Harmonic superspace approach to the effective action in six-dimensional supersymmetric gauge theories

We review the recent progress in studying the quantum structure of $6D$, ${\cal N}=(1,0)$ and ${\cal N}=(1,1)$ supersymmetric gauge theories formulated through unconstrained harmonic superfields. The harmonic superfield approach allows one to carry out the quantization and calculations of the quantum corrections in a manifestly ${\cal N}=(1,0)$ supersymmetric way. The quantum effective action is constructed with the help of the background field method that secures the manifest gauge invariance of the results. Although the theories under consideration are not renormalizable, the extended supersymmetry essentially improves the ultraviolet behavior of the lowest-order loops. The ${\cal N}=(1,1)$ supersymmetric Yang--Mills theory turns out to be finite in the one-loop approximation in the minimal gauge. Also some two-loop divergences are shown to be absent in this theory. Analysis of the divergences is performed both in terms of harmonic supergraphs and by the manifestly gauge covariant superfield proper-time method. The finite one-loop leading low-energy effective action is calculated and analyzed. Also in the abelian case we discuss the gauge dependence of the quantum corrections and present its precise form for the one-loop divergent part of the effective action.


Introduction
The higher-dimensional supersymmetric gauge theories attract the significant interest due to their remarkable properties in classical and quantum domains and profound links with string/brane theory. The various aspects of quantum structure of such theories were intensively investigated for a long time (see, e.g., [1][2][3][4][5][6][7][8] and references therein). Although these theories are not renormalizable because of the dimensionful coupling constant [9,10], it is very interesting to understand, to which extent a large number of (super)symmetries can improve the ultraviolet behaviour. It is expected that supersymmetries sometimes can help cancelling divergences in the lowest loops, but in higher orders the divergences still appear even in the maximally extended supersymmetric models [11]. This looks very similar to what happens in the case of the supergravity theories, but from the technical point of view the calculations in higher-dimensional gauge theories are much simpler.
If we wish to understand how the given symmetry improves the ultraviolet properties of some theory, it is obviously of importance to use a regularization and the quantization procedure which preserve this symmetry. For the higher-dimensional supersymmetric Yang-Mills (SYM) theories with matter it is highly desirable to keep unbroken the gauge invariance and off-shell supersymmetry. For example, quantizing 4D, N = 1 supersymmetric theories in superspace, we ensure a manifest gauge invariance and supersymmetry at all steps of quantum calculations [9,10]. Unfortunately, sometimes it is impossible to quantize a theory in such a way that all supersymmetries are off-shell and manifest. For example, 4D, N = 4 SYM theory cannot be quantized in an N = 4 supersymmetric manner since the manifest N = 4 formulation of this theory is yet lacking. However, 4D, N = 2 supersymmetry can be kept manifest within the harmonic superspace formalism [12][13][14][15][16][17]. This approach can be generalized to 6D case with N = (1, 0) supersymmetry as a manifest symmetry [18][19][20][21][22][23]. Note that, although 6D, N = (1, 0) supersymmetric theories look very similar to their 4D, N = 2 counterparts, there is an essential difference between the two: in the generic case 6D, N = (1, 0) theories are anomalous [24][25][26][27]). However, for the 6D, N = (1, 1) theory the anomalies are canceled. The manifest gauge symmetry is ensured within the background field method formulated in harmonic superspace [16,28].
In this paper we briefly review some recent results [29][30][31][32][33][34] concerning the structure of the ultraviolet divergences and low-energy effective action in 6D, N = (1, 1) and N = (1, 0) SYM theories in the harmonic superspace approach 1 . The main purpose of this study is to reveal the structure of the off-shell divergences in the harmonic superspace approach and to find them explicitly in the lowest loops following the proposals of Ref. [8]. Such calculations can be done using either the formalism of harmonic supergraphs, or the harmonic superspace generalization of the proper time method of Refs. [37,38]. The proper time method is a powerful tool of performing the one-loop calculations. In particular, it well suits for calculating the finite contributions to effective action in the manifestly gauge invariant and supersymmetric way. We explicitly demonstrate the advantages of the harmonic superspace approach for studying the quantum structure of 6D SYM theories. Though these theories are not renormalizable because of dimensionful coupling constant, we will see that in the one-loop approximation N = (1, 1) SYM theory is finite, if the calculations are performed in the Feynman gauge. The absence of divergences in a minimal gauge and their presence in the non-minimal gauges was already encountered in some other calculation (see, e.g., [39]). The paper has the following structure. In Sect. 2 we introduce 6D, N = (1, 0) harmonic superspace and explain how it can be used for formulating supersymmetric gauge theories. Actually, we consider N = (1, 0) SYM theory interacting with a massless matter hypermultiplet which belongs to an arbitrary representation of the gauge group. The simplest abelian theory of this type is investigated in Sect. 3 at the quantum level. First, in Sect. 3.1, we describe the harmonic superspace quantization, give an account of the Feynman rules, and deduce the Ward identities encoding the gauge invariance at the quantum level. The next Sect. 3.2 is devoted to the calculation of the one-loop divergences and the study of their gauge dependence in the abelian case. In particular, we construct the total divergent part of the one-loop effective action and verify that its gauge-dependent part vanishes on shell. One-loop quantum corrections in the non-abelian case are investigated in Sect. 4. We start, in Sect. 4.2, with the quantization procedure described in Sect. 4.1 and then calculate the one-loop divergences, employing the Feynman gauge. In particular, we demonstrate that in this gauge N = (1, 1) SYM theory is finite in the one-loop approximation. The two-loop divergence of the two-point hypermultiplet Green function (also in the Feynman gauge) is calculated in Sect. 4.3. We show that for N = (1, 1) SYM theory this Green function involves no divergences. The calculation of the one-loop divergences by the harmonic superspace generalization of the proper-time method is given in Sect. 4.4. This method is also applied for calculating the finite contributions to the one-loop effective action in Sect. 4.5, where the leading low-energy structure of this action was found. It is worth pointing out that such an effective action is closely related to the on-shell amplitudes in 6D maximally extended supersymmetric Yang-Mills theories (see, e.g., [2] and references therein) and to the so called little strings [40][41][42].
