Quantum gravity, gauge coupling constants, and the cosmological constant
aa r X i v : . [ h e p - t h ] A ug Quantum gravity, gauge coupling constants, and the cosmologicalconstant
David J. Toms ∗ School of Mathematics and Statistics, Newcastle University,Newcastle upon Tyne, United Kingdom, NE1 7RU (Dated: October 24, 2018)
Abstract
The quantization of Einstein-Maxwell theory with a cosmological constant is considered. Weobtain all logarithmically divergent terms in the one-loop effective action that involve only thebackground electromagnetic field. This includes Lee-Wick type terms, as well as those responsiblefor the renormalization group behaviour of the electric charge (or fine structure constant). Ofparticular interest is the possible gauge condition dependence of the results, and we study thisin some detail. We show that the traditional background-field method, that is equivalent to amore traditional Feynman diagram calculation, does result in gauge condition dependent resultsin general. One resolution of this is to use the Vilkovisky-DeWitt effective action method, andthis is presented here. Quantum gravity is shown to lead to a contribution to the running chargenot present when the cosmological constant vanishes. This re-opens the possibility, suggested byRobinson and Wilczek, of altering the scaling behaviour of gauge theories at high energies althoughour result differs. We show the possibility of an ultraviolet fixed point that is linked directly to thecosmological constant.
PACS numbers: 04.60.-m, 11.15.-q, 11.10.Gh, 11.10.Hi ∗ URL: ; Electronic address: [email protected] . INTRODUCTION Einstein gravity when quantized about a fixed background (for example flat space) is notrenormalizable [1, 2, 3, 4, 5]. The basic reason for this is that the gravitational couplingconstant G has units of inverse mass squared in natural ( ~ = c = 1) units. From thestandard quantum field theory point of view this means that when working to higher ordersin perturbation theory the degree of divergence of diagrams must increase with the orderthat one is working to. Naively, we expect a behaviour like ( G Λ c ) to some positive power,where Λ c is a momentum cutoff with the power increasing with the number of loops.The natural energy scale is set by the Planck mass M P = ( ~ c/G ) / ∼ GeV. Providedthat we restrict ourselves to energies E ≪ M P it is expected that an effective field theorytreatment of Einstein’s theory is valid. Indeed classical general relativity is well tested, sowe know that quantum effects must be very small. The methodology for realizing this is theeffective field theory framework. Its application to gravity was emphasized by Donoghue [6,7]. (See [8] for a comprehensive and readable review.) What is it expected is that anyfundamental theory should give the same results as quantization of Einstein’s theory plusmatter fields at energies below the Planck scale. We will concentrate on quantization ofEinstein-Maxwell theory is an example.Robinson and Wilczek [9] presented a calculation that claimed quantum gravity couldalter the behaviour of running gauge coupling constants in Yang-Mills theory. Their calcu-lation showed that the renormalization group β -function receives a purely quantum gravita-tional contribution that tends to render all theories asymptotically free, irrespective of whathappens in the absence of gravity. The phenomenological consequences of their calculationwere examined in [10], and in addition attracted attention from possible applications [11] tothe weak gravity effect [12, 13]. In view of the potential importance of the Robinson-Wilczekresult a number of independent examinations were undertaken.Doubt was first cast on the Robinson-Wilczek conclusion by Pietrykowski [14] who showedthat their result was gauge condition dependent. By choosing a different gauge no quan-tum gravitational correction to the β -function was found. Because of the question of gaugecondition dependence, a subject that will be studied in depth later in the present paper, weundertook a gauge condition independent calculation [15] and supported the conclusion ofPietrykowski; in pure Einstein-Maxwell theory the β -function receives no contribution from2uantum gravity. Dimensional regularization [16] was used in [15], and this is only sen-sitive to logarithmic divergences. Because the quantum gravity calculation of [9] involvedquadratic divergences, the role of regularization dependence of the result was studied [17] inEinstein-Yang-Mills theory. By using both a momentum space cut-off, and ensuring gaugeinvariance by applying the Taylor-Slavnov-Ward-Takahashi identities [18, 19, 20, 21], it wasshown [17] that the quadratic divergences cancelled and that the result agreed with whatwas found using dimensional regularization. No purely quantum gravitational contributionto the β -function was found in agreement with [14, 15]. A further analysis [22] showedthat it was possible to find a regularization scheme that could result in a non-zero gravita-tional contribution to the β -function, although the relation with previous work mentionedis unclear at this point. More recent work has examined the applications to Yukawa and φ interactions [23] (see also [24]) and to higher dimensions [25]. Implications for the Lee-Wick [26, 27] mechanism for gravity have also been considered [28, 29, 30]. It is also worthnoting that a string calculation [31] in a supersymmetric model results in no gravitationalcorrection to the β -function.In contrast to the negative results found for pure gravity, we showed [32] that if a cos-mological constant was present, then a non-zero quantum gravitational correction to the β -function could be obtained, that was different from what Robinson and Wilczek [9] found,but that still tended to result in asymptotic freedom. One purpose of the present paper isto give more details of the calculation described in [32]. Another is to extend the calculationto the poles in the effective action that involve higher derivatives of the electromagneticfield, including those of the Lee-Wick type. A third is to show that when calculated usingtraditional background-field methods, or equivalently using standard Feynman rules, thepole terms calculated do depend on the choice of gauge condition. This will be illustratedby explicit calculation below. The gauge condition independent background-field methoddue to Vilkovisky [33, 34] and DeWitt [35] will be used, and dimensional regularizationadopted. This method is outlined in Sec. II and applied to Einstein-Maxwell theory in thesubsequent sections. We can make a brief comment on quadratic divergences at this stageto justify the use of dimensional regularization. It is possible to show that the quadraticdivergences are completely independent of the Vilkovisky-DeWitt correction to the tradi-tional background-field formalism. Thus the quadratic divergences will agree with what isfound using a traditional Feynman diagram calculation and cancel as found in [17]. Only3ogarithmic divergences will survive and these are calculable by dimensional regularization. II. THE GAUGE INDEPENDENT EFFECTIVE ACTIONA. Introduction
