Finiteness of the two-loop matter contribution to the triple gauge-ghost vertices in {\cal N}=1 supersymmetric gauge theories regularized by higher derivatives
Mikhail Kuzmichev, Nikolai Meshcheriakov, Sergey Novgorodtsev, Ilya Shirokov, Konstantin Stepanyantz
aa r X i v : . [ h e p - t h ] F e b Finiteness of the two-loop matter contribution to the triplegauge-ghost vertices in N = 1 supersymmetric gauge theoriesregularized by higher derivatives M.D.Kuzmichev, N.P.Meshcheriakov, S.V.Novgorodtsev, I.E.Shirokov, K.V.Stepanyantz
Moscow State University, Faculty of Physics, Department of Theoretical Physics,119991, Moscow, Russia
February 25, 2021
Abstract
For a general renormalizable N = 1 supersymmetric gauge theory with a simple gaugegroup we verify the ultraviolet (UV) finiteness of the two-loop matter contribution to thetriple gauge-ghost vertices. These vertices have one leg of the quantum gauge superfield andtwo legs corresponding to the Faddeev–Popov ghost and antighost. By an explicit calculationmade with the help of the higher covariant derivative regularization we demonstrate that thesum of the corresponding two-loop supergraphs containing a matter loop is not UV divergentin the case of using a general ξ -gauge. In the considered approximation this result confirmsthe recently proved theorem that the triple gauge-ghost vertices are UV finite in all orders,which is an important ingredient of the all-loop perturbative derivation of the NSVZ relation. Possible ultraviolet divergences in supersymmetric theories are restricted by some non-renormalization theorems. For example, it is well known that the superpotential of N = 1supersymmetric gauge theories cannot receive divergent quantum corrections [1]. Consequently,the renormalizations of masses and Yukawa couplings can be related to the renormalization ofchiral matter superfields. However, there are also some other non-renormalization theoremseven in theories with N = 1 supersymmetry. For example, it is reasonable to consider the exactNSVZ β -function [2–5] as a non-renormalization theorem, because it relates the renormalizationof the gauge coupling constant to the renormalization of chiral matter superfields. Moreover,it produces the non-renormalization theorems for N = 2 [6–8] and N = 4 [6, 7, 9, 10] super-symmetric gauge theories [11]. It is important that the non-renormalization theorems hold onlyfor special renormalization prescriptions. Strictly speaking, even the non-renormalization of thesuperpotential requires either a manifestly supersymmetric superfield quantization or speciallimitations on a subtraction scheme. Therefore, it is highly desirable that the regularizationand renormalization procedures be consistent with supersymmetry. Similarly, for deriving thefiniteness of N = 2 supersymmetric gauge theories beyond the one-loop approximation from theNSVZ β -function one should use a manifestly N = 2 quantization procedure [12]. Such a proce-dure can be constructed with the help of the harmonic superspace [13–15] and the correspondinginvariant regularization [16]. However, the NSVZ β -function is valid only for certain renormal-ization prescriptions, called “the NSVZ schemes”, which constitute a continuous set [17, 18]. Itappeared that such popular renormalization schemes as DR and MOM do not enter this set, seeRefs. [19–23] and [24, 25], respectively. An all-loop prescription for constructing at least one of1he NSVZ schemes was given in [26]. The NSVZ scheme is obtained if a theory is regularizedby the higher covariant derivative method [30, 31] (which includes introducing the Pauli–Villarsdeterminants for removing one-loop divergences [32]) in the superfield version [33, 34] and therenormalization is made by minimal subtractions of logarithms. This renormalization prescrip-tion is usually called HD+MSL [35, 36]. Note that actually the NSVZ β -function is valid in theHD+MSL scheme because it holds for RGFs defined in terms of the bare couplings for theoriesregularized by higher derivatives independently of a renormalization