Non-trivial renormalization group fixed points and solutions of string field theory equations of motion

Volume 252, number 4 PHYSICS LETTERS B 27 December 1990

Non-trivial renormalization group fixed points and solutions of string field theory equations of motion

Ashoke Sen Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

Received 3 June 1990; revised manuscript received 17 September 1990

If we perturb a conformal field theory by a nearly marginal operator of dimension ( 1 -h , 1 -h ) , we get a nearby fixed point at a value of order h of the coupling constant. Starting from a string field theory based on the original conformal field theory, we show how to construct a solution of the classical equations of motion of this string field theory that represents the conformal field theory at the new fixed point. We also discuss the construction of time-dependent classical solutions that interpolate between these two fixed points, and the application of these results to the cosmological solutions of Antoniadis et al.

Recent advances towards the construct ion of a gauge-invariant closed-string field theory [ 1 ] based on non- polynomial interactions [2-6 ] have made it possible to start analyzing the relat ionship between the solutions of the classical equat ions of mot ion in string field theory and two-dimensional conformally invariant field theories [ 7-9 ]. In par t icular it was shown in ref. [ 7 ] that for every classical solution of string field theory, we can construct an opera tor QB, which is nilpotent, and hence has the potent ial of being identif iable with the BRST charge of a new conformal field theory. In ref. [8] we constructed a specific class of classical solutions, those corresponding to BRST-invar iant physical states of the conformal field theory ( C F T ) on which the original string theory is based, and showed that the corresponding BRST charge QB may be identified, after a s imilar i ty t ransformation, with the BRST charge QB of a new conformal field theory CFT" , obta ined by perturbing the original conformal field theory CFT by a marginal operator . It was further shown in ref. [9 ] that i f we take the original string field theory action, expand it a round the new classical solution, and calculate the corresponding S-matrix elements, the result is identical to the S-matr ix elements computed in the string theory based on CFT" .

We shall consider a string theory based on a uni tary conformal field theory of central charge 2 6 - d + 3Q 2, together with d free scalar fields, one of them having the wrong signature, and hence representing a t ime-l ike coordinate. In order to keep the discussion general, we shall assume that the stress tensor associated with the t ime-like coordinate has an extra term propor t ional to ½QOZX°, which corresponds to the presence of a background t ime-dependent di laton field, and gives rise to a total central charge 1 - 3 Q 2 associated with the t ime- like coordinate [ 10]. We shall refer to the conformal field theory of central charge 2 6 - d + 3Q 2 as the internal conformal field theory, and assume that it has a nearly marginal operator ~0(z, -~) of d imension (1 - h , 1 - h ) (say) . It is well known [ 1 1,12 ] that if we perturb a conformal field theory by the opera tor - 2 f~0(z, _~) dZz, then there is a non-tr ivial fixed point of the renormal izat ion group equations at a value of 2 of order h / C ~ , where C ~ is the coefficient of ~0 in the opera tor product of ~0 with ~0. Furthermore, the central charge of this new conformal field theory differs from that of the original conformal field theory by a term of order h 3 [ 11 ]. Thus, we see that we can construct a consistent string theory based on this conformal field theory if we ignore terms of order h 3. In this letter we shall show how, starting from the string field theory based on the original conformal field theory CFT, we can construct a classical solution that satisfies the string field theory equations of mot ion to order h 2, and represents the conformal field theory CFT" at this new fixed point.

Although the new conformal field theory has a central charge that differs from the central charge of the original

566 Elsevier Science Publishers B.V. (North-Holland)


conformal field theory by a term of order h 3, and hence, by itself, is not strictly a candidate for a classical string vacuum, this shift in the central charge may be cancelled by introducing a background t ime-dependent di laton field [ 10 ]. To be more concrete, let us consider the case where the total stress tensor of the original CFT is given by

½ (~X°OX°+ QO2X °) + T , ( 1 )

where T is the stress tensor of a CFT with central charge 25 + 3Q 2, which adds to the central charge 1 - 3 Q 2 of the X ° system to give c = 26. Now consider per turbing the CFT described by the stress tensor T by a nearly marginal per turba t ion so that we get a new conformal field theory with central charge c ' , described by the stress tensor T' . Then, a conformal field theory descr ibed by the stress tensor

½ (3X°OX° + Q' 02X 0 ) -4- T ' (2)

is a possible candida te for the classical-string ground state, p rovided the total central charge adds up to 26. This gives Q' = (Q2 + 8c/3 ) ~/2 ~_ Q + Kh 3 since 8c ~ h 3. I f we are looking for a solution of the string field theory equations of mot ion describing this new background, then the effect of the change from Q to Q' will be descr ibed by a background t ime-dependent di la ton field of the form Kh 3t where t is the zero mode of X °. This, however, is of order h 3 and will be invisible at order h 2.

