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= 3 Superconformal N SYM Low-Energy Effective Action I.L. Buchbinder,a E.A. Ivanov,b I.B. Samsonovc,1 and B.M. Zupnikb a Departamento de Fisica, UFJF, Juiz de Fora, MG, Brazil and Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk 634061, Russia 2 b Bogolubov Laboratory of Theoretical Physics, Joint Institutefor NuclearResearch, Dubna, 141980 Moscow Region, Russia 2 1 cINFN, Sezione di Padova, via F. Marzolo 8, 35131 Padova, Italy 0 E-mail: [email protected], [email protected], 2 [email protected], [email protected] n a J Abstract: We construct a manifestly N = 3 supersymmetric low-energy effective action 1 of N = 3 super Yang-Mills theory. The effective action is written in the N = 3 harmonic 1 superspace and respects the full N = 3 superconformal symmetry. On mass shell this ] h action is responsible for the four-derivative terms in the N = 4 SYM effective action, -t such as F4/X4 and its supersymmetric completions, while off shell it involves also higher- p derivative terms. For constant Maxwell and scalar fields its bosonic part coincides, up e h to the F6/X8 order, with the bosonic part of the D3 brane action in the AdS × S5 5 [ background. We also argue that in the sector of scalar fields it involves the correctly 2 normalized Wess-Zumino term with the implicit SU(3) symmetry. v 5 4 Keywords: Extended supersymmetry, Superspaces, Supersymmetric effective theories, 1 4 Supersymmetric gauge theory . 1 1 1 ArXiv ePrint: 1111.4145 1 : v i X r a 1On leave from Tomsk Polytechnic University,634050 Tomsk, Russia. 2Permanent address. Contents 1 Introduction 1 2 N = 3 SYM setup 3 2.1 Superfield strengths in N = 3 harmonic superspace 3 2.2 Superconformal transformations in N = 3 HSS 6 3 Superconformal effective action 8 3.1 Non-superconformal F4 term 8 3.2 Scale and γ invariant F4/X4 term 8 5 3.3 Complete N = 3 superconformal symmetry 11 3.4 Independence of the choice of vacua 14 4 Component structure 15 4.1 F4/X4 term 15 4.2 F6/X8 term 16 4.3 A comment on the Wess-Zumino term 18 5 Summary and discussion 19 A Derivation of scale and γ invariant effective action 20 5 B Harmonic integrals 22 C Wess-Zumino term 23 C.1 Derivation from five-dimensions 23 C.2 Expansion around vacuum 25 D Effective equations of motion 26 1 Introduction It is well known that the superfield formulations of supersymmetric field theories, with the maximal number of the underlying supersymmetries being manifest and off-shell, are extremely useful for studying quantum aspects of these theories. In many cases, such formulations not only drastically reduce the amount of perturbative calculations, but also allow one to make certain conjectures about a possible structure of the final results prior to any calculation. The N = 1 superspace [1] is natural for N = 1, d = 4 supersymmetric models, while the adequate superfield approach to N = 2, d = 4 theories is offered by – 1 – N = 2 harmonic superspace [2]. As for the renowned N = 4 SYM theory, no appropriate formulation of it in terms of unconstrained off-shell N = 4 superfields is known to date. On shell, the N = 4 SYM theory is equivalent to the N = 3 SYM theory (see [2] and refs. therein). The latter possesses an unconstrained superfield formulation