Superfield Research Articles

Quantum properties of gauge theories with extended supersymmetry formulated in N=1 superspace

We analyse quantum properties of ${\cal N}=2$ and ${\cal N}=4$ supersymmetric gauge theories formulated in terms of ${\cal N}=1$ superfields and investigate the conditions imposed on a renormalization prescription under which the non-renormalization theorems are valid. For this purpose in these models we calculate the two-loop contributions to the anomalous dimensions of all chiral matter superfields and the three-loop contributions to the $\beta$-functions for an arbitrary ${\cal N}=1$ supersymmetric subtraction scheme supplementing the higher covariant derivative regularization. We demonstrate that, in general, the results do not vanish due to the scheme dependence, which becomes essential in the considered approximations. However, the two-loop anomalous dimensions vanish if a subtraction scheme is compatible with the structure of quantum corrections and does not break the relation between the Yukawa and gauge couplings which follows from ${\cal N}=2$ supersymmetry. Nevertheless, even under these conditions the three-loop contribution to the $\beta$-function does not in general vanishes for ${\cal N}=2$ supersymmetric theories. To obtain the purely one-loop $\beta$-function, one should also chose an NSVZ renormalization prescription. The similar statements for the higher loop contributions are proved in all orders.

On higher-spin mathcal{N} = 2 supercurrent multiplets

We elaborate on the structure of higher-spin mathcal{N} = 2 supercurrent multiplets in four dimensions. It is shown that associated with every conformal supercurrent {J}_{alpha (m)overset{cdot }{alpha }(n)} (with m, n non-negative integers) is a descendant {J}_{alpha left(m+1right)overset{cdot }{alpha}left(n+1right)}^{ij} with the following properties: (a) it is a linear multiplet with respect to its SU(2) indices, that is {D}_{beta}^{Big(i}{J}_{alpha left(m+1right)overset{cdot }{alpha}left(n+1right)}^{jkBig)}=0 and {overline{D}}_{dot{beta}}^{Big(i}{J}_{alpha left(m+1right)overset{cdot }{alpha}left(n+1right)}^{jkBig)}=0 ; and (b) it is conserved, {partial}^{beta overset{cdot }{beta }}{J}_{beta alpha (m)overset{cdot }{beta}overset{cdot }{alpha }(n)}^{ij}=0 . Realisations of the conformal supercurrents {J}_{alpha (s)overset{cdot }{alpha }(s)} , with s = 0, 1, …, are naturally provided by a massless hypermultiplet and a vector multiplet. It turns out that such supercurrents and their linear descendants {J}_{alpha left(s+1right)overset{cdot }{alpha}left(s+1right)}^{ij} do not occur in the harmonic-superspace framework recently described by Buchbinder, Ivanov and Zaigraev. Making use of a massive hypermultiplet, we derive non-conformal higher-spin mathcal{N} = 2 supercurrent multiplets. Additionally, we derive the higher symmetries of the kinetic operators for both a massive and massless hypermultiplet. Building on this analysis, we sketch the construction of higher-derivative gauge transformations for the off-shell arctic multiplet Υ(1), which are expected to be vital in the framework of consistent interactions between Υ(1) and superconformal higher-spin gauge multiplets.

Constructing operator basis in supersymmetry: a Hilbert series approach

In this paper we introduce a Hilbert series approach to build the operator basis for a N = 1 supersymmetry theory with chiral superfields. We give explicitly the form of the corrections that remove redundancies due to the equations of motion and integration by parts. In addition, we derive the maps between the correction spaces. This technique allows us to calculate the number of independent operators involving chiral and antichiral superfields to arbitrarily high mass dimension. Using this method, we give several illustrative examples.

