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Towards string theory in massive background fields
UCLEAR PHYSIC~
EI.SEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 49 (1996) 133-138
Towards string theory in massive background fields I.L. B u c h b i n d e r ~
a Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk 63,~041, Russia A review of recent progress in string theory coupled to massive background fields is given. The problems of formulation are discussed. The general renormalization structure of bosonic string theory in background fields of massive modes is studied. Linear approximation in open string theory in background fields of first massive modes is investigated.
1. I N T R O D U C T I O N As is well known the spectrum of free bosonic string consists of tachionic mode, massless modes and infinite number of massive modes (see f. e. [1 - Z]). These string modes correspond to conventional free fields in 26 - dimensional Minkowski space-time. The tachion mode corresponds to scalar field and the other massive modes correspond to higher spin, higher rank massive fields. Taking into account a set of space-time fields associated with string modes we can consider a string coupled to these fields depending on string coordinates. As a result we get a two-dimensional interacting field theory where a role of fields is played by string coordinates and a role of couplings is played by background fields of string modes. Such a model can be treated as a theory describing the string interactions. This idea has been realized in the sigma-model approach where a string coupled to background fields of massless modes has been investigated [4 - 6] (see also the reviews [7, 81). A natural development of this approach leads to consideration of string theory in massive background fields. Recently several attempts were undertaken to construct such a theory. Unfortunately, this theory faced a problem of nonrenormafizability if you take into account a finite number of different massive mode contributions and the problem of derivation of non-hnear corrections to equations
of motion for background fields. Moreover, the whole clarity is absent even in linear approximation with respect of massive background fields. In resent papers [9, 10] we suggested an approach allowing in principle by-pass the above problems. The aim of this paper is a brief review of the main statements and results of our approach.
2.
PROBLEMS
OF
FORMULATION
Let us consider a closed bosonic string. As is well known the action describing the interaction of the strings with massless background fields includes the terms with two derivatives with respect to world sheet coordinates. Hence to construct an interaction with massive background fields we should add the terms containing the derivatives of forth, sixth, eights and so on orders. This procedure leads to the following terms in Lagrangian 1 (0z)4F +
1
02z(Ox)2T+
+R(2)(Ox)2W + a' R(~)R(2)C + ...] + + [l (ox)°B + ~
O2x(Ox)4D+
+(a'):R(2)R(2)R(:)G + ...] +
0920-5632/96/S 15.00 © 1996 Elsevier Science B.V. All rights reserved. PII: S0920-5632(96)00326-X
(1)
I.L. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
134
where F ~ , ~ , Tuw, Wu~, C, Bu~t~a~, , Du~=~a , G, L,,~,~x~p,~,... are some background fields depending on string coordinates. R(2) is world sheet curvature.
Let us consider now an open bosonic string. In this case the interaction with massless background fields is concentrated on the boundary and includes the term with first derivative with respect to parameter along the boundary. To include the interaction with massive background fields we should consider Lagrangian of the form
[ A ~ + B~ + K ~ + . . . ] + + [Z~a~5~ + Y k ~ + . . . ] + . . .
(2)
where Auv, Bu, ~o, Zu~a, Y~,~,... are some background fields and K is the extrinsic curvature of the boundary. The explicit form of terms in Lagrangian corresponding to first massive level contribution both for closed and for open string has been given in ref [9, 10]. Further it is convenient to rescale the string coordinates x u ---* ~ x u. Taking into account a general idea leading to eqs. (1, 2) we can write a complete action containing the contributions from all string modes as follows OO
M N.
i~=-1 O~
k=0
Nf × E Bo(k)tt i~ , , &(t)) BB~)(x(t) )
(3)
it,=l
Here B~n) are the background fields corresponding to the n-th massive level of closed string, O~_n) are the corresponding composite operators construeted
from
o?)= 1 gab OaXUObXv
B~°) = G~v
and so on. The first integral in S (3) is taken over the whole world sheet. The second integral in (3) is taken over the boundary. Here SB!k) are the background fields ,h belonging to the k-th massive level of open string. The composite operators Bo!k) are constructed from ~ , ~t,, . . . , K, K , . . . . All background fields and composite operators are dimensionless in eq. (3). The only dimensional parameter is or'. Moreover, following the power o f a ' we can foUow really the contributions of given level. The form of action (3) is very convenient for perturbative consideration. We will call this action as the general action. If we start to study the theory with the action (3) we immediately face the problems. We enumerate some of them. • Whether we are able to write down all operators O and all corresponding background fields B for each given massive level? • Let we have found all background fields corresponding to given massive level contribution in sense of given number of derivatives. Can we be sure that their number is equal the number of fields corresponding to given string modes?
