Gauge dependence of the one-loop divergences in $6D$, ${\cal N} = (1,0)$ abelian theory

We study the gauge dependence of the one-loop effective action for the abelian $6D$, ${\cal N}=(1,0)$ supersymmetric gauge theory formulated in harmonic superspace. We introduce the superfield $\xi$-gauge, construct the corresponding gauge superfield propagator, and calculate the one-loop two-and three-point Green functions with two external hypermultiplet legs. We demonstrate that in the general $\xi$-gauge the two-point Green function of the hypermultiplet is divergent, as opposed to the Feynman gauge $\xi =1$. The three-point Green function with two external hypermultiplet legs and one leg of the gauge superfield is also divergent. We verified that the Green functions considered satisfy the Ward identity formulated in ${\cal N}=(1,0)$ harmonic superspace and that their gauge dependence vanishes on shell. Using the result for the two- and three-point Green functions and arguments based on the gauge invariance, we present the complete divergent part of the one-loop effective action in the general $\xi$-gauge.


Introduction
Gauge theories with extended supersymmetries in higher dimensions attract a considerable attention for a long time [1,2,3,4,5,6,7,8]. On the one hand, such theories are nonrenormalizable due to the dimensionful coupling constant (see, e.g., [9,10]). On the other hand, one can expect an improvement of the ultraviolet behavior due to the extended supersymmetry. It is very interesting to check this conjecture on the explicit examples of higher-dimensional supersymmetric theories. To be more realistic, one can expect that the full canceling of divergences is presumably possible only in the lowest loops even in the maximally extended theories (see, e.g., [11]). The problem reveals clear analogies with the most interesting case of gravity. However, the analysis in supersymmetric gauge theories is much simpler.
In order to fully display the underlying properties of theories with some symmetries it is highly desirable to be aware of the regularization and quantization schemes which do not break these symmetries. In the case of extended supersymmetries these purposes can be achieved within the harmonic superspace approach [12,13,14,15,16,17]. For 6D supersymmetric gauge theories (which will be the subject of the present paper) this formalism [18,19,20,21,22,23] ensures manifest N = (1, 0) supersymmetry. With the use of the background field method in harmonic superspace [16,24], gauge symmetry can also be made manifest. For these reasons the harmonic superspace formalism seems to be most suitable for quantum calculations in 6D supersymmetric theories (note that 6D , N = (1, 0) theories are in general anomalous, see, e.g., [25,26,27,28]).
Recently, some explicit calculations based on the harmonic superspace method were done for N = (1, 0) and N = (1, 1) gauge theories [29,30,31,32,33], following the general pattern of Ref. [4]. These calculations were basically performed in the Feynman gauge ξ = 1, which ensures the simplest form of the propagator of the gauge superfield. This considerably simplifies the calculation of quantum corrections. However, the gauge dependence of the results obtained by the harmonic superspace technique has not been yet analyzed. Meanwhile, the calculations in the non-minimal gauges are frequently rather useful as compared to those in the Feynman gauge, because they are capable to make manifest divergences in the lower loops. For example, for N = 1 supersymmetric gauge theories in the one-loop approximation ghosts are not renormalized in the Feynman gauge, while divergences appear for ξ = 1 [35]. For calculations in higher orders, the knowledge of gauge dependence in the lower-order approximations is also essential, see, e.g., [36]. These are the reasons why a vast literature is devoted to calculations in non-minimal gauges. As a characteristic example, let us mention a recent paper [37].
In the present paper we consider the simplest 6D , N = (1, 0) supersymmetric gauge theory, namely, N = (1, 0) supersymmetric electrodynamics, and investigate the structure of the gauge-dependent contributions to the effective action by the harmonic superspace technique. In particular, we demonstrate that (unlike the case of the Feynman gauge considered, e.g., in [29]) the two-point Green function of hypermultiplets is divergent already at the one-loop level. The gauge-dependent divergences are also present in the gauge multiplet -hypermultiplet Green functions. In this paper we explicitly calculate the one-loop three-point Green function and find its divergent part. Moreover, we derive the Ward identity in the harmonic superspace and verify that the Green functions obtained by calculating harmonic supergraphs satisfy this identity, as expected. This result is a non-trivial verification of the correctness of our calculations. One more test, which has also been done in this paper, is the demonstration of the property that the gauge dependence of the effective action vanishes on shell (this is a consequence of the general theorem, see Refs. [38,39,40,41,42,43]). Using the results for the two-and three-point Green functions, we also restore the complete result for the one-loop divergences, based on the gauge invariance of the theory under consideration.
The paper is organized as follows: In Sect. 2 we recall some basic points of the formulation of 6D , N = (1, 0) supersymmetric electrodynamics in harmonic superspace. We present the superfield action for this theory, write down the Ward identity, and formulate the harmonic superspace Feynman rules. In particular, we construct the propagator of the gauge superfield in the non-minimal gauges which are analogs of the ξ-gauges in the usual electrodynamics. In Sect. 3, using these Feynman rules, we investigate the gauge dependence of the one-loop two-point Green functions of the gauge superfield and the hypermultiplet. We also calculate the one-loop three-point gauge superfield -hypermultiplet Green function. Checking the Ward identities for these Green functions is the subject of Sect. 4. The vanishing of the gauge dependence on shell in the approximation we are considering is demonstrated in Sect. 5. The total divergent part of the one-loop effective action (which is an infinite series in V ++ ) is constructed in Sect. 6, by invoking the arguments based on the gauge invariance. Also we verify that the gauge dependence of the expression obtained vanishes on shell. Some technical details are collected in two Appendices.

