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Page 1: Massive fields dynamics in open bosonic string theory

ELSEVIER

30 March 1995

Physics Letters B 348 ( 1995) 63-69

PHYSICS LETTERS B

Massive fields dynamics in open bosonic string theory

I.L. Buchbinder a, V.A. Krykhtin b, V.D. Pershin b a Department of Theoretical Physics, Tomsk State Pedagogical Institute, Tonrsk 634041, Russia

b Department of Quantum Field Theory, Tomsk State University, Tomsk 634050, Russia

Received 13 December 1994 Editor: PV. Landshoff

Abstract

We consider the theory of the open bosonic string in massive background fields. The general structure of renonnalization is investigated. A general covariant action for a string in background fields of the first massive level is suggested and its symmetries are described. Equations of motion for the background fields are obtained by demanding that the renotmalized operator of the energy-momentum tensor trace vanishes.

1. Introduction

The theory of a string in massless background fields provides a possibility to consider string interactions in the low energy approximation [ l-41. Such a theory is remarkable due to its close connection with the two- dimentional quantum field theory and possible gen- eralizations (see review [ 51) . Namely, this approach has led to the equations of string gravity which are now widely used for finding new cosmological solu- tions (see review [ 61) .

The crucial point of the string theory in massless background fields is the fundamental concept of Weyl invariance. In quantum theory it means that the renor- malized operator of the energy-momentum tensor trace must vanish resulting in equations of motion for background fields. The general analysis of the renor- malized operator for a bosonic string interacting with background metric, antisymmetric tensor and dilaton was performed in Refs. [ 7,8] (see also review [ 51) .

A natural development of the approach leads to the consideration of string in background fields which are

connected with massive modes in the string spectrum. Unfortunately, the construction of a consistent quan- tum theory in this case appears to be quite difficult. A string interacting with any finite number of mas- sive background fields is non-renormalizable theory and requires infinite number of countertermes. So we have to deal with the theory containing infinite num- ber of terms in classical action which describe interac- tion with background fields of all the massive modes. The only massive field that does not require infinite number of counterterms is the field of tachyon but in this case non-perturbative effects play a crucial role [9-131 (see also review [ 141).

Recently several attempts were undertaken to de- scribe string in massive background fields [ 12-231. All these investigations (excluding work on the ta- chyon problem) mainly concerned only linear equa- tions of motion for background fields. The linear ap- proximation is of great importance since the equations for background fields in this approximation should correspond to the known equations defining the string spectrum. It turned out that even at the linear level all

0370-2693/95/!$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00127-l

Page 2: Massive fields dynamics in open bosonic string theory

64 I.L. Buchbinder et al. /Physics Letters B 348 (1995) 63-69

clarity is absent and the very possibility of going be- yond the linear approximation presents difficulties.

In Ref. [ 241 we proposed an approach to the string theory in massive background fields that represents a direct generalization of the g-model approach to

the string theory in massless background fields. For

a closed string interacting with background fields of all the massive modes we showed that the renormal-

ization has a special structure. Namely, the renormal-

ization of background fields of any massive level re-

quires consideration only of background fields of this

level and of all the lower ones, but is not affected by the infinite number of background fields belonging to

higher levels. In principle, our approach allows to go

beyond the linear approximation.

In the same paper [24] we examined in detail a closed bosonic string in background fields of the first massive level and received linear equations of motion, though the problem of agreement with the string spec-

trum was not solved completly. It is notable that the

lagrangian of a closed bosonic string in massive back-

ground fields constructed in Ref. [ 241 appeared then to be useful for the formulation of a generalized model of two-dimentional dilaton gravity [ 251.

This paper is devoted to further development of our

approach [24] and to its application to the theory of open string in massive background fields. The theory

of an open string is a field theory in space-time with boundary. Various aspects of quantum calculations in

the theory of open strings were discussed in Ref. 1291.

As was noted in pioneer works [ 21, interaction of an

open string with background fields corresponding to

the open string spectrum is completely concentrated at the boundary of the world sheet. Detailed investi- gation of open string in massless background fields

was conducted in Refs. [ 26-281. Questions of quan- tum field theory on a manifold with boundary were

studied in recent works [ 30,3 1 ] . The paper is organized as follows. Section 2 deals

with the investigation of renormalization in the theory of a string interacting with arbitrary background fields both on the world sheet and its boundary. It is showed that the renormalization has the same structure as in Ref. [ 241. In Section 3 we consider an open string in background fields of the first massive level, introduce the most general action, discuss its symmetries and

conduct the renormalization of background fields. The renormalization of composite operators defining the

energy-momentum tensor trace is carried out and lin-

ear equations of motion for background fieIds are de-

rived in Section 4. Complete agreement with the equa- tions specifying the string spectrum is established.