n=0
×
+OMf
to 2n + 2. N~ is the total number of all independent operators O~:) belonging to the n-th level. For example, on massless level we have
gab, cab, OX, Z)OX, . . . , R (2), OR (2),
:DOR (2), . . . . The derivative 79~ is a covariant one under reparametrization both on the world sheet and in the target space. For each n a dimension of all O!I n'~) in two-dimensional derivatives is equal
• It is evident that the theory with general action (3) is nonrenormalizable if we take into account any finite number of massive level contributions. The only we can expect the theory will renormalizable in some generalized sense when infinite number of terms ineluded into the action together with infinite number of corresponding types of eounterterms. It means we have to work with action containing the infinite number of terms. • A central point of above approach in a whole is the principle of quantum Weyl invariance. If we demand a quantum Weyl
I,L. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133-138
invariance of the theory under consideration we can expect as in standard sigmamodel approach [3 - 8] some equations of motion for background fields. May we be sure that these equation will be consistent with known equations defining each given massive mode? It is clear to answer this question it is sufficient to consider only linear approximation with respect to massive background fields. The background fields corresponding to massive modes are the higher spin fields. One can expect that the principle of quantum Weyl invariance will lead to equations of motion for lower and higher spin fields interacting with each other. Thus the approach under consideration can allow to understand how the interacting higher spin theories arise within the string theory. 3. G E N E R A L
GOALS AND
HOPES
To realize the q u a n t u m Weyl invariance principle we should construct a renormalized trace of energy-momentum tensor for string interacting with massive background fields and then to vanish it. We hope like in standard sigma-model approach the quantum Weyl invarianee principle will lead to equation of motion for massive higher spin fields. How to construct the renormalized trace? It is evident that the operator of trace should be a linear combination of the same coraposite operators Oz that present in general action (3) with some coefficients depending on background fields.
T
o,
:
~z(B)
(4)
1
Construction of renormalized operator IT] consists of two steps. • The first step is the renormalization of effective action and as a result the renormalization of background fields. • The second step is the renormalization of the composite operators O1.
135
As a result we should obtain the renormalized operator [T] in the form [T] = Z [ O l ]
E(O(B)
(5)
I
where [Ol] are the renormalized composite operators and are some functions of background fields and their derivatives. The quantum Weyl invariance principle means IT] = 0 and we obtain the equations of motion for background fields in the form
E(O(B)
E ( 0 ( B ) -- 0
(6)
All above looks rather like general program. From general point of view the sum over l in eq (5) should include the infinite number of terms since we have the infinite number of terms in the general action (3). It is evident we do not know an explicit forms of all possible terms in eq (3). Therefore we have no possibility in principle to find all contribution to IT] (5) and hence to obtain the equations (6). To clarify the situation we should investigate a renormahzation structure of general theory. 4. R E N O R M A L I Z A T I O N
STRUCTURE
An essential observation allowing to understand a renormalization structure of general theory (3) connected with the fact that the power of a ' in (3) directly corresponds to the number of given massive level. Hence following the power of (~' we can really follow the given massive level contribution to any quantities. Another evident but essential observation connected with the fact that the counterterms in perturbation theory should be the local functionals and have the form of power serieses in c~~. The third essential observation consists in that the counterterms can be expanded in the same set o f composite operators O !'~),,,, BO~) as the general action (3). Taking into account above three observations we can write down immediately the counterterms for the theory under consideration
o,. <.. (B)+ n=0
M
i,,=1
136
LL. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
(7) k=0
OM
i~=1
where T:(n)(B] and ~/~--'k)(B] --" are some dimensionless functions of background fields and regularization parameters. Let us consider the terms with fixed number n in eq. (7) and determine what massive levels can contribute to these fixed terms. We know that each massive level is characterized by quite definite power of a ' . It is evident to get the power (a') n with given n or the power (ct') k12 with given k in the counterterm (7) we can use the contributions of massive levels with the numbers not bigger then n or k since in processes of calculation of counterterms in perturbation theory the corresponding powers of a ' can be only summed. This consideration leads to conclusion that the counterterms containing the given power o f a ' can depend only on background fields corresponding to the given power of a ' and on background fields corresponding to massive levels with the numbers lesser then given. Thus, to renormalize the background fields of the n-th massive level it is sufficient to calculate the divergences generated by the terms in general action (3) corresponding to this n-th level and by the terms corresponding to all levels with lesser numbers. To solve this problem we do not need in any information on structure of general action (3) in higher level sectors. For example, to renormalize the background fields of first massive level it is sufficient to study the divergences generated by these fields and by massless ones. The structure of second massive level contribution to general action (3) has no significance. Since the whole picture under discussion is quite general and is based in fact only on dimensional considerations we can be sure that the same conclusions will be fulfilled for renormalized trace of energym o m e n t u m tensor as well. As a result, although the general theory will be renormalized on the whole only if the infinite number of counterterms are taken into account we can renormalize the theory step by step, level by level. The previous levels will eontribute to all following levels but the following levels can
not influence on the previous ones. Renormalization of any given quantities demands only the finite number of counterterm types. Presence or absence of higher level terms in general action (3) does not influence on the renormalization of given level quantities. Hence we have a possibility to study a theory with the truncated number of terms in general action (3). Taking into account the above result we see that the infinite system of equations (6) should has a very special structure. The equations defining the given massive level background fields include these fields and all fields of lower levels but never include the fields of higher levels.
