One-loop divergences in the 6D, N=(1,0) abelian gauge theory
I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz
aa r X i v : . [ h e p - t h ] N ov One-loop divergences in the D , N = (1 , abeliangauge theory I.L. Buchbinder a,b , E.A. Ivanov c B.S. Merzlikin a,d , K.V. Stepanyantz ea Department of Theoretical Physics, Tomsk State Pedagogical University,634061, Tomsk, Russia b National Research Tomsk State University, 634050, Tomsk, Russia c Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia d Department of Higher Mathematics and Mathematical Physics,Tomsk Polytechnic University, 634050, Tomsk, Russia e Department of Theoretical Physics, Moscow State University, 119991, Moscow, Russia
Abstract
We consider, in the harmonic superspace approach, the six-dimensional N = (1 ,
0) su-persymmetric model of abelian gauge multiplet coupled to a hypermultiplet. The superfi-cial degree of divergence is evaluated and the structure of possible one-loop divergences isanalyzed. Using the superfield proper-time and background-field technique, we computethe divergent part of the one-loop effective action depending on both the gauge multi-plet and the hypermultiplet. The corresponding counterterms contain the purely gaugemultiplet contribution together with the mixed contributions of the gauge multiplet andhypermultiplet. We show that the theory is on-shell one-loop finite in the gauge multipletsector in agreement with the results of [1]. The divergences in the mixed sector cannot beeliminated by any field redefinition, implying the theory to be UV divergent at one loop. [email protected] [email protected] [email protected] [email protected] Introduction
The higher-dimensional supersymmetric gauge models are of interest mainly because theydescribe low-energy limits of the superstring/brane theory and inherit many remarkable prop-erties of the latter. In particular, one can expect the existence of various non-renormalizationtheorems governing their ultraviolet (UV) behavior. In this letter we study the UV divergencestructure of the six-dimensional abelian N = (1 ,
0) gauge theory interacting with hypermulti-plets. The analysis of this simplest model can be conducive for the further study of quantumproperties of more complicated non-abelian N = (1 ,
0) and N = (1 ,
1) gauge theories.An analysis of the UV divergences in the higher-dimensional supersymmetric gauge theorieshas been initiated by the paper [1] and continued in the subsequent papers [2], [3], [4], [5], [6], [7],[8], [9] (and references therein). In particular, it was found that in the sector of gauge (or vector)multiplet the divergences at different loops reveal a universal structure and in many cases somecounterterms can be completely eliminated by the field redefinitions. The counterterms in thehypermultiplet sector have never been calculated.As is well known, the most efficient way to describe the quantum aspects of supersymmetrictheories is to use the off-shell superfield formulations (see e.g. [10] for 4 D, N = 1 theoriesand [11] for 4 D, N = 2 theories). An arbitrary ( n, m ) representation of the six-dimensionalsupersymmetry is labeled by the numbers of left ( n ) and right ( m ) independent supersymmetries(see, e.g., [12]). In the case of 6 D, N = (1 ,
0) supersymmetry, the models of vector multipletand hypermultiplet can be formulated off shell in terms of superfields defined on 6 D, N = (1 , N = (1 ,
0) supersymmetric Yang-Mills theory in 6 D, N = (1 , N = (1 ,
0) vector multiplet and hypermultiplet. Using the appropriate set of N = (1 ,
0) harmonic superfields, one can construct N = (1 ,
1) supersymmetric Yang-Millstheories (see e.g. [8]), as well as the free gauge models with N = (2 ,
0) supersymmetry [6].It is worth pointing out that, in many aspects, 6 D, N = (1 ,
0) SYM theory is analogousto 4 D, N = 2 SYM theory, and 6 D, N = (1 ,
1) SYM theory to 4 D, N = 4 SYM theory.These 6 D theories and their 4 D counterparts have equal numbers of supercharges, 8 and 16,respectively. Like 4 D, N = 4 SYM theory possesses manifest off-shell N = 2 supersymmetryand an on-shell hidden N = 2 supersymmetry, 6 D, N = (1 ,