2. General analysis of renormalization

Our aim consists in construction of a u-model type action describing interaction of an open string with

massive background fields and deriving from the quan-

tum Weyl invariance condition effective equations of motion for these fields. So we have to build up the

renormalized operator of the energy-momentum ten-

sor trace and to demand that it vanishes. As shown in Ref. [ 24 J , to make the theory renor-

malizable in the case of a closed string one has to con- sider an action comprising infinite set of terms describ- ing interaction with all possible background fields. Af-

ter resealing string coordinates xp --+ &.P the total

action takes the form

B/,r) are background fields corresponding to the n-th

level in the closed string spectrum, 0:,“’ are con-

structed from gb, enb, dx, Ddx, . . ., R(*), dR’*), Dc?Rc2’, . . . . Here the derivative D, is covariant under reparametrizations both on the world sheet and in the D-dimensional space-time:

DaabXp = aaabXp - r;b(g)acxp + r&(G)aaXAabXP. (2)

For each n dimension of all U!,y’ in two-dimensional derivatives equals 2n + 2. N,, is the total number of all independent 0::’ belonging to the n-th level. The integral in ( 1) is taken over the whole world sheet, and we use a euclidian metric.

In case of an open string there appears a possibility to introduce interaction at the boundary of the world sheet dM and the total action should be of the form

Page 3: Massive fields dynamics in open bosonic string theory

I.L. B&binder et al. /Physics Letters B 348 (1995) 63-69 65

S= d2z& s

M2

+ dte s

dM

Here BICk) are background fields belonging to the n-th

massive level of the open string, C?tCk) are constructed

from kp = 3, RN, . . ., the external curvature of the

boundary K(t) = em2n, (s + I:, gg) and its derivatives. t is a parametr along the boudary, n, is a

unit vector normal to it and e2( t) = &b( z ( t) ) $ $

is one-dimensional metrics. OtCk) may contain any powers of derivatives with respect to r, that is why the second integral in (3) is expanded in powers of (u”/~.

To construct the renormalized operator of the energy-momentum tensor trace one has to renormal- ize both the background fields Bi, and the composite operators c3,,. Renormalization of fields is constructed by demanding that divergences of the quantum ef- fective action vanish. In one-loop approximation it appears as

(4)

where S is the classical action and XFI,, represents its second functional derivative:

(5)

In our case (3) XH,, is an operator of the following form:

D2 = gabD,Db. (6)

Coefficients Pol...@ and vk are functions of back- ground fields and can be presented as series in powers of ff’:

n=o m=o

where each term of expansions depends only on fields of the given massive level.

Divergences in (4) appear from the expression

TrlnH N Trln(-D2) 00 . /co .

-ctTr(CP”1,,,“2kD,,...D.Li~ kl

+gv, (;‘b)k$)i. (8)

Being local constructions, the divergences must be ex- panded in the same set of c3,, as the action (3)

n=o $2 i,,=l

Ni +~a’ki2SdreCO:(k)~~B’k)(B).

k=o dM

it=1

(9)

where 7,:” (B), qkBCk) (B), are some dimensionless functions of background fields.

It is obvious from dimensional considerations that counterterms of some given power in a’ can depend only on background fields of the corresponding mas- sive level and of all the lower ones. Therefore to renor- malize background fields of the n-th massive level of the closed string it is sufficient to calculate divergences generated by background fields from the n-th and all the lower levels of the closed string spectrum and by fields from the 2n-th and all the lower levels of the open string spectrum. Similarly, renormalization of k- th level fields of the open string requires to consider the open string spectrum fields of the k-th and all the lower levels and the closed string spectrum fields of the [k/2]-th and all the lower levels ([ ] means the integer part of a number).

For example, to renormalize the fields of the first massive open string level one is to study divergences generated by these fields and by the massless ones.

Page 4: Massive fields dynamics in open bosonic string theory

66 I.L. Buchbinder et al. /Physics Letters B 348 (1995) 63-69

Moreover, if we are interested merely in linear ap- proximation it is suffitient to restrict ourselves to the

study of divergences generated by fields of the only

given massive level. Contributions of all other levels

contain products of several background fields and so

are beyond the linear approximation.

