Two-index Representations Research Articles

Exploring the orthosymplectic zoo

We study the Higgs branch of the SCFT limit of 5d SO(6) and SO(8) gauge theory with hypermultiplets in the spinor and vector representations. In the case of SO(6) gauge theories, we contrast the magnetic quivers obtained with those of SU(4) gauge theory with hypermultiplets in the fundamental and second rank antisymmetric representations. Since SU(4) gauge theories admit several different values of the Chern-Simons level, we make some observations about how to distinguish those theories from the brane webs of the SO(6) theories. In the case of SO(8) gauge theories, we use SO(8) triality to propose (naively) inequivalent magnetic quivers, which will turn out to have the same moduli spaces of vacua, at least locally around their most singular loci. We encounter several interesting new phenomena occurring in the magnetic quivers, such as hypermultiplets between neighbouring symplectic gauge nodes and matter in two-index representations of unitary nodes. We also give a prescription for computing the local Coulomb branch Hilbert series for quivers involving bad USp(2) gauge nodes.

Conformal window from conformal expansion

We study the conformal window of asymptotically free gauge theories containing ${N}_{f}$ flavors of fermion matter transforming to the vector and two-index representations of $SO(N),SU(N)$ and $Sp(2N)$ gauge groups. For $SO(N)$ we also consider the spinorial representation. We determine the critical number of flavors ${N}_{f}^{\mathrm{cr}}$, corresponding to the lower end of the conformal window, by using the conjectured critical condition on the anomalous dimension of the fermion bilinear at an infrared fixed point, ${\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}}=1$ or equivalently ${\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}}(2\ensuremath{-}{\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}})=1$. To compute ${\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}}$ we employ the (scheme-independent) Banks-Zaks conformal expansion up to the 4th order in ${\mathrm{\ensuremath{\Delta}}}_{{N}_{f}}={N}_{f}^{\mathrm{AF}}\ensuremath{-}{N}_{f}$ with ${N}_{f}^{\mathrm{AF}}$ corresponding to the onset of the loss of asymptotic freedom, where we show that the latter critical condition provides a better performance along with this conformal expansion. To quantify the uncertainties in our analysis, which potentially originate from nonperturbative effects, we propose two distinct approaches by assuming the large order behavior of the conformal expansion separately, either convergent or divergent asymptotic. In the former case, we take the difference in the Pad\'e approximants to the two definitions of the critical condition, whereas in the latter case the truncation error associated with the singularity in the Borel plane is taken into account. Our results are further compared to other analytical methods as well as lattice results available in the literature. In particular, we find that $SU(2)$ with six and $SU(3)$ with ten fundamental flavors are likely on the lower edge of the conformal window, which are consistent with the recent lattice results. We also predict that $Sp(4)$ theories with fundamental and antisymmetric fermions have the critical numbers of flavors, approximately ten and five, respectively.

Dynamics of QCD3 with rank-two quarks and duality

Three-dimensional gauge theories coupled to fermions can develop interesting nonperturbative dynamics. Here we study in detail the dynamics of SU(N ) gauge theories coupled to a Dirac fermion in the rank-two symmetric and antisymmetric representation. We argue that when the Chern-Simons level is sufficiently small the theory develops a quantum phase with an emergent topological field theory. When the Chern-Simons level vanishes, we further argue that a baryon condenses and hence baryon symmetry is spontaneously broken. The infrared theory then consists of a Nambu-Goldstone boson coupled to a topological field theory. Our proposals also lead to new fermion-fermion dualities involving fermions in two-index representations. We make contact between our proposals and some recently discussed aspects of four-dimensional gauge theories. This leads us to a proposal for the domain wall theories of non-supersymmetric gauge theories with fermions in two-index representations. Finally, we discuss some aspects of the time-reversal anomaly in theories with a one-form symmetry.

Scalar gauge dynamics and Dark Matter

We consider theories with one gauge group (SU, SO or Sp) and one scalar in a two-index representation. The renormalizable action often has accidental symmetries (such as global U(1) or unusual group parities) that lead to one or more stable states, providing Dark Matter candidates. We discuss the confined phase(s) of each theory and compute the two Higgs phases, finding no generic dualities among them. Discrete gauge symmetries can arise and accidental symmetries can be broken, possibly giving pseudo-Goldstone Dark Matter. Dark Matter candidates can have a complicated sub-structure characteristic of each group and can be accompanied by extra dark radiation.

