Multiplicative renormalization of Yang–Mills theories in the background-field formalism

In the paper, within the background field method, the structure of renormalizations is studied as for Yang–Mills fields interacting with a multiplet of spinor fields. By extending the Faddeev–Popov action with extra fields and parameters, one is allowed to establish the multiplicative character of the renormalizability. The renormalization of the physical parameters is shown to be gauge-independent.


Introduction
When quantizing non-Abelian gauge field theories [1], whose gauge transformations form a group, one is naturally based on the Faddeev-Popov method [2]. It is a characteristic property of the Faddeev-Popov gauge-fixed action that the latter is invariant under global BRST supersymmetry [3,4], which, in turn, can be expressed in the form of the Zinn-Justin equation [5] for the Faddeev-Popov action. At the quantum level, the BRST symmetry as expressed in terms of the effective action, implies the Slavnov-Taylor identities [6,7] to hold. Further generalization as to the quantization of gauge theories, including the cases of field-dependent structure coefficients, as well as open and/or reducible gauge algebra, is described by the field-antifield BV formalism [8,9]. In that formalism, the effective action is BRST invariant by construction, and thus satisfies the master equation which provides for the gauge invariance of the physical sector of the theory [8,9].
An interest to the gauge dependence problem did appear from the study of the effective potential, which appeared to be gauge-dependent in Yang-Mills theories with spontaneous breaking of the symmetry, when calculating physicallya e-mail: batalin@lpi.ru b e-mail: lavrov@tspu.edu.ru c e-mail: tyutin@lpi.ru sensible results (the energy of the ground state, the masses of the physical particles, and so on) [10,11]. In Refs. [12,13] it was established that the energy of the ground state was gaugeindependent. Later, it was proved [14,15] that in Yang-Mills theories the dependence of gauge parameters in the effective action could be described in terms of gauge-invariant functional whose arguments (fields) were gauge-dependent (see also recent Refs. [16,17] devoted to that problem as resolved via the procedure of redefinition of the field variables, found in [14,15]). Notice that in the general case of gauge theories, a variation in gauge condition is described in the form of certain change of the field variables (in terms of anticanonical transformations) [18,19].
Although there are many papers devoted to various aspects of renormalizability of Yang-Mills theories, gauge dependence of renormalization constants has been studied explicitly only as for the gauge field sector [20]. In the present paper, within the background field formalism, it is studied a multiplicative renormalization procedure and gauge dependence as for Yang-Mills fields interacting with a multiplet of spinor fields. It is shown that renormalizations of physical parameters of the theory are gauge-independent.
The paper is organized as follows. In Sect. 2, it is discussed the action of Yang-Mills fields and spinor fields in the standard approach and in the background field method; it is also introduced extended action, which leads in the background field method to a multiplicative renormalizable theory of the fields considered; it is also studied the symmetry of the extended action. In Sect. 3, it is established the structure and the arbitrariness is described as for any local functional with the quantum numbers of the extended action that satisfies the same set of equations as the extended action. In Sect. 4, the equations are derived for the generating functional of vertexes (effective action), as a consequence at the quantum level, of the symmetry property of the extended action; and it is shown that the generating functional of vertexes satis-fies the same equations as the extended classical action. In Sect. 5, it is studied the renormalization procedure of the theory considered when using the loop expansion technique and the minimal subtraction scheme; and thus the multiplicative renormalizability of the theory is proved. In Sect. 6, the relations are found between the parameters of the renormalized action and the standard renormalization constants of fields and vertexes of the interaction, and renormalized physical parameters are shown to be gauge-independent. Concluding remarks are given in Sect. 7.
Condensed DeWitt's notations [21] are used through the paper. Functional derivatives with respect to field variables are understood as the left. Right derivatives of a quantity f with respect to the variable ϕ are denoted as f ← − δ δϕ .

