Low-energy effective action in two-dimensional SQED: a two-loop analysis

Journal of High Energy Physics, Jul 2017

We study two-loop quantum corrections to the low-energy effective actions in \( \mathcal{N}=\left(2,2\right) \) and \( \mathcal{N}=\left(4,4\right) \) SQED on the Coulomb branch. In the latter model, the low-energy effective action is described by a generalized Kähler potential which depends on both chiral and twisted chiral superfields. We demonstrate that this generalized Kähler potential is one-loop exact and corresponds to the \( \mathcal{N}=\left(4,4\right) \) sigma-model with torsion presented by Roček, Schoutens and Sevrin [1]. In the \( \mathcal{N}=\left(2,2\right) \) SQED, the effective Kähler potential is not protected against higher-loop quantum corrections. The two-loop quantum corrections to this potential and the corresponding sigma-model metric are explicitly found.

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Low-energy effective action in two-dimensional SQED: a two-loop analysis

HJE Low-energy e ective action in two-dimensional SQED: I.B. Samsonov 0 1 Lower Dimensions, Superspaces 0 141980 Dubna , Moscow region , Russia 1 Bogoliubov Laboratory of Theoretical Physics , JINR We study two-loop quantum corrections to the low-energy e ective actions in N = (2; 2) and N = (4; 4) SQED on the Coulomb branch. In the latter model, the low-energy e ective action is described by a generalized Kahler potential which depends on both chiral and twisted chiral super elds. We demonstrate that this generalized Kahler potential is one-loop exact and corresponds to the N = (4; 4) sigma-model with torsion presented by Rocek, Schoutens and Sevrin [1]. In the N = (2; 2) SQED, the e ective Kahler potential is not protected against higher-loop quantum corrections. The two-loop quantum corrections to this potential and the corresponding sigma-model metric are explicitly found. Extended Supersymmetry; Supersymmetric Gauge Theory; Field Theories in - 1 Introduction and summary 2 Exact propagators on constant vector multiplet background Classical action and loop expansion of the e ective action One-loop e ective action and the Wess-Zumino term Vanishing of two-loop corrections to generalized Kahler potential 5 A Conclusions N = (2; 2) superspace conventions 2.1 2.2 2.3 2.4 2.5 3.1 3.2 4.1 4.2 4.3 3.2.1 3.2.2 3.2.3 3.3.1 3.3.2 3.3.3 1 Introduction and summary Two-dimensional supersymmetric gauge theories have a wide range of applications in physics and geometry. In eld theory, 2d gauged linear sigma-models in N = (2; 2) superspace serve as canonical examples which provide very useful insights on low-energy dynamics of four-dimensional supersymmetric gauge theories [2, 3]. Geometrically, twodimensional non-linear sigma-models with extended supersymmetry appear very reach because of existence of numerous twisted-chiral multiplets [4] which possess no analogs in generalized Kahler geometry. It is natural to expect that some of these geometries may arise as low-energy e ective actions in two-dimensional gauge theories in N = (2; 2) superspace. The study of low-energy e ective action in Abelian gauge theories in N = (2; 2) superspace was initiated long ago [2]. The authors of this work showed that the eld strength of N by = (2; 2) vector multiplet is given by a twisted chiral super eld which we denote throughout this work. The e ective action for may have a superpotential W ( ) and a Kahler potential K( ; ), as well as higher-derivative terms which form together the Euler-Heisenberg-type e ective action in two-dimensional SQED. The structure of one-loop quantum corrections to these potentials was found in [2]: W (1)( ) / ln K(1)( ; ) / ln ln : ; (1.1) (1.2) HJEP07(21)46 At leading order, one can discard higher-derivative terms in the e ective action and treat the low-energy theory as a (2; 2) sigma-model with the Kahler potential (1.2) and superpotential (1.1). The superpotential (1.1) is known to be one-loop exact and its form is completely determined by the anomaly of U(1) U(1) R-symmetry [2]. The Kahler potential can, however, receive higher-loop quantum corrections. This paper aims to trigger the study of quantum corrections to the e ective Kahler potential K( ; ) and corresponding sigmamodel geometry beyond one-loop order. We consider two-loop e ective action in N = (2; 2) and (4; 4) SQED on the Coulomb branch. In the (2; 2) case, the Coulomb branch is known to exist only when the U(1) charges of chiral multiplets sum to zero [3, 5]. This is typically satis ed for the SQED with two chiral multiplets which carry opposite charges with respect to the gauge group. For this theory we explicitly compute two-loop quantum corrections to the e ective Kahler potential K(2)( ; ). An important feature of two-dimensional gauge theories is that Feynman graphs with internal (super)photon lines su er from IR divergencies. We show that for supersymmetric gauge theories in the N = (2; 2) superspace it is possible to introduce gauge invariant mass term for the vector multiplet which naturally regulates such IR divergencies. This mass term may be obtained by the dimensional reduction from the three-dimensional (super) Chern-Simons action which is also known to be responsible for the gauge-invariant mass of the vector multiplet in three dimensions. In our case, the two-loop quantum corrections to the e ective action explicitly depend on the vector multiplet mass and are singular in the limit when this mass vanishes. The (4; 4) vector multiplet in the N = (2; 2) superspace is described by the pair ( ; ) where is a twisted chiral multiplet [4]. At leading order in the derivative expansion, the low-energy e ective action in the N = (4; 4) SQED is described by a generalized Kahler potential K( ; ; ; ). Performing explicit quantum computations we demonstrate that this potential does not receive two-loop quantum corrections and is one-loop exact. At one-loop order, this function coincides with the potential for the (4; 4) { 2 { sigma-model with torsion studied in [1]1 1 4 loop exact and quantizes (see, e.g., [9]). This con rms the non-renormalization of the potential (1.3) claimed in [10]. Qualitatively, the presence of the Wess-Zumino term in the low-energy e ective action out the massive chiral multiplets and consider e ective action for the light vector multiplet. However, the total contribution to the anomaly should be the same at low and high energies since the anomaly cannot depend on the energy scale. Thus, the low-energy e ective action must include the Wess-Zumino term compensating the contribution to the anomaly from the fermions that were integrated out. This statement is well known as the 't Hooft anomaly matching argument [11]. It is pertinent to mention here the amazing analogy of the e ective potentials (1.1), (1.2) and (1.3) with certain terms in low-energy e ective actions of fourdimensional N = 2 and N = 4 gauge theories. Recall that the 4d N = 2 gauge multiplet may be described by an N = 2 chiral super eld W. The superpotential (1.1) is somewhat similar to the so-called holomorphic potential [ 12 ] F (W) / W2 ln W while the Kahler potential (1.2) formally coincides with the non-holomorphic potential [13] H(W; W) / ln W ln W. This analogy is not accidental: both F (W) and the superpotential (1.1) appear as a result of integration of the anomaly of U(1) R-symmetry (see [ 12 ] and [2], correspondingly). Surprisingly, the potential (1.3) nicely correlates with the lowenergy e ective action of 4d N = 4 SYM e ective action in the N = 2 superspace [14]. Indeed, the rst term in the right-hand side of (1.3) formally coincides with the nonholomorphic potential H(W; W) while the last term in (1.3) is very similar to the hypermultiplet completion of the non-holomorphic potential that was constructed in [14]. This analogy is even more striking. Indeed, in [15] it was demonstrated that the lowenergy e ective action in N = 4 SYM theory contains the Wess-Zumino term for scalar elds which originates from the 't Hooft anomaly matching for the R-symmetry. This Wess-Zumino term implies the non-renormalization of the coe cient in front of the nonholomorphic potential beyond one loop. As we show in this paper, the potential (1.3) is also responsible for the Wess-Zumino term for two-dimensional scalars, and exactly the same arguments provide its non-renormalization. One of the results of this paper is the illustration of the deep interplay between the two-dimensional N = (4; 4) SQED and 4d N = 4 SYM theory at low energies, although they are very di erent in general. 