For Q = 0 , the same analysis shows that we need a di la ton field D of order Kh3/2t ~. The order h 3/2 dilaton field, however, will not contr ibute to the equations of mot ion to order h2; this can be seen as follows. The only term involving D that can contr ibute to the equat ion of mot ion up to order h 2 is the term linear in the di laton field. This term involves second and higher der ivat ives of the di la ton field with respect to the space and t ime coordinates [ 13,14 ]. For a di la ton field varying l inearly with t ime, the contr ibut ion to the equat ion of mot ion from such terms vanishes identical ly ~2. This shows that the contr ibut ion to the equations of mot ion from terms involving the di laton field is of order higher than h 2, and hence there is no surprise that we can get a consistent solution to the equat ions of mot ion to order h 2 without including the di la ton field.

Since at the new fixed point the coefficient of per turba t ion 2 is of order h, we can use 0 ( 2 ) and O ( h ) interchangeably. To order Z ° we can treat the field ~0 (z, g) to have d imension ( 1,1 ). F rom the analysis of ref. [8 ] we know that the classical solution representing the corresponding conformal field theory is of the form

17%, > = - - x/~2 CoC(O)g(O)~o(O)10> + Is> , (3) g

where g is the string coupling constant , c(z) , g(g) (b(z) , g(~) ) are the ghost (ant ighost) fields, cff = (Co + go)/ x/2, b~ = (bo+go)/x/~, and Is> is a state of order 22. Since we need to f ind Is> to order 22, we now write down the classical equat ions of mot ion [ 7 ] to order 22:

Qsb~ PI 7'c, > + ½gl [ 7%, 7%, ] > = 0 , (4)

where P is the project ion opera tor into the L o = L o state, Qn is the BRST charge constructed from the Virasoro generators of CFT and the ghost fields, and I [AB] > for any two states A and B is defined by

bff I [AB] > = ( L o - / S o ) l [AB] > = 0 , (5)

(C[ [AB] > = ( - 1 ),c(f~ ob6- C(O)f2obffA(O)f3obffB(O) > , (6)

~ In this case, in order to get a real dilaton background, we need h to be negative. Examples of such cases are not uncommon; in fact in the types of theories being discussed here, if perturbing CFT by an operator of dimension ( 1 -h , 1 - h ) produces the conformal field theory CFT', then CFT may be obtained by perturbing CFT' in the reverse direction by the same amount by an operator of dimension ( 1 + h, 1 + h ), (In computing the fl-function in these cases one must expand various terms in powers of h before taking the ultraviolet cut-off to infinity)

~2 If Q # 0, however, we have terms proportional to Q3oD in the equation of motion, which shows that in this case an order h 3 dilaton field does contribute a term of order h 3 to the equations of motion.

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for any three states A, B and C. Here nc is the ghost number of the state I C>, f~, f2 and f3 are three known conformal maps (see ref. [ 7 ] for details), and)Co q) for any field q) denotes the conformal transformation of the field ~3 q) under the conformal m a p f Substituting eq. (3) into eq. (4) and keeping only terms of order 22 we obtain

22 xS2,~g QBc(O)e(O)~o(O)lO> +QBbff PIs> + g [ [ (co-cgq~)(co- cg~o) ] > = 0 . (7)

Now if we set A and B to be equal to co- c(~o in eq. (6), it is clear that there is an infinite set of Lo,/5o eigenstates C for which the right-hand side of this equation will be non-vanishing. These states include various excited states from the ghost as well as the matter sector. In other words, if we expand the term [ (co- cg~o) (co- cgq~) ] in the basis of Lo,/5o eigenstates, it will contain a linear combination of an infinite number of terms, and hence in this basis it will be difficult to solve eq. (7) directly. In order to simplify our analysis, we introduce the operator K, which satisfies

( Lo -/5o)K=bo- K=O

and

(q), Ico- [K, QB] Iq~2>

= x/2 <flo,~, (o)A oCe~o(O)Ao q~ (o) > - < q~ I c~

( 8 )

for any two states I q~ ) and I (/)2) satisfying

(Lo- /5o ) I ~i > = b6- [ ~i > = 0 f o r i = l , 2 .