2 Harmonic superspace formulation of 6D supersymmetric gauge theories The conventional 6D, N = (1, 0) superspace is parametrized by the coordinates z ≡ (x M , θ a i ), where x M with M = 0, . . . 5 are the ordinary space-time coordinates, and θ a i with a = 1, . . . 4 and i = 1, 2 are the Grassmann (i.e., anticommutaing) variables forming a 6D left-handed spinor. The harmonic superspace is obtained from the N = (1, 0) superspace just defined by adding to its coordinates the harmonic variables u ±i , such that u +i u − i = 1 and The basic novel feature of the harmonic superspace is the existence of an analytic subspace in it, with the coodinates This subspace is closed under 6D, N = (1, 0) supersymmetry transformations.
For the integration measures on the harmonic superspace and its analytic subspace we will use the notation Also we introduce the spinor covariant derivatives The integration measures are related by the useful identity An important ingredient of the approach is the harmonic derivatives which constitute the algebra SU (2), In the analytic basis (x M A , θ ±a , u ± i ) the harmonic derivatives acquire some additional terms, the precise form of which can be found in [43].
The harmonic superspace analog of the gauge field is the analytic superfield V ++ (z, u) which satisfies the condition and is real with respect to the "tilde" conjugation, V ++ = V ++ . Geometrically, this object is the gauge connection covariantizing the harmonic deivative D ++ , The pure 6D, N = (1, 0) SYM theory is described by the harmonic superspace action [20] S SYM = 1 . (2.10) In this expression f 0 is the bare coupling constant. The crucial difference of 6D case from the similar 4D case is that the coupling constant f 0 in six dimensions is dimensionful, Obviously, this gives rise to lacking of good renormalization properties at the quantum level.
In the notation accepted in this paper we will always assume that the gauge superfield in the pure Yang-Mills action (2.10) is decomposed over the generators of the fundamental representation, V ++ (z, u) = V ++A t A . The generators t A satisfy the conditions where f ABC are the gauge group structure constants. Just as in the non-supersymmetric case, only terms quadratic in the gauge superfield V ++ survive in the action (2.10) for the abelian gauge group G = U(1).
General 6D, N = (1, 0) gauge theories also involve the hypermultiplets minimally coupled to the gauge superfield V ++ . In the harmonic superspace approach the hypermultiplets are described by analytic superfields q + and their tilde-conjugated q + , The full action of the gauge theory with hypermultiplets reads Note that the covariant harmonic derivative in the second piece of this action, includes the generators T A corresponding to the representation R to which the hypermultiplet superfields q + belong. These generators satisfy the relations analogous to (2.11): Assuming that the gauge group G is simple, we also define C 2 and C(R) i j as Note that C(R) i j is proportional to δ j i only for an irreducible representation R . In particular, for the adjoint representation of a simple group we have If the hypermultiplet belongs to the adjoint representation, R = Adj, the action (2.13) describes N = (1, 1) SYM theory which possesses a hidden N = (0, 1) supersymmetry in addition to the manifest N = (1, 0) one. This theory is 6D analog of 4D, N = 4 SYM theory. The 4D, N = 4 SYM theory is known to possess unique properties in the quantum domain since it is a completely finite quantum field theory [44][45][46][47]. One can expect that the quantum 6D, N = (1, 1) SYM theory possesses some remarkable properties as well.
The general N = (1, 0) gauge theory described by the action (2.13) is invariant under the gauge transformations q + → e iλ q + ; q + → q + e −iλ (2.18) parametrized by an analytic superfield λ, such that λ = λ A t A for V ++ = V ++A t A (in the gauge part of the total action), and λ = λ A T A for V ++ = V ++ T A , q + , and q + (in the hypermultiplet part).