prescription. This state-ment has been verified by numerous multiloop calculations, see, e.g., [38–44], and can be usedfor simple calculation of the β -function in higher orders [45]. The all-loop proof has been donein Refs. [26, 46–48]. An important ingredient of this proof is a non-renormalization theorem forthe triple gauge-ghost vertices, which has been derived in [46] for N = 1 supersymmetric gaugetheories under the assumption of the superfield quantization in a general ξ -gauge. According tothis theorem the triple gauge-ghost vertices in which one line corresponds to the quantum gaugesuperfield and two others correspond to the Faddeev–Popov ghost and antighost are UV finite inall orders. Earlier similar statements were known for theories formulated in terms of usual fieldsin the Landau gauge ξ → ξ -gauge the UV finiteness of the above mentionedvertices in the supersymmetric case was demonstrated by an explicit one-loop superfield calcu-lation in Ref. [46] made with the help of the higher covariant derivative regularization. In thispaper we partially verify that this statement is also true in the two-loop approximation. Namely,we will prove that a part of the two-loop contribution to the triple gauge-ghost vertices comingfrom superdiagrams which contain a matter loop is UV finite for theories regularized by highercovariant derivatives. Note that we will use this regularization because it naturally produces theNSVZ scheme and reveals some interesting features of quantum corrections in supersymmetrictheories, see [42] and references therein. However, calculations of quantum corrections with thisregularization are rather complicated and to a certain degree are similar to the ones for higherderivative theories (see, e.g., [52–54]).The paper is organized as follows. In Sect. 2 we recall the superfield formulation of N = 1supersymmetric gauge theories together with some aspects of their regularization by higherderivatives and superfield quantization. The structure of the triple gauge-ghost vertices is dis-cussed in Sect. 3. The calculation of the two-loop superdiagrams containing a matter loop isdescribed in Sect. 4, where we prove that their overall contribution is not UV divergent. N = 1 supersymmetric gauge theories and the regularizationby higher covariant derivatives We will consider a general renormalizable N = 1 supersymmetric gauge theory with a singlegauge coupling constant. In the superfield formulation its classical action is written in the form S = 12 e Re tr Z d x d θ W a W a + 14 Z d x d θ φ ∗ i ( e V ) ij φ j + (cid:26) Z d x d θ (cid:16) m ij φ i φ j + 16 λ ijk φ i φ j φ k (cid:17) + c.c. (cid:27) , (1)where e and λ ijk are the bare gauge and Yukawa couplings, respectively. The Hermitian gaugesuperfield and its superfield strength are denoted by V and W a , respectively. The chiral matter In the Abelian case a similar prescription has been found earlier [27] on the base of the results of [28, 29]. For N = 1 SQED the on-shell scheme appears to be another all-loop NSVZ renormalization prescription [37]. φ i lie in a certain representation R of the gauge group G . In our notations thegenerators of the fundamental representation denoted by t A are normalized by the conditiontr( t A t B ) = δ AB /
2, while the generators of the representation R are denoted by T A and satisfythe equations tr( T A T B ) = T ( R ) δ AB ; ( T A T A ) ij = C ( R ) ij . (2)The theory is gauge invariant if the bare masses and Yukawa couplings are chosen in such a waythat m ik ( T A ) kj + m kj ( T A ) ki = 0; (3) λ ijm ( T A ) mk + λ imk ( T A ) mj + λ mjk ( T A ) mi = 0 . (4)Below we will always assume that these equations are satisfied. Also we will always assume that m ik m ∗ kj = m δ ij . (5)Note that these conditions can be satisfied only for anomaly free theories. Really, using Eqs.