Effective actions in \( \mathcal{N}=1 \) , D5 supersymmetric gauge theories: harmonic superspace approach

Journal of High Energy Physics, Nov 2015

We consider the off-shell formulation of the 5D, \( \mathcal{N}=1 \) super Yang-Mills and super Chern-Simons theories in harmonic superspace. Using such a formulation we develop a manifestly supersymmetric and gauge invariant approach to constructing the one-loop effective action both in super Yang-Mills and super Chern-Simons models. On the base of this approach we compute the leading low-energy quantum contribution to the effective action on the Abelian vector multiplet background. This contribution corresponds to the ‘F 4’ invariant which is given in 5D superfield form.

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Effective actions in \( \mathcal{N}=1 \) , D5 supersymmetric gauge theories: harmonic superspace approach

HJE 1, D5 supersymmetric gauge theories: harmonic superspace approach Tomsk 1 Russia 1 Novosibirsk 1 Russia 1 I.L. Buchbinder 1 2 4 N.G. Pletnev 0 1 3 Novosibirsk, Russia 0 National Research Novosibirsk State University 1 Tomsk , 634061 Russia 2 National Research Tomsk State University 3 Department of Theoretical Physics, Sobolev Institute of Mathematics 4 Department of Theoretical Physics, Tomsk State Pedagogical University We consider the o -shell formulation of the 5D, N = 1 super Yang-Mills and super Chern-Simons theories in harmonic superspace. Using such a formulation we develop a manifestly supersymmetric and gauge invariant approach to constructing the one-loop e ective action both in super Yang-Mills and super Chern-Simons models. On the base of this approach we compute the leading low-energy quantum contribution to the e ective action on the Abelian vector multiplet background. This contribution corresponds to the `F 4' invariant which is given in 5D super eld form. Extended Supersymmetry; Superspaces; Supersymmetric E ective Theories - N E Dedicated to the memory of Boris Zupnik 1 Introduction 2 Review of the 5D, N = 1 harmonic superspace approach 2.1 5D SYM theory in harmonic superspace 2.2 The super Chern-Simons model 2.3 5D N = 1 hypermultiplet in harmonic superspace 3 The background eld formulation for quantum N 3.3 The leading contribution to e ective action of the 5D SCS multiplet 4 The leading contribution to the e ective action of a 5D SYM multiplet 19 5 The leading and next-to-leading contributions of the ghosts and matter super elds 6 Summary 1 Introduction The study of the quantum structure of the supersymmetric ve-dimensional eld theories attracts recently considerable attention, mainly due to attempts to nd the e ective world-volume action for multiple M5-branes [1]. It was conjectured in [2, 3] that the sixdimensional (2,0) superconformal eld theories on a stack of M5-branes are equivalent to ve-dimensional super Yang-Mills (SYM) theories. To establish this correspondence, the Kaluza-Klein reduction of the general 6D ( 1,0 ) pseudo-action with a non-Abelian gauge group G was performed in the series of papers [4, 5] and the 5D e ective action for the Kaluza-Klein zero-modes was derived. This duality serves as an important constraint on the models for multiple M5-branes. Of course, not all consistent ve-dimensional theories arise in such a circle compacti cation. On the other hand, the 5D theory has a global U( 1 ) symmetry and the current j = ?F ^ F is always conserved. The corresponding conserved charge is the instanton number. Such a conserved current can be coupled to vector super eld what allows us to identify the scalar component of this vector super eld as the gauge coupling < > Using this observation the authors of papers [8{13] proposed that the maximally extended 1 gS2YM [6, 7]. { 1 { 5D supersymmetric gauge theory describes the 6D, (2; 0) superconformal eld compacti ed on circle without introducing the Kaluza-Klein reduction. This proposition was based on the observation that the Kaluza-Klein momentum along the circle can be identi ed with instanton charge in the 5D theory. The latter is a topological charge carried by soliton con gurations, which are analogous to monopole and dyon con gurations in 4D. Such an attractive hypothesis means, in fact, that adding the 1/2-BPS particle soliton states with instanton number k and a mass formula M full nonperturbative particle spectrum and detegrS2YmMines the nonperturbative completion of R5 k to 5D SYM theory gives us the 4 k the theory under consideration. This might be an argument in favor of UV niteness of 5D SYM perturbative theory and thus be an argument for the consistent quantum theory. In the strong coupling limit the 5D SYM should de ne the fully decompacti ed 6D, (2; 0) theory, which, in its turn, is expected to describe the low-energy dynamics of multiple M5-branes. An additional important motivation to study 5D supersymmetric gauge theories comes from the existence of the corresponding super Chern-Simons (SCS) theory. This theory is interesting since it has a conformal xed point in ve dimensions and can admit a holographic duality [6, 7]. There are several reasons why the 5D supersymmetric Chern-Simons theory can be interesting in quantum domain. First of all, the Chern-Simons terms can be generated by integrating out the massive hypermultiplets in the SYM theory when the hypermultiplets transform in complex representations of the gauge group [2, 3]. If we consider the masses of the hypermultiplets as the UV cut-o , then this leads to the generation of the Chern-Simons term in the one-loop correction to the classical theory. Hence inclusion of the SCS term into the action can be useful in some cases if we want to have a complete description of the theory. SCS theory can also be important in the relationship between 5D SYM and 6D, (2,0) theories. In particular, one can argue [14, 15] that the 5D Chern-Simons term can be generated by the anomaly terms in the six-dimensional theory. By focusing on a certain class of anomaly-free six-dimensional theories the authors of [14, 15] formulated the explicit constraints on the spectrum and supersymmetry content of the six-dimensional theory in terms of the ve-dimensional Chern-Simons couplings. Therefore it would be interesting to compute the perturbative quantum corrections in such 5D theories. In particular, it was demonstrated that massive fermions running in the loop generate constant corrections to the 5D Chern-Simons terms of the form kABC AA^F B^F C + AAA^tr (R^R), where AA denotes collectively the graviphoton and the vectors from the vector multiplet, F A are the corresponding eld strengths, and R is the curvature two-form. Though 5D and 3D Chern-Simons theories share some interesting properties, such as quantization of the level k, nevertheless there are also