Gauge Invariance of the Background Average Effective Action

Home , Background field method

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In what follows we consider both gauge invariance and gauge fixing dependence for the effective average action. The paper is organized as follows. In Sec. 2 we give a brief description of the background field formalism in non-Abelian gauge theories and the gauge independence of vacuum functional in this method. In Sec. 3 the background field symmetry is analyzed within the background field method - based functional renormalization group approach. The regulator functions are dependent on the external (background) fields but are chosen not to be invariant under gauge transformations of external vector field. In Sec. 4 we present a solution to the background field symmetry of background average effective action with regulator functions which are invariant under gauge transformations of the external field. In Sec. 5 the gauge dependence problem of the background average effective action is considered. Finally, we discuss the results and draw our conclusions in Sec. 6. Our notations system mainly follows the DeWitt’s book [16]. Also, the Grassmann parity of a quantity A is denoted ε(A).

2 Background ﬁeld formalism

We start by making a brief review of the background ﬁeld formalism for a gauge theory i i describing by an initial action S0(A) of ﬁelds A = {A }, ε(A ) = εi invariant under gauge transformations

i i i α S0,i(A)Rα(A)=0, δA = Rα(A)ξ , (1)

i i α α where Rα(A), ε(Rα(A)) = εi + εα are the generators of gauge transformations, ξ , ε(ξ ) = εα are arbitrary functions. In general, a set of fields Ai = (Aαk, Am) includes fields Aαk of the gauge sector and also fields Am of the matter sector of a given theory. We assume that the i i γ generators Rα = Rα(A) satisfy a closed algebra with structure coefficients Fαβ that do not depend on the fields,

i j εαεβ i j i γ Rα,jRβ − (−1) Rβ,jRα = −Rγ Fαβ, (2)

i where we denote the right functional derivative by δrX/δA = X,i. The structure coefficients γ εαεβ γ satisfy the symmetry properties Fαβ = −(−1) Fβα. We assume as well that the generators i i i j i are linear operators in A , Rα(A)= tαjA + rα. We apply the background field method (BFM) [17, 18, 19] replacing the field Ai by Ai +Bi in the classical action S0(A),

S0(A) −→ S0(A + B) . (3)

Here Bi are external (background) vector ﬁelds being not equal to zero only in the gauge sector. The action S0(A + B) obeys the gauge invariance in the form

i i α δS0(A + B)=0, δA = Rα(A + B)ξ . (4) Through the Faddeev-Popov quantization [20] the ﬁeld conﬁguration space is extended to

A i α α α A φ =(A , B ,C , C¯ ), ε(φ )= εA, (5) where Cα, C¯α are the Faddeev-Popov ghost and antighost ﬁelds, respectively, and Ba is the auxiliary (Nakanishi-Lautrup) ﬁeld. The Grassmann parities distribution are the following

α α α ε(C )= ε(C¯ )= εα +1, ε(B )= εα . (6)

2 The corresponding Faddeev-Popov action SFP (φ, B) in the singular gauge ﬁxing has the form [20]

SFP (φ, B)= S0(A + B)+ Sgh(φ, B)+ Sgf (φ, B) (7) where

¯α i β Sgh(φ, B)= C χα,i(A, B) Rβ(A + B)C , (8) α Sgf (φ, B)= B χα(A, B). (9)

In the last expression χα(A, B) are functions lifting the degeneracy for the action S0(A + B). The standard background ﬁeld gauge condition in the BFM is linear in the quantum ﬁelds

i χα(A, B) = Fαi(B)A . (10)

The action (7) is invariant under the BRST symmetry [21, 22]

A A A δBφ = s (φ, B)µ, ε s (φ, B) = εA +1 , (11) where 1 sA(φ, B)= Ri (A + B)Cα, 0, − F α CγCβ(−1)εβ , (−1)εα Bα (12) α 2 βγ and µ is a constant Grassmann parameter with ε(µ) = 1. One can write (12) as generator of BRST transformations, ←− δ sˆ(φ, B)= sA(φ, B). (13) δφA Then, the action (7) can be written in the form