5. LINEAR A P P R O X I M A T I O N Now we want to demonstrate how whole program can be realized in linear approximation for open string interacting with background fields of first massive level. The linear approximation means that we consider a theory with the action S= S0+&,.
M
a'1/2 / Sint = 2~r
d t e [Au~(x) k u k~+
OM ~-B# (T);~,u _[_ g 2 ~1
(S)
(x)-]-
+/f ~2(z)+ K ~' ~ ( x ) ] and we take into account only linear terms in all background fields A~,v, B u, ~1, ~2, ~u in renormalized trace of energy-momentum tensor. The first step is to find the divergences of effective action. We use the dimensional regularization and some special technique allowing to make calculation of divergences in the theories with boundary (see the details in Refs. [11]). The final result for one-loop divergent part of effective action looks like this r(1)
+OB o
(~,)1/2 u" f
+ K D o+
+ / ~ [ ~ 0 + K i~,[]~o}
(9)
137
I.L. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
where e = d - 2 is a renormalization p a r a m e t e r , # is an a r b i t r a r y p a r a m e t e r of mass dimension. This relation allows to write down the renormalization t r a n s f o r m a t i o n s of the background fields
, ( * ) - 1 R(2) V(2) (~OM (o.) +g ~)*oM
1 ¢I'° = # - ' ( @ + - QO)
÷4V(0) 6eM (o')]
(10)
Here (I, -- (Auu, Bu,~Ol,~O2,!ou). T h e (I,° are the bare fields and the • are the tenormalized ones. We see the renormalization has quite universal structure in linear a p p r o x i m a t i o n . T h e next step is renormalization of composite operators forming the energy - m o m e n t u m tensor. T h e classical trace on 2+~-dimensional world sheet is
K
s
V(2)$OM(~) + ~
From the left side we have the bare operators and from the right side we have the renormalized ones. The V(2 ) and V(0) are some functions of background fields and their derivatives. T h e explicit form of this functions is given in Ref. [10]. Taking into account the eqs. (10, 11, 14) we can obtain finally
IT] = (~,)1/~ ~oM(~)[R(~) E(0) + ~ ~ E ( ~ +
+ ( a , ) l / : 6~M ((r) E(S)
1
E(O) _- 1 4--~ ( A ~ - c9~,B~')
+
+o(~(~') 1/~)
1 ( 2 O A ~ - 2A;~ -
(11)
+OrBs, - OuB,~)
Hun = Ot, B u - O~B u - 2At, ~
E< ~) : (12)
~u
T h e function 6ou(~r) is defined as follows [10]: 6(t
-
¢') 60M(O/) 6 ~
(13)
We see t h a t the trace o p e r a t o r is a linear combination of the t e r m s g~b OaxUObxU~?Uu, Hu~,(x)&u£~ , K ~ , ( x ) , K k ~ ' + u ( x ) , : ~ O u ( x ). Their renormalization can be fulfilled by s t a n d a r d methods. Taking into account the renormalization of background fields (10) one gets
(C)0 : [~],
(+o ~ ) 0 - [+. ~ ]
(#~ ao~,' o ~ ~ ~.~)o : [#~ ao~,' + ('~')~/~ u ~ [-vh> ~g'M(,~)+
a ~ ~ ~,~]+
c9~,c9~,A~-
-2c%,O~,A~ - 2c%,OuA ~ + 30,c9~,0~,B'~ +
The terms O(e(a')~/~) do not contribute to renormalized o p e r a t o r [T]. Here
--
(15)