some di erences. The most important di erence is the presence of local degrees of freedom in the higher dimensional case. This peculiar fact makes it attractive and interesting to perform a more detailed analysis of 5D SCS theories. It is known that non-trivial observables exist in supersymmetric gauge theories which are not very sensitive to details of the UV completion. Quantum e ects which non-trivially contribute to such BPS observables are often highly constrained. With using the procedure of localization of the path integral were studied of the various observables in 5D { 2 { supersymmetric theories [16{21]. It was shown that the partition function for the maximally extended SYM on S5 captures the physical aspects of the 6D, (2; 0) theory in a surprisingly accurate and detailed manner. In particular, the N 3 behavior of this partition function in 5D supersymmetric gauge theory is in agreement with the important results obtained for 6D, (2; 0) theory from the supergravity duals and conformal anomaly [22, 23]. The study of integral invariants in half-maximally and maximally extended supersymmetric theories such as supergravity and SYM attracts an attention because they can be viewed, on the one hand, as possible higher order corrections to the string or brane e ective actions and, on the other hand, as quantum eld counterterms. It is well known on the base of the power counting arguments that the 5D SYM is perturbatively non-renormalizable. Therefore we should expect an in nite number of divergent structures at any loop what leads to in nite number of counterterms. However, the superspace arguments and the requirements of on-shell supersymmetry rule out the rst divergences in D-dimensional SYM theory. Actually the divergences can appear at L loops where D=4+6/L.1 Construction of the various supersymmetric, gauge invariant functionals in quantum eld theory is conveniently formulated in the framework of the e ective action. The low-energy e ective action can be represented as a series in supersymmetric and gauge invariants with some coe cients. In general, the supersymmetry together with conformal symmetry imposes rigid constraints on these coe cients. In some cases, they can be determined exactly [33, 34]. For example, the leading term in the low-energy e ective action on the Abelian vector eld background is F 4 which is generated only at one loop and is not renormalized at higher loops. A possible new non-renormalization theorems for Abelian F n was conjectured in [35]. Recently the authors of [36{38] systematically analyzed the e ective action on the moduli space of (2,0) superconformal eld theories in six dimensions, as well as their toroidal compacti cation to maximally SYM theories in ve and four dimensions. They presented an approach to non-renormalization theorems that constrain this e ective action. The rst several orders in its derivative expansion are determined by a one-loop calculation in ve-dimensional SYM theory. In general, the functional form of the e ective action at the rst several orders in the derivative expansion can be obtained by integrating out the massive degrees of freedom in the path integral. However, it is di cult enough to perform exactly such an analysis for supersymmetric models in the component formulation. Construction of the background eld method in extended supersymmetric gauge theories faces a fundamental problem. The most natural and proper description of such theories should be formulated in terms of a suitable superspace and unconstrained super elds on it. Some time ago a systematic background eld method to study the e ective actions in 4D, N = 2 supersymmetric eld theories was developed in a series of papers [39{44]. This method is based on formulation of N = 2 theories in harmonic superspace [45, 46] and guarantees the manifest N = 2 supersymmetry and gauge invariance at all stages of calculations. The method under consideration gives the possibility to calculate in a straightforward manner not only the holomorphic and non-holomorphic contributions to the low-energy e ective action but also to study the full structure of the e ective action. 1For a discussion of this issue see [24{32] and references therein. { 3 { HJEP1(205)3 Evaluation of the e ective action within the background eld method is often accompanied by using the proper time or heat kernel techniques. These techniques allow us to sum up e ciently an in nite set of Feynman diagrams with increasing number of insertions of the background elds and to develop a background eld derivative expansion of the e ective action in a manifestly gauge covariant way. The 5D SCS theories are superconformally invariant and, hence, their e ective actions must be independent of any scale. Unlike the 4D, N = 2; 4 supersymmetric theories where holomorphy allows one to get the chiral contributions to e ective action [47], in the 5D case the contributions to the e ective action can be written only either in full or in analytic superspaces. Then, taking into account the mass dimensions of the harmonic potential V ++ and super eld strength W as well as the dimensions of the superspace measures d5xd4 + and d5xd8 , one obtains that the most general low-energy e ective U( 1 )-gauge invariant action in the analytic superspace is the SCS action [48, 49]. The next-to-leading e ective Abelian 5D action can be written only at full superspace in terms of the manifestly gauge invariant functional = R d5xd8 W H( W ); where is some scale and H( W ) is the dimensionless function of its argument. The requirement of scale invariance means an equation d d R d5zd8 WH( W ) = 0, where the only solution is H = c ln W : Any perturbative or non-perturbative quantum corrections should be included into a single constant c. The component Lagrangian in the bosonic sector corresponding to the above e ective action is 13 (F 4 + (@ )4 + : : :), where F is the Abelian strength of the component vector eld from 5D, N = 1 vector multiplet and is the corresponding scalar component. In this paper we derive the leading contribution to the low-energy e ective action in the 5D SCS theory using the harmonic superspace description of the theory and propertime techniques. The result precisely corresponds to the above analysis and has the form R d5xd8 W ln W . Besides the e ective action in the 5D SCS theory, we calculate the leading one-loop low-energy contribution in the 5D SYM theory. Although this theory is not superconformal and is characterized by the dimensional coupling constant, its leading contribution to the e ective action has the same functional form as in the 5D SCS theory and does not depend on the scale and coupling constant. Also, we consider the e ective action in the 5D hypermultiplet theory coupled to a background 