[dzg(g)~o(z,g)-dgc(z)q~(z,g)llqb2>+O(2) (9) Izl =~

(10)

Here ~ is an arbitrary small number, which should be taken to zero at the end of the calculation. The operator K was first introduced in ref. [ 8 ], and the existence of a solution of eq. (9) was also proved there. In that analysis, of course, we have taken ~0 to be a dimension ( 1,1 ) primary field in the matter sector, and an exact solution of eq. (9), without the 0 ( 2 ) term, was found. In this case, since (0 is not an exact marginal operator, the solutions for K, given in ref. [ 8 ] satisfy eq. (9) up to terms of order 2. As we shall see, this is sufficient for our purpose. We now define a new state ] t ) through the equation

22 Is> = - - c j Kc(O)e(O)~(O)lO) + It> • (11)

Since c (0) d(0 ) q~ (0) ] 0 ) is BRST invariant to order h (or, equivalently, to order 2), we obtain

22 QBbjPIs> = ~ [QB, K]c(O)e(O)~o(O)10> +QBb~PIt) + 0 ( 2 3 ) . (12)

We substitute this into eq. (7), and take the inner product ofeq. (7) with an arbitrary state qbl ]Co-, with ] 6b I ) satisfying eq. (10). This gives

22 ,,/22g < cb~ Ic~ Qnc(O)g(O)~o(O)10> + ~ (q)~ Ico- [QB, K]c(O)g(O)~o(O)10> + < q~ Fc~ QBbo- Plt>

22 + - - < ~ l Ic~ I[(co-ce~o)(co-ce~o)] >

g

= 0 ( 2 3) . (13)

~3 Here we are using the words state and field interchangeably, since in conformal field theory [ 15 ] there is a one-to-one correspondence between the local fields q~ (z, g) and the states ] • > through the relation I q)) = ~ (0) ] 0 ) .

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Since all terms in eq. (7) are annihilated by b6- and Lo-/So, eq. ( 13 ) for a complete set of states ( qb, I satisfying eq. (10) contains all the information of eq. (7). Using eqs. (6) and (9) we obtain

(q), I c6- QBbff PI t) + ~ - ( ~, Ic6- QBc(O)g(O)¢o(O) [0 )

+ ~ ~I~111Co [dze (~ , ) e ( z , e ) -d f f , c(z)~o(z,e)lc(O)e(O)~o(O)lO) izl =e

= O ( ~ 3 ) . ( 1 4 )

Since q~ is a dimension ( 1 - h, 1 - h) primary field in the matter sector, we obtain

QBc(O)e(O)~o(O) 10) = -h(co +go)c(O)e(O)¢o(O)10) . (15)

In addition, assuming that the conformal field theory under consideration is unitary, and hence has only oper- ators o f positive dimension, we obtain, using an operator product expansion, and the normalization convention f(dz/z) = 1, f (dZ/g) = - 1,

[dz g(g)Co(z, g) -dgc(z)~o(z, g) ]c(0)g(0)q~(0) I 0 ) Izl =E

z

where Co, ojok is the coefficient of 0~ in the operator product of 0j with 0k, and (h,, hr) is the dimension of the field 0r. Note that in eq. (16) we have used the smallness o fh to expand it in a power series in h. The summation over r denotes the sum over all fields o f dimension less than 1, except ~0; such fields are necessarily all primary in a unitary conformal field theory. The term involving ~0 has been written separately. In addition, we shall assume that all the other 0rfor which C ~ is not zero have hr¢ 1, so that 2( 1 - h r ) is a finite number [ 11 ]. The contribution from fields of dimension hr> 1 vanishes in the E-,0 limit, as can be seen from eq. (16). This, in turn, shows that all the fields 0r that contribute to eq. (16) are primary fields. Using eqs. (15) and (16), eq. (14) can be written as

x[5~" ( h - ½,~C~) (Co + eo)c( O )e( O )~o( O ) {O > QBbff P l t ) - ~ -

22 + - ~ (Co+eo) o,~,~E Co,~lelZh'-2c(O)e(O)O,(O)lO>

= O ( h3 ) • ( 1 7 )

A solution of this equation is given by

2= 2h /C~ , (18)

2 2 Co,~o 2hr-- 2Cff I t > = - ~ - - g ~ ( h , - 1 ~ lel C(0)g(0)0r (0) ]0) + O ( ~ 3 ) • (19)

Note that the point 2 = 2h/C~,~,~, is precisely the location of the vanishing of the//-function ~4:

fl~ = 2 h 2 - C~o22 . ( 20 )

~4 The fl-function is calculated with the convention [8 ] that the perturbation term added to the action of the original conformal field theory is -). f d2z ~o(z, z) with dZz_ -- - ( 1/~) dxdy ifz=x+ iy.

