Gauge dependence and multiplicative renormalization of Yang-Mills theory with matter fields

In the paper, within the background field method, the renormalization and the gauge dependence is studied as for an SU(2) Yang-Mills theory with multiplets of spinor and scalar fields. By extending the quantum action of the BV-formalism with an extra fermion vector field and a constant fermion parameter, the multiplicative character of the renormalizability is proven. The renormalization of all the physical parameters of the theory under consideration is shown to be gauge-independent.


Introduction
When constructing modern models of fundamental interactions [1], non-Abelian gauge field theories [2] play a central role. Any gauge theory can be quantized in a covariant way within the BV-formalism [3,4] involving the gauge-fixing procedure as an important tool. Although at the quantum level the gauge symmetry is broken, nevertheless, it causes the existence of fundamental global supersymmetry known as the BRST symmetry [5,6]. The respective conserved fermion nilpotent generator is known as the BRST charge [3,4] responsible for correct construction of physical state space [7,8]. Due to the equivalence theorem [9] and the BRST symmetry, it is succeeded to prove that the physical S-matrix is independent of the choice of gauge fixing, The gauge dependence problem did appear by itself from the study of the effective potential, which appeared to be gauge-dependent in Yang-Mills theories with the spontaneous symmetry breaking, when calculating physically-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 gauge-independent. 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]).
Our investigation of renormalization and gauge dependence in an SU(2) Yang-Mills theory with spinor and scalar fields is based on using the background field formalism [18,19,20]. This formalism is the popular method for quantum studies and calculations in gauge theories because it allows one to work with the effective action invariant under the gauge transformations of the background fields, and to reproduce all usual physical results which can be obtained within the standard quantization approach. Various aspects of quantum properties of Yang-Mills theories and quantum gravity theories have been successfully studied in this technique [21,22,23,24,25,26,27,28] (among recent applications see, for example, [29,30,31,32]).
Although there are many papers devoted to various aspects of renormalizability of Yang-Mills theories, the gauge dependence of the renormalization constants has been studied explicitly only as for the gauge field sector [22]. Recently, we have studied a multiplicative renormalization procedure and a gauge dependence as for an Yang-Mills theory with a multiplet of spinor fields based on an arbitrary simple compact gauge Lie group, and proved the gauge independence of the physical parameters of the theory [31]. The main goal of the present paper is to include scalar fields into a gauge model as to have the possibility of generating masses to physical particles through the spontaneous symmetry breaking [33]. To simplify the presentation and calculations, we restrict ourselves by the simplest Yang-Mills theory with SU(2) gauge group. Generalization of our results to another gauge group can be made straightly.
The paper is organized as follows. In Section 2, the action of SU(2) Yang-Mills theory with multiplets of spinor and scalar fields in the background field method is extended with the help of additional fermion vector field and constant fermion parameter which allows one later to arrive at a multiplicative renormalizable theory. The symmetries of this extended action are studied and presented in a set of equations for that action. In Section 3, it is established the structure and the arbitrariness is described for any local functional with the quantum numbers of the extended action that satisfies the same set of equations as the extended action does. In Section 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 satisfies the same equations as the extended quantum action does. In Section 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 Section 6, the gauge independence of all physical parameters of the theory under consideration to any order of loop expansions is found. Concluding remarks are given in Section 7.
Condensed DeWitt's notations [34] 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 quantum action
Let us consider an SU(2)-gauge theory of non-Abelian vector fields A α µ = A α µ (x), a doublet of spinor fields ψ j = ψ j (x), ψ j = ψ j (x) and a triplet of real scalar fields ϕ α = ϕ α (x) in the d=4 Minkowski space-time, with the action where the notations γ µ are the Dirac matrices, σ α jk are the Pauli matrices and g is a gauge coupling parameter, λ is a coupling constant of the scalar field, ϑ is a coupling constant of the scalar and spinor fields, m and M are mass parameters of the spinor and the scalar fields, respectively.
The action (2.1) is invariant under the SU(2)-gauge transformations with gauge parameters Notice that the form (2.4) of the polynomial S 5 (ϕ) is uniquely determined by the invariance requirement under global SU(2)-transformations.
In the background-field formalism [18,19,20] the gauge field A α µ appearing in the classical where B α µ is considered as an external vector field. Effective action for functional integral in quantum theory is constructed with using the BV-formalism [3,4].
To study a renormalization structure and a gauge dependence of renormalization constants we use an extended action. This action S ext = S ext (Q, Q * , C, B, B, ξ, θ, χ) is constructed by introducing additional fermion fields θ α µ = θ α µ (x) and a constant fermion parameter 4 χ, and has the form where Q means the set of fields {A α µ , ψ j , ψ j , ϕ α , C α }, the symbol Q * is used for the set of corresponding antifields and R Q are generators of the BRST transformations [5,6], (2.10) 4 For the first time such additional variables were used in [22].
The action (2.9) is invariant, under the following global supersymmetry transformations of the variables (here {Π I } is the set of all the variables entering the action, ǫ is a constant fermion parameter of the transformation, ε(ǫ) = 1, ǫ 2 = 0)), Taking into account the right-hand sides of the relations (2.11) -(2.14) and omitting indices of all variables, the invariance condition of the action S ext rewrites in the form of the following equation Also, the action (2.9) satisfies the equation is the operator of the gauge transformations in the sector of the fields B µ , ϕ, ψ, ψ and, at the same time, of the tensor transformations in the sector of the fields A µ , C, C, B, θ µ , A * µ , ϕ * , ψ * , ψ * , C * .
Finally, let us notice the two important relations linear in the fields A µ , B and their derivatives, which the action (2.9) satisfies to, The equation (2.24) means that the action S ext (2.9) depends on the variables A * α µ and C α only in combination A * α µ − D αβ µ (B)C β when θ β µ = 0. The action S ext can be represented in another useful form, are used. Note that the functionals Γ 0|0 − Γ 0|21 are homogeneous with respect to fields Ω and antifields Ω * while the functional Γ 0|22 does not obey this property and in symbolic notation has the form where Γ 0|22,l is homogeneous functional with respect to variable A of order l and T l (B) is a tensor (differential operator) with l gauge and l Lorentz indices. In what follow we do not need in explicit representation of quantities T l (B).
Later on, we will see that the generating functional of vertex functions (effective action), counterterms and renormalized action satisfy the same equations (2.20), (2.21), (2.22), (2.23), (2.24), as the action S ext does. Moreover, the counterterms and the renormalized action are linear combinations of the same vertices as the action S ext is. Explicit form of these vertices will be given in the next section. Now we give the table of "quantum" numbers of fields, antifields, auxiliary fields and constant parameters used in construction of S ext : where "ε" describes the Grassmann parity, the symbol "gh" is used to denote the ghost number, "dim" means the canonical dimension and "ε f " is the fermionic number. Using the table of "quantum" numbers it is easy to establish the quantum numbers of any quantities met in the text.