The harmonic superspace action
The harmonic superspace is very convenient for formulating 6D , N = (1, 0) supersymmetric theories, because it ensures manifest supersymmetry at all steps of quantum calculations. It is parametrized by the coordinate set (x M , θ ai , u ± i ) which will be referred to as the central basis. Here x M with M = 0, . . . 5 are the usual coordinates of the six-dimensional Minkowski space. The Grassmann anticommuting coordinates θ ai with a = 1, . . . 4 and i = 1, 2 form a left-handed 6D spinor. The harmonic variables u ± i satisfy the condition u +i u − i = 1, with u − i = (u +i ) * . The analytic basis of the harmonic superspace is parametrized by the coordinates where γ M are 6D γ-matrices. The coordinate subset (x M A , θ +a , u ± i ) parametrizes the analytic harmonic subspace which is closed on its own under 6D , N = (1, 0) supersymmetry transformations.
It is convenient to define the spinor covariant derivatives and to introduce the notation Also we will need the harmonics derivatives in the central basis They satisfy the commutation relations of the SU (2) algebra. The analytic basis form of these derivatives can be easily found and is given, e.g., in [34]. For constructing the N = (1, 0) invariants we need the invariant superspace integration measures: In this paper we consider N = (1, 0) supersymmetric electrodynamics, which is a particular abelian case of N = (1, 0) supersymmetric Yang-Mills theory with hypermultiplets. The harmonic superspace form of the action of 6D, N = (1, 0) supersymmetric Yang-Mills theory was pioneered in Ref. [20]. As opposed to the analogous 4D, N = 2 construction, the gauge theory coupling constant f 0 in 6D has the dimension m −1 . In the harmonic superspace approach the gauge superfield V ++ (z, u) satisfies the analyticity condition and is real with respect to the special conjugation denoted by , i.e. V ++ = V ++ . The hypermultiplets are described by the analytic superfield q + and its -conjugate q + . Like in the non-supersymmetric case, the action of N = (1, 0) electrodynamics is quadratic in the gauge superfield. It can be written as where and D ++ is taken in the analytic basis. The gauge transformations has the form where λ is an analytic superfield parameter which is real with respect to the -conjugation. It is useful to introduce the non-analytic superfield It satisfies the conditions D ++ V −− = D −− V ++ and transforms as under the gauge transformations. Starting from this superfield, it is possible to construct the analytic superfield F ++ = (D + ) 4 V −− , which is gauge invariant in the abelian case. For further use, we also define the non-analytic superfield q − as a solution of the equation From this definition one can derive that the gauge transformations act on q − as In the explicit form the solution of Eq. (13) can be expressed as the series where subscripts numerate the harmonic "points". For quantizing the theory (8) it is necessary to fix the gauge. This can be done by adding the gauge-fixing term to the action, where ξ 0 is the bare gauge-fixing parameter. This term corresponds to the ξ-gauge in the usual electrodynamics. In particular, the Feynman gauge amounts to the choice ξ 0 = 1. In the abelian case we are considering it is not necessary to introduce the ghosts superfields. Therefore, the generating functional for our theory can be written as where S sources is a sum of the source terms, Here J (+2) is the analytic source for the gauge superfield, while j (+3) and j (+3) denote sources for the hypermultiplet superfields. The effective action is constructed from the generating functional for the connected Green functions W = −i ln Z by making the Legendre transformation, where it is necessary to express the sources in terms of the fields with the help of the equations