1) SYM theory possesses manifest N = (1 ,
0) supersymmetry and an on-shell hidden N = (0 ,
1) supersymmetry (see [8] for detailsand further references).The general analysis of the possible low-energy contributions of different conformal dimen-sions to the effective action of N = (1 ,
0) SYM theories has been carried out in ref. [8]. It wasproved that the superfield counterterm of dimension 6 in N = (1 ,
0) harmonic superspace is alinear combination of three terms, where one depends only on the vector multiplet superfield,another depends only on the hypermultiplet and the third mixed term bears a dependence onboth the vector multiplet and the hypermultiplet. The analysis was based on the N = (1 , N = (1 ,
0) harmonic superfields. Taking into account the results obtained in [8], it wouldbe extremely interesting to demonstrate how these results can in principle be derived in theframework of quantum field theory. Namely this problem is solved in this letter for an abelian N = (1 ,
0) supersymmetric gauge theory, which is an abelian N = (1 ,
0) vector models coupled1o N = (1 ,
0) hypermultiplet.The paper is organized as follows. In section 2 we briefly describe the formulation of abelian6 D, N = (1 ,
0) gauge theory in N = (1 ,
0) harmonic superspace and fix the 6 D notations andconventions. Section 3 presents the harmonic superspace background field method which allowsone to obtain the effective action in a manifestly gauge invariant and N = (1 ,
0) supersymmetricform. In section 4 we derive the superficial degree of divergence in the theory of interactingvector and hyper multiplets and discuss the structure of the one-loop divergences. In particular,we prove that the one-loop counterterms indeed match with the results of [8], except that thepurely hypermultiplet divergent contribution to the effective action is absent in the one-loopapproximation. Section 5 is devoted to the direct calculations of the one-loop divergences. Insection 6 we summarize the results and discuss the problems for further study. D , N = (1 , harmonic superspace Our consideration in this section (including notations, conventions and terminology) willclosely follow ref. [8].The basic objects of the 6 D, N = (1 ,
0) superfield gauge theory are gauge covariant deriva-tives defined by ∇ M = D M + i A M , (2.1)where D M = ( D M , D ia ) are the flat derivatives. Here M = 0 , .., , is the 6 D vector index and a = 1 , .. , is the spinorial one. The superfield A M is the gauge super-connection. The covariantderivatives transform under the gauge group as ∇ ′M = e i τ ∇ M e − i τ , τ † = τ , (2.2)and satisfy the algebra {∇ ia , ∇ jb } = − iε ij ∇ αβ , [ ∇ ic , ∇ ab ] = − ε abcd W i d , (2.3)[ ∇ M , ∇ N ] = iF MN , (2.4)where W i a is the superfield strength and ∇ ab = ( γ M ) ab ∇ M . Further in this paper we consideronly the abelian gauge theory coupled to a hypermultiplet.The constraints (2.3) and (2.4) can be solved in the harmonic superspace framework. Inthe λ -frame [11], the spinor covariant derivatives ∇ + a coincide with the flat ones D + a , while theharmonic covariant derivatives acquire the connections V ++ and V −− , ∇ ±± = D ±± + iV ±± , e V ±± = V ±± , δV ±± = −∇ ±± λ ( ζ , u ) , (2.5)[ ∇ −− , D + a ] = ∇ − a , [ ∇ ++ , ∇ − a ] = D + a , [ ∇ ++ , D + a ] = [ ∇ −− , ∇ − a ] = 0 , (2.6)where ( ζ , u ) stands for the analytic subspace coordinates. The real connection V ++ ( ζ , u ) isanalytic (in virtue of the third constraint in (2.6)) and it is an unconstrained potential of thetheory. The component expansion of V ++ ( ζ , u ) in the Wess-Zumino gauge reads V ++ W Z = θ + a θ + b A ab ( x ( an ) ) + ( θ + ) a λ ia ( x ( an ) ) u − i + 3( θ + ) D ik ( x ( an ) ) u − i u − k . (2.7)It involves the gauge field A ab , the gaugino field λ i a and the auxiliary field D ( ik ) .2he second, non-analytic harmonic connection V −− ( z, u ) is uniquely determined in termsof V ++ as a solution of the harmonic zero-curvature condition [11]. In the abelian case thelatter is [ ∇ ++ , ∇ −− ] = D ⇔ D ++ V −− − D −− V ++ = 0 . (2.8)The equation (2.8) can be solved for V −− as V −− ( z, u ) = Z du V ++ ( z, u )( u + u +1 ) . (2.9)Using the connection V −− , we can construct the spinor and vector superfield connections anddefine the covariant spinor superfield strengths W ± a W + a = − ε abcd D + b D + c D + d V −− , W − a = D −− W + a . (2.10)We also define the Grassmann-analytic superfield [8] F ++ = 14 D + a W + a = ( D + ) V −− , D + a W + b = δ ba F ++ , D ++ F ++ = 0 , (2.11)which will be used for construction of the counterterms.The superfield action of 6 D, N = (1 ,
