One-loop divergences of quantum gravity coupled with scalar electrodynamics
aa r X i v : . [ h e p - t h ] D ec One-loop divergences of quantum gravitycoupled with scalar electrodynamics
Hyun Ju Go
Chung-Ang University (Dated: 10 December 2017)In non-supersymmetric covariant quantum gravity theory, for each system of gravity coupledwith single field is one-loop divergent. Since adding other fields or other interactions to each systemgenerates more possible counter-Lagrangian terms, there is room for improvement to restore renor-malizability. In this paper, we consider Einstein-Maxwell fields coupled with electrically chargedscalar which is the simplest model among the systems of gravity coupled with multiple fields hav-ing their own interaction. First, we introduce how to calculate the possible one-loop diagrams inEinstein-SQED system and show that this system is non-renormalizable.
I. INTRODUCTION
The quantum field theory of gravitation have been de-veloped from Feynman’s pioneer works [1]. Feynmanshowed that the self-consistent spin-2 quantum field the-ory is Einstein’s general relativity. Therefore, Einstein-Hilbert action acts as suitable action for the quantumgravity. From this action, it is possible to calculate everytree-level diagrams by elementary methods. Furthermorehe tried to attack one-loop diagrams and suggested ficti-tious quanta for unitarity of S-matrix.After Feynman’s works, Bryce DeWitt developedFeynman’s results [2, 3]. He formulated manifestly co-variant quantum gravity using background field method.From this formulation, tree theorem was proved and thealgorithm for S-matrix calculations containing arbitraryorder radiative corrections was derived. In this algo-rithm, the fictitious quanta for arbitrary order was in-troduced. DeWitt also analyzed non-renormalizability ofquantum gravity by conventional power counting methodand presented tentative proposals for dealing with thissituation.An algorithm for counter-Lagrangian of one-loop dia-gram was introduced by G. ’t Hooft [4] and this algo-rithm extended to include gravitation [5]. Applying thisalgorithm, one-loop divergences of quantum gravity cou-pled with scalar fields, vector fields or Yang-Mills fieldswere proved explicitly [5–7]. For fermionic field, the sit-uation is quite different. Firstly, one can’t use metricfields as gravitational variables. Instead of this, fermionicfield has to interact with vierbein field. Furthermore,t’Hooft algorithm isn’t applicable for this case becauseof the form of Lagrangian. S. Deser and P. van Nieuwen-huizen solved this problem by explicit calculation of thediagrams with eight external fermions and showed thatEinstein-Dirac system is also non-renormalizable [8].In this paper, we consider Einstein-Maxwell fields cou-pled with electrically charged scalar which is the simplestmodel among the systems of gravity coupled with multi-ple fields having their own interaction. First, we calculatethe possible one-loop diagrams in Einstein-SQED systemand show that this system is non-renormalizable.The rest of this paper is organized as follows. In section2, the Lagrangian for one-loop diagrams of the Einstein- Maxwell fields coupled with electrically charged scalaris obtained using background field method. In section3, the Lagrangian for one-loop diagrams is transformedinto more elegant form and Feynman rules for one-loopdiagrams are derived. Finally, in section 4, the non-renormalizability of Einstein-SQED system is showed us-ing already known results and equation of motion.
