Restricting loop expansions in gauge theories coupled to matter
aa r X i v : . [ h e p - t h ] F e b On Coupling Matter Fields to Gauge Fields With Restrictions on the Loop Expansion
D. G. C. McKeon ∗ Department of Applied Mathematics, The University of Western Ontario, London, Ontario N6A 5B7, Canada andDepartment of Mathematics and Computer Science,Algoma University, Sault Ste. Marie, Ontario P6A 2G4, Canada
F. T. Brandt, † J. Frenkel, ‡ and S. Martins–Filho § Instituto de F´ısica, Universidade de S˜ao Paulo, S˜ao Paulo, SP 05508-090, Brazil (Dated: February 8, 2021)It has been shown that by using a Lagrange multiplier (LM) field to restrict field configurationsthat contribute to the path integral to those that satisfy the classical equations of motion, Green’sfunctions receive no contributions beyond one-loop order. We examine the consequences of this ingreater detail for a scalar model, for Yang-Mills (YM) theory, and for the Einstein-Hilbert (EH)action. For the scalar model, it appears that renormalization of the couplings needed to eliminatedivergences arising at one-loop order leads to results that do not respect unitarity when the LMfield is present. However, this inconsistency is not present for YM theory; it becomes possible tocompute all renormalization group (RG) functions exactly. For the EH action, restricting radiativecorrections to one-loop order means that divergences can be removed by renormalizing the LM field,so that the model becomes consistent with renormalizability and unitarity. It is shown how theseconclusions can be used to find a model in which scalar fields couple to a gauge field (either ametric or YM field) in a way consistent with renormalizability and unitarity, with the scalar fieldpropagating in a background gauge field at all orders in the loop expansion.
PACS numbers: 11.15.-qKeywords: gauge theories; first order, perturbation theory
I. INTRODUCTION
Normally the perturbative calculation of radiative effects in quantum field theory involves the so-called “loopexpansion” with terms with no loops (the “tree diagrams”) being the classical limit. Some models become inconsistentat one-loop order, but others (the EH-action for gravity [1] and massive YM theory [2, 3]) exhibit problems beginningat two-loop order. It has been shown that by using a LM field to restrict paths being considered in the path integralto those satisfying the classical equation of motion, then the usual one-loop perturbative results are doubled and allhigher loop effects are eliminated. We are thus motivated to study in more detail how the use of a LM field affectsthe properties of quantized scalar field, a quantized YM field and a quantized metric when using the EH action.For an O ( N ) scalar model we will find that upon introducing a LM to eliminate diagrams with more than one-loop, divergences that arise at one-loop order can be absorbed by renormalizing the parameters (mass and coupling)that characterize the tree level action, but leave the parameters appearing at one-loop order unrenormalized. Thisrenormalization procedure does not lead to results consistent with unitarity, so the use of the LM field for scalar“matter” fields is not appropriate.However, for YM theory, this problem does not arise for renormalizing the theory. As a result, the renormalizationconstants and hence renormalization group functions can be computed exactly for both the coupling constant andwave function renormalization [4]. Indeed, since the LM field eliminates diagrams beyond one-loop order, it is evenpossible to use a LM field to render a YM gauge model supplemented by a Proca mass term renormalizable, if notunitary [5]. It is of course well known that massless YM theory and massive YM theory when the mass is generated bythe Higgs mechanism are both renormalizable and unitary to all orders in the loop expansion [6]. There is consequentlyno motivation for employing a LM field in conjunction with the YM field.The EH action for General Relativity, at one-loop order, provided it is not interacting, is renormalizable whenthe equations of motion are satisfied [1]. Beyond one-loop order [7] or in interaction with scalar [1], vector [8] or ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: sergiomartinsfi[email protected] spinor [9] fields, this is not the case. Supergravity [10], higher derivative models [11] , string theory [12] and non-perturbative properties of renormalization group functions [13] all have been invoked to resolve this issue. Here we willdemonstrate that using a LM field to restrict configurations of the metric to solutions of the classical field equationswithout matter fields can be used in a straightforward way to obtain a renormalizable, unitary model of the metriceven when it interacts with “matter” fields. Divergences are eliminated in a somewhat unusual way as they areabsorbed into the LM field.In an appendix we discuss some general features of using a LM field in conjunction with the path integral. We alsoshow in a second appendix how the first class constraints, present in a YM action supplemented by a LM field, canbe used to derive the gauge invariances in this model.
II. THE SCALAR FIELD
We will consider now a scalar field φ a with the classical action S [ φ a ] = Z dx (cid:18)
12 ( ∂ µ φ a ) − m φ a ) − G
4! ( φ a φ a ) (cid:19) . (2.1)It possesses a global O ( N ) symmetry as well as the symmetry φ a → − φ a . The path integral quantization procedureleads to the generating functional Z [ j a ] = Z Dφ a exp i (cid:18) S ( φ a ) + Z dxj a φ a (cid:19) . (2.2)If we employ a background field B a , then by Eqs. (A7, A8) the generating function for one particle irreduciblediagrams is Γ where e i Γ[ B a ] = Z DQ a e i ( S [ B a + Q a )+ j a Q a ] (2.3)where now j a = − δ Γ[ B a ] δB a . (2.4)Divergences that arise in the course of performing a loop expression of Eq. (2.3) can be absorbed into a renormalizationof B a , m and G [14].If now we introduce a LM field λ a to impose the equation of motion for φ a , our generating functional becomes, byEq. (A14) Z [ j a , k a ] = Z Dφ a Dλ a exp i Z dx h
12 ( ∂ µ φ a ) − m φ a ) − G
4! ( φ a φ a ) − λ a (cid:18) ∂ φ a + m φ a + G φ a φ b φ b (cid:19) + j a φ a + k a λ a i . (2.5)In Eq. (2.5) we distinguish m and m as well as G and G as the divergences arising in the one-loop contributionto Z in Eq. (2.5) renormalize m and G , but not m and G [4].To see this, we can first of all do a diagrammatic expansion of Z in Eq. (2.5). Since the terms in the exponentialof Eq. (2.5) that are bilinear in φ a and λ a are −
12 ( φ a , λ a ) (cid:18) ∂ + m ∂ + m ∂ + m (cid:19) (cid:18) φ a λ a (cid:19) (2.6)the propagators for φ a , λ a can be found from ∂ + m ∂ + m ∂ + m − = ∂ + m ∂ + m − ∂ + m ( ∂ + m ) ! . (2.7)So also, we have vertices φ - φ - φ - φ and λ - φ - φ - φ . With these Feynman rules, one cannot construct a Feynman diagram a b = 0 a b k = − i k − m ( k − m ) δ ab a b k = ik − m δ ab ac bd = − iG (cid:16) δ ab δ cd + δ ac δ bd + δ ad δ bc (cid:17) ac bd = − iG (cid:16) δ ab δ cd + δ ac δ bd + δ ad δ bc (cid:17) FIG. 1: Feynman rules of the Eq. (2.5). The solid and wavy lines represent the scalar field φ a and the LM field λ a , respectively.FIG. 2: One-loop diagrams with non-amputated external legs. with more than one loop. The divergent Feynman diagrams at one-loop order are in Fig. 2. The external legs have notbeen amputated. The symmetry factor associated with these diagrams ensures that they are twice the correspondingdiagram coming from Eq. (2.2).If we consider amputated one loop diagrams, as well as tree diagrams, then the two- and four-point functions areof the form (with ǫ = 2 − n/ n -dimensions [14]). h φ a φ b i = Z dx (cid:20) − φ a (cid:0) ∂ + m (cid:1) φ a − G m φ a φ a (cid:18) ¯ cǫ + c ln m µ + c (cid:19)(cid:21) (2.8) h φ a φ b φ c φ c i = Z dx (cid:20) − G
4! ( φ a φ a ) − G
4! ( φ a φ a ) (cid:18) ¯ dǫ + d ln m µ + d (cid:19)(cid:21) (2.9)so that as ǫ →
0, all divergences are removed by renormalizing m and G m R = m + G ¯ cm ǫ (2.10) G R = G + G ¯ dǫ . (2.11)There is no need to renormalize the field φ a [14].These results can also be obtained by explicitly performing the functional integrals in Eq. (2.5). By use of Eq.(A17) we find that Z [ j a , k a ] = X ¯ φ a exp i Z dx (cid:20) −
12 ¯ φ a (cid:0) ∂ + m (cid:1) ¯ φ a − G (cid:0) ¯ φ a ¯ φ a (cid:1) + j a ¯ φ a (cid:21) (2.12)det − (cid:0)(cid:0) ∂ + m (cid:1) δ ab + G (cid:0) δ ab ¯ φ c ¯ φ c + 2 ¯ φ a ¯ φ b (cid:1)(cid:1) where ¯ φ a satisfies (cid:0) ∂ + m (cid:1) ¯ φ a + G
3! ¯ φ a ¯ φ b ¯ φ b = k a . (2.13)A perturbative expansion of ¯ φ a in powers of G that follows from Eq. (2.13) has a diagrammatic form given in fig.3 where lines represent ( ∂ + m ) − , −−× denotes a factor of k a and >< is a vertex associated with the coupling G . + · · · + + + + FIG. 3: Diagrammatic form of the perturbative solution of Eq. (2.13).