(3) and (5) after some transformations we obtain m tr( T A T B T C ) = m ∗ ij m jk ( T A ) kl ( T B ) lm ( T C ) mi = − m ∗ ij m mi ( T A ) kj ( T B ) lk ( T C ) ml = − m tr( T A T C T B ) . (6)This implies that the generators T A should satisfy the conditiontr( T A { T B , T C } ) = 0 , (7)which means that the considered theory is not anomalous [55]. Certainly, the absence of theanomalies is also needed for the renormalizability, which will be essentially used in what follows.For quantizing the theory (1) it is convenient to use the background field method. Moreover,one should take into account that the quantum gauge superfield is renormalized in a nonlin-ear way [56–58]. This has been confirmed by explicit calculations in the lowest orders of theperturbation theory [59, 60]. Also explicit calculations demonstrate that without the nonlinearrenormalization the renormalization group equations are not satisfied [61]. To take into accountthe nonlinear renormalization and to introduce the quantum-background splitting, we make thesubstitution e V → e F ( V ) e V . (8)Here V and V denote the quantum and background gauge superfields, respectively. Note thatin this notation the quantum gauge superfield satisfies the equation V + = e − V V e V . After thereplacement (8) the gauge superfield strength takes the form W a = 18 ¯ D (cid:16) e − V e − F ( V ) D a (cid:16) e F ( V ) e V (cid:17)(cid:17) . (9)In this paper we will consider only superdiagrams which do not contain external lines ofthe background superfield. However, for other purposes the background (super)field method isvery useful, so that constructing the generating functional we will keep the dependence on thebackground gauge superfield V . 3ollowing Refs. [62, 63], we introduce the regularization by adding some terms containinghigher derivatives to the action. After this the regularized action can be written in the form S reg = 12 e Re tr Z d x d θ W a h e − V e − F ( V ) R (cid:16) − ¯ ∇ ∇ (cid:17) e F ( V ) e V i Adj W a + 14 Z d x d θ φ ∗ i h F (cid:16) − ¯ ∇ ∇ (cid:17) e F ( V ) e V i ij φ j + (cid:26) Z d x d θ (cid:16) m ij φ i φ j + 16 λ ijk φ i φ j φ k (cid:17) + c.c. (cid:27) , (10)where Λ is the dimensionful cut-off parameter of the regularized theory, and the covariantderivatives are defined as ∇ a = D a ; ¯ ∇ ˙ a = e F ( V ) e V ¯ D ˙ a e − V e − F ( V ) . (11)(The higher derivatives are present inside two regulator functions R ( x ) and F ( x ), which rapidlygrow at infinity and are equal to 1 at x = 0.) In our notations, if f ( x ) = f + f x + f x + . . . ,then the subscript Adj means that f ( X ) Adj Y ≡ f Y + f [ X, Y ] + f [ X, [ X, Y ]] + . . . (12)The gauge fixing term analogous to the background ξ -gauge in the usual Yang–Mills theoryis given by the expression S gf = − ξ e tr Z d x d θ ∇ V K (cid:16) − ¯ ∇ ∇ (cid:17) Adj ¯ ∇ V, (13)which includes the background covariant derivatives ∇ a ≡ D a and ¯ ∇ ˙ a ≡ e V ¯ D ˙ a e − V . Also thegauge fixing term contains one more higher derivative regulator function K ( x ), which has thesame properties as the functions R ( x ) and F ( x ). Then the actions for the chiral Faddeev–Popovghosts c and ¯ c and the chiral Nielsen–Kallosh ghosts b read S FP = 12 Z d x d θ ∂ F − ( e V ) A ∂ e V B (cid:12)(cid:12)(cid:12)(cid:12) e V = F ( V ) (cid:0) e V ¯ ce − V + ¯ c + (cid:1) A × (cid:26)(cid:16) F ( V )1 − e F ( V ) (cid:17) Adj c + + (cid:16) F ( V )1 − e − F ( V ) (cid:17) Adj (cid:16) e V ce − V (cid:17)(cid:27) B ; (14) S NK = 12 e tr Z d x d θ b + (cid:16) K (cid:16) − ¯ ∇ ∇ (cid:17) e V (cid:17) Adj b. (15)To regularize one-loop divergences that survive after introducing the higher derivatives, weshould insert the Pauli–Villars determinants into the generating functional [32]. Accordingto [62, 63], in the supersymmetric case one needs two such determinants. The first one can bepresented as a functional integral over three commuting chiral superfields ϕ , ϕ , and ϕ in theadjoint representation of the gauge group,Det( P V, M ϕ ) − = Z Dϕ Dϕ Dϕ exp( iS ϕ ) , (16)where 4 ϕ = 12 e tr Z d x d θ (cid:16) ϕ +1 h R (cid:16) − ¯ ∇ ∇ (cid:17) e F ( V ) e V i Adj ϕ + ϕ +2 h e F ( V ) e V i Adj ϕ + ϕ +3 h e F ( V ) e V i Adj ϕ (cid:17) + 12 e (cid:16) tr Z d x d θ M ϕ ( ϕ + ϕ + ϕ ) + c.c. (cid:17) . (17)This determinant cancels one-loop divergences generated by the gauge and ghost superfields. Thesecond Pauli–Villars determinant removes one-loop divergences produced by a matter loop. Itis given by the functional integral over the (commuting) chiral superfields Φ i in a representation R PV which admits a gauge invariant mass term such that M ik M ∗ kj = M δ ij , Det(
