The NSVZ beta-function and the Schwinger-Dyson equations for N=1 SQED with N_f flavors, regularized by higher derivatives

The effective diagram technique based on the Schwinger-Dyson equations is constructed for N=1 SQED with N_f flavors, regularized by higher derivatives. Using these effective diagrams, it is possible to derive the exact NSVZ relation between the beta-function and the anomalous dimension of the matter superfields exactly in all loops, if the renormalization group functions are defined in terms of the bare coupling constant. In particular, we verify that all integrals which give the beta-function defined in terms of the bare coupling constant are integrals of double total derivatives and prove some identities relating Green functions.


Introduction
The existence of ultraviolet divergences is a long standing problem of quantum field theory. An important step towards solving this problem is a discovery of supersymmetry [1,2]. It is well known that the behavior of supersymmetric theories in the ultraviolet region is better due to non-renormalization theorems. In particular, the N = 4 supersymmetric Yang-Mills theory is finite [3,4,5,6]. Divergences in N = 2 supersymmetric Yang-Mills theories exist only in the one-loop approximation [7]. Even in N = 1 supersymmetric theories the superpotential is not renormalized [8]. However, the β-function in N = 1 supersymmetric Yang-Mills theories receives quantum corrections in all orders. Nevertheless, this β-function is related with the anomalous dimension of the matter superfields. This relation is called the exact Novikov, Shifman, Vainshtein, and Zakharov (NSVZ) β-function [9,10,11]. In the original papers this β-function was obtained using arguments based on the structure of instanton contributions (for review, see [12]) or on the supermultiplet structure of anomalies. In particular, in the lowest orders the relation between the β-function and anomalies was investigated in [10,13,14,15] and exactly in all orders in [10,16,17]. This was done using the Adler-Bardeen theorem [18] for the axial anomaly, a relation between the anomaly of energy-momentum tensor trace and a β-function, and a supermultiplet structure of anomalies. Another derivation of the exact NSVZ β-function based on anomalies was made in [19]. This β-function was also obtained in [20] using the non-renormalization theorem for the topological term. In [21] the rescaling anomaly is used for explanation of the higher order corrections to the NSVZ β-function.
In this paper we consider the N = 1 supersymmetric electrodynamics (SQED) with N f flavors, for which the NSVZ β-function is written as [22,23] sible to calculate one of the loop integrals analytically and reduce a number of the integrations over loop momentums. At least, in the Abelian case this allows to prove that the β-function and the anomalous dimension of the matter superfields defined in terms of the bare coupling constant satisfy the NSVZ relation [67,68,69]. (For a fixed regularization) these renormalization group functions are scheme independent (see, e.g., [69]), so that the NSVZ β-function is obtained for an arbitrary renormalization prescription. However, if the renormalization group functions are defined by the standard way in terms of the renormalized coupling constant, they depend on the subtraction scheme [70]. In this case the NSVZ β-function is obtained in a special subtraction scheme. If the theory is regularized by higher derivatives, such a scheme can be obtained in all orders by imposing simple boundary conditions on the renormalization constants [68,69]. So far there is no similar prescription in the case of using the dimensional reduction, and the NSVZ scheme should be constructed in each order of the perturbation theory after calculating the renormalization group functions. Thus, using the higher covariant derivative regularization one can naturally construct the scheme in which the β-function coincides with the exact NSVZ β-function at least in the Abelian case. Certainly, it is desirable to generalize the results to the non-Abelian case. However, in the non-Abelian case the calculations with the higher covariant derivative regularization were performed only in the one-and two-loop approximations, where the β-function is schemeindependent. Nevertheless, in both cases the β-function appears to be given by integrals of double total derivatives and coincide with the NSVZ expression. This allows to suggest that the structure of quantum corrections in the non-Abelian case is similar to the case of N = 1 SQED. However, the method used in [67] (which was proposed in [71]) is not convenient for generalizing the results to the non-Abelian case. Possibly, using this method one can prove the factorization of integrands into the double total derivatives, but obtaining the exact β-function by this method seems to be a very complicated problem. Even in N = 1 SQED for this purpose it is necessary to compare coefficients of different Feynman diagrams [67]. From the other side, the NSVZ expression naturally appears in case of using another method proposed in [72]. It is based on substituting solutions of the Ward (or Slavnov-Taylor) identities into the Schwinger-Dyson equations. The Schwinger-Dyson equations can be used for making calculations in a certain approximation as in [73], where the four-loop anomalous dimension of quenched QED was obtained by this method. However, they can also allow to find results which are exact in all orders. In particular, in Abelian supersymmetric theories by using the Schwinger-Dyson equation it is possible to present the two-point Green function of the gauge superfield as a sum of two effective diagrams. One of them is related with the two-point Green functions of the matter superfields and gives the exact NSVZ β-function. The second effective diagram cannot be expressed in terms of these two-point Green functions. However, calculations made in [74,75] show that this effective diagram (in the limit of the vanishing external momentum) is always given by integrals of total derivatives and vanishes, as it was suggested in [72]. This feature was not so far explained within a method based on using the Schwinger-Dyson equations that is the main obstacle for deriving the exact NSVZ β-function by this method. Thus, it is desirable to understand, why the second effective diagram vanishes, especially because it seems that the considered technique can be generalized to the the non-Abelian case. (Vanishing of this diagram can be interpreted as a special identity relating some Green functions.) In this paper we complete derivation of the NSVZ β-function started in [72]. In particular, we directly prove that the second effective diagram vanishes, and the β-function is given by integrals of double total derivatives. The method used in this paper seems to be applicable in the non-Abelian case. That is why throughout the paper we try to use the notation, which can be also used for non-Abelian supersymmetric Yang-Mills theories.
The paper is organized as follows: In Section 2 we describe N = 1 SQED with N f flavors, regularized by higher derivatives, and introduce the notation. In Section 3 we write the Schwinger-Dyson equations for the considered theory and present the β-function (and its derivative with respect to a specially introduced parameter g) as a sum of effective diagrams. In Section 4 we prove that the β-function is given by integrals of total derivatives and is equal to the exact NSVZ β-function. Also in this section we present a direct proof of a special identity for Green functions, which was proposed in [72]. In Section 5 we prove that the β-function is given by integrals of double total derivatives and present the derivation of the exact NSVZ β-function from this fact. A large number of technical details are collected in appendixes.
2 N = 1 SQED with N f flavors, regularized by higher derivatives In this paper we derive the NSVZ β-function for N = 1 SQED with N f flavors in all orders using the technique based on the Schwinger-Dyson equations. It is convenient to write the action for this theory in terms of superfields, because in this case supersymmetry is a manifest symmetry [76,77]. In this notation in the massless limit the action is given by In order to regularize this theory, we modify its action by adding a term with higher derivatives [47,48]: The higher derivatives are included into the function R, which satisfies the conditions R(0) = 1 and R(∞) = ∞. For example, it is possible to choose R = 1 + ∂ 2n /Λ 2n . The term with higher derivatives increases a degree of the momentum in the propagator of the gauge superfield. As a consequence, most loop integrals become convergent in the ultraviolet region. An accurate analysis shows that after introducing the higher derivative term divergences remain only in the one-loop approximation [78]. In order to cancel these remaining one-loop divergences, one should insert the Pauli-Villars determinants det(V, M I ) into the generating functional [79]: For fixing a gauge is added to the classical action, while ghosts can be omitted in the Abelian case. The masses of the Pauli-Villars fields should be proportional to the parameter Λ: where a I are some real constants which do not depend on the bare coupling constant. The coefficients c I should satisfy the conditions n I=1 c I = 1; n I=1 c I M 2 which ensure cancelation of the remaining one-loop divergences. For simplicity, in this paper we use the following choice of this coefficients: where P I is an integer. In this case for even P I we can present the Pauli-Villars determinants as functional integrals over the commuting (chiral) Pauli-Villars superfields. For odd P I the Pauli-Villars superfields are anticommuting. Therefore, P I is a Grassmannian parity of the Pauli-Villars superfields, and n I=1 det(V, M I ) Dφ αI D φ αI exp(iS PV ), (9) where the action for the Pauli-Villars superfields is with In order to simplify subsequent equations and make the calculations similar to a non-Abelian case, we also introduce the notation φ i ≡ (φ αI , φ αI ); φ * i ≡ (φ * αI , φ * αI ), i = 1, . . . 2(n + 1)N f , where the usual fields φ α and φ α by definition correspond to I = 0. The sum of mass terms can be written as where M 0 = 0, because the usual fields, which corresponds to I = 0, are considered in the massless limit. Due to the gauge invariance the mass matrix satisfies the equation where is a generator of the U (1) group in the considered representation and P i is a Grassmanian parity of the superfield φ i .
It is convenient to introduce sources both for the usual superfields V , φ, and φ and for the Pauli-Villars superfields: Eq. (4) is a standard definition of the generating functional for the considered theory, regularized by higher derivatives. However, it is convenient to use the background field method and introduce some auxiliary sources. In the Abelian case for this purpose we make the substitution V → V + V , where V is the background field. Also in the kinetic terms of the matter superfields we introduce the auxiliary real parameter g according to the prescription Then the usual kinetic terms are obtained for g = 1. It is important that this substitution is made only for the quantum gauge field, which is an integration variable in the generating functional. Moreover, we introduce the auxiliary sources φ 0i and φ 0i 1 for each pair of the matter superfields (including the Pauli-Villars fields) according to the prescription From Eq. (18) we see that, by definition, the parameter g is present only in vertices containing internal lines of the gauge superfield. It is important that introducing the parameter g we break the quantum gauge invariance. As a consequence, it is impossible to use Ward identities for Green functions containing external lines of the quantum gauge field. However, the background gauge invariance where A is an arbitrary chiral superfield, is unbroken. Thus, the generating functional is given by the following expression: where Dµ denotes the integration measure and where W a =D 2 D a V /4 is the field strength for the background gauge superfield V . (It is easy to see that terms linear in the quantum field V can be omitted. Also, it is not necessary to introduce the regulator in the part of the action which depends only on the background field.) The effective action is defined by the standard way as where the sources should be expressed in terms of fields through solving the equations Differentiating the effective action we obtain Below we will see that it is not convenient to consider (φ i , V ) as independent variables. A more convenient choice is (φ i , J), where J is a source for the quantum gauge superfield V . That is why below instead of the effective action we will mostly use the Routhian where it is necessary to express only the sources j i in terms of J and φ i . Due to the background gauge invariance (19) the two-point function of the background gauge superfield is transversal: where α 0 = e 2 0 /4π is a bare coupling constant and the supersymmetric transversal projector is given by In this paper we will calculate the β-function defined in terms of the bare coupling constant where α = α(α 0 , Λ/p) is a renormalized coupling constant. It is determined by the requirement that the function d(α 0 (α, Λ/µ), Λ/p) is finite in the limit Λ → ∞. The anomalous dimension can be defined similarly: which are scheme-dependent. However [68,69], the functions (28) and (29) can be obtained from the renormalization group functions (30) by imposing the boundary conditions on the renormalization constants, where x 0 is an arbitrary fixed value of ln Λ/µ. 2 In order to find the β-function (28) we calculate the expression where Λ and α are considered as independent variables. This expression is well defined if the right hand side is expressed in terms of the bare coupling constant α 0 . The left hand side of the expression (32) can be obtained from the two-point Green function of the background gauge superfield after the substitution Strictly speaking, the part of the effective action corresponding to the two-point function of the gauge superfield is infinite after this substitution, because it is proportional to However, this procedure can be rigorously formulated by inserting a regulator I(x) which is approximately equal to 1 at finite x µ and tends to 0 at the large scale R → ∞. Then in the leading order in R the considered part of the effective action is proportional to All terms containing the derivatives of the regulator I are suppressed as 1/RΛ → 0 and can be omitted. That is why below we do not explicitly write the regulator I, but assume that V 4 is finite and tends to infinity. Actually this corresponds to taking the limit of the vanishing external momentum p ∼ R −1 ∼ (V 4 ) −1/4 → 0: where The expressions for the two-point Green functions of the matter superfields can be found using arguments based on the chirality. Taking into account that the two-point functions of the matter superfields constructed from Γ and γ evidently coincide, they can be written as where the fields are set to 0, G i j and (M J) ij are functions of ∂ 2 , and in our notation (2 × 2 matrixes correspond to the fields φ and φ. The function J is real as a consequence of the CP -invariance.) By definition the matrix A −1 constructed from the inverse Green functions satisfies the condition An explicit expression for the matrix A −1 can be easily found: where the operator ∂ 2 G 2 + |M J| 2 is defined by the following prescription:

Schwinger-Dyson equations
Making the change of variables φ i → φ i + A i , where A i are arbitrary chiral superfields, in the generating functional (20), we obtain the equation This equation can be considered as a Schwinger-Dyson equation for the matter superfields.
Similarly, the Schwinger-Dyson equation for the two-point Green function of the gauge superfield can be written as [72] δ(∆Γ) where and the sources should be expressed in terms of the fields using Eq. (23). Because in this paper we use the background field method, Eq. (45) can be simply obtained by differentiation of the effective action with respect to the background field V . It is easy to see that the Schwinger-Dyson equation (45) can be equivalently rewritten in terms of derivatives with respect to the sources where the derivatives with respect to the sources are constructed according to the prescription Below we present Feynman rules (for simplicity, in the massless case). In the massive case the effective diagrams are the same.
In order to verify this equation it is necessary to apply it to (j k ) z taking into account Eqs. (24) and (41). Differentiating Eq. (47) with respect to V y and setting all fields (including φ 0 ) to 0, we obtain the Schwinger-Dyson equation for the two-point Green function of the gauge superfield. Details of this calculation are presented in Appendix A. Following [72] it is convenient to formulate the result in the graphical form. It is presented in Fig. 1. In this figure external lines correspond to the gauge superfield V , and, for simplicity, expressions for vertices and propagators are written for the massless case. In the analytical form the Schwinger-Dyson equation presented in Fig. 1 is written as where all fields are set to 0, and we use the notation (50) Note that all fields here are set to 0 in contrast to Eq. (48). Due to this condition the derivatives δ/δj (anti)commute. However, below we will not usually write explicitly the condition φ, φ 0 , V = 0 as in Eq. (50).
The Schwinger-Dyson equation (49) can be simplified after the substitution (33). (As we already mentioned above, this substitution automatically gives p = 0.) For this purpose it is convenient to use the identity which is proved in Appendix B. It is convenient to define the operator which contains all terms of the first degree inθ in Eq. (51): We will graphically denote this operator by a green effective line with the ends 1 and 2. In Fig. 2 the identity (51) is presented in the graphical form. The indexes 1 and 2 in the right hand side of Eq. (52) point the vertices to which the corresponding derivatives act. (Sometimes we will omit these indexes if they coincide.) Actually this expression can be considered as a modification of the effective propagator (multiplied by two derivatives with respect to the φ 0 ). Also we will also use other effective lines. Our notation is presented in Fig. 3. Note that in the case of using color lines we do not sometimes explicitly draw the external lines. Instead of them we draw a small circle, to which we attach corresponding θ-s. Using the identity (51) it is possible to rewrite the Schwinger-Dyson equation (49) in a different form. The result is presented in Fig. 4. In the analytical form it can be written as where The green effective line can be presented as a sum of the blue and yellow lines, see Fig. 5: where Using Eq. (55) and the identity it is possible to split the effective diagram presented in Fig. 4 into two parts. This is shown in Fig. 6. In this figure we use the notation The result for the effective diagram with the yellow effective line (including δ 1 ) can be expressed in terms of the anomalous dimension of the matter superfield [72]. 3 For this purpose it is necessary to substitute the solution of the Ward identity for the effective vertex. The result is also presented in Fig. 6. It is given by an integral of a total derivative with respect to a loop momentum. Such a structure allows to reduce a number of momentum integrations and relate the β-function with the anomalous dimension of the matter superfield. Really, calculating the integral in the four-dimensional spherical coordinates we obtain where we take into account that for a function f (q 2 ) which rapidly decreases at the infinity 3 In [72] only the case N f = 1 is considered.
14 (The functions considered here rapidly decrease at the infinity due to the higher derivative regularization.) As a consequence, the contribution of the diagram with the yellow effective line gives the exact NSVZ β-function However, using this method it is impossible to calculate the diagram with the blue effective line in Fig. 6. Some explicit calculations in the lowest (three-and four-) loops [75] show that this diagram plus δ 2 is also given by an integral of a total derivative and vanishes. In the graphical form this is presented in Figs. 6 and 7. This equality can be considered as a nontrivial relation between Green functions [72]. It was proved indirectly in [67] using a method proposed in [71]. In particular, it is possible to prove that the integrand corresponding to this diagram is a total derivative. In the analytical form the equality presented in Fig. 7 can be written as  [72]. In this paper this equality is proved.
In this paper we prove this identity directly. Moreover, we prove that the β-function is given by integrals of double total derivatives. In order to do this, it is necessary to use two ideas.
1. First, it is necessary to rewrite the effective vertices in the diagrams presented in Fig. 4 or Fig. 7 using the Schwinger-Dyson equation one more time. This procedure was first proposed in [81]. Let us, for example, start with Eq. (53) and substitute the expression for δΓ/δV y from the Schwinger-Dyson equation (47). It is convenient to write the result in terms of the Routhian γ, because in this case the number of effective diagrams is less. Details of this calculation are presented in Appendix C. The result can be written in the following form: where (∆ can be graphically interpreted as a one-loop effective diagram. However, in order to avoid too large number of effective diagrams we write this term explicitly.) Note that this expression does not contain infrared divergences due to the differentiation with respect to ln Λ, which should be made before the momentum integration.  Fig. 6, then a β-function will be determined by the two-loop effective diagram presented in Fig. 9. As earlier, it is convenient to split this effective diagram into two parts using Eq. (55). The result is graphically presented in Fig. 9, where (see Appendix C.2) In Appendix C.2 the expression ∆ 2 is also written in terms of the functions G and M J. ∆ 1 can be easily found using Eq. (64). After simple transformations we obtain 2. Now let us proceed to the second idea. An attempt to present all two-loop effective diagrams in Fig. 9 as integrals of total derivatives encounters considerable problems. The reason can be understood from the results of [67]. The matter is that the total derivative in this case nontrivially depends on the number of vertices in a diagram. Therefore, it seems impossible to write the total derivatives in the form of effective diagrams. However, the solution can be found. For this purpose we introduce the parameter g according to the prescription (17).
Let us differentiate the upper diagram in Fig. 9 with respect to the parameter g using the identity  4) and (9) in this figure come from the derivative of the green effective line. Actually, it is necessary to differentiate the inverse Green functions inside δ/δj. All terms corresponding to the two-loop effective diagrams are included into diagrams (1), (2), and (9). Similarly, terms corresponding to three-loop effective diagrams are included into diagrams (3) and (4). A green effective line with a cross (in diagrams (4) and (9)) corresponds to the operator which is obtained from the operator GreenLine by keeping only terms proportional to the Pauli-Villars masses. Integrating by parts and using Eq. (14) it is easy to see that this operator is symmetric with respect to the permutation of the points 1 and 2: Due to this symmetry of arguments we do not mark out one of the ends by a circle. The lines with crosses appear, because terms with the masses are quadratic in δ/δj, while the other terms are linear in δ/δj. A line with a dot corresponds to the operator where In Eq. (73) the subscript [3] means that only the derivatives δ/δ 3 φ 0 nontrivially act to the argument of the bracket. Similarly, diagram (2) in Fig. 10 contains the operator Diagrams (5)- (8) correspond to differentiation of the four-point function (the large black circle in Fig. 9). Technical details of the corresponding calculation can be found in Appendix D.2.
Below we try to avoid writing large analytical expressions corresponding to effective Feynman diagrams. Instead of this, we write numerical coefficients for all diagrams, so that the analytical expression can be unambiguously constructed using the definitions of the effective lines. As an example, here we present an analytical expression for the sum of diagrams presented in Fig. 10: Expressions for all differential operators corresponding to the various lines can be found in Fig.  3. Let us remind that if ends of an effective line coincide, we sometimes omit numbers which numerate them. In Sect. 5 we prove that the sum of diagrams presented in Fig. 10 in the momentum representation is given by integrals of double total derivatives. The method considered in this paper also allows to prove the identity presented in Fig. 7 directly. This is made in Sect. 4.2. In particular, we prove that the contribution of this effective diagram to the β-function is given by a vanishing integral of a total derivative. In order to do this, it is convenient to differentiate the diagram with the blue effective line presented in Fig. 9 (or, equivalently, in Fig. 7) with respect to the parameter g. In the graphical form the result is given by a sum of diagrams presented in Fig. 11. The corresponding analytical expression can be unambiguously constructed using the expressions for the effective lines presented in Fig. 3.