The exponential in Eq. (2.12) represents the sum of all tree-level Feynman diagrams and the functional determinantis the square of the contribution coming from one-loop Feynman diagrams such as those of fig. 2.The elimination of divergences through Eqs. (2.10) and (2.11) results in having to use distinct masses and couplingsfor tree-level and one-loop diagrams, as given in the Feynman rules of Fig. 1. This situation results in it not beingpossible to compute an S -matrix using the generating functional of Eq. (2.5) in a way consistent with unitarity [15].As a result, a Lagrange multiplier cannot be used to consistently eliminate diagrams beyond one-loop order whenconsidering the scalar model of Eq. (2.1).We now will turn our attention to using a LM field in conjunction with YM theory. III. YANG-MILLS THEORY
The second order YM action S Y M [ A ] = Z dx (cid:18) − f aµν ( A ) f aµν ( A ) (cid:19) (3.1)where f aµν ( A ) = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν (3.2)possesses the local gauge invariance δA aµ = D abµ ( A ) ξ b (3.3) (cid:0) D abµ ( A ) ≡ ∂ µ δ ab + gf apb A pµ (cid:1) . (3.4)By introducing an auxiliary field F aµν so that we have the first order YM action S Y M [ A, F ] = Z dx (cid:18) − F aµν f aµν ( A ) + 14 F aµν F aµν (cid:19) (3.5)the interaction vertices for S Y M are simplified [16–18]We now consider the generating functional Z Y M (cid:2) j aµ , J aµν (cid:3) = Z DA aµ DF aµν D ¯ c a Dc a exp i Z (cid:16) − F aµν f aµν + 14 F aµν F aµν + j aµ A aµ + J aµν F aµν − α ( ∂ · A a ) + ¯ c a ∂ µ D abµ c b (cid:17) (3.6)where we have employed the gauge fixing ∂ · A a = 0 (3.7)and have the ghost Lagrangian L ghost = ¯ c a ∂ · D ab ( A ) c b (3.8)which is the usual Faddeev-Popov ghost Lagrangian [19].A background field B aµ will be introduced for A aµ , and G aµν for F aµν , following Eqs. (A2) to (A8). If W Y M (cid:2) j aµ , J aµν (cid:3) = − i ln Z Y M (cid:2) j aµ , J aµν (cid:3) (3.9)and Γ Y M (cid:2) A aµ , G aµν (cid:3) = W Y M (cid:0) j aµ , J aµν (cid:1) − Z dx (cid:0) j aµ A aµ + J aµ F aµν (cid:1) (3.10)where B aµ = δW Y M δj aµ (3.11) G aµν = δW Y M δJ aµν (3.12)then the one-particle irreducible Feynman diagrams are generated byexp i Γ Y M (cid:2) B aµ , G aµν (cid:3) = Z Dq aµ DQ aµν D ¯ c a Dc a exp i Z dx h −
12 ( G + Q ) aµν f aµν (cid:0) B aµ + q aµ (cid:1) + 14 (cid:0) G aµν + Q aµν (cid:1) ( G aµν + Q aµν )+ j aµ q aµ + J aµν Q aµν − α (cid:0) D abµ ( B ) q bµ (cid:1) + ¯ c a D abµ ( B ) D bcµ ( B + q ) c a i . (3.13)Following refs. [20], we have used the gauge fixing D abµ ( B ) q bµ = 0 (3.14)in place of Eq. (3.7).We now will introduce LM fields to restrict radiative corrections to one-loop order. If we start from the action S Y M of Eq. (3.1), then this means considering S λ Y M [ A, λ ] = Z dx (cid:18) − (cid:0) f aµν ( A ) (cid:1) + λ aν D abµ ( A ) f bµν ( A ) (cid:19) . (3.15)By Eqs. (A32) and (A34), the gauge invariance of Eq. (3.3) is now accompanied by δλ aµ = gf abc λ bµ ξ c (3.16)as well as δλ aµ = D abµ ( A ) ζ b . (3.17)We now consider the generating functional that is derived following Eq. (A46) Z λ Y M = Z DA aµ Dλ aµ Z DN a DL a Z Dc a D ¯ c a Dd a D ¯ d a exp i Z dx h − f aµν ( A ) f aµν ( A ) + λ aν D abµ ( A ) f bµν ( A ) + j aµ A aµ + k aµ λ aµ (3.18)+ ¯ c a ∂ · D ab ( A ) d b + ¯ d a ∂ · D ab ( A ) c b + ¯ c a ∂ · D ab ( A + λ ) c b + (cid:16) α N a N a − N a ∂ · ( A a + λ a ) + αN a L a − L a ∂ · A a (cid:17) i if we accompany Eq. (3.7) with the gauge condition ∂ · λ a = 0 . (3.19)We now introduce background fields B aµ and Λ aµ for A aµ and λ aµ respectively so that A aµ = B aµ + Q aµ and λ aµ = Λ aµ + q aµ .If Γ[ B aµ, Λ aµ ] is the one-particle irreducible generating functional then e i Γ λ Y M [ B, Λ] = Z DQ aµ, Dq aµ det (cid:18) D abµ ( B ) D bcµ ( B + Q ) D ab ( B ) D bc ( B + Q ) D abµ ( B ) D bcµ ( B + Q + Λ + q ) (cid:19) exp i Z dx h − f aµν ( B + Q ) f aµν ( B + Q ) + (cid:0) Λ aµ + q aµ (cid:1) (cid:0) D abν ( B + Q ) f bµν ( B + Q ) (cid:1) − α (cid:0) D abµ ( B ) Q bµ (cid:1) − α (cid:0) D abµ ( B ) Q bµ (cid:1) ( D acν ( B ) q cν ) + j aµ Q aµ + k aµ q aµ i (3.20)with the gauge conditions D abµ ( B ) Q bµ = 0 = D abµ ( B ) q bµ . (3.21)With this gauge fixing we have maintained the “background gauge invariance” δB aµ = D abµ ( B ) ξ b (3.22a) δQ aµ = gf abc Q bµ ξ c (3.22b) δ Λ aµ = gf abc Λ aµ ξ c (3.22c) δq aµ = gf abc q aµ ξ c (3.22d) δ Λ aµ = D abµ ( B ) ζ b (3.22e) δq aµ = gf abc Q bµ ζ c (3.22f)(3.22g)but have broken the gauge invariances δB aµ = 0 (3.23a) δQ aµ = D abµ ( B + Q ) ξ b (3.23b) δ Λ aµ = 0 (3.23c) δq aµ = gf abc (cid:0) Λ bµ + q bµ (cid:1) ξ c (3.23d) δq aµ = D abµ ( B + Q ) ζ b (3.23e)(3.23f)that were present in S λ Y M ( B + Q, Λ + q ) of Eq. (3.15).One could find the Feynman rules associated with Eq. (3.20) and perform a diagrammatic expansion of Γ λ Y M .However, it is possible perform all the functional integrals that occur, and following the steps that lead to Eq. (A26)we obtain (since det (cid:18) AA B (cid:19) = det (cid:18) AA A + B (cid:19) = det A ) e i Γ λ Y M [ B aµ , Λ aµ ] = X ¯ Q aµ exp i Z dx h − f aµν ( B + ¯ Q ) f aµν ( B + ¯ Q ) (3.24) − α (cid:0) D ab ( B ) · ¯ Q b (cid:1) + Λ aµ (cid:0) D abν ( B + ¯ Q ) f bµν ( B + ¯ Q ) (cid:1) + j aµ ¯ Q aµ i det (cid:0) D