On generalised D1-D5 near horizons and their spectra

There has been considerable recent interest in type II string compactifications on AdS3×S3×M4 due to both new progress in the AdS3/CFT2 correspondence and an independent search for new mathcal{N} = (4, 0) supergravity solutions. In this work we report a new family of small mathcal{N} = (4, 0) preserving warped AdS3×S3×CY2 compactifications of type IIB supergravity. The S3 in this family is non trivially fibered over the CY2. We show how these new solutions provide a IIB embedding of minimal ungauged d = 5 supergravity coupled to an Abelian vector multiplet. We also make a few initial steps towards the identification of dual CFT2 candidates. Specifically, we study the Kaluza-Klein spectrum of the spin two and certain vector fluctuations that are expected to be dual to the stress-energy tensor, SU(2) R current and several other BPS operator families in the dual CFT2.

More on massive gravitino scattering amplitudes and the unitarity cutoff of the new Fayet-Iliopoulos terms

We extend the 2 → 2 gravitino scattering amplitude computed in [1] to an arbitrary mathcal{N} = 1 supergravity model of one chiral and one vector multiplet, in a Minkowski background with supersymmetry breaking driven by both F- and D-terms. We find that the cancellation of the leading term in mathcal{O} (κ2E4/|m3/2|2), that would lead to a breakdown of perturbative unitarity at a scale Λ ~ MSUSY, is a consequence of the vanishing of the scalar potential at its minimum, which is implied by the flat background. We then analyse the inclusion of the new Fayet-Iliopoulos (FI) terms. We find that, since they modify the scalar potential without contributing to the amplitudes, they generically lead to uncanceled leading terms in the latter and a perturbative cutoff at the supersymmetry breaking scale, except for particular cases where the new FI term does not modify the potential at its minimum and the cutoff is pushed up to the Planck scale.

Linearizing the BPS equations with vector and tensor multiplets

We analyse the BPS equations of mathcal{N} = (1, 0) supergravity theory in six dimensions coupled to a vector and tensor multiplet. We show how these BPS equations can be reduced to a set of linear differential equations. This system is triangular in that each layer of equations, while linear, is quadratically sourced by the solutions of the previous layers. We examine several explicit examples and discuss the construction of new families of microstate geometries. We expect that the result presented here will open up new branches of superstrata in which the momentum is encoded in a new class of charge carriers.

On unfolded approach to off-shell supersymmetric models

We construct and analyze unfolded off-shell systems for chiral and vector supermultiplets using multispinor formalism and external currents. We find that auxiliary variables of multispinor formalism allow for the interesting reorganization of the unfolded off-shell modules: one can unify dynamical and auxiliary scalars in the Wess-Zumino model, as well as an auxiliary pseudoscalar and a zero-helicity subsector in the vector supermultiplet, so that the resulting unfolded equations get simplified, while various constraints like chirality, electric current conservation and P-oddness of the pseudoscalar can be formulated entirely in terms of these auxiliary variables.

AdS_5 vacua and holographic RG flows from 5D N=4 gauged supergravity

We study five-dimensional N=4 gauged supergravity coupled to five vector multiplets with SO(2)_Dtimes SO(3)times SO(3) gauge group. There are four supersymmetric AdS_5 vacua in the truncation to SO(2)_{text {diag}} invariant scalars. Two of these vacua preserve the full N=4 supersymmetry with SO(2)_Dtimes SO(3)times SO(3) and SO(2)_Dtimes SO(3)_{text {diag}} symmetries. These have an analogue in N=4 gauged supergravity with SO(2)times SO(3)times SO(3) gauge group. The other two AdS_5 vacua preserve only N=2 supersymmetry with SO(2)_{text {diag}}times SO(3) and SO(2)_{text {diag}} symmetry. The former has an analogue in the previous study of SO(2)_Dtimes SO(3) gauge group while the latter is a genuinely new N=2AdS_5 vacuum. These vacua should be dual to N=2 and N=1 superconformal field theories (SCFTs) in four dimensions with different flavour symmetries. We give the full scalar mass spectra at all of the AdS_5 critical points which provide information on conformal dimensions of the dual operators. Finally, we study holographic RG flows interpolating between these AdS_5 vacua and find a new class of solutions. In addition to the RG flows from the trivial SO(2)_Dtimes SO(3)times SO(3)N=4 critical point, at the origin of the scalar manifold, to all the other critical points, there is a family of RG flows from the trivial N=4 critical point to the new SO(2)_{text {diag}}N=2 critical point that pass arbitrarily close to the SO(2)_Dtimes SO(3)_{text {diag}}N=4 critical point.