1This sigma-model can be considered as a particular case of the N = (4; 4) super-Liouville theory which a short review of the gauge theory in N = (2; 2) superspace and consider basic properties of the parallel displacement propagator which is a key ingredient of the technique of gauge-covariant perturbative computations (for 4d gauge theories in N = 1 superspace this technique was developed in [16{18] and for eld theory on the supergravity background in [19{22]). Making use of this propagator, we construct exact Green's functions for chiral super elds on covariantly constant vector multiplet background. In section 3, we compute the low-energy e ective action in N = (2; 2) SQED with di erent numbers of chiral multiplets. We start with a review of old results [2] of one-loop quantum contributions to the e ective action and show how they can be naturally reproduced by taking advantage of the technique of covariant perturbative computations in the N = (2; 2) superspace. This technique is then applied to compute two-loop quantum corrections to the e ective action of N = (2; 2) SQED on the Coulomb branch. In section 4, we study the structure of low-energy e ective action in N = (4; 4) SQED to the two-loop order in perturbation theory and discuss its interplay with the 4d N = 2 and N = 4 SYM e ective actions. The Conclusions section is devoted to discussions of possible extensions of the results of this work. In appendix we give a summary of our superspace conventions. 2 Exact propagators on constant vector multiplet background In this section we consider two-dimensional non-Abelian gauge theory in N = (2; 2) superspace and, following [18], we introduce parallel displacement propagator which is a key ingredient of gauge-covariant technique of multi-loop quantum computations. Using the properties of this propagator we construct exact heat kernels for basic Green's functions on covariantly constant vector multiplet background. In the Abelian case, we apply these heat kernels in the subsequent sections to compute two-loop quantum correction to the e ective action. We hope that the results of this section will be of use also for the study of e ective action in non-Abelian gauge theories which will be considered elsewhere. We keep the structure of this section close to the corresponding presentation in [23] to facilitate the comparison with the three-dimensional gauge theory in N = 2 superspace. 2.1 Gauge theory in N = (2; 2) superspace We consider the two-dimensional N = (2; 2) superspace with coordinates zA = (xm; ; ), where xm, m = 0; 1, are the Minkowski space coordinates, = 1; 2, are Grassmann coordinates and = ( ) are their complex conjugate. Our superspace conventions are summarized in appendix. They are chosen to be close to the ones employed in the series of papers [23{27] devoted to the study of super eld theories in three-dimensional N = 2 superspace. The (non-Abelian) gauge theory in the N = (2; 2) superspace is described by the set of gauge-covariant superspace derivatives rA = (rm; r ; r ) = DA + VA ; (2.1) { 4 { where DA = (@m; D ; D ) are super-covariant derivatives, see (A.3), and VA = (Vm; V ; V ) are gauge connections subject to the constraints fr ; r g = [r ; rm] = 2i( m ) ( m) [rm; rn] = iFmn : rm + 2i" G + 2 3 H ; W ; [r ; rm] = ( m) W ; Here G, H, W and Fmn are super eld strengths with the following conjugation properties Gy = G ; Hy = H ; (W )y = W ; (Fmn)y = Fmn : HJEP07(21)46 In its turn, the tensor eld strength Fmn is expressed via W and W , Fmn "mnf = 1 4 "mn( 3) (r W r W ) : Another important relation appears by commuting (2.2a) with the super eld G and applying properties (2.5) transformations r W + r W = 2i m rmG + 2 3 [H; G] + " r W : The algebra of covariant derivatives (2.2) is invariant under the -gauge rA ! ei (z) rAe i (z) ; with (z) being real gauge super eld parameter, y = . The gauge connections VA may be expressed via a prepotential. In this paper we will use the real super eld prepotential V which is introduced in such a way that the gauge-covariant spinor derivatives acquire the form (chiral representation) In two dimensions, the antisymmetric tensor Fmn has only one independent component, Fmn = "mnf , where for the antisymmetric "-tensor "mn = "nm with vector indices we use the convention "01 = r W = r W : r = e 2V D e2V ; r = D : { 5 { (2.2a) (2.2b) (2.2c) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) i 4 1 4 In this case, as a consequence of (2.2a), the scalar super eld strengths are expressed via the prepotential as G = D (e 2V D e2V ) ; H = The expressions of the other super eld strengths in terms of V can be obtained using (2.5) and (2.8). Note that all these super eld strengths transform covariantly under the -gauge transformation of the prepotential e2V ! ei e2V e i ; with a chiral . The super eld strengths G and H can be considered as the real and imaginary parts of a complex super eld and its (Hermitian) conjugate = G + iH ; = G iH : From (2.5) it is easy to deduce twisted chirality properties of these super elds r+ = r HJEP07(21)46 (2.14) (2.15) where (r+; r ) (r1; r2). The existence of such twisted chiral super elds is an important feature of two-dimensional gauge theory in superspace as compared with the higherdimensional cases. These super eld strengths play central role in super eld description of gauge theories in the N = (2; 2) superspace. 2.2 Parallel displacement propagator in N = (2; 2) superspace In superspace, the parallel displacement propagator was introduced in the work [18] as a key ingredient which provides correct transformation properties of Green's functions and corresponding heat kernels under gauge transformations. This allowed the authors of [18] to develop a gauge-covariant procedure of perturbative computations of e ective actions in supersymmetric gauge theories. In particular, this technique appeared very fruitful in the study of low-energy e ective actions in di erent four-dimensional gauge theories in N = 1 and N = 2 superspaces [28{34]. For three-dimensional gauge theories this method was generalized in [23, 27]. This section is aimed at extending the basic concepts of the procedure of covariant perturbative computations to two-dimensional gauge theories in the N = (2; 2) superspace. Let us consider a super eld in some representation R of the gauge group, and its Hermitian conjugate transforming in the representation R, (z) ! 0(z) = ei (z) (z) ; (z) ! 0(z) = (z)e i (z) ; where = y is Hermitian, but otherwise arbitrary gauge super eld parameter. Correspondingly, Green's function for these elds G(z; z0) = ih (z) (z0)i has the transformation property G(z; z0) ! ei (z)G(z; z0)e i (z0) : (2.16) (2.17) { 6 { In a similar way, the parallel displacement propagator I(z; z0) is, by de nition, a twopoint superspace function which transforms under the gauge group as Moreover, it is required to obey the di erential equation and the boundary condition I(z; z0) ! ei (z)I(z; z0)e i (z0) : A rAI(z; z0) = A(DA + VA)I(z; z0) = 0 ; The latter means that at coincident superspace points I(z; z0) reduces to the identity operator in the gauge group. In eq. (2.19), A interval with the components ( m; ; ) is the N = (2; 2) supersymmetric m = (x x0)m i( 0) m 0 + i 0 m ( 0) ; = ( 0) ; = ( It is possible to show that the properties (2.18) and (2.20) imply the important relation I(z; z) = 1 : The algebra of covariant derivatives (2.2) can be represented in the condensed form [rA; rBg = TABC rC + iFAB ; where TABC is the supertorsion and FAB is the eld strength for gauge super eld connection (2.1). The non-vanishing components of these tensors can be read o from (2.2). They appear in the following important relation for the derivative of the parallel displacement propagator [18] rBI(z; z0) = i X 1 ( 1)n n=1 (n + 1)! An : : : A1 rA1 : : : rAn 1 FAnB(z) + (n 2 1) An TAnB C An 1 : : : A1 rA1 : : : rAn 2 FAn 1C (z) I(z; z0) : This identity shows that any covariant derivative of the parallel displacement propagator may be expressed in terms of the parallel displacement propagator itself and covariant derivatives of the super eld strength together with the torsion tensor. This identity appears crucial in perturbative computations of low-energy e ective action which is a functional of these tensors. In general, (2.25) is an in nite series over covariant derivatives of the eld strength FAB. It is natural to expect that for certain eld con gurations this series terminates. { 7 { In particular, it is possible to show that for the covariantly constant vector multiplet the identity (2.25) reduces to rm = rm rmW = rmW Green's function can be represented as a proper-time integral of the corresponding heat kernel Kv(z; z0js) i Z 1 0 As we will show in the following subsections, these identities appear very useful in computing heat kernels of Green's functions of various operators in the N = (2; 2) superspace. 