The effective diagram with the yellow line
In order to prove that the β-function (defined in terms of the bare coupling constant) is determined by integrals of total derivatives, it is convenient to use the coordinate representation. In the coordinate representation an integral of a total derivative can be written as where Figure 11: Diagrams obtained after differentiating the effective diagram with a blue effective line presented in Fig. 9 with respect to the parameter g. Here the blue effective line corresponds to the operator Blueline˙b[θ a θ aθḃ ] and the line with a dot corresponds to the operator LineWithDot [1]. and tr denotes the usual matrix trace. We will try to present (the sum of) expressions for the effective diagrams as such traces of commutators. First, as a simple example, we consider a diagram with the yellow effective line presented in Fig. 9 and verify that the sum of the this diagram and ∆ 1 is given by an integral of a total derivative. This sum (for N f = 1) has been already calculated in [72] by substituting solutions of the Ward identities for the effective vertices. In this paper we reobtain the result by a different method, which is also used for calculation of the other diagram (which has not been calculated in [72].) The expression for the considered diagram is written as where we used the identity proved in Appendix F.1 and the notation This operator is very useful, because by the help of this notation commutators of various Green functions with can be written in a very compact form. The details of the corresponding calculations are given in Appendix E.1. Here we present only the results. First, we introduce the following notation: e.t.c. Commutators with other Green functions (with an arbitrary number of indexes) can be constructed similarly. (Each index gives a term in the sum; for upper indexes the sign is "−", and for lower indexes the sign is "+".) Then the result of Appendix E.1 can be written as Using the identities (83) we can present the expression (84) in the form (85) In order to write Eq. (85) as an integral of a total derivative in the momentum space we (anti)commute the generators with the Green functions. Using the results of Appendix E.3 we obtain (86) Then we add ∆ 1 to Eq. (86) and write the result in the momentum representation substituting explicit expressions for the (inverse) Green functions. Details of this calculation are presented in Appendix G. Taking into account Eqs. (40) and (43) the result for the considered contribution (written in the momentum representation after the Wick rotation in the Euclidian space) can be written as where we separate the main contribution of the fields φ α and φ α (corresponding to I = 0) and contributions of the Pauli-Villars fields (corresponding to I ≥ 1). This expression agrees with the result obtained in [72] by a different method for N f = 1. (In [72] the considered contribution was calculated by substituting expressions for vertices obtained by solving the Ward identities.) Thus, the sum of the considered effective diagram and ∆ 1 is given by the integral of a total derivative, which can be easily calculated using the identity where f (q 2 ) is a function which rapidly decreases at the infinity, and S 3 ε is a 3-sphere in the momentum space surrounding the point q = 0 with the radius ε → 0.
Assuming that the other contributions vanish (we prove this statement in the next section) we obtain the NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant. Really, terms containing the Pauli-Villars masses are convergent and finite 22 beyond the one-loop approximation, because these masses are proportional to the parameter Λ. Therefore, Thus, for N = 1 SQED with N f flavors we obtain the NSVZ β-function