In the quantization of any gauge theory there are two main problems to be addressed.The first is that the results must be invariant under the underlying gauge transforma-tions that define the theory. Within the background-field method this is relatively easy todo [36, 37, 38, 39, 40]. A classic paper showing how this works in Yang-Mills theory isAbbott’s [41] calculation of the β -function to two-loop order. Within a more traditionalFeynman diagram calculation, gauge independence is guaranteed by the Slavnov-Taylor-Ward-Takahashi identities satisfied by the various n -point functions [18, 19, 20, 21]. It istherefore possible to ensure gauge invariance of the calculation, even after regularization.The second problem that must be overcome concerns the possible dependence of theresults on the choice of gauge condition. Within the context of the functional integral ap-proach to the background-field method, the gauge condition must be introduced to avoidover-counting field configurations that are related by gauge transformations in the integra-tion over the space of all fields. This is usually dealt with by the imposition of a gaugecondition and the associated ghost fields, the Faddeev-Popov [42] method. The choice ofgauge condition is arbitrary, and it is at this stage that the dependence on this arbitrarychoice can enter the calculation. If we focus on the computation of the effective action us-ing the background-field method, then the effective action can become dependent upon thechoice of gauge condition.An early example that illustrates the dependence of the effective action on the gaugecondition is the calculation of the effective potential (a special case of the effective action) inscalar quantum electrodynamics at one-loop order by Dolan and Jackiw [43]. The one-loopeffective potential was shown to depend explicitly on parameters used to implement thegauge condition. A later computation by Dolan and Jackiw [44] showed that the one-loopeffective potential computed in the unitary gauge differed from that previously calculated.This gauge condition dependence can affect physically measurable quantities, such as thecritical temperature in finite-temperature field theories, so is not a problem that can be4gnored. Often the gauge condition independence is obscured in calculations because aconvenient choice of gauge condition is made to expedite the calculations, and all trace ofthe parameters disappears. This does not solve the problem, merely hides it.A key feature of the background-field method that leads to a possible dependence onthe gauge conditions at one-loop order is that it is necessary to expand the field aboutan arbitrary background field that is not the solution to the classical equations of motion.(After all, one motivation for the use of the effective potential in gauge theories was tostudy symmetry breaking due to radiative corrections by minimizing the effective potentialto determine the ground state [45]. This is not the same as the effective potential evaluatedat a classical solution.) It is possible to modify the background-field method as discussed byNielsen [46] for scalar electrodynamics to obtain a result for the effective potential that doesnot depend on the choice of gauge condition, thereby ensuring that physical consequences ofthe theory do not depend on this choice. However, another approach is more direct: modifythe background-field method at the start to ensure that the effective action is independentof gauge condition. This modification was suggested originally by Vilkovisky [33, 34] andrefined by DeWitt [35] to apply to all orders in the loop expansion, and it is this approachthat we will adopt here. A brief outline of some of the more important features for thecalculations needed in this paper follow in the next section. (A more pedagogical review canbe found in [47].) B. Vilkovisky-DeWitt effective action
The use of DeWitt’s condensed index notation [36] is almost indispensable here. We willconsider only bosonic gauge fields denoted by the generic symbol ϕ i . Here i stands for allof the normal gauge indices, spacetime indices, as well as the dependence on the spacetimecoordinates. Repeated indices are summed over in the usual way in the Einstein summationconvention, but in addition carry an integration over the included spacetime coordinates.Let S [ ϕ ] represent the classical action functional for the theory. We assume that the theoryhas a gauge invariance that can be described using infinitesimal parameters δǫ α . (Again α is a condensed index.) We will assume that the infinitesimal gauge transformation can bewritten as δϕ i = K iα [ ϕ ] δǫ α (2.1)5or some functional K iα [ ϕ ] that can be regarded as the generator of gauge transformations.(We will be more explicit about what K iα [ ϕ ] is in the next subsection.) Invariance of theaction functional S [ ϕ ], that is S [ ϕ + δϕ ] = S [ ϕ ] holds to first order in δǫ α , results in K iα [ ϕ ] S ,i [ ϕ ] = 0 (2.2)where S ,i [ ϕ ] denotes the functional derivative of S [ ϕ ] with respect to ϕ i . Hamilton’s principleof stationary action tells us that S ,i = 0 are the classical equations of motion; thus, (2.2)expresses the fact that these equations are invariant under a gauge transformation.We have already mentioned the problem of quantization of gauge theories using theintegration method over the space of all fields (the Feynman path integral). If we naivelyintegrate over the space of all gauge fields we will include fields as different even though theyare physically equivalent under the gauge transformation (2.1). We can think of all fieldsrelated by gauge transformations as belonging to the same equivalence class and we wish tointegrate in the functional integral only over distinct equivalence classes. The first step inthe implementation of this is to introduce a gauge condition (sometimes call the gauge-fixingcondition) χ α [ ϕ ] = 0 . (2.3)We require χ α [ ϕ + δϕ ] = χ α [ ϕ ] hold only if δǫ α = 0. The consequence of this is that Q αβ [ ϕ ] δǫ β = 0 (2.4)has only the solution δǫ β = 0 where we have defined Q αβ [ ϕ ] = χ α,i [ ϕ ] K iβ [ ϕ ] . (2.5)Provided that det Q αβ = 0, (2.4) does imply that δǫ β = 0 is the only solution as required.(det Q αβ is the Faddeev-Popov [42] factor that we will return to later.) Note also that thegauge condition can depend on the background field, although we will not indicate thisdependence explicitly.The next step in the Vilkovisky-DeWitt effective action relies on assuming that the spaceof all fields is equipped with a metric tensor g ij [ ϕ ]. We can write a line element as usual. Inthe case of Yang-Mills theory and gravity there are natural choices that do not involve theintroduction of dimensional parameters as we will discuss below. (For gravity, the result isthe DeWitt metric [48].) For both gravity and Yang-Mills theory it is possible to show that6 iα [ ϕ ] can be viewed as components of a set of vector fields that form a Lie algebra and areKilling vectors for the field space metric g ij . (See [47] for details.)The central part of the Vilkovisky approach is the choice of connection. One way tocalculate the appropriate connection is by first considering a general displacement in thespace of fields dϕ i . This will not be generated by a gauge transformation in general, butwill be expressible as a linear combination dϕ i = ω i k + ω i ⊥ , (2.6)where ω i k = K iα dǫ α , (2.7)and ω i ⊥ satisfies g ij ω i ⊥ ω j k = 0 . (2.8)To obtain ω i ⊥ we can define a projection operator P ij = δ ij − K iα γ αβ K βj , (2.9)where K βj = g ji K iβ as usual, and γ αβ is the inverse of γ αβ = K iα g ij K jβ . (2.10)It is easy to verify that P ij K jα = 0 , (2.11)and that P ij P jk = P ik . (2.12)Because of (2.11), P ij has the property of projecting vectors perpendicular to the generatorsof gauge transformations. This results in the line element ds = g ij dϕ i dϕ j = g ⊥ ij ω i ⊥ ω j ⊥ + γ αβ dǫ α dǫ β (2.13)that exhibits the local product structure with the first term on the right hand side repre-senting the line element on the space of orbits and the second term representing that on thegauge group. In (2.13) we have g ⊥ ij = P ki P lj g kl (2.14)7nterpreted as the metric on the space of distinct gauge orbits.Because it is the space of distinct gauge orbits that is integrated over in the Feynmanfunctional integral, the natural choice of connection ¯Γ kij is determined from the requirementthat ¯ ∇ i g ⊥ jk = 0 = g ⊥ jk,i − ¯Γ lij g ⊥ lk − ¯Γ lik g ⊥ jl . (2.15)( ¯ ∇ i denotes the covariant derivative with respect to the connection.) This leads to¯Γ lij g ⊥ lk = 12 (cid:0) g ⊥ jk,i + g ⊥ ki,j − g ⊥ ij,k (cid:1) . (2.16)Normally we would introduce the inverse to g ⊥ lk and multiply both sides of (2.16) by thisinverse; however, g ⊥ lk is not invertible on the full field space since g ⊥ ij K jα = 0. Because of this,¯Γ kij is only determined up to an arbitrary multiple of K kα that vanishes when contracted with g ⊥ lk . It can be shown that ¯Γ kij takes the form¯Γ kij = Γ kij + T kij + K kα A αij , (2.17)where Γ kij is the Christoffel connection for the metric g ij , T kij is a complicated expressionthat involves g ij , K iα and their first derivatives, and A αij is completely arbitrary.At this stage we note that the effective action can be computed in the loop expansionwhere in place of the normal derivatives that occur we use