General structure of renormalized action
It will be shown below that the renormalized action is a local functional of field variables with quantum numbers of the action S ext (2.9), which satisfies the same equations (2.20) -(2.24) as the action S ext does. In this section, we find a general solution to the equations (2.20) -(2.24) under the conditions pointed out above.
So, let P be a functional of the form where P (x) is a local polynomial of all variables Q, Q * , C, B, B, ξ, θ, χ with dim(P (x)) = 4. Let the functional P satisfy the equations (2.20) -(2.24) (with substitution S ext → P ), and we represent it in the form where the functional Γ 0|0 is defined in (2.26), the functionals P (1) , P (2) do not depend on χ and obey the properties From the equation (2.23) for P and the presentation (3.2) it follows that P (1) and P (2) do not depend on fields B α , With the help of new variables A * α µ (x), A α µ (2.37), we define new functionalsP (k) by the rulẽ and find thatP (k) do not depend on fields C α , Omitting the indices of all variables in relations (3.6) and (3.7), the following notations are used. Independence of the functionalsP (k) of the fields C α and the relations allow-down us to write the following set of equations forP (k) When studying the structure of functionalsP (k) and in further research, it is helpful to have a consequence from equation (3.12) that corresponds to the case when ω α = const, We refer to the equation of the form (3.14) as the ones of the T -symmetry for the corresponding functionals.
Using the properties of the functionalP (2) : its locality and (3.4) as well as axial symmetry, Poincare-and T -symmetries, we find the general representation ofP (2) where Z i , i = 1, 2, 3, 4, 5 and Z ′ 1 are arbitrary constants and the functionals Γ 0|k , k = 1, 2, 3, 4, 5 were introduced in (2.27), (2.28). Later on any quantities "Z" with any set of indices do not depend on coordinates x and field variables.
Further, when using the equation (3.12) forP (2) , we get that Z ′ 1 = 0. The final expression for theP (2) has the formP Notice that the functionalP (2) does not depend on the fields θ α µ and B α µ . By taking (3.17) into account the equation (3.11) reduces to the following one describing the dependence of renormalization constants on the gauge parameter ξ. We refer to the equation (3.10) as the extended master-equation and to (3.18) as the gauge dependence equation.

