 ~2002!.

@10# Yuji Okawa ~Caltech!, J. High Energy Phys.07, 003 ~2002!.@11# Ashoke Sen, J. High Energy Phys.12, 027 ~1999!.@12# Leonardo Rastelli, Ashoke Sen, and Barton Zwiebach, A

Theor. Math Phys.5, 353 ~2002!.

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