in N = 3 harmonic superspace [3, 4], such that three out of four supersymmetries of the original theory are manifest and off-shell within this framework. This approach proved to be very fruitful for establishing quantum finiteness of N = 3 SYM theory [5], as well as for constructing the N = 3 supersymmetric Born-Infeld theory [6].1 The basic goal of the present paper is to provide an evidence that the N = 3 harmonic superspace approach is also useful for studying the low-energy effective actions in the N = 3 and N = 4 SYM theories. It is known that the leading terms in the N = 4 SYM effective action in the N = 2 superfield formulation (which manifests only two out of four supersymmetries) are de- scribed by the non-holomorphic potential [8–16]. The hypermultiplet completion of the non-holomorphic potential, such that it ensures the on-shell N = 4 supersymmetry, was found in [17] (and further elaborated on in refs. [18–20]). The full action contains a scale- invariant and SU(4) symmetric F4/X4 term, as well as some other terms related to this leading one by N = 4 supersymmetry. Here F is the Maxwell field strength and X2 is mn the square of SU(4) invariant norm of scalar fields. In the present paper we develop the N = 3 harmonic superspace description of the leading terms in the N = 4 SYM effective action to the order F4/X4. We seek this action as an integral over the analytic subspace of the N = 3 harmonic superspace, with the Lagrangian density beinga local functional of theanalytic superfieldstrengths without derivatives onthem. Weshowthattherequirementsofthescaleandγ invarianceuniquely 5 fix the form of this functional. We check that the action constructed respects the full SU(2,2|3) superconformal symmetry and, in components, yields the F4/X4 term, where X2 = ϕiϕ¯ is the bilinear SU(3) (and in fact SU(4)) invariant of the involved scalar fields. i We stress that the obtained action is essentially defined on the Coulomb branch of the theory, when the scalar fields acquire non-vanishing vevs, ci = hϕii 6= 0. These constants ci explicitly appear in the effective action, so that the effective Lagrangian is singular at ci = 0. However, we show that the action is in fact independent of any particular choice of ci, ci 6= 0. This is entirely analogous to what happens in the N = 2 harmonic superspace formulation oftheN = 2improvedtensor multipletmodelgiven in[21]. Itwas emphasized there that the presence of such constants in the action has a topological origin. In accord with this interpretation, the low-energy N = 4 SYM effective action contains a topological term given by the Wess-Zumino action for the scalar fields [22, 23]. Therefore the presence of such constants in the effective Lagrangian is not surprising. One of the advantages of the N = 3 superspace formulation is the possibility to go off shell due to the existence of unconstrained gauge prepotentials. Varying with respect to these prepotentials, we obtain the effective equations of motion corresponding to the effective action. Likeinthenon-scale-invariantN =3Born-Infeldtheory[6],eliminationof 1A possible scale-invariant generalization of the N =3 Born-Infeld theory was discussed in [7]. – 2 – some of the auxiliary fields from the effective equations of motion allows one to reproduce not only F4/X4 term, but also the F6/X8 term in the effective action, which precisely matches with that appearing in the component expansion of the conformally-invariant Born-Infeld action. Therefore, the effective action obtained reproduces the worldvolume action of D3 brane on the AdS ×S5 background up to the order F6/X8. 