5D vector multiplet. The leading low-energy contribution to e ective action in this theory was calculated in paper [49], where it was shown that this contribution is the 5D SCS action. In the given paper we calculated the rst next-to-leading term in the low-energy e ective action for the theory under consideration and found that this term again has the same functional form as the leading term in 5D SCS theory. The paper is organized as follows. Section 2 is devoted to a brief review of harmonic superspace formulation of the 5D, N = 1 supersymmetric eld models such as the SYM theory, the hypermultiplet theory and the SCS theory. In section 3 we consider the Abelian 5D SCS theory and develop the manifestly supersymmetric and gauge invariant procedure for calculating the e ective action. This procedure is based on the background eld method and proper-time technique. We nd the exact expression for one-loop e ective action in terms of functional determinants of di erential operators in analytic subspace of harmonic superspace and calculate the leading low-energy contribution to this e ective action. In { 4 { section 4 we develop the analogous procedure for 5D SYM theory and calculate the leading low-energy contribution to one-loop e ective action. Section 5 is devoted to the study of the rst next-to-leading contribution to e ective action in the 5D hypermultiplet theory coupled to a 5D vector multiplet background. The last section is devoted to the summary of the results. 2 Review of the 5D, N = 1 harmonic superspace approach Various supersymmetric theories with eight supercharges admit the o -shell super eld formulations in terms of formalism of the harmonic superspace. The harmonic superspace approach for the 4D, N = 2 theories was originally developed in [45]. The formulation for the 5D, N = 1 models has been given in [48, 50, 51]. The harmonic superspace approach for the 6D, (1; 0) SYM theories was considered in [52{55] and for the 6D, (1; 1) SYM in [56]. The construction the super-de Rham complex in ve-dimensional, N = 1 superspace and its relationship to the complex of six-dimensional, N = ( 1, 0 ) superspace via dimensional reduction was considered in [57]. All these harmonic superspace formulations in space-time dimensions 4, 5 and 6 look almost analogous modulo some details. In a series of papers [58{62] an extensive program of constructing the manifestly supersymmetric formulation for the 5D, N = 1 supergravity-matter models was realized and the universal procedures to construct manifestly 5D supersymmetric action functionals for a di erent supermultiples were developed. These super eld results are in agreement with the earlier component considerations of the same models [63{65]. The study of the structure of the low-energy e ective action in the 5D superconformal theories looks useful from the point of view of the classi cation of theories consistent at the quantum level. In ref [49] a manifestly 5D, N = 1 supersymmetric, gauge covariant formalism for computing of the one-loop e ective action for a hypermultiplet coupled to a background vector multiplet was developed. It was demonstrated, as a simple application, that the SCS action is generated at the quantum level on the Coulomb branch. The above paper was the only one where explicit one-loop harmonic superspace calculations were done. We believe that until now the possibilities of the covariant 5D harmonic superspace methods were non-su ciently explored to study the e ective action in 5D supersymmetric gauge theories. The aim of this paper is to develop the general manifestly supersymmetric and gauge invariant methods for 5D quantum supersymmetric gauge theories and apply these methods to calculate the low-energy e ective action in 5D SCS and SYM theories. In this section we brie y review the superspace description of a vector multiplet in 5D supersymmetric gauge theories. Our aim here is to x our basic notation and conventions (for details see [50]). The 5D gamma-matrices a^ are de ned as follows f a^ ; ^bg = f 1; 1; : : : ; 1g: The matrices f1; a^ ; a^^bg form a basis in the space of 4 charge conjugation matrix C = ("^ ^) and its inverse C 1 = ("^ ^) are used to raise and lower the spinor indices. The matrices "^ ^ and ( a^)^ ^ are antisymmetric, while the matrices ( a^^b)^ ^ are symmetric. The anticommuting variables i^ are assumed to obey the pseudo2 a^^b1; with a^^b = 4 matrices. The { 5 { ^ i: They obey the anti-commutation relations by the coordinates zM = (x^ ^; i^) where i = 1; 2 and x^ ^ = ( a^)^ ^xa^. Majorana reality condition ( i^)? = i^ = "^ ^"ij j^. 5D, N = 1 superspace is parameterized The N = 1 harmonic superspace R5j8 S2 extends the conventional 5D, N Minkowski superspace R5j8 with the coordinates zM SU(2)=U( 1 ) parameterized by harmonics, i.e., group elements = (xm^ ; i^); by the two-sphere D0; D = 2D D++; D = D0: The main conceptual advantage of harmonic superspace is that the N = 1 vector multiplet as well as hypermultiplets can be described by unconstrained super elds on the analytic subspace of R5j8 S2 parameterized by the coordinates M = (xAm^ ; +^; ui ); where xaA^ = xa^ + i +^ a^ ^ ^ ^ ; = ui i : (xa^; ^; ui ) satisfying the constraints D+^ = 0 is an analytic super eld ( ; u): It The important property of the coordinates M is that they form a subspace closed un der N = 1 supersymmetry transformations. In the coordinates derivatives D+^ = ui+Di^ have a short form (see eqs. (2.5)) and therefore the super eld M the spinor covariant leads to reducing the number of the anticommuting coordinates and, hence, to reducing the number of independent components in comparison with general super elds. However, all component elds depend now on extra bosonic coordinates ui . In harmonic superspace a full set of gauge covariant derivatives includes the harmonic derivatives which form a basis in the space of left-invariant vector elds of SU(2): nDi^; Dj^o = u i 2 SU(2); u+i = ui ; u+iu i = 1 : D (2.1) = 1 (2.2) (2.3) (2.4) ^ (2.5) (2.6) The generator of the SU(2) algebra D0 is an operator of harmonic charge, D0 (q) = q (q). In the analytic basis, parameterized by the coordinates derivatives: D^ = ui Di^ and the harmonic derivatives take the form M (2.2) the spinor covariant D+^ = In this basis eqs. (2.1), (2.4) leads to + +^ D n D+^; D +^o D+^; D ^ o D^ = ^ ; ^ + ; D^ = 0; D ; D^ = D^ ; { 6 { These relations are necessary integrability conditions for the existence of the analytic super elds. Since the eld in the harmonic superspace depend on the additional bosonic variable ui, we must de ne rules of integration over harmonics (that is, over the group manifold SU(2)=U( 1 )). The basic harmonic integrals have the form [46] Z du = 1; Z duu(+i1 : : : ui+n uj1 : : : ujm) = 0; n + m > 0 : (2.7) It means that the harmonic integrals are nonzero only if the integrand has the zero U(1) charge. 