SFP (φ, B) = S0(A + B)+Ψ(φ, B)ˆs(φ, B) , (14) where

α Ψ(φ, B)= C¯ χα(A, B), (15) is the gauge ﬁxing functional. The transformation (11) is nilpotent, that meanss ˆ2 = 0. Taking into account that S0(A+B)ˆs(φ, B) = 0, the BRST symmetry of SFP (φ, B) follows immediately

SFP (φ, B)ˆs(φ, B)=0 . (16)

Due to the presence of external vector ﬁeld Bi, the Faddeev-Popov action obeys an additional local symmetry known as the background ﬁeld symmetry,

δωSFP (φ, B)=0, (17) which is related to the background ﬁeld transformations

(c) i i α δω B = Rα(B)ω , (q) i i i α δω A = Rα(A + B) − Rα(B) ω , (q) α α β γ δω B =−FγβB ω , (18) (q) α α β γ εγ δω C = −FγβC ω (−1) , (q) ¯α α ¯β γ εγ δω C = −FγβC ω (−1) .

3 Here the subscript (c) is used to indicate the background field transformations in the sector of external (classical) fields while the (q) in the sector of quantum fields (integration variables in functional integral for generating functional of Green functions). The symbol δω means the (c) (q) combined background field transformations δω = δω + δω . Note that in deriving (17) the transformation rule for the gauge fixing functions (10)

β γ δωχα(φ, B) = −χβ(φ, B)Fαγω , (19) under the background ﬁeld transformations (18) is assumed. It is useful to introduce the generator of the background ﬁeld transformation Rˆ ω(φ, B), ←− ←− ˆ δ (c) a δ (q) i Rω(φ, B)= dx a δω Bµ + i δω φ δBµ δφ Z ˆ (c) ˆ (q) = Rω (B)+ Rω (φ), (20)

j ˆ (q) ˆ j where φ Rω (φ)= Rω(φ) and ˆ j (q) (q) (q) (q) ¯ Rω(φ)= Rω (A), Rω (B), Rω (C), Rω (C) . (21) Using the new notations (20), the background ﬁeld invariance of the Faddeev-Popov action (17) rewrites as

SFP (φ, B) Rˆ ω(φ, B)=0. (22)

The symmetries (16) and (22) of the Faddeev-Popov action lead to the two very important properties at the quantum level. In order to reveal these consequences we have to introduce the extended generating functional of Green functions in the background ﬁeld method in the form of functional integral

∗ i ∗ i ∗ Z(J, φ , B)= Dφ exp ~ [SFP (φ, B)+ φ (φsˆ)+ Jφ] = exp ~W (J, φ , B) , (23) Z where W = W (J, φ∗, B) is the extended generating functional of connected Green functions and

B ¯ JA = Ji, Jα , Jα, Jα (24)

A are the external sources to the fields φ (ε(JA)= εA). Furthermore, the new quantities (anti- ∗ ∗ fields) φA, with ε(φA)= εA + 1, are the sources of the BRST transformations. The introduction of antifields enable one to simplify the use of the BRST symmetry at the quantum 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) − JΦ, (25) where

A δlW δrΓ Φ = and A = −JA. (26) δJA δΦ

From one hand, one can prove that the BRST symmetry (16) of SFP results in the Slavnov- Taylor identity [23, 24]

δrΓ δlΓ A ∗ =0. (27) δΦ δφA

4 On the other hand, the background field symmetry (22) of SFP leads to the symmetry of the effective action under the background field transformations,

∗ Γ(Φ˜ , B)Rˆω(Φ, B)=0, Γ(Φ˜ , B) = Γ(Φ,φ =0, B). (28)

The fundamental object of the background field method is the background effective action i Γ(B) ≡ Γ(Φ˜ = 0, B). Thanks to the linearity of Rˆ ω(Φ, B) with respect to the mean fields Φ , from (28) it follows