Also we will need the non-analytic gauge superfield , (2.19) which covariantizes the harmonic derivative D −− and satisfies the "harmonic flatness condition" An important object is the analytic superfield strength which obeys the off-shell constraint as a consequence of (2.20) and the analyticity of V ++ . One more useful quantity is a non-analitic superfield q − which is defined by the equation The solution of this equation is given by the series The gauge transformations of the superfields V −− , F ++ and q − defined above are as follows The simplest particular case of the theory (2.13) corresponds to the gauge group U(1). The corresponding abelian gauge theory is 6D, N = (1, 0) supersymmetric analog of QED, and it is described by the action with ∇ ++ = D ++ + iV ++ . In the abelian case the gauge transformations acquire the form (2.27) and the expression for V −− is considerably simplified, (2.28) 3 Quantum corrections in 6D, N = (1, 0) supersymmetric electrodynamics

Quantization, Feynman rules, and Ward identities in the abelian case
We will start investigating quantum properties of 6D, N = (1, 0) gauge theories in harmonic superspace by considering the simplest abelian theory with the action (2.26). The quantization procedure in the abelian case requires fixing the gauge. The harmonic superspace analog of the well-known ξ-gauges in QED is obtained by adding, to the original action, the gauge-fixing term, where ξ 0 is the gauge parameter. As usual, the normalization was chosen so that the Feynman gauge corresponds to ξ 0 = 1. Taking into account the absence of the Faddeev-Popov ghosts in the abelian case, the generating functional of the theory under consideration has the form Z = exp(iW ) = DV ++ D q + Dq + exp i(S + S gf + S sources ) (3.2) (as is well known, W = −i ln Z is the generating functional for the connected Green functions). In harmonic superspace, the source term can be written as where the analytic superfields J (+2) , j (+3) and j (+3) are the sources for V ++ , q + , and q + , respectively.
The 1PI Green functions are generated by the effective action with the sources being expressed in terms of the basic superfields by the equations Using the standard technique and starting from the functional (3.2), one can construct the Feynman rules for the considered theory. 2 Namely, we represent the total classical action as a sum of the free part S (2) which is quadratic in the involved superfields and the interaction part S I which encompasses all terms of the higher orders, This allows us to write the generating functional in the form where the generating functional of the free theory is given by the Gaussian integral Then the expression for S I produces the vertices, while all propagators are encoded in Z 0 .
For the theory (2.26), the free part of the action and the interaction term read From the interaction (3.10) we conclude that there is only one interaction vertex in the theory. It is depicted in Fig. 1. For calculating the Gaussian integral in (3.8) we solve the free equations of motion (see [31] for details) and substitute the result into the argument of the exponential. This gives Here, the propagators of the gauge superfield and of the hypermultiplet are given, respectively, by the expressions (3.12) Graphically, the V ++ propagator is denoted by a wavy line, while the hypermultiplet propagator by a solid line. They are depicted on the left and the right sides of Fig. 2, respectively. It is obvious that the Feynman diagrams containing closed loops are divergent. Their superficial degree of divergence has been found in Ref. [29]. It is defined by the equation where the number of loops is denoted by L, the number of external hypermultiplet lines by N q , and N D denotes the number of spinor covariant derivatives acting on the external legs. From Eq. (3.15) one can directly conclude that in the one-loop approximation divergent diagrams should either contain two external hypermultiplet lines or not contain such external lines at all.
At the quantum level the gauge invariance of the given theory leads to some relations between the Green functions. In the abelian case these are the Ward identities [48]. Their non-abelian generalization is the Slavnov-Taylor identities [49,50]. The harmonic superspace Ward identities were constructed in [33] by making the transformation (2.18) in the generating functional (3.2). Using the notation the generating Ward identity amounts to the equation The adjective "generating" refers to the fact that in this equation the (super)field arguments are not put equal to zero in advance. Therefore, Eq. (3.17) encompasses an infinite set of identities which relate the longitudinal parts of the (n + 1)-point Green functions to the n-point Green functions.
The lowest-order Ward identity leads to the transversality of quantum corrections to the two-point function of the gauge (super)field. In the harmonic superspace language it can be obtained by differentiating Eq. (3.17) twice with respect to V ++ : where the superfield arguments have been set equal to zero at the end.
Similarly, differentiating Eq. (3.17) with respect to q + 2 and q + 3 and again setting the superfields equal to zero afterwards, we obtain a Ward identity which relates three-and two-point Green functions, The Ward identities are a very convenient tool for checking the correctness of various quantum calculations.

One-loop divergences and their gauge dependence
According to the relation (3.15), divergent diagrams should have either N q = 0 or N q = 2 of the external hypermultiplet lines (evidently, odd values of N q are forbidden). However, the number of external gauge lines can be arbitrary and the degree of divergence of the diagram is independent of this number. Nevertheless, the total divergent part of the effective action can be restored by applying to the arguments based on the gauge invariance encoded in the Ward identities. With this in mind, it is actually enough to calculate the lowest divergent Green functions. For example, the (quadratically divergent) two-point function of the gauge superfield V ++ in the one-loop order is determined by the only supergraph presented in Fig. 3.
