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Quantum gravity near two dimensions
Nuclear Physics B133 (1978) 417-434 © North-Holland Publishing Company
QUANTUM GRAVITY NEAR TWO DIMENSIONS R. GASTMANS *, R. KALLOSH and C. T R U F F I N ** P.N. Lebedev Physical Institute, Academy o f Sciences of the USSR
Received 22 August 1977 (Revised 14 November 1977)
Quantum gravity is shown to be renormalizable near two dimensions. The charge renormalization constant is calculated in the one-loop approximation in the framework of the background functional method. At positive e, a non-trivial ultraviolet stable fixed point is present. Matter field contributions are also computed.
1. I n t r o d u c t i o n
The renormalization of theories where quantized gravitation couples to matter fields is one of the outstanding problems in elementary particle theory. Detailed analyses of the interactions between gravitons and various particle systems resulted every time in proofs of unrenormalizability of the theory [ 1 - 5 ] , except for pure gravity at the one-loop level [4,5]. Supersymmetric theories, on the other hand, have been shown to be at least one-loop renormalizable [6,7]. It is generally believed that the source of the renormalizability problems lies with the dimensionality of Newton's constant G, which makes amplitudes grow as the square of the energy. However, in two space-time dimensions (n = 2), this coupling constant is dimensionless and general relativity becomes formally renormalizable. Thus, by studying quantum gravity as a double series expansion in n - 2 and G, one might hope to shed some light on the theory for n = 4. In recent years Polyakov [8], Migdal [9], Brezin and Zinn-Justin [10], Bardeen, Lee and Shrock [ 11 ] have studied a number of theories near two dimensions, at n - 2 = e. The theory was represented as a double series expansion in the coupling constant and in e. It turned out [ 8 - 1 1 ] that non-linear chiral dynamics, which is non-renormalizable in the physical 4-dimensional space, is renormalizable near two dimensions and is asymptotically free at n = 2. Calculation of the one-loop approxi marion in this theory has shown that at e > 0 there exists a non-trivial ultraviolet
* Bevoegdverklaard navorser, NFWO, on leave from University of Leuven, Belgium. ** Chargd de recherches, FNRS, Universit6 Libre de BruxeUes, Belgium.
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418 stable fixed point tc -
n-2 d
2
+ O((n - 2)2),
where d is the number of degrees of freedom of the scalar field. The existence of an ultraviolet stable point has led the authors to ref. [10] to the statement that the theory is renormalizable not only at small e but up to n = 4 as well. Since non-linear chiral dynamics has many properties that are close to quantum gravity, it is natural to try to carry out an anlysis analogous to that of refs. [ 8 - 1 1 ] for quantum gravity. A similar program, called "asymptotic safety" has been suggested by S. Weinberg in his 1976 Erice lectures, to be published. An essential difference between gravity and chiral dynamics (except purely technical complications) in that at e = 0, i.e. in a strictly two-dimensional case, chiral dynamics does exist and is renormalizable (by power counting) since it possesses a dimensionless coupling constant. In quantum gravity at n = 2 the action is an integral of a total divergence since at n -- 2 there exists a topological invariant
(See e.g. appendix B in ref. [12]). Similarly if at an arbitrary n, the action is formally expanded in huv, where guy = 6uv + huu, i.e. if the metric is expanded about the flat one, the propagator (or the vertices) contains poles 1/(n - 2) ([12], [13]). At sufficiently small n - 2 = e, both non-linear chiral dynamics and quantum gravity are renormalizable, i.e. there exists a limited number of counterterms which are invariant local functions containing no more than two derivatives. However in contrast with the theories studied up to now where the poles in e originate from the internal loop-momentum integrations, extra poles in e appear in quantum gravity at 2 + e dimensions, as mentioned before. Thus the problem of representing the theory as a series in e even after renormalization deserves a separate treatment. We shall find that the extra poles disappear from the gauge-invariant part of the oneloop background functional, and we shall argue that this will be the case in higher orders also. Another essential difference between quantum gravity and chiral dynamics must be pointed out. In two dimensions both theories have not only ultraviolet, but also infrared divergences. In chiral dynamics, infrared difficulties are eliminated by introducing in the Lagrangian an additional term H 1V/i--2- ~ 2 = HO that gives a mass to the n field but breaks chiral symmetry. In quantum gravity where there exists a local gauge invariance under the group of general covariant transformations, the infrared problem cannot be eliminated by introducing a graviton mass *. We shall investigate local counterterms corresponding to ultraviolet divergences, as is done for the case of the massless Yang-Mills field theory [14] at n = 4. * It may be that the analogue of the Ha term for quantum gravity will be a cosmological term Xx/g; in this case however the metric cannot be asymptotically flat.
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The question of the number of physical degrees of freedom is only very important. In chiral dynamics there exist d scalar fields, the number d being not at all connected with the dimensionality of the space. In gravity, however, there exist d = ½ n(n + 1) - 2n physical degrees of freedom, i.e. at n = 2, d = - 1 ( ! ) , at n = 3, d = 0(!) and, finally, at n = 4, d = 2. This counting of the number of physical degrees of freedom arouses some skepticism toward the attempt to consider quantum gravity (as distinct from chiral dynamics) near two dimensions. To avoid this difficulty we shall consider the background functional W[guv ] near small e = n 2, and only after continuation (if possible) to larger e(e -+ 2), shall we construct from W[g~,v] the S-matrix in the physical space according to the usual rule. We shall further on understand expressions of the type x / r g R / ( n - 2) as power expansions in huv , the tensor indices ranging in all the sums from 1 to n. If our approach proves applicable for rather small e only, this work should be treated as a model calculation of pure academic interest. It will be an interesting physical problem in the favourable case where the theory can be extended to larger e after renormalization. In such a perspective it is not pointless to recall that the e-expansion was an efficient technique in phase transition theory [ 15]. We shall use De Witt's background W-functional [16] since in recent years it has been shown to have some advantages over the more familiar functionals that depend on external sources [ 12,16,17]. The paper is organized as follows. Sect. 2 presents the main properties of the W-functional; the use of the equations of motion and the general structure of the counterterms are investigated. In sect. 3, we examine the one-loop theory for pure gravity. The 't Hooft-Veltman algorithm is adapted to the case e = n - 2. In sect. 4, the one-loop counterterms and the 3-function * of the renormalization group are computed for e 4: 0. In sect. 5, contributions of matter fields to the/3-function are calculated and we show how to perform the limit e ~ 0 when studying the twodimensional model of black hole evaporation. In conclusion, we discuss the limits e ~ 0 and e -+ 2 and the possibility of applying our results.
2. W - f u n c t i o n a l in q u a n t u m gravity at ~ = n - 2 The q u a n t u m gravity Lagrangian is 1
.c= - ~- {.,/FR - ~ ~ . .
a.av~.v)),
(2.1)
where guy = 6 , v + huv ,
(2.2)
* The problem of asymptotic freedom in quantum gravity at n = 4 was investigated by Fradkin and Vilkovisky [ 18 ].
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i.e..~begins with terms quadratic in h. The action corresponding to ~Oin (2.1) is invariant under general convariance gauge transformations 5guy = 6huv = Du [g] ~u (x) + D v[g] ~,(x).
(2.3)
To construct the W-functional we expand the metric gu, = guy + huv near some metric guy. The non-renormalized W-functional is [12,16,17] exp(iW[guv]} = f
dh d~
d~* exp i(S[g + h] - S[g]
6S 6g.v huv + v ~g ~ ~(D"h.. + ~D.h) 2 + v ~ ~*~Qo~ [g, g + hi
~,~) ,
(2.4)
where
sk] =f dnx~,
(2.5)
and £ is defined in eq. (2.1); 5S ---
~ g~ R)
(2.6)
and ~ * ~ Q ~ 5 ~ is the action of the ghost particles. The main properties of the W-functional are [ 12,16,17] : (i) G-invariance W[guv]a,~ = W[guv + aguvl~,~ ,
(2.7)
with 6guy from eq. (2.3). (ii) Mass shell gauge independence 6S W[guv]a,~ = W[guv]~+aa,~+a~ - i - (6huv). 6guv
(2.8)
In eq. (2.8) ( ) means the averaging in the sense of the functional integral (2.4). It follows from (2.8) that the W-functional is gauge invariant for the fields guy satisfying the classical equation of motion (2.6) Ruv - ½ guuR = 0
and its traced form R = 0 .