(c) (c) i i α δω Γ(B)=0, δω B = Rα(B)ω , (29) i.e. the background effective action is a gauge invariant functional of the external field Bi. The last important feature of the Faddeev-Popov quantization is related to the universality of 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 independent vacuum functional. In the background field formalism this functional is defined starting from (23) as

∗ i ZΨ(B)= Z(B,J = φ =0)= Dφ exp ~ SFP (φ, B) . (30) Z Regardless this object depends on the background field, it is constructed for a certain choice of gauge Ψ(φ, B). However, it can be shown to be independent on this choice. Without the presence of background field, the discussion of this issue in usual QFT and in the FRG approach can 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)+δΨ(φ, B), we get i ZΨ+δΨ(B)= Dφ exp ~ SFP (φ, B)+ δΨ(φ, B)ˆs(φ, B) . (31) Z n h io Then, after a change of variables in the form of BRST transformation (11) but with replacement of the constant parameter µ by the functional i µ(φ, B)= ~ δΨ(φ, B), (32) one can show that

ZΨ+δΨ(B)= ZΨ(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 the Yang-Mills theory look in the framework of the FRG approach.

3 Background average eﬀective action

In this section we shall discuss the use of the BFM applied to the FRG, following the original publication on this subject by Reuter and Wetterich [13] for the case of pure Yang-Mills theory with the action 1 S (A)= − F a (A)F a (A), (34) 0 4 µν µν

5 a a a abc b c where Fµν (A)= ∂µAν − ∂νAµ + gf AµAν is the ﬁeld strength for the non-Abelian vector ﬁeld Aµ and g is coupling constant. The correspondence with the notations used in Sec. 2 reads

i a i a α abc A → Aµ, B → Bµ, Fβγ → f , i ab ab acb c Rα(A) → Dµ (A)= δ ∂µ + gf Aµ. (35) Here the structure coefficients f abc of the gauge group are constant. The action (34) is invariant ab under the gauge transformations defined by the generator Dµ (A) with an arbitrary gauge function ωa with ε(ωa) = 0. In the Faddeev-Popov quantization, the Grassmann parity of the A a a a ¯a fields φ =(Aµ, B ,C , C ) is, respectively, εA = (0, 0, 1, 1). The background field formalism for Yang-Mills theory comprises the definition of the background field transformation

(c) a ab b (q) a abc b c δω Bµ = Dµ (B)ω , δω Aµ = gf Aµω , (q) a abc b c (q) a abc b c δω B = gf B ω , δω C = gf C ω , (q) ¯a abc ¯b c δω C = gf C ω . (36)

a Note that the generator of the transformation in the sector of ﬁelds Aµ reads

ab ab acb c Dµ (A + B) − Dµ (B)= gf Aµ, (37) and thus all the quantum ﬁelds transform according the same rule. The standard choice of the gauge-ﬁxing function is

a ab b χ (A, B)= Dµ (B)Aµ. (38) It leads to the tensor transformation rule for χa(A, B) under the background ﬁeld transformation,

a abc b c δωχ (A, B)= gf χ (A, B)ω . (39) The main point of the FRG approach is the introduction of the scale-dependent regulator action Sk(φ, B), in the framework of the background field method. Let us choose the regulator a a ¯a action for the quantum fields Aµ and C , C in the form 1 S (φ, B)= Aa R(1) ab(D (B)) Ab + C¯a R(2) ab(D (B)) Cb. (40) k 2 µ k µν T ν k S The regulator functions depend on the external field through the covariant derivatives of tensor DT and scalar DS fields

ab 2 ab acb c 2 ab ac cb (DT (B))µν = − ηµν(D ) +2gf Fµν(B), (D ) = Dρ (B)Dρ (B), (41) ab 2 ab (DS(B)) = − (D ) . (42) The form of these functions can be chosen e.g. as in [13],

