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AVERAGE EFFECTIVE POTENTIAL

arXiv:hep-th/9305172v1 29 May 1993SISSA 71/93/EPhep-th/9305172AVERAGE EFFECTIVE POTENTIALFOR THE CONFORMAL FACTORR. Floreanini ∗Istituto Nazionale di Fisica Nucleare, Sezione di TriesteDipartimento di Fisica Teorica, Universit`a di TriesteStrada Costiera 11, 34014 Trieste, ItalyR.

Percacci ∗∗International School for Advanced Studies, Trieste, Italyvia Beirut 4, 34014 Trieste, ItalyandIstituto Nazionale di Fisica Nucleare, Sezione di TriesteAbstractIn a four dimensional theory of gravity with lagrangian quadratic in cur-vature and torsion, we compute the effective action for metrics of theform gµν = ρ2δµν, with ρ constant. Using standard field-theoretic meth-ods we find that one loop quantum effects produce a nontrivial effectivepotential for ρ.

We explain this unexpected result by showing how ourregularization procedure differs from the one that is usually adopted inQuantum Gravity. Using the method of the average effective potential,we compute the scale dependence of the v.e.v.

of the conformal factor.∗florean@ts.infn.it∗∗percacci@tsmi19.sissa.it1

In quantum field theory the vacuum expectation value (v.e.v.) of the fields is usu-ally determined by the effective potential (the nonderivative part of the effective action).In classical theories of gravity the possible form of a potential for the metric is severelyconstrained by general covariance: the only allowed local term in the Lagrangian depend-ing on the metric but not on its derivatives is the cosmological term.

We have suggestedelsewhere that in Quantum Gravity the v.e.v. of the metric could be fixed by an effectivepotential [1].

The particular dynamics that we employed there was based on a bimetricLagrangian, which one could think of as a mean field approximation to an ordinary gravi-tational Lagrangian quadratic in curvature and torsion. We observed that in the presenceof two metrics one could obtain a genuine potential term whose minimum fixes the v.e.v.of the metric.One could think that this result was due to the unconventional dynamics that westarted with.

The main point we want to make in this note is that the same result canbe obtained starting from an ordinary Lagrangian quadratic in curvature and torsion andusing the familiar background field method. We will restrict our attention to the conformalsector and writegµν = ρ2γµν(1)where γµν is a fixed fiducial metric.

In order to simplify the discussion as much as possiblewe will present calculations only for the case γµν = δµν, but our results hold more generally.The effective dynamics of the conformal factor ρ induced by the conformal anomaly ofmatter fields has been the subject of recent investigations [2,3].In this work we willdiscuss the effective potential for ρ in the framework of a gauge theory of gravity.From standard Quantum Gravity arguments, one would expect to find only a cosmo-logical term, i.e. a potential proportional to ρ4.

Instead, we find an effective potential ofthe Coleman–Weinberg form, with the minimum occurring for nonzero ρ. We will explainthe origin of this result: it lies in the way in which the regularization is defined.We then discuss the renormalization group flow of the minimum of the potential.

Wedo this by computing the average effective potential for ρ. The average effective action isa continuum version of the block-spin action of lattice theories, which has been recentlyapplied to scalar and gauge theories [4,5].

We find that the v.e.v. of ρ2 (and thereforeof the metric) is essentially constant up to Planck’s energy, and scales according to itscanonical dimension (mass squared) above Planck’s energy, up to logarithmic corrections.In the conclusion we offer some speculations on the physical meaning of this behavior.In the model we shall consider, the independent dynamical variables are the vierbeinθaµ and an O(4) gauge field Aµab (we shall concentrate on the Euclidean theory, wherea, b = 1, 2, 3, 4 are internal indices and µ, ν = 1, 2, 3, 4 are spacetime indices).