Minimal Nambu-Goldstone-Higgs model in supersymmetric SU(5) revisited

We revisit the minimal Nambu-Goldstone (NG) Higgs supersymmetric (SUSY) SU(5) grand unified model and study its phenomenological implications. The Higgs sector of the model possesses a global SU(6) symmetry, which is spontaneously broken and results in the Higgs doublets of the minimal SUSY Standard Model (MSSM) as NG chiral superfields. Therefore, the model naturally leads to light Higgs doublets and solves the doublet-triplet splitting problem. Because of the SU(6) symmetry, the couplings of the Higgs sector are tightly restricted, and thus the model is more predictive than the minimal SUSY SU(5). We determine all the grand-unified-theory parameters via the matching conditions of the gauge coupling constants at the unification scale and calculate proton lifetime, confronting this with current experimental bounds. We discuss that this model is incompatible with the constrained MSSM, whilst it has a large viable parameter space in the high-scale SUSY scenario. The perturbativity condition on the trilinear coupling of the adjoint Higgs field imposes an upper (lower) limit on the wino (gluino) mass, implying a hierarchical mass pattern for these gauginos. Future proton-decay searches can probe a large part of the parameter space, especially if the SUSY-breaking scale is $\lesssim 100$~TeV.

Hyperdilaton Weyl multiplets of 5D and 6D minimal conformal supergravity

By extending the recent analysis [G. Gold, S. Khandelwal, W. Kitchin, and G. Tartaglino-Mazzucchelli, J. High Energy Phys. 09 (2022) 016] for $\mathcal{N}=2$ conformal supergravity in four dimensions, we define new hyperdilaton Weyl multiplets for five-dimensional $\mathcal{N}=1$ and six-dimensional $\mathcal{N}=(1,0)$ conformal supergravities. These are constructed by coupling the five- and six-dimensional standard Weyl multiplets to on-shell hypermultiplets and reinterpreting the systems as new multiplets of conformal supergravity. In the five-dimensional case, we also construct a new hyperdilaton Poincar\'e supergravity by coupling to an off-shell vector multiplet compensator. As in four dimensions, a $BF$ coupling induces a nontrivial scalar potential for the five-dimensional dilaton that admits ${\mathrm{AdS}}_{5}$ vacua.

Higher derivative invariants in four dimensional \U0001d4a9 = 3 Poincaré supergravity

In this paper we use the superconformal approach to derive the higher derivative action for \U0001d4a9 = 3 Poincaré supergravity in four space-time dimensions. We first study the coupling of \U0001d4a9 = 3 vector multiplets to conformal supergravity. Thereafter we combine it with the pure \U0001d4a9 = 3 conformal supergravity action and use a minimum of three vector multiplets as compensators to arrive at Poincaré supergravity with higher derivative corrections. We give a general prescription on how to eliminate the auxiliary fields in an iterative manner and obtain the supergravity action order by order in derivatives. We also show that the truncation of the action at fourth order in derivatives is a consistent truncation.

Maximally symmetric nuts in 4d \U0001d4a9 = 2 higher derivative supergravity

We initiate a systematic study of supersymmetric backgrounds in 4d \U0001d4a9 = 2 Euclidean supergravity in the presence of infinite towers of higher derivative corrections. Adopting a Gibbons-Hawking view towards the evaluation of the action in terms of nuts and bolts, we consider the two maximally symmetric vacua ℝ4 and ℍ4 (Euclidean AdS4) and their unique supersymmetric deformations with (anti-) self-dual Maxwell tensors corresponding to a single nut at the center. These are the Omega background of Nekrasov-Okounkov, Ωℝ4, and its generalization with a cosmological constant of Martelli-Passias-Sparks, denoted Ωℍ4 (also known as the gravity dual of the U(1) × U(1) squashed sphere). We write down the BPS configurations in the superconformal formalism in the presence of vector multiplets and derive the corresponding off- and on-shell actions. Our results provide a rigorous proof for important parts of the conjecture in [1] and its holographic corollary in [2], which we discuss in detail along with extensions such as the addition of hypermultiplets and the presence of conical defects.