5 The paper is organized as follows. Section 2 contains a brief summary of the basic ingredients of the N = 3 harmonic superspace formalism. In particular, we give the representation of the N = 3 superconformal group SU(2,2|3) on the N = 3 SYM superfield strengths. In Section 3, employing the scale and γ invariance, we derive the N = 3 SYM 5 low-energy effective action and then show its invariance under the full N = 3 superconformal group. In Section 4 we derive the F4/X4 and F6/X8 component terms from the superfield action. The last Section is devoted to discussing some open problems deserving further study. In Appendices A and B we collect some technical details concerning the derivation of the superfield action and calculation of SU(3) harmonic integrals. A possible four-dimensional representation of the Wess-Zumino term with manifest SU(3) symmetry is discussed in the Appendix C. In Appendix D we demonstrate that the N = 3 superfield effective action proposed can be used for studying the effective superfield equations of motion. Throughout the paper we follow the N = 3 superspace conventions employed in [7] and [6]. 2 N = 3 SYM setup 2.1 Superfield strengths in N = 3 harmonic superspace The standard N = 3 superspace is parametrized by the coordinates zM = (xm,θα,θ¯iα˙), i where i = 1,2,3 is the SU(3) triplet index. Following [3, 4], we introduce the SU(3) har- monicvariablesuI = (u1,u2,u3)andtheirconjugates, u¯i = (u¯i,u¯i,u¯i),withtheproperties i i i i I 1 2 3 uIu¯i = δI , uIu¯j = δj, εijku1u2u3 = 1. (2.1) i J J i I i i j k These defining relations are the orthogonality and completeness conditions. The harmonic variables allow one to convert the small indices i,j,... on which the R-symmetry SU(3) groupislinearlyrealized, intothecapital indices,I,J,...,whichareinertunderSU(3). For instance, we will make use of the projected Grassmann variables, θα = θαu¯i, θ¯Iα˙ = θ¯iα˙uI. I i I i Some of these projected Grassmann variables parametrize the analytic subspace, {ζ ,u} = {xm,θα,θα,θ¯1α˙,θ¯2α˙,u}, xm = xm−iθ σmθ¯1+iθ σmθ¯3. (2.2) A A 2 3 A 1 3 The analytic superspace (2.2) is closed under the N = 3 supersymmetry [3, 4], and, hence, plays a role similar to that of usual chiral subspace in the N = 1 superspace [1] and of the N = 2 harmonic analytic superspace [2]. – 3 – The harmonic projections of the covariant spinor derivatives2, ∂ ∂ ∂ ∂ Di = +2iθ¯iα˙ , D¯ = − −2iθα , (2.3) α ∂θα ∂xαα˙ iα˙ ∂θ¯iα˙ i ∂xαα˙ i aregiven byDI = DiuI andD¯ = D¯ u¯i. Itisimportantthatintheanalyticcoordinates α α i Iα˙ iα˙ I (2.2) two of these six derivatives become short, ∂ ∂ D1 = , D¯ = − , (2.4) α ∂θα 3α˙ ∂θ¯3α˙ 1 thus demonstrating that the N = 3 analytic superfields (i.e. those living on the analytic superspace (2.2)) can be covariantly defined by the Grassmann Cauchy-Riemann conditions, D1Φ(z,u) = D¯ Φ(z,u) = 0 ⇒ Φ(z,u) =Φˆ(ζ ,u). (2.5) α 3α˙ A The explicit expressions for the other four derivatives in the analytic basis can be found in the appendix of our previous paper [7], where the harmonic derivatives DI are J also written down. Amongthese harmonicderivatives, D1, D2 andD1 commute with (2.4) 2 3 3 and so preserve the analyticity, while the remaining three D2, D3, D3 do not. These six 1 2 1 harmonicderivatives, together withtheU(1) charges S andS , form ansu(3) algebra[24]. 