P V, M ) − = Z D Φ exp( iS Φ ) , (18)where S Φ = 14 Z d x d θ Φ + F (cid:16) − ¯ ∇ ∇ (cid:17) e F ( V ) e V Φ + (cid:16) Z d x d θ M ij Φ i Φ j + c.c. (cid:17) . (19)Then the generating functional of the regularized theory takes the form Z [ V , Sources] = Z Dµ (cid:16) Det(
P V, M ) (cid:17) c Det(
P V, M ϕ ) − × exp (cid:16) iS reg + iS gf + iS FP + iS NK + iS sources (cid:17) , (20)where Dµ denotes the integration measure, c = T ( R ) /T ( R PV ) and S sources includes all relevantsources. Moreover, to obtain a theory with a single dimensionful regularization parameter,we require that the ratios a ϕ ≡ M ϕ / Λ and a ≡ M/ Λ are constants which do not depend oncouplings.The effective action Γ is constructed according to the standard procedure, as a Legendretransform of the generating functional for the connected Green functions W ≡ − i ln Z . We are interested in the 3-point vertices with two external ghost legs and one external leg ofthe quantum gauge superfield. (Note that similar vertices with a leg of the background gaugesuperfield are in general UV divergent.) There are four different vertices of the consideredstructure, namely, ¯ c + V c , ¯ cV c , ¯ c + V c + , and ¯ cV c + depending on the (anti)ghost superfields onthe external lines. According to [46] all these vertices have the same renormalization constant Z − / α Z c Z V , where Z α , Z c , and Z V are the renormalization constants for the gauge couplingconstant α = e / π , the Faddeev–Popov ghosts, and the quantum gauge superfield, respectively,1 α = Z α α ; ¯ c A c B = Z c ¯ c AR c BR ; V A = Z V V AR , (21) As earlier, due to the above requirements the generators of the representation R PV satisfy the equationtr( T A PV { T B PV , T C PV } ) = 0. R denotes renormalized superfields. Certainly, it should be noted that thequantum gauge superfield is renormalized in a nonlinear way. To take this nonlinear renormal-ization into account, we include an infinite set of parameters into the function F ( V ) A . Say, inthe lowest nontrivial order it is given by the expression F ( V ) A = V A + e y G ABCD V B V C V D + . . . , (22)where G ABCD ≡ ( f AKL f BLM f CMN f DNK + permutations of B , C , and D ) /
6, and contains aparameter y , which should also be renormalized. Then the nonlinear renormalization is reducedto linear renormalizations of V A and of the parameters y , . . . The equation describing therenormalization of the parameter y in the lowest nontrivial approximation can be found, e.g.,in [61].The structure of the triple gauge-ghost vertices can be analysed with the help of dimensionaland chirality considerations. Using the same notations as in Ref. [46] we write the correspondingparts of the effective action as∆Γ ¯ c + V c = ie f ABC Z d θ d p (2 π ) d q (2 π ) ¯ c + A ( p + q, θ ) (cid:16) f ( p, q ) ∂ Π / V B ( − p, θ )+ F µ ( p, q )( γ µ ) ˙ ab D b ¯ D ˙ a V B ( − p, θ ) + F ( p, q ) V B ( − p, θ ) (cid:17) c C ( − q, θ ); (23)∆Γ ¯ c + V c + = ie f ABC Z d θ d p (2 π ) d q (2 π ) ¯ c + A ( p + q, θ ) e F ( p, q ) V B ( − p, θ ) c + C ( − q, θ ); (24)where ∂ Π / ≡ − D a ¯ D D a / F ( p, q ) should be distinguished from the regulator function F in Eq.