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Confining N=1 SUSY Gauge Theories from Seiberg Duality

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Confining N=1 SUSY Gauge Theories from Seiberg Duality Trieste Meeting of the TMR Network on Physics beyond the SM PROCEEDINGS Confining N=1 SUSY gauge theories from Seiberg duality Matthias Klein Departamento de F´ısica Te´orica C-XI and Instituto de F´ısica Te´orica C-XVI Universidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain E-mail: [email protected] Abstract: In this talk I review and generalize 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. It is shown how the confining spectrum of a supersymmetric gauge theory can easily be derived when a magnetic dual is known and this method is applied to many models containing fields in second rank tensor representations and an appropriate tree-level superpotential. 1. Introduction For all of the models considered in [5] a dual de- scription in terms of magnetic variables is known Due to holomorphicity properties and non-renor- [6, 8, 9, 10] and the authors of [5] used the fact N malization theorems valid in =1supersym- that the electric gauge theory confines when its metric theories it has become possible to argue magnetic dual is completely higgsed. that some supersymmetric gauge theories with special matter content confine at low energies. Seiberg already used this idea as an addi- The first example is due to Seiberg [1] who found tional consistency check in his original paper es- that supersymmetric quantum chromodynamics tablishing electric-magnetic duality for non-Abel- ian N = 1 supersymmetric gauge theories [11]. (SQCD) with gauge group SU(Nc)andNf quark He showed how the confining superpotential of flavors shows confinement when Nf = Nc or Nf = SQCD with Nf = Nc + 1 could be obtained by Nc + 1. This has been generalized to more com- plicated models. All N = 1 supersymmetric a perturbative calculation in the completely bro- gauge theories with vanishing tree-level super- ken magnetic gauge theory. Under duality the potential which confine at low energies could be fields of the magnetic theory (which are gauge classified [2, 3, 4] because they are constrained singlets as the gauge symmetry is completely bro- by an index argument. When a tree-level su- ken) are mapped to the mesons and baryons of perpotential is present the index argument is no the electric theory and the confining superpo- longer valid. Because of the lower symmetry tential is easily shown to be the image of the the non-perturbative superpotential is less con- magnetic superpotential under this mapping [11]. N strained and one expects more confining models This a realization in = 1 supersymmetric gauge to exist. Indeed, Cs´aki and Murayama [5] showed theories of an old idea of ’t Hooft and Mandel- that many of the Kutasov-like [6] models exhibit stam [12] that confinement is driven by conden- confinement for special values of the number of sation of magnetic monopoles. 1 quark flavors Nf . These models contain fields in Now, many gauge theory models have been tensor representations of the gauge group and an found that possess a dual description in terms of appropriate superpotential for these tensor fields. magnetic variables in the infrared. This allows 1Some further confining models with non-vanishing us to predict many new examples of confining tree-level superpotential are discussed in [7]. gauge theories. The idea described in the previ- Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein ous paragraph was first used by the authors of Nf = Nc: In this case the superpotential [13] to determine the confining spectrum of the vanishes even at the non-perturbative level [1]. model proposed by Kutasov [6] and has been ap- As a consequence the flat directions that param- plied by Cs´aki and Murayama [5] to six further etrize the moduli space of vacua are not lifted in models that confine in the presence of an appro- the quantum theory. The low-energy spectrum priate superpotential. is given by the mesons M ij defined above and In this talk I review the original example of the baryons B =detQ,B¯=detQ¯(the quarks Seiberg [11] and explain how electric-magnetic Q, Q¯ are viewed as (Nf × Nc)-matrices). The duality is used to obtain the low-energy spectrum physical degrees of freedom at low energies being and the form of the non-perturbative superpo- gauge invariant means that the theory confines. tential of confining gauge theories [14]. One finds The classical constraint