where
1 n : ~ ~o~(~) + n ~, ~g~(:)+
6it'(t) 6xU(o.,)
(14)
+ ( a ' ) ' / : f~M ((r)[& ~' E (6) + K E(7)]+
(~ H,~ ~ ~ ~ou(~)-
~u = 0t,~°~ -
, ,
OaM(O')+
+ ~ E(2) + K ~ E<2) + K ~ E(+ + k E<~)]+
T = ~ g~b OaXU ObX~ ~1~,,+ 87r ÷ (Ot¢)1/2 1
1 +-~
d2V(2) ,
-
1 (213Bu + 30~,3~,B ~ _
O~,A~-
-40~,A~)
1
E(2) : ~ ( a . ~ - ~ + D~. - a ~ a ~ ~) E ( 4 ) = _ _ l (2D~ 1 - 2 ~ 1 - 0 c ` B 4~r E(~) 1 = 4~ (O~2 - 0 ~ )
~+A%)
E(6> = __1
(~,~ _ Ou~: )
E (7) =
--1
(O~B ~' - A ~ ÷ 4~,1)
E ( S ) _-
1 A~:-c%'B'~) 8~(
47r 8~r
(16)
T h e principle of q u a n t u m Weyl invariance leads to [T] -- 0. As a result we have the equations of motion for background fields in the form E (i) = 0 ; i -- 1, 8. Let us notice the action (8) is not Weyl invariant classically. Therefore the equation
138
LL. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
[T] = 0 can be fulfilled only under quite definite combinations of classical and quantum contributions. One can show that the theory under consideration possesses the symmetry transformations
6B~ = A t,
6~v2 = A
where A~, and A are the arbitrary functions. It allows to impose the gauge conditions B~, = 0, ~2 = 0. Returning to initial string coordinates x ~' ---* (a') -1/2 a¢' one can rewrite the equations E (i) : 0 as B~,=0,
~2=0,
~ol = 0 ,
~I, = 0 ,
(17)
3. D.Lust, S.Theisen. Lectures on String Theory. Lecture Notes in Physics. SpringerVerlag, 1989. 4. E.S.Fradkin,
A.A.Tseytlin, Phys.Lett. B158 (1987) 316; Nucl.Phys. B261 (1985) 1.
D.Friedan, E.Martinec, 5. C.G.Callan, M.J.Perry, Nucl.Phys. B262 (1985) 593. 55 (1985) 1846; 6. A.Sen, Phys.Rev.Lett. Phys.Rev. D32 (1985) 2102. 7. A.A.Tseytlin, Int.J.Mod.Phys. 1257.
A4 (1989)
8° H.Osboru, Ann.Phys. 200 (1990) 1.
[~A#~ - m 2 A~,v = 0 0,A" = 0
(18)
A~=O
where rn is the mass of first massive mode of open string. The equation for A~,v (18) are just the correct equations of motion for spin two massive field. As is well known the analysis of the spectrum of open string leads to the same equations.
ACKNOWLEDGMENTS
I am very grateful to E.S.Fradkin, V.A.Krykhtin, S.L.Lyakhovieh and V.D.Pershin for fruitful collaboration. It is a pleasure to thank S.J.Gates, H.Osborn, B.A.Ovrut, J.Schnittger and A.A.Tseytlln for useful discussions on the various aspects of the paper. Work supported in parts by RFBR, project No. 94 - 02 - 03234 and ISF under the grant RI 1300.
REFERENCES
1. M.B.Green, J.H.Schwarz, E.Witten. Superstring Theory. Cambridge Univ. Press. 1987, Vol. 1. 2. L.Brink, M.Henneaux. Principles of String Theory. Plenum Press, 1988.
9. I.L.Buchbinder, E.S.Fradkin, S.L.Lyakhovich, V.D.Pershin, B304 (1993) 239.
Phys.Lett.
10. I.L.Buchbinder, V.A.Krykhtin, V.D.Pershin, Phys.Lett. B348, (1995) 63. 11. D.M.McAvity, H.Osborn, Class. Quant. Gray. 8 (1991) 603; 8 (1991) 1445; Nucl.Phys. B394 (1993) 728; B406 (1993) 655.