II. THE BACKGROUND FIELD METHOD FORONE-LOOP DIAGRAMS
We start with gravitational field ¯ g µν , scalar field ¯ ϕ andelectromagnetic potential ¯ A µ . From these variables, theLagrangian for Einstein-Maxwell fields coupled with elec-trically charged scalar is L = − ( − ¯ g ) / ( ¯ R + ( D µ ¯ ϕ ) ∗ ¯ g µν D ν ¯ ϕ + 14 ¯ F µν ¯ F αβ ¯ g µα ¯ g νβ )(II.1)where, √ ¯ g =(det ¯ g µν ) / , ¯ R is the scalar curvature, D µ ¯ ϕ = ∂ µ ¯ ϕ − i ¯ A µ ¯ ϕ and ¯ F µν ≡ ∂ µ ¯ A ν − ∂ ν ¯ A µ .The fields (¯ g µν , ¯ ϕ, ¯ A µ ) are splitted into backgroundfields ( g µν , ˜ ϕ, A µ ) plus quantum fields ( h µν , ϕ, a µ ) to ap-ply background field method. Then the equation of mo-tion and one-loop amplitudes are calculated by expand-ing (II.1) various functions of field variables with respectto quantum fields up to second order. For the scalar cur-vature and field strength tensor, calculation results canbe found in many literatures such as [5, 6]. The Interac-tion Lagangian in Einstein-SQED up to 2nd order is L I ≡ √ ¯ g ¯ g µν D µ ¯ ϕ ∗ D ν ¯ ϕ =(1 + 12 h αα − h αβ h βα + 18 ( h αα ) )( g µν − h µν + h µα h αν )(( ˜ D µ ˜ ϕ ) ∗ + ( ˜ D µ ϕ ) ∗ + ia µ ˜ ϕ ∗ + ia µ ϕ ∗ )(( ˜ D ν ˜ ϕ ) + ( ˜ D ν ϕ ) − ia ν ˜ ϕ − ia ν ϕ ) (II.2)here, we define ˜ D µ = ∂ µ − iA µ to distinct ¯ D µ . By in-cluding the result of scalar curvature and field strengthtensor, the L can be expanded by collecting the termscontaining any two quantum fields as follows, L = ( − g ) / [ −
12 ( D ν h αβ ) P αβρσ ( D ν h ρσ ) + 12 ( h µ − D µ h ) −
12 ( D ν a µ ) + 12 ( D µ a ν )( D ν a µ ) − ( ∂ ν ϕ ) ∗ ∂ ν ϕ − ϕ ∗ A ν A ν ϕ + i∂ µ ϕ ∗ A µ ϕ − i∂ µ ϕA µ ϕ ∗ + 12 h αβ ( X g + X e + X s ) αβρσ h ρσ + h αβ Q αβρσ D ρ a σ − a µ ( g µν ˜ ϕ ∗ ˜ ϕ ) a ν + h αβ B αβρ ∂ ρ ϕ ∗ + ih αβ B αβρ A ρ ϕ ∗ + h αβ C αβρ ∂ ρ ϕ + ih αβ C αβρ A ρ ϕ + ih αβ ( B αβρ − C αβρ ) a ρ − i ( ∂ µ ϕ ) ∗ ( g µν ˜ ϕ ) a ν + i∂ µ ϕ ( g µν ˜ ϕ ∗ ) a ν + ia ν g µν ( ˜ D µ ˜ ϕ − iA µ ˜ ϕ ) ϕ ∗ − ia ν g µν ( ˜ D µ ˜ ϕ ∗ + iA µ ˜ ϕ ∗ ) ϕ ] (II.3)and symbols for gravitational fields in the equation arecalculated from the expansion as listed in below, P αβρσ = 12 g αρ g βσ − g αβ g ρσ (II.4a) X gαβρσ = P αβρσ R − g αρ R βσ + g αβ R ρσ + R αρβσ (II.4b) X eαβρσ = P αβρσ F − F αρ F βσ − g αρ F βσ + 12 g αβ F ρσ (II.4c) X sαβρσ = − g σβ ( ˜ D ρ ˜ ϕ ) ∗ ( ˜ D α ˜ ϕ ) + g αβ ( ˜ D ρ ˜ ϕ ) ∗ ( ˜ D σ ˜ ϕ ) − g αβ g ρσ ( ˜ D ν ˜ ϕ ) ∗ ( ˜ D ν ˜ ϕ ) + 12 g σβ g ρα ( ˜ D ν ˜ ϕ ) ∗ ( ˜ D ν ˜ ϕ )(II.4d)similarly, symbols for gravitational field coupled toMaxwell field or scalar field in the equation are Q αβρσ = 2 g αρ F βσ − g αβ F ρσ (II.5a) B αβρ = − g αβ ˜ D ρ ˜ ϕ + g ρα ˜ D β ˜ ϕ (II.5b) C αβρ = − g αβ ( ˜ D ρ ˜ ϕ ) ∗ + g ρα ( ˜ D β ˜ ϕ ) ∗ (II.5c)where F µν ≡ F νµ ≡ F µα F να and F µµ ≡ F . On the other hand, quadratic part of our Lagrangianis modified to obtain the Feynman rules for