On extended supersymmetry of 4d Galileons and 3-brane effective actions

We use on-shell amplitude methods to systematically analyze the possibility of extended supersymmetry for 4d Galileon models, expanding on previous mathcal{N} = 1 results. Assuming spins ≤ 1, we prove that there exists no mathcal{N} = 4 supersymmetric extension of 4d Galileons with a single vector multiplet. Thus the Galileons cannot be part of the effective action of a single flat maximally supersymmetric D3-brane, and that explains why such terms do not appear in the α′-expansion of the abelian open superstring amplitude. For mathcal{N} = 2 Galileons, we show that the complex scalar Z = ϕ + iχ of the vector supermultiplet cannot have ϕ and χ both enjoy enhanced shift symmetry; instead, χ can at best be an R-axion with constant shift symmetry. Using the soft bootstrap, we demonstrate that the quartic DBI-Galileon is incompatible with mathcal{N} = 2 supersymmetry. A similar analysis performed at 7-point shows that a 2-parameter family of mathcal{N} = 2 supersymmetric quintic Galileons coupled with DBI passes the soft bootstrap. Finally, we show how supersymmetric couplings between Galileons and gravitons arise in generalizations of our constructions, and we conclude with a discussion of Galileons and DBI-Galileons in the context of UV-completability vs. the Swampland.

Functional renormalization group flows of mathcal {N}=1 supersymmetric abelian gauge model with one chiral and one vector superfield

We apply the functional renormalization group approach to a mathcal {N}=1 supersymmetric gauge model with one chiral superfield coupled to a vector U(1) superfield. We find that the nonrenormalization theorem still works at leading order in the supercovariant derivative expansion of the fields. We also find the beta functions and we study the behavior of its fixed points in the local potential approximation. Regulators are also discussed.

Exact strong coupling results in mathcal{N} = 2 Sp(2N) superconformal gauge theory from localization

We apply the localization technique to compute the free energy on four-sphere and the circular BPS Wilson loop in the four-dimensional mathcal{N} = ∈ superconformal Sp(2N) gauge theory containing vector multiplet coupled to four hypermultiplets in fundamental representation and one hypermultiplet in rank-2 antisymmetric representation. This theory is unique among similar mathcal{N} = ∈ superconformal models that are planar-equivalent to mathcal{N} = ∆ SYM in that the corresponding localization matrix model has the interaction potential containing single-trace terms only. We exploit this property to show that, to any order in large N expansion and an arbitrary ’t Hooft coupling λ, the free energy and the Wilson loop satisfy Toda lattice equations. Solving these equations at strong coupling, we find remarkably simple expressions for these observables which include all corrections in 1/N and 1/ sqrt{lambda } . We also compute the leading exponentially suppressed mathcal{O}left({e}^{-sqrt{lambda }}right) corrections and consider a generalization to the case when the fundamental hypermultiplets have a non-zero mass. The string theory dual of this mathcal{N} = ∈ gauge theory should be a particular orientifold of AdS5 × S5 type IIB string and we discuss the string theory interpretation of the obtained strong-coupling results.

