Gauge invariance of the background average effective action
Peter M. Lavrov, Eduardo Antonio dos Reis, Tibério de Paula Netto, Ilya L.Shapiro
aa r X i v : . [ h e p - t h ] A ug Gauge invariance of the background average effectiveaction
Peter M. Lavrov a,b,c , Eduardo Antonio dos Reis a , Tib´erio de Paula Netto d , Ilya L.Shapiro a,b,c (a) Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora,36036-330 Juiz de Fora, MG, Brazil(b) Department of Mathematical Analysis, Tomsk State PedagogicalUniversity, 634061, Tomsk, Russia(c) National Research Tomsk State University, 634050 Tomsk, Russia(d) Department of Physics, Southern University of Science and Technology,Shenzhen 518055, China Abstract
Using the background field method for the functional renormalization groupapproach in the case of a generic gauge theory, we study the background fieldsymmetry and gauge dependence of the background average effective action,when the regulator action depends on external fields. The final result is thatthe symmetry of the average effective action can be maintained for a wideclass of regulator functions, but in all cases the dependence of the gauge fixingremains on-shell. The Yang-Mills theory is considered as the main particularexample.
One of the most prospective non-perturbative approaches in quantum field theory (QFT)is the functional (or exact) renormalization group (FRG), which is based on the Wetterichequation for the average effective action [1, 2] (see the reviews [3, 4, 5, 6] and textbook [7]for an introduction to the subject). The application of FRG to gauge theories was extensivelydiscussed, including in the recent work [8]. The considerations in the last and many otherpapers are based on the background field method, which enables one to maintain the gaugeinvariance for the Yang-Mills (or gravitational) field explicitly in the effective action. Thebackground field method is, in general, a useful formalism in the theory of gauge fields, andthat is why it attracted a very special attention recently, see e.g. [9, 10, 11, 12]. The applicationof this method to the average effective action has been done long ago [13] (see also the recentwork [14]), but in our opinion there are some important aspects of the problem which shouldbe explored in more details.The main problem of FRG applied to the gauge theories is that the dependence on thechoice of the gauge fixing condition does not disappear on-shell [15], as it is the case in theusual perturbative QFT. As a result of the on-shell gauge fixing dependence, the S -matrix ofthe theory is not well defined, except at the fixed point, where the effective average actioncoincides with the usual effective action. One can expect that the renormalization group flowin the Yang-Mills theory will also manifest a fundamental gauge dependence, and this certainlyshadows the interpretation of the results obtained within the FRG approach in the gaugetheories.In order to better understand the situation with the gauge symmetry at the quantum leveland with the gauge dependence, it is important to analyze the mentioned problems in the E-mails: [email protected] (Peter M. Lavrov), [email protected] (Eduardo Antonio dos Reis),[email protected] (Tib´erio de Paula Netto), shapiro@fisica.ufjf.br (Ilya L.Shapiro). A is denoted ε ( A ). We start by making a brief review of the background field formalism for a gauge theorydescribing by an initial action S ( A ) of fields A = { A i } , ε ( A i ) = ε i invariant under gaugetransformations S ,i ( A ) R iα ( A ) = 0 , δA i = R iα ( A ) ξ α , (1)where R iα ( A ), ε ( R iα ( A )) = ε i + ε α are the generators of gauge transformations, ξ α , ε ( ξ α ) = ε α are arbitrary functions. In general, a set of fields A i = ( A αk , A m ) includes