Table Of ContentRadiative Corrections and the Palatini Action
F. T. Brandt∗
Instituto de F´ısica, Universidade de Sa˜o Paulo, S˜ao Paulo, SP 05508-090, Brazil
D. G. C. McKeon†
Department of Applied Mathematics, The University of Western Ontario, London, ON N6A 5B7, Canada and
Department of Mathematics and Computer Science,
Algoma University, Sault St.Marie, ON P6A 2G4, Canada
(Dated: April 5, 2016)
By using the Faddeev-Popov quantization procedure, we demonstrate that the radiative effects
computed using the first-order and second-order Einstein-Hilbert action for General Relativity are
the same, provided one can discard tadpoles. In addition, we show that the first order form of
this action can be used to obtain a set of Feynman rules that involves just two propagating fields
and three three-point vertices; using these rules is considerably simpler than employing the infinite
6
number of vertices that occur in the second-order form. We demonstrate this by computing the
1
one-loop, two-point function.
0
2
PACSnumbers: 11.15.-q
r Keywords: gaugetheories;firstorder;perturbationtheory
p
A
4
I. INTRODUCTION
]
h
t
- In the Einstein-Hilbert action
p
e (cid:90) √
h S = ddx −ggµνRµν(Γ), (1.1)
[
where
4
v
1
4 Γλ = gλσ(g +g −g ) (1.2)
4 µν 2 µσ,ν νσ,µ µν,σ
9
and
4
0
R (Γ)=Γρ −Γρ −Γσ Γρ +Γρ Γσ (1.3)
. µν µρ,ν µν,ρ µν σρ µσ νρ
1
0
it is usual to take the metric g to be the independent variable and the affine connection Γλ to be dependent; this
6 µν µν
1 is the second-order Einstein-Hilbert action. Classically, it is possible to treat both gµν and Γλµν as being independent;
: the equation of motion for Γλ in this first order action yields Eq. (1.2). It was Einstein who first noted this, though
v µν
i the first-order Einstein-Hilbert (1EH) action is often attributed to Palatini [1].
X
Although the 1EH and 2EH actions are equivalent at the classical level, it has as yet not been established that
r the two forms of the EH action result in the same quantum effects. We first show this quantum equivalence of the
a
1EH and 2EH actions when using the Faddeev-Popov procedure in conjunction with the quantum mechanical path
integral, provided that tadpole integrals can be set equal to zero. This is of some consequence, as it has been noted
[2, 3] that the first order form of gauge theory actions is considerably simpler than the second order form. This is
true both in Yang-Mills theory (where two complicated vertices are replaced by one simple one that is independent
of momentum) and in General Relativity (where a single momentum independent vertex replaces an infinite series
of momentum dependent vertices). The only disadvantage of using the first order action is that there are now two
propagating fields; in the 1EH case, these two fields have rather involved mixed propagator.
Oursecondresultisthatitispossibletoshiftvariablesofintegrationinthe1EHactionwithinthepathintegralto
eliminate this mixed propagator. We are then left with a relatively simple set of Feynman rules; there are now just
two propagating fields (that do not mix) and three vertices. This is an improvement over the situation that occurs in
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
the 2EH action where there is one propagating field and an infinite number of vertices with an arbitrary number of
external fields.
We then demonstrate the utility of our result by computing the two-point function to one-loop order using an
arbitrary gauge fixing parameter. In the limiting case in which this parameter equals one, we reproduce the result of
Ref. [4].
ThefirstorderformalismhasalsobeenusedfordoingloopcalculationsingravityinRef. [5], thoughinthemodels
considered there it is not clear if the first and second order formalisms are equivalent.
We begin by considering the 1YM action.
II. THE FIRST ORDER YANG-MILLS ACTION
It is evident that the 1YM Lagrangian
L =−1Fa (cid:0)∂µAaν −∂νAaµ+g(cid:15)abcAbµAcν(cid:1)+ 1Fa Faµν (2.1)
1YM 2 µν 4 µν
is classically equivalent to the 2YM Lagrangian
L =−1(cid:0)∂ Aa−∂ Aa +g(cid:15)abcAbAc(cid:1)2 (2.2)
2YM 4 µ ν ν µ µ ν
as upon substitution of the equation of motion for Fa that follows from (2.1) back into L , L follows.