Ward identity
In the abelian gauge theory at the quantum level the gauge invariance is encoded in the Ward identity [44], which is a particular case of the Slavnov-Taylor identities [45,46]. The harmonic superspace analog of this identity can be formulated, using the standard technique. For this purpose we make the transformation (10) in the generating functional (17) which evidently remains invariant. Taking into account that the classical action is gauge invariant, in the lowest order in λ we obtain where we used the notation Integrating in Eq. (21) by parts with respect to the derivatives D ++ , using an arbitrariness of λ, and expressing the result in terms of superfields, we obtain where Γ is the effective action defined by Eq. (19), and we also took into account that the gauge-fixing term is quadratic in the gauge superfield. Introducing the Ward identity can be written in a more compact form, It is important that this equation is valid for arbitrary non-zero values of the involved superfields. Differentiating Eq. (25) with respect to various superfields we derive an infinite set of identities relating the longitudinal part of the (n+1)-point Green functions to the n-point Green functions. For example, differentiating with respect to V ++ 2 and setting all fields equal to zero at the end, we obtain that quantum corrections to the two-point Green function of the gauge superfield are transversal, Differentiating Eq. (25) with respect to q + 2 and q + 3 and setting the fields equal to zero at the end give an analog of the usual Ward identity relating three-and two-point Green functions: When deriving this equation, we have taken into account the property implied by the Grassmann analyticity where It is convenient to multiply the identity (27) with the analytic superfields λ 1 , q + 2 , and q + 3 , and integrate the expression obtained over both analytic arguments, where dµ = dζ This form of the Ward identity is most convenient, when checking it for one or another particular class of diagrams.

The Feynman rules
For the explicit calculation of quantum correction it is necessary to formulate the relevant Feynman rules. This can be accomplished quite similarly to the 4D, N = 2 case considered in detail in Refs. [13,14]. To find the propagator of the gauge superfield in the ξ-gauge, we consider the sum of the gauge superfield action and the gauge-fixing term where we made use of the identity and took into account that, when acting on the analytic superfields, Following Ref. [31], we consider the free theory and solve the equation of motion for the superfield V ++ in the presence of the source term, The solution can be presented as whence one extracts the ξ-gauge form of the propagator of the gauge superfield The second term vanishes in the Feynman gauge ξ 0 = 1. Such a choice considerably simplifies calculation of quantum corrections. However, the purpose of the present paper is to investigate the ξ 0 -dependence of various Green functions for the generic choice of ξ 0 .
In left part of Fig. 1, the propagator (37) is depicted by the wavy line ending on the points 1 and 2.
For completeness, we also present the expression for the hypermultiplet propagator, which is denoted by the solid line on the right. The only vertex of the theory (8) is presented in Fig. 2 and stands for the interaction of the hypermultiplet with the gauge superfield The superficial degree of divergence in the theory under consideration has been calculated in Ref. [29]: Here L is a number of loops, N q is a number of external hypermultiplet legs, and N D is a number of spinor supersymmetric covariant derivatives acting on external legs. This formula implies that in the one-loop approximation only diagrams without external hypermultiplet legs or with two such legs can be divergent.
3 Gauge dependence of the one-loop divergences

Two-point function of the gauge superfield
In the one-loop approximation the two-point function of the gauge superfield V ++ is divergent. In the abelian case this divergence comes only from the diagram pictured in Fig. 3. However, this diagram does not contain propagators of the gauge superfield and is therefore gauge-independent. Thus, in the one-loop approximation this Green function in the ξ-gauge is the same as in the Feynman gauge. It is given by the expression [29] (41) The corresponding divergent part of the effective action is gauge-independent and in the dimensional reduction scheme 1 can be written as where ε = 6 − D.