0) abelian model of gauge multiplet interacting withhypermultiplet has the form S [ V ++ , q + ] = 14 f Z d z du du ( u +1 u +2 ) V ++ ( z, u ) V ++ ( z, u ) − Z dζ ( − du ˜ q + ∇ ++ q + , (2.12)where f is a dimensionful coupling constant ([ f ] = − q + ( x, θ )has a short expansion q + ( z ) = f i ( x ) u + i + θ + a ψ a ( x )+ . . . , with a doublet of massless scalars fields f i ( x ) and the spinor field ψ α as the physical fields. It also involves an infinite tail of auxiliaryfields coming from the harmonic expansions. Both the superfield q + ( ζ , u ) and its e -conjugate˜ q + [11] obey the analyticity constraint, D + α q + = D + α ˜ q + = 0 . The action (2.12) is invariantunder the gauge transformation V ++ ′ = − ie iλ D ++ e − iλ + e iλ V ++ e − iλ , q + ′ = e iλ q + , (2.13)where λ = λ ( ζ , u ) is the analytic gauge parameter introduced in (2.5). Using the zero curva-ture condition (2.8), one can derive a useful relation between arbitrary variations of harmonicconnections [8] δV −− = 12 ( D −− ) δV ++ − D ++ ( D −− δV −− ) . (2.14)Classical equations of motion following from the action (2.12) read δSδV ++ = 0 ⇒ f F ++ − i ˜ q + q + = 0 , δSδ ˜ q ++ = 0 ⇒ ∇ ++ q + = 0 . (2.15)Note that the superfield F ++ is real under the e conjugation, e F ++ = F ++ . The e - realityof the first equation in (2.15) (as well as of the action (2.12)) is guaranteed by the conjugationrule f ˜ q + = − q + [11]. 3 Background field method
In this section we outline the background field method for the model (2.12). The construc-tion of gauge invariant effective action in the model under consideration is very similar to thatfor 4 D, N = 2 supersymmetric gauge theories [15], [16] (see also the reviews [17]) .We split the superfields V ++ , q + into the sum of the “background” superfields V ++ , Q + andthe “quantum” ones v ++ , q + , V ++ → V ++ + f v ++ , q + → Q + + q + , (3.1)and then expand the action in a power series in quantum fields. As a result, we obtain theclassical action as a functional of background superfields and quantum superfields.To construct the gauge invariant effective action, we need to impose the gauge-fixing con-ditions only on quantum superfields. As in the four-dimensional case [15], we introduce thegauge-fixing function in the form F (+4) = D ++ v ++ . (3.2)We consider the abelian gauge theory, where gauge-fixing function (3.2) is background-fieldindependent. This means that the Faddeev-Popov ghosts are also background-field independentand so make no contribution to the effective action. According to (3.2), the gauge-fixing partof the quantum field action has the form S GF = − Z d zdu du v ++ (1) v ++ (2)( u +1 u +2 ) + 18 Z d zduv ++ ( D −− ) v ++ . (3.3)A formal expression of the effective action Γ[ V ++ , Q + ] for the theory under consideration isconstructed by the Faddeev-Popov procedure (see the reviews [17] for details).In the one-loop approximation, the first quantum correction to the classical actionΓ (1) [ V ++ , Q + ] is given by the following path integral [15, 16]: e i Γ (1) [ V ++ ,Q + ] = Z D v ++ D q + D ˜ q + e iS [ v ++ , q + ; V ++ , Q + ] . (3.4)Here, the full quadratic action S is the sum of the classical action (2.12), with the background-quantum splitting accomplished, and the gauge-fixing action (3.3) S = 14 Z dζ ( − du v ++ (cid:3) (2 , v ++ − Z dζ ( − du (cid:8) ˜ q + ∇ ++ q + + f ˜ Q + iv ++ q + + f ˜ q + iv ++ Q + (cid:9) , (3.5) There are two approaches for constructing the background field method for 4 D, N = 2 SYM theories. One isformulated in the conventional N = 2 superspace [18], while another in 4 D, N = 2 harmonic superspace [15], [19](see also the reviews [17]). The first formulation faces some troubles basically related to an infinite numberof FP ghosts. The second approach is free of such difficulties and provides a convenient tool for manifestlycovariant loop calculations. In this paper we generalize it to 6 D, N = (1 ,