The effective diagram with the blue line
In order to prove that the β-function defined in terms of the bare coupling constant is given by integrals of total derivatives it is also necessary to present the expression for the last diagram (with the blue effective line) in Fig. 6 plus δ 2 as a trace of a commutator. Calculations in the lowest orders allow to suggest that this contribution is always given by integral of a total derivative and vanishes [72]. An indirect proof of this fact is actually given in [67] by a different method. In this section we present a direct proof. In order to do this, it is necessary to differentiate the generating functional with respect to the auxiliary parameter g, introduced in Eq. (17). Next, we prove the identity presented in Fig. 13 in a graphical form. In this figure an arc with an arrow denotes a trace of a commutator with y * µ . Certainly, the corresponding analytical expression can be easily constructed: Figure 13: This identity is needed for proving that the β-function (defined in terms of the bare coupling constant) is given by integrals of total derivatives and satisfies the NSVZ relation.
Similarly, for an upper index i this expression includes For example, applying this rule we can easily obtain Eq. (82). However, it is necessary to define the operation [(T )y * µ , . . .] (and other similar operations) more accurately for diagrams containing closed loops of the matter superfields. First, let us explain, how to construct the expression in the right hand side of Eq. (91). It can be schematically written as where dµ denotes the integration measure, and we omitted indexes for simplicity. The whole expression in the right hand side (for simplicity, without the derivative d/d ln Λ) can be written as (It is easy to see that all terms in this sum cancel each other.) The operation [(T )y * µ , . . .] in Eq. (91) is constructed formally according to Eq. (95). In the momentum representation the expression (95) is given by an integral of a total derivative, because (The last equation is written in the Euclidian space after the Wick rotation.) Therefore, the equation presented in Fig. 13 implies that the derivative of the effective diagram in the left hand side with respect to ln g is given by an integral of a total derivative. Moreover, the result is 0, because the integrand does not contain singularities. Then we integrate the considered equality over ln g from g = 0 to g = 1. The theory corresponding to g = 0 does not contain quantum gauge field, and quantum corrections to the Green function of the gauge superfield are given only by one-loop diagrams. It is easy to see that in the one-loop approximation the effective diagram in the left hand side of Fig. 13 vanishes. Therefore, (because the original theory corresponds to g = 1) this effective diagram is also given by an integral of a total derivative and is equal to 0 for g = 1.
Thus, taking into account the results of the previous section, the identity presented in Fig.  13 allows to prove that the β-function defined in terms of the bare coupling constant is given by integrals of total derivatives and satisfies the NSVZ relation. Let us proceed to proving this identity. The expressions (and other similar expressions) can be calculated using the Schwinger-Dyson equations. The details of the corresponding calculations are presented in Appendix E. It is convenient to add the (vanishing) diagrams presented in Fig. 14, where Figure 14: We add these terms to the diagram in the right hand side of Fig. 13.
to the diagram presented in the right hand side of Fig. 13. It is easy to see that the diagrams presented in Fig. 14 In the graphical form the result for the sum of all commutators is presented in Fig. 15. The expressions for the effective lines are collected in Fig. 3. Let us briefly explain, how the diagrams presented in Fig. 15  2. It is also necessary to calculate commutators with the inverse Green functions which are contained inside the derivatives δ/δj. The calculation of these commutators is described in Appendix E.3. Commuting (T )y * µ with the operator LineWithDot gives diagrams (6), (8), and (9). In diagrams (8) and (9) the left effective line is given by the operators .
(100) respectively. As earlier, the subscript [3] means that only the derivatives δ/δ 3 φ 0 nontrivially act on the expression in the brackets. Similarly, commuting (T )θ˙b with the operator LineWithDot in the first diagram in Fig. 14 gives diagrams (15), (18), and (19). In diagrams (18) and (19) we use the operators (with α = θ 4 and β = 1) respectively. For example, the operator WhiteBlueLine can be explicitly written as (The other similar operators have much more complicated form.) 3. Commuting (T )y * µ with the inverse Green functions inside the operator BlueLine in a diagram presented in Fig. 13 we obtain diagrams (7) and (23). The effective line with two color disks in diagram (23) encodes the operator Moreover, we also obtain the commutator Note that the last term evidently vanishes. It is included for the convenience, because due to its presence this commutator cancels the corresponding contribution from {(T )θȧ, PinkLineȧ}. Moreover, commuting (T )θȧ with the inverse Green functions inside the operator PinkLineȧ in the first diagram in Fig. 14, we obtain diagrams (16), (17) and (22). The effective line with two color disks in diagram (22) corresponds to 27 Similar commutators in the second diagram in Fig. 14 give diagrams (20) and (21). 4. Some terms (for example, the effective one-loop diagrams) do not have simple graphical interpretation. We write their sum explicitly in Fig. 15.
In order to prove identity (91), it is necessary to verify that the sum of the diagrams presented in Fig. 15 coincides with the sum of the diagrams presented in Fig. 11. (Certainly, the explicitly written terms should be also taken into account). This is made in Appendix H, using the identity 2 · BlueLine˙b[θ a θ aθḃ ; 1, 2] · GreenLine [3,4]  This identity is proved in Appendix F.3. Its graphical version is presented in Fig. 16. According to this figure the sum of four diagrams with the same topology containing the lines given in the left hand side of this figure is equal to the sum of two diagrams containing the lines in the right hand side of this figure. An example of applying this identity is presented in Fig.17.
Thus, we have obtained that the β-function (28) is determined by the integrals of total derivatives and coincides with the NSVZ β-function. Also we have proved the identity (62) directly.
5 Double total derivatives 5.1 Factorization of integrands into double total derivatives 28 In the previous section we have proved that a β-function of N = 1 SQED, regularized by higher derivatives, is given by integrals of total derivatives. This allows to calculate one of the loop integrals and obtain the exact NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant. However, according to [63,64,65,66,67] the βfunction is given by integrals of double total derivatives. In this section we prove this statement for N = 1 SQED with N f flavors, regularized by higher derivatives, in all orders. For this purpose we differentiate the two-point Green function of the background gauge superfield with respect to ln g. The required statement follows from the identity where and "− singularites" means that singular contributions containing δ-functions (see below) should be subtracted from this expression. (The expression in the right hand side should be accurately defined for diagrams which include closed loops of the matter superfields. We will discuss this definition below.) Note that the last equality in Eq. (109) evidently follows from Eq. (77).
In the graphical form the identity (109) is presented in Fig. 18. According to Eq. (109) the β-function of N = 1 SQED with N f flavors, regularized by higher derivatives, in the momentum representation is given by not only by integrals of total derivatives, but by integrals of double total derivatives. (It is necessary to take into account that θ 4 and y * µ commute.) Then the commutators in the right hand side of this equation can be calculated using identities obtained in Appendix E. First, it is necessary to calculate the inner commutator. The right hand side of Eq. (111) can be equivalently rewritten in the form (This equation and Eq. (95) accurately define the expression in the right hand side for diagrams with closed loops of the matter superfields.) The terms containing the operator BrownLine µ in this equality can be easily presented in a graphical form. The result is shown in Fig. 19. In order to avoid too large number of effective lines some terms are written explicitly. Then it is necessary to calculate the second commutator. This can be done similarly to the calculation made in the previous section. As earlier, it is convenient to add some terms to the expression in the right hand side of Fig. 19. They are presented in Fig. 20. These effective diagrams correspond to the analytical expression where the operator is denoted by the red line. The commutators are calculated using equations derived in Appendix E. In a graphical form the result is presented in Fig. 21. (The singular contributions will be calculated in the next section.) Let us describe this calculation in details. 3. The other 1/2 of diagram (5) and diagrams (6) and (11) in Fig. 21 are obtained if (T )y * µ is commuted with the inverse Green functions coming from the derivatives δ/δj which are contained in the brown effective line (in the first effective diagram in Fig. 19). In diagram (11) the effective line with two brown disks denotes the operator Similarly, diagrams (12), (18), and (19) in Fig. 21 are obtained from the first effective diagram in Fig. 20, if (T )θȧ is commuted with the inverse Green functions contained in the red effective line. In diagram (12) the effective line with two color disks denotes the operator In addition to these diagrams the considered commutators give terms containing the operator LineWithDot[θ 4 ] which are explicitly written in Fig. 21. 4. It is also necessary to commute (T )y * µ with the operators Figure 21: These diagrams are obtained by calculating two commutators in Eq. (109). Constructing expressions corresponding to these diagrams we assume that the first spinor index is lower and the second one is upper. The expression One-Loop is given by Eq. (122).
which are contained in the brown effective line. Taking into account that we obtain As a consequence, the terms which do not contain δ-functions cancel the corresponding terms coming from the diagrams presented in Fig. 20. 5. Commuting (T )y * µ with the operator LineWithDot[θ 4 ] we obtain diagrams (7), (8), and the other 1/2 of diagram (9) in Fig. 21. These diagrams are constructed using the effective lines defined by Eqs. (100) and (101). Also the considered commutators and the commutators written in Fig. 19 explicitly give the terms containing the operator BrownLine µ in Fig. 21.
7. The second effective diagram in Fig. 19 gives 1/2 of diagram (9) and diagram (10). Similarly, diagrams (23) and (24) are obtained from the second effective diagram in Fig. 20. In these diagrams we use the notation ; (120) 8. Also the considered diagrams and terms written explicitly in Fig. 19 give some contributions which can be graphically presented as one-loop effective diagrams. In Fig. 21 they are denoted by One-Loop. This expression has the following form: The sum of diagrams presented in Fig. 21 should be compared with the sum of diagrams presented in Fig. 10. (Certainly, the explicitly written terms should be also taken into account.) For this purpose it is necessary to use the identity derived in Appendix F.4. This identity is graphically presented in Fig. 22. As earlier, this figure should be understood as follows: we find a sum of three diagrams with the same topology which contain effective lines presented in the left hand side of Fig. 22 and θ 4 in an auxiliary (but fixed) position. Then this sum can be replaced by a single diagram with the same topology containing two green effective lines. Using this identity we see that 1. The sum of diagrams (1) and (13) in Fig. 21 gives diagram (5) in Fig. 10.
3. The sum of diagrams (3) and (15) in Fig. 21 gives diagram (7) in Fig. 10. 4. The sum of diagrams (7) and (20) in Fig. 21 gives diagram (8) in Fig. 10. 5. The sum of the expression One-Loop and the terms containing the operator LineWithDot explicitly written in Fig. 21 is equal to ∂∆/∂ ln g, which is calculated in Appendix D.4. (It is evident that θ 4 in the terms with the operator LineWithDot can be shifted to an arbitrary point of the diagram.) 6. The sum of diagrams (5), (17), and (18) in Fig. 21 is equal to diagram (a) in Fig. 23. Although this diagram is absent in Fig. 10, it is equal to diagram (4) in this figure (see the first string in Fig. 23). In order to see this, we note that the left part of this diagram is proportional to In both cases there is the projector D 2 x acting on the remaining part of the green effective line: Therefore, the part of the green line containing δ/δφ 0 vanishes. The remaining part of the green line is denoted by the green line with a cross. Thus, we prove the identity presented in the first string of Fig. 23. Using this identity we see that the sum of the considered diagrams gives diagram (4) in Fig. 10. 7. The sum of diagrams (11) and (12) is equal to diagram (b) in Fig. 23, where the line with two green disks corresponds to the operator Using the above arguments it is easy to prove that this diagram vanishes. Really, its left side is proportional to This Green function can contain parts proportional to δ 8 xy , D 2 δ 8 xy ,D 2 δ 8 xy , (D 2D2 ) x δ 8 xy , and (D 2 D 2 ) x δ 8 xy . However, it is easy to see that all these structures give 0 if they act on the product (It is necessary to take into account the integral over d 8 x d 8 y and note that terms which do not contain θ 4 vanish.) Therefore, the sum of diagrams (11) and (12) vanishes.
8. The sum of diagrams (9), (22), and (23) gives diagram (c) in Fig. 23. Using the identity (125) we obtain that the considered sum of diagrams is equal to diagram (2) in Fig. 10. This equality is presented in Fig. 23.
9. Let us consider a sum of diagrams (4), (6), (16), and (19) in Fig. 21. First, it is necessary to note that diagrams (4) and (6) are equal (see the third string in Fig. 23). In order to see this, we consider terms which do not contain the masses in the left brown line. In these terms the right vertex with the right brown line has the following structure xD 2 x δ 8 xy .
(129) In order to verify the last equality we note that from dimensional arguments and the Feynman rules Therefore, this part of the diagram contains the projector D 2 x acting on the left brown line. Taking into account that we see that the part of the left brown line containing δ/δφ 0 vanishes. The remaining part of the brown line is equal to the brown line with a cross. Moreover, diagram (19) is equal to diagram (e) in Fig. 23 multiplied by −1/2. In order to prove this identity we again consider terms which do not contain masses in the red effective line of diagram (e). Then the right vertex with the blue effective line is proportional to (132) (The last equality can be also verified using Eq. (130).) Therefore, the part of the red line containing δ/δφ 0 vanishes. The remaining part of the red line is equal to the red line with a cross.
Taking into account identities presented in the third and forth lines of Fig. 23, we obtain the sum of effective diagrams to which we can apply the identity (255). The result is given by diagram (3) in Fig. 10.
10. The sum of diagrams (8), (10), (21), and (24) is investigated similarly to the previous group of diagrams. For this purpose it is necessary to take into account identities presented in the last two strings of Fig. 23, which can be proved exactly as in the previous case. As usually, the red effective line in diagram (g) corresponds to the operator RedWhiteLineȧ[θ 4 ], and the red line with a cross in diagram (24) corresponds to the operator .
Using these identities we see that the sum of the considered diagrams is equal to diagram (1) in Fig. 10.
11. It is easy to see that terms proportional to (T M * ) explicitly written in Fig. 21 cancel each other. For this purpose it is necessary to use the algebraic identity and its consequence, Eq. (232), which is proved in Appendix F.1. 12. The term proportional to (M T ) and containing the operator BrownLine µ explicitly written in Fig. 21 can be presented in the form 4 · GreenLine · GreenWithCross · γ (135) using the identity (134). This expression coincides with diagram (9) in Fig. 10.
Collecting the results we see that the sum of diagrams presented in Fig. 21 is equal to the sum of diagrams presented in Fig. 10. This completes the proof of the identity (109).