2.3 Real super eld Green's function and its heat kernel The real super eld d'Alembertian is de ned by either expression v = = 1 1 By making use of the algebra (2.2), this operator may be brought to the form Green's function Gv(z; z0) of this operator is de ned as a solution of the equation where m is a mass parameter and 2j4(z z0) is the full superspace delta function, ( v + m2)Gv(z; z0) = 2j4(z z0) ; 2j4(z z0) = 2(x x0) 4( 0) : where ! +0 implements standard boundary condition for the propagator. The equation for the propagator (2.30) is satis ed when the heat kernel obeys the conditions (i d ds v)Kv(z; z0js) = 0 ; s!0 lim Kv(z; z0js) = 2j4(z z0) : (2.33) In general, it is very hard to solve these equations explicitly. Nevertheless, it is possible to nd the exact solution for the heat kernel when the background gauge super eld obeys the following two constraints: i) Gauge multiplet obeys super Yang-Mills equations of motion (on-shell background) HJEP07(21)46 ii) Field strengths are covariantly constant r W = 0 ; rm = rm rmW = rmW = 0 : W (s) (s) (s) m(s) O(s)W O( s) = W (esN ) ; O(s) O(s) O( s) = O( s) = O(s) mO( s) = m + i( m ) + W ((esN 1)N 1 ) W ((e sN 1)N 1 ) ; ; dt W (t) (t) + W (t) (t) ; It is important to note that the compatibility condition for the constraint (2.35) requires that the background gauge super eld belongs to the Cartan subalgebra of the Lie algebra of the gauge group. This means that the background gauge super elds are (anti)commuting. The procedure of solving the heat kernel equation for the covariantly constant vector multiplet background was developed in the four-dimensional case in [17{20] and successfully applied for three-dimensional gauge theories in [23{25]. In the two-dimensional case the same procedure yields where 2 = , 2 = Kv(z; z0js) = 1 sf and O(s) is the `shift' operator O(s) = es(W r W r ) : Within quantum loop computations, it is often necessary to know the value of the heat kernel at coincident superspace points. For this aim, it is useful to have such a representation for the heat kernel (2.36) where the operator O(s) appears on the right and hits the parallel displacement propagator, Kv(z; z0js) = 1 sf e 4i (f coth sf) m(s) m(s) 2(s) 2(s)I(z; z0js) : Here the operator O(s) is used to de ne the s-dependent super eld strengths and components of the superspace interval { 9 { where and (z; z0) solves for In formulas (2.39) we have introduced the notation The s-dependent parallel displacement propagator (2.40) can be represented in the Making use of (2.27a) and (2.27b) we nd (z; z0) = i( W + W )( G + i( 3) H) ih (z) T(z0)i = ( + + m2)G+(z; z0) = ( + m2)G (z; z0) = +2j2(z; z0) ; (2.40) (2.41) (2.42) (2.43) (2.44) (2.46) (2.47a) (2.47b) (2.48a) (2.48b) (z; z0js) can be found from the above formula just by replacing all super eld strengths and components of the superspace interval by the corresponding sdependent quantities from (2.39). 2.4 Heat kernel for chiral super eld Green's function Consider gauge-covariant chiral super eld , r = 0, and its Hermitian conjugate . The d'Alembertian operators acting in the space of such elds are de ned in the standard way + = 1 16 r r sentations for these operators + = = 1 1 16 r r 2 2 = r 16 r r 2 2 = r m m rm + rm + 1 2 1 2 i i f ; g + 2 (r W ) + iW f ; g 2j2(z; z0) are (anti)chiral delta-functions which are related to the full superspace For Green's functions (2.48) there are the associated heat kernels G (z; z0) = ds K (z; z0js)e s( +im2) ; ! +0 : It is known [18, 28] that for the on-shell vector multiplet background (2.34) the chiral It should be noted that the identities (2.53) hold only for the on-shell vector multiplet background (2.34). The equations (2.51) imply similar relations for the corresponding heat kernels K+(z; z0js) = 41 r2Kv(z; z0js) ; K (z; z0js) = 14 r2Kv(z; z0js) : Thus, the computation of the heat kernels K is reduced to nding the result of the action of the operators r 2 and r 2 on the heat kernel (2.36). It is possible to show that upon acting by r 2 2I(z; z0) since the factor in front of this function originates from e is v . The latter operator commutes with r2 owing to the identities (2.53). Thus, for K+ we have K+(z; z0js) = 1 sf Applying (2.27b) we compute the action of the operator r 2 on the parallel displacement propagator 2 on (2.36), this operator hits only 1 2r2( 2I(z; z0)) = 2 e 21 ( m) m W I(z; z0) : It is easy to check these relations using the identities G+(z; z0) = 41 r2Gv(z; z0) ; G (z; z0) = 41 r2Gv(z; z0) : r 2 + = r 2 r 2 + = v = 2 r ; e is e 4i (f coth sf) m(s) m(s) 12 ( m) m(s) (s)W (s) Substituting this identity into (2.55) we nd K+(z; z0js) = e is e 4i (f coth sf) m(s) m(s) 12 ( m) m(s) (s)W (s) Here we pushed the operator O(s) through on the right that resulted in making all objects s-dependent according to (2.39) and (2.40). In a similar way we nd the antichiral heat kernel K (z; z0js) = (2.50) (2.51) (2.52) (2.53) (2.54) (2.56) (2.57) (2.58) We point out that the expressions for the (anti)chiral heat kernels are very similar to the ones in the four-dimensional supersymmetric gauge theory [28]. Finally, we consider the propagators among chiral and antichiral super elds With Green's functions (2.59) are associated the corresponding heat kernels 41 r2G+ (z; z0) + m2G (z; z0) = 41 r2G +(z; z0) + m2G+(z; z0) = ds K+ (z; z0js)e s( +im2) ; ds K +(z; z0js)e s( +im2) : This subsection aims to nd explicit solutions for these heat kernels on the covariantly constant vector multiplet background. First of all, we point out that, as a consequence of the de nitions of covariantly (anti)chiral d'Alembertian operators (2.47), Green's functions (2.59) are related to the (anti)chiral ones (2.48) as G+ (z; z0) = 41 r2G (z; z0) ; G +(z; z0) = Analogous relations hold for the corresponding heat kernels Thus, the problem is reduced to nding the action of the operators r2 and r2 on the heat kernels (2.57) and (2.58). Let us consider the derivation of the heat kernel K+ in some details. It appears upon acting by the operator r operator commutes with the expression e is from e is v . Thus, we need to properties of the parallel displacement operator (2.27b) 2 on the heat kernel (2.58). Note that, owing to (2.53), this e 4i (f coth sf) m m since the latter originates nd the action of this operator on the rest using the 4 r 1 2 e 21 ( m) m W 2I(z; z0) = eR(z;z0)I(z; z0) ; (2.59) (2.60a) (2.60b) (2.61a) (2.61b) (2.62) (2.63) (2.64) (2.65) where R(z; z0) = i i 2 6 G + ( 3) H W + ( m) ~ m 1 2 1 2 ( m) ~m( r W : W + W ) + W 2i 2 3 is a modi cation of the supersymmetric interval which is chiral with respect to the rst argument and antichiral with respect to the other ~ m = m + i m D0 ~m = D ~ Given the function R(z; z0) in the form (2.65), we have the following representation for the heat kernel K+ HJEP07(21)46 As the nal step, in (2.68) we have to push the operator O(s) through on the right and hit the parallel displacement propagator according to eq. (2.40). This procedure e ectively makes the super eld strengths and components of supersymmetric interval s-dependent according to (2.39) We point out that, as follows from (2.65), the function R(z; z0) vanishes at coincident Grassmann coordinates, R(z; z0)j !0 ! 0. However, the contribution from (2.71) is nontrivial at coincident points. 3 3.1 Low-energy e ective action in N = (2; 2) SQED General remarks In general, Abelian gauge theories in N = (2; 2) superspace may include the following terms in the classical action: where R(z; z0js) = O(s)R(z; z0)O( s), and (s) is given by (2.45). For practical computations, it is useful to represent the heat kernel (2.69) in the equivalent form Here, the function R0(t) can be found explicitly from (2.65) using R0(t) = O(t)[W r r ; R]O( t) and combined with (2.45): R0(t) + (t) = O(t) 2i W G 2( 3) W H + W W 5i 3 i 2W 2 + 2i( 3) ( 3) W W +2i W W 11i 2 12 6 1 2 r W 2 + ( m) ~ m r W 2 O( t) : (2.71) The kinetic term for the vector multiplet V SV = 1 Z 2e2 d2j4z : Here e is the dimensional gauge coupling, [e] = 1, and d2j4z is the measure in the full N = (2; 2) superspace (see appendix for our superspace conventions). The mass term for the vector multiplet Sm = i m Z where the integration goes over the twisted chiral subspace and m is, in general, complex mass parameter. Without loss of generality, we can set it to be real, m = m, just to simplify some formulas below. Note that the sum of actions (3.1) and (3.2) amounts to the massive Wess-Zumino model for the twisted chiral multiplet . It should be noted that the mass term (3.2) may be obtained by the dimensional reduction from the 3d N = 2 Chern-Simons action which plays role of the topological mass term in three-dimensional electrodynamics. Fayet-Iliopoulos (FI) term where SFI = In (3.3), the real part of the FI parameter r couples with the auxiliary eld D of the vector multiplet while the imaginary part corresponds to the topological f -term and quantizes [3, 35]. N charged chiral multiplets Qi with charges qi and mass matrix mij SQ = d2j4z Qie2qiV Qi d2j2z mij QiQj + c:c: : (3.5) XN Z i=1 N X i;j=1 Z Needless to say that the mass matrix mij should be such that the gauge invariance is preserved. Chiral multiplets may also have real mass which originates from vevs of scalars in the vector multiplet V . More generally, it is also possible to study quantum dynamics of twisted chiral multiplets as well as semi-chiral ones [4, 8, 36, 37], but such models are beyond the scope of this paper. Depending on the number of chiral multiplets and on the values of all mentioned above parameters, Abelian gauge theories in the N = (2; 2) superspace exhibit di erent phases which are thoroughly investigated in [3]. In this paper, we are interested in the e ective action on the Coulomb branch. It is known that the necessary condition for existence of the Coulomb branch at the quantum level is that the charges of all chiral multiplets should sum to zero N Indeed, when this condition is not satis ed, the following two e ects occur: i) There are UV-divergent tadpole Feynman graphs which result in the renormalization of the FI parameter. These quantum corrections lift the Coulomb branch. ii) The e ective twisted superpotential for the super eld strength is generated at one loop [2]. This e ective superpotential may also be interpreted as a functional reproducing correct transformation properties of the e ective action under anomalous R-symmetry. Correspondingly, when the condition (3.6) is satis ed, there are no divergent quantum contributions to the FI parameter and classical Coulomb branch is preserved at the quantum level. The latter case is of primary importance for our studies as we are interested in the two-loop quantum contributions to the e ective action on the Coulomb branch. However, in this section, for the sake of completeness we will shortly consider a model for which the condition (3.6) is not satis ed and will give a super eld derivation of the e ective twisted superpotential obtained originally in [2] by component eld quantum computations. The typical example of the models for which the constraint (3.6) is violated is the supersymmetric electrodynamics with one chiral multiplet while the well-known case when this constraint is satis ed is the supersymmetric electrodynamics with two chiral multiplets carrying opposite charges under the U(1) gauge symmetry. The latter will be studied in section 3.3 while the former is considered just below. 3.2 3.2.1 SQED with one chiral avor Classical action In this section, we consider the supersymmetric electrodynamics with one chiral multiplet carrying charge +1 S = Z d2j4z 1 2e2 Qe2V Q m 2e2 Obviously, in the limit e ! 1 the classical action becomes scale invariant and superconformal, though this symmetry is known to be broken by quantum corrections [2]. Recall that the N = (2; 2) vector multiplet contains a complex scalar associated with the lowest component of the super eld i Z 2 j ; ' m e2 Re where the bar-projection means vanishing -variables. Denoting the scalar elds in the chiral multiplet by it is not hard to nd the scalar potential which appears after elimination of auxiliary elds e 2 2 ' Qj ; Qj ; 2 V = '' t + '' : (3.6) (3.8) (3.9) (3.10) The Coulomb branch is parametrized by the vev of the scalar eld in the vector multiplet while the scalars from the chiral multiplet must have vanishing vevs Coulomb branch: h i = const; h'i = 0 : The vanishing of the scalar potential (3.10) for such values of scalars is possible only for special value of the FI parameter t = m The even part of the one-loop e ective action may be found by evaluating trace of logarithm of square of this operator Here we have taken into account the de nition (2.46) of the chiral covariant d'Alembertian in terms of covariant spinor derivatives. Associated with this operator is the Green function G+(z; z0) de ned in (2.48a) and the corresponding heat kernel K+(z; z0js), see eq. (2.50). Thus, for the e ective action (3.16) we have the following proper-time representation where Treatment of these parts in the e ective action requires slightly di erent computational methods. Therefore, we will consider them separately. 3.2.2 Even part of the one-loop e ective action Let H be the operator which appears in the matrix of second variational derivatives of S with respect to the chiral super elds, In this case the chiral multiplet acquires real mass proportional to the vev of while the vector multiplet (`photon') has a small mass m. Naively, one could study the e ective action for the vector multiplet which appears by integrating out the massive chiral multiplet. However, the constraint (3.12) appears to be ruined by one-loop quantum corrections and the Coulomb branch is lifted at the quantum level [3]. Although this scenario is well-known, we will demonstrate it explicitly by computing one-loop e ective action in the model (3.7). The details of these computations will be of use in subsequent sections. In general, the e ective action for the vector multiplet V may have odd and even parts with respect to the re ection V ! V , H = i 4 0 (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) 1 4 0 where Tr+K+(s) means the heat kernel K+(z; z0js) at coincident superspace points, z0 = z, and integrated over the chiral subspace Z Tr+ K+(s) = d2j2z K+(z; zjs) : This reduces the problem of computation of the even part of the e ective action to evaluating the limit of coincident superspace points for the heat kernel, limz0!z K+(z; z0js). Recall that we consider the low-energy e ective action for the on-shell, constant vector multiplet background speci ed by the constraints (2.34) and (2.35). For this background, the heat kernel K+ was found in the form (2.57). This formula involves di erent HJEP07(21)46 s-dependent objets de ned in (2.39) and (2.40). For the one-loop e ective action we need the values of these objets at coincident superspace points when all components of the superspace interval vanish, A ! 0. In particular, simple calculations yield 2(s) A=0 = s2W 2 sinh2 sf 2 ; (sf =2)2 where f is the component of the super eld strength tensor, Fmn = "mnf , which can be regarded as a constant for the considered background. It is important to note that the formula (3.19) contains W 2 that prevents any other contributions from the other sdepended objets in (2.57). Thus, this kernel acquires simple form at coincident superspace points K+(z; zjs) = sW 2e is Substituting this expression into (3.17) we nd the even part of the one-loop e ective action (3.18) (3.19) (3.20) (3.21) even = It is instructive to rewrite the functional (3.21) in the full superspace The term in the rst line here speci es the e ective Kahler potential for the twisted chiral super eld . The term in the last line in (3.22) takes into account all higher-derivative corrections with respect to the gauge super eld which can be considered as the EulerHeisenberg e ective action. We point out that the e ective action (3.22) was found for the rst time in [2] using component led one-loop computations and in [38] by means of super eld methods. Here we just gave a derivation of this e ective action by taking advantage of the super eld heat kernel technique. This result will be useful in the study of low-energy e ective action in the model with two chiral avors which will be considered in section 3.3. background hJ i = 2hQe2V Qi = 2iG+ (z; z) ; where hJ i is the e ective current. In the one-loop approximation, this e ective current receives contributions only from the chiral super eld propagator in the vector multiplet HJEP07(21)46 Thus, the computation of (3.23) is reduced to nding the trace of the heat kernel K+ (z; z0js). The problem of evaluating the trace of the heat kernel K+ (z; z0js) is rather technically involved. However, to nd the odd part of the e ective action we don't actually need to now the full expression for K+ (z; zjs). Indeed, the full expression for K+ (z; zjs) contains di erent terms which are responsible both for odd and even parts of the e ective action. Since the even part of the e ective action has been fully studied in the previous subsection, here we have to focus only on possible contributions to odd from K+ (z; zjs). For this goal it is su cient to approximate K+ (z; zjs) by the terms with no derivatives of , K+ (z; zjs) e is : Substituting (3.26) into (3.25) we have UV-divergent integral over the proper time s. Introducing a small regularization parameter this integral may be evaluated hJ i = 1 2 1 + ln( ) + O( ) ; where is the Euler-Mascheroni constant. Thus, we see that the odd part of the e ective action is the sum of divergent and nite contributions odd = div + n : The divergent part of the e ective action can be immediately read o from the rst The odd part of the e ective action cannot be found upon squaring of the operator (3.15). Instead, to catch up the odd contributions we have to consider the general variation of the e ective action with respect to the vector multiplet = Z d2j4z V hJ i ; where G+ (z; z0) is de ned in eq. (2.60a), with the mass parameter set to zero, m = 0. It is useful also to represent this e ective current via the heat kernel K+ using (2.61a) Z 1 0 hJ i = 2 ds K+ (z; zjs) : term in (3.27) of the FI parameter (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) 1 4 s 1 2 8 div = 2 1 Z d2j4z V = d2j2z~ + c:c: This expression, being added to the classical action (3.7), leads to in nite renormalization This means that even if we switch o the FI parameter classically, it is always generated by one-loop divergent tadpole diagram. This is the origin of lifting of the classical Coulomb branch by quantum corrections advocated in [3]. Substituting (3.27) into (3.23) we get the variation of nite terms in the odd part of the e ective action Integrating this variation we uncover the e ective twisted potential for n = 2 d2j4z V ln( ) : n = i Z 4 d2j2z~ (ln 1) + c:c: div = n = 8 i i 4 N X q i Z i=1 N X qi ln qi i=1 d2j2z~ This e ective twisted potential was found originally in [2] using component eld quantum computations. Here we reproduced the same result using the method of covariant perturbative computations in the N = (2; 2) superspace. The above results can be readily generalized to the case of electrodynamics with N chiral avors Qi, i = 1; : : : ; N , with charges qi, see eq. (3.5). For the odd part of the e ective action we have the following modi cation of formulas (3.29) and (3.32): This is the su cient condition of existence of the Coulomb branch on the quantum level [5]. To summarize, we have shown that the FI parameter in the model (3.7) receives in nite one-loop quantum contributions (3.30) which lift the classical Coulomb branch. Such in nite contributions may cancel among each other in the generalization of the model (3.7) which involves N charged chiral avors (3.5). This happens when all charges of chiral The equation (3.33) implies that there is no in nite renormalization of the FI parameter when the condition (3.6) is satis ed. This is the necessary condition for existence of the Coulomb branch. This condition is also su cient for vanishing of the e ective twisted potential in (3.34). However, even when the condition (3.6) is satis ed, the rst term in the right-hand side of (3.34) remains non-vanishing and yields a nite shift of the complex FI parameter The main e ect of this nite quantum contribution is the shift of the imaginary part of the complex FI parameter (3.4). To compensate this shift, one has to add the corresponding value to the classical FI parameter t ! t0 = t + 1 2 N i=1 X qi ln qi : t = m e2 hRe i 1 2 N i=1 X qi ln qi : (3.31) (3.32) multiplets sum to zero (3.6). However, there is still a nite shift of the imaginary part of the FI parameter as in (3.35). Therefore, quantum Coulomb branch exists when the classical FI parameter is tuned to a special value (3.36). Since we are interested in the e ective action in the Coulomb branch, in subsequent sections we will always assume that the condition (3.36) is satis ed. We stress that all results of this subsection are not new; they are well-known owing to [2, 3, 5]. Here we just summarized them for the sake of completeness of our consideration. Let us consider supersymmetric electrodynamics with two chiral multiplets Q+ and Q carrying charges 1, respectively, HJEP07(21)46 S = Sgauge[V ] + Smat[Q; V ] ; Sgauge[V ] = Smat[Q; V ] = 2e2 Z i Z 2 d2j4z where m is the mass of the chiral multiplet while m is the vector multiplet mass. The latter is assumed to be small as compared to the former, m 2 m2 + h i : In this regime, we can study the e ective action for the light eld which appears upon integrating out the heavy chirals Q . In what follows, without loss of generality we will assume that both m and m are real, though, in general, they may be complex. Let ' be scalar elds in the chiral multiplets (3.37a) (3.37b) (3.38) (3.39) (3.40) (3.41) ' = Q j ; ' = Q j : After elimination of auxiliary elds, one can readily nd the scalar potential e 2 2 V = + m2)('+'+ + ' ' ) : m e2 Re )2 + ( Similarly as in the model (3.7), the classical Coulomb branch (3.11) is possible at the special value of the FI parameter (3.12). However, as is explained in the previous section, the imaginary part of the FI parameter receives nite one-loop contributions as in eq. (3.35). To compensate this contribution, we have to set up the corresponding value to the classical FI parameter This allows us to study the low-energy e ective action for the light vector multiplet which appears by integrating out heavy chiral multiplets beyond one-loop order. t = i 2 m In the framework of the background eld method, we split the gauge super eld V into background V and quantum v parts2 Upon this splitting, the actions (3.37a) and (3.37c) decompose as are covariantly (anti)chiral super elds with respect to the background V ! V + e v : 1 8 Sgauge[V ] ! Sgauge[V ] + d2j4z v(iD W + 2m + 2e2t) + Z d2j4z v D D2D + imD D v ; Smat[Q; V ] ! Smat[Q; v] ; Q+ = Q+e2V ; Q+ = Q+ ; Q = Q e 2V ; Q = Q : The operators D D2D and D D in (3.43a) are degenerate and require gauge xing. The gauge xing is implemented by adding to the action (3.43a) the following term Z Sgf = d2j4z v 1 16 fD2; D2g + im 4 (D2 + D2) v : This gauge- xing action appears upon inserting the standard delta-functions [f [f iD2v] into the functional integral over Dv and averaging them with appropriate weight (see [39] for details of this procedure in the three-dimensional case). After gauge xing, we get the action for `quantum' elds Squant = S2 + Sint ; S2 = where This operator obeys the important property ; d2j4z [e(Q+Q+ Q Q )v + e2(Q+Q+ + Q Q )v2] + O(e3) ; im 4 H = (2D D + D2 + D2) : H2 = m (3.42) (3.43a) (3.43b) (3.44) (3.45) iD2v] (3.46a) (3.46b) (3.46c) (3.47) (3.48) HJEP07(21)46 This identity allows us to represent the propagator for the super eld v in the form 2The background gauge super eld is denoted by the same letter as the original super eld V . This should not lead to any confusions as the original unsplit gauge super eld does not show up after the backgroundquantum splitting. 2ihv(z)v(z0)i (1) = iTr ln( + + m2) ; Z sections. 3.3.2 One-loop e ective action important features. to the re ection V ! ihQ+(z)Q (z0)i = are de ned by the equations (2.48a) and (2.60a), respectively. Using the form of cubic and quartic interaction vertices for quantum elds in (3.46c), we deduce the formal decomposition of the e ective action up to two-loop order where Here m, of chiral super elds K0(z; z0js) = (3.50) and are the components of the supersymmetric interval (2.21). In addition to the photon propagator (3.49), the action (3.46b) yields the propagators (2) = 2e2 d2j4zd2j4z0[G+ (z; z0)G+ (z0; z) + m2G+(z; z0)G (z; z0)]Gv(z; z0) : (3.52c) Here (1) is the one-loop e ective action while (2) takes into account two-loop quantum corrections. These quantum contributions will be calculated separately in the subsequent The computation of the one-loop e ective action in the model (3.37) is very similar to the one for SQED with one chiral avor considered in section 3.2. However, it has some First of all, the e ective action in the model (3.37) possesses no odd part with respect V . As is demonstrated in section 3.2.3, the odd contributions to the e ective action cancel against each other in the model where the charges of avors sum to zero, (3.6). Thus, we have to focus only on the even part of the one-loop e ective action. The computation of the even part of the e ective action goes along the same lines as in section 3.2.2. Following these steps, one arrives at the expression (3.21), with two simple modi cations: (i) The result (3.21) should be multiplied by 2 as we have contributions from two chiral avors now; (ii) the mass parameter m should be inserted, (1) = It is an instructive exercise to rewrite the functional (3.53) in the full superspace. We give the details of this procedure for the chiral part of (3.53); the antichiral part can be analyzed in the same way. (3.51) HJEP07(21)46 (3.52a) (3.52b) At the rst step, we identically rewrite the chiral part of (3.53) as the sum of two terms = i Z i Z 8 8 i Z 8 where, in the last line, we have inserted the unity, 1 = 4f12 D2W 2. In this identity, the operator D2 can