0) gauge theory. Though in our casethe problem of ghosts is absent at all because we deal with the abelian theory, the harmonic background fieldmethod looks most preferable like in 4 D case. (cid:3) (2 , = ( D + ) ( D −− ) transforms the analytic superfields v ++ intoanalytic superfields. The Green function, associated with (cid:3) (2 , , G (2 , ( z , u | z , u ) = i h v ++ ( z , u ) v ++ ( z , u ) i , is given by the expression similar to that in the 4 D, N = 2 case [11] G (2 , τ (1 |
2) = − D +1 ) (cid:3) δ ( z − z ) δ ( − , ( u , u ) . (3.6)The action S (3.5) is a quadratic form of quantum fields, with the coefficients dependingon background fields. For further use, it is convenient to diagonalize this quadratic form, thatis to decouple the quantum superfields v ++ and q + . To achieve this, one performs the specialchange of the quantum hypermultiplet variables in the path integral, such that it removes themixed terms, q +1 = h +1 − f Z dζ ( − du G (1 , (1 | iv ++2 Q +2 , (3.7)with h + being the new independent quantum superfield. It is evident that the Jacobian of thevariable change (3.7) is unity. Here G (1 , ( ζ , u | ζ , u ) = i h ˜ q + ( ζ , u ) q + ( ζ , u ) i is the superfieldhypermultiplet Green function in the τ -frame ( G (1 , (1 |
2) = − G (1 , (2 | ∇ ++1 G (1 , τ (1 |
2) = δ (3 , A (1 | ⇒ G (1 , τ (1 |
2) = ( ∇ +1 ) ( ∇ +2 ) ⌢ (cid:3) δ ( z − z )( u +1 u +2 ) , (3.8)where δ (3 , A (1 |
2) is the covariantly-analytic delta-function and ⌢ (cid:3) is the covariantly-analyticd’Alembertian [6] which acts on analytic superfields as follows ⌢ (cid:3) = 12 ( D + ) ( ∇ −− ) = (cid:3) + iW + a ∇ − a + iF ++ ∇ −− − i D − a W + a ) , (3.9)with (cid:3) = ε abcd ∇ ab ∇ cd = ∇ M ∇ M . Note that the covariant d’Alembertian transforms theanalytic superfields into analytic superfields. After some algebra, the quadratic part of theaction S (3.5) splits into the vector-multiplet dependent part S V ect [ V ++ , Q + ] = 14 Z dζ ( − du v ++1 × Z dζ ( − du n (cid:3) δ (2 , A (1 | − f e Q +1 G (1 , (1 | Q +2 o v ++2 , (3.10)and the hypermultiplet part S Hyp [ V ++ ] = − Z dζ ( − du ˜ h + (cid:0) D ++ + iV ++ (cid:1) h + . (3.11)We see that the quadratic part of the action in the vector multiplet sector S V ect is an ana-lytic nonlocal functional of quantum field v ++ . It also contains an interaction between back-ground vector multiplet and hypermultiplet through the background-dependent Green function G (1 , ( V ++ ). A similar substitution was used in [16], [20] and [21] for computing one- and two-loop effective actions insupersymmetric theories, and in [22] for non-local change of fields in non-supersymmetric QED. (1) [ V ++ , Q + ] = i n (cid:3) − f e Q + G (1 , Q + o + i Tr ln ∇ ++ . (3.12)The expression (3.12) is the starting point for studying the one-loop effective action in themodel (2.12). In the next sections we will calculate the divergent part of (3.12). In this section we analyze the superficial degree of divergence in the model under consider-ation. The formal structure of Green functions of the vector multiplet (3.6) and the hypermul-tiplet (3.8) in 6 D, N = (1 ,
0) gauge theory is analogous to that in the four dimensional N = 2case. Hence, we can directly make use of the similar analysis in four dimensional N = 2 the-ory [19]. As in the four-dimensional theory, the Green functions in the case under considerationcontain enough number of Grassmann δ -function to prove the non-renormalization theorem ac-cording to which the loop contribution to the supergraphs defining the effective action can bewritten as a single integral over the total 6 D, N = (1 ,