Derivation of the NSVZ β-function
Diagrams presented in Fig. 21 (the sum of which is equal to the sum of diagrams presented in Fig. 10) are obtained after calculating commutators with y * µ . However, As a consequence, the sum of diagrams presented in Fig. 10 is equal to the sum of terms containing the δ-singularities in Eq. (109) with an opposite sign. Now, let us calculate these singular contributions starting from Eq. (112), which can be written in the form The singular part of this expression is calculated using Eq. (118). The result with the opposite sign (which is equal to the sum of the considered diagrams) is given by This expression contains explicit dependence only on θ 4 . Commutation of θ 4 produces terms of the third and lower degrees of θ. All these terms vanish after integrating over the anticommuting variables. Therefore, it is possible to shift θ 4 to an arbitrary point of the diagram. Let us shift θ 4 to the point x: Next, we use the equalities which are proved in Appendix D. It is easy to see that all terms which do not contain the derivatives with respect to ln g vanish. Really, according to Eq. (158) (141) (The first term in this expression vanishes after differentiation with respect to ln Λ.) Similarly, Collecting all terms, the considered singular contribution can be presented as (143) Using the equation for the derivative of the inverse matrix it is possible to rewrite this expression in a simpler form 38 Then we express the derivatives with respect to sources in terms of the derivatives with respect to fields and substitute explicit expressions for the Green functions, for example, After calculating the integrals over d 4 θ in the Euclidean space the result can be presented in the following form: (146) This result is very similar to Eq. (87), which was found earlier by a different method. The only difference is the presence of the derivative with respect to ln g. As earlier, we obtain Let us integrate this equation over ln g from g = 0 to g = 1. The considered theory coincides with N = 1 SQED with N f flavors for g = 1. Therefore, at the upper limit For g = 0 the considered theory does not contain the quantum gauge field. Therefore, for g = 0 only one-loop diagrams contribute to the two-point Green function of the gauge superfield, and where Thus, after the integration we obtain the NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant.