(2.9)
,/,R/,,1ocal3, ~ guu and Moreover, on dimensional grounds, one sees that at n - 2 = e, ,v,,uv therefore the gauge non-invariant local counterterms are ~(SS/6gtav)g~v ~ x/g R. It is important to note that S[g] defined by (2.5) does not vanish in the background functional when the equations of motion (2.9) are used; this is easily understood if one notes that the tree approximation of quantum gravity is given up to
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gauge terms, by the functional *: 6S 1 J{'dn x (4 - n) S[g] - --6guv guy = K-5 2-- ~-R
Wtree
- ~ h~u
~u~vlt~v) • (2.10)
For instance, in four dimensions, all tree graphs are generated by
fd4x(~ h.. - O.avh.v), where
Cjd.x,
c
'
- x )
,v,
+ glf-2interaction ~
~gu'~'
]x' '
(2.1 l)
and Ju~ is the source of hay. (2.11) shows that the last term of (2.10) is not zero despite the fact it is a total divergence, since []xDC~,u,~,(x - x') ~ 64(x - x'). Substituting (2.11) into the "surface term" of (2.10) therefore leads to contributions corresponding to Feynman graphs with the appropriate number of external gravitons (see e.g. [25]). Let us consider the divergences of the theory (2.4). For e = n - 2 small enough, the theory can be renormalized by subtracting twice each Green function. Therefore, by power counting of the primitive divergences, the most general counterterms are arbitrary local functions of guy with at most two derivatives. In dimensional regularization these divergences give poles in e. It follows from (2.7), i.e. from general covariance, that only two counterterms, depending on the background guy are possible: a/2,
bv~R
,
(2.12)
with/2 defined in eq. (2.1). The coefficient a is gauge independent and corresponds to the charge renormalization; b is gauge dependent as discussed before and corresponds to the wave function renormalization. The appearance of two possible counterterms, differing by a non-vanishing "surface term" only is clearly a peculiarity of the theory near two dimensions; this feature however is essential in obtaining a non-trivial result in sect. 4. In the renormalized theory we should write: 1
/./n-2
= K-~ ~ 0 [ g + h ] = Z---~-/2o [Z(g + h ) ] .
(2.13)
/a is an arbitrary normalization momentum and K is the dimensionless renormalized ~' This result has been obtained earlier by one of us (R.K.) together with I. Tyutin (unpublished).
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coupling constant. In the one-loop approximation we obtain the counterterms: Z
1
/~n 2
./~=/1" 2(l -- 1 ) ~ - S[g] + - ~ - { S [ Z g ] _
tl n-2 Kz { ( 1 - Z 1 ) S [ g ]
6S +--guy(Z6guy
S[g]} 1)).
(2.14)
•
It will be shown in a separate publication [19] that Z depends on the gauge parameter/3 of (2.4) in such a way that there exists a number/3 for which Z = 1. This clarifies the statement, that we can use strict classical equations of motion
xfff(Ruv - ~-guuR) = O,
(2.15)
and the traced equation: (n - 2) x/g-R = 0 ,
(2.16)
when calculating the S-matrix: we have the possibility to choose a gauge condition where the counterterms proportional to the equations of motion are absent. Therefore if we look for gauge-invariant quantities in any gauge it is sufficient to determine only the counterterms which are not zero when the classical equations of motion are used. As already discussed in the introduction, quantum gravity at 2 + e dimensions will exhibit poles in e due to non-existence of the theory at n = 2. The propagator is therefore a singular operator at n = 2. Indeed, since quantum gravity at n = 2 is conformal invariant, i.e. 6S
guu 6guv
(n
2),
(2.17)
it follows from (2.17) that 62S
6S
guy 6guu(x ) 6go~(x') ~ 6g~(x') 6(n)(x - x') ~ (n - 2).
(2.18)
Eq. (2.18) considered at guy = 6u~ gives 62S
,
~ (n - 2 ) .
(2.19)
Therefore when calculating [19] the propagator in an arbitrary gauge we find poles at e = 0 in the graviton propagator c-/)uv,xo:
1 +213
CDuv, Xo (,P) =- ~ 1
Q11 +2/3 (
-n--L--~ Q4 + (l +~-~
(n - 2)(1 +/3) 2/3+3 1+-2o~
n_2]Qs
03
}
,
(2.20)
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where the Qi's are the kinematical structures: Q1 tav,Xe = 1 (SuXSv ° + (StaoSvX) , 1
Q2 uv, Xo = 2p--g (5uxPvPo + 6uoPvPx + 6vxpuPo + 6wPuPx) , 1
Q3 ,v,Xa = - ~ (PuPv6xa 4-pxpo6uv) , 1
Q4 uv.Xo = 6.v6Xo ,
Q 5 uv,Xo =-2~ p upvpxP o • P
It follows from (2.20) that some of the poles depend on the gauge condition, therefore one might hope that these extra poles give no contribution to the gauge invariant quantities. In the one-loop approximation this will be demonstrated explicitly.
3. One-loop approximation In this section we follow the paper of 't Hooft and Veltman [12] with extra attention to the dependence on n. The gauge condition is defined in eq. (2.4) by a = 1,/3 = -21-. The one-loop effective Lagrangian is (3.1) where ~ i s the quadratic part of the expansion of./2[g + h] in huv. Cu is the gauge condition, and.t2¢ is the ghost Lagrangian.
=
-~,v.¢
+ ~ h~,vh~'" + ~ h#X~vh u } ,
(3.2)
6vR~ u + 6a~,U O~*'V + R~U v
(3.3)
XCuu = _ ! ~4 u~ ORc V V * " +L2 ~ Rt *_* ~ vu vO
A?~ = V~-~ *u {Do~D°~guv - Ruv } d/v .
(3.4)
After the doubling trick [12] we get (g
1 C~)d = x/-g{h~Pa~UVDn, D3'h.v + ½ h ~ ( X °~"v + X U ~ ) h u v } ,
eo~.v = ~ g~. g~V _ i g~e g~V .
The theory (3.5) can be reduced to a simpler form by replacing the integration
(3.5) (3.6)
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variable *,
* ho~
~ h *uv p uvo~ -1 ,
(3.7)
where 2
-1
(3.8)
P u v ~ = guc~ gvo + gu~ gv~ - -n -- 2 g u r g l e .
Instead of (3.5) we get (.~0_ _
g1 C ~2 ) d - x_ / g ( h u # D* ~ , D
t'v~po
3, h~# + 1 h" *u v "p -umO~,A 1
+ Xpoc~)hpa .
(3.9)
Notice that some terms in (3.9) contain the pole 1/e connected with the last term in (3.8). The corresponding terms take the form 2
(3.10)
_n - _ 2 h*g(RUV _ g1 g u vR ) h w , ,
i.e. the terms containing 1/e are proportional to the equation of motion, as was claimed in sect. 2. Further treatment of (3.9) requires replacing the covariant derivatives D 7 acting on huv as a second-rank tensor by derivatives which do not "see" the indices t~, v, i.e. act on huv as a scalar. In eq. (3.9) we designate the fields hc~ = ( h l b h12 .... ) as ~9i = {q01 . . . . . ~On(n+l)[2}. Then (3.9) can be presented as follows:
,/~((Vu~0)*(V"~) + ~otx~},
(3.11)
Vt2 = Ott6ij + 7712ij ,
(3.12)
where
~uiJ =
2I"
8~ + s y m ,
i = (c~,/3),
] = (a',/3'),
(3.14)
lD--1 X i j = ~* o~ I~vk(vUv~'O' a + X~'~ ' lau) .
The ghost Lagrangian (3.4) can also be rewritten in the form (3.11) with ~oi = (~o1..... ~on} and F
~uii=-I'ua~
,
i=a,
j=a'
,
(3.15)
i Xij : -R~
,
(3.16)
Thus, the original theory is reduced to a simpler form (3.1 1), where -~ n ( n + 1) The replacement (3.7) is not a covariant operation, but, as a change of variables, it does not violate the main property of the functional, i.e. its invariance with respect to the fieldg~v.
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scalar particles ~o interact with the external fields guy, ~u (3.13) and X (3.14), and n complex scalar particles of Fermi statistics (fictitious particles) interact with guy and with r~u and X determined in (3.15) and (3.16), respectively. We now investigate the theory of the type (3.11) with the arbitrary external fields ~Tu and X and in the arbitrary external metric g~v. We want to find local ultraviolet divergences at small e. Consider the invariance properties of the one-loop functional. (a) The one-loop functional of the classical fields guy, ~u, X, is invariant under a gauge group of general covariant transformations if r~u transforms as a vector and X as a scalar. The counterterms of the theory also possess this property. (b) The following transformation [20] can be performed in (3.11) 6~0 = A(x) ~0, 6X= AX-
XA ,
6~7u = - D ~ A + Ar/u - r~uA,
(3.17)
where A(x) is some anti-Hermitian matrix A(x) = - A * (x).