With θ andA we can construct metric, curvature and torsion fields:gµν = θaµ θbν δab ,(2a)Fµνab = ∂µAνab −∂νAµab + eAµacAνcb −eAνacAµcb ,(2b)Θµaν = ∂µθaν −∂νθaµ + eAµabθbν −eAνabθbµ ,(2c)where e is the gauge coupling constant. As an action we takeS(θ, A) = 14Zd4xp| det g| gµρgνσ δacδbdFµνab Fρσcd + δabΘµaνΘρbσ.

(3)2

It is manifestly invariant under local O(4) and general coordinate transformations. Notethat θaµ and Aµab have canonical dimension of mass, and gµν of mass squared.

In (1) wetake ρ to carry dimension of mass. Let us note right away that gµν can not be the “geo-metric” metric, which has to be dimensionless (since we are assuming that the coordinateshave dimensions of length).

We shall return to this point later.We will evaluate the one-loop effective potential for the conformal factor ρ using thebackground field method.We first expand S up to second order around the classicalsolution of the field equations A(cl)µab = 0, θ(cl)aµ = ρ δaµ, with ρ constant. The linearizedaction has the formS(2) = 12Zd4xhδAµabδacδbd −δµρ∂2 + ∂µ∂ρ+ e2ρ2 δbdδµρ −δbρδdµδAρcd−2eρ δθaµδdµ∂ρ −δµρ∂dδacδAρcd + δθaµδac−δµρ∂2 + ∂µ∂ρδθcρi.

(4)In this expression, indices are raised and lowered with δµν. This linearized action is invari-ant under the linearized gauge transformations and linearized coordinate transformations.We add to the linearized action the gauge-fixing termsZd4x 12α(∂µδAµab)2 + 12β (∂µδθaµ)2.

(5)The effective action is one half the logarithm of the determinant of the differential operatorappearing in (4), taking into account gauge fixing and ghost terms. The operator can bediagonalized using the method of the spin projectors, which is discussed for example in [6].More details will be given elsewhere.

We findΓ(ρ) = 12Zd4xZd4q(2π)4(5 + 3) ln(q2 + 12e2ρ2) + 3 ln(q2 + e2ρ2) + ln(q2 + 2e2ρ2)(6)plus terms independent of ρ (we used the notation q2 = δµνqµqν). The first term comesfrom the modes with spin 2−and 1−, the second from those with spin 1+, the last fromthose with spin 0−.

The ghost contribution turns out to be independent of ρ.The integral can be regularized with a simple cutoffΛ. Adding suitable countertermsof the form Λ2ρ2 and ρ4 ln Λ one arrives at the renormalized effective potentialΓren(ρ, γ) =Zd4x √γ V0(ρ) ,(7a)V0(ρ) =964π2 e4ρ4lne2ρ2µ2−12,(7b)where µ is a renormalization constant with dimensions of mass; we have written the resultfor an arbitrary constant γµν.

This potential has the same form of the one we computedpreviously in the mean field approach [1]. It has a minimum for ρ = ρ0 = µ/e.The potential (7) is not simply a cosmological constant.

This result is surprising. Theclassical theory depends on ρ and γµν only through the combination gµν given in (1).

Thisgives rise to invariance under the Weyl transformationsγ′µν = ω2γµν ,ρ′ = ω−1ρ . (8)3

The theory can be quantized in such a way that this symmetry is preserved [7]. As aconsequence, also the quantum effective action should depend on γµν and ρ only throughthe combination gµν, and this is not the case for (7).It is the regularization proce-dure that we have chosen that breaks this invariance.

In fact we have integrated overthe range of momenta γµνqµqν < Λ2; this introduces a dependence of the theory on γµνalone, not accompanied by a factor of ρ, and is ultimately responsible for the appearanceof the logarithm of ρ in (7). Note that one could add to Γ(ρ, γ) the local counterterm(9/64π2)Rd4x √γ e4ρ4 ln(γ1/4).