EI.SEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 49 (1996) 133-138
Towards string theory in massive background fields I.L. B u c h b i n d e r ~
a Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk 63,~041, Russia A review of recent progress in string theory coupled to massive background fields is given. The problems of formulation are discussed. The general renormalization structure of bosonic string theory in background fields of massive modes is studied. Linear approximation in open string theory in background fields of first massive modes is investigated.
1. I N T R O D U C T I O N As is well known the spectrum of free bosonic string consists of tachionic mode, massless modes and infinite number of massive modes (see f. e. [1 - Z]). These string modes correspond to conventional free fields in 26 - dimensional Minkowski space-time. The tachion mode corresponds to scalar field and the other massive modes correspond to higher spin, higher rank massive fields. Taking into account a set of space-time fields associated with string modes we can consider a string coupled to these fields depending on string coordinates. As a result we get a two-dimensional interacting field theory where a role of fields is played by string coordinates and a role of couplings is played by background fields of string modes. Such a model can be treated as a theory describing the string interactions. This idea has been realized in the sigma-model approach where a string coupled to background fields of massless modes has been investigated [4 - 6] (see also the reviews [7, 81). A natural development of this approach leads to consideration of string theory in massive background fields. Recently several attempts were undertaken to construct such a theory. Unfortunately, this theory faced a problem of nonrenormafizability if you take into account a finite number of different massive mode contributions and the problem of derivation of non-hnear corrections to equations
of motion for background fields. Moreover, the whole clarity is absent even in linear approximation with respect of massive background fields. In resent papers [9, 10] we suggested an approach allowing in principle by-pass the above problems. The aim of this paper is a brief review of the main statements and results of our approach.
2.
PROBLEMS
OF
FORMULATION
Let us consider a closed bosonic string. As is well known the action describing the interaction of the strings with massless background fields includes the terms with two derivatives with respect to world sheet coordinates. Hence to construct an interaction with massive background fields we should add the terms containing the derivatives of forth, sixth, eights and so on orders. This procedure leads to the following terms in Lagrangian 1 (0z)4F +
1
02z(Ox)2T+
+R(2)(Ox)2W + a' R(~)R(2)C + ...] + + [l (ox)°B + ~
O2x(Ox)4D+
+(a'):R(2)R(2)R(:)G + ...] +
0920-5632/96/S 15.00 © 1996 Elsevier Science B.V. All rights reserved. PII: S0920-5632(96)00326-X
(1)
I.L. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
134
where F ~ , ~ , Tuw, Wu~, C, Bu~t~a~, , Du~=~a , G, L,,~,~x~p,~,... are some background fields depending on string coordinates. R(2) is world sheet curvature.
Let us consider now an open bosonic string. In this case the interaction with massless background fields is concentrated on the boundary and includes the term with first derivative with respect to parameter along the boundary. To include the interaction with massive background fields we should consider Lagrangian of the form
[ A ~ + B~ + K ~ + . . . ] + + [Z~a~5~ + Y k ~ + . . . ] + . . .
(2)
where Auv, Bu, ~o, Zu~a, Y~,~,... are some background fields and K is the extrinsic curvature of the boundary. The explicit form of terms in Lagrangian corresponding to first massive level contribution both for closed and for open string has been given in ref [9, 10]. Further it is convenient to rescale the string coordinates x u ---* ~ x u. Taking into account a general idea leading to eqs. (1, 2) we can write a complete action containing the contributions from all string modes as follows OO
M N.
i~=-1 O~
k=0
Nf × E Bo(k)tt i~ , , &(t)) BB~)(x(t) )
(3)
it,=l
Here B~n) are the background fields corresponding to the n-th massive level of closed string, O~_n) are the corresponding composite operators construeted
from
o?)= 1 gab OaXUObXv
B~°) = G~v
and so on. The first integral in S (3) is taken over the whole world sheet. The second integral in (3) is taken over the boundary. Here SB!k) are the background fields ,h belonging to the k-th massive level of open string. The composite operators Bo!k) are constructed from ~ , ~t,, . . . , K, K , . . . . All background fields and composite operators are dimensionless in eq. (3). The only dimensional parameter is or'. Moreover, following the power o f a ' we can foUow really the contributions of given level. The form of action (3) is very convenient for perturbative consideration. We will call this action as the general action. If we start to study the theory with the action (3) we immediately face the problems. We enumerate some of them. • Whether we are able to write down all operators O and all corresponding background fields B for each given massive level? • Let we have found all background fields corresponding to given massive level contribution in sense of given number of derivatives. Can we be sure that their number is equal the number of fields corresponding to given string modes?