(10).) Differentiating these expressions with respect to superfields we present the consideredGreen functions in the form δ Γ δ ¯ c + Ax δV By δc Cz = − ie f ABC Z d p (2 π ) d q (2 π ) (cid:16) f ( p, q ) ∂ Π / − F µ ( p, q )( γ µ ) ˙ ab ¯ D ˙ a D b + F ( p, q ) (cid:17) y (cid:16) D x δ xy ( q + p ) ¯ D z δ yz ( q ) (cid:17) ; (25) δ Γ δ ¯ c + Ax δV By δc + Cz = − ie f ABC Z d p (2 π ) d q (2 π ) e F ( p, q ) D x δ xy ( q + p ) D z δ yz ( q ) , (26)where δ xy ( q ) ≡ e iq µ ( x µ − y µ ) δ ( θ x − θ y ).The one-loop expressions for the functions f , F µ , F , and e F can be found in Ref. [46]. Forexample, the sum of the tree and one-loop contributions to F is given by the following functionof the Euclidean momenta P and Q F ( P, Q ) = 1 + e C Z d K (2 π ) ( − ( Q + P ) R K K ( K + P ) ( K − Q ) − ξ P K K K ( K + Q ) ( K + Q + P ) + ξ Q K K K ( K + P ) ( K + Q + P ) + (cid:18) ξ K K − R K (cid:19) (cid:18) − Q + P ) K ( K + Q + P ) + 2 K ( K + Q + P ) − K ( K + Q ) − K ( K + P ) (cid:19) ) + O ( α , α λ ) , (27) In our conventions Euclidean momenta are always denoted by capital letters. R K ≡ R ( K / Λ ) etc. We see that this expression is finite in the UV region independentlyof the value of the gauge parameter ξ , although some terms inside it are logarithmically diver-gent. The function e F is given by a similar UV finite expression. In the one-loop approximationthe UV finiteness of the functions F µ and f immediately follows from the fact that they havethe dimensions m − and m − , respectively.In this paper we would like to verify that a part of the two-loop contribution to the Greenfunctions (25) and (26) coming from supergraphs containing a matter loop is UV finite. Thestraightforward calculation is rather complicated, especially due to the use of the regularizationby higher covariant derivatives. However, it is possible to make some simplifications. First,we know that all 4 three-point gauge-ghost vertices have the same renormalization constants.Therefore, it is sufficient to calculate only one of them. In this paper we will consider the func-tion (25). Moreover, the integrals giving the functions F µ and f have the superficial degree ofdivergence − −
2, respectively. This implies that the corresponding divergences can comeonly from the divergent subdiagrams. For the considered renormalizable theory these subdi-vergences are evidently removed by the renormalization in the previous orders. Therefore, tofind the two-loop contribution to the renormalization constant Z − / α Z c Z V , we need to calculateonly the function F . This function can be extracted from the corresponding part of the effectiveaction (given by Eq. (23)) by a formal substitution V → ¯ D B, (28)where B is a Hermitian superfield, because after this substitution the expression (23) takes theform ∆Γ ¯ c + V c = ie f ABC Z d θ d p (2 π ) d q (2 π ) ¯ c + A ( p + q, θ ) F ( p, q ) ¯ D B B ( − p, θ ) c C ( − q, θ ) . (29)Moreover, the calculation of the function F can be done in the limit of the vanishing externalmomenta. Really, terms proportional to external momenta are given by integrals with a negativesuperficial degree of divergence, so that after removing subdivergences by the renormalizationin the previous orders we will obtain UV finite contributions.Thus, we will extract the function F with the help of the formal substitution (28) andcalculate it in the limit of the vanishing external momenta, P, Q →
0. Details of this calculationwe describe in the next section.
We will investigate a part of the two-loop contribution to the three-point gauge-ghost verticescoming from supergraphs containing a matter loop. They are presented in Fig. 1. Some ofthe diagrams presented in this figure include a gray disk which encodes a sum of two diagramsdepicted in Fig. 2. Actually, it corresponds to that part of the one-loop quantum gauge superfieldpolarization operator which comes from diagrams with a matter loop. The analytic expressionfor it has been found in Ref. [63]. In the considered massive case it is written as This substitution is needed only for extracting a certain part of the Green function, so that it is not essentialthat the expression ¯ D B is not Hermitian. We will consider only two-loop supergraphs in which a matter loop corresponds to the usual superfields φ i and the Pauli–Villars superfields Φ i . The supergraphs with a loop of the Pauli–Villars superfields ϕ , , producecontributions proportional to C , so that it is natural to investigate them together with the two-loop supergraphswithout matter loops.
