Z. Phys. C 59, 617-620 (1993) ZEITSCHRIFT FOR PHYSIK C �9 Springer-Verlag 1993

One-loop counterterms in 2d dilaton-Maxweli quantum gravity

E. Elizalde x'*, S. Naftulin 2, S.D. Odintsov 3'**

~Department E.C.M., Faculty of Physics, University of Barcelona, Diagonal 647, E-08028 Barcelona, Spain 2Institute for Single Crystals, 60 Lenin Ave., 310141 Kharkov, Ukraine 3Department of Physics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 724, Japan

Received 24 February 1993

A b s t r a c t . The renormalization structure of two-dimen- sional quantum gravity is investigated in a covariant gauge. One-loop divergences of the effective action are calculated. All the surface divergent terms are taken into account thus completing previous one-loop calculations of the theory. It is shown that the on-shell effective action contains only surface divergences. The off-shell renor- malizability of the theory is discussed and classes of renor- malizable dilaton and Maxwell potentials are found.

The current activity in the study of the two-dimensional (2d) quantum gravity is motivated by the following (main) reasons. First of all, it is extremely difficult to study the physics of black holes and of the early universe in the ambitious frames of the modern models of four-dimen- sional quantum gravity. Solvable toy models like the 2d gravity simplify the considerations and hence can help in the study of more realistic situations. From another point of view, the 2d quantum gravity is interesting on its own as ~a good laboratory for developing formal methods of the quantum field theory. Moreover, a connection with very fundamental physics may be developed because some models of the 2d gravity are string-inspired models.

Recently, an investigation of the renormalization struc- ture of the 2d dilaton gravity has been started [1-43. The one-loop counterterms in covariant gauges have been calculated [1-4] and it has been shown that the theory may be renormalizable off-shell for some choices of the dilatonic potential (in particular, for the Liouville poten- tial). The only one-loop divergences on shell are surface terms [1].

In the present paper we address the question of the calculation of the one-loop counterterms which appear in the dilaton-Maxwell 2d quantum gravity as given by the

* E-mail address: eli @ ebubecml.bitnet ** On sabbatical leave from Tomsk Pedagogical Institute, 634041 Tomsk, Russia. E-mail address: odintsov @ theo.phys.sci.hiroshima- u.ac.jp

following action* [5, 6].

S = - ~ d2x x//g [�89 ~pt~, ~p + c, R~0 + V(q~)

where F~v=OuA,-t~,A u is the electromagnetic field strength, ~p is the dilaton, V(~o) is a general dilatonic potential andf(~0) is an arbitrary positive valued function. Notice that for some choices o f f and V this action corres- ponds to the heterotic string effective action. The theory admits black hole solutions. Properties of the 2d black holes associated with different variants of the theory (1) were studied in [5,9, 10] along the same lines as de- veloped for the pure dilaton gravity (without the Maxwell term) [7, 8]. The model given by (1) is also connected (via some compactification) with the four-dimensional Ein- stein-Maxwell theory which admits charged black hole solutions [14].

The one-loop counterterms for the theory with the action (1) have been already calculated in [6, 9 ] - b u t not taking into account the contributions of the divergent surface terms. In this paper we generalize those calcu- lations in order to take into account all such terms in the effective action. It is well known that they are relevant, e.g. in the context of the Casimir effect calculations (see, for example, [15]).

The classical field equations corresponding to (1) are r 1 , 2 Vu(fFU~)=O, --Aq)+ClR + V (~o)+~f (q~)F,~=0,

- l(W~o)(VPq~) + l g ~ V'~pV, q~ + cl (WVP -g~PA)q~ (2)

~__i ~f117• !~fllc~ 2 _I ~C It? ~t K ' f l # ~ 0 " 2 ~ / v rs~ J l l ~ v - - 2 J ~ l i ~

We will use (2) below, in the analysis of the divergences of the theory.