(3.18)
The Lagrangian (3.11) is invariant under the transformations (3.17), (3.18). Thus, the structure of the counterterms near n = 2 is determined by general covariance, invariance under transformations (3.17), (3.18) and considerations of dimensionality The counterterms to be added to the initial Lagrangian to eliminate the divergences of the theory have the form (in dimensional regularization) z5./2= al~?+ b 1v~-Sp X ,
(3.19)
J2being defined in (2.1). Note, that in (3.19) there is no r/u dependence, though at n = 4, zSA?contains a r/u dependent term of the form @ ( n - 4) SpCffgv [12] where: cff uv = DurTv - D v ~ u + rlurlv - ~v~u •
(3.20)
Near n = 2 the corresponding counterterm would have the form n-2
Sp cffuv
(3.21)
which is equal to zero, since ~[uv is antisymmetric.
4. The calculation of the one-loop charge renormalization Before determining the coefficients in (3.19) we would like to make sure that calculations will confirm the structure (3.19), where there is no 7h, dependence.
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QB
.... o
.... (11
i::© (2)
..... © (3)
(.¢1
15)
(61
Fig. 1. Wiggly line: gravitational field g#v; dotted line: vector field n#; solid line: scalar field ~o; dashed line: scalar field X.
Consider the diagrams with two external ?7u fields, figs. 1(1) and (2). The self-energy diagram of fig. 1(1) is calculated in dimensional regularization and gives a non-zero result for the counterterm
v7
(4.1)
n - 2 rlurTu "
The diagram in fig. 1(2) determined by the vertex x/~(r/u¢)t(r/u~o ) gives also a counterterm of the form (4.1), which should compensate the corresponding term of the diagram in fig. 1(1). In this case the formal property of the symmetry under transformations (3.17), (3.18), leading to the independence of the ultraviolet counterterm on r/u near n = 2, will be confirmed by calculation. A massless tadpole in dimensional regularization is commonly thought to be zero, but this result f d2q/q 2 = 0 stems from a cancellation of UV and IR divergences at n = 2. If, however, we turn to ref. [20], we shall see that I = (dnk= d k2
7rn/2f(ln)n/z-lr(1
- In)
(4.2)
In (4.2) f(½ n) is a function used for analytic continuation and possessing a number of properties, one of which is j"(~n) = 0 ,
n = O, 1, 2 . . . . .
(4.3)
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427
and in particular, at n ~ 2 f ( ~ n ) ~ (n - 2) 2 .
(4.4)
From (4.3), (4.4) it follows that the tadpole determined in (4.2) is zero at all integer n, except n ~ 2, where I=
[(n -
2rr
2)21 (~-2)/2
+
. . .
2-n =
e(n_2)ln(n_2 )
277" 2 -n
- -
+ ...-
27r 2 -n
+ ....
(4.5)
i.e. near two dimensions the tadpole cannot be considered to be zero. It turns out that if one calculates the self-energy diagram (fig. 1(1)by the dimensional regularization formulae and if for the seagull diagram (fig. 1(2)) one uses the value from (4.2), (4.5), one does obtain the compensation of the counter terms of the form (4.1). No other r/u dependence can appear from general covariance and considerations of dimensionality. Thus, dimensional regularization keeps the symmetry (3.1 7)-(3.1 8) of the initial functional integral. We now discuss the term b l x / ~ S p X in eq. (3.19). It is easy to calculate Sp X, where X is determined in (3.14) for gravitons and in (3.1 6) for fictituous particles. The extra poles 1/(n - 2) in X (see eq. (3.10)) will disappear as soon as the trace is taken, since Sp(R "v - ½ g~VR) = R (1 - 1 n). For gravitons and fictitious particles Sp X is proportional to R. The coefficient is determined by a diagram like fig. 1(3). The result is b~rx/g-Sp X -
x/g-R 47r(n - 2)
(4.6)
x/g-R 27r(n - 2)
(4.6')
for gravitons and b]~x/g-Sp X -
for ghost particles. We shall now discuss the determination of the coefficient al in (3.19). For this purpose we shall consider (3.1 1) with r/u = X = 0, i.e. a simple theory of a complex scalar field in the external gravitational field guy: £?= -v~gUV3u~p*3v~O = -3u~o*3u~o + Suv3~O*3v~O + . . . ,
(4.7)
Suv = huv - ~ 6uvh ~ .
(4.8)
where
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The term linear in the gravitational field (fig. 1(4)) is zero, since the diagram appears with two derivatives and
nk - ~ - kuk v = 0
fd
(4.9)
in dimensional regularization. The term quadratic in the gravitational field arises due to the self-energy diagram (fig. 1(5)) alone, since the diagram with two external gravitons (fig. 1(6)) has also the form (4.9) and is zero. Consider now the diagram in fig. 1(5). It is equal to
2 •2
("dnk ku(k +P)v k,~(k +p)~
2 i t'(-2~)n J
k 2 (k + p)=
4rrn/2p 2n = (SuvSag + 8uaSv~ + 8ufiv'~) 2(2rr) n 16(n 2 -- 1) X [P(n - 2)1-1['(2 - ½ n)[17(½ n - DI e + ....
(4.10)
Dots mean other structures containing PuPo~,etc. It follows from (4.10) that the counterterm, which eliminates the one-loop divergence of one real scalar field in external guy is equal to g2"t2 zX£Os- 24rr(n - 2) '
(4.11)
12being defined in (2.1). This result has also been obtained elsewhere [22]. Note that the second term in the action corresponding to (2.1) does not violate general covariance, since the parameters of the general covariant transformation ~u (x) in (2.3) are regarded to be well decreasing functions of x, and the integral
f dnx([E] 8uv - 8uav) 8huv = 0 ,
(4.12)
even if
f dnx( D 8uv - OuOv) "huu4=0 ,
(4.13)
as is the case for the fields buy, which are solutions of the classical equations of motion. If we remember now that in the initial gravitational problem there are n(n + 1)/2 degrees of freedom connected with huv and 2n degrees of freedom connected with
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fictitious particles, we have
al = ( 2 n +n(n + l)) 2
K2
(4114)
24rr(n - 2 ) '
and at n = 2 a 1.12=
- K 2 ~° 24rr(n - 2) '
b 1x/g-Sp X =
18x/g-R 247r(n - 2)
(4.15)
We have also performed a detailed calculation without the use of the 't Hooft-Veltman algorithm. The background field was expanded around the flat space and A ~ was calculated to second order in the background. It appears that general covariance is maintained only with the prescription (4.5). The result is - ( 1 + 18)•2 A ~ = 24rr(n - 2) (-/~)uv,Kx ,
(4.16)
where (.L-°)uv,Kx is the second-order expansion of -(1/K 2) x/g-R. This result coincides with the second-order expansion of both terms in the previously obtained result (4.15). As (4.16) is the result of a standard Feynman rule calculation, performed in the gauge defined at the beginning of sect. 3, it does not allow us to separate the gauge dependent from the gauge independent contributions. This would require a recalculation o f , ~ i n an arbitrary gauge, which appears to be a technically formidable task. This indicates the superiority of the background field method on the standard field theoretic ones for this application. As discussed in sect. 2, charge renormalization is given by the counterterm of the form al./2; moreover, it will be shown in a forthcoming publication [19] that the variation of the one-loop counterterms in the background functional with the gauge condition/3 (2.4) is proportional to -
)2vUR,
where
1 +2/3 ~= 2(1 + fi) Thus we conclude that the one-loop charge renormalization in (2.11) is K2 Z~ = 1 +
24rr(n - 2)
(4.17)
After charge renormalization, the gauge invariant radiative corrections of the oneloop approximation contain no singularities in e. At small enough e, the whole
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procedure can be repeated for the next approximations. As it was shown in sect. 2 only the two counterterms (2.10) appear and the theory is renormalizable. Thus • we have shown in the one-loop approximation that the extra poles 1/e disappear from the charge renormalization. This fact suggests in general that the extra poles in e disappear from gauge invariant quantities also in higher approximations. If this is the case, the theory will be represented as a double series in e and in the coupling constant, i.e. in this respect quantum gravity will repeat the case [ 8 - 1 1 ] investigated already in non-linear chiral dynamics at e = n - 2. The action will be expressed through a renormalized dimensionless charge as in eq. (2.11), where (4.18)
K2 = Z 1 K 2 1 a 2 - n •
Invariance of a non-renormalized theory under the variations of/a at fixed K gives the usual renormalization group equations: /~ ~-~ +/3 ~ - ~ -
1 N~ l-'U(p, K2,/l) : 0 ,
/3(K 2) = (n - 2) K2 + K 2 u - ~ l n Z1
(4.19)
I
In the one-loop approximation from eq. (4.17) it follows that K4
/3(K 2) : (n - 2) K 2 - - - + O ( K 6 ) , 24n
(4.20)
i.e. there exists an ultraviolet stable fixed point at Kc2 = 247re.