abµ ( B ) D bcµ ( B + ¯ Q ) (cid:1) × det − n (cid:0) D ab ( B ) η νµ + D ba ( B ) η νµ (cid:1) − (cid:18) − α (cid:19) (cid:0) D apµ ( B ) D pbν ( B ) + D bpν ( B ) D paµ ( B ) (cid:1) + (cid:0) f apb f pµν ( B ) + f bpa f pνµ ( B ) (cid:1) + 12 h D apλ ( B ) (cid:0) gf pqb ¯ Q qλ (cid:1) η µν + D bpλ ( B ) (cid:0) gf pqa ¯ Q qλ (cid:1) η νµ i + 12 (cid:2) D apν ( B ) (cid:0) gf pbq ¯ Q qµ (cid:1) + D bpµ ( B ) (cid:0) gf paq ¯ Q qν (cid:1)(cid:3) − (cid:2)(cid:0) D pqµ ( B ) ¯ Q qν (cid:1) (cid:0) gf pab (cid:1) + (cid:0) D pqν ( B ) ¯ Q qµ (cid:1) (cid:0) gf pba (cid:1)(cid:3) + 12 (cid:2) D bpµ ( B ) (cid:0) gf par ¯ Q rν (cid:1) + D apν ( B ) (cid:0) gf pbr ¯ Q rµ (cid:1)(cid:3) − (cid:2)(cid:0) D pqν ( B ) ¯ Q qµ (cid:1) (cid:0) gf pba (cid:1) + (cid:0) D pqµ ( B ) ¯ Q qν (cid:1) (cid:0) gf pab (cid:1)(cid:3) + 12 h η µν D bqλ ( B ) (cid:0) gf pqa ¯ Q qλ (cid:1) + η νµ D aqλ ( B ) (cid:0) gf pqb ¯ Q qλ (cid:1)i − g h f mab f mrs ¯ Q rµ ¯ Q sν + f mba f mrs ¯ Q rν ¯ Q sµ + f maq f mbs ¯ Q qλ η µν ¯ Q sλ + f mbq f mas ¯ Q qλ η νµ ¯ Q sλ + f maq f msb ¯ Q qν ¯ Q sµ + f mbq f msa ¯ Q qµ ¯ Q sν io where ¯ Q aµ satisfies D ab ( B ) ¯ Q bµ − (cid:18) − α (cid:19) D apµ ( B ) D pbλ ( B ) ¯ Q bλ + 2 gf abc f bµν ( B ) ¯ Q cν + D abλ ( B ) (cid:0) gf bpq ¯ Q pλ ¯ Q qµ (cid:1) − g (cid:0) D pqµ ( B ) ¯ Q qλ (cid:1) (cid:0) f paq ¯ Q qλ (cid:1) − g (cid:0) D pqλ ( B ) ¯ Q qµ (cid:1) (cid:0) f pqa ¯ Q qλ (cid:1) − g f maq f mrs ¯ Q qλ ¯ Q rµ ¯ Q sλ = k aµ . (3.25)If k aµ = ¯ Q aµ = 0, then Eq. (3.24) reduces to e i Γ λ Y M [ B, Λ] = exp i Z dx (cid:20) − f aµν ( B ) f aµν ( B ) + Λ aµ D abν ( B ) f bµν ( B ) (cid:21) det (cid:0) D abµ ( B ) D bcµ ( B ) (cid:1) det − (cid:18) D ab ( B ) η µν − (cid:18) − α (cid:19) D apµ ( B ) D pbν ( B ) + 2 gf apb f pµν ( B ) (cid:19) . (3.26)The exponential in Eq. (3.26) is the sum of all tree diagrams and the determinants are the sum of all one-loop diagramscontributing to Γ λ Y M ; no higher loop contributions occur. As noted in the appendix, the equation of motion for thebackground field (Eq. (A12)) does not have to be satisfied.We now consider renormalization of g and B aµ needed to remove divergences arising in the perturbative evaluationof the functional determinants appearing in Eq. (3.26) [4]. Maintaining the gauge invariance of Eq. (3.22) ensuresthat the product gB aµ is invariant under renormalization, so that the renormalized coupling g R and renormalized field B aRµ satisfy [20] g R B aRµ = gB aµ . (3.27)Consequently, if the two-and three-point functions determine the renormalization of g and B aµ , we use the result that h BB i = 12 B aµ (cid:0) p η µν − p µ p ν /p (cid:1) B aν (cid:20) g (cid:18) ¯ Dǫ + D ln p µ + D (cid:19)(cid:21) (3.28)and h BBB i = f abc B aµ B bν B cλ V µνλ (cid:20) g + g (cid:18) ¯ Eǫ + E ln p µ + E (cid:19)(cid:21) . (3.29)One does not have separate couplings and fields at tree and one loop order, as with scalar fields in Eq. (2.5), in orderto eliminate divergences occurring in Eqs. (3.28, 3.29) as ǫ →
0. One defines in this case B aR µ B bRν = B aµ B bν (cid:18) g ¯ Dǫ (cid:19) (3.30) B aR µ B bR ν B cR λ g R = B aµ B bν B cλ (cid:18) g + ¯ g ¯ Eǫ (cid:19) . (3.31)By Eq. (3.30) B aR µ = B aµ (cid:18) g ¯ Dǫ (cid:19) / (3.32)and so by Eq. (3.31) g R = (cid:18) g + ¯ g ¯ Eǫ (cid:19) (cid:18) g ¯ Dǫ (cid:19) − / . (3.33)In order that Eq. (3.27) is satisfied, we must have ¯ D = ¯ E, (3.34)and so by Eq. (3.33) g R = g (cid:18) g ¯ Eǫ (cid:19) − / . (3.35)Eqs. (3.28) and (3.29) now become h BB i = 12 B aR µ (cid:0) p η µν − p µ p ν /p (cid:1) B aR ν (cid:20) g R (cid:18) D ln p µ + D (cid:19)(cid:21) (3.36) h BBB i = f abc B aR µ B bR ν B cR λ V µνλ (cid:20) g R + g R (cid:18) E ln p µ + E (cid:19)(cid:21) . (3.37)Dimensionally, g = g µ ǫ (3.38)where µ is an arbitrary dimensionful constant so Eq. (3.35) implies that µ ∂g R ∂ µ = ǫg R − ¯ Eg R . (3.39)As ǫ →
0, we arrive at the exact results for the β function associated with g , β ( g R ) = − ¯ Eg R (3.40)which is twice the usual one-loop result for the β -function in YM theory [21]. Since gB aµ is independent of µ , Eqs.(3.27) and (3.40) show that µ ∂B aRµ ∂µ = ¯ Eg R B aRµ . (3.41)Having established how YM theory with a LM field is renormalized, we now consider how a matter field, in theform of a scalar field φ a , is coupled to A aµ . This involves supplementing S Y M in Eq. (3.1) with S Aφ = Z dx (cid:20) (cid:0) D abµ ( A ) φ b (cid:1) − m φ a φ a − G
4! ( φ a φ a ) (cid:21) (3.42)where D abµ ( A ) is defined in Eq. (3.4). If now we were to make use of LM fields to eliminate diagrams beyond one-looporder, we would be using the classical action S λσAφ = Z dx h − f aµν ( A ) f aµν ( A ) + 12 (cid:0) D abµ ( A ) φ b (cid:1) − m φ a φ a − G