This would restore the invariance under the transforma-tions (8) but would break diffeomorphisms. In fact, our effective action is invariant underdiffeomorphisms if gµν and γµν are transformed simultaneously, and ρ is treated as a scalarfield.There is an alternative way of regulating the theory: integrate over the range ofmomenta gµνqµqν = ρ−2γµνqµqν < λ2, with λ a dimensionless cutoff.

Redefining the inte-gration variables as q′µ = ρ−1qµ and discarding a term proportional to δ(0), the integralsin (6) are reduced to the general form ρ4 Rd4q′ ln(γµνq′µq′ν + c), with c a dimensionlessconstant independent of ρ, the integration being now over the range γµνq′µq′ν < λ2. Theimportant point is that the integral does not depent on ρ anymore.

Thus after renor-malization, the effective action would be of the form Γ ∼Rd4x √γ ρ4 =Rd4x √g, i.e. acosmological term.

From the point of view of quantum field theory it is unusual to havea regularization which itself depends on the dynamical variable. Nevertheless, this is thechoice which is tacitly made in most works on Quantum Gravity.The difference between the two ways of implementing the cutoffis that in the formercase the domain of integration is independent on the dynamical variable θaµ, while inthe latter it depends on it.

Both procedures are mathematically correct and the choicebetween the two has to be dictated by physical arguments. One could argue that the correctquantization is the one that preserves the Weyl invariance (8), but in the present case itseems that breaking this invariance would not violate any physical principle.

Since formingthe modulus squared of the four-momentum is a geometric construction, the distinctionbetween the two procedures has to do with what metric is taken to represent the geometryof spacetime. As already remarked, the geometry cannot be given directly by the compositeoperator gµν, since it is dimensionful.

It will be related to it by a constant of proportionalityℓ2 having dimension of length squared.If the geometry is given by the metric ℓ2 gµν, the cutoffdepends on the metric andthe effective potential will be just a cosmological term, as shown above. In this case thenondegeneracy of the metric has to be imposed from the outside, and the parameter ℓisundetermined.

On the other hand the geometry could be given by the metric ℓ2 ⟨gµν⟩=ℓ2⟨ρ2⟩γµν = ℓ2 ρ20 γµν. In this case it is natural to identify ℓ−1 = ρ0, in which case thegeometric metric coincides with γµν.

This is the point of view implicit in our calculations,and it leads to the effective potential (7). In this approach the nondegeneracy of gµν is aresult of the quantum dynamics of the theory.

It also has the advantage that it does notnecessitate the introduction of an external dimensionful parameter. We note here that thepresence of a nontrivial effective potential for the conformal factor is also relevant to theproblem of the cosmological constant [8].We turn now to another definition of the effective action which has been applied4

recently to scalar and gauge field theories: the so-called average effective action [4,5]. Thiswill allow us to compute the scale dependence of the effective potential, and hence of thev.e.v.

of the operator gµν. The average effective action depends on a momentum scale k.To define it, one begins by adding to the action (3) quadratic terms which constrain theaverages of the fields θ and A in volumes of size k−4 centered around the point x to takecertain values ¯θ(x) and ¯A(x) (up to small fluctuations):Sconstr =Zd4x √γh14( ¯Fµνab −fkFµνab) γµργνσ1 −f 2k( ¯Fρσab −fkFρσab)+ 12αγµν ¯∇µ(A −¯A)νab11 −f 2kγρσ ¯∇ρ(A −¯A)σab+12¯∇µ(¯θ −fkθ)aνγµργνσ1 −f 2k¯∇ρ(¯θ −fkθ)aσi.

(9)In this formula fk = fk(−γµν ¯∇µ ¯∇ν), where ¯∇µθaν = ∂µθaν + e ¯Aµabθbν −Γ λµνθaλ and Γare the Christoffel symbols for the metric γµν. The differential operator fk(−γµν ¯∇µ ¯∇ν)will perform the desired averaging operation if we take fk(x) = exp(−a(x/k2)b), with a,b constant parameters.