fields A αk of thegauge sector and also fields A m of the matter sector of a given theory. We assume that thegenerators R iα = R iα ( A ) satisfy a closed algebra with structure coefficients F γαβ that do notdepend on the fields, R iα,j R jβ − ( − ε α ε β R iβ,j R jα = − R iγ F γαβ , (2)where we denote the right functional derivative by δ r X/δA i = X ,i . The structure coefficientssatisfy the symmetry properties F γαβ = − ( − ε α ε β F γβα . We assume as well that the generatorsare linear operators in A i , R iα ( A ) = t iαj A j + r iα .We apply the background field method (BFM) [17, 18, 19] replacing the field A i by A i + B i in the classical action S ( A ), S ( A ) −→ S ( A + B ) . (3)Here B i are external (background) vector fields being not equal to zero only in the gauge sector.The action S ( A + B ) obeys the gauge invariance in the form δS ( A + B ) = 0 , δA i = R iα ( A + B ) ξ α . (4)Through the Faddeev-Popov quantization [20] the field configuration space is extended to φ A = ( A i , B α , C α , ¯ C α ) , ε ( φ A ) = ε A , (5)where C α , ¯ C α are the Faddeev-Popov ghost and antighost fields, respectively, and B a is theauxiliary (Nakanishi-Lautrup) field. The Grassmann parities distribution are the following ε ( C α ) = ε ( ¯ C α ) = ε α + 1 , ε ( B α ) = ε α . (6)2he corresponding Faddeev-Popov action S F P ( φ, B ) in the singular gauge fixing has the form [20] S F P ( φ, B ) = S ( A + B ) + S gh ( φ, B ) + S gf ( φ, B ) (7)where S gh ( φ, B ) = ¯ C α χ α,i ( A, B ) R iβ ( A + B ) C β , (8) S gf ( φ, B ) = B α χ α ( A, B ) . (9)In the last expression χ α ( A, B ) are functions lifting the degeneracy for the action S ( A + B ).The standard background field gauge condition in the BFM is linear in the quantum fields χ α ( A, B ) = F αi ( B ) A i . (10)The action (7) is invariant under the BRST symmetry [21, 22] δ B φ A = s A ( φ, B ) µ, ε (cid:0) s A ( φ, B ) (cid:1) = ε A + 1 , (11)where s A ( φ, B ) = (cid:16) R iα ( A + B ) C α , , − F αβγ C γ C β ( − ε β , ( − ε α B α (cid:17) (12)and µ is a constant Grassmann parameter with ε ( µ ) = 1. One can write (12) as generator ofBRST transformations, ˆ s ( φ, B ) = ←− δδφ A s A ( φ, B ) . (13)Then, the action (7) can be written in the form S F P ( φ, B ) = S ( A + B ) + Ψ( φ, B ) ˆ s ( φ, B ) , (14)where Ψ( φ, B ) = ¯ C α χ α ( A, B ) , (15)is the gauge fixing functional. The transformation (11) is nilpotent, that means ˆ s = 0. Takinginto account that S ( A + B ) ˆ s ( φ, B ) = 0, the BRST symmetry of S F P ( φ, B ) follows immediately S F P ( φ, B ) ˆ s ( φ, B ) = 0 . (16)Due to the presence of external vector field B i , the Faddeev-Popov action obeys an additionallocal symmetry known as the background field symmetry, δ ω S F P ( φ, B ) = 0 , (17)which is related to the background field transformations δ ( c ) ω B i = R iα ( B ) ω α ,δ ( q ) ω A i = (cid:2) R iα ( A + B ) − R iα ( B ) (cid:3) ω α ,δ ( q ) ω B α = − F αγβ B β ω γ , (18) δ ( q ) ω C α = − F αγβ C β ω γ ( − ε γ ,δ ( q ) ω ¯ C α = − F αγβ ¯ C β ω γ ( − ε γ . c ) is used to indicate the background field transformations in the sectorof external (classical) fields while the ( q ) in the sector of quantum fields (integration variablesin functional integral for generating functional of Green functions). The symbol δ ω means thecombined background field transformations δ ω = δ ( c ) ω + δ ( q ) ω . Note that in deriving (17) thetransformation rule for the gauge fixing functions (10) δ ω χ α ( φ, B ) = − χ β ( φ, B ) F βαγ ω γ , (19)under the background field transformations (18) is assumed. It is useful to introduce thegenerator of the background field transformation ˆ R ω ( φ, B ),ˆ R ω ( φ, B ) = Z dx (cid:16) ←− δδ B aµ δ ( c ) ω B aµ + ←− δδφ i δ ( q ) ω φ i (cid:17) = ˆ R ( c ) ω ( B ) + ˆ R ( q ) ω ( φ ) , (20)where φ j ˆ R ( q ) ω ( φ ) = ˆ R jω ( φ ) andˆ R jω ( φ ) = (cid:0) R ( q ) ω ( A ) , R ( q ) ω ( B ) , R ( q ) ω ( C ) , R ( q ) ω ( ¯ C ) (cid:1) . (21)Using the new notations (20), the background field invariance of the Faddeev-Popov action (17)rewrites as S F P ( φ, B ) ˆ R ω ( φ, B ) = 0 . (22)The symmetries (16) and (22) of the Faddeev-Popov action lead to the two very importantproperties at the quantum level. In order to reveal these consequences we have to introducethe extended generating functional of Green functions in the background field method in theform of functional integral Z ( J, φ ∗ , B ) = Z D φ exp (cid:26) i ~ [ S F P ( φ, B ) + φ ∗ ( φ ˆ s ) + J φ ] (cid:27) = exp (cid:26) i ~ W ( J, φ ∗ , B ) (cid:27) , (23)where W = W ( J, φ ∗ , B ) is the extended generating functional of connected Green functionsand J A = (cid:0) J i , J Bα , ¯ J α , J α (cid:1) (24)are the external sources to the fields φ A ( ε ( J A ) = ε A ). Furthermore, the new quantities (anti-fields) φ ∗ A , with ε ( φ ∗ A ) = ε A + 1, are the sources of the BRST transformations.The introduction of antifields enable one to simplify the use of the BRST symmetry at thequantum level. The next step is to introduce the extended effective action Γ = Γ(Φ , φ ∗ , B )through the Legendre transformation of W ( J, φ ∗ , B )Γ(Φ , φ ∗ , B ) = W ( J, φ ∗ , B ) − J Φ , (25)where Φ A = δ l WδJ A and δ r Γ δ Φ A = − J A . (26)From one hand, one can prove that the BRST symmetry (16) of S F P results in the Slavnov-Taylor identity [23, 24] δ r Γ δ Φ A δ l Γ δφ ∗ A = 0 . (27)4n the other hand, the background field symmetry (22) of S F P leads to the symmetry of theeffective action under the background field transformations,˜Γ(Φ , B ) ˆ R ω (Φ , B ) = 0 , ˜Γ(Φ , B ) = Γ(Φ , φ ∗ = 0 , B ) . (28)The fundamental object of the background field method is the background effective actionΓ( B ) ≡ ˜Γ(Φ = 0 , B ). Thanks to the linearity of ˆ R ω (Φ , B ) with respect to the mean fields Φ i ,from (28) it follows δ ( c ) ω Γ( B ) = 0 , δ ( c ) ω B i = R iα ( B ) ω α , (29)i.e. the background effective action is a gauge invariant functional of the external field B i .The last important feature of the Faddeev-Popov quantization is related to the universalityof the S -matrix, that is independent on the choice of the gauge fixing. According to the well-known result [25], the universality of the S -matrix is equivalent to the gauge fixing independentvacuum functional. In the background field formalism this functional is defined starting from(23) as Z Ψ ( B ) = Z ( B , J = φ ∗ = 0) = Z D φ exp (cid:26) i ~ S F P ( φ, B ) (cid:27) . (30)Regardless this object depends on the background field, it is constructed for a certain choiceof gauge Ψ( φ, B ). However, it can be shown to be independent on this choice. Without thepresence of background field, the discussion of this issue in usual QFT and in the FRG approachcan be found in Ref. [15]. Here we generalize it for the background field method case.Taking an infinitesimal change of the gauge fixing functional, Ψ( φ, B ) → Ψ( φ, B )+ δ Ψ( φ, B ),we get Z Ψ+ δ Ψ ( B ) = Z D φ exp n i ~ h S F P ( φ, B ) + δ Ψ( φ, B )ˆ s ( φ, B ) io . (31)Then, after a change of variables in the form of BRST transformation (11) but with replacementof the constant parameter µ by the functional µ ( φ, B ) = i ~ δ Ψ( φ, B ) , (32)one can show that Z Ψ+ δ Ψ ( B ) = Z Ψ ( B ) , (33)which is the starting point for the proof of the gauge fixing independence of the S -matrix[25, 26]. In the next sections we shall see how this and other features for the case of theYang-Mills theory look in the framework of the FRG approach. In this section we shall discuss the use of the