Conclusion
In this paper we present a derivation of the NSVZ relation for N = 1 SQED with N f flavors, regularized by higher derivatives, using a method based on the effective diagram technique and the Schwinger-Dyson equations. We prove that with this regularization the exact NSVZ β-function relates the renormalization group functions defined in terms of the bare coupling constant. (If the renormalization group functions are defined in terms of the renormalized coupling constant, the NSVZ scheme can be easily constructed by imposing the simple boundary conditions (31) on the renormalization constants [68,69].) The technique based on the Schwinger-Dyson equations seems to be more convenient for generalization of the results to the non-Abelian case than another method discussed in [67].
The method considered in this paper allows to easily calculate a contribution to the βfunction proportional to the anomalous dimension of the matter superfields. For this purpose expressions for the effective vertices are found by solving the Ward (or Slavnov-Taylor) identities [72]. However, in order to prove that the other contributions vanish for the considered theory, it is necessary to essentially modify the method. First, a β-function should be written in terms of two-loop effective diagrams. Moreover, it is necessary to introduce an auxiliary parameter g and perform a differentiation with respect to ln g. The derivative of the two-point function of the gauge superfield with respect to ln g can be presented as a sum of three-loop effective diagrams. After these modifications it is possible to find the remaining contribution to the β-function defined in terms of the bare coupling constant. In this paper we obtain that this contribution vanishes. Moreover, we prove that the β-function is given by integrals of double total derivatives in agreement with the results of [63,67]. Such a structure allows to calculate one of the loop integrals and obtain the NSVZ β-function in all orders. The origin of the exact NSVZ β-function can be easily explained, because taking one of loop integrals we relate the β-function in a certain order with the anomalous dimension in the previous order.
The results obtained in this paper can be verified by explicit calculations in the lowest loops. The three-loop calculation will be described in the forthcoming paper.

Acknowledgements.
The author is very grateful to A.L.Kataev for valuable discussions. The work was supported by RFBR grant No 14-01-00695.

A The Schwinger-Dyson equation for the two-point Green function of the gauge superfield
We are interested in the expression where the effective action Γ is given by Eq. (22) and all fields are set to 0. In order to calculate this expression we differentiate the Schwinger-Dyson equation (47) with respect to V y and set all fields to 0. Then the result is multiplied by V x V y /2. After integrating over d 8 x d 8 y we rewrite Eq. (152) as 40 In order to present this expression as a sum of effective diagrams, it is convenient to commute δ/δV and δ/δj. Differentiating inverse Green functions inside the derivatives δ/δj j and simplifying the result using Eq. (44), it is easy to see that where the derivatives with respect to sources should be expressed in terms of the derivatives with respect to fields. Really, for example, due to the chirality of the derivative with respect to the source j. All vertices containing odd degrees of the matter superfields vanish after setting the fields to 0. Two-point Green functions of the matter superfields constructed from the functionals Γ and γ evidently coincide (again, after setting the fields to 0). Using these facts Eq. (153) can be written in the form where the derivative δ/δj i is defined by Eq. (50). It differs from the derivative δ/δj i , because all fields in the inverse two-point Green functions are set to 0. The graphical interpretation of this result is presented in Fig. 1.

B The identity for the effective lines
In order to simplify the calculations we use the substitution V → θ 4 . Then we find a sum of effective lines presented in the left hand side of Fig. 2. In the analytical form this sum corresponds to the expression where the fields in the two-point functions are set to 0. Note that in our notation effective lines include derivatives which act on the vertices attached to the line. This allows to considerably simplify expressions for the multiloop effective diagrams.
Using Eq. (44) and arguments based on chirality it is easy to verify that (158) (The fields are set to 0.) In order to prove these equations, it is necessary to apply to them the operator −D 2 x /2. Substituting the Green functions (158) into Eq. (157) after some simple transformations we rewrite the considered expression as The first term in this expression can be transformed using the identity The degree of θ in the last term of Eq. (159) can be also decreased by the help of Eq. (14): Then integrating by parts gives Therefore, the expression (157) can be rewritten in the following form: C The Schwinger-Dyson equation in terms of two-loop effective diagrams C.1 β-function in terms of two-loop effective diagrams In order to rewrite the Schwinger-Dyson equation as a sum of two-loop effective diagrams, let us start with Eq. (53) and substitute δΓ/δV y from Eq. (47) [81]: (Certainly, all fields in this expression should be set to 0 after calculation of the derivatives.) First, let us find a contribution of the second term in the last square brackets (or, equivalently, terms containing φ l + φ 0l ). We will denote this contribution by ∆. After calculating the derivatives nontrivial terms can be written as Here instead of the effective action Γ we use the Routhian γ, because this considerably simplifies the calculations. In order to rewrite all equations in terms of γ, it is necessary to take into account that Note that some terms in Eq. (165) vanish, because they are proportional to θ in a more than the forth degree. For example, all terms in the second string of this equation vanish due to this reason. Really, taking into account that we can commute θ b with this Green function. Then the result will be proportional to (θ 4 ) y (θ b ) y = 0. In order to simplify the remaining terms we use Eq. (158). Taking into account that some terms vanish after differentiation with respect to ln Λ we obtain Shifting (θ a θ a ) x and (θȧ(γ µ )ȧ b θ b ) x to the point y this expression can be rewritten as It is important that this expression does not contain infrared singularities due to the differentiation with respect to ln Λ, which should be made before calculating the momentum integral.
Let us now consider the remaining terms in Eq. (164) (which are obtained from the first term in the last square brackets). In order to rewrite them in a more convenient form, we commute the derivatives δ/δj and δ/δφ 0 (including δ/δφ 0 which is contained in δ/δj i ) taking into account that Note that so far we did not set the fields to 0. Because the Routhian γ is used instead of the effective action Γ, the expression in the right hand side does not contain derivatives with respect to the gauge superfield V . (This is the main reason, why in the subsequent equations we use the functional γ.) As a consequence, setting all fields equal to 0 we obtain whereÂ is an operator acting on the coordinates y which does not containθȧθȧ. Its explicit form in the considered case can be found from Eq. (164). Then it is possible to substitute explicit expressions for the two-point Green functions and repeat all transformations made in Appendix B. It is easy to see that the result can be written as where we take into account that only terms proportional toθ 2 nontrivially contribute to the result. (In particular, a term withoutθ in Eq. (163) gives a vanishing contribution.) Therefore, finally we obtain where ∆ is given by Eq. (169). The first term in this expression can be graphically presented as a two-loop effective diagram, while the second one corresponds to a one-loop effective diagram.