n=0
×
+OMf
to 2n + 2. N~ is the total number of all independent operators O~:) belonging to the n-th level. For example, on massless level we have
gab, cab, OX, Z)OX, . . . , R (2), OR (2),
:DOR (2), . . . . The derivative 79~ is a covariant one under reparametrization both on the world sheet and in the target space. For each n a dimension of all O!I n'~) in two-dimensional derivatives is equal
• It is evident that the theory with general action (3) is nonrenormalizable if we take into account any finite number of massive level contributions. The only we can expect the theory will renormalizable in some generalized sense when infinite number of terms ineluded into the action together with infinite number of corresponding types of eounterterms. It means we have to work with action containing the infinite number of terms. • A central point of above approach in a whole is the principle of quantum Weyl invariance. If we demand a quantum Weyl
I,L. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133-138
invariance of the theory under consideration we can expect as in standard sigmamodel approach [3 - 8] some equations of motion for background fields. May we be sure that these equation will be consistent with known equations defining each given massive mode? It is clear to answer this question it is sufficient to consider only linear approximation with respect to massive background fields. The background fields corresponding to massive modes are the higher spin fields. One can expect that the principle of quantum Weyl invariance will lead to equations of motion for lower and higher spin fields interacting with each other. Thus the approach under consideration can allow to understand how the interacting higher spin theories arise within the string theory. 3. G E N E R A L
GOALS AND
HOPES
To realize the q u a n t u m Weyl invariance principle we should construct a renormalized trace of energy-momentum tensor for string interacting with massive background fields and then to vanish it. We hope like in standard sigma-model approach the quantum Weyl invarianee principle will lead to equation of motion for massive higher spin fields. How to construct the renormalized trace? It is evident that the operator of trace should be a linear combination of the same coraposite operators Oz that present in general action (3) with some coefficients depending on background fields.
T
o,
:
~z(B)
(4)
1
Construction of renormalized operator IT] consists of two steps. • The first step is the renormalization of effective action and as a result the renormalization of background fields. • The second step is the renormalization of the composite operators O1.
135
As a result we should obtain the renormalized operator [T] in the form [T] = Z [ O l ]
E(O(B)
(5)
I
where [Ol] are the renormalized composite operators and are some functions of background fields and their derivatives. The quantum Weyl invariance principle means IT] = 0 and we obtain the equations of motion for background fields in the form
E(O(B)
E ( 0 ( B ) -- 0
(6)
All above looks rather like general program. From general point of view the sum over l in eq (5) should include the infinite number of terms since we have the infinite number of terms in the general action (3). It is evident we do not know an explicit forms of all possible terms in eq (3). Therefore we have no possibility in principle to find all contribution to IT] (5) and hence to obtain the equations (6). To clarify the situation we should investigate a renormahzation structure of general theory. 4. R E N O R M A L I Z A T I O N
STRUCTURE
An essential observation allowing to understand a renormalization structure of general theory (3) connected with the fact that the power of a ' in (3) directly corresponds to the number of given massive level. Hence following the power of (~' we can really follow the given massive level contribution to any quantities. Another evident but essential observation connected with the fact that the counterterms in perturbation theory should be the local functionals and have the form of power serieses in c~~. The third essential observation consists in that the counterterms can be expanded in the same set o f composite operators O !'~),,,, BO~) as the general action (3). Taking into account above three observations we can write down immediately the counterterms for the theory under consideration
o,. <.. (B)+ n=0
M
i,,=1
136
LL. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
(7) k=0
OM
i~=1
where T:(n)(B] and ~/~--'k)(B] --" are some dimensionless functions of background fields and regularization parameters. Let us consider the terms with fixed number n in eq. (7) and determine what massive levels can contribute to these fixed terms. We know that each massive level is characterized by quite definite power of a ' . It is evident to get the power (a') n with given n or the power (ct') k12 with given k in the counterterm (7) we can use the contributions of massive levels with the numbers not bigger then n or k since in processes of calculation of counterterms in perturbation theory the corresponding powers of a ' can be only summed. This consideration leads to conclusion that the counterterms containing the given power o f a ' can depend only on background fields corresponding to the given power of a ' and on background fields corresponding to massive levels with the numbers lesser then given. Thus, to renormalize the background fields of the n-th massive level it is sufficient to calculate the divergences generated by the terms in general action (3) corresponding to this n-th level and by the terms corresponding to all levels with lesser numbers. To solve this problem we do not need in any information on structure of general action (3) in higher level sectors. For example, to renormalize the background fields of first massive level it is sufficient to study the divergences generated by these fields and by massless ones. The structure of second massive level contribution to general action (3) has no significance. Since the whole picture under discussion is quite general and is based in fact only on dimensional considerations we can be sure that the same conclusions will be fulfilled for renormalized trace of energym o m e n t u m tensor as well. As a result, although the general theory will be renormalized on the whole only if the infinite number of counterterms are taken into account we can renormalize the theory step by step, level by level. The previous levels will eontribute to all following levels but the following levels can
not influence on the previous ones. Renormalization of any given quantities demands only the finite number of counterterm types. Presence or absence of higher level terms in general action (3) does not influence on the renormalization of given level quantities. Hence we have a possibility to study a theory with the truncated number of terms in general action (3). Taking into account the above result we see that the infinite system of equations (6) should has a very special structure. The equations defining the given massive level background fields include these fields and all fields of lower levels but never include the fields of higher levels.