µν 1YM 2YM
The 1YM and 2YM Lagrangians have the gauge invariance
δFa =g(cid:15)abcFb θc (2.3a)
µν µν
δAa =∂ θa+g(cid:15)abcAbθc; (2.3b)
µ µ µ
we are led to the path integral for L
1YM
(cid:90) (cid:90)
Z = DAaDFa ∆ (A)expi ddx(L +L ) (2.3c)
µ µν FP 1YM gf
where ∆ (A) is the Faddeev-Popov determinant associated with the gauge fixing Lagrangian L . (More than one
FP gf
gauge fixing may occur [6–8].) The field Aa (but not Fa ) interacts with other “matter” fields.
µ µν
If in Eq. (2.3c) we perform the shift
Fa →Fa +(cid:0)∂ Aa−∂ Aa +g(cid:15)abcAbAc(cid:1) (2.4)
µν µν µ ν ν µ µ ν
then we find that
(cid:90) (cid:90) (cid:20)1 (cid:21)
Z = DAaDFa ∆ (A)expi ddx Fa Faµν +L +L . (2.5)
µ µν FP 4 µν 2YM gf
the integral over Fa decouples and the usual generating functional associated with L is recovered with its three-
µν 2YM
point and four-point vertices. (In its unshifted form, Eq. (2.3c) results in the three propagators (cid:104)AA(cid:105), (cid:104)FF(cid:105) and
(cid:104)AF(cid:105) and the vertex (cid:104)FAA(cid:105) [2, 3].)
We can also make the shift
Fa →Fa +(cid:0)∂ Aa−∂ Aa(cid:1) (2.6)
µν µν µ ν ν µ
leaving us with
Z = (cid:90) DAaDFa ∆ (A)expi(cid:90) ddx(cid:20)1Fa Faµν − 1(cid:0)∂ Aa−∂ Aa(cid:1)2
µ µν FP 4 µν 4 µ ν ν µ
(cid:21)
−1(cid:0)Fa +∂ Aa−∂ Aa(cid:1)(cid:0)g(cid:15)abcAbAc(cid:1)+L . (2.7)
2 µν µ ν ν µ µ ν gf
WhenthegeneratingfunctionalZ iswritteninthisformweseethattherearenowtwopropagators(cid:104)FF(cid:105)and(cid:104)AA(cid:105)as
well as two three point functions (cid:104)FAA(cid:105) and (cid:104)AAA(cid:105) (but no mixed propagators (cid:104)AF(cid:105) or four point vertex (cid:104)AAAA(cid:105).)
This possibility of altering the Feynman rules in YM theory will now be exploited when examining the first order
(Palatini) form of the Einstein-Hilbert action.
3
III. THE FIRST ORDER EINSTEIN-HILBERT ACTION
Rather than using g and Γλ as independent fields in the 1EH Lagrangian of Eq. (1.1), it proves convenient to
µν µν
use [9]
√
hµν = −ggµν (3.1a)
and
Gλ =Γλ − 1(cid:0)δλΓσ +δλΓσ (cid:1) (3.1b)
µν µν 2 µ νσ ν µσ
so that now we have
(cid:18) (cid:19)
1
L =hµν Gλ + Gλ Gσ −Gλ Gσ . (3.2)
1EH µν,λ d−1 µλ νσ µσ νλ
The canonical structure of this action has been examined in refs. [9, 10] and the resulting path integral in ref. [11].
Here, we will consider using the Faddeev-Popov path integral [12]
(cid:90) (cid:90)
Z = DhµνDGλ ∆ (h)expi ddx[L +L ]. (3.3)
1EH µν FP 1EH gf
DirectlyusingtheformofEq. (3.2)makesitimpossibletodefineapropagatorforhµν andGλ . (Thisiseasilyseenif
µν
onewereattempttofindapropagatorforfieldsφandVλ withtheLagrangianL=φVλ.) Inref. [4],hµν isexpanded
,λ
about a flat metric ηµν =diag(+,+,+,...,−) so that
hµν(x)=ηµν +φµν(x); (3.4)
the propagators (cid:104)φφ(cid:105), (cid:104)GG(cid:105), (cid:104)φG(cid:105) and the vertex (cid:104)φGG(cid:105) are given in ref. [3]. However, it is not immediately evident
how this form of Z yields results consistent with those that follow from the 2EH Lagrangian L .