Simplified models of vector SU(4) leptoquarks at the TeV

Assuming confirmation of the anomalies in semi-leptonic B-decays, their explanation in terms of the exchange of a massive vector leptoquark field, U_{mu }^a, of charge 2/3, appears to require the inclusion of U_{mu }^a in the vector multiplet of the adjoint of Pati–Salam SU(4), {mathcal {G}}_{mu }^A, as well as the introduction of vector-like fermions, F_j, in the fundamental of SU(4). We consider simplified models characterised by the nature of the SU(4) symmetry (global or local), by the number of vector-like fermions, and by the couplings of the Higgs boson to SM fermions (direct or induced by mixing with the vector-like fermions). In all cases, we implement a minimal breaking of a U(2)^n_f flavour symmetry, with a single motivated exception. We then perform a global fit including the main observables sensitive to exchanges of the {mathcal {G}}_{mu }^A at tree level and in loops dominated by logs insensitive to the UV completion.

Renormalization of supersymmetric chiral theories in rational spacetime dimensions

We renormalize models with scalar chiral superfields with an odd superpotential to several orders in perturbation theory. These extensions of the cubic Wess–Zumino model are renormalizable in spacetime dimensions which are rational. When endowed with an O(N) symmetry it is shown that they share the same property as their non-supersymmetric counterparts in that at a particular fixed point there is an emergent OSp(1|n-1) symmetry, where n is the power of the superpotential. This is shown at a loop order beyond that for which it was established in the parallel non-supersymmetric theory.

Confinement on the moose lattice

In this work we present a new class of mathcal{N} = 1 supersymmetric confining gauge theories, with strikingly simple infrared theories that descend from intricate interconnected networks of product gauge groups. A diagram of the gauge groups and the charged matter content of the ultraviolet theory has the structure of a triangular lattice, with SU(N) or SU(3N) gauge groups at each of the vertices, connected by bifundamental chiral superfields. This structure admits a U(1)R conserving superpotential with marginal (trilinear) operators. With the introduction of this superpotential, the SU(3N) and SU(N) gauge groups confine: in the far infrared limit of the supersymmetric theory, the relevant degrees of freedom are gauge invariant “mesons” and “baryons.” In this paper we show how the properties of the infrared degrees of freedom depend on the topology and shape of the moose/quiver “lattice” of the original gauge theory. Between the local structure of the moose lattice at higher energies, and the boundary dependence that emerges in the infrared, the 4d moose lattice theory reproduces features of a 6d spacetime with two compact dimensions. We investigate various deformations of the theory, and propose some phenomenological applications for BSM models.

Novel counterexample to the Nelson-Seiberg theorem

We present a new type of counterexample to the Nelson-Seiberg theorem. It is a generic R-symmetric Wess-Zumino model with nine chiral superfields, including one field of R-charge 2 and no R-charge 0 field. As in previous counterexamples, the model gives a set of degenerate supersymmetric vacua with a non-zero expectation value for a pair of oppositely R-charged fields. However, one of these fields appears quadratically in the superpotential, and many other fields with non-zero R-charges gain non-zero expectation values at the vacuum, and so this model escapes the sufficient condition for counterexamples established in previous literature. Thus there are still open problems in the relation of R-symmetries to supersymmetry breaking in generic models.

Three-loop β-functions and two-loop anomalous dimensions for MSSM regularized by higher covariant derivatives in an arbitrary supersymmetric subtraction scheme

Three-loop β-functions of the Minimal Supersymmetric Standard Model regularized by higher covariant derivatives are obtained for an arbitrary supersymmetric subtraction scheme. For this purpose we first calculate two-loop anomalous dimensions for all MSSM chiral matter superfields defined in terms of the bare couplings. Then we use the NSVZ equations for the renormalization group functions defined in terms of the bare couplings, which are valid in all orders in the case of using the higher covariant derivative regularization. This gives the three-loop β-functions defined in terms of the bare couplings. After that, we construct the three-loop β-functions and the two-loop anomalous dimensions standardly defined in terms of the renormalized couplings for an arbitrary subtraction scheme. As a nontrivial correctness test, we verify that for a certain renormalization prescription the general results reproduce the ones obtained earlier in the overline{mathrm{DR}} scheme. Also this can be considered as an indepedent confirmation of the overline{mathrm{DR}} results.