Gauging 1-form center symmetries in simple SU(N) gauge theories

Consequences of gauging exact {mathbb{Z}}_k^C center symmetries in several simple SU(N) gauge theories, where k is a divisor of N, are investigated. Models discussed include: the SU(N) gauge theory with Nf copies of Weyl fermions in self-adjoint single-column antisymmetric representation, the well-discussed adjoint QCD, QCD-like theories in which the quarks are in a two-index representation of SU(N), and a chiral SU(N) theory with fermions in the symmetric as well as in anti-antisymmetric representations but without fundamentals. Mixed 't Hooft anomalies between the 1-form {mathbb{Z}}_k^C symmetry and some 0- form (standard) discrete symmetry provide us with useful information about the infrared dynamics of the system. In some cases they give decisive indication to select only few possiblities for the infrared phase of the theory.

Vacuum misalignment and pattern of scalar masses in the SU(5)/SO(5) composite Higgs model

Composite Higgs models based on SU(5)/SO(5) are characterised by the presence of custodial triplets, like the Georgi-Machacek model. We classify all the operators giving rise to the top mass and Higgs potential in presence of fermion partial compositeness, with top partners in two-index representations of SU(5). A detailed study of each operator allows us to find correlations in the couplings of Higgs and non-Higgs pseudo-Nambu-Goldstone bosons, which depend only on one, two or three independent parameters. We also analyse the Higgs potential, finding that a misalignment along the custodial invariant triplet direction is forbidden by CP conservation. Avoiding custodial breaking allows us to select a handful of feasible models, which feature universal patterns in the scalar masses related to their transformation properties under the custodial symmetry. Finally, we briefly study the LHC phenomenology of the scalars, which are always below 1 TeV even for multi-TeV condensation scales, and find promising same-sign lepton final states enriched by hard photons.

QCD3 with two-index quarks, mirror symmetry, and fivebrane anti-BIons near orientifolds

We consider a non-supersymmetric Hanany-Witten type IIB brane configuration that realises a three dimensional USp(2N) gauge theory with quarks in the two-index antisymmetric representation and SO(4) flavour symmetry. Using type IIB S-duality we find the mirror dual, an SO(2N-1) field theory with scalars in the antisymmetric representation. Analysing the magnetic dual we study the vacuum structure of the USp(2N) model and propose that the SO(4) global symmetry is unbroken. In order to support our proposal we present an SO(4) symmetric BIon configuration that describes anti-D3 branes polarising into fivebranes in the S-dual Hanany-Witten setup. We also comment on dynamical flavour symmetry breaking in other QCD3 theories with quarks in two-index representations.

Effective potential in ultraviolet completions for composite Higgs models

We consider a class of composite Higgs models based on asymptotically free $SO(d)$ gauge theories with $d$ odd, with fermions in two irreducible representations, and in which the Higgs field arises as a pseudo Nambu-Goldstone boson and the top quark is partially composite. The Nambu-Goldstone coset containing the Higgs field, or Higgs coset, is either $SU(4)/Sp(4)$ or $SU(5)/SO(5)$, whereas the top partners live in two-index representations of the relevant flavor group ($SU(4)$ or $SU(5)$). In both cases, there is a large number of terms in the most general four-fermion lagrangian describing the interaction of third-generation quarks with the top partners. We derive the top-induced effective potential for the Higgs coset together with the singlet pseudo Nambu-Goldstone boson associated with the non-anomalous axial symmetry, to leading order in the couplings between the third-generation quarks and the composite sector. We obtain expressions for the low-energy constants in terms of top-partner two-point functions. We revisit the effective potential of another composite Higgs model that we have studied previously, which is based on an $SU(4)$ gauge theory and provides a different realization of the $SU(5)/SO(5)$ coset. The top partners of this model live in the fundamental representation of $SU(5)$, and, as a result, the effective potential of this model is qualitatively different from the $SO(d)$ gauge theories. We also discuss the role of the isospin-triplet fields contained in the $SU(5)/SO(5)$ coset, and show that, without further constraints on the four-fermion couplings, an expectation value for the the Higgs field will trigger the subsequent condensation of an isospin-triplet field.