The background field method is employed. The fields are split into their quantum and background parts,

~o~qS=q~+q~, A~,~A,=Au+Qu , g~O~,~=g~,~+hu~, (3)

where the second terms ~o, Q, and h,, are the quantum fields. In what follows we shall use the dynamical variables h = gU~h,~ and/~.v i =h,,-~hg~, rather than h,,.

* For a higher derivative generalization, see [12]

618

In order to make contact with [4] where the surface divergences in the absence of the Maxwell term are cal- culated, the following term is added to the initial action

AS = -c~ ~ ~ d2x~/g[r "~- �89 (4)

where ~ is an arbitrary parameter. Owing to the 2d ident- ity Ru~-�89 (4) is obviously zero. (Notice that AS is very similar to a kind of Wess-Zumino topological term. The observation that the parameter ~ only appears in the surface on-shell counterterms and is in fact related to the trace anomaly, hints in this direction.) However, the sec- ond variation of (4) may be important for the removal of non-minimal contributions from the differential operator corresponding to S ~2). A possible interpretation of this term is given in [4].

To fix both the abelian and general covariant invari- ances we use the following covariant gauge conditions

S C j F = --�89 I d 2 x { z A c A B ) ~ B } '

z*=-vvo,,

'~ "~---" e l ( ~ - 1 ) / 2 . ( 5 )

The index A={*,kt} r u n s over the three-dimensional space of gauge transformations. The most convenient form for the matrix Can is

CA. = v/gdiag(f , 27r

If (4) is absent then ~ = 0 and y= - q / 2 . As is well known, the divergences of the one-loop effective action are given by

F i div = ~ Tr In ~ - - iTr In ~/lgh I d~, (6)

where M//gh is the ghost operator corresponding to S~v (5) and the second variation of the action defines the operator 9vg:

�9 ^ �89 ~Ot~Pij~O J = S (2) --] - A S (2) + Sr v . (7)

Here r stands for the set of the quantum fields, {Qu, r h,/~u~ }. Making use of the background vs quantum fields splitting (3) one easily finds that

+ L ' v , + is)

where

f gua

0

o o o

2 0 -- 0

/ ~ - 1 C1

0 2 (4+2 77) 0

1 fiuv,~,,a \ o o o t

the projector l 6"~'an being t$~*'~n-2v~-~*~n~ , and s , a .z , ~ aa , a ~a

= f (V q~)g - f ( V q~)g - f ( V q~)g ,

^ ^ ^ 1 LZ12=-s ~a, LZ~a=--L~ ,=-~ fF "a,

(9)

^ ^ ^ C 2

L~4 = - L~41 =fF~P "#'uto-fF~fi "~'ato, L~2 = ~ ~-~ , ~ (VZr z~q~-

s163 ^z ^~ ^aa xto ^ c~ L24= - L42 =(Vtor ' , L~s = - ~ - (V*q~),

= - s = 2 (% (10)

~ - - ~

3e~ (VZr 2

flii=�89 M 2 3 = M 3 2 ~_ �89 l/-t 1 s ^ 1 2 - - - s J -u, , M33 = ~fF~,~,

^ 3 to ^#v , , l , r *aft

to ^ l iv , J,r ^a s + (Va~b)(V q~)n n~,

- - 8 J ~ / ~ v * T J I t o . l z * Z p l r

.F _ r r ' c o K ~ ,~p ~ u v ~ctl~ 2 J ~ 1 I t a 2 a r p "

Now we can start the computation of the contribution of the gravitational-Maxwell part to the effective action divergences. First of all, one easily sees that

Tr In 3v~ Idiv= Tr In (-- / ( - 1 ~ ) Idi~, (11)

because T r l n ( - / s gives a contribution proportional to 6(0) which is zero in the dimensional regularization. How- ever, in order to apply the standard algorithm

i ~ div i 2 / ~ V z + / l ) div Fdiv=~ Trln =~ Trln(1A +

=--2e / 7 + g i - V Z / ~ - / ~ z / ~ a , (12)

where e = 27r(n- 2), one should take care that the operator 3r be hermitean, properly symmetrized, etc. Imprudent integrations by parts actually change the matrix elements which may lead to the breakdown of these compulsory properties.