Obviously, the expression for it is gauge-independent due to the absence of the gauge propagators. The result obtained in [29] can be presented in the form (3.20) When using the dimensional reduction [51] to regularize the theory, the divergent part of this expression is where ε = 6 − D. However, the regularization by dimensional reduction allows calculating only the logarithmical divergences, while the considered supergraph diverges quadratically. For finding these quadratic divergences one needs to use another type of regularization. For example, one could use a special modifications of the Slavnov higher covariant derivative regularization [52,53] (its harmonic superspace version for 4D, N = 2 supersymmetric theories was worked out in [54]). In the one-loop approximation it suffices to use the simplest ultraviolet cut-off procedure. If the loop momentum is cut at the scale Λ, the divergence of the considered contribution to the effective action can be written as [31] This expression is gauge invariant, so there appear no further divergent contributions coming from the diagrams with larger numbers of external gauge lines. Indeed, it is easy to see that the gauge invariant structures proportional to (F ++ ) n with n ≥ 3 correspond to the finite part of the effective action.
Next, let us consider the divergent part of the Green functions with N q = 2. The simplest one is the two-point Green function of the hypermultiplet . In the one-loop order it is given by the logarithmically divergent supergraph presented in Fig. 4. The result calculated in [33] is given by the gauge-dependent expression which is logarithmically divergent in agreement with Eq. (3.15). The corresponding divergent part (calculated using the regularization by dimensional reduction) is written as If applying the cut-off regularization, it is necessary to replace 1/ε by ln Λ. We see that the divergence disappears only in the Feynman gauge ξ 0 = 1.
Surely, the expression (3.24) is not gauge invariant. To obtain the gauge invariant answer, it is necessary to take into account divergent contributions corresponding to Green functions with N q = 2 and an arbitrary number of the external gauge superfield lines. If the number of the external V ++ lines is equal to 1, then the corresponding Green function in the one-loop order is contributed to by the only superdiagram presented in Fig. 5. The relevant expression was calculated in [33], and it has the form It is logarithmically divergent. The divergent part calculated within the dimensional reduction technique reads [33] , (3.26) where the subscripts denote the harmonic arguments.
To verify the results presented above, it is possible Both these checks have been done in [33], thereby confirming the correctness of the calculations.
However, so far we have not yet considered all the divergent one-loop diagrams. Even the sum of the expressions (3.21), (3.24) and (3.26), is not gauge invariant. The full gauge invariant result can be restored, without further calculations, solely on the ground of gauge invariance considerations. Below we will show that in the hypermultiplet sector the gauge invariant result is given by an infinite series in V ++ . The expression (3.27) is merely a sum of the lowest terms in the V ++ expansion of the full gauge invariant expression.
In order to construct the gauge invariant expression for the one-loop divergences we recall the V ++ series representation (3.28) for the non-analytic superfield q − defined in (2.23). The first terms of this series read This representation implies that the total one-loop divergences for 6D, N = (1, 1) supersymmetric electrodynamics in the general ξ 0 -gauge are written in the form where we also made use of the definition (2.21) and the precise form (2.28) of V −− in the abelian case.
Note that (in agreement with the general theorems [38,[55][56][57][58][59]) the effective action appears to be gauge independent on shell. To demonstrate this, we make use of the on-shell property whence Using this relation, we conclude that all ξ 0 -dependent terms in the expression (3.29) disappear, Let us proceed to investigating the non-abelian case. There are two main differences of the quantization procedure in this case as compared to the abelian one: 1. It is convenient to use the background (super)field method for constructing the manifestly gauge invariant effective action; 2. The gauge-fixing procedure requires adding ghosts.
According to the background field method, we split the gauge (super)field into the background and quantum parts, so that the theory becomes invariant under two types of gauge transformations. Namely, the background gauge invariance remains unbroken and so is still a manifest symmetry of the effective action. On the contrary, the quantum gauge invariance is broken by gauge fixing, although its remnant, the so called BRST symmetry [60,61], survives as a symmetry of the total gauge-fixed action.
Within the harmonic superspace formalism the background-quantum splitting is linear. The original superfield V ++ is presented as a sum of the background gauge superfield V ++ and the quantum gauge superfield v ++ , (4.1) The background gauge superfield is treated as an external superfield, for which reason it can appear only on the external legs. We denote the external legs corresponding to V ++ by the bold wavy lines. The internal and external legs of the quantum gauge superfield will be denoted by the standard wavy lines.
The background-quantum splitting for the hypermultiplets is also possible, but not necessary. The point is that the gauge-fixing term is chosen to be independent of the hypermultiplet superfields, so the effective action depends only on a sum of the quantum and background hypermultiplet superfields. For this reason here we do not split the hypermultiplets into the background and quantum parts.
After the background-quantum splitting (4.1), the gauge invariance (2.18) produces the background gauge invariance and the quantum gauge invariance Clearly, if we wish to preserve the background gauge invariance as a manifest symmetry of the effective action, it is necessary to arrange the gauge-fixing term to be invariant under the background transformations. To construct such a term, we introduce the background bridge superfield related to the superfields V ++ and V −− as where a new gauge parameter τ = τ (x, θ) does not depend on the harmonic variables. With the help of the bridge superfield the background gauge invariant gauge-fixing term is constructed as It is analogous to the usual ξ-gauge fixing term for non-supersymmetric Yang-Mills theory, the Feynman (minimal) gauge corresponding to the choice ξ 0 = 1 . Note that in the abelian case the dependence on the bridge superfield in (4.6) is canceled out, and for 6D, N = (1, 0) electrodynamics we recover the expression (3.1).