3. Action, symmetries, renormalization

In this section we will consider an open string in-

teracting with all the massive fields of the first level, describe its symmetries and construct one-loop renor- malization of background fields. The action should

be the sum of the free string action So and the action

SI describing interaction with the fields of the first massive level and containing all possible terms of the

second order in derivatives:

sl~xl =

S/[Xl = (yw-J - /d&(t) [A,,(x)kpiY + B,(x)2 27T

aM

+ K2Spl (x> + kq2(x) + K-++,(x) I. (10)

Massless fields do not contribute to renormalization

of massive fields in the &near approximation and so

are not considered here. A theory similar to (IO)

was investigated in [ 161, but it did not contain all

possible background fields and calculations were not

explicitly covariant. The possibility to add an arbitrary total derivative

to the lagrangian in (IO) yields the symmetry of the theory under the following transformations:

{ SA,, = a(,A,,

1 %% = aPA

SB, = A, ’ 6ql7~ = h ’ (11)

where A,(X) and A(x) are arbitrary functions play- ing the role of transformation parameters. As one can see, the field Bp( x) and (Pi are Stuckelberg ones and the symmetry allows to choose them to be equal to zero. Thus the essential background fields

are APy(x),~,(x),~~(x). To renormalize background fields it is necessary to

calculate divergences of the action

*‘l/2 i TrM&p + zV,p),

where we denote

(12)

S2So[xl cu’w s -_ Off@ = sxa(yxp ’

” ZT 4

_ S2S,[xl ~ SxasxP . (13)

( 12) is expanded into series in powers of (a’) i12:

(yf1/2 4 Trlnthp + 2a~s)

= $TrlnSo,p+2a ’ “‘2TrV,yG”P+O(a’), (14)

where the Green function Gap of a free string is de- termined by the equation

~ITS~,~G~~ = ay a’ (15)

Divergences of the first term in (14) are cancelled by

the corresponding contribution of the ghosts provid- ing that D = 26. Terms 0( a’) contribute to renormal-

ization of background fields of the second and all the higher levels and so will be omitted.

Choose coordinates (t, y) on the world sheet such that t is a parameter along the boundary c3M and y equal the distance between the point z’ = (t, y) and the boundary along a geodesic line targent to the in- ternal normal vector n,. The metrics in terms of these

coordinates is

ds2 =e2(t,y)dt2 +dy2. (16)

Specifying the coordinate t so that e( t, y) Iv=0 = 1 we get the following expansion [ 301:

e(t,y) = 1 - K(t)y - SR(t,0)y2 + 0(y3>.

Calculation of V& in such coordinates gives

V,p(t,Y;f’vY’)

(17)

(18)

(19)

Page 5: Massive fields dynamics in open bosonic string theory

I.L. Buchbindcr et al. /Physics Letters B 348 (199s) 63-69 67

and whereQ, = (A,,,B,,PI,P~,(PJ.

&p = B,,fi + &.a - 2&p,

C cc(+) = AQ,,L - A,,p - A,p,, + B,_+

@‘[/A = %..cL - 4PlWr

Q/L f P2,l.l - SD/L. (20)

Here the function 6,~ (z ) is defined as follows:

4. Renormalized trace of energy-momentum tensor and equations of motions

In classical theory the trace of the energy- momentum tensor for the theory (10) on the 2 + E- dimensional world sheet is

Gxc’(t)

Sx”fz’) = s(t - t’)&&.f(z’)Sf. (21)

Considering ( 18), the divergences of ( 14) are con- tained in the expression

where we used that the divergences of Green function in coincident points in the framework of dimensional renormalization are [ 30,3 I]

Gyt,O;t’,O)(;e+,, = -$7y,

$G’“(t,O;t’,O)

div

= ~G”“(t,O;t’,O)

div

t’-+t dt2 F-+2 = 0. (23)

Omitting in (22) total derivatives we arrive at the following one-loop effective action:

&2 c r(l) = - P

2?r ahf

-lo i,,) E

+k(& -iCl&) +KP(G, -iO G,) +(fin), >

cl E ?j@a,ap, (24)

where o denotes bare background fields and (fin) stands for a finite part of the one-loop correction.

To cancel the divergences in (24) renormalization of all the background fields should be of the form

& jL_‘(Q + 1 !J@), (25) E

- -F’@/&.f (z ) -t O( t3x”‘2). 47r

(26)

Terms 0( &/2) will not contribute to the renormal- ized trace of energy-momentum tensor .

