SUSY Gauge Theories Research Articles

Approximate analytic solutions of the renormalization group equations for Yukawa and soft couplings in SUSY models

We present simple analytical formulae which describe solutions of the RG equations for Yukawa couplings in SUSY gauge theories with the accuracy of a few per cent. Performing the Grassmannian expansion in these solutions, one finds those for all the soft couplings and masses. The solutions clearly exhibit the fixed point behaviour which can be calculated analytically. A comparison with numerical solutions is made.

Strings, susy gauge theories, and integrable systems

We consider supersymmetric Yang-Mills theories in the general framework of string theory (M-theory) and describe particular compactifications leading to the Seiberg-Witten (SW) exact effective theories. We show strong parallels between SW and integrable system theories.

More confining N = 1 SUSY gauge theories from non-Abelian duality

We expand on an idea of Seiberg that an N = 1 supersymmetric gauge theory shows confinement without breaking of chiral symmetry when the gauge symmetry of its magnetic dual is completely broken by the Higgs effect. This has recently been applied to some models involving tensor fields and an appropriate tree-level superpotential. We show how the confining spectrum of a supersymmetric gauge theory can easily be derived when a magnetic dual is known and we determine it explicitly for many models containing fields in second rank tensor representations. We also give the form of the confining superpotential for most of these models.

Erratum: Matching conditions and duality inN=1 SUSY gauge theories in the conformal window [Phys. Rev. D53, 4526 (1996)

We discuss duality in $N=1$ SUSY gauge theories in Seiberg's conformal window, $(3N_c/2)<N_f<3N_c$. The 't Hooft consistency conditions -- the basic tool for establishing the infrared duality -- are considered taking into account higher order $\alpha$ corrections. The conserved (anomaly free) $R$ current is built to all orders in $\alpha$. Although this current contains all orders in $\alpha$ the 't Hooft consistency conditions for this current are shown to be one-loop. This observation thus justifies Seiberg's matching procedure. We also briefly discuss the inequivalence of the ``electric" and ``magnetic" theories at short distances.

Aspects of confinement and screening in M-theory

Confinement and screening are investigated in SUSY gauge theories, realized by an M5-brane configuration, extending an approach applied previously to N = 1 SYM theory, to other models. The electric flux tubes are identified as M2-branes ending on the M5-branes and the conserved charge they carry is identified as a topological property. The group of charges carried by the flux tubes is calculated and the results agree in all cases considered with the field-theoretical expectations. In particular, whenever the dynamical matter is expected to screen the confining force, this is reproduced correctly in the M-theory realization.

Implications of exact SUSY gauge couplings for QCD

The phase structure of SUSY gauge theories can be very different from their nonsupersymmetric counterparts. Nonetheless, there is interesting information which might be gleaned from detailed investigation of these theories. In particular, we study the precise meaning of the strong interaction scale $\Lambda$. We ask whether one can meaningfully apply naive dimensional analysis and also ask whether the study of supersymmetric theories can shed light on the apparent discrepancy between the perturbative scale $\Lambda_{QCD}$ and the ``chiral lagrangian'' scale $\Lambda_\chi$. We show that in N=1 supersymmetric Yang Mills theory, ``naive dimensional analysis'' seems to work well, with $\Lambda_\chi$ consistently equal to the scale at which the perturbatively evolved physical coupling becomes of order $4 \pi$. We turn to N=2 theories to understand better the effect of instantons in accounting for the QCD discrepancy between scales. In N=2 supersymmetric SU(2) the instanton corrections are known to all orders from the Seiberg-Witten solution and give rise to a finite scale ratio between the scale at which the perturbatively evolved and ``nonperturbatively evolved'' couplings blow up. Correspondingly, instanton effects are important even when the associated perturbatively evolved gauge coupling only gives $\alpha$ of order 1 (rather than $4 \pi$). We compare the N=2 result to instanton-induced corrections in QCD, evaluated using lattice data and the instanton liquid model, and find a remarkably similar behavior.