5. LINEAR A P P R O X I M A T I O N Now we want to demonstrate how whole program can be realized in linear approximation for open string interacting with background fields of first massive level. The linear approximation means that we consider a theory with the action S= S0+&,.
M
a'1/2 / Sint = 2~r
d t e [Au~(x) k u k~+
OM ~-B# (T);~,u _[_ g 2 ~1
(S)
(x)-]-
+/f ~2(z)+ K ~' ~ ( x ) ] and we take into account only linear terms in all background fields A~,v, B u, ~1, ~2, ~u in renormalized trace of energy-momentum tensor. The first step is to find the divergences of effective action. We use the dimensional regularization and some special technique allowing to make calculation of divergences in the theories with boundary (see the details in Refs. [11]). The final result for one-loop divergent part of effective action looks like this r(1)
+OB o
(~,)1/2 u" f
+ K D o+
+ / ~ [ ~ 0 + K i~,[]~o}
(9)
137
I.L. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
where e = d - 2 is a renormalization p a r a m e t e r , # is an a r b i t r a r y p a r a m e t e r of mass dimension. This relation allows to write down the renormalization t r a n s f o r m a t i o n s of the background fields
, ( * ) - 1 R(2) V(2) (~OM (o.) +g ~)*oM
1 ¢I'° = # - ' ( @ + - QO)
÷4V(0) 6eM (o')]
(10)
Here (I, -- (Auu, Bu,~Ol,~O2,!ou). T h e (I,° are the bare fields and the • are the tenormalized ones. We see the renormalization has quite universal structure in linear a p p r o x i m a t i o n . T h e next step is renormalization of composite operators forming the energy - m o m e n t u m tensor. T h e classical trace on 2+~-dimensional world sheet is
K
s
V(2)$OM(~) + ~
From the left side we have the bare operators and from the right side we have the renormalized ones. The V(2 ) and V(0) are some functions of background fields and their derivatives. T h e explicit form of this functions is given in Ref. [10]. Taking into account the eqs. (10, 11, 14) we can obtain finally
IT] = (~,)1/~ ~oM(~)[R(~) E(0) + ~ ~ E ( ~ +
+ ( a , ) l / : 6~M ((r) E(S)
1
E(O) _- 1 4--~ ( A ~ - c9~,B~')
+
+o(~(~') 1/~)
1 ( 2 O A ~ - 2A;~ -
(11)
+OrBs, - OuB,~)
Hun = Ot, B u - O~B u - 2At, ~
E< ~) : (12)
~u
T h e function 6ou(~r) is defined as follows [10]: 6(t
-
¢') 60M(O/) 6 ~
(13)
We see t h a t the trace o p e r a t o r is a linear combination of the t e r m s g~b OaxUObxU~?Uu, Hu~,(x)&u£~ , K ~ , ( x ) , K k ~ ' + u ( x ) , : ~ O u ( x ). Their renormalization can be fulfilled by s t a n d a r d methods. Taking into account the renormalization of background fields (10) one gets
(C)0 : [~],
(+o ~ ) 0 - [+. ~ ]
(#~ ao~,' o ~ ~ ~.~)o : [#~ ao~,' + ('~')~/~ u ~ [-vh> ~g'M(,~)+
a ~ ~ ~,~]+
c9~,c9~,A~-
-2c%,O~,A~ - 2c%,OuA ~ + 30,c9~,0~,B'~ +
The terms O(e(a')~/~) do not contribute to renormalized o p e r a t o r [T]. Here
--
(15)