1EH 2EH
To show this equivalence, we start by writing Eq. (3.2) as
(cid:16) (cid:17) 1
L =Gλ −hµν + Mµνπτ(h)Gλ Gσ , (3.5)
1EH µν ,λ 2 λ σ µν πτ
where
(cid:20)
1 1
Mµνπτ(h) = (δνδτhµπ+δµδτhνπ+δνδπhµτ +δµδπhντ)
λ σ 2 d−1 λ σ λ σ λ σ λ σ
(cid:105)
−(δτδνhµπ+δτδµhνπ+δπδνhµτ +δπδµhντ) (3.6)
λ σ λ σ λ σ λ σ
From Eq. (3.5) we obtain the equation of motion
hµν =Mµνπτ(h)Gσ (3.7)
,λ λ σ πτ
from which we see that (upon using Eq. (3.6) and with h hλν =δν)
µλ µ
H ≡ −h h hµν +h h hµν +h h hµν
πτ,λ πµ τν ,λ τµ λν ,π λµ πν ,τ
(cid:18) (cid:19)
1
= 2 h Gσ −h Gσ . (3.8)
d−1 πτ λσ λσ πτ
Upon contracting Eq. (3.8) with hτλ we see that
d−1
Gσ =− h hµν (3.9)
πσ 2(d−2) µν ,π
and so by Eq. (3.8)
(cid:18) (cid:19)
1 1
Gρ = hρλ − h h hµν −H . (3.10)
πτ 2 d−2 πτ µν ,λ πτ,λ
4
From Eq. (3.8) it is apparent that
(cid:0)M−1(cid:1)ρ λ (h) = −1 hρλh h + 1hρλ(h h +h h )
πτµν 2(d−2) πτ µν 4 πµ τν πν τµ
− 1(cid:0)h δρδλ+h δρδλ+h δρδλ+h δρδλ(cid:1) (3.11)
4 τµ ν π πµ ν τ τν µ π πν µ τ
(We have
(cid:0)M−1(cid:1)ρ λ Mµνγδ =∆γδδρ ≡ 1(cid:16)δγδδ +δδδγ(cid:17)δρ(cid:19) (3.12)
αβµν λ σ αβ σ 2 α β α β σ
In the Lagrangian of Eq. (3.5) we insert Eq. (3.10) and obtain
L =−1hµν(cid:0)M−1(cid:1)λ σ (h)hπτ (3.13)
1EH 2 ,λ µνπτ ,σ
which is just the second-order EH Lagrangian L . This demonstrates that classically, L and L are equiv-
2EH 1EH 2EH
alent.
We now make the shift
Gλ →Gλ +(cid:0)M−1(cid:1)λ σ (h)hπτ (3.14)
µν µν µνπτ ,σ
in the path integral of Eq. (3.3). We then find that
Z =(cid:90) DhµνDGλ ∆ (h)expi(cid:90) ddx(cid:20)1Gλ Mµνπτ(h)Gσ + 1hµν(cid:0)M−1(cid:1)λ σ (h)hπτ +L (cid:21). (3.15)
1EH µν FP 2 µν λ σ πτ 2 ,λ µνπτ ,σ gf
The expansion of Eq. (3.4) can now be made in Eq. (3.15). Since M is linear in hµν, it follows that
Mµνπτ(η+φ)=Mµνπτ(η)+Mµνπτ(φ). (3.16)
λ σ λ σ λ σ
Consequently, any Feynman diagrams contributing to Green’s functions with only the field φµν on external legs and
which involve the field Gλ on internal lines, necessarily will have the field Gλ appearing in a closed loop. But the
µν µν
propagator for the field Gλ is independent of momentum (see Eq. (3.11)) and hence the loop momentum integral
µν
associated with any loop coming from the field Gλ is of the form
µν
(cid:90)
ddkP(kµ), (3.17)
where P(kµ) is a polynomial in the loop momentum kµ. If we use dimensional regularization [13, 14] then such loop
momentum integrals vanish.
Consequently, for Green’s functions involving only the field φµν on external legs, the only contribution to Feynman
diagrams come from the last two terms in the argument of the exponential in Eq. (3.15); from Eq. (3.13) we see that
this is just the generating functional associated with −L and so these Green’s functions can be derived by using
2EH
either the first order or the second order form of the EH action.