Two-point hypermultiplet Green function
In the one-loop approximation the two-point Green function of the hypermultiplet is contributed to by the single logarithmically divergent diagram presented in Fig. 4. In the Feynman gauge this superdiagram vanishes. However, it includes the propagator of the gauge superfield, for which reason we can expect that the result for it is in fact gauge-dependent. Using the Feynman rules defined above, the expression for this diagram in the generic ξ-gauge can be written as The derivatives (D + 1 ) 4 (D + 2 ) 4 in the hypermultiplet propagator can be used to convert the integrations over dζ (−4) into those over d 14 z, Taking into account the identities we find that the first term in this expression vanishes, reducing (44) to the form This expression can be rewritten in the momentum representation as We observe that this expression is logarithmically divergent and does not vanish, unless the Feynman gauge is chosen. If the theory is regularized by dimensional reduction, the corresponding contribution to the divergent part takes the form

Three-point gauge-hypermultiplet Green function
According to Eq. (40), all diagrams containing two external hypermultiplet legs are logarithmically divergent, irrespective of the number of the external gauge legs. That is why in calculating the one-loop divergences it is necessary to take into account such Green functions. The simplest of them is the three-point gauge superfield -hypermultiplet. In the one-loop approximation, it is contributed to by the single supergraph depicted in Fig. 5.
To work out this expression, we, first, convert the integrals over dζ (−4) in it into integrals over d 14 z using Eq. (6): Next, we integrate by parts with respect to (D + 2 ) 4 (assuming that D + 2 acts on z 1 ), taking into account that In the term containing the harmonic δ-function we further integrate over du 3 . Integrating also over θ 2 , we finally obtain for (50): As the further step, we use the identities (45), (46) together with in order to do the integrals over the Grassmann coordinate θ 2 . After renaming θ 1 → θ, the expression for the diagram in question in the momentum representation is written as where ( γ M ) ab = ε abcd (γ M ) cd /2. The divergent part of this expression can now be found after the Wick rotation. There remains only one divergent integral which, after regularizing it by dimensional reduction, is reduced to where the capital letters denote Euclidean momentums. Thus, the divergent part of the diagram in Fig. 5 can be presented as where the subscripts on the superfields refer to the relevant harmonic arguments.

Verification of the Ward identities
To be convinced of the correctness of the results obtained in the previous sections, let us check that the two-and three-point Green functions derived above satisfy the Ward identities.
First, for completeness, we verify the Ward identity (26). The two-point Green function of the gauge superfield is obtained by differentiating Eq. (41) with respect to V ++ , using Eq. (28). This gives where Therefore, Thus, we have verified that the Ward identity (26) is indeed satisfied. The two-point Green function of the hypermultiplet is obtained by differentiating Eq. (48) with respect to q + and q + . These derivatives are calculated with the help of Eq. (28). We obtain where The three-point gauge superfield -hypermultiplet Green function can be constructed quite similarly, starting from Eq. (56), but we prefer not to present the expression for it explicitly. Instead, we will check for it the Ward identity in the form (30). From Eq. (56) we obtain Next, we integrate by parts with respect to the harmonic derivatives D ++ 1 , taking into account the identity (66) After some algebra (described in Appendix A), this gives The right-hand side of this equation can be rewritten as i dζ thus demonstrating that the Green functions (48) and (56) satisfy the Ward identity (30), as it should be. Obviously, they also satisfy the Ward identity in the original form (27). This completes checking the correctness of our calculation.

The vanishing of the gauge dependence on shell
According to the general theorem of Refs. [38,39,40,41,42,43], the gauge-dependent terms should disappear on shell. Let us verify that our results are in agreement with this statement.
It is convenient to represent the effective action in the form where is the effective action in the Feynman gauge and stands for the gauge-dependent remainder of the effective action. The purpose of this section is to demonstrate, by an explicit calculation, that in the approximation considered, Γ indeed vanishes on shell. To this end, we use the equations of motion for the hypermultiplets following from the action (8), In Appendix B (after some lengthy calculations) we demonstrate that, with these equations taken into account, the gauge-dependent part of the one-loop effective action can be cast in the form On shell, where q 2 = 0 and (q + p) 2 = 0, 2 this expression vanishes. Thereby we have proved that the gauge dependence is vanishing on shell. Note that, while deriving this result, we ignored all terms proportional to (V ++ ) k for k ≥ 2, because in this paper we limit our attention only to the diagrams without external gauge superfield legs at all, and to those having a single gauge superfield leg. In this approximation, terms of higher orders in V ++ are irrelevant. 6 The total divergent part of the one-loop effective action So far we investigated gauge dependence of the two-and three-point Green functions only. In particular, we demonstrated that the corresponding one-loop divergences are gauge-dependent. However, according to Eq. (40), the Green functions with an arbitrary number of external gauge legs (and two external hypermultiplet legs) are also divergent. Nevertheless, the total divergent part of the one-loop effective action can be found using the reasoning based on the gauge invariance. Actually, the one-loop divergences corresponding to the two-and three-point Green functions (see Eqs. (42), (49), and (59)) have the form The first term in this equation is gauge invariant. The expression corresponding to the first term in the curly brackets can also be rewritten in the explicitly gauge invariant form, According to Eq. (15), the remaining two terms in Eq. (74) are the lowest terms in the series expansion of the gauge invariant expression in powers of V ++ . Thus, the divergent part of the one-loop effective action can be written as Acting on the latter equation by ∇ ++ it is easy to find In deriving these relations, we made use of the well known properties D ++ ω −n = 0 → ω −n = 0, D −− ω +m = 0 → ω +m = 0 for n ≥ 1, m ≥ 1. As a consequence of (79), we obtain that on shell Thus, on shell, the one-loop divergence (77) takes the form We see that this expression does not depend on the parameter ξ and, hence, on the gauge choice.