7Π = − πα T ( R ) Z d L (2 π ) h ( K, L ) , (30)where h ( K, L ) ≡ K + L ) − L ) (cid:18) F L L F L + m ) − F K + L K + L ) F K + L + m ) − m F ′ L Λ F L ( L F L + m ) + m F ′ K + L Λ F K + L (( K + L ) F K + L + m ) − F L L F L + M )+ F K + L K + L ) F K + L + M ) + M F ′ L Λ F L ( L F L + M ) − M F ′ K + L Λ F K + L (( K + L ) F K + L + M ) (cid:19) , (31)(1)¯ c + c (2) (3) (4)(5) (6) (7) (8)(9) (10) (11) (12)(13) (14) (15)Figure 1: The superdiagrams giving a matter contribution to the three-point gauge-ghost verticesin the two-loop approximation. A gray disk denotes the sum of two subdiagrams presented inFig. 2.and primes denote derivatives with respect to the arguments, e.g., F ′ L ≡ F ′ ( L / Λ ). Thepolarization operator Π is related to the two-point Green function of the quantum gauge super-field. Taking into account that quantum corrections to this function are transversal due to theSlavnov–Taylor identities [64, 65], we can present the corresponding part of the effective actionin the formΓ (2) V − S (2)gf = − π tr Z d k (2 π ) d θ V ( − k, θ ) ∂ Π / V ( k, θ ) d − q ( α , λ , k / Λ ) . (32)8 +Figure 2: Superdiagrams producing a matter contribution to the one-loop polarization operatorof the quantum gauge superfield.Then the polarization operator is defined by the equation d − q ( α , λ , k / Λ ) − α − R ( k / Λ ) ≡ − α − Π( α , λ , k / Λ ) . (33)From the above equations it is possible to write the exact propagator of the quantum gaugesuperfield in terms of the function Π,2 i (cid:18) R − Π) ∂ − ∂ (cid:16) D ¯ D + ¯ D D (cid:17)(cid:16) ξ K − R − Π (cid:17)(cid:19) δ xy δ AB , (34)where δ xy ≡ δ ( x µ − y µ ) δ ( θ x − θ y ). Using this equation it is easy to construct expressions forvarious superdiagrams containing insertions of the polarization operator and, in particular, forthe superdiagrams with a gray disk in Fig. 1.To construct vertices containing ghost lines, it is necessary to take into account Eq. (22)and use the equations V − e V = −
12 + 12 V − V + 190 V + O ( V ); (35) V − e − V = 12 + 12 V + 16 V − V + O ( V ) . (36)Then we see that vertices with ghost legs are generated by the expression S FP = Z d x d θ (cid:18) c + A ¯ c A + 14 ¯ c + A c A + ie f ABC (¯ c A + ¯ c + A ) V B ( c C + c + C ) − e f ABC × f CDE (¯ c A + ¯ c + A ) V B V D ( c E − c + E ) − e f ABC f CDE f EF G f GHI (¯ c A + ¯ c + A ) V B V D V F × V H ( c I − c + I ) − e y G ABCD (¯ c A + ¯ c + A ) V C V D ( c B − c + B ) + . . . (cid:19) . (37)The term containing y is essential even for calculating the two-loop anomalous dimension ofthe ghost superfields, see Ref. [61] for details. Therefore, it is certainly needed for calculatingthe two-loop contribution to the triple gauge-ghost vertices. However, it is not essential forobtaining its part proportional to C T ( R ) (which we are interested in), so that the effects of thenonlinear renormalization can be ignored in this paper. Also we see that the vertices with twoghost and three gauge lines are absent and, therefore, we need not include the correspondingsuperdiagrams into Fig. 1.It is convenient to divide the superdiagrams presented in Fig. 1 into three groups.1. The superdiagrams (1), (5), and (6), in which an external gauge leg is attached to a gaugeinternal line.2. The superdiagrams (13), (14), and (15) containing an insertion of the one-loop polarizationoperator (30) and an external gauge leg attached to a ghost line.9. The other superdiagrams (2), (3), (4), (7), (8), (9), (10), (11), and (12) in which anexternal gauge line is attached to a matter loop.Let us demonstrate that the sum of superdiagrams in each of these groups is UV finite.1. First, we consider the superdiagrams (1), (5), and (6). They include a triple gauge vertexin which (after the replacement (28)) one leg corresponds to ¯ D B . The original expression forthe triple gauge vertex is written as∆ S V = ie f ABC Z d x V A D a V B R ( ∂ / Λ ) ¯ D D a V C + ie f ABC ∞ X n =1 r n n − X α =0 