Using the second order form with the Lagrangian of Eq. (3.13) results in an infinite series of vertices involving the
field hµν (see ref. [4]). To obtain them, we note that when Eq. (3.16) is substituted into Eq. (3.12), we schematically
obtain
(cid:0)M−1(cid:1)(η+φ)=M−1(η)−M−1(η)M(φ)M−1(η)+M−1(η)M(φ)M−1(η)M(φ)M−1(η)−.... (3.18)
ThefirstterminEq. (3.18)isassociatedwiththepropagatorfortheφµν fieldinthesecondorderformalismwhileeach
subsequent term is associated with a vertex. This means that direct use of the 2EH Lagrangian becomes exceedingly
complicated if more than the one-loop two-point Green’s function is to be computed [15, 16].
We now will show that the 1EH generating functional can be used to compute Green’s functions with only the two
propagators (cid:104)φφ(cid:105), (cid:104)GG(cid:105) and the three point functions (cid:104)GGφ(cid:105), (cid:104)Gφφ(cid:105) and (cid:104)φφφ(cid:105). First the expansion of Eq. (3.4) is
made and then the shift occurs
Gλ →Gλ +(cid:0)M−1(cid:1)λ σ (η)hπτ (3.19)
µν µν µνπτ ,σ
5
(This is the shift of Eq. (3.14) with h being replaced by η.) This leads to Eq. (3.3) becoming
(cid:90) (cid:90) (cid:20)1 1
Z = DhµνDGλ ∆ (h)expi ddx Gλ Mµνπτ(η)Gσ − φµνM−1λ σ (η)φπτ
1EH µν FP 2 µν λ σ πτ 2 ,λ µνπτ ,σ
+1(cid:16)Gλ +φαβ(cid:0)M−1(cid:1)ρ λ (η)(cid:17)(Mµνπτ(φ))(cid:16)Gσ +(cid:0)M−1(cid:1)σ ξ (η)φγδ(cid:17)+L (cid:21). (3.20)
2 µν ,ρ αβµν λ σ πτ πτγδ ,ξ gf
The contributions coming from the various terms in the argument of the exponential appearing in Eq. (3.20) that
lead to the Feynman rules can be immediately seen to be:
1
G-G: Gλ Mµνπτ(η)Gσ : (3.21a)
2 µν λ σ πτ
1 1
φ-φ:− φµνM−1λ σ (η)φπτ − (φµν)2 : (3.21b)
2 ,λ µνπτ ,σ 2α ,ν
1
G-G-φ: Mµνπτ(φ)Gλ Gσ : (3.21c)
2 λ σ µν πτ
G-φ-φ:Gλ Mµνπτ(φ)M−1σ ξ (η)φγδ : (3.21d)
µν λ σ πτγδ ,ξ
1
φ-φ-φ: φαβM−1ρ λ (η)Mµνπτ(φ)M−1σ ξ (η)φγδ :. (3.21e)
2 ,ρ αβµν λ σ πτγδ ,ξ
In Eq. (3.21b) we have used the gauge fixing Lagrangian
1
L=− (φµν)2. (3.22)
2α ,ν
Withthisgaugefixing, thecontributioncomingfromtheFaddeev-Popovdeterminant∆ inEq. (3.20)involvesthe
FP
Feynman rules that follow from [3, 4]
L = d¯ (cid:2)∂2ηµν +(φρσ)∂ ηµν −(φρµ)∂ν
ghost µ ,ρ σ ,ρ
+φρσ∂ ∂ ηµν −(∂ ∂νφρµ)]d , (3.23)
ρ σ ρ ν
where dµ and d¯µ are Fermionic vector ghost fields. These are found to be
d¯-d:d¯ ∂2d : (3.24a)
µ ν
d¯-d-φ:d¯ (cid:2)(φρσ)∂ ηµν −(φρµ)∂ν +φρσ∂ ∂ ηµν −(∂ ∂νφρµ)(cid:3)d : (3.24b)
µ ,ρ σ ,ρ ρ σ ρ ν
Let us now consider an explicit calculation of an one-loop radiative correction. From Eqs. (3.21) and (3.24) we
readily find the following momentum space Feynman rules (all vertex momenta are inwards and p+q+r =0)
(1−α)(pνpσηµρ+pνpρηµσ+pµpσηνρ+pµpρηνσ−2pρpσηµν −2pµpνηρσ)