4! ( φ a φ a ) + λ aν (cid:0) D abµ ( A ) f bµν ( A ) + gf apq φ p D qrν ( A ) φ r (cid:1) − σ a (cid:18) D abµ ( A ) D bcµ ( A ) φ c + m φ a + G φ a φ b φ b (cid:19) i . (3.43)This action is invariant under the gauge transformations of Eqs. (3.3), (3.16) and (3.17) along with (using Eqs. (A32)and (A34) δφ a = gf abc φ b ξ c (3.44) δσ a = gf abc σ b ξ c (3.45)and δσ a = gf abc φ b ζ c . (3.46)As in Eq. (2.5), renormalization of divergences arising at one-loop order in radiative effects involving scalars makesit necessary to have distinct masses and couplings at one-loop order ( m and G ) and two-loop order ( m and G ).This again leads to results inconsistent with unitarity, and so all terms in Eq. (3.43) proportional to σ a are to bediscarded.However, eliminating the term σ a D abµ ( A ) D bcµ ( A ) φ c (3.47)from Eq. (3.43) breaks the invariance of Eq. (3.17) unless the term λ aν ( gf apq φ p D qrν ( A ) φ r ) (3.48)is also removed. We are then left with the action S λAφ = Z dx h − f aµν ( A ) f aµν ( A ) + 12 (cid:0) D abµ ( A ) φ b (cid:1) − m φ a φ a − G
4! ( φ a φ a ) + λ aν D abµ ( A ) f bµν ( A ) i . (3.49)This is invariant under the gauge transformations of Eqs. (3.3, 3.16, 3.17, 3.44).The generating functional is, by Eq. (A46), Z λAφ (cid:2) j aµ , k aµ , J a (cid:3) = Z Dφ a DA aµ Dλ aµ Z Dc a D ¯ c a Dd a D ¯ d a exp i h S λAφ + Z dx (cid:16) ¯ c a ∂ · D ab ( A + λ ) c b + ¯ d a ∂ · D ab ( A ) c b + ¯ c a ∂ · D ab ( A ) d b − α ( ∂ · A a ) − α ∂ · A a ∂ · λ a + j aµ A aµ + k aµ λ aµ + J a φ a (cid:17)i (3.50)if we use the gauge fixing of Eqs. (3.7) and (3.19).The fields φ a , A aµ and λ aµ are expanded about backgrounds Φ a , B aµ , Λ aµ so that φ a = Φ a + ψ a (3.51) A aµ = B aµ + Q aµ (3.52) λ aµ = Λ aµ + q aµ . (3.53)Using the same steps used to arrive at Eq. (3.26) when there is no scalar “matter” field φ a , we find that e i Γ λAφ [ B, Λ , Φ] = Z Dψ a exp i Z dx h − f aµν ( B ) f aµν ( B ) + Λ aµ D abν ( B ) f bµν ( B )+ 12 (cid:0) D abµ ( B ) (cid:0) Φ b + ψ b (cid:1)(cid:1) − m a + ψ a ) (Φ a + ψ a ) − G
4! ((Φ a + ψ a ) (Φ a + ψ a )) i det (cid:0) D abµ ( B ) D bcµ ( B ) (cid:1) det − (cid:16) D ab ( B ) η µν − (1 − α ) D apµ ( B ) D pbν ( B ) + 2 gf apb f pµν ( B ) (cid:17) . (3.54)From Eq. (3.54) it follows that a perturbative expansion of Γ λAφ [ B, Λ , Φ] has the following contributions:01. loops involving scalar fields φ a propagating in the presence of a background scalar field Φ a and a backgroundvector gauge field B aµ
2. all tree level diagrams involving the vector gauge field, given by the exponentialexp i R dx (cid:0) − f aµν ( B ) f aµν ( B ) (cid:1) [22].3. twice the contribution of all one-loop diagrams arising in normal YM theory when a background field is used,coming from the functional determinants in Eq. (3.54).4. no higher loop contributions involving the propagation of the gauge field.As a result of this, the renormalization group functions for g R and B aRµ in Eqs. (3.40,3.41) can receive loopcontributions of higher order in G R . So also, the usual renormalization group functions for G R , m A and φ R areunaltered.We now will consider how the arguments leading to Eq. (3.54) can be used when there is a gravitational fieldinteracting with a scalar field. IV. GRAVITY
At one-loop order, the EH action is renormalizable as divergences vanish when the metric g µν satisfies the classicalequations of motion [1]. However, at higher loop order [7], or if the metric couples to matter fields [1, 8, 9], thenrenormalizability is lost. This has made it interesting to examine the consequences of using a LM field to limitradiative corrections to the EH action to one-loop order [23]. In this section we will further consider use of a LM fieldin conjunction with the EH action supplemented by matter fields which couple to the metric.The first-order form of the EH action is useful, as in this form, the EH action has only a three-point vertex [24].Here however we will use the second order form of the EH action, treating the metric g µν as a gauge field. This actionis S EH = Z dx (cid:18) − κ √− g g µν R µν (Γ) (cid:19) (4.1)where κ = 16 πG N , g = det g µν , Γ λµν = g λρ ( g µρ,ν + g νρ,µ − g µν,ρ ) and R µν = − (cid:16) Γ λµν,λ − Γ λµλ,ν + Γ λµν Γ σλσ − Γ σµλ Γ λνσ (cid:17) . The metric is coupled to a scalar matter field φ with the action S gφ = Z dx (cid:0) √− g (cid:1) (cid:18) − g µν ∂ µ φ∂ ν φ − m φ − G φ (cid:19) . (4.2)Due to diffeomorphism invariance, S EH + S gφ is invariant under the infinitesimal transformation δg µν = g µα ξ α,ν + g να ξ α,µ + ξ α g µν,α (4.3) δφ = ξ α φ ,α . (4.4)The nature of the EH action makes it impossible to perform any perturbative expansion of the generating functionalwithout using a background field ¯ g µν for the metric g µν, so that [25] g µν = ¯ g µν + κh µν . (4.5)Quite often, this background field is chosen to be flat¯ g µν = η µν . (4.6)An expansion of the geometric quantities found in Eqs. (4.1) and (4.2) in powers of h µν is found in ref. [1].If V α ; ¯ β denotes a covariant derivate of V α using the background field ¯ g µν , then the gauge transformation of Eq.