2 ze−z/k R (z)= Z 2 , (43) k k 1 − e−z/k with Zk corresponding to the wave function renormalization. Let us consider the variation of the regulator action (40) under the background field transformations (18) in the first order approximation, Rk(z) = Zkz. The first term in (41) can be rewritten through integration by parts, as follows

a 2 ab b c c − Aµηµν (D ) Aν = χρµ(A, B) χρµ(A, B), (44)

6 where

a ab b χρµ(A, B) ≡ Dρ (B)Aµ. (45)

a The transformation rule for χρµ(A, B) under the background ﬁeld transformation is very close to (19). It has the form

a acb c b δω χρµ(A, B)= gf χρµ(A, B)ω . (46)

As consequence, we ﬁnd the ﬁrst term invariance

a 2 ab b a a δω(−Aµηµν (D ) Aν)= δω(χρµ(A, B)χρµ(A, B)) acb a c b =2gf χρµ(A, B)χρµ(A, B)ω =0 . (47)

Furthermore, taking into account that

acb a c b a b c ace ebd abe edc ade ecb δω f AµFµν (B)Aν =gAµAνFµν f f ++f f + f f =0, because of the Jacobi identity. The invariance holds also for the ghost regulator, as one can easily verify. In this approximation the scale-dependent action Sk(φ, B) obeys the background ﬁeld symmetry, δωSk(φ, B) = 0. The same consideration can be done for the terms of the higher orders in z. Thus, we can ensure that the invariance is maintained in all orders. With these results the action (40) is invariant under the background ﬁeld transformations,

δωSk(φ, B)=0. (49)

The full action Sk FP = Sk FP (φ, B) is constructed by the rule

Sk FP (φ, B)= SFP (φ, B)+ Sk(φ, B) , (50) where SFP (φ, B) is the Faddeev-Popov action (7). Using the action (50), the generating functional of Green function is given by the following functional integral:2

Zk(J, B)= Dφ exp i[SFP (φ, B)+ Sk(φ, B)+ Jφ] = exp iWk(J, B) , (51) Z n o where Wk = Wk(J, B) is the generating functional of connected Green functions. The main object of the FRG approach in the background field method is the background average effective action Γk =Γk(Φ, B), defined through the Legendre transform of Wk,

Γk(Φ, B) = Wk(J, B) − JΦ, (52) where δ W ΦA = l k δJA 2From here we adopt units in which ~ = 1.

7 and δ Γ r k = −J . δΦA A The eﬀective average action can be presented as a sum of the regulator action of the mean ﬁeld and the quantum correction,

Γk(Φ, B)= Sk(Φ, B)+ Γ¯k(Φ, B). (53)

The functional Γ¯k satisﬁes the ﬂow equation, or the Wetterich equation [1, 13],

¯ i ∂tRk(B) ∂tΓk(Φ, B)= sTr ¯′′ . (54) 2 ( Γk(Φ, B) + Rk(B) )

d In (54) ∂t = k dk and the symbol sTr means the functional supertrace, this last is necessary due a a ¯a to the presence of quantum fields Aµ and C , C , with different Grassmann parity. Another important notation is ¯ ¯′′ δl δrΓk(Φ, B) Γk(Φ, B) = (55) AB δΦA δΦB 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),

δωSk FP (φ, B)= δωSk(φ, B)= Sk(φ, B)Rˆ ω(φ, B)=0. (56) At the quantum level (56) provides the invariance of the background average effective action a Γk(Φ, B). Indeed, variation of Zk(J, B) with respect to the external field Bµ reads δ Z δ(c)Z (J, B)= iJ RA l k . (57) ω k A ω iδJ In terms of the functional Wk(J, B) the relation (57) rewrites δ W δ(c)W (J, B)= J RA l k . (58) ω k A ω δJ As a consequence of (58), the background average effective action is invariant under the background field transformations,

δωΓk(Φ, B)=0. (59)

In terms of the functional Γ¯k(Φ, B) the relation (59) becomes

δωΓ¯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 as well due to linearity of the background field symmetry (c) ¯ δω Γk(B)=0, (61) in agreement with (29). In particular this means that the flow equation for Γ¯k(B),

i ∂tRk(B) ∂ Γ¯ (B)= sTr ′′ , (62) t k 2 ¯ ( Γk(Φ, B) Φ=0 + Rk(B) ) maintains the background ﬁeld symmetry.