covariant ones. (That is, acovariant Taylor expansion of the classical action is used.) When this is done, it is possibleto show that the terms arising from A αij in (2.17) vanish as a consequence of gauge invariance;thus, the arbitrariness of the connection is not a problem. Only the Christoffel connectionΓ kij and the term T kij make a contribution to the result.When we perform the integration over the space of fields, the natural measure followsformally from (2.13) as dµ [ ϕ ] = Y i ω i ⊥ ! Y α dǫ α ! (cid:0) det g ⊥ ij (cid:1) / (det γ αβ ) / . (2.18)As a consequence of Killing’s equation and the anti-symmetric property of the structureconstants it is possible to show that (cid:0) det g ⊥ ij (cid:1) / and (det γ αβ ) / are both gauge invariant(independent of the gauge parameters ǫ α ). Thus, if we integrate any gauge invariant expres-sion using the measure (2.18) the integration over the gauge group parameters ǫ α may befactored out leaving only an integration over the orbit space as required. (By orbit space we8ean the full field space factored out by the group of gauge transformations.) Note howeverthat the factor (det γ αβ ) / remains. This geometric observation [49] is the basis of the usualFaddeev-Popov “ansatz” [42]. It is now possible show that we can take (with the integrationover ǫ α dropped but (det γ αβ ) / kept) dµ [ ϕ ] = Y i dϕ i ! (cid:0) det g ⊥ ij (cid:1) / (det Q αβ ) δ [ χ α ] , (2.19)since the space of orbits is also fixed by the gauge condition χ α = 0. This correspondsexactly to the usual Faddeev-Popov [42] construction.It is now possible to prove three things about the effective action. The first is thatif we define the standard functional integral expression, expressed in a suitably covariantformulation, it does not depend on the choice of field variables ϕ i that are chosen. The secondis that the effective action is a gauge invariant functional of the background field. The thirdis that the effective action is not dependent on the choice made for the gauge condition.In proving this last property, the Vilkovisky-DeWitt connection (2.17) is essential, and inparticular the role of T kij is crucial. This has been verified explicit calculations [50, 51, 52].The basic idea now is to pick a gauge choice, compute all the geometric arsenal describedabove, and calculate the effective action. Because we are guaranteed that the result does notdepend on the choice of gauge condition we can make the calculations simpler by adoptinga suitable gauge choice. This choice was called the Landau-DeWitt gauge by Fradkin andTseytlin [50] and is sometimes called the background field gauge. It begins by expressingthe field ϕ i = ¯ ϕ i + η i (2.20)where ¯ ϕ i is the background field. The Landau-DeWitt gauge condition reads χ α = K αi [ ¯ ϕ ] η i = 0 . (2.21)Because of the form taken by T kij it is possible to show that it makes no contributionto the effective action at one-loop order if the Landau-DeWitt gauge is used. Any otherchoice of gauge requires the inclusion of T kij . This leads to considerable technical simpli-fications. Beyond one-loop order this is no longer the case in general. For certain classesof theories, including Yang-Mills theory but not gravity, it is possible to prove that T kij makes no contribution to the effective action to all orders in the loop expansion for the9andau-DeWitt gauge. Thus the correct gauge invariant and gauge condition independenteffective action for Yang-Mills theory can be calculated from the usual formalism providedthat we adopt only the Landau-DeWitt gauge; for any other choice of gauge we must use thefull Vilkovisky-DeWitt expression [47, 50, 53]. We will only use the Landau-DeWitt gaugecondition here.The aim of this paper is to study only quantum corrections to quantum gravity at one-loop order. This involves an expansion of the classical action in a covariant Taylor series toquadratic order in the quantum field η i defined in (2.20) followed by a Gaussian functionalintegral. The complication due to the presence of the δ -function in the measure (2.19) canbe dealt with by use of the familiar identity δ [ χ α ] = lim ξ → (4 πiξ ) − / exp (cid:18) i ξ χ α χ α (cid:19) (2.22)suitably generalized to the case of functions. The result for the effective action to one-looporder may be taken asΓ[ ¯ ϕ ] = S [ ¯ ϕ ] − ln det Q αβ [ ¯ ϕ ] (2.23)+ 12 lim ξ → ln det (cid:18) ∇ i ∇ j S [ ¯ ϕ ] + 12 ξ K iα [ ¯ ϕ ] K αj [ ¯ ϕ ] (cid:19) . We work in the Landau-DeWitt gauge as discussed. Here, ∇ i ∇ j S [ ¯ ϕ ] = S ,ij [ ¯ ϕ ] − ¯Γ kij S ,k [ ¯ ϕ ] (2.24)gives the covariant derivative computed using the connection (2.17). It should be clear whythe arbitrary third term in (2.17) does not matter at one-loop order (since K kα S ,k = 0 is theexpression of gauge invariance). It is not immediately obvious that the term T kij makes nocontribution in the Landau-DeWitt gauge but it can be shown not to. (See the pedagogicaltreatment in [47].) We may therefore replace ¯Γ kij in (2.24) with the Christoffel connectionΓ kij .At this stage it should be clear why it is significant to know whether or not we areexpanding about a background field that is the solution to the classical equations of motion.If we are, then S ,i = 0 and the terms in the effective action that arise from the connectionvanish. The formalism reduces to the usual one. As we will see, we must not assumethat this is the case in what follows. Another observation that can be made is that if theChristoffel connection vanishes, then by adopting the Landau-DeWitt gauge there is no10istinction between covariant and ordinary derivatives, and the usual traditional effectiveaction formalism can be used. (This occurs in the case where the metric g ij on the space offields does not depend on the fields.) III. EINSTEIN-MAXWELL THEORY
The interest of the present paper is to study the one-loop quantization of Einstein-Maxwelltheory as a simple model of a gauge theory coupled to gravity. The classical action functionalmay be chosen to be S = S M + S G , (3.1)where S M = 14 Z d n x | g ( x ) | / F µν F µν , (3.2)is the Maxwell field action, and S G = − κ Z d n x | g ( x ) | / ( R − , (3.3)is the gravitational Einstein-Hilbert action with the inclusion of a cosmological constant Λ.We have defined κ = 32 πG, (3.4)with G Newton’s gravitational constant, allowed the spacetime dimension to be n , andadopted the curvature conventions of [54] but with a Riemannian (as opposed to aLorentzian) metric chosen. There is no deep significance to be attached to this last choice;it merely avoids factors of i .In Sec. II condensed notation has been used with ϕ i standing for all of the fields. Althoughconvenient for discussing basic formalism, for practical calculations normal notation mustbe resorted to. We will make the association ϕ i = ( g µν ( x ) , A µ ( x )) . (3.5)Here A µ is the electromagnetic gauge field with the convention F µν = ∂ µ A ν − ∂ ν A µ . (3.6)The Vilkovisky-DeWitt formalism has been set up to be completely covariant. Any choiceof field variables ( g µν , A µ , | g | / g µν etc.) may be made in place of (3.6) without affecting theresults. We have merely adopted the simplest, and perhaps most natural, choice here.11he action (3.1) is invariant under combined spacetime coordinate changes and U (1)gauge transformations. If we let δǫ λ be the infinitesimal parameters describing spacetimecoordinate transformations, and δǫ be the infinitesimal parameter for the U (1) gauge trans-formation, then the fields (3.5) behave like δg µν = − δǫ λ g µν,λ − δǫ λ,µ g λν − δǫ λ,ν g µλ , (3.7) δA µ = − δǫ ν A µ,ν − δǫ ν,µ A ν + δǫ ,µ . (3.8)These last two results are represented by δϕ i = K iα δǫ α in condensed notation. (See (2.1).)We will make the condensed index association δǫ α = ( δǫ λ ( x ) , δǫ ( x )). The indices in (2.1)can be uncondensed by writing δg µν ( x ) = Z d n x ′ (cid:8) K g µν ( x ) λ ( x, x ′ ) δǫ λ ( x ′ ) + K g µν ( x ) ( x, x ′ ) δǫ ( x ′ ) (cid:9) , (3.9) δA µ ( x ) = Z d n x ′ (cid:8) K A µ ( x ) λ ( x, x ′ ) δǫ λ ( x ′ ) + K A µ ( x ) ( x, x ′ ) δǫ ( x ′ ) (cid:9) . (3.10)Here we use the actual field as a component label as in [55]. By comparing (3.9) and (3.10)with (3.7) and (3.8) we can read off K g µν ( x ) λ ( x, x ′ ) = − g µν,λ ( x ) δ ( x, x ′ ) − g µλ ( x ) ∂ ν δ ( x, x ′ ) − g λν ( x ) ∂ µ δ ( x, x ′ ) , (3.11) K g µν ( x ) ( x, x ′ ) = 0 (3.12) K A µ ( x ) λ ( x, x ′ ) = − A µ,λ ( x ) δ ( x, x ′ ) − A λ ( x ) ∂ µ δ ( x, x ′ ) , (3.13) K A µ ( x ) ( x, x ′ ) = ∂ µ δ ( x, x ′ ) . (3.14)Here δ ( x, x ′ ) is the symmetric Dirac δ -distribution defined by R d n x ′ δ ( x, x ′ ) F ( x ′ ) = F ( x ) forscalar test function F ( x ). δ ( x, x ′ ) transforms like | g ( x ′ ) | / at x ′ and a scalar at x .The natural line element for the space of fields is ds = Z d n xd n x ′ (cid:8) g g µν ( x ) g λσ ( x ′ ) dg µν ( x ) dg λσ ( x ′ ) + g A µ ( x ) A ν ( x ′ ) dA µ ( x ) dA ν ( x ′ ) (cid:9) (3.15)where we choose g g µν ( x ) g λσ ( x ′ ) = 12 κ | g ( x ) | / (cid:0) g µλ g νσ + g µσ g νλ − g µν g λσ (cid:1) δ ( x, x ′ ) , (3.16)to be the DeWitt metric [48], and g A µ ( x ) A ν ( x ′ ) = | g ( x ) | / g µν ( x ) δ ( x, x ′ ) . (3.17)12he factor of κ − in (3.16) ensures that both terms in (3.15) have the same units and resultsin ds in (3.15) having units of length squared.Given the metric components in (3.16) and (3.17), the Christoffel connection can becomputed. The non-zero components turn out to beΓ g λτ ( x ) g µν ( x ′ ) g ρσ ( x ′′ ) = (cid:20) − δ ( µ ( λ g ν )( ρ δ σ ) τ ) + 14 g µν δ ρ ( λ δ στ ) − − n ) (cid:26) g λτ g µ ( ρ g σ ) ν − g λτ g µν g ρσ (cid:27) + 14 g ρσ δ µ ( λ δ ντ ) (cid:21) δ ( x ′′ , x ) δ ( x ′′ , x ′ ) (3.18)Γ g µν ( x ) A λ ( x ′ ) A τ ( x ′′ ) = 12 δ ( λµ δ τ ) ν δ ( x, x ′ ) δ ( x ′ , x ′′ ) , (3.19)Γ A µ ( x ) A ν ( x ′ ) g αβ ( x ′′ ) = 14 (cid:0) δ νµ g αβ − δ αµ g νβ − δ βµ g να (cid:1) δ ( x, x ′ ) δ ( x, x ′′ ) (3.20)= Γ A µ ( x ) g αβ ( x ′′ ) A ν ( x ′ ) . The round brackets around indices denote a symmetrization over the indices enclosed alongwith a factor of 1 /
2. The Christoffel connection components in (3.18–3.20) will be used tocompute the second term of (2.24).At this stage we may choose a background. If we keep the background metric generalthen we would be repeating the monumental calculation of [3] using the Vilkovisky-DeWittformalism. Although this would be interesting and challenging to do, we will focus instead onthe quantum gravity corrections to the running value of the electric charge, or fine structureconstant, as well as computing the pole terms of the Lee-Wick type. This means that wedo not need to consider terms in the effective action that involve the curvature and we willchoose the background spacetime to be flat. We therefore choose ¯ ϕ i to be¯ ϕ i = ( δ µν , ¯ A λ ( x )) , (3.21)where we keep the background gauge field ¯ A λ ( x ) general. If we are only interested inthe terms in the effective action that can affect the electric charge then we can take thebackground electromagnetic field ¯ F µν to be constant as in our earlier work [15, 32]; however,this would miss out any poles that involve derivatives of the electromagnetic field that couldbe of the Lee-Wick type. We do not make any assumptions about ¯ A λ ( x ) at this stage. Animportant feature of the background is that it is not a solution to the classical Einstein-Maxwell equations, and therefore the inclusion of the connection term in (2.24) is crucial if13he result for the effective action is to be gauge condition independent. We will illustratethat this is so by an explicit calculation showing how a gauge condition dependent result isobtained using the traditional effective action method.The results for S ,i [ ¯ ϕ ] can be computed from appropriate functional derivatives of (3.1–3.3)with respect to ϕ i in (3.5) followed by setting ϕ i = ¯ ϕ i in (3.21). The results are δSδg µν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ ϕ = 2 κ Λ δ µν + 18 δ µν ¯ F αβ ¯ F αβ −
12 ¯ F µλ ¯ F νλ , (3.22) δSδA µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ ϕ = ∂ ν ¯ F µν . (3.23)The last result (3.23) vanishes if we restrict ¯ F µν to be constant, so would not contribute tothe charge renormalization, but can contribute to Lee-Wick type terms. The second andthird terms of (3.22) are just those involved in the stress-energy-momentum tensor of theelectromagnetic field.It is worth explaining at this stage why we can concentrate on pole terms in the effectiveaction that involve only the electromagnetic field to deduce the charge renormalization. Thebasic reason is the Ward-Takahashi identity that relates the charge and field renormalizationfactors. The calculation of Abbott [41] showed how this works within the background fieldmethod. Let e B and e R be the bare and renormalized charges respectively. Using dimensionalregularization [16, 56] we have e B = ℓ n/ − Z e e R , (3.24)where ℓ is an arbitrary unit of length (the reciprocal of ‘t Hooft’s [56] unit of mass) and Z e isthe charge renormalization factor. Similarly, let ¯ A µB and ¯ A µR be the bare and renormalizedbackground gauge fields respectively. Then,¯ A µB = ℓ − n/ Z / A ¯ A µR , (3.25)with Z A the field renormalization factor. A consequence of the gauge invariant backgroundfield method is e B ¯ A µB = e R ¯ A µR . (3.26)(Think of the gauge covariant derivative written in terms of the bare quantities in thebare classical action. This must be rewritten in terms of the renormalized ones in a gaugeinvariant way.) From (3.24–3.26) we find Z e Z / A = 1 , (3.27)14s the Ward-Takahashi identity tells us. The standard ‘t Hooft [56] approach to the renor-malization group relates the running value of the charge to the pole terms in Z e . This willbe outlined in Sec. VI. The identity (3.27) allows us to deduce the pole terms in Z e fromthose in Z A , and Z A is determined by the renormalization of the background gauge field.It is for this reason that we can concentrate on the pole parts of the effective action thatinvolve the background gauge field.The calculation become simpler to deal with if we re-express the last two terms of theone-loop effective action (2.23) as functional integrals. The ln det Q αβ term can be expressedas an integration over ghost fields, whereas the last term can be written as an integrationover the quantum field η i defined in (2.20). We will writeΓ G = 12 ln det (cid:26) ∇ i ∇ j S [ ¯ ϕ ] + 12 ξ K iα [ ¯ ϕ ] K αj [ ¯ ϕ ] (cid:27) = − ln Z [ dη ] e − S q , (3.28)where S q = 12 η i η j (cid:18) S ,ij − Γ kij S ,k + 12 ξ K α i K αj (cid:19) , (3.29)with the limit ξ → ϕ defined in (3.21). The ghost contribution isΓ GH = − ln det Q αβ = − ln Z [ d ¯ ηdη ] e − ¯ η α Q αβ η β , (3.30)where ¯ η α and η β are anti-commuting ghost fields.The aim now is to identify terms in the arguments of the exponentials of the functionalintegrals that depend on the background gauge field, treat these terms interactions, andexpand in powers of the interactions up to a given order. Simple power counting shows thatthere will be poles that involve two, three, or four powers of the background electromagneticfield. The terms with three powers of the field would be expected to vanish because theclassical theory is symmetric under ¯ F µν → − ¯ F µν , and we will verify that this is the casebelow. We will first concentrate on the gravity and gauge field contribution in Γ G (3.28,3.29)in the next section. The ghost contribution will be studied in the subsequent section.15 V. GRAVITY AND GAUGE FIELD CONTRIBUTIONA. Expansion of the effective action