FunctionalP
(1) AB do not depend on antifields Ω * . The functionalP (1) Ω * can be presented in the form (3.23) Using the arguments similar to those that led us to establish the form of the functionalP depends on the variables Ψ, A, B, and is quadratic in Ψ. Taking into account the axial symmetry as well as the Poincare-and B-gauge invariance, we find the general structure ofP (3.26)

FunctionalP
(1) ϕ depends on the variables ϕ, A, B, and vanishes when ϕ = 0. Taking into account the Poincare-and the B-gauge invariance, we establish the general form of the functionalP fulfils then the functionalP (1) ϕ18 satisfies the equation (3.12). FunctionalP (1) ψϕ is an interaction vertex of the fields ϕ and Ψ. Taking into account the axial, Poincare-and T -symmetries, this functional has the following general form: .P The functionalP  .10) is reduced to solutions to the subequations which follow from the requirement for independent polynomial structures appearing in the left-hand side of the equation (3.10), to be equal to zero. In their turn, these subequations are reduced to algebraic equations for coefficients "Z" or, in two cases, to variational differential equations for the functionalP (1) AB (A, B). We explain the results obtained by using an example for the block This block should be understood as the following: the requirement for the structure θA * C to be equal to zero leads to equation (3.31), from which it follows the relation (3.32). Further Multiplying the equality (3.35) by A α µ and integrating then over x, we find that the relation (3.29) is satisfied and the functionalP Requirement for the rest structures in the left-hand side of the equation (3.10) to be equal to zero is satisfied identically.
Notice that the functionalP (1) can be represented as a linear combination of independent polynomials Γ 0|k ,P (3.44)

Solution to the gauge dependence equation
Consider now a solution to the equation (3.18) describing the dependence of constants "Z" on the gauge parameter ξ appearing in the general solution to the extended master-equation (3.10) for the functionalP (1) . For the functionalP (1) we will use the representation (3.44). Notice that every functional Γ 0|k , k = 6, ..., 21 is eigen for the operatorL (3.19), The equations (3.46) for independent constants Z k lead to the following relations and consequences, The last relation is equivalent to (3.48).
Below, in section 5, we find that all constants "Z" can be interpreted as renormalization constants which are uniquely defined from the conditions of reducing divergences. Now we formulate the results obtained in this subsection as the following lemma. Then the functional P has the form It should be noted that the functional P does not contain vertices additional to that from which the action S ext (2.9) is built up. Namely, the obvious relation is valid, (3.66) In its turn, the functional P (3.60) can be represented in the form analogous to (2.9) for the action S ext ,