S-matrix.First, consider the following gauge transformations, h ′ µν = h µν + ( g µα D ν + g να D µ ) η α + κ [( h µα D ν + h να D µ ) η α + η α D α h µν ] (II.6a) a ′ µ = a µ + η α F αµ + D µ η + κ ( a α D µ η α + η α D α a µ )(II.6b)Our original action is then invariant under these trans-formations, Z d x ′ L (¯ g ′ , ¯ A ′ , ¯ ϕ ′ , ( ¯ ϕ ∗ ) ′ ) = Z d x L (¯ g, ¯ A, ¯ ϕ, ¯ ϕ ∗ ) (II.7)To obtain Feynman rules, it is needed to choose gaugefixing terms − C µ for gravitational fields and vectorfields respectively and include ghost Lagrangian in ourcalculations. From the form of (II.3), one can choose C µ as follows, C a = ( − g ) e µα ( h µ − D µ h ) (II.8a) C = ( − g ) D µ a µ (II.8b)where e µα is a square root of a metric field which is calleda vierbein field. With these gauge fixing terms, we canfinally write quadratic Lagrangian for non-ghost parts : L NG = ( − ¯ g ) / [ −
12 ( D ν h αβ ) P αβρσ ( D ν h ρσ ) −
12 ( D ν a µ ) − ( ∂ ν ϕ ) ∗ ∂ ν ϕ − ϕ ∗ A ν A ν ϕ + i∂ µ ϕ ∗ A µ ϕ − i∂ µ ϕA µ ϕ ∗ + 12 h αβ ( X g + X e + X s ) αβρσ h ρσ + h αβ Q αβρσ D ρ a σ − a µ ( − R µν + g µν ˜ ϕ ∗ ˜ ϕ ) a ν + h αβ B αβρ ∂ ρ ϕ ∗ + ih αβ B αβρ A ρ ϕ ∗ + h αβ C αβρ ∂ ρ ϕ + ih αβ C αβρ A ρ ϕ + ih αβ ( B αβρ − C αβρ ) a ρ − i ( ∂ µ ϕ ) ∗ ( g µν ˜ ϕ ) a ν + i∂ µ ϕ ( g µν ˜ ϕ ∗ ) a ν + ia ν g µν ( ˜ D µ ˜ ϕ − iA µ ˜ ϕ ) ϕ ∗ − ia ν g µν ( ˜ D µ ˜ ϕ ∗ + iA µ ˜ ϕ ∗ ) ϕ ] (II.9)Here, the Ricci identity is used( D α D β − D β D α ) A µ = R µγαβ A γ ( D µ D β − D β D µ ) A µ = − R µβ A µ (II.10)On the other hand, the ghost Lagragian L G can be cal-culated by subjecting C µ to the gauge transformations (II.6a),(II.6b). From (II.8a) and (II.8b), we find L G = ( − g ) ( φ ∗ α , χ ∗ ) (cid:18) e αβ D ν D ν − R αβ − ( D λ F λβ ) − F λβ D λ D ν D ν (cid:19) (cid:18) φ β χ (cid:19) (II.11)where, φ α is a vector ghost and χ is a scalar ghost. III. FEYNMAN RULES
In this section, our Lagrangian is transformed intomore elegant form and Feynman rules are derived. Letus consider the following form of Lagrangian, L = ( − g ) / ( φ ∗ i D µ W µνij D ν φ i + 2 φ ∗ i N µij ∂ µ φ j + φ ∗ i M ij φ j )(III.1)where W µνij = g µν δ ij . In our case, the Lagrangian istransformed according to the following procedure. First,we introduce complex fields h ≡ ( h + ih )2 / and a ≡ ( a + ia )2 / where h , h , a , a are identical with h , a . To fit into the (III.1), integral by parts should beperformed for the terms containing Dφ ∗ as follows, hQ ( Da ) ∗ = − DhQa ∗ − h ( DQ ) a ∗ (III.2a) hA ( Dϕ ) ∗ = − DhAϕ ∗ − h ( DA ) ϕ ∗ (III.2b) a ˜ ϕ ( Dϕ ) ∗ = − Da ˜ ϕϕ ∗ − a ( D ˜ ϕ ) ϕ ∗ (III.2c) ϕA ( Dϕ ) ∗ = − DϕAϕ ∗ − ϕ ( DA ) ϕ ∗ (III.2d)Second, we replace h ∗ αβ P αβρσ → h ∗ ρσ and a ∗ α g αβ → a ∗ β which are not change counter Lagrangian according to lemma in [5]. And finally, double-derivative terms areexpressed in terms of ˜ D which is not work on