In evaluating the result for S q in (3.29) it can be noted initially that the first term, η i η j S ,ij [ ¯ ϕ ] is just the quadratic term in the Taylor series expansion of S [ ¯ ϕ + η ] . Thisis the term (along with that from the gauge condition) that is present in the traditionaleffective action. The term that involves the connection is only present in the Vilkovisky-DeWitt approach. In order to trace the effect of including the connection, we will includea parameter v in ¯Γ kij that when set to zero gives us the traditional result, and when set tounity gives us the (correct) Vilkovisky-DeWitt result.To deal with the gauge-fixing condition it can be noted first that the condensed indexexpression for the gauge-fixing condition S GF = 14 ξ η i η j K αi K αj = 14 ξ ( χ α ) (4.1)where χ α is the Landau-DeWitt gauge condition (2.21). In our case there are two gaugeconditions, one for the graviton field, and one for the electromagnetic field. With the gaugetransformation generators given in (3.11–3.14) we find the Landau-DeWitt gauge conditionsspecified by χ λ = 2 κ ( ∂ µ h µλ − ∂ λ h ) + ω ( ¯ A λ ∂ µ a µ + a µ ¯ F µλ ) , (4.2) χ = − ∂ µ a µ , (4.3)where we have set η i = ( κh µν , a µ ) , (4.4)so that g µν = δ µν + κh µν , (4.5) A µ = ¯ A µ + a µ , (4.6)and defined h = h µµ = δ µν h µν . (4.7)The factor of κ in (4.5) is a standard convenience that removes a factor of κ − presentin the Einstein-Hilbert action (3.3) from the quadratic part of the action that defines the16ropagators. The factor of ω in (4.3) is included in order to show the gauge conditiondependence present in the traditional effective action. ω should be taken to be unity in theVilkovisky-DeWitt result. By keeping ω present we can compare the use of a de Donder (orharmonic) gauge ( ω = 0) with the Landau-DeWitt gauge ( ω = 1).One important comment is that the formalism of the Vilkovisky-DeWitt effective actionensures that the results are independent of the choice made for ω ; however, this will not beshown in the present calculation because we have restricted attention to the Landau-DeWittgauge for expediency. If we wish to keep ω general, then the neglect of the terms in theconnection denoted by T kij in (2.17) is not justified; it is the presence of such terms thatensures the result for general ω agrees with that for ω = 1 in the Landau-DeWitt gauge.It should be possible to show this explicitly, although the calculations will be much moreinvolved than those presented in the present paper and will be given elsewhere.Another comment worth making is that we can use the gauge condition (4.3) to setthe ∂ µ a µ term in (4.2) to zero. (This is true because the gauge conditions appear as δ -functions in the functional integral before they are promoted to exponentials.) We will dothis later because it simplifies the calculations, although we will keep it present for themoment. We have checked explicitly that the ∂ µ a µ term in (4.2) makes no contribution tothe electromagnetic field renormalization to verify this formal conclusion.Because we have two gauge conditions we will have two terms arising from uncondensing(4.1). We will call the gauge parameters ξ and ζ and take S GF = 14 ξ Z d n xχ λ + 14 ζ Z d n xχ . (4.8)The Landau-DeWitt gauge condition is specified by taking the ξ → ζ → ξ general to show the gauge condition dependence of the traditional background-field result, but take ζ → S q in (3.29) can be written as S q = S + S + S , (4.9)17here S = Z d n x ( − h µν (cid:3) h µν + 14 h (cid:3) h + (cid:18) κ ξ − (cid:19) (cid:18) ∂ µ h µν − ∂ ν h (cid:19) − Λ (cid:18) h µν h µν − h (cid:19) (cid:20) v (cid:18) n − − n (cid:19)(cid:21) + 12 a µ ( − δ µν (cid:3) + ∂ µ ∂ ν ) a ν + 14 ζ ( ∂ µ a µ ) − v δ µν a µ a ν (cid:27) , (4.10) S = κ Z d n x (cid:0) ¯ F µν h∂ µ a ν − F αν h µα ∂ µ a ν + 2 ¯ F αν h µα ∂ ν a µ (cid:1) − κv Z d n x (cid:0) δ λσ δ µν − δ µσ δ λν − δ νσ δ λµ (cid:1) ∂ τ ¯ F στ h µν a λ + ωκξ Z d n x (cid:18) ∂ µ h µν − ∂ ν h (cid:19) (cid:0) ¯ A ν ∂ λ a λ + a λ ¯ F λν (cid:1) , (4.11) S = κ Z d n x ¯ F µν ¯ F αβ (cid:0) δ µα h νλ h λβ + h µα h νβ − δ µα hh νβ (cid:1) − κ (cid:18) v ( n − − n (cid:19) Z d n x ¯ F αβ ¯ F αβ (cid:18) h µν h µν − h (cid:19) + κ v Z d n x (cid:16) − ¯ F µγ ¯ F σγ δ νλ + 14 ¯ F λγ ¯ F σγ δ µν + 14 ¯ F µγ ¯ F νγ δ λσ − − n ) δ µλ δ σν ¯ F αβ ¯ F αβ + 14(2 − n ) δ µν δ λσ ¯ F αβ ¯ F αβ (cid:17) h µν h λσ − κ v Z d n x (cid:18) δ µν ¯ F αβ ¯ F αβ −
12 ¯ F µλ ¯ F νλ (cid:19) a µ a ν + ω ξ Z d n x ¯ F µλ ¯ F νλ a µ a ν (4.12)In these expressions the subscript 0 , , S denotes the order in the background gaugefield ¯ A µ and we have shown explicitly the Vilkovisky-DeWitt terms with the factor v asdescribed above. The traditional result is obtained using v = 0. The spacetime dimension n has been kept general at this stage, although we will be interested ultimately in the limit n →
4. Because our concern here is only with pole terms in the effective action, it can beseen that as n → n −
4) , such as occur in (4.10) and (4.12),will not contribute. Another observation is that the Vilkovisky-DeWitt connection leads toa term in S that acts like a photon mass if Λ = 0.The graviton and photon propagators follow from S in the usual way. The terms in S and S will be treated as interactions. We can write the photon propagator as G µν ( x, x ′ ) = Z d n p (2 π ) n e ip · ( x − x ′ ) G µν ( p ) , (4.13)and the graviton propagator as G ρσλτ ( x, x ′ ) = Z d n p (2 π ) n e ip · ( x − x ′ ) G ρσλτ ( p ) . (4.14)18sing the result for S leads to G µν ( p ) = δ µν p − v Λ + (2 ζ − p µ p ν ( p − v Λ)( p − ζ v Λ) . (4.15)and, G ρσλτ ( p ) = δ ρλ δ στ + δ ρτ δ σλ − n − δ ρσ δ λτ p − λ ) + 12 ( κ ξ − δ ρλ p σ p τ + δ ρτ p σ p λ + δ σλ p ρ p τ + δ στ p ρ p λ ( p − λ ) ( p − κ ξλ ) , (4.16)where we have defined λ = Λ + v Λ (cid:18) n − − n (cid:19) . (4.17)In our calculations of the pole terms, the Vilkovisky-DeWitt correction in (4.17) will makeno contributions to the poles when n →
4, and we may set λ → Λ in this limit. This will notbe true for the finite part of the effective action or in spacetimes of dimension other thanfour. (Of course the Vilkovisky-DeWitt correction enters the calculation in other placesthrough the interaction terms in any case.)As explained we will treat the terms S + S as an interaction. Simple power countingshows that the divergent part of the effective action can involve ¯ F µν up to and includingterms of fourth order. (In more than four spacetime dimensions, higher powers of ¯ F µν mustbe considered.) We can write Γ G = h e − S − S − i (4.18)where h· · · i means to evaluate the enclosed expression using Wick’s theorem and the basicpairings h a µ ( x ) a ν ( x ′ ) i = G µν ( x, x ′ ) , (4.19) h h ρσ ( x ) h λτ ( x ′ ) i = G ρσλτ ( x, x ′ ) , (4.20)If we drop terms of order ¯ F and higher, use of Wick’s theorem shows thatΓ G = Γ G + Γ G + · · · , (4.21)where Γ Gk is of order, ¯ F k . There is no cubic term in ¯ F present as claimed earlier because suchterms can only arise from those in the expansion of Γ G that involve odd numbers of gravitonand photon fields; these vanish upon use of Wick’s theorem. ( e.g. h S S i = 0 = h S i .) Wefind Γ G = h S i − h S i , (4.22)Γ G = − h S i + 12 h S S i − h S i . (4.23)19he (correct) Vilkovisky-DeWitt result is obtained by taking the parameters v = 1 , ω =1 , ξ = 0 , ζ = 0 in these expressions. We now examine these two terms separately, and thenturn to the possible ghost contributions. B. Evaluation of Γ G We first of all use Wick’s theorem to evaluate both h S i and h S i . This will give us theresults in terms of the graviton and photon propagators. The momentum space representa-tions (4.13–4.16) can be used and the resulting integrals evaluated using standard methods.(We give the basic results in the appendix.)For h S i we find, after use of Wick’s theorem, h S i = h S i + h S i , (4.24)where h S i = κ Z d n x n (2 − v ) (cid:16) ¯ F µγ ¯ F βγ δ να − δ µν ¯ F αγ ¯ F βγ (cid:17) + ¯ F µα ¯ F νβ + 14 ( v −
1) ¯ F (cid:16) δ µα δ νβ − δ µν δ αβ (cid:17)o G µναβ ( x, x ) , (4.25) h S i = κ Z d n x n v (cid:16) ¯ F µγ ¯ F νγ −
14 ¯ F δ µν (cid:17) + ω κ ξ ¯ F µγ ¯ F νγ o G µν ( x, x ) . (4.26)We have abbreviated ¯ F = ¯ F µν ¯ F µν here and in the following. The two terms (4.25) and(4.26) involve only the coincidence limit of the Green functions and no derivatives of theelectromagnetic field strength ¯ F . Any pole terms will contribute to the charge renormaliza-tion.We use dimensional regularization with only the logarithmic divergences present as de-scribed earlier. Because we are only interested in pole terms of the effective action comingfrom logarithmic divergences we can adopt the method described in the appendix. Theresults, after some calculation, turn out to be given by (where L stands for the basic loga-rithmic divergence defined in (A.10,A.12)) h S i = 38 κ Λ( κ ξ + 1) L Z d x ¯ F , (4.27) h S i = ω ξ v Λ L Z d x ¯ F . (4.28)We have let ζ → ∂ λ a λ term in S to shorten the expressions obtained and simplify the calculation and it would be inconsistent20o retain ζ . The first term (the one multiplied by v ) of h S i in (4.26) does not contribute tothe result since G µν ( x, x ) ∝ δ µν and the contraction of this with the field strength vanishes.(This can be recognized as involving the trace of the stress-energy-momentum tensor for theelectromagnetic field which vanishes for n = 4.) The Vilkovisky-DeWitt parameter does notenter (4.27) although this was not obvious from (4.25). Combining the two results (4.27)and (4.28) results in h S i = 38 κ (cid:18) κ ξ + ω v κ ξ (cid:19) Λ L Z d x ¯ F , (4.29)as the relevant pole part. For the Vilkovisky-DeWitt result we take ω = v = 1 and tryto let ξ →