Extended action for Yang-Mills theories
Let us consider a gauge theory of non-abelian vector fields A α μ = A α μ (x) and spinor fields ψ where the notations Here γ μ are the Dirac matrices, g and m are the coupling constant of gauge interaction and the mass parameter of spinor field, respectively. The action (2.1) is invariant under gauge transformations with gauge parameters The corresponding Faddeev-Popov action [2] S (1) F P = S (1) F P (A, , C, C, B, ξ) in the Feynman gauge has the form where ξ is a constant gauge parameter, are auxiliary fields introducing a gauge fixing condition. The action (2.5) is invariant under global supersymmetry (BRST symmetry) [3,4], where λ is a constant anticommuting parameter.
In the background field formalism [22,23] a gauge field A α μ entering the classical action (2.1) is replaced by A α μ +B α μ , where B α μ is considered as an external vector field. The Faddeev-Popov action is constructed by using the modified Feynman gauge (the background gauge condition), and reads This action is invariant under BRST transformations of the form (2.6) with the following modification of the transformation law in the gauge field sector, The invariance property of Faddeev-Popov actions (2.5) and (2.8) under BRST transformations can be described in the form of non-linear functional equations for the extended action S ext with the help of additional variables (antifields) A * α μ , ψ * j , ψ * j , C * α , C * α , being sources to the generators of BRST transformations, where Q means the set of the fields {A α μ , ψ j , ψ j , C α } and the symbol Q * is used to indicate the set of the corresponding antifields for fields Q, wherein the BRST transformations (2.4), (2.6) are presented as δ λ Q = R (a) Q λ, a = 1, 2. Then, as a consequence of the BRST symmetry, the actions S  To study the structure of renormalizations it is convenient to extend the original set of the variables with extra fields and auxiliary quantities. An initial action, we proceed from, when studying the structure of renormalizations and dependence of renormalization constants on gauge fixing is the extended action S ext = S ext (Q, Q * , C, B, B, ξ, θ, χ), where θ α μ = θ α μ (x) are anticommuting extra fields and χ is a constant nilpotent parameter. 1 The action (2.12) is invariant (δS ext = 0) under the following transformations of the quantities entered, 14) 1 These extra variables have been used first in Ref. [20].
Due to the variations (2.13)-(2.20), the invariance condition of the action rewrites Also, the action (2.12) satisfies the equation where the notation is used for the operator describing the gauge transformations of the variables B μ , ψ, ψ and simultaneously the tensor transformation of fields and antifields A μ , C, C, B, θ μ , A * μ , ψ * , ψ * , C * . Finally, we notice that the action (2.12) satisfies the two important relations linear in fields A μ , B and also in derivatives of variables B, C, A * μ , The Eq. (2.25) means that the action S ext (2.12) depends on variables A * α μ "Z ", " p" C α in combination A * α μ − D αβ μ (B)C β only when θ β μ = 0. We give the table of "quantum" numbers of fields, antifields, auxiliary fields and constant quantities which have been used in constructing S ext : where "ε" describes the Grassmann parity, the symbol "gh" is used for the ghost number, "dim" denotes the canonical dimension and "ε f " means the fermion number. Using this table of "quantum" numbers it is easy to establish quantum numbers of any quantities found in the text.

General structure of renormalized action
It is to be proved below that the renormalizable action is a local functional of field variables, carries the quantum number of the action S ext ( where P(x) is a local polynomial in all variables Q, Q * , C, B, B, ξ, θ, χ with dim(P(x)) = 4. Require the functional P to satisfy the Eqs. (2.21)-(2.25) with substitution S ext → P, and let P be of the form P = P 00 + P (1) + χ P (2) , (3.2) where ε(P (1) ) = 0, gh(P (1) ) = 0, dim(P (1) ) = 0, ε f ((P (1) )) = 0, (3.4) ε(P (2) ) = 1, gh(P (2) ) = −1, dim(P (2) ) = −1, ε f ((P (2) )) = 0, (3.5) and the functionals P (1) and P (2) do not depend on χ . It follows from the Eq. (2.24) for P, and representation (3.3) that P (1) and P (2) do not depend on B α , "Z ", " p", By introducing new variables A * α μ (x), we define new functionalsP (k) by the rulẽ to find thatP (k) do not depend on the fields C α , In the relations (3.8) and (3.9) the following notations are used. Independence of functionalsP (k) of the fields C α and (3.11) allow one to write down the following set of equations as for When studying the structure of functionals and further investigating it appears useful a consequence of the Eq. (3.13) at ω α = const, We refer to equations of the form (3.15) as the ones of the T -symmetry for the corresponding functional.
Using the properties of the functionalP (2) (3.5), its locality as well as axial symmetry, Poincare-and T -symmetries we find the general representation, where Z i , i = 1, 2, 3, 4, "Z ", " p" Z 1 are arbitrary constants. Further, when using the Eq. (3.13) forP (2) we get that Z 1 = 0. The final expression forP (2) has the form Notice that the functionalP (2) does not depend on the fields θ α μ . By taking (3.18) into account the Eq. (3.12) reduces to the following one 2ξ ∂ ∂ξP (1) describing the dependence of renormalization constants on the gauge parameter ξ . We refer to the Eq. (3.12) as an extended master-equation and to (3.19) as a gauge dependence equation.