(4.21)
From (4.20) we can conclude also that/3(K 2) has a limit for n -+ 2 and it is negative in this limit, i.e. we deal with asymptotic freedom at n = 2. From the renormalization group equations one finds that, at large momenta, the asymptotic behaviour is determined not by perturbation theory but that there exists a power behaviour determined by the fixed point (4.21): IN(x/),K 2) ~
k n-N(n-2+n)/2
.
(4.22)
The existence of a fixed point may indicate renormalizability of the theory also for finite e, as is argued in chiral dynamics [ 8 - 1 1 ].
5. Matter fields contributions
We now investigate the problem of computing the/3 function when matter fields also contribute to the renormalization of the gravitational coupling constant. The
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431
calculations follow closely sects. 3 and 4: charge renormalization is given exclusively by diagrams of the type of fig. 1(5) with different internal matter loops, i.e. by terms proportional toZ?. All other contributions will yield counterterms proportional to vrg-R, which are connected to the matter Lagrangian by using the equations of motion. For scalar and vector particles, no additional calculations are required; the result stems directly from eq. (4.11) and from counting the number of degrees of dreedom of the fields, Every real scalar field gives a counterterm
zx&-
K2£9 247re '
and thus tends to suppress the presence of the ultraviolet stable fixed point. For one vector field, n degrees of freedom have to be taken into account, and per vector field, we find a contribution g2n.~O zS~?v = nZk2s -
24he
to charge renormalization. Every complex ghost field associated with the gauge condition for the vector fields yields _2K2~o k a ~ v = _2z3ZOs -
247re
In other words, there is no charge renormalization counterterm for the quantized vector fields: K2~2 ZkL~ + Z k ~ v - 24rr '
finite.
Our result disagrees with that of [23], where charge renormalization has not been extracted from the other counterterms. For fermions, an additional calculation, already performed in [23,24] shows that the (massless) fermion contribution equals that of the scalar field. Finally, we have also studied the supergravity Lagrangian [6] and computed the hemitrion contribution to charge renormalization at 2 + e dimensions. There is no contribution from the supersymmetry ghosts, but the spin -3 field gives -5K2~2 ~-~3/2 - 16n(n
2) '
thus pointing into the direction of ultraviolet stability. In summary, we find that the vector-gravity and supergravity systems have an
R. Gastmans et al. / Quantum gravity
432
ultraviolet stable fixed point at 2 + e dimensions, whereas this is n o t the case for scalar and fermion fields. The total c o u n t e r t e r m is
-K2~ A.£~=
-
-
24he
(1 + ~
N3/2 - Nil 2 - No),
where N s represents the n u m b e r of particles of spin s. We n o w turn to the problem of two-dimensional black-hole evaporation. Consider the two-dimensional reduced Schwarschild metric [22]
ds2=-I1-2~Mr l dt2 + I1-2~Mr l - l dr 2 .
(5.1)
In two dimensions every metric satisfies the identity:
Ruv - ½ g~vR ~ 0. If at first we consider the e q u a t i o n of m o t i o n (2.13), (2.14) at n - 2 = e 4: 0, where they are meaningful, and then consider the limit e ~ 0, we see that R should n o t be zero at n = 2, as is the case for the metric (5.1). It follows from our eq. (4.11) that for the divergent part of the c o n t r i b u t i o n of the scalar field to the background we have --K2.~ WSaiv [guv] =
z ~ = 247r(n - 2) + ....
where dots m e a n the c o n t r i b u t i o n non-singular in (n - 2). Therefore r d i v .v _ __2
w
_ - 2 ( R " - ½ g "R) 24n(n - 2)
V ~ 6g~v
....
R Tdiv ~ la = g,uvTdiV ~v _ 24n " It can be shown [24] that T~u = T d i V # + T f i n i t e t a = 0 . Therefore T~nite u u
- -R 24~
1 M 67r r 3 "
Using our c o u n t e r t e r m for fermions in the background, or that of [23,24], we find wdiv _ _g2~ 1/2 [guv] 24n(n - 2) '
,rfinite 11/2
Uu -
1
6•
M
r3 '
Thus, when we calculate by means of dimensional regularization and perform the
R. Gastmans et al. / Quantum gravity
433
limit e -+ 0, the result gives identical scalar and fermion contributions to the twodimensional black-hole evaporation. In [22] this result seems to depend on the details of the calculations and is not stated definitely.
6. Conclusion It is thus shown that it is possible to work with quantum gravity near n = 2 and to calculate the background functional. It turns out to be renormalizable, at least at the one-loop level, i.e. the counterterms have a structure coinciding with the initial Lagrangian and therefore can be eliminated by charge renormalization. Moreover, the theory exhibits an ultraviolet stable fixed point, even in the presence of vector or spin -3 fields. What can be said about the limit e ~ O? It seems natural to expect that the theory of gravitons becomes trivial in this limit. Indeed, if the renormalized background functional Wig] is represented as a series in e, then the coefficients (functions of guu) may well become trivial at n = 2 when the gauge invariant part of Wig] is calculated for guy satisfying the classical equations of motion. But if we consider for example a scalar field in a background such as the two-dimensional collapsing-shell metric [22], the theory is non-trivial; for the trace of the energy-momentum tensor, we obtain by the correct limiting procedure e ~ 0 the same result as in [22] where the calculation was performed directly at n = 2 by means of the point splitting method. We also find that the contribution of fermions equals exactly the scalar field contribution in the two-dimensional black-hole evaporation. In connection with a more interesting limit to the physical space, e ~ 2, we can only hope that due to the presence of the ultraviolet stable fixed point at e > 0, the representation of quantum gravity as a double series in e and charge K 2 may prove useful. We are thankful to E. Brdzin, J. Zinn-Justin and S. Weinberg for drawing our attention to this problem and to E. Fradkin, A. Linde, A. Polyakov, A. Starobinsky, I. Tyutin, G. Vilkovisky and B. de Wit for very useful discussions.
References [1] S. Deser and P. van Nieuwenhuizen, Phys. Rev. Lett. 32 (1974) 245; Phys. Rev. D10 (1974) 401. [2] S. Deser and P. van Nieuwenhuizen, Phys. Rev. D10 (1974) 411. [3] S. Deser, H.-S. Tsao and P. van Nieuwenhuizen, Phys. Lett. 50B (1974) 491; Phys. Rev. D10 (1974) 333. [4] G. 't Hooft, Nucl. Phys. B62 (1973) 444. [5] G. 't Hooft and M. Veltman, Ann. Inst. Henri Poincar6 20 (1974) 69. [6] D.Z. Freedman, P, van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; D14 (1976) 912;
434
[7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17]
[18] [19] [20] [21] [22]
[23] [24] [25]
R. Gastmans et al. / Quantum gravity S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335; S. Ferrara and P. van Nieuwenhuizen, ITP-SB-76-48. M.T. Grisaru, P. van Nieuwenhuizen and J.A.M. Vermaseren, ITP-SB-76-52; S. Deser, J.H. Kay and K.S. Stalle, Brandeis University preprint (1976). A.M. Polyakov, Phys. Rev. Lett. 59B (1975) 79. A.A. Migdal, ZhETF (USSR) 69 (1975) 1457. E. Br~zin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; E. Br~zin, J. Zinn-Justin and J.C. Le Guillou, Phys. Rev. D14 (1976) 2615. W.A. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D15 (1976) 985. G. 't Hooft and M. Veltman, Ann. Inst. H. Poincar~ 20, 1 (1974) 69. D.M. Capper, G. Leibbrandt and M.R. Medrano, Phys. Rev. D8 (1973) 4320. T.D. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3121. K.G. Wilson and J. Kogut, Phys. Reports 12 (1974) 75; E. Br~zin, J.C. Le Guillou and J. Zinn-Justin, Phase transitions and critical phenomena ed. G. Domb and M.S. Green (Academic Press, New York, 1975) vol. 6. B.S. DeWitt, Phys. Rev. 162 (1967) 1195, 1239; Phys. Reports 19 (1975) 295. J. Honerkamp, Nucl. Phys. B48 (1972) 269; R. Kallosh, Nucl. Phys. B78 (1974) 293; M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 (1975) 3203. B.S. Fradkin and G.A. Vilkovisky, University of Berne, preprint (1976). R. Kallosh, O. Tarasov and I. Tyutin, Nucl. Phys. B, to be published. G. 't Hooft, Nucl. Phys. B62 (1973) 444. D.M. Capper and G. Leibbrandt, J. Math. Phys. 15 (1974) 82. P.C.W. Davies, S.A. Fulling and W.G. Unruh, Phys. Rev. D13 (1976) 2720; W.G. Unruh, Phys. Rev. D14 (1976) 870; S.M. Christensen and S.A. Fulling, Trace anomalies and the Hawking effect, preprint, University of London King's College (1976). M.J. Duff, Observations on conformal anomalies, University of London, Queen Mary College preprint (March 1977). S. Deser, M.J. Duff and C.J. Isham, Nucl. Phys. B l l l (1976) 45; D.M. Capper and M.J. Duff, Nuovo Cim. 23A (1974) 173. M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 (1975) 3203.