As is well known, for quantizing non-abelian theories one should introduce the Faddeev-Popov ghosts. In the background superfield method the Nielsen-Kallosh ghosts are also needed. In the harmonic superspace language, the Faddeev-Popov ghost action is written as (4.7) Here the ghosts c and the antighosts b are the Grassmann analytic superfields in the adjoint representation of the gauge group. Correspondingly, the background covariant derivative of the ghost superfield takes the form In the background superfield method the functional integral after quantization includes determinants which are usually written as functional integrals over the Nielsen-Kallosh ghosts. Within the harmonic superspace approach such determinants are given by the expression Here we introduced the notation where ϕ are the commuting Nielsen-Kallosh ghosts, analytic Grassmann-even superfields in the adjoint representation. The determinant Det ⌢ ✷ in (4.8) can also be cast in the form of a functional integral by introducing the Grassmann-odd analytic superfields ξ (+4) and σ in the adjoint representation, The sources for the gauge and hypermultiplet superfields differ from the abelian case basically by the presence of the internal symmetry indices, It is necessary to take into account that only the quantum gauge superfield v ++ is present in the term (4.12). In principle, if necessary, it is also possible to introduce sources for ghosts.
The propagators of the quantum gauge superfield and those of the hypermultiplet are similar to those in the abelian case: (4.14) In the explicit calculations in the non-abelian case we will use only the Feynman gauge ξ 0 = 1, because under this choice the gauge propagator (4.13) has the simplest form. The propagators (4.13) and (4.14) will be graphically denoted, as in the abelian case, by the wavy and solid lines (see Fig. 6). Also we will need the ghost propagators. They have the same form for both the Faddeev-Popov and the Nielsen-Kallosh ghosts, and will be depicted by the dashed and dotted lines, respectively.
(1) Finally, the propagator of the superfields ξ (+4) and σ introduced in (4.10) has the form (4.16) The interaction vertices can be easily read off from the interaction terms in the action. It is important that in the non-abelian case on the external legs there can appear the background gauge superfield. Such legs will be denoted by the bold wavy lines. Due to the linear background-quantum splitting (4.1) all vertices can contain both quantum and background wavy lines. Precisely as in the N = (1, 0) supersymmetric electrodynamics, in the non-abelian theory only the triple vertex describing the interaction of the hypermultiplet with the gauge superfield is present (the gauge superfield can be either background or quantum).
From the action (2.10) we observe that there are infinitely many vertices with the number n ≥ 3 of the gauge superfield lines (and with no lines of any other superfields). Note that the gauge-fixing term (4.6) also contributes to these vertices (in this case the legs of the background gauge superfield come from the bridge).
Due to the presence of two super-background covariant derivatives in the ghost action (4.7), there are triple and quartic vertices containing two ghost lines. These vertices can have no more than one line of the quantum gauge superfield v ++ and no more than two lines of the background gauge superfield V ++ .
The superfields ϕ, ξ (+4) and σ interact with the background gauge superfield only. For the superfield ϕ only the triple and quartic vertices are possible, while the vertices involving ξ (+4) and σ can also contain an arbitrary number of the background gauge superfields coming from the superfield V −− concealed in the operator ⌢ ✷.

One-loop divergences in harmonic superspace
In order to calculate the divergent part of the one-loop effective action, we again start from calculating divergences of the lowest-order Green functions and then restore the full result by the reasoning based on the unbroken background gauge invariance. This can be done as follows. According to [43], on shell the one-loop logarithmic divergences have the structure where c i with i = 1, 2, 3 are real numerical coefficients and the regularization by dimensional reduction is assumed. The coefficients c i can be obtained by calculating the divergences of the two-point function of the background gauge superfield (c 1 ) and of the three-point gaugehypermultiplet function (c 2 ). The coefficient c 3 vanishes, because the corresponding four-point hypermultiplet Green function is finite. Actually, in the non-abelian case the degree of divergence for diagrams without external ghost legs is also given by the expression (3.15). In the case of L = 1, N q = 4, N D = 0 we obtain ω = −2, for which reason the one-loop four-point hypermultiplet Green function is given by the convergent integrals.
For calculating the coefficient c 1 in the expression (4.17) we consider the two-point Green function of the background gauge superfield. In the one-loop order it is contributed to by the superdiagrams presented in Fig. 7, in which the external bold wavy lines correspond to the background gauge superfield. They were calculated in Ref. [31]. The following result for the sum of the corresponding contribution to the effective action has been obtained there: This expression is divergent, the leading divergence being quadratic. However, the dimensional reduction can catch only the logarithmical divergences which can be written as (4.20) (1) To calculate the quadratic divergences, one is led to use a regularization with an ultraviolet cut-off Λ. Then the leading quadratically divergent terms are represented by the expression while the logarithmical ones are obtained from (4.20) via the substitution 1/ε → ln Λ.