Z d x (cid:16) ∂ Λ (cid:17) α D ¯ D D a V A V B (cid:16) ∂ Λ (cid:17) n − − α ¯ D D a V C , (38)where V A are components of the quantum gauge superfield, d x ≡ d x d θ , and the coefficients r n are defined by the equation R ( x ) = 1 + ∞ X n =1 r n x n . (39)After making the replacement (28) for one gauge leg we obtain the expression ie f ABC Z d x (cid:16) ¯ D B A D a V B R ( ∂ / Λ ) ¯ D D a V C + V A D a ¯ D B B R ( ∂ / Λ ) ¯ D D a V C (cid:17) + ie f ABC ∞ X n =1 r n n − X α =0 Z d x (cid:16) ∂ Λ (cid:17) α D ¯ D D a V A ¯ D B B (cid:16) ∂ Λ (cid:17) n − − α ¯ D D a V C . (40)The function F in Eq. (23) is determined by logarithmically divergent integrals. Therefore, anycontribution in which more than two supersymmetric covariant derivatives act on the superfield B is UV finite and vanishes in the limit of the vanishing external momenta. (In the consideredrenormalizable theory all subdivergences are removed by the renormalization in the previousorders.) This implies that the term containing D a ¯ D B can be omitted. Integrating the super-symmetric covariant derivatives by parts, omitting terms with more than 2 derivatives acting on B , and using the identity ¯ D D ¯ D = −
16 ¯ D ∂ , the vertex under consideration can be rewrittenas ie f ABC Z d x (cid:18) ¯ D B A D a V B R ( ∂ / Λ ) ¯ D D a V C − ∞ X n =1 r n n − X α =0 (cid:16) ∂ Λ (cid:17) α D a V A × ¯ D B B (cid:16) ∂ Λ (cid:17) n − α ¯ D D a V C (cid:19) . (41)In the second term we integrate the usual space-time derivatives by parts and omit terms pro-portional to the external momentum − p µ (of the superfield B ). Then with the help of theequation ∞ X n =1 r n nx n = xR ′ ( x ) (42)we present the vertex in the limit p → ie f ABC Z d x ¯ D B A D a V B (cid:18) R (cid:16) ∂ Λ (cid:17) + 2 ∂ Λ R ′ (cid:16) ∂ Λ (cid:17)(cid:19) ¯ D D a V C . (43)10gain integrating by parts (with respect to the last derivative ¯ D and the usual derivativesinside the round brackets) and omitting terms vanishing in the limit p → B and C ) tensor and the antisymmetric structure constants f ABC . Certainly, such a product vanishes.Therefore, the vertex in which an external ¯ D B -leg is attached to an internal line of the quantumgauge superfield is equal to 0 in the limit of vanishing external momenta. This implies that alldiagrams containing such vertices are finite. In particular, we see that the superdiagrams (1),(5), and (6) in Fig. 1 are finite.2. Superdiagrams (13), (14), and (15) also give UV finite contributions. Indeed, the ghostpropagator is proportional to either D x ¯ D y ∂ δ xy , or ¯ D x D y ∂ δ xy (44)depending on a sequence of the ghost vertices. As we have already discussed above, if at leastone supersymmetric covariant derivative (certainly, except for ¯ D inside ¯ D B ) acts on externallines, then a superdiagram evidently vanishes in the limit of the vanishing external momenta.¯ c + c | ¯ D | ¯ c + c | D | Figure 3: For the superdiagrams (13), (14), and (15) in the limit of the vanishing externalmomenta with the help of the integration by parts one can achieve that the supersymmetriccovariant derivatives act on the gauge superfield propagator.Let us consider a triple vertex with an external ghost leg and integrate by parts with respectto D or ¯ D coming from ghost propagator (44). For the superdiagram (15) this is illustrated inFig. 3. All possible terms in which covariant derivatives act on the external ghost leg are finite.Therefore, divergences can arise only if D or ¯ D act on the propagator of the quantum gaugesuperfield producing the expressions D (cid:18) R∂ − ∂ (cid:16) D ¯ D + ¯ D D (cid:17)(cid:16) ξ K − R (cid:17)(cid:19) = ξ D ∂ K ;¯ D (cid:18) R∂ − ∂ (cid:16) D ¯ D + ¯ D D (cid:17)(cid:16) ξ K − R (cid:17)(cid:19) = ξ ¯ D ∂ K . (45)The remaining derivatives D or ¯ D act on the polarization operator, which is transversal dueto Eq. (32). Therefore, from the equations D ∂ Π / = 0; ¯ D ∂ Π / = 0 (46)we conclude that the considered