Solution to the extended master-equation
Now we consider a solution to the extended master-equation (3.12) for the functionalP (1) as presented it in the form The functionalP (1) θ rewrites as (3.21) and the functionalsP ψ ,P (1) * do not depend on the fields θ α μ . By taking into account the properties dim(P α μθ ) = 2, gh(P α μθ ) = −1, ε(P α μθ ) = 1, ε f (P α μθ ) = 0, as well as the Poincare-and T -symmetries of the functionalP (1) θ , we find that where Z 5 is an arbitrary constant. The functionalP (1) * is linear in the antifields * (3.10), and the functionalsP (1) AB andP (1) ψ do not depend on the antifields * . The functionalP (1) * can be represented in the form By using the arguments analogous to those led us to the structure of the functionalP Taking into account the gauge symmetry in the external field B (see the Eq. (3.13)), we find that Z α 7βγ = 0. The quantities "Z " introduced in (3.24)-(3.27) are constants that satisfy the equations, In its turn, taking into account the axial symmetry, the Poincare-and the T -invariance we determine the general structure of the functionalP where constants Z α 12 jk satisfy the equations The contribution to theP so that it follows from the Eq. (3.13) that the equalities Z α 11 jk = 0 and hold. Notice that in the case Z α 12 jk = Z 12 t α k j the Eq. (3.33) are fulfilled and the functionalP (1) ψ (3.35) satisfies the Eq. (3.13). Insert the representation for the functionalP (1) in the form (3.20) into the Eq. (3.12). Then, analysis of the θψψ components in the extended master-equation (3.12) yields and the possibility to represent the functionalP where the notation and to the representatioñ Consideration of the A * ACC components in the Eq. (3.12) gives the relations and the representation for the functionalP (1) C * in the form Studying the ψψ∂C, ψψBC and mψψC components in the Eq. (3.12) lead to the relations and, as a consequence, to the representation of the functionals P The functionalP (1) AB depends on the fields A and B only. The θ AB components in the Eq. (3.12) allow us to conclude that the functionalP (1) AB depends on the fields A and B only in combination (3.38), Finally, consideration of the ABC components in the Eq. (3.12) leads to equations for the functional X (U ) (3.46) The required solution to the Eq. (3.47) can be written in the form Thus the general solution to the extended master-equation, P (1) , is constructed. It is defined by fifth independent arbitrary constants Z 5 , Z 6 , Z 11 , Z 13 , Z 14 and has the form , the equality (the initial condition), Henceforth we use the notatioṅ (3.54) Considering the ψγ D ψ (U )ψ components in the Eq. (3.19), we obtain Analyzing the mψψ components in the Eq. (3.19), we find By making use of the change of constants "Z " Analysis of the ψ * tψC, ψ * t t ψC and C * f CC components in the Eq. (3.19) gives no new information. Below, in Sect. 5 find that all constants "Z " can be interpreted as renormalization constants which are uniquely defined from the conditions of divergence elimination.
Let us formulate the results obtained in that Section in the form of a lemma. C, B, B, ξ, θ, χ),

Lemma Let
where P 00 is given by the formula (3.3), P (1) and P (2) do not depend on B α and χ and are functionals of arguments , * , B, C, ξ , θ ,

Generating functional of vertex functions
It is convenient to define the generating functional of Green functions by making use of the action functional P constructed in the previous Section as the action yields then a finite theory certainly. In what follows we re-denote the functional P, P ≡ S R , and, respectively, The generating functional of Green functions is given by the functional integral, so that all the functionals S R,l are linear combination of a single set of monomials, where {S i , i = 1, . . . , I } is a sub-set of monomials which the action S ext is expanded in, and a l,i are constant coefficients for l-loop order. The generating functional of vertex Green functions (effective action) is defined by the Legendre transformation has the quantum numbers ε( ) = 0, gh( ) = 0, dim( ) = 0, ε f ( ) = 0, and satisfies the relations (4.5) Functional average of the Eqs. (2.21)-(2.25) with substitution S ext → S R yields the corresponding equations for the functional = ( m| , L), copying the equations for S R , where ←− H α m| ω α is given by the expression (2.23) with the replacement → m| , Represent the functional in the following form where (4.11) and the functionals (1) and (2) do not depend on the parameter χ . Due to the structure chosen for the functional (4.11) it follows from the Eqs. (4.8) and (4.9) that the functionals (1) and (2) do not depend on the fields B α m| , and satisfy the equations In its turn, the Eq. (4.6) splits in the two, one of which is closed as for the functional (1) , −θ α μ δ δB α μ (1) = 0, (4.14) and the second includes both the functionals and describes their dependence on the gauge parameter ξ , 2ξ ∂ ∂ξ (1) The Eq. (4.7) rewrites now in the form of the two equations as for the functionals (1) and (2) , (4.17) As for the Eq. (4.13), it is convenient to introduce the variables (4.18) and to use the following convention as for the sake of uniformity. Also, introduce the new func-tionals˜ (k) by the rule, (4.20) where the notation = {Q, ψ * , ψ * , C * , ξ, θ}, is used. With the definitions (4.18)-(4.20) taken into account, we have Then, we find from the Eqs. 2ξ ∂ ∂ξ˜ (1) = (˜ (1) ,˜ (2) ) (4.29) where the notation for the antibracket [8,9] is used, where the operator ← − h α m| ω α is defined in the equality (3.14) with the replacement → m| , * → * m| . Then, when studying the tensor structure of divergence parts of the generating functional of vertexes, it is convenient to use a consequence of the Eq. (4.32) in particular case ω α (x) = const, i.e. as to a global T m| -symmetry: where the operators ← − T α m| are defined by the Eq. (3.16) with the replacement → m| , * → * m| .