Effective action
It is useful to define the generating functional of Green functions with the help of the functional P constructed in the previous subsection, because it allows one to obtain a finite theory from the beginning. In what follows we redefine the functional P , P ≡ S R , and, respectively, The generating functional of Green functions is given by the following functional integral and all functionals S R,l are linear combinations of the same set of polynomials {Γ 0|k , k = 0, 1, ..., 22} with the help of which the functionals S ext and P are presented.
The generating functional of vertex functions (effective action) is defined with the help of the Legendre transformation, with quantum numbers ε(Γ) = 0, gh(Γ) = 0, dim(Γ) = 0, ε f (Γ) = 0 and satisfies the relations Functional averaging of equations (2.20) -(2.24) with substitution S ext → S R leads to the equations for the functional Γ = Γ(Φ m| , L) copying the equations for S R , Represent the functional Γ in the form where the functionals Γ (1) and Γ (2) do not depend on the parameter χ. Thanks to the structure chosen for the functional (4.9), from the equations (4.7) and (4.8), it follows that the functionals Γ (1) and Γ (2) do not depend on the fields B α m| , δ δB m| Γ (k) = 0, Γ (k) = Γ (k) (Q m| , C m| , B, Q * , ξ, θ), k = 1, 2, (4.10) and satisfy the equations (4.11) In its turn, the equation (4.5) splits into two ones, one of them is closed with respect to Γ (1) , (4.12) and the second includes both functionals and describes their dependence on the gauge parameter ξ, Now, the equation (4.6) rewrites in the form of the two equations for the functionals Γ (1) and Γ (2) , Due to the equations (4.11) it is useful to introduce new variables 16) and to use the following agreement for sake of uniformity of further notations Also, let us introduce new functionalsΓ (k) by the rulẽ where the following notations were used. Taking into account the definitions (4.16) -(4.18) we have  Here and below we will use the notations Now, thanks to (4.18), (4.24), (4.25), the equations (4.12) and (4.13) rewrite as where the notation for antibracket [3,4] (F, is used. Further, taking into account (4.18), (4.24) and we find that where the operator ← − h α m| ω α is defined in the equality (3.13) with the substitution Ω → Ω m| , Ω * → Ω * m| .
Later, on when studying the tensor structure of divergent parts of the generating functional of vertex functions, it is useful to use the consequence of the equation (4.30) corresponding to the particular case of T m| -symmetry when ω α (x) = const, where the operators ← − T α m| are defined by the equalities (3.15) with substitution Ω → Ω m| , Ω * → Ω * m| .

Multiplicative renormalization
In this section, we study a structure of renormalization in the model under consideration, and find multiplicative character of renormalizability. The main role is played by solving the extended master-equation (3.10) and the equation describing the gauge dependence (3.18). We will show that the renormalized quantum action and the effective action satisfy this equation exactly to every order of loop expansions. The structure of the renormalized action is determined by the same monomials in fields and antifields as the non-renormalized quantum action does but with constants defined by the divergences of the effective action. For simplicity of notations, we will often omit the lower index m| in arguments of functional Γ.