2.1 5D SYM theory in harmonic superspace To describe a Yang-Mills supermultiplet in 5D conventional superspace we introduce the gauge covariant derivatives DA = DA + iAA where DA = (@a^; Di^) are the at covariant derivatives and AA is the gauge connection taking values in the Lie algebra of the gauge group. The operator DA satis es the gauge transformation law DA ! ei (z) DAe i (z); y = with a super eld gauge parameter (z). The gauge covariant derivatives are required to obey some constraints [66] HJEP1(205)3 iFa^^b = 1 a richer gauge freedom than the original -group eiv0(z;u) = ei (z;u)eive i (z); D+^ = 0: The - and - transformations generate, respectively, the so-called - and -groups. In the -frame the spinor covariant derivatives D+^ coincide with the at ones, while the harmonic 2The properties of these matrices are discussed in [50, 61]. n i D^; Dj^o = i D^; Da^ = 2i"ij (D^ ^ + "^ ^iW); ^ i i( a^)^ D ^W; Da^; D^b = 1 4 ^ ^ i a^^b D^D ^iW = iFa^^b; with the matrices a^ and Wy = W and obeys the Bianchi identity a^^b de ned above.2 Here the eld strength W is Hermitian, (i j) covariant derivatives acquire connections D is an analytic super eld, D+^V ++ = 0 and its transformation law is = D + iV . The real connection V ++ V ++ 0 = ei V ++e i iei D++e i : (2.12) The gauge freedom the form can be used to impose the Wess-Zumino gauge in In this gauge, the super eld V ++ contains the real scalar eld , the Maxwell eld Am^ , the isodoublet of spinors i^ and the auxiliary isotriplet Y (ij). The analytic super eld V ++ turns out to be the single unconstrained potential of the pure 5D, N = 1 SYM theory and all other quantities, associated with this theory, are expressed in its terms. In particular, the other harmonic connection V using the zero-curvature condition [D++; D turns out to be uniquely determined in terms of V ++ ] = D0: The result looks like [69] V (z; u) = 1 X( i)n+1 Z n=1 du1 : : : dun u+u1+ V ++( ; uu11+) u::2+: V: +::+(un+;uun+) ; where (u1+u2+) = u1+iu2+i. The details of the harmonic analysis on S2 = SU(2)=U( 1 )) were designed in the pioneering works [45, 46, 67{69]. In the -basis the connections A^ , Am^ and the eld strength can be expressed in terms of V using the relations (2.10): A^ = D+^V ; W = D+^D+^V ; F^ ^ = i 8 i 2 D(^D ^)W: + i[V ++; W ] = 0: In the Abelian case, the super eld W The super eld strength W satis es the following constraints D ++ W = D++ W = 8i R du(D )2V ++ does not depend on harmonics and moreover, in this case there is no distinction between W W. The Bianchi identity (2.9) takes the form + + where we have used the notation (D+)2 = D of [50] built an important covariantly analytic super eld G +^ + ++: D^ : Using these identities, the authors This super eld can be transformed to the form G ++ = D+ 4 (V W); ) 2 ( ) = where (D )4 = 312 (D )2(D )2 and 12 (D+)4(D Unlike the 4D, N = 2 case, the chiral superspaces are not Lorentz-covariant in the case of 5D, so that to construct the super eld actions we can use only the full superspace 1 ^ ^D^ ^ ( ): 4 D { 8 { (2.14) (2.15) + and (2.16) (2.17) (2.18) or else the analytic superspace. In the full harmonic superspace the 5D SYM action has the universal form [69] = ' + i^ i^ + ^ i ^ i H^ ^ + : : : ; subject to the constraint D(^iDj^) = 0: The -expansion of is The SYM equations of motions (D+)2W = 0; 2W = 0 have a vacuum Abelian solution )2Z where Z is the linear combination of the Cartan generators of the gauge group [48]. This vacuum solution spontaneously breaks the gauge symmetry, but it conserves the 5D supersymmetry with the central charge and produces BPS masses of the Z-charged elds. In addition, let us note that on-shell, the degrees of freedom (1 + 3)B + 4F in W match the degrees of freedom in a 5D tensor multiplet described by the super eld (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) SSYM V ++ = 1 X1 ( 1 )n 2 gSYM n=1 n tr Z d13zdu1 : : : dun V ++(z; u1) : : : V ++(z; un) ; u1+u2+ : : : un+u1+ where gS2YM is the coupling constant of dimension [gS2YM] = 1. The SYM action in terms of the component elds de ned by (2.13) is SSYM = 1 2 gY M Z d ( 4)i + 2 tr GS+Y+M: Z ( a^^b)^ ^Ha^^b is dual to the 3-form eld strength Fa^^bc^ of the 2-form gauge eld Ba^^b. In this situation the vector massless representation is equivalent under duality to a tensor representation. The other universal procedure to construct 5D manifestly supersymmetric actions is based on ideas developed in [70, 71]. Let us consider two vector multiplets, namely a U( 1 ) vector multiplet V ++ and a Yang-Mills vector multiplet VS+Y+M. They can be coupled in a gauge-invariant way, using the interaction where GSYM corresponds to a non-Abelian multiplet and is de ned in eg. (2.17). If we assume that the physical scalar eld in V ++ possesses a non-vanishing expectation value < V ++( ; u) >= i( +)2 gY21M , then one gets The integration in (2.22), (2.23) is carried out over the analytic subspace of the harmonic superspace Let us consider the 5D harmonic superspace formulation of a supersymmetric Chern-Simons (SCS) model. The o -shell non-Abelian SCS action in ve dimensions was rst constructed in [65]. SCS model possesses remarkable properties, which makes this theory very interesting for various applications. First of all, in ve dimensions the pure U(N ) and SU(N ) Chern-Simons theories have superconformal xed points [6, 7]. Another reason to study the SCS models is that they can be generated by integrating out the massive hypermultiplets in the SYM theory [2, 3, 14, 15] when the hypermultiplets transform in complex representations of the gauge group. For example, if we consider the masses of the hypermultiplets as the UV cut-o , we obtain the Chern-Simons term in the one-loop correction to the classical theory. Hence inclusion of this term can be important in some cases to get a complete description of the theory. In a manifestly supersymmetric setting, where the entire vector supermultiplet is taken into account, the corresponding one-loop calculation was given in [49], both in the Coulomb and non-Abelian phases. Using the covariant harmonic supergraphs and the heat kernel techniques in harmonic superspace [39{44], it was shown that the hypermultiplet e ective action contains the SCS term. Finally ChernSimons theory can be important in the relation between the 5D SYM and 6D (2; 0) theories. In particular, one can argue [14, 15] that the 5D Chern-Simons term can be generated by the anomaly terms in the six-dimensional theory. In general, in space-time with dimension (2n 1), the action of the Chern-Simons theory can be constructed using the Chern-Simons form 2n 1, de ned as d (2n 1) = tr [F n] where F = dA + iA ^ A is the gauge eld strength two-form and its powers F n are de ned by the wedge product. In three space-time dimensions this form gives rise to the famous 3D Chern-Simons action. In the 5D Chern-Simons theory the action is given by SCS = k Z 12 d5x"a^^bc^d^e^tr Aa^F^bc^Fd^e^ iAa^F^bc^Fd^e^ 2 5 Aa^A^bAc^Ad^Ae^ ; where k is the Chern-Simons level which plays the role of the coupling constant. Here Aa^ is the gauge eld with the gauge group SU(N ) or U(N ). The eld Aa^ transforms under the gauge transformation g as Aa^ ! g 1 Aa^g ig 1@a^g: Under this transformation, SCS gets an additional term SCS given, modulo a total derivative, by SCS = 2 ik Q(g); where Q(g) takes only integer values. Like in the case of three-dimensional Chern- Simons theory, gauge invariance of the partition function leads to k 2 Z. However, despite the fact that the action of the theory CS does not depend on the metric and thus the theory is topological, the ve (and, generally, any higher odd) dimensional case admits local propagating degrees of freedom [72, 73]. Unlike the component construction of [65], a closed-form expression for the non-Abelian SCS action has never been given in terms of the super elds. Here there exists only a unique de nition of the variation of the SCS action with respect to an in nitesimal deformation of the gauge potential V ++. However, as it was noted in [62], it is unknown how to integrate this variation in a closed form to obtain the action as an integral over the superspace. The component formulation of the non-Abelian SCS model can be constructed in the framework of the superform approach [62], where a closed-form expression for the non-Abelian SCS action was given. The super eld analysis in the Abelian case is more transparent and the SCS action was derived in the 5D N = 1 harmonic [48, 50] and projective superspaces [61]. The approach of constructing the manifestly supersymmetric actions developed in [48, 50] leads to Abelian SCS action in the form The action (2.25) is invariant under the gauge transformations equations of motions for the model with such an action are V ++ = D++ : The The Abelian SCS theory with the super eld action (2.25) leads to the following component action: F a^^bFa^^b i 2 Yij 1 2 Y ij Yij (2.27) i j : (2.25) (2.26) The action (2.27) clearly shows that the ve dimensional Abelian SCS theory is a non-trivial interacting eld model (see e.g. [74]). The theory (2.27) is superconformal at the classical level and the coupling constant g2 is dimensionless. The latter can mean a renormalizability of the theory. However, the action (2.27) does not involve a quadratic part and perturbative calculations, based on the weakness of the interaction term with respect to the free part, are impossible. However, we can use the presence of the vacuum moduli space < >= m in the Lagrangian (2.27). This allows one to decompose the Lagrangian into the free and interaction parts and construct the S matrix in a conventional way. However in this case, conformal symmetry of the original Lagrangian is broken spontaneously and the mass parameter in the action makes the theory nonrenormalizable. 2.3 5D N = 1 hypermultiplet in harmonic superspace Here we brie y discuss the hypermultiplet formulation in harmonic superspace. On mass-shell, the 5D, N = 1 massless hypermultiplet contains two complex scalar elds forming an isodoublet of the automorphism group SU(2)A of the supersymmetry algebra (2.1) and an isosinglet Dirac spinor eld. The description of the o -shell hypermultiplet in terms of the analytical supspace of harmonic superspapce is completely analogous to the description of a 4D, N = 2 hypermultiplet. Like in the four-dimensional 4D, N = 2 supersymmetric theory [45], the o -shell hypermultiplet coupled to the vector supermultiplet is described by a super eld q+( ) and its conjugate q+( ) with respect to the analyticity preserving conjugation [46]. The classical action for a massless hypermultiplet coupled to the background 5D, N = 1 vector multiplet is (2.28) Z Shyper = d ( 4)q+D++q+: The hypermultiplet that transforms in a real representation of the gauge group, onshell has scalars in the representations (1 + 3) of SU(2)A and an SU(2)A doublet of pseudoMayorana fermions. It can be described by a real analytic super ed !( ). Such a super eld is called the !-hypermultiplet. The action describing an interaction of this hypermultiplet with a vector supermultiplet is written in the form In this section we will construct the background eld method for the super eld theory with The harmonic super eld background eld method (see construction of this method for 4D, N = 2 SYM theory in [39{41]) is based on the so-called background-quantum splitting of the initial gauge eld into two parts: the background eld V ++ and the quantum eld v+ To quantize the theory, one imposes the gauge xing conditions only on the quantum eld, introduces the corresponding ghosts and considers the background eld as the functional argument of the e ective action. Then, the original in nitesimal gauge transformations (2.12) can be realized in two di erent ways: as the background transformations and as the quantum transformations V ++ = 0; v++ = i v++; : To construct an e ective action as a gauge-invariant functional of the background super eld V ++ we will use another form of writing the action (2.25)3 (3.1) (3.2) (3.3) d ( 4)V ++ ++ = G d5xd8 du V ++V W ; (3.4) and expand the action S[V ++ + v++] in powers of the quantum eld v++: d13z1du1 : : : dzndun n! v++( 1 ) : : : v++(n) v++=0 v++( 1 ) : : : v++(n): (3.5) 1 Here W , and v++ denote the - and -frame forms of W, and v++ respectively W = eiv W e iv; v++ = e ivv++eiv: Each term in the action (3.5) is manifestly invariant with respect to the background gauge transformations. The rst variation of the action is where v++ = e iv V ++eiv. It depends on V ++ via the dependence of v++ on the bridge v. The term linear in v++ determines the equations of motion. This term should be dropped when one considers the e ective action. To determine the second variation of the action it is necessary to express V via V ++. A variation of V ++ = ieiv(D++e iv) can be represented as e iv V ++eiv = iD++(e iv eiv) = V ++: This equation is solved in the form e iv eiv = i Z u+u 1 du1 u+u1+ V ++(u1): Z u+u 1 du1 u+u1+ du1du2 1 u1+u2+ 2 g2(W) 1 g2(W) = W : V ++(u1) = du1 Z Vu+++u1+(u12) : V ++(u1) V ++(u2) : V = iD e iv eiv = D 2SSCS = d5xd8 Calculating the second variation of the action yields the result These two expressions illustrate the speci c feature of 5D, N = 1 SCS theory as a theory with local coupling constant (since there is a non-trivial background-dependent factor in the vector eld kinetic operator) [39{41]. In the framework of the background eld method, we should x only the quantum eld