1 2 The conventional N = 3 SYM superfield strengths in the standard N = 3 superspace aredescribedbytheantisymmetricSU(3)tensorsuperfieldsWij = −Wji. Inthelinearized approximation, these superfields obey the constraints [25], 1 DiW = (δiDkW −δiDkW ), D¯ W +D¯ W = 0, (2.6) α jl 2 j α kl l α kj iα˙ jk jα˙ ik which eliminate all non-physical components in these superfieldsand put the physical ones on shell. Projecting these superfield strengths on the harmonic variables, we obtain the following six superfields, W¯ 12 = u1u2W¯ ij, W¯ 23 = u2u3W¯ ij, W¯ 13 = u1u3W¯ ij, i j i j i j W = u¯iu¯jW , W = u¯iu¯jW , W = u¯iu¯jW . (2.7) 12 1 2 ij 23 2 3 ij 13 1 3 ij It is straightforward to find the harmonic projections of the constraints (2.6), which gives rise to a number of differential relations among the superfields (2.7). Consider, for instance, W¯ 12 and W . They obey the following (on-shell) constraints [24]3: 23 (i) First-order analyticity constraints, D1W¯ 12 = D2W¯ 12 = D¯ W¯ 12 = 0, α α 3α˙ D1W = D¯ W = D¯ W = 0; (2.8) α 23 2α˙ 23 3α˙ 23 2 We use the following rules of converting the vector and bi-spinorial indices into each other, xαα˙ = (σm)αα˙xm, xm= 12(σ˜m)α˙αxαα˙, ∂αα˙ = 21(σm)αα˙∂m, ∂m =(σ˜m)α˙α∂αα˙. 3The constraints (2.8)–(2.10) can also be derived by quantizing a massless superparticle moving in the N =3 harmonic superspace [26]. – 4 – (ii) First-order harmonic shortness constraints, D2W¯ 12 = D1W¯ 12 = D2W¯ 12 = D1W¯ 12 = 0, 1 2 3 3 D1W = D2W = D1W = D3W = 0; (2.9) 2 23 3 23 3 23 2 23 (iii) Second-order Grassmann linearity constraints, (D3)2W¯ 12 = (D¯ )2W¯ 12 = (D¯ )2W¯ 12 = (D¯ D¯ )W¯ 12 = 0, 1 2 1 2 (D2)2W = (D3)2W = (D2D3)W = (D¯ )2W = 0. (2.10) 23 23 23 1 23 Altogether, the constraints (2.8), (2.9) and (2.10) kill all non-physical (auxiliary) field components inW¯ 12 andW andputthephysicalonesonshell4. As aresult, thesuperfield 23 strengths W¯ 12 and W have the following component structurein theanalytic coordinates 23 (2.2), W = ϕ1+2iθαθ¯2α˙∂ ϕ1 −4iθαθ¯1α˙∂ ϕ2−4iθαθ¯1α˙∂ ϕ3 23 2 αα˙ 2 αα˙ 3 αα˙ +4iθαθβF +θ¯1α˙λ¯ +θαλ −θαλ 2 3 αβ α˙ 2 3α 3 2α +2iθαθ¯2α˙θ¯1β˙∂ λ¯ +2iθβθαθ¯2α˙∂ λ +4iθβθαθ¯1α˙∂ λ 2 αα˙ β˙ 2 3 αα˙ 2β 2 3 αα˙ 1β +8θαθβθ¯1α˙θ¯2β˙∂ ∂ ϕ3, 2 3 αα˙ ββ˙ W¯ 12 = ϕ¯ −2iθαθ¯2α˙∂ ϕ¯ +4iθαθ¯1α˙∂ ϕ¯ +4iθαθ¯2α˙∂ ϕ¯ 3 2 αα˙ 3 3 αα˙ 1 3 αα˙ 2 +4iθ¯1α˙θ¯2β˙F¯ +θαλ −θ¯2α˙λ¯1 +θ¯1α˙λ¯2 α˙β˙ 3 α α˙ α˙ +2iθαθβθ¯2α˙∂ λ +2iθ¯1α˙θ¯2β˙θα∂ λ¯2 +4iθ¯1α˙θ¯2β˙θα∂ λ¯3 2 3 αα˙ β 2 αα˙ β˙ 3 αα˙ β˙ +8θ¯1α˙θ¯2β˙θαθβ∂ ∂ ϕ¯ . (2.11) 2 3 αα˙ ββ˙ 1 Here ϕI = uIϕi, ϕ¯ = u¯iϕ¯ , (2.12) i I I i and ϕi is a triplet of physical scalars, (cid:3)ϕi = 0. The four spinor fields are comprised by the SU(3) singlet λ and the triplet λ = u¯iλ which obey free equations of motion, α Iα I iα ∂αα˙λ = ∂αα˙λ = 0. The fields F = F and F¯ = F¯ are spinorial components of α iα αβ (αβ) α˙β˙ (α˙β˙) the Maxwell field strength F = ∂ A −∂ A , ∂mF = 0. mn m n