It is worth to note that the gauge invariant result in the non-abelian case also contains higher degrees of V ++ , which are encoded in (4.17). Comparing the expression(4.20) with we obtain which implies that, in the case of employing the dimensional reduction regularization, the divergent part of the one-loop effective action can be written as As for the quadratic divergences (4.21), they correspond to the lowest term in the power expansion of the gauge invariant object (4.25) where S SYM is given by (2.10).
The two-point Green function of the hypermultiplet is calculated similarly to the abelian case already considered earlier. For non-abelian theories it is also determined by a single logarithmically divergent supergraph presented in Fig. 4. The only novelty is the presence of the hypermultiplet indices and the factor C(R) i j . Exactly as in the abelian case, in the Feynman gauge ξ 0 = 1 the two-point Green function of the hypermultiplet vanishes (recall (3.23)).
The coefficient c 2 in the expression (4.17) can be found by calculating the one-loop contribution to the three-point gauge-hypermultiplet Green function, which is determined by two harmonic supergraphs presented in Fig. 8. The details of the calculation can be found in Ref. [31], while here we provide only the answers: (1) (2) ( (4.26) (4.27) Obviously, both these expressions are logarithmically divergent. When using the regularization by dimensional reduction [51], the divergent part of their sum is written as is the lowest (linear) term in the expansion of V −− in powers of V ++ .
Rewriting the expression (4.28) in the coordinate representation, we can cast it in the form where the linear part of F ++ is denoted by The expression (4.30) is the lowest term in the expansion of the gauge invariant expression in powers of V ++ . Comparing it with (4.17), we conclude that Thus, when using the regularization by dimensional reduction, the total divergent part of the one-loop effective action for an arbitrary 6D, N = (1, 0) gauge theory can be written as This is a final result for one-loop divergences. We see that in the N = (1, 1) theory, where T (Adj) = C 2 and C(Adj) i j = C 2 δ i j , the all one-loop divergences are absent off-shell. This result was obtained in the framework of the symersymmetric dimensional regularization.
However, it is interesting to understand how such a result depends on the regularization This is the reason why it is instructive to study the one-loop divergences in the framework of some another regularization. Here we present the corresponding result in the regularization by an ultraviolet cut-off Λ. In this case it is possible to calculate both quadratic and logarithmical one-loop divergences, (4.35) Now we get the additional divergent term S SY M [V ++ ] in comparison with divergences within the dimensional regularization. Nevertheless in the N = (1, 1) theory this divergent terms also vanishes. Note that using of the cut-off regularization can lead to some problems in higher loops. Actually, because of a possible violation of the BRST invariance, the Slavnov-Taylor identities [49,50] can be broken at the quantum level (see, e.g., the calculation for supersymmetric theories in Ref. [62]). However, these identities can be restored with the help of a special subtraction scheme, similar to the one constructed in [63,64]. Moreover, the BRST symmetry guarantees the stability of the background-quantum splitting (4.1). For non-invariant regularizations this equation can receive some quantum corrections. Nevertheless, in the oneloop approximation for the considered part of the effective action all these problems are not essential. To overcome them in higher loops, it is necessary to use an invariant regularization, e.g., some versions of the higher covariant derivative regularization [52,53] in the harmonic superspace (see [54]).
As we already pointed out, with taking into account the relations (2.17) we get that in 6D, N = (1, 1) SYM theory all the divergences (including the quadratic ones) vanish. 3 In the gauge sector this occurs, because both quadratic and logarithmical divergences are proportional to C 2 −T (R). This result agrees with the calculation made earlier in [65,66], where the divergences in the gauge sector have been found using the component formulation of the theory. However, we also demonstrated that the divergences in the hypermultiplet sector vanish as well, if the theory is quantized in the manifestly N = (1, 0) supersymmetric and gauge invariant way, and the Feynman gauge condition is used.

Two-loop divergent part of the hypermultiplet two-point Green function of 6D SYM theories
The calculation of quantum corrections in the two-loop approximation is a much more complicated problem. To date, the two-loop divergences in the harmonic superspace formalism have been found only for the two-point Green function of the hypermultiplet. It is determined by the diagrams depicted in Fig. 9. In the diagram (5) in Fig. 9 the gray disk corresponds to the insertion of the one-loop polarization operator of the quantum gauge superfield. It is given by the sum of the one-loop superdiagrams presented in Fig. 10. The details of the two-loop calculations can be found in Ref. [32]. The formal result for the Green function under the consideration (without a regularization) is given by the expression (written in the Minkowski space before the Wick rotation) (4.36) (1) (4) (5) Figure 9: Supergraphs representing the two-point hypermultiplet Green function in the twoloop approximation.
= + + + Figure 10: In Fig. 9 the gray circle corresponds to the one-loop polarization operator which is given by the sum of the harmonic supergraphs depicted here.