be used to restore the full superspace measure due to (A.8). Then, after evaluation of the proper-time integral in the second line of (3.54), we have = + i Z 8 8 d2j2z sf =2 1 : (3.55) Next, we have to restore the full superspace measure in the rst line of (3.55) using the operators D from W = D G = i( 3) D H, see (2.5). Making use of properties of the super eld strengths (2.4) and (2.5), one can prove the identity Z Z d2j4z F (X) = d2j2z W 2[(X m2)F 00(X) + F 0(X)] ; for some function F (X) and X the rst line of (3.55), one nds the following di erential equation for this function + m2. Comparing the right-hand side of (3.56) with with the general solution (X m2)F 00(X) + F 0(X) = F (X) = c1 + c2 ln(X m2) + 1 2 ln2 X m2 m2 m2 X m2 1 X ; + Li2 where c1 and c2 are arbitrary constants of integration. The terms with these constants drop out upon integration over the full superspace owing to the properties (2.14) and (2.15). The remaining two terms in (3.58) allow us to get the full-superspace representation for the rst term in (3.55) 1 Z 8 d2j2z W 2 + m2 = 8 1 Z Note that the last term in (3.59) vanishes in the limit m = 0 owing to the identity Li2(0) = 0. In this limit, the expression (3.59) coincides with the non-holomorphic potential in (3.22). : (3.56) (3.57) (3.58) (3.59) i Z 4 + Li2 sf =2 1 : (3.60) The term in the rst line here can be interpreted as the one-loop quantum correction to the e ective Kahler potential for the twisted chiral super eld K(1) = ln ln + Li2 m2 : The second line in (3.60) is responsible for higher-derivative corrections in the one-loop Euler-Heisenberg-type action. Two-loop e ective Kahler potential In principle, starting from (3.52c) it is possible to determine two-loop quantum corrections to the Euler-Heisenberg-type action.3 However, the form of the resulting expression appears not very illuminating as it involves numerous proper-time integrations and may have very limited applications. Therefore, in this section we restrict ourself to studying two-loop quantum corrections only to the e ective Kahler potential for the twisted chiral super eld . To this aim, it is su cient to consider the gauge super eld background constrained by W = 0 ; W = 0 ; while super elds and are constant and non-vanishing. In this approximation, the heat kernels (2.57) and (2.68) reduce to Recall that we considered here the chiral part of (3.53). It can be shown that the antichiral part gives the same contribution as (3.55), so that (1) = 2 . Thus, substituting (3.59) into (3.55), we end up with the representation for the one-loop e ective action in the full superspace action was studied in [40] up to the two-loop order. We point out that beyond one loop this e ective action in the supersymmetric QED cannot be found by simple composition of non-supersymmetric results in scalar and spinor electrodynamics. The two-loop e ective action (3.52c) is given by the sum of two terms, which we denote by A and B, respectively, K+(z; z0js) K+ (z; z0js) 1 4 s 1 4 s e is e i 42s 2I(z; z0) ; 2 e is e i 4s I(z; z0) : (2) = A = B = A + 2e2 Z B ; Z d2j4zd2j4z0 G+ (z; z0)G+ (z0; z)Gv(z; z0) ; 2e2m2 d2j4zd2j4z0 G+(z; z0)G (z; z0)Gv(z; z0) : (3.61) (3.62) (3.63) (3.64) (3.65a) (3.65b) (3.65c) Q+ v Q u + u v Type A v u Q These two terms correspond to the Feynman graphs of types A and B in gure 1. They have slightly di erent structure and need to be considered separately. Note that the twoloop graph of the topology `8' vanishes identically and, thus, does not show up in (3.65). It is possible to show that the part of e ective action (3.65c) does not contribute to the e ective Kahler potential. In this formula, we express Green's functions via the corresponding heat kernels (2.50) and (3.49) Z B = 2ie2m2 ds dt du e i(s+t)m2 K+(z; z0js)K (z; z0jt) K0(z; z0ju) : The operator H in the last line contains the covariant spinor derivatives (see eq. (3.47)), which can be integrated by parts B = 2ie2m2 Z where we have taken into account the explicit form of the heat kernel (3.50). The last line in (3.67) contains the terms of the following three types: K+(z; z0js)K (z; z0jt) K+(z; z0js)r2K (z; z0jt) !0 ; !0 ; (3.66) (3.67) (3.68a) (3.68b) (3.68c) K+(z; z0js)K (z; z0jt) r2K+(z; z0js)K (z; z0jt) !0 ; !0 r K+(z; z0js)r K (z; z0jt) ; !0 : The terms involving (3.68a) cannot contribute to the e ective Kahler potential since the expression (3.20) vanishes in the approximation (3.62). For the same reason there are no contributions from the terms (3.68b). In a similar way it is easy to argue that the expressions (3.68c) cannot contribute to the e ective Kahler potential owing to the properties r K+(z; z0js)j !0 / W ; r K (z; z0jt)j !0 / W : (3.69) These properties follow from the explicit form of the heat kernels (2.57) and (2.58). Thus, in the approximation (3.62) Now let us consider the contributions to the e ective Kahler potential from the part of the e ective action (3.65b). Making use of the identities (2.61) and (3.49), this e ective action can be cast to the form (3.70) Z A = 2ie2 d2j4z d2j4z0 It is easy to argue that the terms with the operator H in the last line of (3.71) give no contributions to the e ective Kahler potential. Indeed, covariant spinor derivatives in this operator can hit the heat kernels yielding the terms r2K+ (z; z0js)K +(z; z0jt) r K+ (z; z0js)r K +(z; z0jt) !0 ; !0 : For such terms one can derive the following identities K+ (z; z0js)r2K +(z; z0jt) !0 ; 41 r2K+ (z; z0js) 41 r2K +(z; z0js) !0 !0 = i = i d ds d ds K (z; z0js) r K+ (z; z0js)j !0 / W ; r K +(z; z0jt)j !0 / W : Thus, contributions from these terms to the e ective action vanish in the approximation (3.62). Non-trivial contributions to (3.71) appear only from the terms without the operator H A = ie2 Z d2j4z d2 Z 1 ds dt du 0 s t u e i(s+t)(m2+ )e ium2 e i42 (s 1+t 1+u 1) : Here, the Gaussian integral over d2 can be easily evaluated A = ie2 Z Finally, it is possible to perform integration over one of the parameters, say u, and to represent the corresponding contribution to the e ective action in the form s + t e i(s+t)(m2+ )ei ms2+stt E1 im2s t s + t ; (3.72) (3.73) (3.74) (3.75) (3.76a) (3.76b) where E1(z) is the exponential integral E1(z) = Z 1 dt 1 t e tz : The expression (3.76b) represents the two-loop quantum correction to the e ective Kahler potential for the twisted chiral super eld . This formula involves integration over the parameters s and t which are hard to evaluate for generic values of masses m and m. However, it is possible to nd explicitly the leading contributions to the e ective Kahler potential for small photon mass, i.e., in the regime (3.38). In this case, we can use the asymptotics of the function E1(ix) for small x, The integrals over s and t reduce to where a = i(m2 + ). This yields the simple expression for (3.76b) This formula is a good approximation for the two-loop quantum correction to the e ective Kahler potential for in the regime (3.38), i.e., when the photon possesses a small but non-vanishing mass m. Obviously, (3.80) is singular in the limit m ! 0. It emphasizes a feature of two-dimensional electrodynamics that the quantum loop diagrams with internal photon lines su er from IR singularities unless the photon possesses a mass. In conclusion of this section, let us consider the full e ective Kahler potential K( ; ) which contains the classical part 21e2 corrections (3.61) and (3.80) and includes both one- and two-loop quantum K( ; ) = + ln ln + 1 2e2 + e 2 4 2 m2 + 8 2 m2 + + e 2 1 4 Li2 1 m2 ln m2 + m2 : : (3.77) (3.78) (3.79) (3.80) (3.81) (3.82) The corresponding sigma-model metric reads 1 2e2 + 1 1 4 zz + m2 e 2 m2 e 2 zz m2 8 2 (m2 + zz)3 + 8 2 (m2 + zz)3 ln m2 + zz m2 dzdz : For vanishing mass of the chiral multiplet, m = 0, this metric acquires a simple form ds2jm=0 = 1 2e2 + 1 1 4 zz e 2 1 + 8 2 (zz)2 ln zz m2 dzdz : (3.83) Z 1 ds dt 0 s + t E1(ix) = ln x + O(x) : Z 1 ds dt s + t e (s+t)a ln 0 e 2 i 2 e (s+t)a = s t s + t + = e 2 1 a ; 2 + a 1 1 a ln ln a ; m2 + m2 1 1 4 1 We stress that this metric makes sense for a small but non-vanishing photon mass m. The two-loop Kahler potential (3.81) and the corresponding metric (3.82) are new results obtained here by direct quantum computations in the N = (2; 2) superspace. Though the one-loop quantum corrections to this metric were found long ago in [2], to the best of our knowledge the two-loop quantum corrections have never been presented before. 