BFM applied to the FRG, following the originalpublication on this subject by Reuter and Wetterich [13] for the case of pure Yang-Mills theorywith the action S ( A ) = − F aµν ( A ) F aµν ( A ) , (34)5here F aµν ( A ) = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν is the field strength for the non-Abelian vector field A µ and g is coupling constant. The correspondence with the notations used in Sec. 2 reads A i → A aµ , B i → B aµ , F αβγ → f abc ,R iα ( A ) → D abµ ( A ) = δ ab ∂ µ + gf acb A cµ . (35)Here the structure coefficients f abc of the gauge group are constant. The action (34) is invariantunder the gauge transformations defined by the generator D abµ ( A ) with an arbitrary gaugefunction ω a with ε ( ω a ) = 0. In the Faddeev-Popov quantization, the Grassmann parity of thefields φ A = ( A aµ , B a , C a , ¯ C a ) is, respectively, ε A = (0 , , , δ ( c ) ω B aµ = D abµ ( B ) ω b , δ ( q ) ω A aµ = gf abc A bµ ω c ,δ ( q ) ω B a = gf abc B b ω c , δ ( q ) ω C a = gf abc C b ω c ,δ ( q ) ω ¯ C a = gf abc ¯ C b ω c . (36)Note that the generator of the transformation in the sector of fields A aµ reads D abµ ( A + B ) − D abµ ( B ) = gf acb A cµ , (37)and thus all the quantum fields transform according the same rule. The standard choice of thegauge-fixing function is χ a ( A, B ) = D abµ ( B ) A bµ . (38)It leads to the tensor transformation rule for χ a ( A, B ) under the background field transforma-tion, δ ω χ a ( A, B ) = gf abc χ b ( A, B ) ω c . (39)The main point of the FRG approach is the introduction of the scale-dependent regulatoraction S k ( φ, B ), in the framework of the background field method. Let us choose the regulatoraction for the quantum fields A aµ and C a , ¯ C a in the form S k ( φ, B ) = 12 A aµ R (1) abk µν ( D T ( B )) A bν + ¯ C a R (2) abk ( D S ( B )) C b . (40)The regulator functions depend on the external field through the covariant derivatives of tensor D T and scalar D S fields( D T ( B )) abµν = − η µν ( D ) ab + 2 gf acb F cµν ( B ) , ( D ) ab = D acρ ( B ) D cbρ ( B ) , (41)( D S ( B )) ab = − ( D ) ab . (42)The form of these functions can be chosen e.g. as in [13], R k ( z ) = Z k ze − z/k − e − z/k , (43)with Z k corresponding to the wave function renormalization.Let us consider the variation of the regulator action (40) under the background field trans-formations (18) in the first order approximation, R k ( z ) = Z k z . The first term in (41) can berewritten through integration by parts, as follows − A aµ η µν ( D ) ab A bν = χ cρµ ( A, B ) χ cρµ ( A, B ) , (44)6here χ aρµ ( A, B ) ≡ D abρ ( B ) A bµ . (45)The transformation rule for χ aρµ ( A, B ) under the background field transformation is very closeto (19). It has the form δ ω χ aρµ ( A, B ) = gf acb χ cρµ ( A, B ) ω b . (46)As consequence, we find the first term invariance δ ω ( − A aµ η µν ( D ) ab A bν ) = δ ω ( χ aρµ ( A, B ) χ aρµ ( A, B ))= 2 gf acb χ aρµ ( A, B ) χ cρµ ( A, B ) ω b = 0 . (47)Furthermore, taking into account that δ ( c ) ω F aµν ( B ) = gf acb F cµν ( B ) ω b , (48)for the second term in (41), we have δ ω (cid:0) f acb A aµ F cµν ( B ) A bν (cid:1) = gA aµ A bν F cµν (cid:0) f ace f ebd + + f abe f edc + f ade f ecb (cid:1) = 0 , because of the Jacobi identity. The invariance holds also for the ghost regulator, as one caneasily verify. In this approximation the scale-dependent action S k ( φ, B ) obeys the backgroundfield symmetry, δ ω S k ( φ, B ) = 0.The same consideration can be done for the terms of the higher orders in z . Thus, we canensure that the invariance is maintained in all orders. With these results the action (40) isinvariant under the background field transformations, δ ω S k ( φ, B ) = 0 . (49)The full action S k F P = S k F P ( φ, B ) is