µν ρσ
p :
p4
ηµσηνρ+ηµρηνσ−(2−α)ηµνηρσ
− (3.25a)
p2
(cid:18) (cid:19)
λµν ρπτ : 14ηλρ ηµτηνπ+ηµπηντ − d−2 2ηµνηπτ
−1(cid:0)δλδρη +δλδρη +δλδρη +δλδρη (cid:1)≡Dλ ρ (3.25b)
4 τ µ νπ τ ν µπ π ν µτ π µ ντ µνπτ
6
αβ
λ
(cid:40)(cid:34)(cid:32) (cid:33) (cid:35) (cid:41)
1 δβδδδαδγ
: µ ν λ σ −δβδδδαδγ +µ↔ν +α↔β +γ ↔δ
µν 8 d−1 µ ν σ λ
γδ
σ
+ (λ,α,β)←→(σ,γ,δ) (3.25c)
q
µν
(cid:26)(cid:20)(cid:18) (cid:19) (cid:21) (cid:27)
ir 1
γδp : θ δγδδDθ ρ −δγDθ δ +µ↔ν +α↔β +γ ↔δ
σ 4 d−1 µ σ αβνρ µ αβνσ
r
αβ
+ (q,α,β)←→(r,µ,ν) (3.25d)
q
αβ
(cid:26)(cid:20)(cid:18) (cid:19) (cid:21) (cid:27)
q r 1
p : κ θ Dκ π Dθ σ − Dκ σ Dθ π +µ↔ν +α↔β +γ ↔δ
µν 8 αβµσ γδνπ d−1 αβµσ γδνπ
r
γδ
+ six permutations of (p,µ,ν) (q,α,β) (r,γ,δ) (3.25e)
d¯ d ηµν
µ ν : − (3.25f)
p p2
q
α
1
p : [η (q r +q r )−q (p η +p η )] (3.25g)
µν 2 αβ µ ν ν µ β µ αν ν αµ
r
β
The one-loop contributions to the two-point function (cid:104)φφ(cid:105) are given by the Feynman diagrams of fig. 1. After loop
integration, the result can only depend (by covariance) on the five tensors shown in table I, so that each diagram in
figure 1 can be written as
5
(cid:88)
ΠI (k)= Ti (k)CI(k); I =a, b, c and d. (3.26)
µναβ µναβ i
i=1
The coefficients CI can be obtained solving the following system of five algebraic equations
i
5
(cid:88)
Ti (k)Tjµναβ(k)CI(k)=ΠI (k)Tjµναβ(k)≡JIj(k); j =1,...,5. (3.27)
µναβ i µναβ
i=1
Using the Feynman rules for ΠI (k) the integrals on the right hand side have the following form
µναβ
(cid:90) ddp
JIj(k)= sIj(p,q,k). (3.28)
(2π)d
where q = p+k; p is the loop momentum, k is the external momentum and sIj(p,q,k) are scalar functions. Using
the relations
p·k =(q2−p2−k2)/2, (3.29a)
q·k =(q2+k2−p2)/2, (3.29b)
p·q =(p2+q2−k2)/2, (3.29c)
7
the scalars sIj(p,q,k) can be reduced to combinations of powers of p2 and q2. As a result, the integrals JIj(k) can
be expressed in terms of combinations of the following well known integrals
(cid:90) ddp 1 (k2)d/2−a−bΓ(a+b−d/2)Γ(d/2−a)Γ(d/2−b)
Iab ≡ = (3.30)
(2π)d(p2)a(q2)b (4π)d/2 Γ(a)Γ(b) Γ(d−a−b)
(thishasalsobeenconsideredin[17]). Theonlynon-vanishing(ienontadpole)integralsaretheoneswithbotha>0
a b
and b > 0. As we have pointed out earlier the integrals J (k) and J (k), associated respectively with the diagrams
i i
(a) and (b) of figure 1, are tadpole like and will not contribute (either a or b is not positive). For a general gauge
parameter, α(cid:54)=1 the diagram (c) in figure 1 involves the following three kinds of integrals
(k2)d/2−2Γ(cid:0)2− d(cid:1)Γ(cid:0)d −1(cid:1)2
I11 = 2 2 (3.31a)
2dπd/2 Γ(d−2)
(3−d)I11
I12 = I21 = (3.31b)
k2
(3−d)(6−d)I11
I22 = . (3.31c)
k4
The ghost loop diagram only involves I11.