4 Low-energy e ective action in N = (4; 4) SQED Classical action and loop expansion of the e ective action HJEP07(21)46 The (4; 4) vector multiplet may be described by the N = (2; 2) vector multiplet V and a chiral multiplet . The hypermultiplet is described by the pair of chiral elds (Q+; Q ). Let us consider the following action for these multiplets S = Sgauge[V; ] + Smat[Q; V; ] ; Sguage[V; ] = 2e2 it Z 2 Z im 4e2 ) Z Z Z + d2j2z 2 d2j2z~ 2 For m = 0 this action is invariant under `hidden' (2; 2) supersymmetry with anticommuting parameters and (4.1a) (4.1b) V = 1 2 = i = W ; ) ; Q ) ; Q = i = W ; 1 2 4 r ( Q ) ; (4.2) where Q are as in (3.44). For non-vanishing photon mass, m 6= 0, the action (4.1) is invariant under (4.2) only for the real supersymmetry parameter = . This means that for generic m the model (4.1) describes the N = (3; 3) supersymmetric electrodynamics while for m = 0 the supersymmetry extends up to N = (4; 4). This scenario is completely analogous to the three-dimensional N = 4 electrodynamics which can have only reduced N = 3 supersymmetry when the topological Chern-Simons mass term is turned on [41, 42]. In our case, in (4.1) we keep non-vanishing photon mass m in order to get rid of IR singularities of Feynman graphs beyond one loop. The one-loop contributions to the e ective action, however, are independent of m and have the same form for both N = (3; 3) and N = (4; 4) cases. For quantizing the theory, we perform the background-quantum splitting V ! V + e v ; + e ; (4.3) while the hypermultiplet (Q+; Q ) is considered as the `quantum' super eld which will be integrated out in the path integral. The background gauge super eld V is constrained by (2.34) and (2.35) while is simply constant D = 0 ; D = 0 : After adding the gauge xing term (3.45), the `quantum' elds are described by the action Squant = S2 + Sint ; S2 = d2j2z im 2 4 Q+ Q d2j4z[e(Q+Q+ Q Q )v + e2(Q+Q+ + Q Q )v2] + e Z + O(e3) : As compared with (3.46), in (4.5) there are two essential modi cations: (i) in (4.5b) we have the background chiral super eld in place of the mass m; (ii) in the last line in (4.5c) there are two additional vertices with the quantum chiral super eld and its conjugate . Taking these features into account, one can readily generalize the N = (2; 2) e ective action (3.52a) to the N = (4; 4) (or, rather, N = (3; 3)) case where = ) ; (2) = A + B + C + D ; A = B = D = 2e2 2e2 Z Z Z ie2m Z + 4 ie2m Z 4 C = 2e2 d2j4zd2j4z0 G+ (z; z0)G+ (z; z0)G0(z; z0) ; d2j4zd2j4z0 G+ (z; z0)G+ (z0; z)Gv(z; z0) ; d2j2zd2j2z0 2 G+(z; z0)G+(z0; z)D2G0(z; z0) d2j2zd2j2z0 2 G (z; z0)G (z0; z)D2G0(z; z0) : Here Gv(z; z0) is given by (3.49) while G0(z; z0) is simply G0(z; z0) = 2j4(z z0) = 1 + m2 i Z 1 0 ds e ism2 K0(z; z0js) : Below, we compute separately the one- and two-loop contributions to the e ective action (4.6). (4.4) (4.5a) (4.5b) (4.5c) (4.6a) (4.6b) (4.6c) (4.7a) (4.7b) (4.7c) (4.7d) (4.8) Recall that we consider the approximation (4.4) which means that we discard any derivatives of (anti)chiral super eld ( ). In this case the procedure of computation of the one-loop e ective action (4.6b) is exactly the same as in section 3.3.2 for the N = (2; 2) SQED. Thus, we can readily generalize the result (3.53) to the case of N = (4; 4) SQED (1) = i Z 8 + ) tanh(sf =2) sf =2 (4.9) The e ective action (4.9) is represented as a functional in (anti)chiral superspace. It HJEP07(21)46 is instructive to rewrite it in the full N = (2; 2) superspace. Following the same procedure as in section 3.3.2, we nd 4 (1) = + Li2 i Z 4 + ) W 2W 2 f 2 sf =2 (4.10) The terms in the rst line in (4.10) are leading in the derivative expansion of the e ective action while the terms in the second line correspond to higher-derivative corrections. The leading terms Z 1 4 leading = d2j4z K(1)( ; ; ; ) ; deserve several comments. First of all, we point out the similarity of the super eld expression (4.11) with the lowenergy e ective action of four-dimensional N = 4 SYM theory in N = 2 superspace which was constructed in [14]. Indeed, (4.11b) contains the term ln ln which is analogous to the non-holomorphic potential for N = 2 4d super eld strength while the other terms are very similar to the hypermultiplet completion of the non-holomorphic potential which was constructed in [14]. Surprisingly, such terms in 2d and 4d cases are described by the same Li2 function and have very similar form although they are given in very di erent superspaces and for di erent models. Recall that the N = 4 susy-complete e ective action in 4d N = 4 SYM theory was derived originally in [14] by imposing the requirement of invariance under hidden supersymmetry while in subsequent works this e ective action was found by direct quantum computations in superspace [43{45] (see also [46] for a review). In our case, we obtained (4.11) as the leading part of the one-loop e ective action in N = (4; 4) SQED although originally it was found in [1] as a susy completion of the non-holomorphic potential (1.2). The mentioned above similarity of (4.11) with the low-energy e ective action in 4d N = 4 SYM theory is even deeper. As was demonstrated in [47, 48] (see also [49] for a review), the structure of leading terms in the low-energy e ective action in N = 4 SYM theory can be recovered from the fact that it contains the Wess-Zumino term for scalar elds. This Wess-Zumino term is known to appear in the low-energy theory as a result of 't Hooft anomaly matching for SU(4) R-symmetry of N = 4 SYM theory [15]. Surprisingly, the e ective action (4.11) may be given exactly the same interpretation. Indeed, in [1] the action of the form (4.11) was proposed as a super eld generalization of a two-dimensional sigma-model with the Wess-Zumino term. In our case, the appearance of this term in the low-energy e ective action is well understood. Classically, the N = (4; 4) electrodynamics (4.1) respects the SU(2) SU(2) symmetry which is the R-symmetry of N = (4; 4) Poincare superalgebra. However, because of 't Hooft anomaly, this symmetry cannot be realized explicitly in the low-energy theory but is still the symmetry of the e ective Lagrangian up to full derivative terms. Recall that the e ective action (4.11) is obtained upon integrating out the hypermultiplet (Q+; Q ) which contains chiral fermions with respect to the R-symmetry group. Thus, in the low-energy theory the Wess-Zumino term must appear as a response to the change of the number of chiral fermions since the total contribution to the anomaly should be the same regardless of the energy scale. This is the essence of the 't Hooft anomaly matching argument [11]. Let us derive the Wess-Zumino term for scalar elds from the super eld action (4.11). The scalars appear in the component eld expansion of and as follows + i i 1 2 2 4 + : : : ; 1 2 2 4 + : : : ; where dots stand for other component elds. Substituting these expressions into (4.11) and integrating over the Grassmann variables one readily nds in the component eld expansion the Wess-Zumino term for the scalar elds (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) where and are phases of the complex scalars and SWZ = = j jei ; + = j jei : SWZ = (XAXA)2 { 31 { The action (4.14) is explicitly invariant under U(1) U(1) symmetry which shifts the phases and . This symmetry is the subgroup of the full SU(2) SU(2) R-symmetry group of the theory. It is possible to show that (4.14) is implicitly invariant under SU(2) SU(2) ' SO(4) since this is the symmetry of the Wess-Zumino term modulo total derivative terms. To show this, let us introduce real scalars XA = (X1; X2; X3; X4) which transform as a vector under SO(4) space for details) = X1 + iX2 ; = X3 + iX4 : Then, the action (4.14) can be rewritten in the form of integral over a three-dimensional which has standard 2d Minkowski space as its boundary, @ = R1;1 (see e.g. [47] and h i . Type C Q u Q u + u Q+ Q Type D u Q+ where "ABCD and "mnp are antisymmetric tensors. The Wess-Zumino term in the form (4.17) has explicit SO(4) symmetry. One can reverse the arguments: once we know that the Wess-Zumino term (4.17) appears in the low-energy e ective action of N = (4; 4) SQED, we can immediately nd (4.11) as its supersymmetric generalization. However, performing perturbative quantum computations we uncover not only the leading term (4.11) in the low-energy e ective action, but also higher-derivative corrections which are encoded in the second line of (4.10). We point out once more that (4.14) is explicitly invariant under U(1) U(1) ' SO(2) SO(2) which is one of the maximal subgroups of the full R-symmetry group SO(4). However, there are two more inequivalent maximal subgroups: SO(3) ' SU(2) and SU(2) U(1). We speculate that these subgroups may be made manifest in other supereld descriptions of the N = (4; 4) gauge theory such as the harmonic superspace [50{54]. Recall that in the 4d N = 4 SYM theory the careful account of all maximal subgroups of the SU(4) R-symmetry group resulted in di erent but equivalent super eld descriptions of the low-energy e ective action [47{49, 55]. It is tempting to develop similar ideas for the low-energy e ective action in 2d supersymmetric gauge theories. 