QUANTUM GRAVITY NEAR TWO DIMENSIONS R. GASTMANS *, R. KALLOSH and C. T R U F F I N ** P.N. Lebedev Physical Institute, Academy o f Sciences of the USSR
Received 22 August 1977 (Revised 14 November 1977)
Quantum gravity is shown to be renormalizable near two dimensions. The charge renormalization constant is calculated in the one-loop approximation in the framework of the background functional method. At positive e, a non-trivial ultraviolet stable fixed point is present. Matter field contributions are also computed.
1. I n t r o d u c t i o n
The renormalization of theories where quantized gravitation couples to matter fields is one of the outstanding problems in elementary particle theory. Detailed analyses of the interactions between gravitons and various particle systems resulted every time in proofs of unrenormalizability of the theory [ 1 - 5 ] , except for pure gravity at the one-loop level [4,5]. Supersymmetric theories, on the other hand, have been shown to be at least one-loop renormalizable [6,7]. It is generally believed that the source of the renormalizability problems lies with the dimensionality of Newton's constant G, which makes amplitudes grow as the square of the energy. However, in two space-time dimensions (n = 2), this coupling constant is dimensionless and general relativity becomes formally renormalizable. Thus, by studying quantum gravity as a double series expansion in n - 2 and G, one might hope to shed some light on the theory for n = 4. In recent years Polyakov [8], Migdal [9], Brezin and Zinn-Justin [10], Bardeen, Lee and Shrock [ 11 ] have studied a number of theories near two dimensions, at n - 2 = e. The theory was represented as a double series expansion in the coupling constant and in e. It turned out [ 8 - 1 1 ] that non-linear chiral dynamics, which is non-renormalizable in the physical 4-dimensional space, is renormalizable near two dimensions and is asymptotically free at n = 2. Calculation of the one-loop approxi marion in this theory has shown that at e > 0 there exists a non-trivial ultraviolet
* Bevoegdverklaard navorser, NFWO, on leave from University of Leuven, Belgium. ** Chargd de recherches, FNRS, Universit6 Libre de BruxeUes, Belgium.
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R. Gastmans et al. / Quantum gravity
418 stable fixed point tc -
n-2 d
2
+ O((n - 2)2),
where d is the number of degrees of freedom of the scalar field. The existence of an ultraviolet stable point has led the authors to ref. [10] to the statement that the theory is renormalizable not only at small e but up to n = 4 as well. Since non-linear chiral dynamics has many properties that are close to quantum gravity, it is natural to try to carry out an anlysis analogous to that of refs. [ 8 - 1 1 ] for quantum gravity. A similar program, called "asymptotic safety" has been suggested by S. Weinberg in his 1976 Erice lectures, to be published. An essential difference between gravity and chiral dynamics (except purely technical complications) in that at e = 0, i.e. in a strictly two-dimensional case, chiral dynamics does exist and is renormalizable (by power counting) since it possesses a dimensionless coupling constant. In quantum gravity at n = 2 the action is an integral of a total divergence since at n -- 2 there exists a topological invariant
(See e.g. appendix B in ref. [12]). Similarly if at an arbitrary n, the action is formally expanded in huv, where guy = 6uv + huu, i.e. if the metric is expanded about the flat one, the propagator (or the vertices) contains poles 1/(n - 2) ([12], [13]). At sufficiently small n - 2 = e, both non-linear chiral dynamics and quantum gravity are renormalizable, i.e. there exists a limited number of counterterms which are invariant local functions containing no more than two derivatives. However in contrast with the theories studied up to now where the poles in e originate from the internal loop-momentum integrations, extra poles in e appear in quantum gravity at 2 + e dimensions, as mentioned before. Thus the problem of representing the theory as a series in e even after renormalization deserves a separate treatment. We shall find that the extra poles disappear from the gauge-invariant part of the oneloop background functional, and we shall argue that this will be the case in higher orders also. Another essential difference between quantum gravity and chiral dynamics must be pointed out. In two dimensions both theories have not only ultraviolet, but also infrared divergences. In chiral dynamics, infrared difficulties are eliminated by introducing in the Lagrangian an additional term H 1V/i--2- ~ 2 = HO that gives a mass to the n field but breaks chiral symmetry. In quantum gravity where there exists a local gauge invariance under the group of general covariant transformations, the infrared problem cannot be eliminated by introducing a graviton mass *. We shall investigate local counterterms corresponding to ultraviolet divergences, as is done for the case of the massless Yang-Mills field theory [14] at n = 4. * It may be that the analogue of the Ha term for quantum gravity will be a cosmological term Xx/g; in this case however the metric cannot be asymptotically flat.
R. Gastmans et al. / Quantum gravity
419
The question of the number of physical degrees of freedom is only very important. In chiral dynamics there exist d scalar fields, the number d being not at all connected with the dimensionality of the space. In gravity, however, there exist d = ½ n(n + 1) - 2n physical degrees of freedom, i.e. at n = 2, d = - 1 ( ! ) , at n = 3, d = 0(!) and, finally, at n = 4, d = 2. This counting of the number of physical degrees of freedom arouses some skepticism toward the attempt to consider quantum gravity (as distinct from chiral dynamics) near two dimensions. To avoid this difficulty we shall consider the background functional W[guv ] near small e = n 2, and only after continuation (if possible) to larger e(e -+ 2), shall we construct from W[g~,v] the S-matrix in the physical space according to the usual rule. We shall further on understand expressions of the type x / r g R / ( n - 2) as power expansions in huv , the tensor indices ranging in all the sums from 1 to n. If our approach proves applicable for rather small e only, this work should be treated as a model calculation of pure academic interest. It will be an interesting physical problem in the favourable case where the theory can be extended to larger e after renormalization. In such a perspective it is not pointless to recall that the e-expansion was an efficient technique in phase transition theory [ 15]. We shall use De Witt's background W-functional [16] since in recent years it has been shown to have some advantages over the more familiar functionals that depend on external sources [ 12,16,17]. The paper is organized as follows. Sect. 2 presents the main properties of the W-functional; the use of the equations of motion and the general structure of the counterterms are investigated. In sect. 3, we examine the one-loop theory for pure gravity. The 't Hooft-Veltman algorithm is adapted to the case e = n - 2. In sect. 4, the one-loop counterterms and the 3-function * of the renormalization group are computed for e 4: 0. In sect. 5, contributions of matter fields to the/3-function are calculated and we show how to perform the limit e ~ 0 when studying the twodimensional model of black hole evaporation. In conclusion, we discuss the limits e ~ 0 and e -+ 2 and the possibility of applying our results.
2. W - f u n c t i o n a l in q u a n t u m gravity at ~ = n - 2 The q u a n t u m gravity Lagrangian is 1
.c= - ~- {.,/FR - ~ ~ . .
a.av~.v)),
(2.1)
where guy = 6 , v + huv ,
(2.2)
* The problem of asymptotic freedom in quantum gravity at n = 4 was investigated by Fradkin and Vilkovisky [ 18 ].
420
R. Gastmans et al. / Quantum gravity
i.e..~begins with terms quadratic in h. The action corresponding to ~Oin (2.1) is invariant under general convariance gauge transformations 5guy = 6huv = Du [g] ~u (x) + D v[g] ~,(x).