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Thus the classical solution I q~cl ) of the string field theory, obtained from eqs. (3 ) , ( 11 ) and ( 19 ):

( Co'~° 2hr-2~r(O))c(O (21) I T d ) = x/22c6- ( l+½2K)q~(0) -½2 Z ~ lel ) e ( 0 ) [ 0 ) g 0r :P: ~

for 2 given in eq. ( 18 ), represents the conformal field theory corresponding to the non-trivial fixed point of the renormalization group equation. Note that although the solution given in eq. (21 ) appears to depend on e, there cannot really be any dependence on ~, since it is a fictitious parameter introduced through eq. (9). In fact the explicit e-dependent terms in eq. (21) must cancel against the e-dependence of K appearing in the same equation.

The operator K plays an important role in obtaining the solutions in that eq. ( 11 ) transforms the non-local equation (7) on I s ) to a local equation (14) on I t ) , where the only non-vanishing terms come from singular terms in the operator product expansion of ~0 (z, ~) with ~0 (0). As a result, eq. ( 14 ) contains only a finite number of terms and can be solved easily. In addition, the contribution now comes from short-distance effects, the connection between the equations of motion and the renormalization group fl-functions becomes quite clear. This role of the operator K is also apparent in the analysis of ref. [ 8 ], where the similarity transformation by the operator S = 1 + 2 K converts an apparently non-local operator (~n to a local operator 0n, which can then be identified with the BRST operator of a new local conformal field theory.

Once we have identified the classical solution (eq. (21) ) corresponding to the perturbed conformal field theory CFT", we can now go ahead and construct the corresponding nilpotent charge QB from the string field theory (see refs. [7,8] ) and compare it with the BRST charge (~n of the perturbed conformal field theory. One can also expand the original string field theory action around the new classical solution, compute the S-matrix elements from the corresponding field theory, and compare them with the S-matrix elements calculated in string theory based on the new conformal field theory CFT". Presumably, the analysis may be carried out in the same manner as in refs. [8,9]. We hope to return to these questions in the future.

One can also investigate the existence of t ime-dependent classical solutions interpolating between these two classical solutions, one corresponding to I ~d) = 0 and the other given in eq. (21). The analysis can be done along the lines of the analysis of ref. [ 16]. Let us take a trial solution of the form

I ~ c , ) - g co [l+½2(t)g]~o(O)-½2(t) o)~ ~ l~12m-2Or(O) c(0)e(0)10>, (22)

where t=-x °, the zero mode of the time-like coordinate X °. We assume that 2(t) is a slowly varying function of ! so that d ' 2 / d t " < 22 for all n >/1. If we now plug eq. (22) into the equations of motion (4) we get an extra term arising from the fact that

C +C . / l d22 dZ) [Q~,2(t)]c(O)e(O)~o(O)lO>=( o : o ) ~ c ( O ) e ( O ) + ½ Q ~ ~0(0)10). (23)

As a result, eq. ( 18 ) for 2 is now modified to

2 h 2 ( t ) - dZ2(t) d2 dt 2 Q ~ -C~,~o[2(t)]2=0. (24)

We want to study whether there exists a solution 2 ( 0 that tends to zero and 2h/C~,~o as t ~ ~ Go. To do this, note that [ 16 ] the equation of motion of 2 is equivalent to that of a point particle moving in a potential,

1/(2) = - h 2 2 + ~ C ~ 2 3 , (25)

with a damping (antidamping) term if the coefficient Q is positive (negative) [ 16] ~5. For h > 0 the potential has a local maximum #6 at 2 = 0 where it takes the value zero, and a local minimum at 2 = 2h/C~,~, where it takes

~5 The sign of Q, in turn, depends on whether we are considering expanding or contracting cosmological solutions. In particular, we obtain damping for an expanding universe and antidamping for a contracting universe.

~6 This clearly indicates the instability of the original string theory arising from the existence of tachyons.

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the value - 4 h 3/3 (C~o~o)2. I f Q is posi t ive then there is certainly a solution that starts at 2 = 0 ( top of the potential well) at t = - o ~ and rolls down to 2 = 2 h / C ~ as t ~ . On the other hand, if Q is negative, then one can take the "rol l ing down" solution corresponding to the paramete r - Q and t ime reverse it to get a "roll ing up" solution that goes from the point 2 = 2 h / C ~ at t = - ~ to 2 = 0 at t = ~ . F rom the conservat ion of energy we see that for Q = 0 there cannot be any solution that approaches the points 2 = 0 and 2 = 2h / C ~ asymptotically.