(4.3) can be written as δ ¯ g µν = 0 (4.7)1 δh µν = 1 κ ( ξ µ ;¯ ν + ξ ν ;¯ µ ) + ξ λ h µν ;¯ λ + h µλ ξ λ ;¯ ν + h νλ ξ λ ;¯ µ . (4.8)Indices are raised and lowered using ¯ g µν .Upon varying g µν in S EH + S gφ , we find that δ ( S EH + S gφ ) = Z dx √− gδg µν (cid:20) +1 κ (cid:18) R µν − g µν R (cid:19) + 12 T µν (cid:21) (4.9)where T µν = g µα g νβ φ ,α φ ,β − g µν (cid:18) g αβ φ ,α φ ,β + m φ + G φ (cid:19) . (4.10)If a LM field λ µν is used to impose the equation of motion for g µν that follows from S EH , we have the action S λEH = 1 κ Z dx √− g (cid:20) − g µν R µν + λ µν (cid:18) R µν − g µν R (cid:19)(cid:21) . (4.11)It now is possible to follow the steps which led to the generating functional Γ λ Y M in Eq. (3.20). If we havebackground fields ¯ g µν and ¯ λ µν for g µν and λ µν so that g µν = ¯ g µν + κh µν (4.12) λ µν = ¯ λ µν + κσ µν (4.13)then S λEH in Eq. (4.10) is invariant under the gauge transformation of Eq. (4.3) combined with δλ µν = λ µα ξ α,ν + λ να ξ α,µ + ξ α λ µν,α (4.14)and δg µν = (cid:0) g µα ζ α,ν + g να ζ α,µ + ζ α g µν,α (cid:1) (4.15)as can be seen from Eqs. (A32) and (A34). Eq. (4.14) follows from the fact that λ µν is a tensor under a diffeomorphismtransformation while Eq. (4.15) follows from the fact the S EH in Eq. (4.1) is invariant under a diffeomorphismtransformation.Just as Eqs. (4.7) and (4.8) follow from Eq. (4.3), we see that Eqs. (4.14) and (4.15) result in δ ¯ λ µν = 0 (4.16) δσ µν = (cid:18) κ ¯ λ µα + σ µα (cid:19) ξ α ;¯ ν + (cid:18) κ ¯ λ να + σ να (cid:19) ξ α ;¯ ν + ξ α (cid:18) κ ¯ λ µν + σ µν (cid:19) ; α . (4.17) δσ µν = 1 κ ( ζ µ ;¯ ν + ζ ν ;¯ µ ) + ζ λ h µν ;¯ λ + h µλ ζ λ ;¯ ν + h νλ ζ λ ;¯ µ . (4.18)It is now possible to break the invariance of Eqs. (4.7, 4.8, 4.16, 4.17, 4.18) by the gauge fixing conditions h ¯ νµν ; − kh νν ;¯ µ = 0 = σ ¯ νµν ; − kσ νν ;¯ µ . (4.19)We now can follow the steps used to find the generating functional Γ λgφ (cid:2) ¯ λ, ¯ g, ¯ φ (cid:3) for one-particle irreducible graphswhen a scalar φ = Φ + ψ with background Φ is in the presence of background ¯ g µν and ¯ λ µν . In analogy with Eq.(3.54), this leads toexp i Γ λgφ (cid:2) ¯ λ, ¯ g, Φ (cid:3) = Z Dψ Z D ¯ c µ Dc µ D ¯ d µ Dd µ exp i Z dx √− ¯ g n κ h − ¯ g µν R µν (¯ g ) + ¯ λ µν (cid:16) R µν (¯ g ) −
12 ¯ g µν R (¯ g ) (cid:17)i + h ¯ c µ (cid:0) d ;¯ νµ ;¯ ν − d ¯ νν ; ;¯ µ (cid:1) + ¯ d µ (cid:0) c ;¯ νµ ;¯ ν − c ¯ νν ; ;¯ µ (cid:1) i + h −
12 ¯ g µν ∂ µ (Φ + ψ ) ∂ ν (Φ + ψ ) − m ψ ) − G
4! (Φ + ψ ) io det − n δ δh πτ δh γδ h − g µν R µν ( g ) − α (cid:0) h ¯ αµα ; − kh αα ;¯ µ (cid:1)(cid:16) h µ ¯ ββ ; − kh β ¯ µβ ; (cid:17) i h =0 o . (4.20)2We find from Eq. (4.20) that a perturbative expansion of Γ λgφ [¯ g, ¯ λ, Φ] leads to a structure much like that which followsfrom Eq. (3.54) for YM theory. We find that the following diagrams contribute to Γ λgφ :1. all tree diagrams with external metric ¯ g µν
2. twice all one-loop diagrams that follow from the EH action alone, but no diagrams beyond one-loop order3. all loop diagrams for the scalar Φ in the presence of a background metric ¯ g µν .Since the one loop diagrams involving a loop of metric, scalar or vector fields involves a divergence proportionalto R µν (¯ g ) [1, 8] we see that such one-loop divergences can be absorbed by ¯ λ µν [23]. The divergences in diagramsinvolving no external metric field can be eliminated by renormalizing m , G and Φ. A one-loop diagram involvingthe scalar field in a background metric field ¯ g µν may receive radiative corrections proportional to G N . We assumethat divergences in such higher-loop diagrams can be removed by renormalizing ¯ λ µν , m , G and Φ, as summing thesediagrams leads to using a propagator that follows from −√− ¯ g (cid:0) ¯ g µν φ ;¯ µ φ ;¯ ν − m φ (cid:1) and a vertex that follows from − G √− ¯ gφ , with the metric not altering the short distance behaviour of divergent diagrams involving φ . V. DISCUSSION
We have shown that by use of a LM field all radiative corrections beyond one loop order can be eliminated. Althoughthis necessitates using a renormalization procedure that is inconsistent with unitarity in the case of scalar fields, itis possible to use LM fields to restrict radiative corrections to the YM and EH actions to one-loop order and obtainresults that, after using renormalization to remove divergences, are consistent with unitarity. Furthermore, a scalarfield without a LM field can be coupled in a gauge invariant way to a vector or metric gauge field which has anassociated LM field.For a pure YM gauge theory, having a LM field results in it being possible to compute the renormalization groupfunctions exactly. Though this is interesting, one can in fact compute with a YM action without a LM field, as it isboth unitary and renormalizable to all orders in the loop expansion.However, radiative corrections to the EH action alone beyond one-loop order lead to divergences that cannotbe removed through renormalization. It is of interest then to see that a LM field can be used to eliminate thosediagrams that result in these higher-loop divergences. For some time, quantized matter fields have been consideredpropagating on a curved background whose dynamics is determined by the EH action, possibly supplemented byone-loop corrections [26]; it may even be possible to resolve the Hawking information paradox for black holes usingthis approach [27]. We have shown that when one uses a LM field, it is unnecessary to invoke additional fields tocancel higher-loop divergences, as is done for example in supergravity.
Acknowledgments
Discussions with Roger Macleod were quite helpful. F. T. B. and J. F. thank CNPq (Brazil) for financial support.S. M.–F. thanks CAPES (Brazil) for financial support. This study was financed in part by the Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001. This work comes as an aftermathof an original project developed with the support of FAPESP (Brazil), grant number 2018/01073-5.