C.2 The identity (62) in terms of two-loop effective diagrams
In order to prove the identity (62), it is necessary to present its left hand side as a sum of two-loop effective diagrams. For this purpose we use the Schwinger-Dyson equation (47). The first term in Eq. (62) can be presented as a two-loop effective diagram similarly to the calculation made in the previous section. In the graphical form this diagram (containing one blue effective line and one green effective line) is shown in Fig. 9. All terms which could be interpreted as one-loop effective diagrams in this case vanish. However, the second term in Eq. (62) can be written as a one-loop effective diagram. Let us remind that a contribution of this term is denoted by δ 2 . In order to write all two-loop contributions in the same form we also use the notation ∆ 2 = δ 2 . Using the Schwinger-Dyson equation (47) δ 2 can be written as The first term in the square brackets vanishes, because it is proportional to the first degree of θ. This follows from Eq. (172). Calculating the derivatives with respect to φ 0 it is easy to see that the only nontrivial term is Commuting θ-s with the covariant derivatives and using the identity This expression can be written in terms of the functions G and J. For this purpose we substitute the explicit expression for the inverse Green function from Eq. (42). Then, it is necessary to use Eq. (44), which allows to express the remaining Green function in terms of J: Calculating the integrals over θ after the Wick rotation we obtain D Derivatives with respect to the parameter g

D.1 The derivative of the Routhian
Let us set the background gauge superfield to 0, V = 0, and differentiate the Routhian γ with respect to the parameter g using the identity The result can be written as In terms of the Routhian γ the right hand side of this equation can be presented in the form where Let us consider the first term in this expression. Integrating by parts and using Eq. (44) we obtain where we take into account that δγ/δφ i = −j i . Note that the first term in this expression vanishes, because due to the first equality in Eq. (7). The third term in Eq. (183) can be considered similarly. Thus, we obtain

D.2 Derivatives of effective vertices
Using the derivative of the Routhian γ with respect to ln g given by Eq. (187) we can easily calculate the derivatives of various Green functions. For example, Substituting δγ/δ ln g from Eq. (187), differentiating, and then setting all fields to 0, we obtain Similar expressions can be written for the derivatives of the other two-point functions. In order to rewrite the result in a more compact form, we use the notation (72). Then the derivatives of the two-point functions can be written as (The second equality is derived using the same method.) Derivatives of the other two-point functions can be written in a similar form.
In order to calculate the derivatives of the four-point functions we again use this method. It is convenient to introduce the notation Then it is possible to write the derivatives of the four-point functions in a rather simple form. For example, taking into account that any Green function with an odd number of legs corresponding to the matter superfields vanishes, we obtain In this expression the derivatives δ/δ 1 j act on γ [1], and the derivatives δ/δ 2 j act on γ [2]. The derivatives of the other four-point functions with respect to ln g can be written in a similar form.

D.3 Derivatives of effective lines
Expressions for the derivatives of effective vertices obtained in the previous section allow to find derivatives of effective diagrams. The effective diagrams include effective lines. Therefore, it is desirable to find derivatives of these effective lines, which, in particular, contain inverse Green functions inside the derivatives δ/δj. The derivatives of the inverse Green functions can be easily calculated using Eq. (190): Similarly, From these equations we obtain We can use this equation for differentiating the operator GreenLine (or other similar operators) with respect to ln g. Let us consider, for the definiteness, the derivative of the green effective line. More exactly, let us differentiate 48 It is convenient to write the result in the following form: Let us briefly explain the derivation of this identity. In this expression all terms without the operator LineWithDot [1] are included into the effective lines and GreenWithCrossWhite [1,2] These expressions can be written in a more compact form, Eqs. (73) and (75), respectively. The term with the operator GreenWithCross appears when we differentiate the factor g −1 in the mass term. The other terms containing the operator LineWithDot are obtained from the derivatives of g δ/δj. However, it is necessary to take into account that the terms without masses are linear in this derivative, while the terms containing the masses are quadratic. That is why the result contains the sum of the operators GreenLine and GreenWithCross. Derivatives of the other effective lines can be constructed similarly.

D.4 Derivative of ∆
The derivative of the additional contribution ∆, given by Eq. (64), with respect to ln g can be calculated using Eq. (193). The result is The terms containing the operator LineWithDot [1] can be graphically presented as two-loop effective diagrams. The other terms correspond to one-loop effective diagrams.

E Calculation of commutators
In this section we calculate commutators of (T )y * µ with various Green functions. For this purpose we can use the Schwinger-Dyson equation (47), which is valid for any values of g. Let us multiply this equation by y * µ and integrate the result over d 8 x: The first term in this expression vanishes. Really, using antichirality of the derivative with respect to j * and integrating by parts this term can be rewritten as (202) Taking into account Eq. (44) we obtain (The term with the derivative of j * vanishes due to the evident identity T i i = 0. The last equality can be obtained by integrating D 2 by parts and taking into account Eq. (14).) Note that the fields here are not yet set to 0. From the other side, Comparing Eq. (201) and Eq. (204) we obtain Differentiating Eq. (205) with respect to various fields, it is possible to find (T )y * µ , and other similar expressions. (All fields in this commutators are set to 0.) For example, differentiating Eq. (205) with respect to (φ 0k ) y and (φ 0l ) z and setting all fields to 0 we obtain (T )y * µ , Commuting the derivative with respect to the source j j with the derivatives with respect to the fields φ 0 and taking into account that all Green functions with an odd number of φ-lines vanish, this expression can be presented in the form After substituting the two-point Green functions from Eq. (158) the result is written as In order to simplify this expression we note that and Using these identities we finally obtain (T )y * µ , where we have defined the operator Similarly, we can differentiate Eq. (205) with respect to (φ 0k ) y and (φ * l 0 ) z and set all fields equal to 0. Then, repeating the same operations as above, we find The commutators of (T )y * µ with four-point Green functions can be calculated by the same method. Differentiating Eq. (205) with respect to (φ 0i ) x , (φ 0j ) y , (φ 0k ) z , and (φ 0l ) w and setting all fields equal to 0, we obtain (T )y * µ , Similar identities can be written for the derivatives with respect to φ * 0 . In this case in the left hand side for each φ * 0 it is necessary to replace T i j by −(T ) j i .