0. However, there is a term in 1 /ξ present that prohibits this limit to be takencompletely. Because the Vilkovisky-DeWitt formalism ensures that ξ → h S i (and the higher order terms in Γ G ) it proves convenient towrite S in (4.11) as S = Z d n x (cid:0) P αβµν h αβ ∂ µ a ν + P αβλ h αβ a λ (cid:1) , (4.30)where P αβµν = κ (cid:26)(cid:16) − ωκ ξ (cid:17) (cid:2) ¯ F µν δ αβ − ¯ F βν δ µα − ¯ F αν δ µβ (cid:3) + ¯ F βµ δ να + ¯ F αµ δ νβ (cid:27) , (4.31) P αβλ = κv (cid:0) δ λα ∂ τ ¯ F βτ + δ λβ ∂ τ ¯ F ατ (cid:1) − ω κξ ( ∂ α ¯ F λβ + ∂ β ¯ F λα )+ (cid:18) ω κξ − κv (cid:19) δ αβ ∂ τ ¯ F λτ . (4.32)Wick’s theorem gives us h S i = Z d n x Z d n x ′ n P αβµν ( x ) P λσγδ ( x ′ ) G αβλσ ( x, x ′ ) ∂ µ ∂ ′ γ G νδ ( x, x ′ )+2 P αβµν ( x ) P λσγ ( x ′ ) G αβλσ ( x, x ′ ) ∂ µ G νγ ( x, x ′ )+ P αβγ ( x ) P λσδ ( x ′ ) G αβλσ ( x, x ′ ) G νδ ( x, x ′ ) o . (4.33)The products of Green functions may be evaluated using the momentum space representa-tions and results of the Appendix. After considerable calculation it may be shown that h S i = κ α L Z d x ¯ F + κ βL Z d x (cid:0) ∂ µ ¯ F µν (cid:1) , (4.34)21here α = 3Λ vω κ ξ −
34 Λ vω + 38 Λ v + 34 Λ ω + 34 Λ+ 38 Λ vκ ξ −
32 Λ ωκ ξ + 34 Λ κ ξ , (4.35) β = −
112 + 23 ω + 316 v + 14 v + 12 vω + 14 κ ξ + 316 v κ ξ − vκ ξ. (4.36)In writing down this expression we have chosen to write the term that involves derivativesof ¯ F as shown. In the calculation we also find a term ¯ F µν (cid:3) ¯ F µν that when integrated byparts is equivalent to − ∂ µ ¯ F µν ) .We can now find the complete pole part of the effective action that is quadratic in ¯ F andcomes from the gauge field and graviton. (We still need to find the ghost contribution andwe will do this in the next section.) From (4.22,4.29,4.34–4.36) we haveΓ G = κ α Λ L Z d x ¯ F − κ βL Z d x (cid:0) ∂ µ ¯ F µν (cid:1) , (4.37)where α = 38 vω − v − ω − vκ ξ + 34 ωκ ξ, (4.38)and β is given in (4.36).There are several comments to be made at this stage. The first is that although termsthat are singular as ξ → ω and ξ even if we take the Vilkovisky-DeWittparameter v = 0 corresponding to the use of the standard background-field method. Unlessspecial care is taken when using the traditional background-field method, or equivalently thenaive Feynman rules, results will be obtained for the effective action that are gauge conditiondependent. This is completely obscured in calculations that fix any of these parametersat the start for calculational convenience. A final comment is that if the cosmologicalconstant vanishes then there is no contribution to the term in ¯ F that is responsible forthe electromagnetic field, and hence the charge renormalization, in agreement with earlierresults of [14, 15, 17]. The result for Λ = 0 was first given in [32].22 . Evaluation of Γ G We begin with each of the three terms that comprise the contributions to Γ G . Ondimensional grounds there can be no derivatives of the background electromagnetic field,so we may safely take ¯ F µν to be constant. This simplifies the calculation. There are twoindependent invariants that are gauge invariant and we take them to be ( ¯ F ) and ¯ F where¯ F = ¯ F µν ¯ F µν , (4.39)¯ F = ¯ F µν ¯ F νλ ¯ F λσ ¯ F σµ . (4.40)We will write Γ G = κ L Z d x (cid:8) A ( ¯ F ) + B ¯ F (cid:9) , (4.41)for some coefficients A and B . Nether A nor B can depend on the cosmological constant (ondimensional grounds); thus, the result that we will obtain for the pole part of the effectiveaction indicated in (4.41) will apply equally well to Einstein-Maxwell theory without acosmological constant.We begin by noting that the term called P αβλ in (4.30,4.32) cannot contribute to the poleterms in (4.41) as it vanishes when we set ¯ F µν to be constant. We can write S in (4.12) ina convenient way as S = Z d n x (cid:0) R µναβ h µν h αβ + R µν a µ a ν (cid:1) , (4.42)where R µναβ and R µν can be read off by comparison of (4.42) with (4.12) and the resultssymmetrized in the obvious way. Both R µναβ and R µν may be taken to be constant for ourpurposes. Application of Wick’s theorem gives (cid:10) ( S ) (cid:11) = 2 Z d n xd n x ′ n R µναβ R ρσλτ G µνρσ ( x, x ′ ) G αβλτ ( x, x ′ )+ R µν R ρσ G µρ ( x, x ′ ) G νσ ( x, x ′ ) o . (4.43)The products of Green functions are evaluated as described in the Appendix and the resultsare then contracted with R µναβ and R µν in (4.43). The result takes the form on the righthand side of (4.41) where A = A and B = B with A = ω κ ξ − vω κ ξ + v − v , (4.44) B = 7 ω κ ξ + 7 vω κ ξ − v v
384 + 132 . (4.45)23he next term of order ¯ F involves (cid:10) ( S ) S (cid:11) = 2 Z d n xd n x ′ d n x ′′ P αβµν P λσγδ n R ǫρθφ ∂ µ ∂ ′ γ G νδ ( x, x ′ ) G αβǫρ ( x, x ′′ ) G λσθφ ( x ′ , x ′′ )+ R ǫρ G αβλσ ( x, x ′ ) ∂ µ G νǫ ( x, x ′′ ) ∂ ′ γ G δρ ( x ′ , x ′′ ) o . (4.46)The products of Green functions are evaluated as before, and we again find a result takingthe form on the right hand side of (4.41) where this time A = A and B = B with A = ω κ ξ − vω κ ξ + 5 ω κ ξ − ω κ ξ − vκ ξ
384 + 5 vω
64 + 3 v
128 + 5 ω
192 + 3 κ ξ − ωκ ξ
32 + vκ ξ − vωκ ξ
24 + vω − ω
16 + κ ξ
64 + 364 , (4.47) B = 7 ω κ ξ + 7 vω κ ξ − ω κ ξ − ω κ ξ + 23 vκ ξ − vω − v
32 + 13 ω − κ ξ − ωκ ξ − vκ ξ
12 + vωκ ξ − vω
12 + 3 ω κ ξ . (4.48)The third and final piece of Γ G involves h ( S ) i . The Wick reduction leads to (cid:10) ( S ) (cid:11) = 3 Z d n xd n x ′ d n x ′′ d n x ′′′ P αβµν P λσρδ P θφψχ P κτǫι G αβλσ ( x, x ′ ) G θφκτ ( x ′′ , x ′′′ ) × (cid:2) ∂ µ ∂ ′′ ψ G νχ ( x, x ′′ ) ∂ ′ ρ ∂ ′′′ ǫ G δι ( x ′ , x ′′′ ) + ∂ µ ∂ ′′′ ǫ G νι ( x, x ′′′ ) ∂ ′ ρ ∂ ′′ ψ G δχ ( x ′ , x ′′ ) (cid:3) . (4.49)Evaluating the products of Green functions leads to a result taking the form on the righthand side of (4.41) where A = A and B = B with A = ω κ ξ + 5 ω κ ξ − ω κ ξ + 3 ω − ωκ ξ κ ξ − ω κ ξ , (4.50) B = 7 ω κ ξ − ω κ ξ − ω κ ξ + 21 ω − ωκ ξ κ ξ ω − κ ξ . (4.51)We have kept ω, v and ξ present to demonstrate that individual terms are singular as ξ →
0, and that the results computed using the standard background-field method are gaugecondition dependent. The net result for Γ G follows as (4.41) with A = − A + 12 A − A = − ω − ω
24 + 1128 + 13 κ ξ − ωκ ξ
192 + κ ξ v (cid:18) ω − κ ξ
768 + 23 v ω
128 + κ ξ − ωκ ξ − (cid:19) , (4.52) B = − B + 12 B − B = − ω
12 + ω − − κ ξ
96 + ωκ ξ
48 + 5 κ ξ v (cid:18) − ω
24 + 23 κ ξ − v − ω − κ ξ
24 + ωκ ξ
12 + 164 (cid:19) . (4.53)24s with our earlier calculation, individual contributions to the effective action contain singu-lar terms as ξ →
0; however, when all terms of the same order are combined all such singularbehaviour cancels to leave a finite result as ξ →
0. We again see that if v = 0, the traditionalresult for the effective action is gauge dependent. The correct, gauge condition independentresult can be found from ω = v = 1 and ξ = 0. There is still the ghost contribution toconsider and this is the subject of the next section. V. GHOST CONTRIBUTIONA. Expansion of the effective action