4.3 Vanishing of two-loop corrections to generalized Kahler potential In the previous section we computed one-loop e ective action which contains the term (4.11) as the leading part in the derivative expansion. In [10] it was claimed that this potential is non-renormalized by higher-loop quantum corrections. This section aims to demonstrate explicitly that two-loop quantum corrections to the generalized Kahler potential (4.11b) cancel among each other. In the two-loop expansion of e ective action (4.6c), the terms A and B can be represented by Feynman graphs which have the same structure as those in the N = (2; 2) SQED given in gure 1. The terms C and D are new since they involve the propagators for the (anti)chiral super eld ( ). These terms are represented by the Feynman graphs in gure 2. To nd the contributions to the e ective action from these terms it is su cient to consider the vector multiplet background constrained by (3.62) and (4.4). The details of computations of contributions to the e ective action (4.7a) and (4.7b) are exactly the same as those in section 3.3.3. We can immediately generalize the results (3.70) and (3.75) to the N = (4; 4) case A = ie2 Z e ium2 e i(s+t)( + It is easy to argue that the contribution to the e ective action (4.7d) vanishes. Indeed, the propagator G0 (4.8) contains the delta-function which implies that we need to consider at coincident points. As follows from (3.20), K+(z; z0js) D = 0 : It remains to consider the contribution C to the e ective action. Substituting here the heat kernels K+ and K K+(z0; zjt)j =0 = 0. Thus, the propagators (3.64) and (4.8) we have Z 1 0 0 s t u 2ie2 Z (4 )3 C = 2ie2 )e i 4 (s 1+t 1+u 1)e ium2 : (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) 1 2e2 K( ; ; ; ) = + Li2 (4.24) After integration over d2 it becomes evident that this expression contributes to the e ective action with the opposite sign to (4.18) C = ie2 Z 8 2 st + su + tu e ium2 e i(s+t)( + Thus, we conclude that the sum of the terms (4.18), (4.19), (4.20) and (4.22) vanishes (2) = A + B + C + D = 0 : We stress that this does not mean that the complete two-loop e ective action vanishes, but just implies that there are no two-loop quantum corrections to (4.11b). The non-renormalization of the generalized Kahler potential K( ; ; ; ) in (4,4) gauge theories was claimed in [10]. In this section, we have explicitly demonstrated the absence of two-loop quantum corrections to this potential. There are also purely eldtheoretical arguments that all higher-loop quantum correction to this function vanish and K( ; ; ; ) is one-loop exact. Indeed, in the previous section it was demonstrated that this potential is responsible for the Wess-Zumino term for scalar elds (4.14). It is wellknown that appearance of Wess-Zumino terms in low-energy e ective action is strictly one-loop e ect associated with the 't Hooft anomaly matching [11]. The form of the WessZumino term (4.17) as well as the coe cient in front of this action are rigidly xed by topological arguments (see e.g. [9]). Therefore, the function (4.11b) cannot receive any higher-loop corrections, and we end up with the exact result + d d ) : (4.25) As is demonstrated in [4], this sigma-model corresponds to the generalized Kahler geometry with N = (4; 4) extended supersymmetry. This geometry possesses non-trivial torsion which can also be read o from (4.11b). Here we have demonstrated that this generalized Kahler potential naturally arises as the leading term in the low-energy e ective action in N = (4; 4) gauge theory on the Coulomb branch. 1 2e2 + 1 + 5 In this paper, we have studied two-loop quantum corrections to the low-energy e ective actions in the N = (2; 2) and N = (4; 4) SQED. On the Coulomb branch, leading terms in the e ective action in N = (2; 2) SQED are represented by a superpotential and a Kahler potential for super eld strengths described by a twisted chiral super eld . Although, at one-loop order, these potentials were studied long ago [2], to the best of our knowledge the two-loop quantum corrections to the e ective Kahler potential (3.80) have not been presented before. The corresponding sigma-model metric in the two-loop approximation is given by (3.83). We point out that this metric depends on the vector multiplet mass m and is singular in the limit m ! 0. This is a new feature of the two-dimensional case as compared with the low-energy e ective action of three-dimensional [23] and four-dimensional [28, 34] SQED where it was well-de ned for massless vector multiplet. In the N = (4; 4) SQED, the leading part of the low-energy e ective action is described by the generalized Kahler potential K( ; ; ; ) where is a chiral and is a twisted chiral super elds. We show that this potential is one-loop exact and is given by (4.24). This potential was introduced for the rst time in [1] in the study of two-dimensional sigma models with torsion which originates from the Wess-Zumino term for scalar elds. In our case, we demonstrate that the Wess-Zumino term is the integral part of the low-energy e ective action associated with the 't Hooft anomaly matching for the SU(2) SU(2) Rsymmetry. The from of the potential K appears surprisingly similar to the low-energy e ective action in 4d N = 4 SYM theory obtained in [14]. We have studied two-loop e ective action in two-dimensional SQED on the Coulomb branch. In is also very interesting to investigate the structure of the e ective action on the Higgs branch. Some of the leading terms in this e ective action were discussed in the recent work [56] but the form of higher-loop quantum correction is unknown. Another interesting problem is the study of quantum aspects of two-dimensional gauge theories with semichiral multiplets [8, 36, 37]. Only some limited results in this direction are available [57], but the structure of quantum corrections to the generalized Kahler potential remains unknown. Acknowledgments I am very grateful to E.A. Ivanov for useful discussions. I acknowledge the support from the RFBR grant No 15-02-06670. may be 0 = i 2, 1 = the Cli ord algebra In this paper, we use the two-dimensional N = (2; 2) superspace which appears by the dimensional reduction from the three-dimensional N = 2 superspace or from the fourdimensional standard N = 1 superspace. Therefore, we employ the super eld notation and conventions which are very close to the ones used in the series of papers [23{27] devoted to the study of the low-energy e ective actions in three-dimensional super eld theories. The 2d N = (2; 2) superspace is parametrized by the coordinates zA = (xm; ; ), where xm = (x0; x1) are the Minkowski space coordinates and = ( 1; 2) are Grassmann coordinates ( = ( ) are their complex conjugate). The spinor indices are raised and lowered by means of the antisymmetric "-tensor, = " , = " , "12 = "21 = 1. One of the possible choices for the two-dimensional gamma-matrices ( m) = ( m) 1, where i are the Pauli matrices. The gamma-matrices obey f m ; n g = 2 mn12 2 ; mn = diag(1; 1) ; and possess the following orthogonality and completeness relations ( m ) ( n) = 2 mn ; ( m ) ( m) = " " + ( 3) ( 3) ; where 3 = 1 0 is the chirality matrix. The chiral projectors are P The covariant spinor derivatives may be chosen in the form = 12 (1 3). with the following anti-commutation relation + i ( m i ( m fD ; D g = The integration measure in the full N = (2; 2) superspace is de ned as d2x f (x) = d2j4z 2 2f (x) ; for some eld f (x). Here we adopt the following conventions for contractions of spinor indices D2 = D D ; D2 = D D ; 2 = 2 = The chiral and antichiral subspaces are parametrized by the coordinates z+ = (x+m; ) and z = (xm; ), correspondingly, where xm = xm i m The integration measure in the chiral subspace d2j2z d2xd2 is related to the full superspace measure (A.5) as d2j4z = 4 1 d2j2zD2 = 4 1 d2j2zD2 ; so that Z Z d2x f (x) = d2j2z 2f (x+) : (A.8) (A.1) (A.3) (A.4) (A.5) (A.6) (A.7) Given a two-component spinor , we can consider each of its components independently as they are Lorentz-covariant and appear as the P projections of , ( +; ) ; = P : In a similar way, for a spinor with the upper spinor index, , we have ( + ; ; +). As a consequence, it is possible to introduce twisted chiral coordinates z~+ = ) and the twisted antichiral ones z~ = (x~m; ; +), where x~m = xm i( m The integration measures over these subspaces are denoted by d2j2z~ and d2j2z~. 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I. B. Samsonov. Low-energy effective action in two-dimensional SQED: a two-loop analysis, Journal of High Energy Physics, 2017, 146, DOI: 10.1007/JHEP07(2017)146