det M = BB¯ is modified that all of the gauge theory models based on sim- in the quantum theory [1] to ple gauge groups considered in [10, 15] confine det M − BB¯ =Λ2Nc . (2.2) when the gauge groups of their magnetic duals are completely broken by the Higgs effect. For The expectation values of the mesons and baryons nine of these theories the confining phase has not that satisfy this constraint span the quantum been discussed before. moduli space. The observation that the expec- tation values of det M and BB¯ cannot vanish 2. Phase structure of SQCD simultaneously tells us that the chiral symmetry is spontaneously broken. Let us briefly review the well-known phase struc- Nf = Nc + 1: There is again a quantum N ture of SQCD [16]. By this we mean an =1 moduli space, but now the classical constraints SU N N supersymmetric ( c) gauge theory with f are not modified by quantum effects. In the low- N quark flavors, i.e. f chiral matter supermulti- energy theory they can be derived from the non- Q plets transforming in the fundamental repre- perturbative superpotential [1] sentation of the gauge group and the same amount ¯ BMB¯ − det M of matter multiplets Q transforming in the anti- W = , (2.3) np 2Nf −3 fundamental representation. Consider first the Λ case of vanishing tree-level superpotential. De- where the baryons Bi are defined as the deter- pending on the relative values of Nf and Nc the minant of the quark matrix Q with the i-th line the low-energy theory resides in different phases. omitted. This describes confinement without Nf = 0: This is pure super Yang-Mills the- breaking of the chiral symmetry. N ≤N ≤ 3N ory. It is believed to show confinement. Accord- c +2 f 2 c: The low-energy energy ing to an index argument by Witten [17] there theory is rather complicated and more appropri- are Nc distinct supersymmetric vacua. ately described in terms of dual magnetic vari- 0 <Nf <Nc: There is a non-perturba- ables. The dual magnetic theory is infrared free tive superpotential generated by gluino conden- for this range of parameters. N <N − 3 N <N < N sation (for f c 1) or by instantons (for 2 c f 3 c: At low energies the the- Nf = Nc − 1), as was shown by Affleck, Dine ory is driven to an infrared fixed point of the and Seiberg [18]: renormalization group [11] and resides in a non- 1 Abelian Coulomb phase. Seiberg found a dual N −N Λ3 c f Nc−Nf W N −N , description of this model [11] by an SU(Nf −Nc) np =( c f) M (2.1) det gauge theory with Nf (magnetic) quark flavors q, q N 2 M ij where Λ is the dynamically generated scale of the ¯ and f additional singlets mag which couple theory and the meson matrix M is defined by to the magnetic quarks via the superpotential ij αi j M = Q Q¯α, i, j =1,...,Nf, α =1,...,Nc. Wmag = Mmagqq.¯ (2.4) The minimum of the potential lies at infinite field expectation values and therefore the theory has This magnetic theory flows to the same infrared no stable vacuum for this range of parameters. fixed point. The gauge invariant operators of 2 Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein both theories are in one-to-one correspondence: are denoted byq ˆ, ˆq¯. They couple to the me- son singlets Mˆ mag via the tree-level superpoten- M ←−→ µM , (2.5) q mag tial (2.4). Due to non-renormalization theorems N c − N f 3 N c − N f this is not corrected in perturbation theory. But B ←−→ − ( − µ ) Λ B mag . there are non-perturbative corrections generated The mass scale µ had to be introduced by dimen- by instantons. The full superpotential of the low- energy magnetic theory reads [11] sional analysis. The three scales Λ, Λmag and µ are related by 3−Nf Wˆ mag = Mˆ magqˆˆq¯+Λmag det Mˆ mag. (3.1) N −N −N 3Nc−Nf 3( f c) f Nf −Nc Nf Λ Λmag =(−1) µ . From the fact that all physical degrees of free- (2.6) dom of the effective magnetic theory are gauge Nf > 3Nc: In the infrared the theory flows invariant one expects that, as a consequence of to the trivial fixed point of free quarks and glu- the duality mapping, the degrees of freedom of ons. the electric theory are gauge singlets as well. We will see that this intuition is right. The mapping (2.5) gives 3. Confinement from duality M ←−→ µM , p mag (3.2) Let us consider deformations [11] of the theory B ←−→ µ − 1 Λ 2 N f − 3 q,ˆ described in the previous section by mass terms W =Tr(mM), where m is an (Nf × Nf )-matrix and the scale matching (2.6) now reads p −N of rank . By the duality mapping (2.5) this cor- 2Nf −3 3 f Nf Λ Λmag = −µ .
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