0) superspace.Let us consider L -loop supergraph G with P propagators, V vertices, N Q external hyper-multiplet legs, and an arbitrary number of gauge superfield external legs. We denote by N D the number of spinor covariant derivatives acting on the external legs as a result of integrationby parts in the process of transforming the contributions to a single integral over d θ . The su-perficial degree of divergence ω ( G ) of the supergraph G can be found by counting the degreesof momenta in the loop integrals.The supergraph G involves L integrals over 6-momenta, which contribute 6 L to the degreeof divergence. Each of the hypermultiplet vertices contains one integration over d θ + . Prop-agators of the gauge superfields contribute the factors 1 /k , ( D + ) , as well as the Grassmann δ -functions. Similarly, propagators of the hypermultiplet superfields contribute 1 /k , ( D + ) foreach of two harmonic arguments of propagator (3.8) (eight D + - factors on a whole), and alsothe Grassmann δ -functions. From each hypermultiplet propagator we take the operator ( D + ) and so complete d θ + to d θ in all hypermultiplets lines, except for the number N Q of them.Then we consider the corresponding vertices and we take N Q operators ( D + ) off the prop-agators, which allow us to restore the integrations over d θ . After calculating the supergraphwe will end up with a single d θ integration. The other V − V is a totalnumber of vertices, are done due to the Grassmann δ -functions. The remaining P − V + 1 = L Grassmann δ -functions survive. Each of them is killed by eight D + a . Therefore, the number ofremaining D + a is 4 P − N Q − L . This implies that the superficial degree of divergence is ω ( G ) = (6 L − P ) + (2 P − N Q − L ) − N D = 2 L − N Q − N D , (4.1)where N D is the number of the spinor covariant derivatives acting on the external lines.Equivalently, the degree of divergence can be calculated, using dimension reasonings. Eachgauge propagator brings f , [ f ] = m − . The external gauge superfields are dimensionless,[ V ] = m , while the dimension of hypermultiplets is [ q ] = m . The effective action also6ontains a single integration over the full superspace. Taking into account that [ d x ] = m − and [ d θ ] = m we see that − ω ( G ) = − − P V + 2 N Q + 12 N D , (4.2)where P V is the number of gauge propagators. For hypermultiplets N Q = 2( − P Q + V Q ), so that ω ( G ) = 2 − V + 2 P − N Q − N D = 2 L − N Q − N D . (4.3)Our aim is to calculate a divergent part of the one-loop effective action, in this case thenumber of loops in Eq.(4.3) is L = 1. Due to the analyticity of the hypermultiplet superfield, D + q + = 0 , the number N D of spinor covariant derivatives acting on the external legs is equalto zero, N D = 0. Thus, in our case the superficial degree of divergence ω ( G ) (4.3) is reduced to ω − loop ( G ) = 2 − N Q . (4.4)Let us apply the relation (4.4) to the analysis of the one-loop divergences. According tothe general consideration of ref. [8], the possible contributions to divergent part of the effectiveaction of abelian theory is given by the following integral over the analytic subspace of harmonicsuperspace: Γ div = Z dζ ( − du n c ( F ++ ) + ic e Q + F ++ Q + + c ( e Q + Q + ) o . (4.5)Here, the coefficients c , c , c depend on the regularization parameters .Let N Q = 0, then ω = 2. The corresponding divergent structure has to be quadratic inmomenta and given by the full N = (1 ,
0) superspace integral. The unique possibility isΓ (1)1 ∼ Z d z du V −− (cid:3) V ++ , (4.6)where (cid:3) = ( D + ) ( D −− ) . Integrating in (4.6) by parts, we can transfer the factor ( D + ) from d’Alembertian on V −− and use the definition of superfield F ++ (2.11). Then we take onefactor D −− off the second multiplier and make use of the zero-curvature condition (2.8). Moreprecisely, Γ (1)1 ∼ Z d z du F ++ ( D −− ) V ++ = − Z d z du D −− F ++ D −− V ++ = − Z d z du D −− F ++ D ++ V −− = Z d z du D ++ D −− F ++ V −− . (4.7) In this paper we use the proper-time regularization (see [6], [23] and references therein) preserving thesupersymmetry at least at one loop and are interested in the logarithmic divergences only. One-loop logarithmicdivergences are known to be not susceptible to such subtleties of quantum field theory as, e.g., presence ofanomalies. We emphasize that the regularization aspects of six dimensional theories deserve a special attention,like those in four dimensional theories (see e.g., discussion in [24]). However, various choices of regularizationscheme do not affect the form of one-loop logarithmic divergences which are the subject of our paper. D ++ and D −− , use the property D ++ F ++ = 0 and obtain D ++ D −− F ++ = D F ++ = 2 F ++ . Finally, passing to the analytical subspace, we haveΓ (1)1 = c Z dζ ( − du ( F ++ ) . (4.8)The coefficient c is divergent in the limit of removing the regularization.Let N Q = 2, then ω = 0. The unique candidate divergent term involving no dependence onmomenta and representable as an integral over the full N = (1 ,