Summary
In this paper, using the 6D , N = (1, 0) harmonic superspace formalism, we studied the gauge dependence of the one-loop effective action for N = (1, 0) supersymmetric quantum electrodynamics. As compared to the case of the Feynman gauge, in the general ξ-gauge some new divergences appear. In particular, we demonstrated that in the general case the hypermultiplet Green function is divergent already in the one-loop approximation, as opposed to the case of the Feynman gauge, in which this divergence vanishes. Moreover, we calculated the three-point gauge -hypermultiplet Green function in the general ξ-gauge. To check the correctness of the calculation, we have verified the relevant Ward identity. Also it was checked that the gauge dependence vanishes on shell. Taking into account the gauge invariance, we also restored the divergent part of the one-loop effective action with terms of higher orders in the gauge superfield V ++ . It is given by Eq. (77) and contains a new term which is absent in the Feynman gauge. We demonstrated that the gauge dependence of this general expression also vanishes on shell.
It would be interesting to investigate the gauge dependence in the non-abelian case. In particular, from the results of this paper we can expect that in the general ξ-gauge the 6D , N = (1, 1) sypersymmetric Yang-Mills theory is not finite even in the one-loop approximation, while the divergent terms are vanishing on shell.

A Ward identity in harmonic superspace
Let us show how to pass from Eq. (65) to its equivalent form (67). After integrating by parts with respect to the derivatives D ++ 1 and using the identity (66), we obtain Then we integrate by parts with respect to the derivatives D −− and take off one harmonic integral with the help of the delta functions. Taking into account that the first term vanishes as a consequence of the analyticity of the superfields λ, q + , and q + , the expression (82) can be further rewritten as Once again, integrating by parts and taking into account that on the analytic superfields, this expression can be cast in the form where we have also used the relations (q + p) 2 + k 2 + 2k M (q + p) M = (q + k + p) 2 , B Gauge-dependent part of the effective action and the hypermultiplet equations of motion In this appendix we verify that the gauge-dependent part of the effective action vanishes on shell. This is an important non-trivial check of the correctness of our calculations.
First, we consider the two-point Green function of the hypermultiplet given by Eq. (48). Using the identity we rewrite it as The second term in this expression vanishes due to the analyticity of the hypermultiplet superfield, (89) After integrating by parts with respect to the harmonic derivatives, the considered contribution to the effective action can be represented as (90) Using the equations of motion for the hypermultiplets 0 = ∇ ++ q + = (D ++ + iV ++ )q + ; 0 = ∇ ++ q + = (D ++ − iV ++ ) q + , we see that on shell the expression (90) is proportional to q + (V ++ ) 2 q + . However, in this paper we do not consider terms quadratic in the gauge superfield V ++ . This implies that, within the accuracy of our approximation, the part of the one-loop effective action corresponding to the hypermultiplet two-point function vanishes on shell. Next, we consider the gauge dependent part of the three-point gauge superfield -hypermultiplet Green function. It corresponds to the terms proportional to q + V ++ q + in the expression (71). We will demonstrate that Γ (3) vanishes on shell (in the approximation when all terms with more than one V ++ are omitted).
For the on-shell hypermultiplets the relations q 2 = 0 and (q +p) 2 = 0 are valid, so this expression vanishes. The conclusion is that the gauge-dependent contributions to the effective action are indeed canceled on shell in the approximation we stick to.