To calculate the trace of the energy-momentum ten- sor in quantum theory we should define renormalized values for composite operators. Consider, for exam- ple, the vacuum average for one of these operators:

(H,,(x)?Y') = s DxedSLX1 H,,( x) k~i” s Dxe-Sl+l

Making the shift xp = X” + JP in the functional in- tegral (xP are solutions of the classical equations of motion) and using (23) we get in linear approxima- tion:

(H,,(x)f”k”) = ,u’ipkY(H&) - i Cl H,,(X))

+ (fin). (27)

Renormalized operators should have a finite average value

(WpvW~p~i-‘l) = Hp,(R)kpiy + (fin) (28)

hence

=,u”‘[ipY(Hpv(x) - L 0 H,,(x))]

+ (fin). (29)

Page 6: Massive fields dynamics in open bosonic string theory

68 I.L. Buchbinder et al. /Physics Letters B 348 (1995) 63-69

or, using (25),

G P#_Y)D = [ H,,f~k”].

In the same way one can get

(30)

(G I>0 = Ip11, & .P”>o = I@,kc”]. (31)

A similar but more tedious calculation gives the renor-

malization of the operator $b ( z ) daXPdbXYr)P,:

+ --v2 adM(z) +4V(O,~dM(Z)], dt2 ( )

V(2) = %2)pJY> V(O) = vgl)pv?7p”. (321

As a result, the renormalized operator of the energy-

momentum tensor trace has the form

[T] = Cu”f%(y)[RE(“)(n)

+ i”iW’(x) + f”E(2’(X) P P

+ KI’E;) + K2Ec4’(x) -j-kEc5~(x)]

+ &/28/ (y) [kpEc6)(x) + KIT(~)(X)]

+ CY’~/~~“( y) Ec8’ (x)

with

(33)

L$~‘(_x) = - ’ s,‘B; - A:),

E”‘(X) fiu = & (2 q A,,, - 2A, - A&, - 2A;,,,

- 2A;,,, + 3BP,,, + B,,, + B,,) t

Ec2’(x) = CL $2 0 B, + 3B;, - A& - 4A;,,),

Ec3’(x) = P ~(~2,1i-9p+O~p-iP~J7

E’4’(X) = &(2& -BP,+A:-2401),

C’ (x) = &P2 - rpp,),

_@‘(x) = - P

Ec7’(x) = -I_(Ba _ 4TT ,a

A:+4w),

E’8’(x) = -&(B^, - A,“). (34)

From the requirement of quantum Weyl invariance it follows that all the coefficients (34) at the linear in-

dependent operators in (33) should be equal to zero. Therefore equations of motion for the background fields of the first massive level are E(x) = 0.

Using the symmetry of the theory under the trans- formations ( 11) to make B, = 0 and ~2 = 0 and re- turning to the dimensional string coordinates XP +

cy’-‘12x~ one can rewrite these equations as

B,=O, (p2=0, PI =o, ppc( =o, 0 A,, - m2ApV = 0, A;=O, d,A;=O, (35)

where m2 is the mass of the first level.

So the equations of motion for the background fields of the first massive level of the open string are equiv- alent to the equations describing a massive field with spin 2. As is well known, the analysis of the spectrum of open string physical states gives the same result.

5. Summary

We have considered the general approach to the the- ory of an open string interacting with massive back-

ground fields. Our analysis has shown that there ex- ist a consistent description of string models in back- ground fields corresponding to a finite number of the massive string modes. We have proposed the most general model describing interaction of an open string with background fields of the first massive level. The

theory is invariant under symmetry transformations of

background fields. The renormalized operator of the energy-momen-

tum tensor trace has been constructed. In linear ap- proximation it has been shown that the requirement of quantum Weyl invariance leads to equations of motion for background fields which are consistent with the struture of the open string spectrum at the first massive level. The approach proposed opens up possibilities for deriving equations of motion for massive fields of higher spins in the framework of the string theory.

Page 7: Massive fields dynamics in open bosonic string theory

I.L. Buchbinder et al. /Physics Letters B 348 (1995) 63-69 69

Acknowledgements

I.L.B. is grateful to S.J. Gates, H. Osborn, B.A. Ovrut, J. Schnittger and A.A. Tseytlin for useful dis-

cussions on a various aspects of the work. The re-

search described in this publication has been supported

in parts by International Science Foundation, grant RI1000 and Russian Foundation for Fundamental Re-

search, project No 94-02-03234.

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