8 4 Background invariant regulator functions

The prove of invariance of Sk under background field transformations (49) is based on the certain 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 (c) (1) ab (c) (2) ab δω Rk µν (DT (B)) =6 0, δω Rk (DS(B)) =6 0. In this section we shall discuss the background field symmetry of the background average effective action and formulate a possible restriction on the regulator functions in the scale-dependent action Sk in the general settings that allow us to arrive at the invariance of the background average effective action under background field transformations. Consider the scale-dependent regulator action Sk = Sk(φ, B) in the background field formalism, including the ghost sector, 1 S (φ, B)= Aa R(1) ab(B)Ab + C¯aR(2) ab(B)Cb, (63) k 2 µ k µν ν k

(1) ab (2) ab where Rk µν (B) and Rk (B) are the regulator functions. We assume that they are local a functions of external ﬁelds Bµ and their partial derivatives. The full action has a standard FRG form

SkFP (φ, B)= SFP (φ, B)+ Sk(φ, B). (64)

Due to the background field symmetry of the Faddeev-Popov action(17), the full action (64) will be invariant under the background field transformations (18), if the scale-dependent regulator action Sk = Sk(φ, B) satisfies the equation

δωSk(φ, B)=0. (65)

Using the explicit form of the background ﬁeld transformations (18) the variation of Sk(φ, B) reads 1 δ S (φ, B)= Aa g f adcωdR(1) cb(B) − R(1) ac(B)f cdbωd + δ(c)R(1) ab(B) Ab ω k 2 µ k µν k µν ω k µν ν h i (66) ¯a adc d (2) cb (2) ac cdb d (c) (2) ab b + dx C g f ω Rk (B) − Rk (B)f ω + δω Rk (B) C . Z h i From Eq. (66) follows that (65) is satisﬁed if

adc d (1) cb (1) ac cdb d (c) (1) ab g f ω Rk µν (B) − Rk µν (B)f ω + δω Rk µν (B)=0, (67)

adc d (2) cb (2) ac cdb d (c) (2) ab g f ω Rk (B) − Rk (B)f ω + δω Rk (B)=0. (68) Any solution of these equations provides the invariance of Sk under background ﬁeld transformations. Let us consider the case when regulator functions are invariant under background a transformations of external ﬁeld Bµ,

Due to the arbitrariness in the choice of the functions ωa(x), from (67), (68) and (69) follow the relations

d (1) µν d (2) µν t , Rk (B) ab =0, t , Rk (B) ab =0, (70)

9 a bac for the generators (t )bc = f of the Lie group. Therefore, we see that the regulator functions commute with all the generators of Lie group. Then, applying the Shur’s lemma we ﬁnd

(1) ab ab (1) Rk µν (B)=δ Rk µν(D(B)), (2) ab ab (2) Rk (B)=δ Rk (D(B)), (71)

(1) (2) where the quantities Rk µν(D(B)) and Rk (D(B)) are scalars with respect to the background a transformations of external field Bµ. It means that the arguments of these quantities should be scalars as well. It is easy to construct an example of such kind of a scalar argument, a ab b a D(B)= Fµν(B)Dµ (B)Bν , where Fµν is defined in (34). So, in the case under consideration, the scale-dependent regulator action has the form 1 S (φ, B)= Aa R(1) (D(B))Aa + C¯a(x)R(2)(D(B))Ca, (72) k 2 µ k µν ν k maintaining the background field symmetry δωSk(φ, B) = 0.