0) superspace readsΓ (1)2 ∼ Z d z du e Q + V −− Q + . (4.9)Passing to the analytic subspace and using (2.11), we immediately obtainΓ (1)2 = ic Z dζ ( − du e Q + F ++ Q + , (4.10)where, once again, the coefficient c is divergent in the limit of removing the regularization.We see that the contributions (4.8) and (4.10) match with the general structure (4.5) of thedivergent part of the effective action.For all other values of N Q the index ω is negative and the corresponding Feynmann integralsare UV finite. In particular, the divergent term of the form ( e Q + Q + ) is absent in the one-loopapproximation. Such divergent terms could appear, starting with two loops. In the previous section we discussed the general structure of the one-loop contributions tothe divergent part of effective action. Here we perform the direct calculation of the coefficients c and c in (4.5).The ( F ++ ) part of the effective action depends only on the background vector multiplet V ++ and is defined by the second term in eq. (3.12). More precisely,Γ (1) F [ V ++ ] = i Tr ln ∇ ++ = − i Tr ln G (1 , . (5.1)Here G (1 , is the superfield propagator for hypermultiplet (3.8). The details of calculationfor (5.1) were discussed in recent works [7], [23]. We consider an arbitrary variation of theexpression (5.1) δ Γ (1) F [ V ++ ] = − i Tr δ iV ++ G (1 , = Z dζ ( − du δV ++ G (1 , (1 | (cid:12)(cid:12)(cid:12) . (5.2)Our aim is to calculate the divergent part of the effective action (5.1). In the proper-timeregularization scheme [6], [23], the divergences are associated with the pole terms of the form ε , ε →
0, where ε = 6 − d with space-time dimension d . Taking into account the expressionfor Green function G (1 , (3.8), one gets δ Γ (1) F [ V ++ ] = Z dζ ( − du δV ++ Z ∞ d ( is )( isµ ) ε e is ⌢ (cid:3) ( D +1 ) ( D +2 ) δ ( z − z )( u +1 u +2 ) (cid:12)(cid:12)(cid:12) . (5.3)8ere s is the proper-time parameter and µ is an arbitrary regularization parameter of massdimension. Like in the four- and five-dimensional cases, one makes use of the identity (see [25]for details)( D +1 ) ( D +2 ) δ ( z − z )( u +1 u +2 ) = ( D +1 ) n ( u +1 u +2 )( ∇ − ) − ( u − u +2 )Ω −− − ⌢ (cid:3) ( u − u +2 ) ( u +1 u +2 ) o δ ( z − z ) , (5.4)where we have introduced the notationΩ −− = i ∇ ab ∇ − a ∇ − b + 4 W − a ∇ − a − ( D − a W − a ) . (5.5)One can show [21] that only the first term in (5.4) gives contribution to the divergent part ofthe one-loop effective action δ Γ (1) F [ V ++ ] = Z dζ ( − du δV ++ (1) ×× Z ∞ d ( is )( isµ ) ε e is ⌢ (cid:3) ( u +1 u +2 )( D +1 ) ( D − ) δ ( z − z ) (cid:12)(cid:12)(cid:12) . (5.6)Those terms in the right hand side of (5.6) which produce the divergent part read e is ⌢ (cid:3) ( u +1 u +2 ) e − is ⌢ (cid:3) (cid:12)(cid:12)(cid:12) = − i ( is ) (cid:3) F ++ ) − i ( is ) n ∂ M ∂ N F ++ ) ∂ M ∂ N o . (5.7)Then we pass to momentum representation of the delta function and calculate the proper-timeintegral. This leads to the expression δ Γ (1) F [ V ++ ] = − π ) ε Z dζ ( − du δV ++ (cid:3) F ++ . (5.8)Let us compare (5.8) with (4.5). Keeping in mind the definition F ++ = ( D + ) V −− , we cantransform the variation (5.8) to the form δ Γ (1) div = 2 c Z dζ ( − du F ++ ( D + ) δV −− . (5.9)Then we use the relation between δV −− and δV ++ (2.14) and the property D ++ F ++ = 0 .After that we restore the full 6 D , N = (1 ,