gauge transformations. As to the super eld W it is invariant under these transformations. Let us introduce the gauge xing function in the form F (+4) = D++v++; F (+4) = D ++ D ++ + i v++; ; Z SFP = tr d ( 4)du b D ++ 2 c: which changes by the law under the quantum gauge transformations. Eq. (3.12) leads to the Faddeev-Popov determinant and to the corresponding ghost action (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) Next we average the (F (+4) f (+4)) with the weight ( 1 Z 2 1 = V ++ exp d13zdu1du2f (+4)(u1)W ghosts and f (+4) = e ivf (+4)eiv is a tensor of the -group. Note that we need the insertion of the super eld W to balance the dimensions. The functional [V ++] is chosen from the condition 1 V ++ = Z Df (+4) exp The background eld dependent operator A^ has the form and act on the space of analytic super elds. Thus A^ = nd DetA^ we represent it by a functional integral over analytic super elds of the form Det 1 A^ = Z and perform the following replacement of functional variables itr Z D (+4) D (+4) exp d 1( 4)d 2( 4) (+4)( 1 )A^(1j2) (+4)(2) ; (3.16) (+4) = D ++ 2 ; Det (+4) ! = Det D ++ 2 : Further, repeating the construction of the e ective action from the work [67, 68] we obtain Z d13zdu1du2 1 (+4) W Z Z d13zdu (+4) d ( 4) (+4) _2 1 W 2 W D ; 2 where is the deformed version of the covariant analytic d'Alembertian [50] _ 2 W = _ 2= 1 (D )2: Eventually we get the representation of [V ++] by the following functional integral V ++ = Det 1=2 D ++ 2 _ Det(14=;20) 2 W _ = Det(14=;20) 2 W D'eiSNK[';V ++] : Z (3.15) (3.17) (3.18) (3.19) (3.20) Here Det(q;4 q)(H) is de ned as follows Det(q;4 q)H^ = eTr (q;4 q) ln H^ ; and the functional trace of operators acting on the space of covariantly analytic super elds of U( 1 ) charges q and 4 q is Tr (q;4 q)H^ = tr Z H d ( 4) (q;4 q)( ; ); where `tr ' is the matrix trace and H (q;4 q)( 1; 2) is the kernel of the operator. Super eld ' in (3.20) is a bosonic real analytic super eld, the Nielsen-Kallosh ghost, with the action SCS[V ++] with the weight (3.14), one gets the following path integral representation of the one-loop e ective action Abelian 5D SCS theory (S1C)S[V ++] for the ei (S1C)S[V ++] = eiSSCS[V ++] Z Dv++ DbDcD'eiSquant[v++;b;c;';V ++] ; where Squant = SSCS v++; V ++ + Sgh v++; V ++ + SFP b; c; v++; V ++ + SNK '; V ++ : Here Sgh[v++; V ++] is the gauge xing contribution to the action of quantum super elds Sgh v++; V ++ = = The sum of the quadratic part special values = 1 has the form SSCS (3.9) in quantum super eld v++ and Sgh (3.26) for 1 Z 2 d ( 4)duv++ D d ( 4)duv++ _2 W v++: (3.27) eqs. (3.24){(3.25) completely determine the structure of the perturbation expansion to calculate the one-loop e ective action manifestly supersymmetric and gauge invariant form.4 The complete one-loop e ective (S1C)S[V ++] of the pure N = 1 SCS theory in a 4We restricted ourselves to the one loop approximation. However, the same construction will be valid at any loop if we replace in (3.24) the quadratic part of the action for the quantum super elds with the general non-quadratic action. (3.21) (3.22) (3.23) (3.24) (3.25) (3.26) action is given by the sum of the contributions coming from the ghost super elds and from the quantum super elds v++. It has the form 1 2 ~(S1C)S = i Here the rst term is the contribution from the Nielsen-Kallosh ghosts, the second term is the contribution from the Faddeev-Popov ghost and the third term is the contribution from the vector multiplet. In the next subsection we will consider the contribution to this e ective action from the SCS multiplet, The contribution from the ghosts will be considered The proper-time representation of the contributions in (3.28) from the The contribution of the Chern-Simons vector multiplet to (3.28) can be written in the proper-time representation (see the analogous representation for 4D, N = 2 SYM theories (S1C)S = (2;2) is the ve-dimensional gauge covariant version of the projector [42{44, 67, 68] on the space of covariant analytic transverse super elds v++: The properties of the (2;2)( 1; 2) were described in the paper [61]. T For later, it is convenient to rewrite the projector Here we follow the work [42{44]. We have5 (2;2)( 1; 2) in a di erent form. T (2;2)( 1; 2) = T _ = A (2;2)(1; 2) A (2;2)(1; 2) L The operator 2 is de ned by (3.19). As a result, T (2;2) can be expressed as (2;2)( 1; 2) = _ T 1 hD1++; _21i _1 D1+ 4 D2+ 4 13(z1 z2) 21 21 1 _ 21 D1+ 4 D2+ 4 13(z1 z2) 1 u1+u2+ 3 (3.29) (3.30) (3.31) Note that the rst term does not vanish since [D++; _2] (q) = 14 (D+2 W)(1 q) (q). 5Here we are using a manifestly analytic form of the delta function: (2;2)(1; 2) = A 1 _ The next step is a representation of a two-point function in the form D1 where expression for T (2;2): = iD D D + W(D )2 + 4(D W)D + (D D W): (3.33) This representation was obtained in the work [49]. As a result one gets the following (2;2)(1j2) = u1 u2 T + D+2 W (3.34) 1 4 1 _ 21 D1 13(z1 z2): 1 2 D The d'Alembertian (3.19) W 1 64 1 64 It is useful to compare this expression with a similar projector in 4D, N = 2 SYM theory [42{44]. 3.2 The deformed covariantly analytic d'Alembertian _2 W The operator in the quadratic part of the action for quantum elds plays a fundamental role in calculations of the e ective action in the framework of the background eld method. In the case under consideration this operator is 2 In the analytic subspace the operator 2 _ W (3.18). W (3.18) is rewritten in the form + 4 W(D )2 = W D + 4+4 D +^ W + D^ D + 2 + D + 2 W D + 2 (D )2: rst term here is, up to the multiplier W, the standard covariant analytic D + 2 + 2 D (D )2 = W Da^Da^ 1 4 D +^ + D^ W D 1 4 D + D +^ W D^ + ^ + D^ W W 2 = W 2 : _ The deformation of the operator de ned by (3.18) is stipulated by the super eld W as follows 1 64 1 64 +^ D W + 2 W D + D^ D + 2 + 2 (D (D )2 = )2 = 1 64 1 64 D D Da^ D+D W 1 W G D+^ W This means that in this case the operator 2 Summing up all together we obtain _ 2 W = W a^ D Da^ + 3 W D^ + " D+^ WD+^W + W 1 2W 1 +^ + D^ W D+^ iD^ ^D # D ^ + D+^ 1 32 D+2 W W ^ + D^ W (3.35) W This is a nal result for the operator 2 W acting on analytic super elds. The deformed covariantly analytic d'Alemberian 2 possesses a useful property [D+^; _2W ] = 0: It is important to note that the coe cient at the harmonic derivative ++ and that on the equations of motion (2.26) this term vanishes! ++ = 0 and therefore D D+ W = D+D W : _ W takes the form The leading contribution to e ective action of the 5D SCS multiplet Our next aim is to calculate the leading low-energy quantum contribution to the oneloop e ective action. To do that it is su cient to evaluate the one-loop e ective action (S1C)S[V ++] for an on-shell background vector multiplet. The representation (3.29) provides a simple and powerful scheme for computing the e ective action in the framework of 5D, N = 1 super eld proper-time technique.6 _ 2 Hence, we get If the background gauge multiplet is on-shell, G W does not involve any harmonic derivative D _ ++ = 0, the analytic d'Alembertian , but 2 contains the factor D +2 WD . 