where
1 n : ~ ~o~(~) + n ~, ~g~(:)+
6it'(t) 6xU(o.,)
(14)
+ ( a ' ) ' / : f~M ((r)[& ~' E (6) + K E(7)]+
(~ H,~ ~ ~ ~ou(~)-
~u = 0t,~°~ -
, ,
OaM(O')+
+ ~ E(2) + K ~ E<2) + K ~ E(+ + k E<~)]+
T = ~ g~b OaXU ObX~ ~1~,,+ 87r ÷ (Ot¢)1/2 1
1 +-~
d2V(2) ,
-
1 (213Bu + 30~,3~,B ~ _
O~,A~-
-40~,A~)
1
E(2) : ~ ( a . ~ - ~ + D~. - a ~ a ~ ~) E ( 4 ) = _ _ l (2D~ 1 - 2 ~ 1 - 0 c ` B 4~r E(~) 1 = 4~ (O~2 - 0 ~ )
~+A%)
E(6> = __1
(~,~ _ Ou~: )
E (7) =
--1
(O~B ~' - A ~ ÷ 4~,1)
E ( S ) _-
1 A~:-c%'B'~) 8~(
47r 8~r
(16)
T h e principle of q u a n t u m Weyl invariance leads to [T] -- 0. As a result we have the equations of motion for background fields in the form E (i) = 0 ; i -- 1, 8. Let us notice the action (8) is not Weyl invariant classically. Therefore the equation
138
LL. Buchbinder/Nuclear Physics B (Proc. Suppl.) 49 (1996) 133 138
[T] = 0 can be fulfilled only under quite definite combinations of classical and quantum contributions. One can show that the theory under consideration possesses the symmetry transformations
6B~ = A t,
6~v2 = A
where A~, and A are the arbitrary functions. It allows to impose the gauge conditions B~, = 0, ~2 = 0. Returning to initial string coordinates x ~' ---* (a') -1/2 a¢' one can rewrite the equations E (i) : 0 as B~,=0,
~2=0,
~ol = 0 ,
~I, = 0 ,
(17)
3. D.Lust, S.Theisen. Lectures on String Theory. Lecture Notes in Physics. SpringerVerlag, 1989. 4. E.S.Fradkin,
A.A.Tseytlin, Phys.Lett. B158 (1987) 316; Nucl.Phys. B261 (1985) 1.
D.Friedan, E.Martinec, 5. C.G.Callan, M.J.Perry, Nucl.Phys. B262 (1985) 593. 55 (1985) 1846; 6. A.Sen, Phys.Rev.Lett. Phys.Rev. D32 (1985) 2102. 7. A.A.Tseytlin, Int.J.Mod.Phys. 1257.
A4 (1989)
8° H.Osboru, Ann.Phys. 200 (1990) 1.
[~A#~ - m 2 A~,v = 0 0,A" = 0
(18)
A~=O
where rn is the mass of first massive mode of open string. The equation for A~,v (18) are just the correct equations of motion for spin two massive field. As is well known the analysis of the spectrum of open string leads to the same equations.
ACKNOWLEDGMENTS
I am very grateful to E.S.Fradkin, V.A.Krykhtin, S.L.Lyakhovieh and V.D.Pershin for fruitful collaboration. It is a pleasure to thank S.J.Gates, H.Osborn, B.A.Ovrut, J.Schnittger and A.A.Tseytlln for useful discussions on the various aspects of the paper. Work supported in parts by RFBR, project No. 94 - 02 - 03234 and ISF under the grant RI 1300.
REFERENCES
1. M.B.Green, J.H.Schwarz, E.Witten. Superstring Theory. Cambridge Univ. Press. 1987, Vol. 1. 2. L.Brink, M.Henneaux. Principles of String Theory. Plenum Press, 1988.
9. I.L.Buchbinder, E.S.Fradkin, S.L.Lyakhovich, V.D.Pershin, B304 (1993) 239.
Phys.Lett.
10. I.L.Buchbinder, V.A.Krykhtin, V.D.Pershin, Phys.Lett. B348, (1995) 63. 11. D.M.McAvity, H.Osborn, Class. Quant. Gray. 8 (1991) 603; 8 (1991) 1445; Nucl.Phys. B394 (1993) 728; B406 (1993) 655.