(2.3)
To construct the W-functional we expand the metric gu, = guy + huv near some metric guy. The non-renormalized W-functional is [12,16,17] exp(iW[guv]} = f
dh d~
d~* exp i(S[g + h] - S[g]
6S 6g.v huv + v ~g ~ ~(D"h.. + ~D.h) 2 + v ~ ~*~Qo~ [g, g + hi
~,~) ,
(2.4)
where
sk] =f dnx~,
(2.5)
and £ is defined in eq. (2.1); 5S ---
~ g~ R)
(2.6)
and ~ * ~ Q ~ 5 ~ is the action of the ghost particles. The main properties of the W-functional are [ 12,16,17] : (i) G-invariance W[guv]a,~ = W[guv + aguvl~,~ ,
(2.7)
with 6guy from eq. (2.3). (ii) Mass shell gauge independence 6S W[guv]a,~ = W[guv]~+aa,~+a~ - i - (6huv). 6guv
(2.8)
In eq. (2.8) ( ) means the averaging in the sense of the functional integral (2.4). It follows from (2.8) that the W-functional is gauge invariant for the fields guy satisfying the classical equation of motion (2.6) Ruv - ½ guuR = 0
and its traced form R = 0 .
(2.9)
,/,R/,,1ocal3, ~ guu and Moreover, on dimensional grounds, one sees that at n - 2 = e, ,v,,uv therefore the gauge non-invariant local counterterms are ~(SS/6gtav)g~v ~ x/g R. It is important to note that S[g] defined by (2.5) does not vanish in the background functional when the equations of motion (2.9) are used; this is easily understood if one notes that the tree approximation of quantum gravity is given up to
R. Gastmans et al. / Quantum gravity
421
gauge terms, by the functional *: 6S 1 J{'dn x (4 - n) S[g] - --6guv guy = K-5 2-- ~-R
Wtree
- ~ h~u
~u~vlt~v) • (2.10)
For instance, in four dimensions, all tree graphs are generated by
fd4x(~ h.. - O.avh.v), where
Cjd.x,
c
'
- x )
,v,
+ glf-2interaction ~
~gu'~'
]x' '
(2.1 l)
and Ju~ is the source of hay. (2.11) shows that the last term of (2.10) is not zero despite the fact it is a total divergence, since []xDC~,u,~,(x - x') ~ 64(x - x'). Substituting (2.11) into the "surface term" of (2.10) therefore leads to contributions corresponding to Feynman graphs with the appropriate number of external gravitons (see e.g. [25]). Let us consider the divergences of the theory (2.4). For e = n - 2 small enough, the theory can be renormalized by subtracting twice each Green function. Therefore, by power counting of the primitive divergences, the most general counterterms are arbitrary local functions of guy with at most two derivatives. In dimensional regularization these divergences give poles in e. It follows from (2.7), i.e. from general covariance, that only two counterterms, depending on the background guy are possible: a/2,
bv~R
,
(2.12)
with/2 defined in eq. (2.1). The coefficient a is gauge independent and corresponds to the charge renormalization; b is gauge dependent as discussed before and corresponds to the wave function renormalization. The appearance of two possible counterterms, differing by a non-vanishing "surface term" only is clearly a peculiarity of the theory near two dimensions; this feature however is essential in obtaining a non-trivial result in sect. 4. In the renormalized theory we should write: 1
/./n-2
= K-~ ~ 0 [ g + h ] = Z---~-/2o [Z(g + h ) ] .
(2.13)
/a is an arbitrary normalization momentum and K is the dimensionless renormalized ~' This result has been obtained earlier by one of us (R.K.) together with I. Tyutin (unpublished).
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R. Gastmans et al. / Quantum gravity
coupling constant. In the one-loop approximation we obtain the counterterms: Z
1
/~n 2
./~=/1" 2(l -- 1 ) ~ - S[g] + - ~ - { S [ Z g ] _
tl n-2 Kz { ( 1 - Z 1 ) S [ g ]
6S +--guy(Z6guy
S[g]} 1)).
(2.14)
•
It will be shown in a separate publication [19] that Z depends on the gauge parameter/3 of (2.4) in such a way that there exists a number/3 for which Z = 1. This clarifies the statement, that we can use strict classical equations of motion
xfff(Ruv - ~-guuR) = O,
(2.15)
and the traced equation: (n - 2) x/g-R = 0 ,
(2.16)
when calculating the S-matrix: we have the possibility to choose a gauge condition where the counterterms proportional to the equations of motion are absent. Therefore if we look for gauge-invariant quantities in any gauge it is sufficient to determine only the counterterms which are not zero when the classical equations of motion are used. As already discussed in the introduction, quantum gravity at 2 + e dimensions will exhibit poles in e due to non-existence of the theory at n = 2. The propagator is therefore a singular operator at n = 2. Indeed, since quantum gravity at n = 2 is conformal invariant, i.e. 6S
guu 6guv
(n
2),
(2.17)
it follows from (2.17) that 62S
6S
guy 6guu(x ) 6go~(x') ~ 6g~(x') 6(n)(x - x') ~ (n - 2).
(2.18)
Eq. (2.18) considered at guy = 6u~ gives 62S
,
~ (n - 2 ) .
(2.19)
Therefore when calculating [19] the propagator in an arbitrary gauge we find poles at e = 0 in the graviton propagator c-/)uv,xo:
1 +213
CDuv, Xo (,P) =- ~ 1
Q11 +2/3 (
-n--L--~ Q4 + (l +~-~
(n - 2)(1 +/3) 2/3+3 1+-2o~
n_2]Qs
03
}
,
(2.20)
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R. Gastmans et al. / Quantum gravity
where the Qi's are the kinematical structures: Q1 tav,Xe = 1 (SuXSv ° + (StaoSvX) , 1
Q2 uv, Xo = 2p--g (5uxPvPo + 6uoPvPx + 6vxpuPo + 6wPuPx) , 1
Q3 ,v,Xa = - ~ (PuPv6xa 4-pxpo6uv) , 1
Q4 uv.Xo = 6.v6Xo ,
Q 5 uv,Xo =-2~ p upvpxP o • P
It follows from (2.20) that some of the poles depend on the gauge condition, therefore one might hope that these extra poles give no contribution to the gauge invariant quantities. In the one-loop approximation this will be demonstrated explicitly.
3. One-loop approximation In this section we follow the paper of 't Hooft and Veltman [12] with extra attention to the dependence on n. The gauge condition is defined in eq. (2.4) by a = 1,/3 = -21-. The one-loop effective Lagrangian is (3.1) where ~ i s the quadratic part of the expansion of./2[g + h] in huv. Cu is the gauge condition, and.t2¢ is the ghost Lagrangian.
=
-~,v.¢
+ ~ h~,vh~'" + ~ h#X~vh u } ,
(3.2)
6vR~ u + 6a~,U O~*'V + R~U v
(3.3)
XCuu = _ ! ~4 u~ ORc V V * " +L2 ~ Rt *_* ~ vu vO
A?~ = V~-~ *u {Do~D°~guv - Ruv } d/v .
(3.4)
After the doubling trick [12] we get (g
1 C~)d = x/-g{h~Pa~UVDn, D3'h.v + ½ h ~ ( X °~"v + X U ~ ) h u v } ,
eo~.v = ~ g~. g~V _ i g~e g~V .
The theory (3.5) can be reduced to a simpler form by replacing the integration
(3.5) (3.6)
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R. Gastmans et al. / Quantum gravity
variable *,
* ho~
~ h *uv p uvo~ -1 ,
(3.7)
where 2
-1
(3.8)
P u v ~ = guc~ gvo + gu~ gv~ - -n -- 2 g u r g l e .
Instead of (3.5) we get (.~0_ _
g1 C ~2 ) d - x_ / g ( h u # D* ~ , D
t'v~po
3, h~# + 1 h" *u v "p -umO~,A 1
+ Xpoc~)hpa .
(3.9)
Notice that some terms in (3.9) contain the pole 1/e connected with the last term in (3.8). The corresponding terms take the form 2
(3.10)
_n - _ 2 h*g(RUV _ g1 g u vR ) h w , ,
i.e. the terms containing 1/e are proportional to the equation of motion, as was claimed in sect. 2. Further treatment of (3.9) requires replacing the covariant derivatives D 7 acting on huv as a second-rank tensor by derivatives which do not "see" the indices t~, v, i.e. act on huv as a scalar. In eq. (3.9) we designate the fields hc~ = ( h l b h12 .... ) as ~9i = {q01 . . . . . ~On(n+l)[2}. Then (3.9) can be presented as follows:
,/~((Vu~0)*(V"~) + ~otx~},
(3.11)
Vt2 = Ott6ij + 7712ij ,
(3.12)
where
~uiJ =
2I"
8~ + s y m ,
i = (c~,/3),
] = (a',/3'),
(3.14)
lD--1 X i j = ~* o~ I~vk(vUv~'O' a + X~'~ ' lau) .