0) superspace measure, δ Γ (1) div = c Z dzdu F ++ ( D −− ) δV ++ + Z duD ++ ( ... ) , (5.10)and integrate by parts with respect to ( D −− ) . Omitting the total derivative terms and passingto the analytic subspace, we obtain δ Γ (1) div = c Z dzdu ( D −− ) F ++ δV ++ (5.11)= c Z dζ ( − du ( D + ) ( D −− ) F ++ δV ++ . (5.12)9he derivatives ( D + ) act only on ( D −− ) F ++ because δV ++ is an analytic superfield. Thenwe use the definition of analytic d’Alambertian (cid:3) = ( D + ) ( D −− ) and finally find δ Γ (1) div = 2 c Z dζ ( − du δV ++ (cid:3) F ++ . (5.13)As is expected, the variation of the divergent part of effective action (5.8) proved to have thesame structure as (5.13). Hence we obtain, up to an unessential additive constant,Γ (1) F [ V ++ ] = − π ) ε Z dζ ( − du ( F ++ ) . (5.14)The hypermultiplet-dependent part e Q + F ++ Q + of the one-loop counterterm arises from thefirst term in (3.12) . In order to calculate this contribution one expands the logarithm in thefirst term (3.12) up to the first order and compute the functional traceΓ (1) QF Q [ V ++ , Q + ] = i n (cid:3) − f e Q + G (1 , Q + o ≈ − if Z dζ ( − du e Q + Q + (cid:3) G (1 , (1 | (cid:12)(cid:12)(cid:12) . (5.15)We again use the identity (5.4) and consider only the first term here, because just this term isresponsible for divergence:1 (cid:3) G (1 , (1 | (cid:12)(cid:12)(cid:12) = 1 (cid:3) ( D +1 ) ( D +2 ) ⌢ (cid:3) δ ( z − z )( u +1 u +2 ) (cid:12)(cid:12)(cid:12) = 1 (cid:3) ( D +1 ) ( D − ) ⌢ (cid:3) ( u +1 u +2 ) δ ( z − z ) (cid:12)(cid:12)(cid:12) . (5.16)Now we observe that the first analytic d’Alembertian (cid:3) in the denominator comes from thepure vector multiplet part and does not contain background fields. The second covariantd’Alembertian ⌢ (cid:3) in the denominator emerges from the Green function for hypermultiplet afternon-local change of variables (3.7). This operator depends on the background vector multipletas in (3.9).To calculate the divergent part of the expression under consideration it suffices to take intoaccount only two first terms in the ⌢ (cid:3) , namely ⌢ (cid:3) = (cid:3) + iF ++ ∇ −− + . . . . Other two terms do not contribute to the divergent part of one-loop effective action in thepoint-coincidence limit. We expand the operator ⌢ (cid:3) up to the first order in iF ++ ∇ −− andact by it on the harmonic distribution ( u +1 u +2 ). Using properties of Grassmann delta-function,( D +1 ) ( D − ) δ ( θ − θ ) (cid:12)(cid:12)(cid:12) = 1, we obtain( D +1 ) ( D − ) (cid:3) ( (cid:3) + iF ++ ∇ −− + .. ) ( u +1 u +2 ) δ ( z − z ) (cid:12)(cid:12)(cid:12) = − iF ++ ( u − u +2 ) (cid:3) δ ( x − x ) (cid:12)(cid:12)(cid:12) . (5.17) It is known that calculations of harmonic supergraphs with hypermultiplet propagators require a certaincare related to coinciding harmonic singularities [26]. As argued in [26] (see [11], [27] as well) this problem canbe avoided in all cases of interest. In our case we also do not face such a problem. δ -function and calculates themomentum integral in the ε -regularization scheme. It leads to1 (cid:3) δ ( x − x ) (cid:12)(cid:12)(cid:12) = i (4 π ) ε , ε → . (5.18)As a result, one gets Γ (1) QF Q [ V ++ , Q + ] = 2 if (4 π ) ε Z dζ ( − du e Q + F ++ Q + . (5.19)Summing up the contributions (5.14) and (5.19), we finally obtainΓ (1) div [ V ++ , Q + ] = − π ) ε Z dζ ( − du n ( F ++ ) − if e Q + F ++ Q + o . (5.20)If the background hypermultiplet vanishes, the divergent part of the effective action is propor-tional to the classical equation of motion F ++ = 0. Therefore the divergence as a whole can beeliminated by a field redefinition ( δV ++ ∼ ε F ++ ) in the classical action and the theory underconsideration is one-loop finite on shell, in accordance with the results of ref. [1]. However, ifthe background hypermultiplet does not vanish, we obtain, after some field redefinition pro-portional to the equation of motion, the divergent part of on-shell effective action in the formΓ (1) div ∼ ε R dζ ( − du ( e Q + Q + ) . Thus, the on-shell divergence in the hypermultiplet sector cannotbe eliminated and the full theory is not finite even at the one-loop level. Let us briefly summarize the results obtained. We have considered the six-dimensional N = (1 ,
0) supersymmetric theory of the abelian vector multiplet coupled to hypermultiplet inthe 6 D, N = (1 ,