Classification of NLO operators for composite Higgs models

We provide a general classification of template operators, up to next-to-leading order, that appear in chiral perturbation theories based on the two flavour patterns of spontaneous symmetry breaking SU($N_F$)/Sp($N_F$) and SU($N_F$)/SO($N_F$). All possible explicit-breaking sources parametrised by spurions transforming in the fundamental and in the two-index representations of the flavour symmetry are included. While our general framework can be applied to any model of strong dynamics, we specialise to composite-Higgs models, where the main explicit breaking sources are a current mass, the gauging of flavour symmetries, and the Yukawa couplings (for the top). For the top, we consider both bilinear couplings and linear ones a la partial compositeness. Our templates provide a basis for lattice calculations in specific models. As a special example, we consider the SU(4)/Sp(4)$\cong$ SO(6)/SO(5) pattern which corresponds to the minimal fundamental composite-Higgs model. We further revisit issues related to the misalignment of the vacuum. In particular, we shed light on the physical properties of the singlet $\eta$, showing that it cannot develop a vacuum expectation value without explicit CP violation in the underlying theory.

Bosonizing three-dimensional quiver gauge theories

We start with the recently conjectured 3d bosonization dualities and gauge global symmetries to generate an infinite sequence of new dualities. These equate theories with non-Abelian product gauge groups and bifundamental matter. We uncover examples of Bose/Bose and Fermi/Fermi dualities, as well as a sequence of dualities between theories with scalar matter in two-index representations. Our conjectures are consistent with level/rank duality in massive phases.

Hidden Topological Angles in Path Integrals.

We demonstrate the existence of hidden topological angles (HTAs) in a large class of quantum field theories and quantum mechanical systems. HTAs are distinct from theta parameters in the Lagrangian. They arise as invariant angles associated with saddle points of the complexified path integral and their descent manifolds (Lefschetz thimbles). Physical effects of HTAs become most transparent upon analytic continuation in n_{f} to a noninteger number of flavors, reducing in the integer n_{f} limit to a Z_{2} valued phase difference between dominant saddles. In N=1 super Yang-Mills theory we demonstrate the microscopic mechanism for the vanishing of the gluon condensate. The same effect leads to an anomalously small condensate in a QCD-like SU(N) gauge theory with fermions in the two-index representation. The basic phenomenon is that, contrary to folklore, the gluon condensate can receive both positive and negative contributions in a semiclassical expansion. In quantum mechanics, a HTA leads to a difference in semiclassical expansion of integer and half-integer spin particles.

S-duality of nonsupersymmetric gauge theories

We propose a method for constructing pairs of nonsupersymmetric gauge theories related by S-duality. Starting from a known S-duality of supersymmetric theories realized on the worldvolume of D3 branes in type IIB string theory, a new duality is obtained by replacing the D3-branes with antibranes. Large classes of dual pairs of nonsupersymmetric theories can be obtained in this way, with different interactions and matter content (chiral and vector-like). The approach is illustrated on gauge theories realized on three-branes and fractional branes probing orbifold singularities. The duality sheds light on the dynamics of gauge theories and their possible infrared phases by providing concrete magnetic dual descriptions of strongly coupled theories. Some of the models share various properties with QCD, including confinement and chiral symmetry breaking. More generally, these theories feature fermions in multiple two-index representations and could realize intriguing phases such as a free magnetic phase with chiral symmetry breaking or mixed phases where an interacting fixed point coexists with a confining phase.