Appendix A: General Formalism for Quantization
In this appendix, some general features of how a Lagrange multiplier field can be used to eliminate higher loopcontributions to Green’s functions are presented [4, 23].In general, a Lagrangian L ( φ i ) defines the dynamics of a field φ i ( x ). On occasion it is advantageous to introducean auxiliary field Φ j so that the Lagrangian for the system becomes L ( φ i , Φ j ). The equation of motion for Φ i leadsto Φ j = Φ j ( φ i ) so that L ( φ i ) = L ( φ i , Φ j ( φ i )) . (A1)The generating functional Z [ j i ] = Z Dφ i exp i ~ Z dx ( L ( φ i ) + j i φ i ) (A2)3has a perturbative expansion that gives rise to multi-loop Feynman diagrams. Only connected diagrams occur in theexpansion of W [ j i ] = − i ~ ln Z [ j i ] (A3)while if B i = δW [ j i ] δj i (A4)then the Legendre transform of W [ j i ] Γ[ B i ] = W [ j i ] − Z dxB i j i (A5)gives rise to one particle irreducible Feynman diagrams [20, 28].Together, Eqs. (A3-A5) lead to e i ~ Γ[ B i ] = Z Dφ i exp ik Z dx ( L ( φ i ) + j i ( φ i − B i ))or, if φ i = B i + Q i (A6) e i ~ Γ[ B i ] = Z DQ i exp i ~ Z dx ( L ( B i + Q i ) + j i Q i ) , (A7)where B i is a “background field” while Q i is a “quantum field”. From Eqs. (A4) and (A5), it follows that j i = − δ Γ[ B i ] δB i . (A8)One can now make the expansions Γ[ B i ] = Γ (0) [ B i ] + ~ Γ (1) [ B i ] + ~ Γ (2) [ B i ] + · · · (A9)and L ( B i + Q i ) = L ( B i ) + 11! L ,i ( B i ) Q i + 12! L ,ij ( B i ) Q i Q j + 13! L ,ijk ( B i ) Q i Q j Q k + . . . . (A10)When Eqs. (A8 - A10) are substituted into Eq. (A7) we find that to lowest order in ~ ,Γ (0) [ B i ] = Z dx L ( B i ) . (A11)From Eq. (A11), terms linear in Q i in the exponential of Eq. (A7) cancel. It is not necessary to impose the equationof motion L ,i ( B i ) = 0 (A12)on the background field B i to eliminate terms linear in Q i . To zeroth order in ~ , we find thatΓ (1) = − i ln det − / L ,jk ( B i ) , (A13)the usual one-loop result. Subsequent powers of ~ show that Γ ( n ) [ B i ] ( n >
1) are associated with n -loop one-particleirreducible Feynman diagrams [20, 28]. In obtaining this result, it is assumed that L ( φ i ) is independent of ~ . Thisis not always the case; it has been pointed out that if L ( φ i ) depends on ~ then loop diagrams can possibly havenon-vanishing contributions in the “classical limit” ~ → λ i to ensure that φ i satisfies theequation of motion (A12). The action of Eq. (A2) now becomes (upon setting ~ = 1) Z [ j i , k i ] = Z Dφ i Dλ i exp i Z dx (cid:18) L ( φ i ) + λ k ∂ L ( φ i ) ∂φ k + j i φ i + k i λ i (cid:19) . (A14)4The functional integral over λ i in Eq. (A14) results in a functional δ -function so that Z [ j i , k i ] = Z Dφ i δ (cid:18) ∂ L ( φ i ) ∂φ k + k κ (cid:19) exp i Z dx ( L ( φ i ) + j i φ i ) . (A15)The functional analogue of Z dxf ( x ) δ ( g ( x )) = X ¯ x i f (¯ x i ) / ( g ′ (¯ x i ) | (A16)reduces Eq. (A14) to Z [ j i , k i ] = X ¯ φ i exp i Z dx (cid:0) L ( ¯ φ i ) + j i ¯ φ i (cid:1) det − (cid:0) L ,jk ( ¯ φ i ) (cid:1) . (A17)In Eq. (A16), ¯ x i is a solution to g (¯ x i ) = 0 (A18)while in Eq. (A17) ¯ φ i ( x ) satisfies L ,k ( ¯ φ i ) + k k = 0 . (A19)If we define W [ j i , k i ] = − i ln Z [ j i , k i ] (A20)and then have B i = δW [ j i , k i ] δj i (A21)Γ[ B i , k i ] = W [ j i , k i ] − Z dxB i j i (A22)we find that e i Γ[ B i ,k i ] = Z DQ i Dλ i exp i Z dx (cid:16) L ( B i + Q i ) + λ k ∂∂Q k L ( B i + Q i ) + j i Q i + k i λ i (cid:17) , (A23)where now j i = − δ Γ[ B i , k i ] δB i . (A24)The integral over λ i in Eq. (A23) now results in e i Γ[ B i ,k i ] = Z DQ i δ (cid:18) ∂∂Q k L ( B i + Q i ) + k k (cid:19) exp i Z dx ( L ( B i + Q i ) + j i Q i ) (A25)so that, much like Eq. (A17), we have= X ¯ Q i exp i Z dx (cid:0) L (cid:0) B i + ¯ Q i (cid:1) + j i ¯ Q i (cid:1) det − ∂ L (cid:0) B i + ¯ Q i (cid:1) ∂Q k ∂Q ℓ ! (A26)where ¯ Q i satisfies ∂ L ( B i + ¯ Q i ) ∂Q k + k k = 0 . (A27)5In the limit k k = ¯ Q i = 0, we see from Eqs. (A11) and (A13) that on the right side of Eq. (3.13) we have theproduct of all “tree” diagrams (the exponential) and the square of all the usual “one-loop diagrams” (the functionaldeterminant), with the field B i on external legs. No contributions beyond one-loop arise. This is consistent with whatresults from a Feynman diagram expansion of Z [ j i , k i ] [4].The action in some models S = Z dx L ( φ i ) (A28)is invariant under a “gauge transformation” φ i → φ ′ i = φ i + H ij ( φ k ) ξ j (A29)so that Z dx ∂ L ( φ i ) ∂φ k H kℓ ( φ i ) ξ ℓ = 0 . (A30)If we now consider the action S λ = Z dx (cid:18) L ( φ i ) + λ k ∂ L ( φ i ) ∂φ k (cid:19) (A31)then by Eq. (A30) it immediately follows that the transformation λ i → λ ′ i = λ i + H ij ( φ k ) ζ k (A32)leaves S λ invariant. Furthermore, if Z dx (cid:18) L ( φ j ) + λ ′ i ∂ L ( φ ′ j ) ∂φ ′ i (cid:19) = Z dx (cid:18) L ( φ j ) + λ i ∂ L ( φ j ) ∂φ i (cid:19) (A33)where φ ′ j = φ i + H ij ( φ k ) ξ j leaves S in Eq. (A28) invariant, then it follows that λ i → λ ′ i = λ k ∂φ ′ i ∂φ k = λ i + λ k ∂H ij ( φ ℓ ) ξ j ∂φ k . (A34)Together, Eqs. (A29) and (A34) as well as Eq. (A32) are invariances of S λ in Eq. (A31) [23].In order for the gauge transformation of Eq. (A29) to close under commutation of two successive gauge transfor-mations, we must have ( δ A δ B − δ B δ A ) φ i = δ A (cid:0) H ij ξ Bj (cid:1) − δ B (cid:0) H ij ξ Aj (cid:1) = ∂H ij ∂φ k H kℓ ξ Aℓ ξ Bj − ∂H ij ∂φ k H kℓ ξ Aj ξ Bℓ = H im f mjℓ ξ aℓ ξ Bj (A35)so that ∂H ij ∂φ k H kℓ − ∂H iℓ ∂φ k H kj = H im f mjℓ . (A36)For a gauge transformation to be consistent, the Jacobi identity([ δ A , [ δ B , δ C ]] + [ δ B , [ δ C , δ A ]] + [ δ C , [ δ A , δ B ]]) φ i = 0 must be satisfied. This implies that f kab f ℓck + f kca f ℓbk + f kbc f ℓak =0, provided f ijk is independent of φ i .The Faddeev-Popov procedure [19] can be used to quantize a model with the action S λ possessing the gaugeinvariances of Eqs. (A29, A34) and (A32). If we want to impose the same gauge restriction on φ and λ i , F ij φ j = 0 = F ij λ j (A37)then we begin by inserting the constant [23]1 = Z Dξ i Dζ i δ " F ij (cid:18)(cid:18) φ j λ j (cid:19) + (cid:18) H jk H jk λ ℓ ∂H jk ∂φ ℓ (cid:19) (cid:18) ζ k ξ k (cid:19)(cid:19) − (cid:18) p i q i (cid:19) det (cid:20) F ij (cid:18) H jk H jk λ ℓ ∂H jk ∂φ ℓ (cid:19)(cid:21) (A38)6into the path integral of Eq. (A14), followed by insertion of the constant Z Dp j Dq i exp − i α Z dx ( p i p i + 2 p i q i ) . (A39)If we then perform the gauge transformations of Eqs. (A29, A34, A32) with ξ i , ζ i replaced by ( − ξ i , − ζ i ) we are thenleft with Z [ j i , k i ] = Z Dφ i Dλ i det (cid:20) F ij (cid:18) H jk H jk λ ℓ ∂H jk ∂φ ℓ (cid:19)(cid:21) exp i Z dx (cid:20) L ( φ i ) + λ k ∂ L ( φ i ) ∂φ k + j i φ i + k i λ i − α ( F ij φ j F ik φ k + 2 F ij φ j F ik λ k ) (cid:21) (A40)upon dropping normalization factors R Dξ i Dζ i .It is possible to impose more than one gauge condition on φ i , λ i . This is useful in spin-two models in which onewants a propagator that is both traceless and transverse [30]. If in addition to the gauge conditions of Eq. (A37) wewish to have G ij φ j = 0 = G ij λ j (A41)then into the path integral of Eq. (A14) we insert not just Eqs. (A38) , but also a constant that is found by replacing( ζ i , ξ i ) → ( ρ i , θ i ) , ( p i , q i ) → ( r i , s i ) , F ij → G ij in Eq. (A38) and a constant Z Dp i Dq i Z Dr i Ds i exp − i α Z dx ( p i r i + p i s i + q i r i ) . (A42)We then are left with ˜ Z [ j i , k i ] = Z Dφ i Dλ i Z D ¯ ξ i D ¯ ζ i det F ij (cid:18) H jk H jk λ ℓ ∂H jk ∂φ ℓ (cid:19) det G ij (cid:18) H jk H jk λ ℓ ∂H jk ∂φ ℓ (cid:19) exp i Z dx " L ( φ i ) + λ k ∂ L ( φ i ) ∂φ k + j i φ i + k i λ i − α ( F ij φ j ) (cid:0) G ik (cid:0) φ k + H kℓ ¯ ξ ℓ (cid:1)(cid:1) + ( F ij φ j ) (cid:18) G ik (cid:18) λ k + H kℓ ¯ ζ ℓ + λ m ∂H kℓ ∂φ m ¯ ξ ℓ (cid:19)(cid:19) + ( F ij λ j ) (cid:0) G ik (cid:0) φ k + H kℓ ¯ ξ ℓ (cid:1)(cid:1) ! (A43)where ¯ ξ i = θ i − ξ i , ¯ ζ i = ρ i − ζ i .Exponentiation of the functional determinants in Eqs. (A40) and (A43) by use ofdet M ij = Z Dc i D ¯ c i exp ¯ c i M ij c j (A44)where c i , ¯ c i are Grassmann leads to Fermionic ghost fields. In Eq. (A43), ¯ ξ i and ¯ ζ i are Bosonic ghost fields.If we use det (cid:18) AA B (cid:19) = det (cid:18) AA A + B (cid:19) (A45)7in Eq. (A40) and then use Eq. (A44), we find that Z [ j i , k i ] = Z Dφ i Dλ i Z DN i DL i Z Dc i D ¯ c i Dd i D ¯ d i exp i Z dx " L ( φ i ) + λ k ∂ L ( φ i ) ∂φ k + j i φ i + k i λ i + ¯ c i F ij H jk d k + ¯ d i F ij H jk c k + ¯ c i F ij (cid:18) H jk + λ ℓ ∂H jk ∂φ ℓ (cid:19) c k + (cid:16) α N i N i − N i F ij ( φ j + λ j ) + αN i L i − L i F ij φ j (cid:17) (A46)where N i , L i are “Nakanishi-Lautrup” fields [31].Provided f mjℓ in Eq. (A36) is independent of φ i , it can be shown that L ( φ i ) + λ k ∂ L ( φ i ) ∂φ k + ¯ c i F ij H jk d k + ¯ d i F ij H jk c k + ¯ c i F ij (cid:18) H jk + λ ℓ ∂H jk ∂φ ℓ (cid:19) c k + (cid:16) α N i N i − N i F ij ( φ j + λ j ) + αN i L i − L i N ij φ j (cid:17) (A47)is invariant under the transformation δφ i = H ij c j ǫ (A48) δλ i = H ij d j ǫ + λ k ∂H ij ∂φ k c k ǫ (A49) δN i = 0 = δL i (A50) δ ¯ d i = − ¯ cN i (A51) δ ¯ d i = − ǫL i (A52) δc i = − f ijk c j c k ǫ (A53) δd i = − f ijk c j d k ǫ, (A54)where ǫ is a Grassmann constant. Eqs. (A48-A54) are the global “BRST” transformations associated with Eq. (A47)[32]. The presence of this invariance ensures that introduction of a Lagrange multiplier field as in Eq. (A14) for agauge theory is consistent with unitarity, provided it is a nilpotent transformation [33]. This requires that f ijk isindependent of φ i . Appendix B: Derivation of the Gauge Symmetries
In this appendix we show how the Dirac constraint formalism [34] can be applied to the action of Eq. (3.15) toderive the gauge invariances of Eqs. (3.3, 3.16, 3.17).With the metric η µν = diag( − , + , + , +), the Lagrangian appearing in Eq. (3.15) can be written L = 12 f i f i − f ij f ij + λ ai D abj f bij + λ a D abi f b i + f a i D ab λ bi . (B1)8The canonical momenta are given by π a = ∂ L ∂ ( ∂ A a ) = 0 (B2) σ a = ∂ L ∂ ( ∂ λ a ) = 0 (B3) π ai = ∂ L ∂ ( ∂ A ai ) = f a i − D abi (cid:0) A b + λ b (cid:1) + D ab λ bi (B4) σ ai = ∂ L ∂ ( ∂ λ ai ) = f a i . (B5)As a result we find that π ai = σ ai − D abi λ b + D ab λ bi . (B6)The canonical Hamiltonian is given by H c = π a ∂ A a + π ai ∂ A ai + σ a ∂ λ a + σ ai ∂ λ ai − L = π ai σ ai − σ ai σ ai + 14 f aij f aij − (cid:0) D abi λ bj (cid:1) (cid:0) f aij (cid:1) − λ a D abi σ bi − A a (cid:0) D abi π bi + gf abc λ bi σ ci (cid:1) . (B7)From Eqs. (B2, B3) we immediately have the primary constraints [34] φ aI = π a (B8) φ aII = σ a . (B9)In order for these primary constraints to be constant in time, their Poisson bracket (PB) with H c must either vanishor else new (secondary) constraints are present. Using the PB (cid:8) A aµ ( ~x, t ) , π bν ( ~y, t ) (cid:9) = δ ab η µν = (cid:8) λ aµ ( ~x, t ) , σ bν ( ~y, t ) (cid:9) (B10)this leads to the secondary constraints Φ aI = D abi π bi + gf abc λ bi σ ci (B11)Φ aII = D abi σ bi . (B12)No tertiary constraints need to be introduced. These constraints are all first class, as their PB with each other eithervanish, or else vanish on the “constraint surface” as (cid:8) Φ aI , Φ bI (cid:9) = gf abc Φ cI (B13) (cid:8) Φ aI , Φ bII (cid:9) = gf abc Φ cII . (B14)The “total Hamiltonian” H T and “extended Hamiltonian” H E are now defined H T = H c + x ai φ aI + x aII φ aII (B15) H E = H T + X aI Φ aI + X aII Φ aII (B16)9with x aI , x aII , X aI , X aII being treated as dynamical variables whose equation of motion simply ensures that theconstraints are satisfied. If we consider the Hamilton equations of motion that follow from the actions S T = Z d x [ π a ∂ A a + σ a ∂ λ a + π ai ∂ A ai + σ ai ∂ λ ai − H T ] (B17) S E = S T − Z d x ( X aI Φ aI + X aII Φ aII ) (B18)then the Hamilton equations of motion that follow from Eq. (B17) have the same dynamical content as the Lagrangianequations of motion that follow from S λ Y M in Eq. (3.15).In general, if we have first class constraints γ a i in the i th generation, then we will show that a generator G of theform G = Z d xρ a i γ a i (B19)will lead to local transformations of dynamical variables φ A and their conjugate momenta π A of the form δF = { F, G } (B20)that leave S T invariant, thereby providing the gauge invariances of the action. Following the HTZ approach [33], δS E = δ Z d xdt ( π A ∂ φ A − H c − U a i γ a i )= Z d xdt " { π a , G } ∂ φ A − { π a , G } ∂ π A − {H c , G } − U a i { γ a i , G } − δU a i γ a i (B21)= Z d xdt " − ∂G∂φ A ∂ φ A − ∂G∂π A ∂ π A − {H c , G } − U a i { γ a i , G } − δU a i γ a i . (B22)If now { U a i , G } = 0 , (B23)and since dGdt = Z d x (cid:20) ˙ ρ a i γ a i + ∂G∂φ A ∂ φ A + ∂G∂π A ∂ π A (cid:21) (B24)then δS E = Z d x [ ˙ ρ a i γ a i + { G, H E } − δU a i γ σ i ] . (B25)If now we specialize to the case where U a i = 0 = δU a i for i >
1, then we obtain δS T = Z d e x [ ˙ ρ a i γ a i + { G, H T } − δU a γ a ] . (B26)If we use Eq. (B26) to solve for ρ a i so that δS T = 0, we have the generator G that leaves the action invariant.With H c of Eq. (B7), G is of the form G = ρ aI φ aI + ρ aII φ aII + R aI Φ aI + R aII Φ aII . (B27)From Eq. (3.13), we see that ρ aI and ρ aII can be found in terms of R aI and R aII ; the PB of Eqs. (B13, B14) show that ρ aI = − D ab R bI (B28)0 ρ aII = − D ab R bII − gf abc λ b R cI (B29)leading to G = R aI (cid:2) D abµ π bν + gf abc λ bµ σ cµ (cid:3) + R aII (cid:2) D abµ σ bµ (cid:3) , (B30)which gives the gauge transformations of Eqs. (3.3, 3.6, 3.17) provided ξ a = R aI (B31) ζ a = R aII . (B32)Using this same approach, it should be possible to derive the transformations of Eqs. (4.3, 4.14, 4.15) for the actionof Eq. (4.11). [1] G. ’t Hooft and M. Veltman, Ann. Inst. H. Poincare XX
69 (1974).[2] M. Veltman,
Nucl. Phys.
B21
288 (1970).A.A. Slavnov,
Theor. Math. Fiz.
201 (1972).[3] J.S. Bell,
Nucl. Phys.
B60
427 (1973).C.H. Llewellyn-Smith,
Phys. Lett.
B46
233 (1973).[4] D.G.C. McKeon and T.N. Sherry,
Can. J. Phys. , 441 (1992).[5] F.A. Chishtie and D.G.C. McKeon, Can. J. Phys. , 164 (2013).[6] G. ’t Hooft, Nucl. Phys.
B33 , 173 (1971);
B35 , 167 (1971).[7] M.H. Goroff and A. Sagnotti,
Nucl. Phys.
B266
709 (1986).A.E.M. van de Ven,
Nucl. Phys.
B378
309 (1992).[8] S. Deser and P. van Niewenhuizen,
Phys. Rev.
D10 , 411 (1974).[9] S. Deser, H.S. Tsao and P. van Niewenhuizen,
Phys. Rev.
D10 , 3337 (1974).[10] D.Z. Freedman and A. Van Proeyen, “Supergravity” (Cambridge University Press, Cambridge, England, 2012).[11] P.D. Mannheim,
Found. Phys. , 388 (2012); K. Stelle, Phys. Rev.
D16 , 953 (1977).[12] P. West, “An Introduction to Strings and Branes” (Cambridge University Press, Cambridge, England, 2012).[13] S. Nagy,
Ann. Phys. (Amsterdam) , 310 (2014).[14] J.C. Collins,
Phys. Rev.
D10 , 1213 (1974).[15] M. Veltman, “Diagrammatica” (Cambridge University Press, Cambridge, England, 1994).[16] D.G.C. McKeon,
Can. J. Phys. , 601 (1994).[17] J. Frenkel and J. C. Taylor, Ann. Phys (Amsterdam) . , 1 (2017).[18] P. M. Lavrov, [arXiv:2101.06868 [hep-th]].[19] L.D. Faddeev and V.N. Popov, Phys. Lett. , 29 (1967).[20] L. Abbott, Act. Phys. Pol.
B13 , 33 (1982);
Nucl. Phys.
B185 , 189 (1981).[21] H.S. Politzer,
Phys. Rev. Lett. , 1346 (1973).D.J. Gross and F. Wilczek, Phys. Rev. Lett. , 1343 (1973).V.S. Vanyashin and M.V. Terentev, JETP , 375 (1966).I.B. Khriplovich, Sov. J. Nucl. Phys. , 325 (1970).[22] D.G. Boulware and L.S. Brown, Phys. Rev. , 1628 (1968).[23] F.T. Brandt, J. Frenkel and D.G.C. McKeon,
Can. J. Phys. , 344 (2020).F.T. Brandt, J. Frenkel, D.G.C. McKeon and G.S.S. Sakoda, Phys. Rev.
D100 , 125014 (2019).F.T. Brandt, J. Frenkel, D.G.C. McKeon and S. Martins-Filho, arXiv: hep-th 2009.09553.[24] I.L. Buchbinder and I.L. Shapiro,
Acta. Phys. Pol.
B16 , 103 (1985).F.T. Brandt and D.G.C. McKeon,
Phys. Rev.
D91 , 105006 (2015);
Phys. Rev.
D93 , 105037 (2016).[25] B.S. DeWitt,
Phys. Rev. , 1195 (1967).[26] J.F. Donoghue,
Phys. Rev.
D50 , 3874 (1994).[27] T. Hartman “Lectures on Quantum Gravity and Black Holes”.[28] I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, “Effective Action in Quantum Gravity” (IOP Press, Bristol, 1992).[29] B.R. Holstein and J.F. Donoghue,
Phys. Rev. Lett. , 201602 (2004).[30] F.T. Brandt, J. Frenkel and D.G.C. McKeon, Phys. Rev.
D76 , 105029 (2007).[31] N. Nakanishi,
Prog. Theor. Phys. , 1111 (1966); B. Lautrup, Kong. Dan. Vid. Sel. Mat. Fys. Med. (1967).[32] C. Becchi, A. Rouet and R. Stora, Ann. Phys. , 287 (1976).I.V. Tyutim, Lebedev Preprint FIAN
39 (1975).N. Nakanishi and I. Ojima, “Covariant Operator Formalism of Gauge Theories and Quantum Gravity” (World Scientific,1990).[33] T. Kugo and I. Ojima,
Prog. Theor. Phys. Suppl. , 1 (1979). [34] P.A.M. Dirac, Can. J. Math. , 129 (1950).M. Henneaux, and C. Teitelboim, “Quantization of Gauge Systems” (Princeton University Press, Princeton, 1992).[35] M. Henneaux, C. Teitelboim and J. Zanelli, Nucl. Phys.