explicitfield indices as follows, h ∗ αβ D ν D ν h αβ = h ∗ αβ ˜ D ν ˜ D ν h αβ + 2 h ∗ αβ N µρσαβ ˜ D µ h ρσ + h ∗ αβ T ρσαβ h αβ (III.3a) a ∗ α D ν D ν a α = a ∗ α ˜ D ν ˜ D ν a α + 2 a ∗ α n µβα ˜ D µ a β + a ∗ α τ βα a β (III.3b)where N µρσαβ = − g µλ Γ ρλα δ σβ (III.4a) T ρσαβ = ( D µ N µ + N µ N µ ) ρσαβ (III.4b) n µβα = − g µλ Γ ρλα (III.4c) τ βα = ( D µ n µ + n µ n µ ) βα (III.4d)Applying these formula, the Lagrangian in the scalarform is obtained in terms of 10+4+1 independent com-plex fields φ i = ( h µν , a µ , ϕ ) with N µNG = N µρσαβ ( P − Q µ ) δαβ P − C αβµ − g γλ Q ρσµλ n µδγ iδ µγ ˜ ϕ ∗ − B αβµ ig µγ ˜ ϕ − iA µ (III.5a) M NG = P − ( X g + X s + X e ) + T iP − ( B αβρ − C αβρ ) iP − C αβρ A ρ − g γλ ∂ µ Q ρσµλ + ig γλ ( B αβλ − C αβλ ) R δγ − δ δγ ˜ ϕ ∗ ˜ ϕ + τ δγ − i ˜ D ν ˜ ϕ ∗ + A ν ˜ ϕ ∗ − ∂ ρ B αβρ + iA ρ B αβρ i ˜ D ν ˜ ϕ − A ν A ν − i∂ ν A ν (III.5b)Since the ghost Lagrangian already has desired form, the N µG and M G is written directly as follows: N µG = (cid:18) n µβα − F λβ (cid:19) (III.6a) M G = (cid:18) − R βα + τ βα − D λ F βλ (cid:19) (III.6b)Note that the factor ( − g ) / e αβ is absorbed into φ ∗ andalso ˜ D ν is applied as non-ghost case.With these { W µνij , M µij , N ij } the known results are thefollowings [4, 5]. First, if W µνij = δ µν δ ij , it is possibleto regard the propagator as δ ij / (2 π ) i ( k − iǫ ) and theexternal vertices are corresponding to the each element of M µij , N ij . Although W µνij = g µν δ ij as our case, the sameconsideration is established by appropriate subtitutionand when we calculate in terms of M µij , N ij , it should benoted that there are more one-loops coming from thatsubstitution to be considered such astr(( M − D µ N µ − N µ N µ ) R ) , tr( R ) , tr( R µν R µν )Second, the tadpole diagrams with one M µij or N ij are zero and the one-loop diagrams with the product of pos-sible combinations of M µij , N ij , R µν , R by power countingis physically meaningful only in diagonal parts. In sum-mary, all possible one-loop diagrams has the one of thefollwoing forms, ∝ tr( MM ) , ... ∝ tr( MN µ N µ ) , ... ∝ tr( N µ N µ N ν N ν ) , ... FIG. 1. One-loop diagrams in Einstein-SQED system note that the number of the ommited legs attached at ver-tices can be up to 4 and when we calculate exact counter-Lagrangian, the integral parts multiplied with each tracemight contain more invariants such as the tr( ∂N ∂N ).Unlike Einstein-Scalar or Einstein-Maxwell case, the M µij , N ij in Einstein-SQED system contains scalar-photon vertices with subindex ij = 23 or ij = 32. Andvertices in Einstein-Scalar and Einstein-Maxwell systemsare corrected by the amount in corresponding M µij , N ij .For example, the one-loop diagram with graviton-photonvertices is M M ha FIG. 2. One-loop diagram with graviton-photon vertices here, the external double lines represent the fuction of ex-ternal fields. The counter-term corresponding to abovediagram is ∆ L = 18 π ( n −