In agreement with Eq. (3.15) this Green function is quadratically divergent. The regularization by dimensional reduction cannot be used for calculating the quadratic divergences, so it is necessary to use different regularization schemes. However, let us consider N = (1, 1) SYM theory, with the hypermultiplet in the adjoint representation, R = Adj. Using Eq. (2.17), we observe that the expression (4.36) for this theory vanishes identically. This implies that the leading quadratic divergences are canceled out and the total divergences can be calculated, based on the dimensional reduction. However, even after the replacement 6 → D the expression (4.36) vanishes. Therefore, the considered Green function for N = (1, 1) SYM theory vanishes identically. Taking into account that N = (1, 1) supersymmetry intertwines the gauge and hypermultiplet superfields, it is reasonable to suggest that all two-point Green functions of this theory also vanish identically.
Nevertheless, two-loop off-shell divergences may arise in the four-point Green functions. To see this, it is sufficient to calculate the four-point Green function of the hypermultiplet. This work is in progress now.

Manifestly gauge covariant analysis
In this Section we briefly discuss how the proper-time method can be used for analysis of divergent contributions in 6D N = (1, 0) SYM theory (2.13). After splitting the superfields V ++ , q + into the sum of the background parts V ++ , Q + and the quantum parts v ++ , q + , we expand the full action in a power series in quantum superfields. In the one-loop order, the first quantum correction to the classical action, Γ (1) [V ++ , Q + ] , is given by the following functional integral [16,67]: In this expression, the full quadratic (with respect to the quantum superfields) action S 2 is the sum of three terms, namely, the classical action (2.13) in which the background-quantum splitting was performed, the gauge-fixing term (4.6) and the ghost actions (4.7) and (4.9). The action S 2 contains the mixed term of quantum vector multiplet and hypermultiplet. After diagonalization we obtain the following one-loop contribution to the effective action where G (1,1) q (1|2) is the background-dependent hypermultiplet Green function (4.14). Also we introduce the covariant d'Alembertian Here (G (1,1) q ) i j is the superfield Green function (4.14) for the operator (∇ ++ ) i j acting on the superfields in the representation R to which the hypermultiplet belongs. Also we denoted the Green function for the operator (∇ ++ ) BA , which acts on superfields in adjoint representation, by (G (1,1) ) BA . The Green function (G (1,1) ) BA has the structure similar to (4.14).

(4.43)
Here s is the proper-time parameter and µ denotes an arbitrary regularization parameter with the dimension of mass. Our aim is to calculate the divergent part of the effective action (4.41).
In the proper-time regularization scheme (see, e.g., [10]) the divergences correspond to the pole terms of the form 1/ε , ε → 0, with D = 6 − ε. Then, calculating the divergences according to the standard technique, after some (rather non-trivial) transformations we obtain Γ (1) where F ++ = F ++A t A , with t A being the fundamental representation generators.
The hypermultiplet-dependent part Q + F ++ Q + of the one-loop counterterm comes from the first term in (4.39). In order to find this contribution, firstly we rewrite it as a sum of two terms, Then, following [29], we decompose the second logarithm up to the first order and compute the functional trace .
Here we use of the explicit form of the Green function (G Summing up the contributions (4.44) and (4.47), we obtain the final result for the total divergent contribution We see that the result (4.48) derived by the manifestly gauge invariant method, coincides with the previous result (4.34) based on supergraph calculations.

Low-energy effective action
The background field method developed in the previous sections is a powerful tool for calculation of the finite contributions to the effective action in a manifestly gauge invariant way. 4 . In this section we evaluate the finite one-loop leading low-energy contribution to the effective action of 6D, N = (1, 1) SYM theory in the N = (1, 0) harmonic superspace formulation. An important aspect of the consideration is the use of omega-hypermultiplet.
First, we formulate 6D, N = (1, 1) SYM theory in terms of 6D, N = (1, 0) analytic harmonic superfields V ++ and ω, which are the gauge supermultiplet and the hypermultiplet, respectively. The action of N = (1, 1) SYM theory in this case reads where Here both V ++ and ω take the values in the adjoint representation. The action (4.49) is invariant under the infinitesimal gauge transformations where Λ(ζ, u) = Λ(ζ, u) is an analytic real gauge parameter.
Our further consideration is based on the background field method in six-dimensional N = (1, 0) harmonic superspace which was developed in the previous subsection. Here we focus only on aspects related to omega-hypermultiplet. As in the previous sections, we represent the original superfields V ++ and ω as a sum of the background superfields V ++ , Ω and the quantum superfields v ++ , ω . In the present case it is convenient to append the coupling constant f 0 in front of quantum fields Then we expand the action in a powers of the quantum fields. The one-loop contribution to the effective action Γ (1) for the model (4.49) is defined by the quadratic part of quantum action S 2 , The action S gh in (4.54) is a sum of the action for Faddeev-Popov ghosts b and c (4.7) and the action for Nielsen-Kallosh ghost ϕ (4.9). The covariantly-analytic operator ⌢ (4.40) depends on the background gauge superfield.