(l+1)-loop approximation (order η l+1 )
The proof of multiplicative renormalizability will be given by the method of mathematical induction in loop expansions of the effective action applying the scheme of minimal subtractions. To this end, let us suppose we have found the parameters Z k ind , η n z k ind ,n , ∀k ind , z kcon,n = 0, ∀k con , 1 ≤ n ≤ l, (5.11) so that the l-loop approximation for Γ, Γ [l] = l n=0 η n Γ n , is a finite functional. We will show that the (l + 1)-loop approximation for Z k ind can be picked up so that, which compensates divergences of the (l + 1)-loop approximation of the functional Γ.
Represent the action S R in the form R is the action S R with independent parameters Z k ind replaced by Z (5.14) In this subsection we will use the short notations for variational derivatives of the type when it will not lead to uncertainties. Now let us study the structure of the functional Γ with taking into account the (l + 1)-loop approximation. It describes by diagrams with vertices of the action S R with parameters z k ind ,n , 0 ≤ n ≤ l + 1, z k ind ,0 = 0, i.e. with vertices of action S R and with vertices of s l+1 . Because we are interested in diagrams of loop order not higher than l + 1, the vertices from s l+1 cannot appear in loop diagrams, i.e. the vertices from s l+1 give "tree" contribution to Γ equal to η l+1 s l+1 . The rest diagrams are generated by the action S  (5.17) Repeating calculations made in section 3 we find that R |Ω m| , Ω * m| , B, ξ, θ), k = 1, 2, (5.18) and functionalsΓ (k) (S R ) are finite to n-loop approximations, 0 ≤ n ≤ l, by proposition, we obtaiñ To simplify the presentations, we introduce the notations Notice that the form of equations (5.22) -(5.25) does not depend on index l.

Solution to equation (5.22)
Consider a solution to the equation (5.22) for the functionalΓ To this end, we will find the general form of the functionalΓ For the functionals linear in antifields we find where the relation q 11,l+1 = q 10,l+1 was used.
Thus, the divergences u (1) l+1 can be represented in the form and the functional M AB,l+1 is given by the formula (5.48). As it was noted above, all functionals considered as polynomials are independent, and polynomials Γ 0|6 − Γ 0|21 are homogeneous with respect to the fields Ω and the antifields Ω * .
It is evident that this method works for any l, in particular for l = 0, so that by using the loop induction method in Feynman diagrams for the functional Γ, we arrive at the following statement: for the l-loop approximation Γ [l] of the functional Γ defined by the relations (4.2),

Gauge independence of physical parameters
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 S R = dx Z 22 Z −2 6 ∂A∂A + gZ 22 Z −3 6 A 2 ∂A + Z 13 ψ∂ψ + mZ 15 ψψ + Z 16 ∂ϕ∂ϕ+ +M 2 Z 19 ϕ 2 + λZ 20 ϕ 4 +ϑZ 21 ϕψψ... , (6.1) where the ellipsis means the rest vertexes. As the propagators of the fields A, ψ and ϕ are finite, they should be considered as renormalized fields. Then, we find: where Z A , Z ψ and Z ϕ are the renormalization constants of the bare fields A 0 , ψ 0 and ϕ 0 . The coefficient of the second vertex in the expression (6.1) gives the renormalization for vertex gA 2 ∂A, Z gA 2 ∂A = Z 22 Z −3 6 . It follows from the equations (5.75) that the renormalization constants of physical parameters g, m, M 2 , λ and ϑ do not depend on gauge, (6.10)

Summary
In the present paper, within the background field formalism [18,19,20], we have studied the renormalization and the gauge dependence of the SU(2) Yang-Mills theory with the multiplets of massive spinor and scalar fields. The corresponding master-action of the BV-formalism [3,4] has been extended with the help of additional fermion vector field θ and fermion constant parameter χ. The action introduced is invariant under global supersymmetry and gauge transformations caused by the background vector field B appearing in the background field formalism. These symmetries allowed one to reduce, at the quantum level, the analysis of the renormalization and the gauge dependence problem for solutions to the extended masterequation and the gauge dependence equation. In comparison with our previous investigations of the multiplicative renormalization of the Yang-Mills theories [31], recent study involves the scalar fields which can be responsible for generating masses to physical particles through the mechanism of spontaneous symmetry breaking [33].
The proofs of multiplicative renormalizability and gauge independence of renormalization constants are based on the possibility to expand the extended effective action in loops, as well as to use the minimal subtraction scheme as to eliminate divergences. In addition, we propose the existence of a regularization preserving the used symmetries. Among the results obtained, we emphasize the rigorous proof of the gauge independence of all the physical parameters of the theory under consideration, to any order of loop expansions.