constructed by the rule S k F P ( φ, B ) = S F P ( φ, B ) + S k ( φ, B ) , (50)where S F P ( φ, B ) is the Faddeev-Popov action (7). Using the action (50), the generating func-tional of Green function is given by the following functional integral: Z k ( J, B ) = Z D φ exp n i [ S F P ( φ, B ) + S k ( φ, B ) + J φ ] o = exp (cid:8) iW k ( J, B ) (cid:9) , (51)where W k = W k ( J, B ) is the generating functional of connected Green functions. The mainobject of the FRG approach in the background field method is the background average effectiveaction Γ k = Γ k (Φ , B ), defined through the Legendre transform of W k ,Γ k (Φ , B ) = W k ( J, B ) − J Φ , (52)where Φ A = δ l W k δJ A From here we adopt units in which ~ = 1. δ r Γ k δ Φ A = − J A . The effective average action can be presented as a sum of the regulator action of the meanfield and the quantum correction,Γ k (Φ , B ) = S k (Φ , B ) + ¯Γ k (Φ , B ) . (53)The functional ¯Γ k satisfies the flow equation, or the Wetterich equation [1, 13], ∂ t ¯Γ k (Φ , B ) = i ( ∂ t R k ( B )¯Γ ′′ k (Φ , B ) + R k ( B ) ) . (54)In (54) ∂ t = k ddk and the symbol sTr means the functional supertrace, this last is necessary dueto the presence of quantum fields A aµ and C a , ¯ C a , with different Grassmann parity. Anotherimportant notation is (cid:16) ¯Γ ′′ k (Φ , B ) (cid:17) AB = δ l δ Φ A (cid:18) δ r ¯Γ k (Φ , B ) δ Φ B (cid:19) (55)for the matrix of the second order functional derivatives with respect to the mean fields Φ.As we have seen above, because of the invariance of the scale-dependent regulator term (40),the full action (50) is invariant under the background field transformations (17), δ ω S k F P ( φ, B ) = δ ω S k ( φ, B ) = S k ( φ, B ) ˆ R ω ( φ, B ) = 0 . (56)At the quantum level (56) provides the invariance of the background average effective actionΓ k (Φ , B ). Indeed, variation of Z k ( J, B ) with respect to the external field B aµ reads δ ( c ) ω Z k ( J, B ) = iJ A R Aω (cid:18) δ l Z k iδJ (cid:19) . (57)In terms of the functional W k ( J, B ) the relation (57) rewrites δ ( c ) ω W k ( J, B ) = J A R Aω (cid:18) δ l W k δJ (cid:19) . (58)As a consequence of (58), the background average effective action is invariant under the back-ground field transformations, δ ω Γ k (Φ , B ) = 0 . (59)In terms of the functional ¯Γ k (Φ , B ) the relation (59) becomes δ ω ¯Γ k (Φ , B ) = 0 . (60)Thus, the background field symmetry is preserved for the background average effective action¯Γ k (Φ , B ), confirming the main statement of the paper [13].For the functional ¯Γ k ( B ) = ¯Γ k (Φ = 0 , B ), the background field symmetry is preserved aswell due to linearity of the background field symmetry δ ( c ) ω ¯Γ k ( B ) = 0 , (61)in agreement with (29). In particular this means that the flow equation for ¯Γ k ( B ), ∂ t ¯Γ k ( B ) = i ( ∂ t R k ( B )¯Γ ′′ k (Φ , B ) (cid:12)(cid:12) Φ=0 + R k ( B ) ) , (62)maintains the background field symmetry. 8 Background invariant regulator functions
The prove of invariance of S k under background field transformations (49) is based on thecertain form of the regulator functions and its arguments. In particular, the regulator functions(43) with argument (41) by itself are not invariant under background field transformations δ ( c ) ω R (1) abk µν ( D T ( B )) = 0, δ ( c ) ω R (2) abk ( D S ( B )) = 0. In this section we shall discuss the backgroundfield symmetry of the background average effective action and formulate a possible restrictionon the regulator functions in the scale-dependent action S k in the general settings that allowus to arrive at the invariance of the background average effective action under background fieldtransformations.Consider the scale-dependent regulator action S k = S k ( φ, B ) in the