5 Gauge dependence of background average eﬀective action

Here the problem of gauge dependence of background average eﬀective action will be discussed in general setting of Sec. 2 The regulator action Sk is invariant under the background transformations (49), but not under the BRST transformations,

Sk(φ, B)ˆs(φ, B) =06 . (73) Let us discuss the implications of this fact for the gauge dependence problem of the background average eﬀective action. Consider the extended generating functional of Green functions ∗ ∗ Zk(J, φ , B), and the extended generating functional of connected Green functions Wk(J, φ , B),

∗ ∗ ∗ Zk(J, φ , B)= Dφ exp{i[SFP (φ, B)+ Sk(φ, B)+ Jφ + φ (ˆsφ)]} = exp{iWk(J, φ , B)}, (74) Z As the first step we derive the modified Ward identity for the FRG in the BFM which is a consequence of the BRST invariance of the action SFP (φ, B) (16). Making use the change of variables in the form of the BRST transformations in the functional integral (74), φA → ϕA(φ) = φA + (ˆsφA)µ, and taking into account the triviality of the corresponding Jacobian if the conditions δ Ri (−1)εi l α +(−1)(εα+1)F β =0 (75) δAi βα are satisfied (for detailed discussion of this point see [27]), we arrive at the relation

A ∗ 0= Dφ JA + Sk,A(φ, B) (ˆsφ ) exp{i[SFP (φ, B)+ Sk(φ, B)+ Jφ + φ (ˆsφ)]} Z δ = J + S l , B Dφ(ˆsφA) exp{i[S (φ, B)+ S (φ, B)+ Jφ + φ∗(ˆsφ)]}. (76) A k,A iδJ FP k Z From (76) it follows the modiﬁed Ward identity for the extended generating functional of Green ∗ functions Zk(J, φ , B) δ δ J + S l , B l Z (J, φ∗, B)=0. (77) A k,A iδJ δφ∗ k A 10 This identity in terms of the extended generating functional of connected Green functions ∗ Wk(J, φ , B) reads

δ W δ δ J + S l k + l , B l W (J, φ∗, B)=0. (78) A k,A δJ iδJ δφ∗ k A ∗ Introducing the generating functional of vertex functions Γk = Γk(Φ,φ , B) with the help of ∗ Legendre transformation of Wk = Wk(J, φ , B)

∗ ∗ A A δlWk Γk(Φ,φ , B)= Wk(J, φ , B) − JAΦ , Φ = , (79) δJA δrΓk δlΓk δlWk A = −JA, ∗ = ∗ , δΦ δφA δφA the modiﬁed Ward identity rewrites in the form

δrΓk δlΓk ˆ δlΓk A ∗ = Sk,A(Φ, B) ∗ , (80) δΦ δφA δφA where the notation

′′ δ Φˆ A = ΦA + i (Γ −1)AB l , (81) k δΦB

′′−1 ′′ has been used. The matrix (Γk ) is inverse to the matrix Γk, the last has elements

′′ δ δ Γ (Γ ) = l r k , i.e., k AB δΦA δΦB ′′−1 AC ′′ A Γk · Γk CB = δ B . (82) Now consider the variation of the extended generating functional of Green functions under infinitesimal variation of the gauge fixing functional, Ψ(φ, B) → Ψ(φ, B)+ δΨ(φ, B). We find

∗ δZk(J, φ , B)=i Dφ δΨ(φ, B)ˆs(φ, B) exp{i[SFP (φ, B) Z ∗ + Sk(φ,B)+ Jφ + φ (ˆsφ)]}. (83)