E.2 Commutators with (T )θ
Commutators with (T )θ can be calculated by the same method as the commutators with (T )y * µ . Exactly as in the previous section we obtain Differentiating this equation we obtain commutators with various Green functions. For example, the commutator with the two-point Green function can be written as (T )θȧ, This equation can be simplified using the relations (The last equality follows from Eq. (14).) Then we obtain where we use the notation As always, the subscripts 1 and 2 denote end points of the line. If these points coincide, we sometimes for simplicity omit these indexes. Other commutators can be found similarly. For example, for the two-point function, or for the four-point function.

E.3 Commutators with propagators
In order to calculate commutators of (T )y * µ or (T )θ with various Feynman diagrams according to the prescription (95), it is necessary to commute (T )y * µ or (T )θ with inverse Green functions. Here we demonstrate, how this can be made. As a starting point we consider the identities (158), which can be presented in the following matrix form: Let us commute this equation with (T )y * µ or (T )θ a , taking into account that the commutators in the right hand side do not vanish. We use the identities For example, let us consider the commutator Commuting (T )y * µ with Eq. (223) after some simple transformations we obtain (T )y * µ , All commutators here can be calculated according to the prescription obtained in the previous section: This expression can be written in a very compact form (The fields here should be set to 0.) The other commutators can be considered similarly. Commutators with (T )θȧ are calculated by the exactly the same method. The result can be obtained by the substitution In particular, there are no terms containing the masses in the commutators with (T )θȧ.
F Identities for effective lines F.1 Proof of the identity (79) In order to present the two-loop effective diagram with the yellow line presented in Fig. 9 (plus ∆ 1 ) as an integral of a total derivative, we use the identity In this section we prove this equality. Using the same method in the subsequent sections we prove more complicated identities relating the effective lines. First, we note that both sides of the considered equation are quadratic inθ. As a consequence, it is possible to shiftθ to an arbitrary point of a diagram, because the integral over d 8 x does not vanish only if the integrand contains θ 4 . Comparing definitions of the operators GreenLine and RedLineȧ and taking into account the possibility of shiftingθȧ we easily obtain (omitting unessential terms which do not containθ) where z is an arbitrary point of the considered supergraph. Therefore, taking into account that θċθȧ = δċȧθ˙bθ˙b/2, we see that the equality (230) is equivalent to the identity Let us compare terms quadratic in θ. In these terms we can make arbitrary shifts of θ, because terms proportional to the first degree of θ (denoted by O(θ)) vanish after integrating over d 4 θ.
Then the considered terms in the left hand side of Eq. (232) can be written as and coincide with the terms quadratic in θ in the right hand side of Eq. (232). Terms proportional to the third degree of θ can be investigated similarly. In the left hand side such terms are given by In a Feynman graph the points z and x are connected by a sequence of vertices and propagators. This allows to write the left part of the above expression in the form (P A = 0) where A is a differential operator, which does not explicitly depend on θ. (This operator encodes the sequence of vertices and propagators which connect the points z and x.) Similarly, for an arbitrary P A Terms O(θ) vanish after integration over d 4 θ. Omitting these terms we see that the last expression in Eq. (235) corresponds to and coincides with the terms cubic in θ in the right hand side of Eq. (232). Thus, we have proved Eq. (232) and Eq. (230).

F.2 Auxiliary identities
In order to compare different groups of effective diagrams it is necessary to use some identities which relate various effective lines. All these identities follow from some simple commutators of θ a with differential operators containing supersymmetric covariant derivatives and usual derivatives. In this subsection we prove some simple algebraic equalities which allow to relate various effective lines: where A and B are differential operators which do not explicitly depend on θ. Actually, these operators correspond to sequences of vertices and propagators in a Feynman graph connecting two fixed points. In order to prove the first identity we rewrite its left hand side as Evidently, [A, θ a } and [B, θ a } do not explicitly depend on θ. Therefore, the whole expression is quadratic in θ, and shifts of θ a θ a can change only the terms O(θ). As a consequence, the considered expression can be rewritten as Using the identities (238) -(240) it is possible to prove the identity presented in Fig. 16 where z is an arbitrary point of the considered supergraph. (Certainly, we assume that all effective lines are included into a connected Feynman graph.) Both sides of the considered identity are quadratic inθ. Therefore, it is possible to shiftθ to an arbitrary point of the supergraph, because the terms O(θ) vanish after the integration over d 4 θ. Using Eq. (231) one can equivalently rewrite the identity (245) in the form (θ a θ a ) z BlueLine˙b[1; 1, 2] · PinkLine˙b [3,4] This identity contains terms cubic in θ and terms quartic in θ, which will be considered separately. We start with the cubic terms. Let A˙b denotes a sequence of lines and vertices connecting the points z and x which also includes terms coming from the operator Similarly, let B denotes a sequence of line and vertices connecting the points z and y. Note that in this operator we do not include terms coming from the operator and coincides with the contribution of the first term in the right hand side of Eq. (246). (So far we discuss only terms cubic in θ.) Using the same method we prove that the second and fourth terms in the left hand side of Eq. (246) give the second term in the right hand side. Therefore, the terms cubic in θ coincide. Let us now verify that terms quartic in θ are also the same in both sides of Eq. (246). In this case the operators A˙b and B are defined exactly as earlier. In particular, B denotes a sequence of line and vertices connecting the points z and y and does not include terms coming from the operator (As earlier, in this case P A = 1, P B = 0.) Then terms quartic in θ can be formally presented in the form 4A˙b θ c θ c B θ d θ d + 4θ c θ c A˙b θ d θ d B + 4(γ µ )ḋ a θ a Aḋ θ c θ c B(γ µ )˙b e θ e .
It is easy to see that the last term in this equation can be equivalently rewritten as 8 θ a A˙bθ c θ c Bθ a . 58 This allows to apply Eq. (240). As a result, we obtain that terms of the forth order in θ are given by the expression which coincides with the corresponding terms in the right hand side of Eq. (246). Taking into account that the cubic terms also coincide, we conclude that the identity (108) is proved.
F.4 Proof of identity presented in Fig. 22 In order to verify that a β-function is given by integrals of double total derivative we use the identity (θ 4 ) z BlueLine˙b[1; 1, 2] · RedLine˙b [3,4]  This equation contains terms quadratic, cubic, and quartic in θ. The quartic terms in both sides are equal to the corresponding quartic terms in Eq. (246), which are considered in the previous section, multiplied by 2. Similarly, terms cubic in θ are obtained by multiplying cubic terms in Eq. (246) by 4. Therefore, it is necessary to consider only terms quadratic in θ. It is evident that in such terms θ-s can be shifted to an arbitrary point of the supergraph. Then the required equality of the quadratic terms follows from the algebraic identity 2 · θ a θ a η µν = − 1 8 · 4(γ µ )˙b c θ c · 4(γ ν )˙b d θ d .
(In the right hand side we shift both θ-s to the same point z.)

G Derivation of the expression (87)
The sum of ∆ 1 , which is given by Eq. (67), and the expression (86) is (288) 18. It is easy to see that the sum of Eq. (271), (272), (286), and (288) is given by 19. The sum of Eqs. (282) and (289) is equal to ∂∆ 2 /∂ ln g (∆ 2 is given by Eq. (68) or Eq. (180)): This completes the prove that the sum of diagrams presented in Fig. 11 is equal to the sum of diagrams presented in Fig. 15.