The ghost Lagrangian (3.4) can also be rewritten in the form (3.11) with ~oi = (~o1..... ~on} and F
~uii=-I'ua~
,
i=a,
j=a'
,
(3.15)
i Xij : -R~
,
(3.16)
Thus, the original theory is reduced to a simpler form (3.1 1), where -~ n ( n + 1) The replacement (3.7) is not a covariant operation, but, as a change of variables, it does not violate the main property of the functional, i.e. its invariance with respect to the fieldg~v.
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R. Gastmans et al. / Quantum gravity
scalar particles ~o interact with the external fields guy, ~u (3.13) and X (3.14), and n complex scalar particles of Fermi statistics (fictitious particles) interact with guy and with r~u and X determined in (3.15) and (3.16), respectively. We now investigate the theory of the type (3.11) with the arbitrary external fields ~Tu and X and in the arbitrary external metric g~v. We want to find local ultraviolet divergences at small e. Consider the invariance properties of the one-loop functional. (a) The one-loop functional of the classical fields guy, ~u, X, is invariant under a gauge group of general covariant transformations if r~u transforms as a vector and X as a scalar. The counterterms of the theory also possess this property. (b) The following transformation [20] can be performed in (3.11) 6~0 = A(x) ~0, 6X= AX-
XA ,
6~7u = - D ~ A + Ar/u - r~uA,
(3.17)
where A(x) is some anti-Hermitian matrix A(x) = - A * (x).
(3.18)
The Lagrangian (3.11) is invariant under the transformations (3.17), (3.18). Thus, the structure of the counterterms near n = 2 is determined by general covariance, invariance under transformations (3.17), (3.18) and considerations of dimensionality The counterterms to be added to the initial Lagrangian to eliminate the divergences of the theory have the form (in dimensional regularization) z5./2= al~?+ b 1v~-Sp X ,
(3.19)
J2being defined in (2.1). Note, that in (3.19) there is no r/u dependence, though at n = 4, zSA?contains a r/u dependent term of the form @ ( n - 4) SpCffgv [12] where: cff uv = DurTv - D v ~ u + rlurlv - ~v~u •
(3.20)
Near n = 2 the corresponding counterterm would have the form n-2
Sp cffuv
(3.21)
which is equal to zero, since ~[uv is antisymmetric.
4. The calculation of the one-loop charge renormalization Before determining the coefficients in (3.19) we would like to make sure that calculations will confirm the structure (3.19), where there is no 7h, dependence.
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R. Gastmans et al. / Quantum gravity
QB
.... o
.... (11
i::© (2)
..... © (3)
(.¢1
15)
(61
Fig. 1. Wiggly line: gravitational field g#v; dotted line: vector field n#; solid line: scalar field ~o; dashed line: scalar field X.
Consider the diagrams with two external ?7u fields, figs. 1(1) and (2). The self-energy diagram of fig. 1(1) is calculated in dimensional regularization and gives a non-zero result for the counterterm
v7
(4.1)
n - 2 rlurTu "
The diagram in fig. 1(2) determined by the vertex x/~(r/u¢)t(r/u~o ) gives also a counterterm of the form (4.1), which should compensate the corresponding term of the diagram in fig. 1(1). In this case the formal property of the symmetry under transformations (3.17), (3.18), leading to the independence of the ultraviolet counterterm on r/u near n = 2, will be confirmed by calculation. A massless tadpole in dimensional regularization is commonly thought to be zero, but this result f d2q/q 2 = 0 stems from a cancellation of UV and IR divergences at n = 2. If, however, we turn to ref. [20], we shall see that I = (dnk= d k2
7rn/2f(ln)n/z-lr(1
- In)
(4.2)
In (4.2) f(½ n) is a function used for analytic continuation and possessing a number of properties, one of which is j"(~n) = 0 ,
n = O, 1, 2 . . . . .
(4.3)
R. Gastmans et al. / Quantum gravity
427
and in particular, at n ~ 2 f ( ~ n ) ~ (n - 2) 2 .
(4.4)
From (4.3), (4.4) it follows that the tadpole determined in (4.2) is zero at all integer n, except n ~ 2, where I=
[(n -
2rr
2)21 (~-2)/2
+
. . .
2-n =
e(n_2)ln(n_2 )
277" 2 -n
- -
+ ...-
27r 2 -n
+ ....
(4.5)
i.e. near two dimensions the tadpole cannot be considered to be zero. It turns out that if one calculates the self-energy diagram (fig. 1(1)by the dimensional regularization formulae and if for the seagull diagram (fig. 1(2)) one uses the value from (4.2), (4.5), one does obtain the compensation of the counter terms of the form (4.1). No other r/u dependence can appear from general covariance and considerations of dimensionality. Thus, dimensional regularization keeps the symmetry (3.1 7)-(3.1 8) of the initial functional integral. We now discuss the term b l x / ~ S p X in eq. (3.19). It is easy to calculate Sp X, where X is determined in (3.14) for gravitons and in (3.1 6) for fictituous particles. The extra poles 1/(n - 2) in X (see eq. (3.10)) will disappear as soon as the trace is taken, since Sp(R "v - ½ g~VR) = R (1 - 1 n). For gravitons and fictitious particles Sp X is proportional to R. The coefficient is determined by a diagram like fig. 1(3). The result is b~rx/g-Sp X -
x/g-R 47r(n - 2)
(4.6)
x/g-R 27r(n - 2)
(4.6')
for gravitons and b]~x/g-Sp X -
for ghost particles. We shall now discuss the determination of the coefficient al in (3.19). For this purpose we shall consider (3.1 1) with r/u = X = 0, i.e. a simple theory of a complex scalar field in the external gravitational field guy: £?= -v~gUV3u~p*3v~O = -3u~o*3u~o + Suv3~O*3v~O + . . . ,
(4.7)
Suv = huv - ~ 6uvh ~ .
(4.8)
where
R. Gastmans et al. / Quantum gravity
428
The term linear in the gravitational field (fig. 1(4)) is zero, since the diagram appears with two derivatives and
nk - ~ - kuk v = 0
fd
(4.9)
in dimensional regularization. The term quadratic in the gravitational field arises due to the self-energy diagram (fig. 1(5)) alone, since the diagram with two external gravitons (fig. 1(6)) has also the form (4.9) and is zero. Consider now the diagram in fig. 1(5). It is equal to
2 •2
("dnk ku(k +P)v k,~(k +p)~
2 i t'(-2~)n J
k 2 (k + p)=
4rrn/2p 2n = (SuvSag + 8uaSv~ + 8ufiv'~) 2(2rr) n 16(n 2 -- 1) X [P(n - 2)1-1['(2 - ½ n)[17(½ n - DI e + ....