We can evaluate the ghost contribution to the effective action in the same way as we didfor the graviton and gauge fields. From (3.30) we identify the ghost action as S GH = ¯ η α Q αβ η β , (5.1)with Q αβ given by (2.5). It can be noted that Q αβ η β = χ α,i K iβ η β = δχ α , (5.2)where δχ α represents the change in the gauge condition under a gauge transformation withthe infinitesimal gauge parameters δǫ β replaced with the anticommuting ghost field η β . Thebackground fields are held fixed when computing Q αβ .In our case we have the two gauge conditions (4.2) and (4.3). We need a vector ghost η µ ( x ) and its antighost ¯ η µ ( x ) for gravity, and a scalar ghost η ( x ) and its antighost ¯ η ( x ) forelectromagnetism. The ghost action will be S GH = Z d n x (cid:0) ¯ η λ δχ λ + ¯ ηδχ (cid:1) . (5.3)Here, δχ λ and δχ denote the changes in the gauge conditions (4.2) and (4.3) under a gaugetransformation of the metric and electromagnetic field ((3.7) and (3.8)) using (4.5) and (4.6)with the gauge parameters δǫ λ ( x ) → η λ ( x ) and δǫ ( x ) → η ( x ). Furthermore, because we areonly working to one-loop order we can neglect all terms in S GH that involve the quantumfields h µν and a µ . (They would be important at higher loop orders.)The result for S GH can be conveniently expressed as a sum of three terms, S GH = S GH + S GH + S GH , (5.4)25ith the subscript 0 , , S GH = Z d n x (cid:18) − κ ¯ η λ (cid:3) η λ − ¯ η (cid:3) η (cid:19) , (5.5) S GH = Z d n x (cid:2) ω ¯ η λ ¯ F µλ η ,µ + ¯ η (cid:0) ¯ A µ,ν + ¯ A ν,µ (cid:1) η ν,µ +¯ η ¯ A µ,µν η ν + ¯ η ¯ A ν (cid:3) η ν (cid:3) , (5.6) S GH = ω Z d n x ¯ η λ ¯ F µλ (cid:0) − ¯ A µ,ν η ν − ¯ A ν η ν,µ (cid:1) . (5.7)We can again treat the terms that involve the background gauge field S GH and S GH asinteraction terms and in place of (4.18) haveΓ GH = −h e − S GH − S GH − i , (5.8)with the overall minus sign due to the ghost statistics. We have the basic pairing relations h η µ ( x )¯ η ν ( x ′ ) i = ∆ µν ( x, x ′ ) , (5.9) h η ( x )¯ η ( x ′ ) i = ∆( x, x ′ ) , (5.10)where ∆ µν ( x, x ′ ) = κ δ µν ∆( x, x ′ ) , (5.11)and, − (cid:3) ∆( x, x ′ ) = δ ( x, x ′ ) , (5.12)follow from (5.5).We find, up to fourth order in the background gauge fieldΓ GH = −h S GH i + 12 h ( S GH ) i + 12 h ( S GH ) i− h ( S GH ) S GH i + 124 h ( S GH ) i . (5.13)Again, the potential cubic terms in the background gauge field do not contribute becausethey involve odd numbers of ghost fields and vanish by application of the Wick reduction. B. Evaluation of Γ GH It is convenient to use the gauge condition for the electromagnetic field to simplify χ λ in (4.2). We can set the term in ∂ µ a µ in (4.2) to zero as before, and this simplifies the26valuation of the ghost contributions. We have checked this by not making this simplificationand replacing the second term of (4.2) by ω ′ ¯ A λ ∂ µ a µ + ωa µ ¯ F µλ . It can then be shown that ω ′ cancels out of Γ G and therefore may be safely taken to vanish without any loss of generality.We find from S GH in (5.7) that h S GH i = ω Z d n x ¯ F µλ (cid:2) ¯ A µ,ν ∆ νλ ( x, x ) + ¯ A ν ∂ µ ∆ νλ ( x, x ′ ) (cid:12)(cid:12) x ′ = x (cid:3) = 0 , (5.14)since the coincidence limit of the Green functions involve massless propagators that getregularized to zero in dimensional regularization.For h ( S GH ) i we find (cid:10) ( S GH ) (cid:11) = − ω Z d n xd n x ′ n ¯ A µ,ν ( x ′ ) ∂ ′ µ ∆ νλ ( x ′ , x ) + ¯ A µ,µν ( x ′ )∆ νλ ( x ′ , x )+ ¯ A ν ( x ′ ) (cid:3) ′ ∆ νλ ( x ′ , x ) + ¯ A ν,µ ( x ′ ) ∂ ′ µ ∆ νλ ( x ′ , x ) o ¯ F σλ ( x ) ∂ σ ∆( x, x ′ ) , (5.15)after Wick reduction using the pairing relations (5.9–5.11) with the ghosts treated as anti-commuting. Evaluating the products of Green functions as before followed by some integra-tion by parts, results in (cid:10) ( S GH ) (cid:11) = − κ ωL Z d x (cid:0) ∂ µ ¯ F µν (cid:1) . (5.16)Although separate terms of (5.15) are not gauge invariant, the net result is that all termswhen combined lead to a gauge invariant answer. This is as it must be since the formalismguarantees that this is so. We therefore findΓ GH = − κ ωL Z d x (cid:0) ∂ µ ¯ F µν (cid:1) . (5.17)This vanishes for ¯ F µν constant, so the ghosts make no contribution to the charge renor-malization. They do however contribute to the pole part of the effective action for generalbackground fields. C. Evaluation of Γ GH The Wick reduction of h ( S GH ) i results in (cid:10) ( S GH ) (cid:11) = − κ ω Z d n xd n x ′ ¯ F µλ ( x ) ¯ F αν ( x ′ ) n ¯ A µ,ν ( x ) ¯ A α,λ ( x ′ )∆( x, x ′ )∆( x ′ , x )+ ¯ A µ,ν ( x ) ¯ A λ ( x ′ )∆( x, x ′ ) ∂ ′ α ∆( x ′ , x ) + ¯ A ν ( x ) ¯ A α,λ ( x ′ ) ∂ µ ∆( x, x ′ )∆( x ′ , x )+ ¯ A ν ( x ) ¯ A λ ( x ′ ) ∂ µ ∆( x, x ′ ) ∂ ′ α ∆( x ′ , x ) o . (5.18)27he calculations of the ghost contribution starts to become extremely messy if we keep thebackground gauge field ¯ A µ general. However we can simplify things enormously by notingthat the final result must be expressible in terms of the two invariants ( ¯ F ) and ¯ F as wehad earlier in (4.41). Because the result must be invariant under gauge transformations ofthe background field, we may simplify with the choice¯ A µ ( x ) = −
12 ¯ F µν x ν , (5.19)so that ∂ µ ¯ A ν = ¯ F µν and ∂ µ ¯ A µ = 0. In order to implement this it is easiest to assume that¯ F µν is constant, and integrate by parts so that all derivatives act on factors of ¯ A µ . Aftersome work it can be shown that (cid:10) ( S GH ) (cid:11) = − κ ω L Z d x ¯ F . (5.20)For h ( S GH ) S GH i we find, after Wick reduction and use of the pairings (5.9–5.11), (cid:10) ( S GH ) S GH (cid:11) = 12 κ ω Z d n xd n x ′ d n x ′′ ¯ F µν ( x ) ∂ µ ∆( x, x ′ ) Y λ ( x ′ )∆( x ′ , x ′′ ) × ¯ F βλ ( x ′′ ) (cid:2) ∂ ′′ ν ¯ A β ( x ′′ ) + ¯ A ν ( x ′′ ) ∂ ′′ β (cid:3) ∆( x ′′ , x ) , (5.21)where we have defined Y λ ( x ) = ∂ λ ∂ σ ¯ A σ + (cid:0) ∂ λ ¯ A σ + ∂ σ ¯ A λ (cid:1) ∂ σ + ¯ A λ (cid:3) . (5.22)The result in (5.21) can be evaluated as we described above for h ( S GH ) i and the resultturns out to be (cid:10) ( S GH ) S GH (cid:11) = κ ω L Z d x (cid:20)
196 ( ¯ F ) −
148 ¯ F (cid:21) . (5.23)Finally we come to h ( S GH ) i that proved to be the most lengthy to evaluate. The Wickreduction yields (cid:10) ( S GH ) (cid:11) = − κ ω Z d n xd n x ′ d n x ′′ d n x ′′′ ¯ F µν ( x ) ¯ F αβ ( x ′ ) ∂ µ ∆( x, x ′′ ) ∂ ′ α ∆( x ′ , x ′′′ ) × Y β ( x ′′ )∆( x ′′ , x ′ ) Y ν ( x ′′′ )∆( x ′′′ , x ) (5.24)= − κ ω L Z d x (cid:20)
14 ( ¯ F ) + 316 ¯ F (cid:21) , (5.25)with the second equality following after some calculation.We can now form the complete ghost contribution to the effective action that is quarticin the background gauge field from the last three terms of (5.13). The result isΓ GH = − κ ω L Z d x (cid:20)
164 ( ¯ F ) + 596 ¯ F (cid:21) . (5.26)28 I. COMPLETE POLE PART OF THE EFFECTIVE ACTION
The behaviour of the coupling constants in quantum field theory at different energy,or length, scales is governed by the Callan-Symanzik [57, 58], or renormalization groupequations. We will use ‘t Hooft’s [56] approach as it is based on dimensional regularization.We can now combine the results for the gauge and ghost fields found above to obtainthe complete pole part of the effective action that involves terms only in the backgroundelectromagnetic field and deduce the necessary renormalization counterterms. From (4.37)and (5.17) we find the quadratic terms to be given byΓ = − κ α Λ8 π ( n − Z d x ¯ F + κ ¯ β π ( n − Z d x (cid:0) ∂ µ ¯ F µν (cid:1) , (6.1)where α was given in (4.38) and ¯ β = β + ω/ β given by (4.36). We have substitutedfor L from (A.12).The quartic pole part of the effective action follows from (4.41) and (5.26) asΓ = − κ π ( n − Z d x (cid:2) A tot ( ¯ F ) + B tot ¯ F (cid:3) , (6.2)where A tot = A − ω , (6.3) B tot = B − ω , (6.4)with A and B given by (4.52) and (4.53) respectively.We summarize what would be obtained in various popular choices, along with the gaugecondition independent result in Table I. The final row of this table contains the gaugecondition independent result. All of the results, including that which gives rise to therunning value of the charge, are seen to be gauge condition dependent when calculatedusing traditional methods.The renormalization of the background field and charge were given in (3.24–3.27). Usingthis in the bare Maxwell action (3.2) gives S M = 14 Z A Z d x ¯ F . (6.5)Since this must absorb the pole coming from the quadratic part of Γ above we find Z A = 1 + κ α Λ2 π ( n −
4) (6.6)29
ABLE I: This shows the results for α and ¯ β in (6.1) and for A tot and B tot in (6.2) for popularchoices of the parameters. The final row shows the correct gauge condition independent result foundwith v = 1 , ω = 1 and ξ = 0. For all rows other than the final one we take v = 0 corresponding tothe traditional background-field expression and Feynman rules. Choosing ω = 0 is usually calledthe de Donder or harmonic gauge. The choice κ ξ = 1 is usually called the Feynman gauge. α ¯ β A tot B tot ω = ξ = 0 , v = 0 0 -1/12 1/128 -1/32 ω = 0 , κ ξ = 1 , v = 0 0 1/6 31/384 -35/192 ω = 1 , ξ = 0 , v = 0 -3/8 3/4 -5/96 0 ω = κ ξ = 1 , v = 0 3/8 1 1/64 -25/192 v = ω = 1 , ξ = 0 -3/16 27/16 1/1024 -163/768 to one-loop order. The standard ‘t Hooft [56] analysis applied to (3.24), starting from ℓde B /dℓ = 0 results in E dedE = 12 ( n − e + 12 (cid:18) E ddE ln Z A (cid:19) e, (6.7)where we have dropped the subscript ‘ R ’ on the renormalized charge, and used the moreconventional energy scale E rather than the length scale ℓ , with E = 1 /ℓ . Because therenormalized charge cannot contain any pole terms the second term of (6.7) must be finiteas n →