n m mn The crucial feature of the N = 3 harmonic superspace approach is that one can relax some of the constraints (2.8), (2.9), (2.10) and express the superfield strengths in terms of unconstrained off-shell gauge superfield potentials [6]. Consider the analytic superfields V1 and V2, D1(V1, V2) = D¯ (V1, V2) = 0. They possess the following gauge transfor- 2 3 α 2 3 3α˙ 2 3 mations δV1 = iD1λ, δV2 = iD2λ, (2.13) 2 2 3 3 4Besides eqs. (2.9), the original constraints (2.6) imply some other relations of the first order in spinor derivatives, connecting W¯12 and W23 with the remaining harmonic projections of Wkl and W¯kl. These extraconstraints can beused todeducethesecond-orderconstraints (2.10) which, togetherwith (2.8)and (2.9), form a closed set of theharmonic superspace constraints on W¯12 and W23 [26]. – 5 – withλbeingananalyticgaugesuperfieldparameter. Usingthesesuperfields,oneconstructs the non-analytic gauge potentials V2 and V3 as solutions of the zero-curvature equations 1 2 [6], D1V2 = D2V1, D2V3 = D3V2. (2.14) 2 1 1 2 3 2 2 3 Finally, the gauge-invariant superfield strengths W¯ 12 and W can be expressed in terms 23 of V2 and V3 as 1 2 1 1 W¯ 12 = − D1αD1V2, W = D¯ D¯α˙V3. (2.15) 4 α 1 23 4 3α˙ 3 2 It should be pointed out that the analyticity constraints (2.8) are valid off shell while the other constraints (2.9) and (2.10) put the superfield strengths on shell, except for the equations D1W¯ 12 = 0 and D2W = 0 which are also satisfied off shell. 2 3 23 2.2 Superconformal transformations in N = 3 HSS TheN = 3superconformalgroupSU(2,2|3), besidestheN = 3superPoincaré transformations,containsdilatation(withtheparametera),γ -transformation(withtheparameterb), 5 conformal boosts (with the parameters k ), conformal supersymmetry (with the param- αα˙ eters ηi, η¯ ) and SU(3) R-symmetry transformations (with the parameters λj, λj = −λi, α iβ˙ i i j λi = 0). The realization of this supergroup on the analytic coordinates (2.8) was found in i [27], δ xαα˙ = axαα˙ +k xαβ˙xβα˙ −4k θβθ¯2α˙θαθ¯2β˙ +4ixαβ˙θ¯1α˙u¯iη¯ sc A A ββ˙ A A ββ˙ 2 2 A 1 iβ˙ +2ixαβ˙ θ¯2α˙u¯iη¯ +4ixβα˙θαu3ηi +2ixβα˙ θαu2ηi A− 2 iβ˙ A 3 i β A+ 2 i β −4iλjθαθ¯1α˙u3u¯i −2iλjθαθ¯1α˙u2u¯i −2iλjθαθ¯2α˙u3u¯i , i 3 j 1 i 2 j 1 i 3 j 2 δ θα = (a/2+ib)θα+k xαβ˙ θβ −4i(θαu2+θαu3)θβηi sc 2 2 ββ˙ A+ 2 2 i 3 i 2 β +xαβ˙ u¯iη¯ +λj(θαu2+θαu3)u¯i , A+ 2 β˙i i 2 j 3 j 2 δ θα = (a/2+ib)θα+k xαβ˙ θβ −4iθαθβu3ηi +xαβ˙ u¯iη¯ +λjθαu3u¯i , sc 3 3 ββ˙ A− 3 3 3 i β A− 3 β˙i i 3 j 3 δ θ¯1α˙ = (a/2−ib)θ¯1α˙ +k xβα˙ θ¯1β˙ +4iθ¯1β˙θ¯1α˙u¯iη¯ +xβα˙ u1ηi −λjθ¯1α˙u¯iu1, sc ββ˙ A+ 1 β˙i A+ i β i 1 j δ θ¯2α˙ = (a/2−ib)θ¯2α˙ +k xβα˙ θ¯2β˙ +4iθ¯2β˙(θ¯1α˙u¯i +θ¯2α˙u¯i)η¯ sc ββ˙ A− 1 2 β˙i +xβα˙ u2ηi −λj(θ¯1α˙u¯i +θ¯2α˙u¯i)u2, (2.16) A− i β i 1 2 j where xαα˙ = xαα˙ ±2iθαθ¯2α˙. For preserving the N = 3 harmonic analyticity, the harmonic A± A 2 variables should transform according to the rules, δ u1 = u2λ1+u3λ1, δ u¯i =0, sc i i 2 i 3 sc 1 δ u2 = u3λ2, δ u¯i =−u¯iλ1, (2.17) sc i i 3 sc 2 1 2 δ u3 = 0, δ u¯i =−u¯iλ2−u¯iλ1, sc i sc 3 2 3 1 3 where λI = −4ik θβθ¯Iβ˙ −4i(η¯ θ¯Iβ˙u¯i +θβηiuI)+uIu¯jλi . (2.18) J ββ˙ J β˙i J J β i i J j In this paper we will use the so-called passive form of superconformal transformations of superfields, when the variation is taken at different points, e.g., δ W ≃ W′(x′)−W(x). sc – 6 – In such an approach, not only the superfields but also their derivatives, as well as the superspace measures, should be varied while computing the superconformal transformations of the superfield actions. It is known [2] that the analytic measure dζ(33)du is invariant under (2.16) and (2.17), 11 ∂(x′ ,θ′,u′) Ber A = 1. (2.19) ∂(x ,θ,u) A Using the coordinate transformations (2.16) and (2.17), it is straightforward to find the superconformal variations of harmonic derivatives: δ D1 = −λ1S , δ D2 = (λ1−λ2)D2, sc 2 2 1 sc 1 1 2 1 δ D2 = −λ2S , δ D3 = (λ2−λ3)D3, sc 3 3 2 sc 2 2 3 2 (2.20) δ D1 = λ1D2−λ2D1−λ1(S +S ), δ D3 = (λ1−λ3)D3+λ2D3−λ3D2, sc 3 2 3 3 2 3 1 2 sc 1 1 3 1 1 2 2 1 δ D1 = δ D2 = δ D3 = 0, δ S = δ S = 0. sc 1 sc 2 sc 3 sc 1 sc 2 Recall that the gauge covariant harmonic derivatives involve the gauge superfield prepotentials ∇I = DI +iVI. (2.21) J J J Requiring the lengthened derivatives (2.21) to be superconformally covariant, with taking into account the transformations (2.20), implies the following transformation laws for the gauge prepotentials: δ V1 = 0, δ V2 = (λ1−λ2)V2, sc 2 sc 1 1 2 1 δ V2 = 0, δ V3 = (λ2−λ3)V3, (2.22) sc 3 sc 2 2 3 2 δ V1 = λ1V2−λ2V1, δ V3 = (λ1−λ3)V3+λ2V3−λ3V2. sc 3 2 3 3 2 sc 1 1 3 1 1 2 2 1 Note that the superconformal variations of the analytic gauge superfields V1, V2 and V1 2 3 3 were earlier given in [2, 27], whilethe transformations of thenon-analytic gauge superfields V2, V3 and V3 were not presented before. 1 2 1 Using (2.16) and (2.17) it is also easy to find the superconformal transformations of the covariant spinor derivatives (2.4), δ D1 = (−a/2−ib−λ1)D1 +BβD1, sc α 1 α α β δ D¯ = (−a/2+ib+λ3)D¯ +B¯β˙D¯ , (2.23) sc 3α˙ 3 3α˙ α˙ 3β˙ where λ1 and λ3 were defined in (2.18) and 1 3 Bβ = −k (xββ˙ +4iθβθ¯1β˙)−4iθβuIηj , α αβ˙ A+ 1 I j α B¯β˙ = −k (xββ˙ −4iθβθ¯3β˙)−4iθ¯Iβ˙u¯jη¯ . (2.24) α˙ βα˙ A− 3 I α˙j It is worth pointing out that the spinor derivatives D1 and D¯ are not mixed under the α 3α˙ superconformal transformations. Finally, using the variations of the gauge prepotentials (2.22) and derivatives (2.23), we can find the superconformal transformations of the superfield strengths (2.15), δ W = AW , δ W¯ 12 = A¯W¯ 12, (2.25) sc 23 23 sc – 7 – where A = −a+2ib+λ2+λ3+B¯α˙ , A¯= −a−2ib−λ1−λ2+Bα. (2.26) 2 3 α˙ 1 2 α One can check that the superfields A and A¯ are analytic, D1(A, A¯) = D¯ (A, A¯) = 0. (2.27) α 3α˙ Hence, the transformations (2.25) preserve analyticity. 3 Superconformal effective action 3.1 Non-superconformal F4 term The N = 3 supersymmetric completion of the fourth-order term in the Born-Infeld action was constructed in [6], 1 (W¯ 12W )2 S = dζ(33)du 23 . (3.1) 4 32 11 (Λ¯Λ)2 Z Here Λis acouplingconstant of dimensionone inmass units, whichis introducedto ensure the correct dimension of the integrand. The analytic measure is defined as follows [6, 7], 1 dζ(33)= d4x (D3)2(D2)2(D¯ )2(D¯ )2. (3.2) 11 162 A 1 2 The analytic measure is dimensionless, [dζ(33)du] = 0, and [W¯ 12] = [W ] = 1. With this 11 23 normalization of the analytic measure, it is straightforward to check that, along with other component terms, the action (3.1) yields the standard F4 term, 1 F2F¯2 S = d4x +... . (3.3) 4 2 (Λ¯Λ)2 Z Consider now the superconformal variation of the action (3.1), 1 (W¯ 12W )2 δ S = dζ(33)du(A+A¯) 23 , (3.4) sc 4 16 11 (Λ¯Λ)2 Z where we have used the variations of the superfield strengths (2.25) and the invariance of the analytic measure (2.19). Here A and A¯ are superfields (2.26) collecting the constant parameters of the superconformal transformations (2.16) and (2.17). The variation (3.4) is non-zero, hence the action (3.1) is not superconformal. 