Renormalization
In that section we study the structure of renormalizations, and show the multiplicative character of the renormalizabiliuty of the model considered. The main role in that study is given to resolving the extended master-equation (3.12) and the one (3.19) describing the gauge dependence. We show that the renormalized quantum action and the effective action satisfy exactly their master equations to each subsequent order in loops. In this resolving, the structure of the renormalized quantum action is determined by the same monomials in fields and antifields as it does for the non-renormalized quantum action with constants determined by the divergencies of the effective action. For the sake of notational simplicity, we omit lower case m| of any arguments of any functionals.

(l+1)-loop approximation
We carry out the proof of the multiplicative renormalizability via the mathematical induction method in the framework of loop expansion of the effective action with the use of the minimal subtraction scheme. To this end we suppose that we managed to find such parameters Z [l] i , η n z i,n , i = 5, 6, 11, 14, 15, 16, (5.12) that the l-loop approximation for , [l] = l n=0 η n n , is a finite functional. We are to show that it is possible to pick up the l + 1-loop approximation for Z i , 6,11,14,15,16, z 14,l+1 =ż 15,l+1 = 0, (5.13) which does compensate the divergences of l +1-loop approximation for the functional . Represent the action S R in the form (5.14) where S [l] R is the action S R with independent parameters Z i replaced by Z [l] i , and satisfying the Eqs. (2.21)-(2.25), and the functional s l+1 reads For the functional s (1) l+1 we use the representation where s θ,l+1 = z 5,l+1 0θ , s ψ,l+1 = z 11,l+1 0ψ|1 − z 5,l+1 A∂ A 0ψ|1 +(z 11,l+1 + z 15,l+1 ) 0ψ|2 , (5.21) In its turn, the functional s (2) l+1 has the form s (2) l+1 = dx 2ξż 5,l+1 A * A + 2ξ(ż 6,l+1 −ż 5,l+1 )C * C +ξ(ż 11,l+1 + z 16,l+1 )ψ * ψ + +ξ(ż 11,l+1 − z 16,l+1 )ψ * ψ . (5.23) Here (and below in this section) we use the abbreviation to denote the variational derivatives of the kind when it does not cause an ambiguity. Let us study the structure of the functional with the accuracy including the (l + 1)-loop approximation. It is described by the diagrams with vertexes from the action S R with parameters z i,n , i = 5, 6, 11, 14, 15, 16, 0 ≤ n ≤ l + 1, or, in other words, by vertexes from the action S [l] R and from the summand s l+1 . As we are interested in diagrams of the loop order not higher than l +1, the vertexes from s l+1 cannot appear in loop diagrams, i.e. vertexes from s l+1 give the "tree" contribution to , equal to η l+1 s m|,l+1 . Other diagrams are generated by the action S [l] R . Let (S [l] R ) be the contribution of those diagrams into the functional , i.e.
By repeating the calculations of Sect. 3 we find that R ) are, by assumption, finite to the n-loop approximations, 0 ≤ n ≤ l, we obtaiñ so that the functionals˜ (k) (S [l] R ) l+1,div are local ones of arguments with the quantum numbers of the action S ext and contain divergent terms only (the minimal subtraction scheme). Then, as a consequence of the Eqs. (4.28), (4.29) (4.32), they satisfy the following equations, l+1,div , Notice that the form of the Eqs. (5.31)-(5.34) does not depend on the label l. By taking into account the quantum numbers, axial-, Poincare-, T -symmetries, the general expression for local functional˜ (S [l] R ) (2) l+1,div , reads where q i,l+1 , i = 1, 2, 3, 4, "Z ", " p" q 1,l+1 are arbitrary constants. Then, by using the Eq. (5.33) for˜ (S [l] R ) (2) l+1,div , we find that q 1,l+1 = 0. The final expression for˜ (S [l] R ) (2) l+1,div has the form Notice that the functional˜ (S [l] R ) (2) l+1,div does not depend on the fields θ and B.