SUSY Gauge Theories and Whitham Integrable Systems from Compactification and SUSY Breaking

We review the Seiberg-Witten construction of low-energy effective actions and BPS spectra in SUSY gauge theories and its formulation in terms of integrable systems. It is also demonstrated how this formulation naturally appears from the compactified version of the theory with partially broken supersymmetry so that the integrable structures arise from the relation between bare and quantum variables and superpotentials of SUSY gauge theories. The Whitham integrable systems, literally corresponding to the uncompactified theory, are then restored by averaging over fast variables in the decompactification limit.

The instanton hunter's guide to supersymmetric SU( N) gauge theories

We present a compendium of results for ADHM multi-instantons in SU( N) SUSY gauge theories, followed by applications to N = 2 supersymmetric models. Extending recent SU(2) work, and treating the N = 1 and N = 2 cases in parallel, we construct: (i) the ADHM supermultiplet, (ii) the multi-instanton action, and (iii) the collective coordinate integration measure. Specializing to N = 2, we then give a closed formula for F k, the k-instanton contribution to the prepotential, as a finite-dimensional collective coordinate integral. This amounts to a weak-coupling solution, in quadratures, of the low-energy dynamics of N = 2 SQCD, without appeal to duality. As an application, we calculate F 1 for all SU( N) and any number of flavors N F ; for N F < 2 N − 2 and N F = 2 N − 1 we confirm previous instanton calculations and agree with the proposed hyper-elliptic curve solutions. For N F = 2 N − 2 and N F = 2 N with N > 3 we obtain new results, which in the latter case we do not understand how to reconcile with the curves.

Picard-Fuchs equation and prepotential of five-dimensional SUSY gauge theory compactified on a circle

Five-dimensional supersymmetric gauge theory compactified on a circle defines an effective N = 2 supersymmetric theory for massless fields in four dimensions. Based on the relativistic Toda chain Hamiltonian proposed by Nekrasov, we derive the Picard-Fuchs equation on the moduli space of the Coulomb branch of SU (2) gauge theory. Our Picard-Fuchs equation agrees with those from other approaches - the spectral curve of the XXZ spin chain and the supersymmetric cycle in compactified M-theory. By making use of a relation to the Picard-Fuchs equation of SU (2) Seiberg-Witten theory, we obtain the prepotential and the effective coupling constant that incorporate both a perturbative effect of Kaluza-Klein modes and a non-perturbative one of four-dimensional instantons. In the weak coupling regime we check that the prepotential exhibits a consistent behavior in large and small radius limits of the circle.

Patterns of duality in N = 1 SUSY gauge theories

We study the patterns in the duality of a wide class of N = 1 supersymmetric gauge theories in four dimensions. We present many new generalizations of the classic duality models of Kutasov and Schwimmer, which have themselves been generalized numerous times in works of Intriligator, Leigh and the present authors. All of these models contain one or two fields in a two-index tensor representation, along with fields in the defining representation. The superpotential for the two-index tensor(s) resembles A k or D k singularity forms, generalized from numbers to matrices. Looking at the ensemble of these models, classifying them by superpotential, gauge group, and “level”, we identify emerging patterns.

Constructing a c-function for SUSY gauge theories

Recently a non-perturbative formula for the RG flow between UV and IR fixed points of the coefficient in the trace of the energy momentum tensor of the Euler density has been obtained for N=1 SUSY gauge theories by relating the trace and R-current anomalies. This result is compared here with an earlier perturbation theory analysis based on a naturally defined metric on the space of couplings for general renormalisable quantum field theories. This approach is specialised to N=1 supersymmetric theories and extended, using consistency arguments, to obtain the Euler coefficient at fixed points to 4-loops. The result agrees completely, to this order, with the exact formula.

Instanton calculus and SUSY gauge theories on asymptotically locally Euclidean manifolds

We study instanton effects along the Coulomb branch of an N=2 supersymmetric Yang-Mills theory with gauge group SU(2) on Asymptotically Locally Euclidean (ALE) spaces. We focus our attention on an Eguchi-Hanson gravitational background and on gauge field configurations of lowest Chern class.

SUSY field theories in higher dimensions and integrable spin chains

Five- and six-dimensional SUSY gauge theories, with one or two compactified directions, are discussed. The 5d theories with the matter hypermultiplets in the fundamental representation are associated with the twisted XXZ spin chain, while the group product case with bi-fundamental matter corresponds to the higher rank spin chains. The Riemann surfaces for 6d theories with fundamental matter and two compact directions are proposed to correspond to the XYZ spin chain based on the Sklyanin algebra. We also discuss the obtained results within the brane and geometrical engineering frameworks and explain the relation to the toric diagrams.