4, and we can identify the renormalization group β -function as β e = lim n → (cid:18) E ddE ln Z A (cid:19) . (6.8)We can write Z A = 1 + Z ( n −
4) + + Z ( n − · · · , (6.9)for some coefficients Z , Z , . . . that will in a general theory depend on e, κ, Λ. (In our casewe have not obtained a dependence on e because we have not coupled the Maxwell field tocharged matter. Our analysis will be general here.) κ and λ will satisfy renormalizationgroup equations of their own; however, the analysis that we have presented is not sufficientto determine this. From the Einstein-Hilbert action (3.3) we can write κ B = ℓ ( n − / ( κ + δκ ) , (6.10)Λ B = Λ + δ Λ , (6.11)30ith the counterterms δκ and δ Λ expressed as a sum of pole terms in (6.9). It can be shown(see [47] for example)
E dκdE = 12 ( n − κ + β κ , (6.12) E d Λ dE = β Λ , (6.13)for renormalization group functions β κ and β Λ .To one-loop order, we find from (6.9) using (6.8,6.12) and (6.13) E ddE ln Z A = 1( n − E ddE Z + · · · = 1( n − (cid:26)(cid:18) E dedE (cid:19) ∂∂e + (cid:18) E dκdE (cid:19) ∂∂κ + (cid:18) E d Λ dE (cid:19) ∂∂ Λ (cid:27) Z + · · · = 12 e ∂∂e Z + 12 κ ∂∂κ Z + · · · (6.14)where in the last line we have dropped terms that vanish as n →
4. Comparison of (6.8)with (6.7) shows that (to one-loop order) β e = 14 e ∂∂e Z + 14 κe ∂∂κ Z . (6.15)The first term is that present in the absence of gravity which arises in standard Minkowskispacetime quantum field theory. The second term is a consequence of quantum gravitycorrections that we will call β grav . Using (6.6) for Z we see that β grav = α π κ e Λ . (6.16)The main calculations presented in this paper show that α = − /
16. (See the final lineof Table I.) This means that β grav has the opposite sign to the cosmological constant Λ.We can conclude that if Λ >
0, as current observations favour [59], then e is a monotonicdecreasing function of E . Thus as E → ∞ , meaning that we look at the high energy (shortdistance) behaviour of the theory, the charge decreases. The quantum gravity correctiontends to make the theory asymptotically free. This is also the conclusion found by Robinsonand Wilczek [9] for Λ = 0, but the scaling behaviour is very different here. Of course if weuse the currently determined values for κ and Λ then the magnitude of β grav is exceptionallysmall, and the observability of the quantum gravity correction to the running charge is highlyunlikely. 31or a more realistic gauge theory, if we assume that the quantum gravity correction isthe same form as that found in the Maxwell case, then the renormalization group equationfor the gauge coupling constant g would be expected to be of the form E dgdE = ag + bg, (6.17)for calculable expressions a and b . a would be the result found in standard Minkowski space-time calculations, and b would be the correction due to quantum gravity. b would dependon κ Λ. Conventionally b = 0 and asymptotic freedom is determined by the sign of a ; a < a > g = g ⋆ away from zero where g ⋆ = − b/a. (6.18)This obviously requires b and a to have opposite signs. If the calculation of the presentpaper applies to matter fields other than Maxwell, it suggests that since b < g ⋆ will exist if a >
0. This corresponds to a theory that in the absence of gravity is notasymptotically free (eg. QED), but becomes so once gravity is quantized.
VII. DISCUSSION AND CONCLUSIONS
We have shown how the presence of a cosmological constant leads to a non-zero resultfor the renormalization group β -function and examined the consequences for the gaugecoupling constant. We have also worked out the pole parts of the effective action thatinvolve higher order curvature terms, including those of the Lee-Wick form. By performingthe calculations in a sufficiently general way we were able to show conclusively that thetraditional background-field result leads to gauge condition dependent results, even thoughthe results are still gauge invariant. One way to ensure that gauge condition independenceis maintained is to use the Vilkovisky-DeWitt formalism, as we did.Notwithstanding our comments concerning the quadratic divergences made in the intro-duction, it is of interest to examine them more fully within the gauge condition independentformalism, and this is currently under investigation. We are also looking at the implicationsfor other matter fields (see also [23]) and will report on this elsewhere [62]. The extension32o higher dimensions with the possible lowering of the energy scale as discussed in [10] forthe Robinson-Wilczek [9] calculation is of interest. It is also of direct interest to see how thegauge condition dependence cancels in a general gauge by inclusion of the term T kij in theconnection, and this is currently under investigation. Acknowledgments
Some of the more tedious calculations in this paper were done using Cadabra [63, 64, 65].I am very grateful to Kasper Peeters for his help in answering my questions about Cadabra.
APPENDIX: EVALUATION OF INTEGRALS
We will give a brief outline of how we may evaluate the products of Green functionsencountered in the calculation of the pole part of the effective action described in the maintext. As an example, we will consider I ( x, x ′ ) = G αβγδ ( x, x ′ ) ∂ µ · · · ∂ µ r G λσρτ ( x, x ′ ) (A.1)where r = 0 , , I ( x, x ′ ) = Z d n p (2 π ) n e ip · ( x − x ′ ) I ( p ) , (A.2)where I ( p ) = Z d n q (2 π ) n ( iq µ ) · · · ( iq µ r ) G λσρτ ( q ) G αβγδ ( p − q ) . (A.3)We can use the momentum space graviton propagator (4.16) for each of the two terms in(A.3). We will end up with momentum integrals that involve factors of q µ in the numeratorand various denominators that involve ( q − λ ) , [( p − q ) − λ ] etc. At this stage the standardprocedure is to introduce Feynman-Schwinger parameters [66] to combine the products offunctions in the dominator into a single term, shift the momentum integration accordingly,compute the momentum integration, and finally evaluate the parameter integration. Thisprocess proves to be extremely complicated as the number of factors in the denominatorincreases when three and four Green functions are present. Although this, or some equivalentprocedure, is necessary for obtaining the finite part of the effective action, a simpler process33ay be used to obtain the pole terms. This is because if we are only after the logarithmicdivergences of the various integrals over momentum q we only require terms in the integrandthat behave like q − for large q . We may therefore expand the momentum integrands inpowers of q − for large q and extract the term that behaves like q − . For example, in (A.3)we use the momentum space expressions for the propagators (4.16) and expand the productof the two Green functions in powers of q − keeping the term of order q − − r (since there are r factors of q in the numerator). All of the resulting integrals are then of the form I µ ··· µ s = Z d n q (2 π ) n q µ · · · q µ s ( q ) n/ s , (A.4)where s = 0 , , , . . . . When the number of factors of q µ in the numerator is odd we regularizethe result to zero since the integrand is an odd function of q . I µ ··· µ s is a symmetric tensor, and we can write I µ ··· µ s = f ( n, s ) δ µ ··· µ s (A.5)for some function f ( n, s ) with δ µ ··· µ s expressible as the sum of products of s Kroneckerdeltas with all possible pairings of indices. For example, δ µ µ µ µ = δ µ µ δ µ µ + δ µ µ δ µ µ + δ µ µ δ µ µ . (A.6)it is easy to see that δ µ µ δ µ ··· µ s = ( n + 2 s − δ µ ··· µ s . (A.7)From (A.4), it is clear that δ µ µ I µ ··· µ s = I µ ··· µ s . (A.8)Using (A.5) and (A.7) shows that( n + 2 s − f ( n, s ) = f ( n, s − . (A.9)This allows us to relate all integrals of the form (A.4) to the basic logarithmically divergentintegral L = Z d n q (2 π ) n q ) n/ . (A.10)If we are interested in the case, n →
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