1 _ 21 u1+u2+ = u1+u2+ 1 _ 21 + 1 _ : 21 Taking into account this relation we obtain that the second term in the operator absent. Therefore the projector is reduced only to (D+)4 13(z1 z2): Then = i Z ds Z 2 s d ( 4)eis_2W D Now we replace the delta-function in (3.37) by its Fourier representation 5(x1 x2) D1 + 4 8 ( 1 2) = eipa^ a^ 4 + 1 2+ ; Z d5p (2 )5 6Note that Tr ln W does not contribute to (3.28). where a^ = (x1 2i( 1+ + 2 ) a^ 2 is the supersymmetric interval. Then we push eipa^ a^ through all the operator factors in (3.37) to the left and then replace it by unity in the coincidence limit. This leads to the following transformation of the covariant derivatives Da^ ! Da^ + ipa^; D^ ! D^ 2ip^ ^( 1 is no enough operators D+^ to annihilate ( 1 2) ^ in D^ vanishes in the coincidence limit since there 2 ). To get the expansion of the e ective action in the background elds and their derivatives we should expand the exponent of the operator and calculate the standard momentum integrals. Thus, we expand the exponent eis_2W in powers of spinor derivatives near e isp2 up to the fourth order in spinor derivatives (it corresponds to expanding up to the fourth order in the proper time) and use the identity 2 (D1 )4 4( 1+ 1 + 2 )j 1= 2 = 1. It remains to perform a standard integration over the momentum variables and over the proper-time. The nal result has the form is some scale.7 One can show, using the methods developed in [51], that the action (3.40) is superconformal. A component form of (3.40) in the bosonic sector corresponds to the Lagrangian F34 . The e ective action (3.40) can be treated as the 5D analogue of the non-holomorphic potential in 4D; N = 2 supersymmetric gauge theories. This e ective action is also similar to the action of the the so-called 4D improved tensor multiplet [75{77]. 4 The leading contribution to the e ective action of a 5D SYM multiplet It is interesting and instructive to compare the one-loop leading low-energy e ective action in 5D SCS theory and in 5D SYM theory. 5D, N = 1 SYM theory is described by the action (see (2.14), (2.19)): where the coupling constant has dimension [gS2YM] = 1: Let us consider the construction of the background eld method for the theory (4.1). It is obvious that the second variation of the action has the form completely analogous to the 4D, N = 2 case 2 S = 1 7It is easy to see that the e ective action (3.40) does not depend on the scale, since Z d13zW ln = ln Z d ( 4) D+ 4 (3.39) (3.40) (4.1) (4.2) Therefore we can simply repeat step by step all the stages of the construction of the e ective action developed in [39{44]. In particular, we can use the same choice of the gauge conditions on the quantum super eld D++v++ = 0 and the same procedure to x gauge as in [39{41]. It gives the sum of the quadratic part of action (4.2) and the corresponding gauge- xed action in the form SGF S2 + Sgh = 1 2 v++ 1 (D 2 1 gSYM v++( 1 )v++(2) )2v++ Z Further we put gauge parameter = The derivation of the general expression for the one-loop e ective action in 5D SYM theory is completely analogous to derivation of the expression (3.28). The nal result is i 2 i 2 ~( 1 ) SYM = Tr (2;2) ln 2 Tr (0;4) ln 2 : _ _ The rst term is the contribution of the ghosts and the second term is the contribution of a SYM multiplet. Here the covariantly analytic d'Alembertian _2 is given by (3.19) and has the form [49] _2 = + 1 64 D 1 )2j = Da^Da^ + D 2 W : Now we consider a calculation of the contribution to the e ective action (4.3) from a SYM multiplet. Contribution of the rst term in (4.3) will be calculated in section 5. In the proper-time representation, the contribution of SYM multiplet has the form analogous to (3.29) ( 1 ) SYM = The only di erence with (3.29) is the operator 2 instead of the operator 2 . To nd the leading low-energy contribution to e ective action it is su cient to consider the onshell background super eld. Classical on-shell equation for the 5D SYM theory look like = 0. In this case the covariantly analytic d'Alembertian _2 and the projection T (2;2) are simpli ed and we have the e ective action in the following form _ W ( 1 ) SYM = Evaluation of this expression is realized completely the same way as it was done in the subsection 3.3 to obtain the (3.40). The nal result has the form (4.3) (4.4) (4.5) (4.6) We see that the functional form of the e ective action (4.6) generated by a SYM multiplet coincides with the one generated by a SCS multiplet (3.40). Although the on-shell conditions for 5D SCS theory and for 5D SYM theory are di erent and, moreover, 5D SYM theory is characterized by a dimensional coupling constant, the leading low-energy contributions in these two theories to the one-loop e ective action turn out to be the same up to a numerical coe cient. 5 The leading and next-to-leading contributions of the ghosts and matter super elds The ghost contribution to the one-loop e ective action in both SCS and SYM theories is de ned by the expression i 2 ( 1 ) ghost = hyper = ( )iTr ln D iTr ln G(1;1) ; obtained variation. ( 1 ) hypermultiplets or from ghosts The contribution from matter hypermultiplet super elds di ers only by a sign and by the choice of the representation of the gauge group. Hence, to nd the leading contribution to (5.1) we can use the results from the previous analysis [49] of the e ective action for a q-hypermultiplet coupled to a background vector multiplet.8 It was shown in that paper that the leading quantum correction is the SCS action. Therefore, further we will focus only on the rst next-to-leading correction. For calculation of the rst next-to-leading contribution to the e ective action (5.1) we will follow the procedure proposed in the papers [42{44, 49]. This procedure is based on calculating the variation of the e ective action with its subsequent restoration given the Let hyper be the one-loop contribution to the e ective action from either matter (5.1) (5.2) (5.3) HJEP1(205)3 where the upper sign corresponds to the contribution from matter and the lower sign to that of ghosts. Further, for simplicity, we will consider for only the plus sign. Here G(1;1)( 1; 2) is the hypermultiplet Green function satisfying the equation D1 ++G(1;1)( 1; 2) = (3;1)( 1; 2) ; A where A(3;1)( 1; 2) is the appropriate covariantly analytic delta-function (3;1)( 1; 2) = (D1+)4 13(z1 A z2) ( 1;1)(u1; u2)1: This equation is similar to the equation for the hypermultiplet Green function in 4D, N = 2 theories and we can use methods developed in [39{41]. The Green function G(1;1)( 1; 2) 8From a formal point of view, the e ective action (5.1) corresponds to a so called !