3.2 Scale and γ invariant F4/X4 term 5 Our aim here is to find a superconformalgeneralization of the action (3.1). In what follows we will denote this superconformal action by Γ (to stress that it is a part of the N = 3 SYM low-energy effective action). The action Γ should meet the following criteria: 1. It should be a local functional defined on the analytic superspace and constructed out of the superfield strengths W¯ 12 and W without derivatives on them, 23 Γ = dζ(33)duH11(W¯ 12,W ). (3.5) 11 33 23 Z – 8 – The analytic Lagrangian density H11 is an arbitrary function of its arguments, such 33 that its external harmonic U(1) charges cancel those of the analytic integration measure. This is the most general form of the superspace action yielding terms with four-derivatives in components, since the analytic measure (3.2) contains just eight spinor derivatives which can produce four space-time ones on the component fields. 2. The action Γ should be invariant under the superconformal transformations (2.25), δ Γ = 0. (3.6) sc As a weaker requirement, in this subsection we will employ only the scale- and γ - 5 transformations out of the full SU(2,2|3) superconformal group. We will show that this is sufficient to uniquely specify the structure of the action. The check of the full superconformal symmetry will be performed in the next subsection. 3. Inthecomponent-fieldexpansiontheaction Γshouldreproducethescale-andSU(3)- invariant F4/X4 term (3.3), F2F¯2 d4x . (3.7) (ϕiϕ¯ )2 i Z 4. We are interested in the low-energy effective action for massless fields, with massive ones being integrated out. The massive fields appear in the Coulomb branch when the gauge symmetry is broken down spontaneously. For instance, the SU(2) gauge symmetry is broken down to U(1) when the scalar field corresponding to the Cartan subalgebra of su(2) acquire non-trivial vevs, ci = hϕii 6= 0, c¯ = hϕ¯ i 6= 0. (3.8) i i However, the effective action should be independent of any particular choice of these constants, Γ(c′i,c¯′) =Γ(ci,c¯ ), cic¯ 6= 0, (3.9) j j i because such a dependence would break superconformal invariance of the action. 5. Finally, we simplify the problem by considering only that part of the action (3.5) which does not vanish on the mass shell, i.e., we will assume that the superfield strengths obey the constraints (2.8)–(2.10). We will neglect all terms in the action Γ which vanish when these constraints are imposed. As a consequence, one is free to add to Γ or to subtract from it the following expressions which vanish on the mass shell, dζ(33)W¯ 12F(W ) ∝ d4x(D3)2(D2)2(D¯ )2[F(W )(D¯ )2W¯ 12]≃ 0, 11 23 1 23 2 Z Z dζ(33)W F(W¯ 12) ∝ d4x(D3)2(D¯ )2(D¯ )2[F(W¯ 12)(D2)2W ]≃ 0. 11 23 2 1 23 Z Z (3.10) Here F(W) is an arbitrary function of its argument. We will frequently employ this property while deriving the action. – 9 –