4) 14 M M (III.7)From the (III.5b) the function of scalar fields B αβρ − C αβρ should be counted in the calculation of counter-term. IV. NON-RENORMALIZABILITY
So far, we have considered how to calculate the one-loops in Einstein-SQED system. In this section, we showthe non-renormalizability of Einstein-SQED system. Thetotal counter-Lagrangian is represented by the sum of allpossible one-loops contribution :∆ L = 1 ǫ ( − g ) / { tr[ 112 Y µν Y µν + 12 X + 160 ( R µν R µν − R )] } (IV.1)where Y µν = ∂ µ N ν − ∂ ν N µ + N µ N ν − N ν N µ (IV.2a) X = M − D µ N µ − N µ N µ − R (IV.2b)Note that the trace is to be taken over the 15 independentfields (10( h µν )+1( ϕ )+4( a µ )) for the non-ghost parts, and5 independent fields (4( φ a )+1( χ )) for the ghost parts. But we use the equation of motion and already knownresult rather than lengthy calculation. The equation ofmotion is obtained by requiring that the action is sta-tionary with respect to variations. Then the equation ofmotion for each the quantum fields h µν , a µ , ϕ are˜ D µ ˜ D µ ˜ ϕ = 0 ˜ D µ ˜ D µ ˜ ϕ ∗ = 0 (IV.3a) D α F αβ = i ( ˜ ϕ ∗ ˜ D µ ˜ ϕ − ˜ ϕ ˜ D µ ˜ ϕ ∗ ) (IV.3b) R µν − g µν R = − T µν (IV.3c)where the energy-momentum tensor is T µν = −
12 (2 ˜ D µ ˜ ϕ ∗ ˜ D ν ˜ ϕ − g µν ˜ D α ˜ ϕ ∗ ˜ D α ˜ ϕ + F µα F αν − g µν F αβ F αβ ) (IV.4)It is also possible to express R µν and R separately bytaking the trace of (IV.3c) as follwing R = − ˜ D α ˜ ϕ ∗ D α ˜ ϕ − F αβ F αβ (IV.5a) R µν = − ˜ D µ ˜ ϕ ∗ D ν ˜ ϕ − F µα F αν (IV.5b)On the other hand, Einstein-Scalar and Einstein-Maxwellsystem are non-renormalizable with αR µν R µν + βR form. Since Einstein-SQED system contains scalar-electromagnetic interaction terms that are not in thesesystems, the candidates that can remove divergent termsare ∂F ∂ϕ and ∂∂ϕF . But these terms can not be trans-formed into the energy-momentum tensor form of Ein-stein equation because the equations are coupled. Hence,the R µν , R can not be eliminated even after applyingthe equation of motions. Therefore the theory of theEinstein-SQED is non-renormalizable. V. CONCLUSIONS
In this paper, the algorithm for one-loop diagrams ofEinstein-SQED is derived in non-supersymmetric covari-ant theory. By adding scalar field to Einstein-Maxwellfields, there are more possible counter-Lagrangian terms,but these do not remove the divergent terms. On theother hand, supersymmetry provides the physical princi-ple to add other fields, namely supersymmetric partners.In that case, there are miraculous cancellations in loopcalculations. [1] R. P. Feynman
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Nuclear Phys. , 62B, 1973, p.444[5] G. ’t Hooft and M. Veltman
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