-hypermultiplet [46]. It was pointed out some time ago [39{41] that this e ective action is equal, up to the coe cient 2; to the e ective action for a q hypermultiplet. Therefore, further we will take into account just the q hypermultiplet To evaluate this expression one can use the proper-time technique. The leading low-energy contribution goes from the terms without derivatives of W. It was calculated in [49] and [1] hyper = 1 (4 )2 sign(W) Z The variation of (5.6) corresponds precisely to the classical action (2.25) of the SCS gauge has the form theory. hyper = Z V ++G(1;1) (5.5) can be written in the form Here 2 is the covariantly analytic d'Alembertian (4.4) and (u1+u2+) 3 is a special harmonic distribution [67, 68]. The variation of the e ective action (5.2) under the background super eld V ++ is written as follows (see [42{44, 49] for details) (5.4) (5.6) (5.7) (5.8) Our main purpose in this section is to nd the rst next-to-leading correction to the SCS action. First of all one considers the Dyson type equation relating the free and full hypermultiplet propagators [39{41] Z G(1;1)(1j2) = G(1;1)(1j2) 0 d 3( 4)G(1;1)(1j3)iV ++(3)G(1;1)(3j2) : 0 Substituting eq. (5.7) into the variation (5.5), one nds Z ( 1 ) hyper = d 1( 4)d 2( 4) V ++( 1 )iV ++(2)G(01;1)(1j2)G(1;1)(2j1) : Taking into account the explicit form of the propagator (5.4), we rewrite this expression as follows ( 1 ) hyper = Z d 1( 4)d 2( 4) V ++( 1 ) 1 21 D1+ 4 D2+ 4 13(z1 z2) Now we use the spinor derivatives from the rst delta-function to restore the full N = 1 superspace measure according to the rule R d ( 4)(D+)4 = R d13z. So far, we did not consider any restrictions for the background super eld, therefore (5.8) is the exact representation for the one-loop hypermultiplet e ective action. In principle it can be a starting point for calculations of di erent contributions to the one-loop e ective action. To compute the rst next-to-leading quantum corrections to (5.6) it is su cient to expand in (5.8) the covariant analytic d'Alembertian in powers of the derivatives of the background super elds. Let us remember that the Green function of the hypermultiplet is antisymmetric with respect to the permutation of its arguments. This allows us to rewrite the expression (5.8) in the form The next step is the expansion of the operator 2 in the denominator in power series of the derivatives of W. In this expansion we keep only the leading terms with two derivatives. It leads to ( 1 ) Now the usual steps lead to Z d13zdu1du2 V ++( 1 ) u1+u2+ 2 D2 2 iV ++(2) + 2 (2 W2)3 D2 4 D2 + 4 13(z2 z1): W 2 + 2 ln W = 1 As a result we obtain the rst next-to-leading contribution to the one-loop hypermultiplet e ective action in the form Z [1] hyper = chyper Here chyper is a numerical coe cient depending on details of the hypermultiplet action (such as the number of components, the representation, whether hypermultiplet super elds are commuting or anticommuting). We see that the e ective action (5.10) has the same functional structure as the e ective actions generated by the SCS multiplet (3.40) and by the SYM multiplet (4.6). 6 Summary We have considered the ve-dimensional N = 1 supersymmetric eld models such as the Abelian Chern-Simons theory, the Yang-Mills theory and the hypermultiplet theory coupled to a background vector multiplet, formulated in the harmonic superspace approach. In all these models we calculated the universal four derivative contributions to the one-loop In the 5D SCS theory we have developed the background eld method in harmonic superspace and represented the one-loop e ective action in terms of functional determinants of the operators acting in the analytic subspace of harmonic superspace (eq. (3.28)). The above expression contains the contributions from the ghost super elds and from a SCS vector multiplet super eld. We studied the structure of the latter contribution to the e ective action and evaluated the leading low-energy e ective action (eq. (3.40)). The same consideration has been realized in 5D SYM theory as well. We developed the background eld method for this theory, found the one-loop e ective action in terms of the functional determinants of the di erential operators acting in the analytic subspace of harmonic superspace (eq. (4.3)). We studied the structure of the one-loop contributions to the e ective action from a SYM vector multiplet and calculated the leading low-energy contribution of this multiplet. Although 5D SYM theory is not superconformal, its coupling constant is dimensional, and the expressions (3.28) and (4.3) are di erent, the leading contribution to the e ective action in 5D SYM theory (eq. (4.6)) has the same functional form as in 5D SCS theory. In 5D hypermultiplet theory in a vector multiplet background eld we have calculated the rst next-to leading contribution to the e ective action (eq. (5.10)). The corresponding leading contribution is the SCS action and was found in the paper [49]. We have shown that this rst next-to leading contribution (5.10) again has the same functional form as the leading contribution in SCS theory. We have found the manifestly 5D, N = 1 superconformal form of the term W the e ective actions of the SCS theory, SYM theory and hypermultiplet theory. The next step of studying of the one-loop e ective action in the theories under consideration is a construction of the full low-energy one-loop e ective actions where all the powers of the Abelian strength are summed up. In other words, the next purpose is to construct the 5D super eld Heisenberg-Euler type of the e ective action. Like in 4D, 3D superconformal 2 = 12 D4 ln W: We hope to study this issue in the forthcoming work. gauge theories [78, 79] it is reasonable to expect that this e ective action will be expressed in the terms of so called superconformal invariants which transform as scalars under the 5D, N = 1 superconformal group. For example, such a scalar invariant can be a super eld F 4 in Acknowledgments The authors are very grateful to E.A. Ivanov and S.M. Kuzenko for useful comments. The authors are thankful to the RFBR grant, project No 15-02-06670-a and LRSS grant, project No 88.2014.2 for partial support. Work of I.L.B was supported by Ministry of Education and Science of Russian Federation, project No 2014/387/122. Also I.L.B is grateful to the DFG grant, project LE 838/12-2. Open Access. 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I. L. Buchbinder, N. G. Pletnev. Effective actions in \( \mathcal{N}=1 \) , D5 supersymmetric gauge theories: harmonic superspace approach, Journal of High Energy Physics, 2015, 130, DOI: 10.1007/JHEP11(2015)130