A straightforward computer algebra code can now be setup in order implement the steps above described and to
c d
obtain the structures C and C . The results are the following
i i
(cid:20)
Cc = 1 1(cid:0)d3−2d2+96d−64(cid:1)−4(α−1)(cid:0)4d2−21d+24(cid:1)
1 8(d−1) 8
+ 4(α−1)2(cid:0)d3−7d2+22d−26(cid:1)(cid:3)I11 (3.32a)
(cid:20)
Cc = 1 1d(cid:0)−7d2+4d+52(cid:1)+4(α−1)(cid:0)2d3−16d2+41d−30(cid:1)
2 8(d−1)(d−2)2 8
− 2(α−1)2(cid:0)d4−12d3+68d2−179d+162(cid:1)(cid:3)k4I11 (3.32b)
(cid:20)
Cc = 1 1(cid:0)4d2+5d−16(cid:1)−16(α−1)(d−4)(d−1)
3 32(d−1) 2
+ 4(α−1)2(cid:0)d3−8d2+30d−43(cid:1)(cid:3)k4I11 (3.32c)
(cid:20)
Cc = 1 1(cid:0)d3−2d2+40d+16(cid:1)−8(α−1)(cid:0)3d2−18d+20(cid:1)
4 16(d−1)(d−2) 4
+ 8(α−1)2(d3−7d2+22d−26)(cid:3)k2I11 (3.32d)
(cid:20)
Cc = 1 1(cid:0)−4d2−5d+20(cid:1)+16(α−1)(d−4)(d−1)
5 32(d−1) 2
− 4(α−1)2(cid:0)d3−8d2+31d−44(cid:1)(cid:3)k2I11 (3.32e)
(d−2)(cid:0)d2+8d+8(cid:1)
Cd =− I11 (3.33a)
1 16(d2−1)
Cd =Cd =− d k4I11 (3.33b)
2 3 16(d2−1)
(cid:0)d2+2d+2(cid:1)
Cd =− k2I11 (3.33c)
4 16(d2−1)
8
Cd =− 1 k2I11 (3.33d)
5 16(d2−1)
In the special case when α=1 the result is in complete agreement with ref. [4]. The final expression for the one-loop
contribution to (cid:104)φφ(cid:105) can now be expressed as
5
Π =(cid:88)(cid:16)Cc+Cd(cid:17)Ti . (3.34)
µναβ i i µναβ
i=1
T1 (u,k)= k k k k
µναβ µ ν α β
T2 (u,k)= η η
µναβ µν αβ
T3 (u,k)= η η +η η
µναβ µα νβ µβ να
T4 (u,k)= η k k +η k k
µναβ µν α β αβ µ ν
T5 (u,k)= η k k +η k k +η k k +η k k
µναβ µα ν β µβ ν α να µ β νβ µ α
TABLE I: The five independent tensors built from η and k , satisfying the symmetry conditions Ti (k) = Ti (k) =
µν µ µναβ νµαβ
Ti (k)=Ti (k).
µνβα αβµν
(a) (b)
(c) (d)
FIG. 1: One-loop contributions to (cid:104)φφ(cid:105).
9
IV. DISCUSSION
Establishing the equivalence between the first and second order forms of the Yang-Mills Lagrangians at both the
classical and quantum levels is straightforward; this was demonstrated in section two above. It is not so easy to show
at both the classical and quantum levels that the first- and second-order forms of the Einstein-Hilbert action are
equivalent. Insectionthreeabovewehaveshownthatthisequivalenceholdsprovideditispossibletodiscardtadpole
diagrams (which are regulated to zero when using dimensional regularization.) (One feature of this demonstration
whose significance is not immediately apparent is the difference in sign between L in (3.13) and the hM−1(h)h
1EH
term in Eq. (3.15).)
We have also shown that by rewriting the 1EH action judiciously, it is possible to have just two propagating fields
and three three-point functions. This may prove to be an advantage when considering higher order diagrams in the
loop expansion in (super-)gravity.
It is quite straightforward to adopt the methods of refs. [15, 16, 18, 19], involving the use of geodesic coordinates
in conjunction with a background field for φµν, to determine counter terms while working with the 1EH Lagrangian.
Itwouldalsobeinterestingtocomputetheoneloopcorrectiontothetwo-pointfunction(cid:104)φφ(cid:105)usingthetransverse-
traceless (TT) gauge of ref. [6].
Acknowledgments
We would like to thank CNPq and Fapesp (Brazil) for financial support. Profs. J. Frenkel and J. C. Taylor have
made helpful comments on this manuscript. An enlightening conversation with Roger Macleod is acknowledged.
10
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