Note that the explicit introduction of the fields ¯θ and ¯A breaksboth coordinate and gauge invariance, so no further gauge fixing is needed. In (9) we havecontracted all spacetime indices with the metric γµν, in line with our assumption thatit is this metric that dictates the geometry.

Other choices are possible but will not beconsidered here.In order to compute the average effective potential we choose the average fields ¯A = 0and ¯θaµ = ρ δaµ with ρ constant. If the parameter b in fk is chosen larger than 2, theAnsatz Aclµab = ¯Aµab and θclaµ = ¯θaµ gives a solution of the classical equations of motionof the total action.

Proceeding as before, we arrive at the average actionΓk(ρ) = 12Zd4xZd4q(2π)4h8 lnPk + 12e2ρ2+ 6 lnPk + 12e2ρ2f 2k+ 3 lnP 2k + e2ρ2Pk1 +α + 12f 2k+ αe4ρ4f 2k+ 3 lnPk + 12αe2ρ2f 2k+ lnPk + 2e2ρ2i(10)where f 2k =fk(q2)2 and Pk(q2) = q2/(1 −f 2k).Note that in the limit k →0, thefunction fk becomes zero and Pk becomes equal to q2. One can then easily check thatup to field-independent terms, Γ0 = Γk=0 reduces to the old effective action (6).

One cansplit Γk(ρ) = Γ0(ρ) + ∆Γk(ρ), where Γ0(ρ) contains the divergences but is independent of5

k and∆Γk(ρ) = 12Zd4xZd4q(2π)4"8 lnPk + 12e2ρ2q2 + 12e2ρ2+ 6 lnPk + 12e2ρ2f 2kq2+ 3 ln P 2k + e2ρ2Pk1 +α + 12f 2k+ αe4ρ4f 2kq2(q2 + e2ρ2)!+ 3 lnPk + 12αe2ρ2f 2kq2+ lnPk + 2e2ρ2q2 + 2e2ρ2#(11)is automatically convergent. The part Γ0(ρ) can be renormalized as before, leading tothe effective potential V0(ρ) given in (7).

Define the average effective potential Vk(ρ) =V0(ρ) + ∆Vk(ρ) by Γk(ρ) =Rd4x √γ Vk(ρ). To find its minimum we have to solve theequation0 = ∂Vk(ρ)∂(ρ2) =e264π218e2ρ2 ln e2ρ2µ2+ k2Fe2ρ2k2,(12)where, using the dimensionless variables x = q2/k2, t = e2ρ2/k2 and ˜P(x) = Pk(q2)/k2 =x/(1 + f 2), f 2 = exp(−2axb), the function F is given byF(t) = 64π2e2k2∂∆Vk(ρ)∂(ρ2)= 2Z ∞0dx x f 2"−4 ˜P( ˜P + t2)(x + t2)+3˜P + 12tf 2+ 3˜P( 12x −˜P) + α( ˜Px + t(2x + t))(x + t) ˜P 2 + ˜Pt(1 + (α + 12)f 2) + αt2f 2+ 32α˜P + α2 tf 2 −2 ˜P( ˜P + 2t)(x + 2t)#.

(13)This function can be studied numerically. Choosing a = 1, b = 3.19 in f 2, (see [4]) andsetting α = 0, F(t) grows from F(0) = −c1 ∼−12 to zero for t ∼5, it reaches a maximumof order 0.2 for t ∼15 and decreases slowly to zero for large t like K/t for K slowly varying.The minimum of the effective potential can be plotted numerically.

One can only studyanalytically the behavior for t very large and very small. Let us denote ρk the minimumof Vk.

For k = 0, ρ0 = µ/e. For t ≫1 (which corresponds to k ≪µ) we can expandρk = ρ0 + ǫ, and use the asymptotic behavior F(t) ∼K/t.

Inserting in (12) one findsρ2k = ρ201 −K18k4µ4. (14)On the other hand for t ≪1 we can expand the function F in Taylor series around t = 0:F(t) = −c1 +c2t+.