background field for-malism, including the ghost sector, S k ( φ, B ) = 12 A aµ R (1) abk µν ( B ) A bν + ¯ C a R (2) abk ( B ) C b , (63)where R (1) abk µν ( B ) and R (2) abk ( B ) are the regulator functions. We assume that they are localfunctions of external fields B aµ and their partial derivatives. The full action has a standardFRG form S kF P ( φ, B ) = S F P ( φ, B ) + S k ( φ, B ) . (64)Due to the background field symmetry of the Faddeev-Popov action (17), the full action (64) willbe invariant under the background field transformations (18), if the scale-dependent regulatoraction S k = S k ( φ, B ) satisfies the equation δ ω S k ( φ, B ) = 0 . (65)Using the explicit form of the background field transformations (18) the variation of S k ( φ, B )reads δ ω S k ( φ, B ) = 12 A aµ h g (cid:16) f adc ω d R (1) cbk µν ( B ) − R (1) ack µν ( B ) f cdb ω d (cid:17) + δ ( c ) ω R (1) abk µν ( B ) i A bν + Z dx ¯ C a h g (cid:16) f adc ω d R (2) cbk ( B ) − R (2) ack ( B ) f cdb ω d (cid:17) + δ ( c ) ω R (2) abk ( B ) i C b . (66)From Eq. (66) follows that (65) is satisfied if g (cid:16) f adc ω d R (1) cbk µν ( B ) − R (1) ack µν ( B ) f cdb ω d (cid:17) + δ ( c ) ω R (1) abk µν ( B ) = 0 , (67) g (cid:16) f adc ω d R (2) cbk ( B ) − R (2) ack ( B ) f cdb ω d (cid:17) + δ ( c ) ω R (2) abk ( B ) = 0 . (68)Any solution of these equations provides the invariance of S k under background field transfor-mations. Let us consider the case when regulator functions are invariant under backgroundtransformations of external field B aµ , δ ( c ) ω R (1) abk µν ( B ) = 0 , δ ( c ) ω R (2) abk ( B ) = 0 . (69)Due to the arbitrariness in the choice of the functions ω a ( x ), from (67), (68) and (69) followthe relations (cid:2) t d , R (1) µνk ( B ) (cid:3) ab = 0 , (cid:2) t d , R (2) µνk ( B ) (cid:3) ab = 0 , (70)9or the generators ( t a ) bc = f bac of the Lie group. Therefore, we see that the regulator functionscommute with all the generators of Lie group. Then, applying the Shur’s lemma we find R (1) abk µν ( B ) = δ ab R (1) k µν ( D ( B )) ,R (2) abk ( B ) = δ ab R (2) k ( D ( B )) , (71)where the quantities R (1) k µν ( D ( B )) and R (2) k ( D ( B )) are scalars with respect to the backgroundtransformations of external field B aµ . It means that the arguments of these quantities shouldbe scalars as well. It is easy to construct an example of such kind of a scalar argument, D ( B ) = F aµν ( B ) D abµ ( B ) B bν , where F aµν is defined in (34).So, in the case under consideration, the scale-dependent regulator action has the form S k ( φ, B ) = 12 A aµ R (1) k µν ( D ( B )) A aν + ¯ C a ( x ) R (2) k ( D ( B )) C a , (72)maintaining the background field symmetry δ ω S k ( φ, B ) = 0. Here the problem of gauge dependence of background average effective action will be dis-cussed in general setting of Sec. 2 The regulator action S k is invariant under the backgroundtransformations (49), but not under the BRST transformations, S k ( φ, B )ˆ s ( φ, B ) = 0 . (73)Let us discuss the implications of this fact for the gauge dependence problem of the back-ground average effective action. Consider the extended generating functional of Green functions Z k ( J, φ ∗ , B ), and the extended generating functional of connected Green functions W k ( J, φ ∗ , B ), Z k ( J, φ ∗ , B ) = Z D φ exp { i [ S F P ( φ, B ) + S k ( φ, B ) + J φ + φ ∗ (ˆ sφ )] } = exp { iW k ( J, φ ∗ , B ) } , (74)As the first step we derive the modified Ward identity for the FRG in the BFM which is aconsequence of the BRST invariance of the action S F P ( φ, B ) (16). Making use the changeof variables in the form of the BRST transformations in the functional integral (74), φ A → ϕ A ( φ ) = φ A + (ˆ sφ A ) µ , and taking into account the