Near the sill of the conformal window: Gauge theories with fermions in two-index representations

We apply Schr\"odinger functional methods to two gauge theories with fermions in two-index representations: the SU(3) theory with ${N}_{f}=2$ adjoint fermions, and the SU(4) theory with ${N}_{f}=6$ fermions in the two-index antisymmetric representation. Each theory is believed to lie near the bottom of the conformal window for its respective representation. In the SU(3) theory we find a small beta function in strong coupling but we cannot confirm or rule out an infrared fixed point. In the SU(4) theory we find a hint of walking---a beta function that approaches the axis and then turns away from it. In both theories the mass anomalous dimension remains small even at the strongest couplings, much like the theories with fermions in the two-index symmetric representation investigated earlier.

Domain walls and metastable vacua in hot “Orientifold Field Theories”

We consider "Orientifold field theories", namely SU(N) gauge theories with Dirac fermions in the two-index representation at high temperature. When N is even these theories exhibit a spontaneously broken Z2 centre symmetry. We study aspects of the domain wall that interpolates between the two vacua of the theory. In particular we calculate its tension to two-loop order. We compare its tension to the corresponding domain wall in a SU(N) gauge theory with adjoint fermions and find an agreement at large-N, as expected from planar equivalence between the two theories. Moreover, we provide a non-perturbative proof for the coincidence of the tensions at large-N. We also discuss the vacuum structure of the theory when the fermion is given a large mass and argue that there exist N-2 metastable vacua. We calculate the lifetime of those vacua in the thin wall approximation.

Spectrum of orientifold QCD in the strong coupling and hopping expansion approximation

We use the strong coupling and hopping parameter expansions to calculate the pion and rho meson masses for lattice Yang–Mills gauge theories with fermions in irreducible two-index representations, namely the adjoint, symmetric and antisymmetric. The results are found to be consistent with orientifold planar equivalence, and leading order 1/Nc corrections are calculated in the lattice phase. An estimate of the critical bare mass, for which the pion is massless, is obtained as a function of the bare coupling. A comparison to data from the two-flavour SU(2) theory with adjoint fermions gives evidence for a bulk phase transition at βc∼2, separating a pure lattice phase from a phase smoothly connected to the continuum.

Discrete symmetry breaking and baryon currents inU(N)andSU(N)gauge theories

In SU(N) gauge theories with fermions in the fundamental or in a two-index (either symmetric or antisymmetric) representation formulated on a manifold with at least one compact dimension with nontrivial holonomy the discrete symmetries C, P, and T are broken at small enough size of the compact direction(s) for certain values of N. We show that for those N in the broken phase a nonzero baryon current wrapping in the compact direction exists, which provides a measurable observable for the breaking of C, P, and T. We prove that in all cases where the current is absent there is no breaking of those discrete symmetries. This includes the limit N{yields}{infinity} of the SU(N) gauge theory with symmetric or antisymmetric fermions and U(N) gauge theory at any value of N. We then argue that the component of the baryon current in the compact direction is the physical order parameter for C, P, and T breaking due to the breaking of Lorentz invariance.

Center symmetry and the orientifold planar equivalence

We study the center symmetry of SU(N) gauge theories with fermions in the two-index representations, by computing the effective potential of the Polyakov loop in the large-mass expansion on the lattice. In the large-N limit and at non-zero temperature, we find that the center symmetry is Z_N for fermions in the adjoint representation and just Z_2 for fermions in the (anti)symmetric representation. We discuss the fact that our results do not contradict the orientifold planar equivalence, which relates a common sector defined by the bosonic gauge-invariant C-even states of theories with fermions in different two-index representations. Our results complement the work of Armoni et al. (2007), who showed how at zero temperature a Z_N center symmetry is dynamically recovered also for fermions in the (anti)symmetric representation, by considering the theories at finite temperature.

Critical exponents for higher-representation sources in 3D SU(3) gauge theory from CFT

We establish an exact mapping between the multiplication table of the irreducible representations of SU(3) and the fusion algebra of the two-dimensional conformal field theory in the same universality class of 3D SU(3) gauge theory at the deconfining point. In this way the Svetitsky-Yaffe conjecture on the critical behaviour of Polyakov lines in the fundamental representation naturally extends to whatever representation one considers. As a consequence, the critical exponents of the correlators of these Polyakov lines are determined. Monte Carlo simulations with sources in the symmetric two-index representation, combined with finite-size scaling analysis, compare very favourably with these predictions.