. ..

Equation (12) shows that ρ2k grows slower than c1k2 and faster thanc1k2/18 ln(c1k2/µ2) + c2. For k ≫µ the denominator becomes large and this justifies aposteriori the approximation t ≪1.

In fact, this can also be checked numerically.6

For α ̸= 0 but not too large, the behavior of the potential is essentially the same. Wenote that the behavior for large and small k agrees with the one found using a mean fieldapproximation and treating k simply as a sharp infrared cutoff[9].We now return to the discussion of the physical implications of our results.Thediscussion on the choice of the geometric metric given earlier for the case k = 0 can berepeated for k > 0.

Assuming that the geometry is given by ℓ2⟨gµν⟩one has the furtheroption of identifying ℓ−1 with ρ0 or ρk. In the first case the geometry is given by thek-dependent metric (ρ2k/ρ20)γµν, while in the second the geometry is fixed, given by γµν.

(This is the choice that was made in writing (9).) Either way, the scale dependence ofthe metric can lead to striking effects.

For example, in the case of a scalar field coupledminimally to the k-dependent geometric metric, the scaling of the metric improves theultraviolet behavior of the propagator and shifts the physical pole [9].Here we shall briefly discuss an alternative approach, in which a (dimensionless) scalarfield is coupled directly to the quantum field gµν [10]:Sscalar(ϕ) = 12Zd4x √ggµν∂µϕ∂νϕ + c2ϕ2. (15)Defining the canonical field φ = ρϕ the action can be rewritten in the formSscalar(φ) = 12Zd4x √γγµν∂µφ∂νφ + c2ρ2φ2 + .

. .,(16)where the dots represent terms containing derivatives of ρ.

When ρ is constant, the La-grangian of φ has the same general form of the linearized Lagrangian (4) and thereforequantum fluctuations of φ contribute to the effective potential for ρ (see also [3]).On the other hand, the running of ρ directly affects the propagator of φ. We assumethat in the propagator for a free particle of four-momentum qµ the “mass” c⟨ρ⟩has tobe taken at scale k = |q| (a similar assumption was discussed recently in a differentcontext [11]).

The inverse propagator of φ would then have the form q2 + c2ρ2q, with ρ2qapproximately constant for q2 < ρ20 and growing roughly like q2 for q2 > ρ20. The physicalpole of the propagator occurs at mass approximately equal to cρ0 for c < 1, but is shiftedto exponentially large values for c > 1.

In fact, a positive anomalous dimension for ρ couldmake the pole disappear altogether. The mass ρ0 has to be identified with Planck’s mass[1,10].

Thus, particles with masses larger than Planck’s mass would essentially disappearfrom the spectrum. One may hope that a mechanism of this type is capable of removingthe ghosts of the gravitational sector.

This seems to be a restatement of the criterion givenin [12].To summarize, we have suggested that there exists an alternative method for quantiz-ing a gauge theory of gravity which produces a nontrivial effective potential for the metric.The difference from the traditional approach lies therein, that the domain of integrationover the momenta is determined by some fixed metric, and not by the dynamical met-ric. In this sense it is a bimetric theory, even though at the level of the starting classicalLagrangian no second metric appears.

In this it differs from the mean-field approach of[1]. General covariance will be preserved provided both metrics are transformed simul-taneously, although this aspect cannot be completely appreciated by looking just at theeffective potential.7

Concerning the scale dependence of the metric, one should perform a more sophisti-cated analysis by taking into account the running of the coupling constants, along the linesof [5]. Nevertheless we believe that our simple minded approach is sufficient to captureat least some qualitative features of this phenomenon.

We plan to return on these openproblems in the future.AcknowledgementsWe benefitted from discussions with D. Anselmi, S. Bellucci, M. Fabbrichesi, L. Griguolo,M. Reuter, A. Schwimmer, E. Spallucci, K.S.

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