contribution vanishes. Certainly, this argumentation is validfor each of the superdiagrams (13), (14), and (15).3. In the remaining supergraphs in Fig. 1 an external gauge leg is attached to a matter loop.By other words, these superdiagrams contain subdiagrams presented in Fig. 4. Investigating11hem we should take into account that the considered theory is free from anomalies, see Eq. (7)and a similar equation for the generators T A PV . After a rather non-trivial calculation we haveobtained that in the limit p → V V ¯ D B V V ¯ D B V V ¯ D B V V ¯ D B Figure 4: The subdiagrams obtained by attaching an external ¯ D B -leg to the matter part ofthe one-loop polarization operator. e f ABC T ( R ) Z d θ ¯ D B A (0 , θ ) Z d k (2 π ) [ ¯ D ˙ a , D b ] V B ( k, θ ) V C ( − k, θ )( γ µ ) ˙ ab Z d q (2 π ) (2 q + k ) µ ( q + k ) − q × F q + k F q (cid:18) q F q − m − q F q − M − q + k ) F q + k − m + 1( q + k ) F q + k − M (cid:19) . (47)(Certainly, here we omitted the integral over d p , because we are interested only in the formof the momentum integral in the limit p →
0, and a numerical coefficient, which will not beessential below.) Evidently, after the Wick rotation the integral in this equation, I µ ≡ Z d Q (2 π ) (2 Q + K ) µ F Q + K F Q ( Q + K ) − Q (cid:18) Q F Q + m − Q F Q + M − Q + K ) F Q + K + m + 1( Q + K ) F Q + K + M (cid:19) , (48)will be proportional to K µ . Taking into account that K ν (2 Q + K ) ν = ( Q + K ) − Q , (49)we see that it can be equivalently presented in the form I µ = K µ K ν K I ν = K µ K Z d Q (2 π ) F Q + K F Q (cid:18) Q F Q + m − Q F Q + M (cid:19) − K µ K Z d Q (2 π ) F Q + K F Q (cid:18) Q + K ) F Q + K + m − Q + K ) F Q + K + M (cid:19) = 0 . (50)The last equality is obtained after the change of the integration variable Q µ → Q µ − K µ , which ispossible, because the above integrals are only logarithmically divergent due to the contributionsof the Pauli–Villars superfields. Moreover, it is necessary to take into account that the firstintegral in Eq. (50) evidently depends only on K .Due to Eqs. (47) and (50) the sums of the superdiagrams (2), (3), (9), (12) and (4), (7), (8),(10), (11) turn out to be UV finite. If higher covariant derivatives are used for a regularization, then vertices with a large number of legs are muchmore complicated than for the original theory.
In this paper we have demonstrated that a part of the two-loop contribution to the three-point gauge-ghost vertices proportional to C T ( R ), which comes from superdiagrams containinga matter loop, is finite in the UV region. These vertices have two external ghost legs and oneleg of the quantum gauge superfield. The calculation has been done with the help of the highercovariant derivative regularization in the limit of the vanishing external momenta for a general ξ -gauge. The result completely agrees with the general statement derived in Ref. [46], accordingto which the considered vertices are finite in all orders of the perturbation theory. Unlike somesimilar previous results [49, 50], it was proved in the case of using the superfield formulationof N = 1 supersymmetric gauge theories for a general ξ -gauge. In the supersymmetric casethis statement turned out to be a very important step for the perturbative derivation of thenon-Abelian NSVZ β -function made in Refs. [26,46,47]. That is why the result of this paper canbe treated as a check of a certain part of this proof. However, it should be noted that the totaltwo-loop contribution to the three-point gauge-ghost vertices also includes terms proportionalto C , which were not considered in this paper. We hope to consider them in the forthcomingpublications. Acknowledgments
This work was supported by Foundation for Advancement of Theoretical Physics and Math-ematics “BASIS”, grants 19-1-1-45-5 (M.K.), 18-2-6-159-1 (N.M.), 18-2-6-158-1 (S.N.), 19-1-1-45-3 (I.S.), and 19-1-1-45-1 (K.S.).
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