In order to recover an operator ~ with the required properties the doubling procedure of 't Hooft and Veltman [13] is very useful. A clear explanation on how to do this in the present context can be found in [2-4] (for more details, see [13]). In fact, using this method amounts to the following redefinitions of the operator ~ f given in (8):

L',=�89 L D - v , R., ^ ^ , 1 ^ ^ T 1 ~ ^ T 1 ^ . / / 4 ~ ' = ~ ( J / / + ~ ' ) - ~ V Lz--2AK. (13)

^ ~ _ • and Now, introd-ucing the notation -~---2 ~ '

H = - K - ~ s / 4 " , we have the operator 3 f of the proper form and thus we can apply the algorithm (12). ̂

The explicit form of the operators E ~ and H can be found from (10) and (13) to be the following

f, E~ ~ V ~ ~ V ~ ~ V a~ ( ) l =~ [ ( r q$)gp+( pr ], z j

619

(i~)~,= vr

( I~ 2'11 1iS"A [~A~a __1 17 3:zfl, am _I IC? 3.o I) eft t~ 13=~p, tL, / 4 - - 2 1 p o x --2 ~ Xpo ~ (i*)I = - f V ~, ( ~ ) ~ =(i*)~ = O,

(i~)I= -�89 ~,~'o

( / ~ ) ~ = ( f + f + f ' ) F , a ' c 2 1 2?4 Ca) ( i~ , )23 274 2ca (V*4),

(i~)~ =o,

(~)~ = ~ (v~4)f~ ~~ (i~)~' =2-~ >-~-~" -~-"""

1 44~ i% 4) Po~, (i*)~ = 2 ~ (V- 4)FL~' (i'%+ = ^*~'

( c a 1 ) ,o %a^x. ^ ^.a,,. (/~)~= ~r f(;b (V 4)(Po.,Po,,-Po,,,o,~P )

1 a ~̂/~.

^ 1 �9 ^ 1 L 2 171= - ~ Rgo, 17~= - V'+4Cl Fur' Cl

(14)

r,3=_'_ c, C1 \2C 2 4?4 4Cl] \C, 274 ]

/i~. = [ ( 3ca ; ) (VxV'~ 1 L\2N ) - ~ (V~O)(w4)

;4

,., o.a PKv 27~

4~4J We)-R

__v iF/,]^,, +274 + 874 ] Po,,"

Applying the algorithm (12) and the matrix elements (14), after some tedious algebra we find the gravitation- Maxwell contribution to the effective action:

1 [ 2 V (2~-1 f ) ro~i~=-2~d~x ~ 2R+ v'-7~+ + FL

+ q~b Ca

/f,, f,2 1 ca Cl z "~(Va4)(V,z4)] ] +(7 s

(15)

To complete the calculations we should add the ghost contributions. The ghost operators corresponding to (5) are

^

^ C 1 ca (V~O)Vu+~(VuV~4)+R~. (16)

Again, using the algorithm (12), we get the divergent ghost contributions of the effective action

1 j, dZxx/g [3R Cl Fgh aiv = - ~ + 2 ~ (Aq $)

+(-2~ 2 87~2 ~(V~q~)(V~gb)l. (17)

Note that the doubling procedure (13) should not be invoked.

Finally, the total divergent part of the one-loop effective action given by the sum of (15) and (17) becomes

Fdiv=-~sId2x~l/gIse+2 V'--~+(f2@l+f ) Flv

I f ' 1 15

fS" S'/ 1 ~ ) ] + t 7 f 2 ~ + (V~4)(V~.~) . (18)

A few remarks are in order. First of all, in the absence of the Maxwell sector ( f=0) and when dropping the surface divergent terms, (18) (no surface divergences) coincides with the results obtained in [2-4] in the same gauge,

, V c 1 z -lId2xSI v- + (v4)(v ) j. ( 1 9 ) raiv=

Notice that the term (4) was not introduced in [2, 3] so that ?= -ca~2. Besides, we actually disagree with [4] in the surface divergent terms. This stems from the fact that the term _~7~/~ corresponding to the algorithm (12) is missing in [4]. This term gives an additional contribution to the surface divergencies.