The action (4.54) includes the background superfields V ++ and Ω which belong to the Lie algebra of gauge group. Let us suppose that the gauge group of the theory (4.49) is SU(N). For simplicity, we will also assume that the background fields V ++ and Ω align in a fixed direction in the Cartan subalgebra of su(N) Here H is a fixed generator of the Cartan subalgebra corresponding to some abelian subgroup U(1). For our choice of the background superfields the symmetry group of classical action SU(N) is broken down to SU(N −1)⊗U(1). It is worth to note that the pair of the background abelian superfields (V ++ , Ω) forms the abelian gauge N = (1, 1) multiplet. In the bosonic sector it contains a single real gauge vector field A M (x) and four real scalar fields φ(x) and φ (ij) (x) , i, j = 1, 2. The fields φ and φ (ij) are scalar components of the hypermultiplet Ω [15]. It is known that the abelian vector field and four scalars describe the bosonic world-volume degrees of freedom of a single D5-brane in in six-dimensional space-time [74,75].
According to the definition (4.55), the classical motion equations for the background superfields V ++ and Ω are reduced to the free ones In our further consideration we assume that the background superfields satisfy the classical equation of motion (4.56) and also are slowly varying in space-time Since we assume that the background vector multiplet solves the free equation of motion, F ++ = 0, the gauge superfield strength W +a becomes an analytic superfield on shell. In the general case of unconstrained background, F ++ = 0, the superfield W +a is non-analytic.
The transformations of the hidden N = (0, 1) supersymmetry for the gauge superfield strength W +a and Ω (4.52), in accordance with the conditions (4.56) and (4.57), have the simple form Using (4.58), one can try to investigate the simplest N = (1, 1) invariants which can be obtained from the abelian analytic superfields W +a and Ω under the assumptions (4.56) and (4.57). It is easy to check that the following gauge-invariant action, is invariant under the transformation (4.58). Here we introduced the fourth power of gauge superfield strength (W + ) 4 = − 1 24 ε abcd W +a W +b W +c W +d . The function F (f 0 Ω) can in principle be arbitrary. The simplest choice, when the coupling constant f 0 is absent in the invariant, is F = 1 f 2 0 Ω 2 in (4.59), which yields (4.60) The numerical coefficient c in (4.60) cannot be fixed only by the symmetry considerations and should be calculated using the quantum field theory methods.
So our next step is to find the constant c by calculating the leading low-energy contribution to the effective action of the theory (4.49). To perform the calculation we choose the Cartan-Weyl basis for the SU(N) generators. In this basis the quantum superfield v ++ is decomposed as As for the second term Γ (1) high in (4.63), we will show that it corresponds to the next-to-leading approximation. Further we demonstrate that the N = (1, 1) invariant action (4.60) can be found as a leading contribution to the one-loop effective action Γ (1) lead (4.64). The action (4.60) includes only the gauge superfield strength W +a and superfield Ω and does not contain terms with D ++ Ω, D − a Ω and D − a W +b . Hence we will systematically neglect such terms in our computations. The contribution Γ (1) high collects terms with D ++ Ω and spinorial derivatives of the background superfields only. Thus, below the contribution Γ (1) high can be ignored. The scheme of calculation of the contribution (4.64) is quite similar to the analogous one in the four-dimensional case [76]. First of all we notice that on shell the harmonic derivative ∇ ++ H commutes with the covariant d'Alembertian. But it is not true for the operator H Ω 2 , ∇ ++ H ] ∼ D ++ Ω. However, all such terms are beyond the scope of our consideration. Thus, in accordance with the method of Ref. [76], the well-defined expression for the contribution Γ (1) lead to the one-loop effective action reads (4.65) Here we have introduced the projection operator on the space of transverse covariantly analytic superfields, Π (2,2) T (ζ 1 , u 1 ; ζ 2 , u 2 ). One can show [76] that where have introduced the notation ∆ . Then we substitute (4.66) in the one-loop contribution Γ (1) lead (4.65) and take the coincident-harmonic points limit u 2 → u 1 . It is easy to see that only the third term in (4.66) survives. As the next steps we collect the terms quartic in the derivative D − a from the exponential in diag(1, .., 1, 1 − N).
As was expected, the N = (1, 1) invariant I 1 (4.60) comes out as the leading low-energy contribution (4.67) to the effective action for the theory (4.49). The coefficient c was calculated and it is equal to It is interesting to note that the same expression for the coefficient c was obtained in 4D N = 4 SYM theory (see, e.g., [77] and references therein). The bosonic part of the effective action (4.67) is where F 4 = 3F M N F M N F P Q F P Q − 4F N M F M R F RS F SN and F M N is the abelian gauge field strength.

Conclusion
Harmonic superspace is a very convenient powerful tool for investigating quantum properties of 6D N = (1, 0) and N = (1, 1) theories, because it allows to keep N = (1, 0) supersymmetry manifest at all steps of calculating quantum corrections. Moreover, this technique considerably simplifies the calculations, because a huge amount of usual Feynman diagrams appear to be included into an essentially smaller number of superdiagrams. Surely, most of the statements and methods related to N = (1, 0) and N = (1, 1) SYM theories can be reformulated within the harmonic formalism. The results obtained in the harmonic superspace approach in the lowest loops agree with those found with the help of other techniques, say, within the component approach. However, the harmonic superspace technique looks certainly more preferable for calculations in the higher loops, where the advantages of the manifestly supersymmetric quantization method are especially essential.