triviality of the corresponding Jacobian ifthe conditions ( − ε i δ l R iα δA i + ( − ( ε α +1) F ββα = 0 (75)are satisfied (for detailed discussion of this point see [27]), we arrive at the relation0 = Z Dφ (cid:0) J A + S k,A ( φ, B ) (cid:1) (ˆ sφ A ) exp { i [ S F P ( φ, B ) + S k ( φ, B ) + J φ + φ ∗ (ˆ sφ )] } = (cid:20) J A + S k,A (cid:18) δ l iδJ , B (cid:19)(cid:21) Z Dφ (ˆ sφ A ) exp { i [ S F P ( φ, B ) + S k ( φ, B ) + J φ + φ ∗ (ˆ sφ )] } . (76)From (76) it follows the modified Ward identity for the extended generating functional of Greenfunctions Z k ( J, φ ∗ , B ) (cid:20) J A + S k,A (cid:18) δ l iδJ , B (cid:19)(cid:21) δ l δφ ∗ A Z k ( J, φ ∗ , B ) = 0 . (77)10his identity in terms of the extended generating functional of connected Green functions W k ( J, φ ∗ , B ) reads (cid:20) J A + S k,A (cid:18) δ l W k δJ + δ l iδJ , B (cid:19)(cid:21) δ l δφ ∗ A W k ( J, φ ∗ , B ) = 0 . (78)Introducing the generating functional of vertex functions Γ k = Γ k (Φ , φ ∗ , B ) with the help ofLegendre transformation of W k = W k ( J, φ ∗ , B )Γ k (Φ , φ ∗ , B ) = W k ( J, φ ∗ , B ) − J A Φ A , Φ A = δ l W k δJ A , (79) δ r Γ k δ Φ A = − J A , δ l Γ k δφ ∗ A = δ l W k δφ ∗ A , the modified Ward identity rewrites in the form δ r Γ k δ Φ A δ l Γ k δφ ∗ A = S k,A ( ˆΦ , B ) δ l Γ k δφ ∗ A , (80)where the notation ˆΦ A = Φ A + i (Γ ′′ − k ) AB δ l δ Φ B , (81)has been used. The matrix (Γ ′′ − k ) is inverse to the matrix Γ ′′ k , the last has elements(Γ ′′ k ) AB = δ l δ Φ A (cid:16) δ r Γ k δ Φ B (cid:17) , i.e., (cid:0) Γ ′′ − k (cid:1) AC · (cid:0) Γ ′′ k (cid:1) CB = δ AB . (82)Now consider the variation of the extended generating functional of Green functions underinfinitesimal variation of the gauge fixing functional, Ψ( φ, B ) → Ψ( φ, B ) + δ Ψ( φ, B ). We find δZ k ( J, φ ∗ , B ) = i Z Dφ (cid:0) δ Ψ( φ, B )ˆ s ( φ, B ) (cid:1) exp { i [ S F P ( φ, B )+ S k ( φ, B ) + J φ + φ ∗ (ˆ sφ )] } . (83)Now take into account that the functional integral of total variational derivative is zero we havethe relation0 = Z Dφ δ r δφ A h(cid:0) δ Ψ s A (cid:1) exp { i [ S F P ( φ, B ) + S k ( φ, B ) + J φ + φ ∗ (ˆ sφ )] } i = Z Dφ h iδ Ψ s A (cid:0) J A + S k,A (cid:1) + (cid:0) δ Ψˆ s (cid:1)i exp { i [ S F P ( φ, B ) + S k ( φ, B ) + J φ + φ ∗ (ˆ sφ )] } , (84)where the BRST invariance of S F P action, the nilpotency of BRST transformations and therelations (75) have been used. From (84) one has i Z Dφ (cid:0) δ Ψ( φ, B )ˆ s ( φ, B ) (cid:1) exp { i [ S F P ( φ, B ) + S k ( φ, B )+ J φ + φ ∗ (ˆ sφ )] } = Z Dφ (cid:0) J A + S k,A φ, B (cid:1) s A ( φ, B ) ×× δ Ψ( φ, B ) exp { i [ S F P ( φ, B ) + S k ( φ, B ) + J φ + φ ∗ (ˆ sφ )] } , (85)11hich allows to present the Eq. (83) in the form closed with respect to Z k ( J, φ ∗ , B ), δZ k ( J, φ ∗ , B ) = − i (cid:20) J A + S k,A (cid:18) δ l iδJ , B (cid:19)(cid:21) δ l δφ ∗ A δ Ψ (cid:18) δ l iδJ , B (cid:19) Z k ( J, φ ∗ , B ) , (86)or, in terms of W k ( J, B ), δW k ( J, φ ∗ , B ) = − (cid:20) J A + S k,A (cid:18) δ l W k δJ + δ l iδJ , B (cid:19)(cid:21) δ l δφ ∗ A δ Ψ (cid:18) δ l W k δJ + δ l iδJ , B (cid:19) . (87)In deriving (87) the modified Slavnov-Taylor identity (78) has been used. The last equationcan be rewritten for the background average effective action, Γ k (Φ , φ ∗ , B ), in the form δ Γ k (Φ , φ ∗ , B ) = δ r Γ k δ Φ A δ l δφ ∗ A δ Ψ( ˆΦ , B ) − S k,A ( ˆΦ , B ) δ l δφ ∗ A δ Ψ( ˆΦ , B ) , (88)where ˆΦ was introduced in (81). From Eq. (88) follows that δ Γ k (Φ , φ ∗ , B ) (cid:12)(cid:12)(cid:12) δ Γ kδ Φ =0 = 0 . (89)As result, the average effective action depends on gauge fixing even on the equations of motion(on-shell) and the S -matrix defined in the framework of the FRG approach is gauge dependent. We considered several