Now take into account that the functional integral of total variational derivative is zero we have the relation δ 0= Dφ r δΨsA exp{i[S (φ, B)+ S (φ, B)+ Jφ + φ∗(ˆsφ)]} δφA FP k Z h i A ∗ = Dφ iδΨs JA + Sk,A + δΨˆs exp{i[SFP (φ, B)+ Sk(φ, B)+ Jφ + φ (ˆsφ)]}, (84) Z h i where the BRST invariance of SFP action, the nilpotency of BRST transformations and the relations (75) have been used. From (84) one has

i Dφ δΨ(φ, B)ˆs(φ, B) exp{i[SFP (φ, B)+ Sk(φ, B)+ Z ∗ A Jφ + φ (ˆsφ)]} = Dφ JA + Sk,Aφ, B s (φ, B)× Z ∗ × δΨ(φ, B) exp{i[SFP (φ, B)+ Sk(φ, B)+ Jφ + φ (ˆsφ)]}, (85)

11 ∗ which allows to present the Eq. (83) in the form closed with respect to Zk(J, φ , B),

δ δ δ δZ (J, φ∗, B)= − i J + S l , B l δΨ l , B Z (J, φ∗, B), (86) k A k,A iδJ δφ∗ iδJ k A or, in terms of Wk(J, B),

δ W δ δ δ W δ δW (J, φ∗, B)= − J + S l k + l , B l δΨ l k + l , B . (87) k A k,A δJ iδJ δφ∗ δJ iδJ A In deriving (87) the modiﬁed Slavnov-Taylor identity (78) has been used. The last equation ∗ can be rewritten for the background average eﬀective action, Γk(Φ,φ , B), in the form

∗ δrΓk δl ˆ ˆ δl ˆ δΓk(Φ,φ , B)= A ∗ δΨ(Φ, B) − Sk,A(Φ, B) ∗ δΨ(Φ, B), (88) δΦ δφA δφA where Φˆ was introduced in (81). From Eq. (88) follows that

∗ δΓk(Φ,φ , B) Γ =6 0. (89) δ k δΦ =0

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.

6 Conclusions

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 background invariance of the regulator actions and background average effective action in the framework of the background field method for a wide class of regulator functions which include (43), but can be generalized to any other functions of the arguments z. As a new technical result we formulated general conditions of regulator actions being invariant with respect to the purely background transformations. The main motivation of this work was to check whether the on-shell dependence of the average effective action [15] holds within the background field method formalism. The answer to this question is given by the relation (89) and is strictly positive. This output does not contradict the recent works [8, 14] because in these publications the subject of study was the gauge invariance of background average effective action, and the question of gauge fixing dependence was not investigated. From our viewpoint, the on-shell gauge dependence of the average effective action is a fundamental principal difficulty of the FRG approach applied to the Yang-Mills theories. We have confirmed that the situation does not improve in the background field 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 results obtained within the FRG formalism applied to Yang-Mills theories, and therefore it makes sense 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 gauge fixing is “physical” or “correct”, such that changing the gauge should be strictly forbidden. As far as FRG provides valuable nonperturbative results, the theoretically inconsistent formulation is 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. For instance, in the fixed point the background average effective action boils down to the standard QFT effective action and then S-matrix, amplitudes and all related observables are well-defined.

12 Unfortunately, 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 group flows and the last are supposed to be gauge-fixing dependent, it is unclear how the fixed-point invariance can be actually used. Finally, there is an alternative formulation of the FRG in gauge theories which is gauge-fixing independent, exactly as a conventional perturbative QFT is [15]. This scheme is technically more difficult, since the regulator actions are constructed in a more complicated way, that includes composite fields. At least by now, the disadvantage of this approach is that there is no 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 supported partially by the Ministry of Education and Science of the Russian Federation, grant 3.1386.2017 and by the RFBR grant 18-02-00153. This work of I.L.Sh. was partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico - CNPq under the grant 303893/2014-1 and Funda¸cão de Amparo àPesquisa de Minas Gerais - FAPEMIG under the project APQ- 01205-16. E.A.R. is grateful to Coordena¸cão de Aperfei¸coamento de Pessoal de N´ıvel Superior - CAPES for supporting his Ph.D. project.

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