Secondly, dropping the surface divergent terms one can see that the divergencies of the Maxwell sector coincide with the ones previously obtained in [6, 9].

Eq. (18) which contains the surface divergent terms of the theory (1), constitutes the main result of the present work, and completes the calculations done in [6, 9]. It is, interesting to notice that there is no dependence on 7 in the terms which result from the Maxwell or the Maxwell- dilaton sectors (only in the pure dilaton sector does such dependence appear). Also to be remarked is the fact that the contribution from the Maxwell terms to the dilaton sector

1

is trivial: it is merely a surface term [6]. Let us now consider the result (18) on shell. Integrating

(18) by parts, keeping all the surface terms and using the classical field equations (2) we obtain the on-shell diver- gences of the effective action

f 2 / "~ f ' "divF .... hell__ 2131 Id2xN/gJ5R+-- lP'+4 - v # 2 v ` ' l c, t )

620

Using the second of the field equations (2), we can rewrite (21) as follows:

ro~ 1 fd2x,,/ div

Hence, one can see that the on-shell divergences of the one-loop effective action are just given by the surface terms. If we are allowed to drop the surface terms we can conclude that the one-loop S-matrix is finite as in the pure dilaton gravity [1]. Note also that the surface counter- terms on shell depend on the parameter 7 (and hence on ~). This is not surprising because by adding the term (4) one introduces a new parameter into the theory. (This parameter apparently can influence only surface counter- terms because the triviality of (4) is a consequence of the Bianchi identities.)

Finally, let us discuss the renormalizability of the theory off shell. Dropping the surface terms in (18) we get

Fdiv = - - ~ f d2x V' - - + + F,~

+ 7f~z (V~b)(V~b)I. (23)

For the case ~ = 0 this result has been obtained previously [9]. Adding to the counterterms (i.e. (23) with the opposite sign) to the initial action (1) we obtain the renormalized action. Choosing the one-loop renormalization of the metric gu~ as

we get the renormalized action in the following form

2 XN//~ [- l .~ 1 SR= --~ d [~ O~vO.c~cvO "~-C 1 R4~+ V(c~)+~ f(~)F2~

V'(q~) f'(q~) F 2 ] " (25) ecl 4ecl J

The dilaton and the coupling cl do not get renormalized in the one-loop approximation.

From (25) it follows that the theory under discussion is one-loop multiplicatively renormalizable for the families of potentials:

V ( ~ ) = e ' ~ / (~b)=e ~O or / (q~)=f l ,

V(40 = A 1 sin ~b + B1 cos ~b,

f(~) = A2 sin ~ + B 2 cos ~b, (26)

where e, A, /3, fa, AI, BI, A2 and B2 are arbitrary constants.

The black hole solutions which appear for the Liouville-like potentials in (26) were discussed in [5,9, 10]. It would be interesting to study the time- dependent solutions (the 2d cosmology) for the theory that has been considered here with the potentials (26) and taking into account the back reaction [11].

Summing up, we have investigated the renormalization structure of Maxwell-dilaton gravity and found all the one-loop counterterms including the surface counter- terms. The on-shell limit of the effective action and the renormalizability off-shell have also been studied in detail.

Acknowledgments. S.D.O. wishes to thank the Japan Society for the Promotion of Science (JSPS, Japan) for financial support and the Particle Physics Group at Hiroshima University for kind hospital- ity. E.E. has been supported by DGICYT (Spain), research project PB90-0022, and by the Alexander von Humboldt Foundation (Germany).

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