We note that as in the case of the t ime- independent solution, the complete solution 7"(t) should include a di laton background of the form Kh 3tf(t) where f ( t ) --, 0 as t-~ - ~ a n d f ( t ) - , 1 as t-* ~ , so that the di laton field has the correct asymptot ic form. This is, however, invisible to order h 2.

The reader may wonder how we can see the flow from one theory to another by restricting ourselves to terms of order h 2, since in the case of renormal iza t ion group flow, the flow from one theory to another occurs only i f the central charge of one theory is smaller than the other, while in this case the central charges o f the two theories are the same to order h 2. This may be unders tood in the following way. First of all, note that in our case even when we take into account higher-order terms in h, the flow is from a c = 2 6 system to a c = 2 6 system, that is there is no change in the total central charge. Despi te the fact that the initial system and the final system both have the same total value of c, the string equations of mot ion tell us that there is a flow from one background to another. This does not contradic t Zamolodch ikov ' s c-theorem [ 12 ], since the flow in our case is in real t ime, and not in the renormal iza t ion group scale. In the study of renormal iza t ion group flow, Zamolodch ikov ' s c- function plays the role o f the action, since the fl-functions that appear in the renormal iza t ion group equations are related to the der ivat ives of c with respect to the various coupling constants. Thus to get an order 22 contri- but ion to fl we need to keep terms to order 23 in c. This is what is needed to see the flow from one CFT to a nearby CFT under the renormal iza t ion group t ransformat ion. In string field theory, the role o fc is played by the action of the string field theory. In order to see the flow (in t ime) between two nearby conformal field theories, we need to keep terms of order 7 '3 in the action. In our analysis we have kept terms of order 7"2 in the equations of motion, which is equivalent to keeping terms of order 7"3 in the action. Thus, it is not at all surprising that we can see the flow from one theory to another within this approximat ion .

Finally, note that eq. (24) may be der ived from the action

f dt e 'F1 e L~ ( ~ t t ) ) 2 - v ( 2 ) l , (26)

which, to order ~3, may be shown to be identical to the act ion of the string field theory evaluated for the field configurat ion given in eq. (22) . The origin of the overall mul t ip l icat ive factor of e Qt can be t raced to the fact that now in a given correlat ion function, the zeroth component of the m o m e n t u m is not conserved, but is vio- lated by a fixed amount iQ [ 10]. This breaks t ime t ranslat ion invariance, and when we express the act ion as an integral over t by going to the Four ier - t ransformed space, we obtain a lagrangian of the form given in eq. (26) .

References

[ 1 ] T. Kugo and K. Suehiro, Nucl. Phys. B 337 (1990) 434. [ 2 ] M. Saadi and B. Zwiebach, Ann. Phys. (NY) 192 ( 1989 ) 213;

T. Kugo, H. Kunitomo and K. Suehiro, Phys. Lett. B 226 (1989) 48. [3] M. Kaku and J. Lykken, Phys. Rev. D 38 (1988) 3067;

M. Kaku, preprints CCNY-HEP-89-6, Osaka-OU-HET 121. [4 ] H. Sonoda and B. Zwiebach, Nucl. Phys. B 331 (1990) 592. [ 5 ] B. Zwiebach, MIT preprints MIT-CTP-1830, 1831. [6 ] M. Saadi, MIT preprint MIT-CTP-1860. [7] A. Sen, Phys. Len. B 241 (1990) 350. [8 ] A. Sen, TIFR preprint TIFR/TH/90-07, Nucl. Phys. B, in press. [9] A. Sen, TIFR preprint TIFR/TH/90-24, Nucl. Phys. B, in press.

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[ 10 ] I. Antoniadis, C. Bachas, J. Ellis and D. Nanopoulos, Nucl. Phys. B 328 (1989) 117. [ 11 ] J. Cardy and C. Ludwig, Nucl. Phys. B 285 [FS 19 ] ( 1987 ) 687. [ 12] A.B. Zamolodchikov, JETP Lett. 43 (1986) 731; Sov. J. Nucl. Phys. 46 (1987) 1090. [ 13 ] C.G. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B 262 (1985) 593. [ 14] T. Banks, D. Nemeschansky and A. Sen, Nuel. Phys. B 277 (1986) 67. [ 15] A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [ 16 ] S. Das, A. Dhar and S. Wadia, Mod. Phys. Lett. A 5 (1990) 799.

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Non-trivial renormalization group fixed points and solutions of string field theory equations of motion