Multiscale N = 2 SUSY field theories, integrable systems and their stringy /brane origin

We discuss supersymmetric Yang-Mills theories with multiple scales in the brane language. The issue concerns N = 2 SUSY gauge theories with massive fundamental matter including the UV finite case of n f = 2 n c , theories involving products of SU( n) gauge groups with bifundamental matter, and systems with several parameters similar to Λ QCD. We argue that the proper integrable systems are, accordingly, twisted XXX SL(2) spin chain, SL( p) magnets and degenerations of the spin Calogero system. The issue of symmetries underlying integrable systems is addressed. Relations with the monopole systems are specially discussed. Brane pictures behind all these integrable structures in the IIB and M-theory are suggested. We argue that degrees of freedom in integrable systems are related to KK excitations in M-theory or D-particles in the IIA string theory, which substitute the infinite number of instantons in the field theory. This implies the presence of more BPS states in the low-energy sector.

Finiteness of soft terms in finite N=1 SUSY gauge theories

Recently proposed relations between the renormalization group functions of the soft supersymmetry breaking terms and those of a rigid theory allow one to consider a possibility of constructing a totally all loop finite N=1 SUSY gauge theory, including the soft SUSY breaking terms. The requirement of finiteness, which can be satisfied in previously constructed finite SUSY GUTs, imposes some constraints on the SUSY breaking parameters which, in the leading order, coincide with those originating from the supergravity and superstring-inspired models. Explicit relations between the soft terms, which lead to a completely finite theory in any loop order, are given.

Duals for SU( N) SUSY gauge theories with an antisymmetric tensor: five easy flavors

I consider N=1 supersymmetric SU ( N c ) gauge theories with matter fields consisting of one antisymmetric representation, five flavors, and enough anti-fundamental representations to cancel the gauge anomaly. Previous analyses are extended to the case of even N c with no superpotential. Using holomorphy I show that the theory has an interacting infrared fixed point for sufficiently large N c . These theories are interesting due to the fact that in going from five to four flavors the theory goes from a non-trivial infrared fixed point to confinement, in contradistinction to SUSY QCD, but in analogy to the behavior expected in non-SUSY QCD.

WDVV Equations in seiberg-witten theory and associative Algebras

This is a short review of the results on the associativity algebras and WDVV equations found recently for the Seiberg-Witten solutions of N=2 4d SUSY gauge theories. The presentation is mostly based on the integrable treatment of these solutions.

Renormalizations in softly broken SUSY gauge theories

The supergraph technique for calculations in supersymmetric gauge theories where supersymmetry is broken in a “soft” way (without introducing quadratic divergencies) is reviewed. By introducing an external spurion field the set of Feynman rules is formulated and explicit connections between the UV counterterms of a softly broken and rigid SUSY theories are found. It is shown that the renormalization constants of softly broken SUSY gauge theory also become external superfields depending on the spurion field. Their explicit form repeats that of the constants of a rigid theory with the redefinition of the couplings. The method allows us to reproduce all known results on the renormalization of soft couplings and masses in a softly broken theory. As an example the renormalization group functions for soft couplings and masses in the Minimal Supersymmetric Standard Model up to the three-loop level are calculated.

Lecture on SUSY Gauge Theories and Integrable Systems

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.

Renormalizations in softly broken SUSY gauge theories

The supergraph technique for calculations in supersymmetric gauge theories where supersymmetry is broken in a “soft” way (without introducing quadratic divergencies) is reviewed. By introducing an external spurion field the set of Feynman rules is formulated and explicit connections between the UV counterterms of a softly broken and rigid SUSY theories are found. It is shown that the renormalization constants of softly broken SUSY gauge theory also become external superfields depending on the spurion field. Their explicit form repeats that of the constants of a rigid theory with the redefinition of the couplings. The method allows us to reproduce all known results on the renormalization of soft couplings and masses in a softly broken theory. As an example the renormalization group functions for soft couplings and masses in the Minimal Supersymmetric Standard Model up to the three-loop level are calculated.