Solution to equation (5.31) for˜ (S
Consider a solution to the Eq. (5.31) for the functional With this aim, we find first the general form of the functional (1) l+1,div , using the locality, the quantum numbers, axial-, Poincare-, T -symmetries and partially the gauge symmetry in the external field B. In fact, all required calculations do copy ones performed in Sect. 3 when constructing the general form of the functionalP (1) [see formulas (3.21)-(3.35) with the obvious replacements likeP (1) θ → M θ ]. Here, we reproduce the final results only. The functional M θ,l+1 has the form M θ,l+1 = q 5,l+1 dxA * α μ (x)θ α μ (x) = q 5,l+1 dxA * θ.
which is not more than linear in antifields.
As the coefficient of the A * D(B)CC vertex of the Eq. (5.56) should be zero, it follows that q α 8,l+1,βγ = q 8,l+1 f αβγ , q 8,l+1 = q 7,l+1 = q 6,l+1 −q 5,l+1 . (5.57) Next, we consider the equations which follow from (5.56) for zero-valued antifields. They split into the two sets of equations. In the first set of equations, all vertexes contain the spinor fields. In the second ones, vertexes are constructed of the fields A, B and their coordinate-derivatives only.

Solution to equation (5.37) for˜ (S [l]
When inserting the representation for the functional (1) (S [l] R ) l+1,div given by (5.64)-(5.71) into the Eq. (5.37), it takes the form of zero value for some linear combinations of structures appeared in the right-hand side of formulas (5.65)-(5.71).

(l+2)-loop approximation
The renormalization of S R to the (l + 1)-loop approximation allows one to construct the effective action , finite to that approximation; however it does not satisfy exactly the extended master-equation and the gauge dependence equation, by itself. We show the possibility to complete the renormalization constants of the action S R with the help of the (l + 2)-loop approximation, so that it will satisfy the equations mentioned to the (l + 1)-loop approximation and, in its turn, the corresponding effective action, finite to the (l + 1)loop approximation, will satisfy the set of Eqs. (2.21)-(2.25) to that approximation.
Indeed, we represent the action S R as R is the actionS R with independent parameters "Z ", " p" Z i replaced by Z [l+1] i , and s l+2 is equal to Further calculations and consequences from them do copy exactly the results of the previous subsection with the natural replacement l + 1 → l + 2.
Also, it is obvious that the procedure of divergence compensations discussed can be applied to the case l = 0 so that by using the loop induction method in Feynman diagrams for the functional , we arrive at the following statement: for the l-loop approximation [l] , where l is arbitrary positive integer, such that the functional [l] does not contain divergences and satisfies the Eqs. (4.6)-(4.9).

Relations between parameters of S R and standard renormalization constants
In that section we find relations between some parameters of the action S R and the standard renormalization constants. Within the expression for S R , we restrict ourselves only by desired vertexes in symbolic notation where the ellipsis means the rest vertexes. As the propagators of fields A and ψ are finite, they should be considered as renormalized fields. Then, we find: where Z A and Z ψ are the renormalization constants of the bare fields A 0 and ψ 0 . The coefficient of the second vertex in the expression (6.1) gives the renormalization for vertex A 3 , 3) The coefficient of the forth vertex in the expression (6.1) gives the renormalization for vertex ψψ, It follows from the Eq. (5.103) that the renormalization constants of physical parameters g and m do not depend on gauge, ∂ ξ Z g = 0, ∂ ξ Z m = 0. (6.5)

Summary
In the present paper, within the background field formalism, it is studied the renormalization procedure and the gauge dependence of the theory of Yang-Mills fields interacting with a multiplet of massive spinor fields. It is shown that the extension of the Faddeev-Popov action with extra fields and parameters allows one to establish the multiplicative character of the renormalizability. The proofs given above are based on the possibility to expand the effective action in loops, as well as to use the minimal subtraction scheme as to eliminate divergences. It is a new and important result that the renormalization constant of the mass parameter is shown to be gauge-independent.