0) harmonic superspace formulation. We have studied the quantum effectiveaction involving dependence on both the vector multiplet and the hypermultiplet superfields.The corresponding background field method in harmonic superspace was formulated, such thatit allows one to preserve manifest gauge invariance and supersymmetry at all stages of cal-culating the effective action. It is important to point out that the superfield propagators inthe theory under consideration have, in the sector of anticommuting variables and harmonics,the same structure as the propagators in 4 D, N = 2 SYM theory. It leads to 6 D, N = (1 , D, N = (1 ,
0) superspace. Using this result, we have calculated the superficial degree ofdivergences and analyzed the structure of one-loop counterterms in both the vector multipletand the hypermultipelt sectors. It was shown, in particular, that one of the possible divergentcounterterms in the purely hypermultiplet sector, which is allowed on the supersymmetry anddimension grounds [8], is actually prohibited at one loop.We have developed an efficient manifestly gauge invariant and N = (1 ,
0) supersymmetrictechnique to calculate the one-loop effective action. As an application of this technique, wefound the one-loop divergences of the theory under consideration. The results completely matchthe analysis of the general structure of divergences based on considering superficial degree of11ivergences. It was shown that, if the background hypermultiplet superfield does not vanish,the one-loop divergences cannot be eliminated by any field redefinition and the theory is notone-loop finite.Let us discuss some possible generalizations and extensions of the results obtained. Asthe next step, it is quite natural to study the structure of the effective action for the non-abelian 6 D, N = (1 ,
0) SYM theories. All such theories admit a formulation in 6 D, N = (1 , F ++ ) and the e Q + F ++ Q + terms in the one-loop divergent part. We expect that the purelyhypermultiplet contribution to the divergent part of the one-loop effective action in non-abeliantheory will be absent as in the abelian theory. Besides the divergent part of effective action, itwould be interesting to study the finite contributions to low-energy effective action, which havenever been considered before.It would be extremely interesting to study the effective action in 6 D, N = (1 ,
1) SYMtheory. Such a theory can be formulated in 6 D, N = (1 ,
0) harmonic superspace in terms of N = (1 ,
0) analytic harmonic superfields, viz. the gauge connection V ++ and the hypermultiplet q + , ˜ q + , both in the adjoint representation [8]. This theory exhibits the manifest off-shell N =(1 ,
0) supersymmetry and an additional hidden on-shell N = (0 ,
1) supersymmetry, and inmany aspects is analogous to 4 D, N = 4 SYM theory [11]. It was shown, based solely uponthe invariance of the effective action under both manifest and hidden supersymmetries, that N = (1 ,
1) SYM theory is one-loop finite. It would be tempting to analyze the divergencesof N = (1 ,
1) SYM theory within the quantum setting and explicitly calculate the one-loopcounterterms (in parallel with constructing the full quantum N = (1 ,
1) SYM effective action).It is well known that the 6 D, N = (1 ,
0) supersymmetric theories are anomalous (seediscussions of chiral anomalies in higher dimensional supersymmetric theories in refs. [28]).It would be interesting to study such anomalies in the harmonic superspace formulation of6 D, N = (1 ,
0) SYM coupled to hypermultiplets and show, by a direct quantum field theoreticalanalysis, that the N = (1 ,
1) SYM theory is anomaly-free. We are going to tackle all theseproblems in the forthcoming works.
Acknowledgements
I.L.B and E.A.I are grateful to Kelly Stelle for useful comments and discussion. They alsothank the organizers of the MIAPP program Higher spin theory and duality (May 2-27, 2016)for the hospitality in Munich at the early stages of this work. The authors acknowledge supportfrom the Russian Science Foundation grant, project No 16-12-10306.
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