aspects of background average effective action in the FRG framework.At the first place we confirmed the well-known classical result of [13] concerning the backgroundinvariance of the regulator actions and background average effective action in the frameworkof the background field method for a wide class of regulator functions which include (43), butcan be generalized to any other functions of the arguments z . As a new technical result weformulated general conditions of regulator actions being invariant with respect to the purelybackground transformations.The main motivation of this work was to check whether the on-shell dependence of theaverage effective action [15] holds within the background field method formalism. The answerto this question is given by the relation (89) and is strictly positive. This output does notcontradict the recent works [8, 14] because in these publications the subject of study wasthe gauge invariance of background average effective action, and the question of gauge fixingdependence was not investigated. From our viewpoint, the on-shell gauge dependence of theaverage effective action is a fundamental principal difficulty of the FRG approach applied to theYang-Mills theories. We have confirmed that the situation does not improve in the backgroundfield method, regardless of the different structure of lifting the degeneracy of the classical action.It is unclear whether one can achieve a reasonable physical interpretation of the resultsobtained within the FRG formalism applied to Yang-Mills theories, and therefore it makessense to discuss the possible ways out from this difficult situation.Certainly the simplest way is to ignore the problem e.g. by deciding that one special gaugefixing is “physical” or “correct”, such that changing the gauge should be strictly forbidden. Asfar as FRG provides valuable nonperturbative results, the theoretically inconsistent formulationis the price to pay for going beyond the well-defined perturbative framework.Another possibility is to look for some observables that may be gauge-fixing invariant. Forinstance, in the fixed point the background average effective action boils down to the standardQFT effective action and then S -matrix, amplitudes and all related observables are well-defined.12nfortunately, even in the vicinity of the fixed point this is not true due to the relation (89).Since the search of the nonperturbative fixed point is based on the renormalization groupflows and the last are supposed to be gauge-fixing dependent, it is unclear how the fixed-pointinvariance can be actually used.Finally, there is an alternative formulation of the FRG in gauge theories which is gauge-fixingindependent, exactly as a conventional perturbative QFT is [15]. This scheme is technicallymore difficult, since the regulator actions are constructed in a more complicated way, thatincludes composite fields. At least by now, the disadvantage of this approach is that there isno method to perform practical calculations. Acknowledgements
P.M.L. is grateful to the Departamento de F´ısica of the Federal University of Juiz de Fora(MG, Brazil) for warm hospitality during his long-term visit. The work of P.M.L. is supportedpartially by the Ministry of Education and Science of the Russian Federation, grant 3.1386.2017and by the RFBR grant 18-02-00153. This work of I.L.Sh. was partially supported by ConselhoNacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - CNPq under the grant 303893/2014-1and Funda¸c˜ao de Amparo `a Pesquisa de Minas Gerais - FAPEMIG under the project APQ-01205-16. E.A.R. is grateful to Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior- CAPES for supporting his Ph.D. project.
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