(4.10)
Dots mean other structures containing PuPo~,etc. It follows from (4.10) that the counterterm, which eliminates the one-loop divergence of one real scalar field in external guy is equal to g2"t2 zX£Os- 24rr(n - 2) '
(4.11)
12being defined in (2.1). This result has also been obtained elsewhere [22]. Note that the second term in the action corresponding to (2.1) does not violate general covariance, since the parameters of the general covariant transformation ~u (x) in (2.3) are regarded to be well decreasing functions of x, and the integral
f dnx([E] 8uv - 8uav) 8huv = 0 ,
(4.12)
even if
f dnx( D 8uv - OuOv) "huu4=0 ,
(4.13)
as is the case for the fields buy, which are solutions of the classical equations of motion. If we remember now that in the initial gravitational problem there are n(n + 1)/2 degrees of freedom connected with huv and 2n degrees of freedom connected with
R. Gastmanset al. / Quantum gravity
429
fictitious particles, we have
al = ( 2 n +n(n + l)) 2
K2
(4114)
24rr(n - 2 ) '
and at n = 2 a 1.12=
- K 2 ~° 24rr(n - 2) '
b 1x/g-Sp X =
18x/g-R 247r(n - 2)
(4.15)
We have also performed a detailed calculation without the use of the 't Hooft-Veltman algorithm. The background field was expanded around the flat space and A ~ was calculated to second order in the background. It appears that general covariance is maintained only with the prescription (4.5). The result is - ( 1 + 18)•2 A ~ = 24rr(n - 2) (-/~)uv,Kx ,
(4.16)
where (.L-°)uv,Kx is the second-order expansion of -(1/K 2) x/g-R. This result coincides with the second-order expansion of both terms in the previously obtained result (4.15). As (4.16) is the result of a standard Feynman rule calculation, performed in the gauge defined at the beginning of sect. 3, it does not allow us to separate the gauge dependent from the gauge independent contributions. This would require a recalculation o f , ~ i n an arbitrary gauge, which appears to be a technically formidable task. This indicates the superiority of the background field method on the standard field theoretic ones for this application. As discussed in sect. 2, charge renormalization is given by the counterterm of the form al./2; moreover, it will be shown in a forthcoming publication [19] that the variation of the one-loop counterterms in the background functional with the gauge condition/3 (2.4) is proportional to -
)2vUR,
where
1 +2/3 ~= 2(1 + fi) Thus we conclude that the one-loop charge renormalization in (2.11) is K2 Z~ = 1 +
24rr(n - 2)
(4.17)
After charge renormalization, the gauge invariant radiative corrections of the oneloop approximation contain no singularities in e. At small enough e, the whole
430
R. Gastmans e t al. / Q u a n t u m gravity
procedure can be repeated for the next approximations. As it was shown in sect. 2 only the two counterterms (2.10) appear and the theory is renormalizable. Thus • we have shown in the one-loop approximation that the extra poles 1/e disappear from the charge renormalization. This fact suggests in general that the extra poles in e disappear from gauge invariant quantities also in higher approximations. If this is the case, the theory will be represented as a double series in e and in the coupling constant, i.e. in this respect quantum gravity will repeat the case [ 8 - 1 1 ] investigated already in non-linear chiral dynamics at e = n - 2. The action will be expressed through a renormalized dimensionless charge as in eq. (2.11), where (4.18)
K2 = Z 1 K 2 1 a 2 - n •
Invariance of a non-renormalized theory under the variations of/a at fixed K gives the usual renormalization group equations: /~ ~-~ +/3 ~ - ~ -
1 N~ l-'U(p, K2,/l) : 0 ,
/3(K 2) = (n - 2) K2 + K 2 u - ~ l n Z1
(4.19)
I
In the one-loop approximation from eq. (4.17) it follows that K4
/3(K 2) : (n - 2) K 2 - - - + O ( K 6 ) , 24n
(4.20)
i.e. there exists an ultraviolet stable fixed point at Kc2 = 247re.
(4.21)
From (4.20) we can conclude also that/3(K 2) has a limit for n -+ 2 and it is negative in this limit, i.e. we deal with asymptotic freedom at n = 2. From the renormalization group equations one finds that, at large momenta, the asymptotic behaviour is determined not by perturbation theory but that there exists a power behaviour determined by the fixed point (4.21): IN(x/),K 2) ~
k n-N(n-2+n)/2
.
(4.22)
The existence of a fixed point may indicate renormalizability of the theory also for finite e, as is argued in chiral dynamics [ 8 - 1 1 ].
5. Matter fields contributions
We now investigate the problem of computing the/3 function when matter fields also contribute to the renormalization of the gravitational coupling constant. The
R. Gastmans et aL / Quantum gravity
431
calculations follow closely sects. 3 and 4: charge renormalization is given exclusively by diagrams of the type of fig. 1(5) with different internal matter loops, i.e. by terms proportional toZ?. All other contributions will yield counterterms proportional to vrg-R, which are connected to the matter Lagrangian by using the equations of motion. For scalar and vector particles, no additional calculations are required; the result stems directly from eq. (4.11) and from counting the number of degrees of dreedom of the fields, Every real scalar field gives a counterterm
zx&-
K2£9 247re '
and thus tends to suppress the presence of the ultraviolet stable fixed point. For one vector field, n degrees of freedom have to be taken into account, and per vector field, we find a contribution g2n.~O zS~?v = nZk2s -
24he
to charge renormalization. Every complex ghost field associated with the gauge condition for the vector fields yields _2K2~o k a ~ v = _2z3ZOs -
247re
In other words, there is no charge renormalization counterterm for the quantized vector fields: K2~2 ZkL~ + Z k ~ v - 24rr '
finite.
Our result disagrees with that of [23], where charge renormalization has not been extracted from the other counterterms. For fermions, an additional calculation, already performed in [23,24] shows that the (massless) fermion contribution equals that of the scalar field. Finally, we have also studied the supergravity Lagrangian [6] and computed the hemitrion contribution to charge renormalization at 2 + e dimensions. There is no contribution from the supersymmetry ghosts, but the spin -3 field gives -5K2~2 ~-~3/2 - 16n(n
2) '
thus pointing into the direction of ultraviolet stability. In summary, we find that the vector-gravity and supergravity systems have an
R. Gastmans et al. / Quantum gravity
432
ultraviolet stable fixed point at 2 + e dimensions, whereas this is n o t the case for scalar and fermion fields. The total c o u n t e r t e r m is
-K2~ A.£~=
-
-
24he
(1 + ~
N3/2 - Nil 2 - No),
where N s represents the n u m b e r of particles of spin s. We n o w turn to the problem of two-dimensional black-hole evaporation. Consider the two-dimensional reduced Schwarschild metric [22]
ds2=-I1-2~Mr l dt2 + I1-2~Mr l - l dr 2 .
(5.1)
In two dimensions every metric satisfies the identity:
Ruv - ½ g~vR ~ 0. If at first we consider the e q u a t i o n of m o t i o n (2.13), (2.14) at n - 2 = e 4: 0, where they are meaningful, and then consider the limit e ~ 0, we see that R should n o t be zero at n = 2, as is the case for the metric (5.1). It follows from our eq. (4.11) that for the divergent part of the c o n t r i b u t i o n of the scalar field to the background we have --K2.~ WSaiv [guv] =
z ~ = 247r(n - 2) + ....
where dots m e a n the c o n t r i b u t i o n non-singular in (n - 2). Therefore r d i v .v _ __2
w
_ - 2 ( R " - ½ g "R) 24n(n - 2)
V ~ 6g~v
....
R Tdiv ~ la = g,uvTdiV ~v _ 24n " It can be shown [24] that T~u = T d i V # + T f i n i t e t a = 0 . Therefore T~nite u u
- -R 24~
1 M 67r r 3 "
Using our c o u n t e r t e r m for fermions in the background, or that of [23,24], we find wdiv _ _g2~ 1/2 [guv] 24n(n - 2) '
,rfinite 11/2
Uu -
1
6•
M
r3 '
Thus, when we calculate by means of dimensional regularization and perform the
R. Gastmans et al. / Quantum gravity
433
limit e -+ 0, the result gives identical scalar and fermion contributions to the twodimensional black-hole evaporation. In [22] this result seems to depend on the details of the calculations and is not stated definitely.
6. Conclusion It is thus shown that it is possible to work with quantum gravity near n = 2 and to calculate the background functional. It turns out to be renormalizable, at least at the one-loop level, i.e. the counterterms have a structure coinciding with the initial Lagrangian and therefore can be eliminated by charge renormalization. Moreover, the theory exhibits an ultraviolet stable fixed point, even in the presence of vector or spin -3 fields. What can be said about the limit e ~ O? It seems natural to expect that the theory of gravitons becomes trivial in this limit. Indeed, if the renormalized background functional Wig] is represented as a series in e, then the coefficients (functions of guu) may well become trivial at n = 2 when the gauge invariant part of Wig] is calculated for guy satisfying the classical equations of motion. But if we consider for example a scalar field in a background such as the two-dimensional collapsing-shell metric [22], the theory is non-trivial; for the trace of the energy-momentum tensor, we obtain by the correct limiting procedure e ~ 0 the same result as in [22] where the calculation was performed directly at n = 2 by means of the point splitting method. We also find that the contribution of fermions equals exactly the scalar field contribution in the two-dimensional black-hole evaporation. In connection with a more interesting limit to the physical space, e ~ 2, we can only hope that due to the presence of the ultraviolet stable fixed point at e > 0, the representation of quantum gravity as a double series in e and charge K 2 may prove useful. We are thankful to E. Brdzin, J. Zinn-Justin and S. Weinberg for drawing our attention to this problem and to E. Fradkin, A. Linde, A. Polyakov, A. Starobinsky, I. Tyutin, G. Vilkovisky and B. de Wit for very useful discussions.
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